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monte carlo methods @xcite appeared about sixty years ago with the need to evaluate numerical values for various complex problems . these methods evolved and were applied early to quantum problems , thus putting within reach exact numerical solutions to non - trivial quantum problems @xcite . many improvements of these methods followed , avoiding critical slowing down near phase transitions and allowing to work directly in the continuous imaginary time limit @xcite . in recent years , interest in methods that work in the canonical ensemble with global updates yet allow access to green functions has intensified @xcite . however , a method that works well for a given hamiltonian often needs major modifications for another . for example , the addition of a 4-site ring exchange term in the bosonic hubbard model required special developments for a treatment by the stochastic series expansion algorithm @xcite , as well as by the wordline algorithm @xcite . this can result in long delays . it is , therefore , advantageous to have at one s disposal an algorithm that can be applied to a very wide class of hamiltonians without requiring any changes . in a recent publication @xcite , the stochastic green function ( sgf ) algorithm was presented , which meets this goal . the algorithm can be applied to any lattice hamiltonian of the form @xmath3 where @xmath1 is diagonal in the chosen occupation number basis and @xmath2 has only positive matrix elements . this includes all kinds of systems that can be treated by other methods presented in ref.@xcite , for instance bose - hubbard models with or without a trap , bose - fermi mixtures in one dimension , heisenberg models ... in particular hamiltonians for which the non - diagonal part @xmath2 is non - trivial ( the eigen - basis is unknown ) are easily treated , such as the bose - hubbard model with ring exchange @xcite , or multi - species hamiltonians in which a given species can be turned into another one ( see eq.([twospecies ] ) and fig . [ density ] and [ momentum ] for a concrete example ) . systems for which it is not possible to find a basis in which @xmath1 is diagonal and @xmath2 has only positive matrix elements are said to have a `` sign problem '' , which usually arises with fermionic and frustrated systems . as other qmc methods , the sgf algorithm does not solve this problem . the algorithm allows to measure several quantities of interest , such as the energy , the local density , local compressibility , density - density correlation functions ... in particular the winding is sampled and gives access to the superfluid density . equal - time n - body green functions are probably the most interesting quantities that can be measured by the algorithm , by giving access to momentum distribution functions which allow direct comparisons with experiments . all details on measurements are given in ref.@xcite . in addition the algorithm has the property of being easy to code , due in part to a simple update scheme in which all moves are accepted with a probability of 1 . despite of such generality and simplicity , the algorithm might suffer from a reduced efficiency , compared to other algorithms in situations where they can be applied . the purpose of this paper is to present a `` directed '' update scheme that ( i ) keeps the simplicity and generality of the original sgf algorithm , and ( ii ) enhances its efficiency by improving the sampling over the imaginary time axis . while the sgf algorithm is not intended to compete with the speed of other algorithms , the improvment resulting from the directed update scheme is remarkable ( see section v ) . but what makes the strength of the sgf method is that it allows to simulate hamiltonians that can not be treated by other methods or that would require special developments ( see eq.([twospecies ] ) for a concrete example ) . the paper is organized as follows : we introduce in section ii the notations and definitions used in ref.@xcite . in section iii , we propose a simplification of the update scheme used in the original sgf algorithm , and determine how to satisfy detailed balance . a generalization of the simplified update scheme is presented in section iv , which constitutes the directed updated scheme . finally section v shows how to determine the introduced optimization parameters , and presents some tests of the algorithm and a comparison with the original version . in this section , we recall the expression of the `` green operator '' introduced in the sgf algorithm , and the extended partition function which is considered . although not required for understanding this paper , we refer the reader to ref.@xcite for full details on the algorithm . as many qmc algorithms , the sgf algorithm samples the partition function @xmath4 the algorithm has the property of working in the canonical ensemble . in order to define the green operator , we first define the `` normalized '' creation and annihilation operators , @xmath5 where @xmath6 and @xmath7 are the usual creation and annihilation operators of bosons , and @xmath8 is the number operator . from ( [ normalizedoperators ] ) one can show the following relations for any state @xmath9 in the occupation number representation , @xmath10 with the particular case @xmath11 . appart from this exception , the operators @xmath12 and @xmath13 change a state @xmath9 by respectively creating and annihilating one particle , but they do not change the norm of the state . using the notation @xmath14 to denote two subsets of site indices @xmath15 and @xmath16 with the constraint that all indices in subset @xmath17 are different from the indices in subset @xmath18 ( but several indices in one subset may be equal ) , we define the green operator @xmath19 by @xmath20 where @xmath21 is a matrix that depends on the application of the algorithm @xcite . in order to sample the partition function ( [ partitionfunction ] ) , an extended partition function @xmath22 is considered by breaking up the propagator @xmath23 , and introducing the green operator between the broken parts , @xmath24 defining the time dependant operators @xmath25 and @xmath26 , @xmath27 and working in the occupation number basis in which @xmath1 is diagonal , the extended partition function takes the form @xmath28 where the sum @xmath29 implicitly runs over complete sets of states @xmath30 . we will systematically use the labels @xmath31 and @xmath32 to denote the states appearing on the left and the right of the green operator , and use the notation @xmath33 to denote the diagonal energy @xmath34 . we will also denote by @xmath35 and @xmath36 the time indices of the @xmath2 operators appearing on the left and the right of @xmath19 . as a result , the extended partition function is a sum over all possible configurations , each being determined by a set of time indices @xmath37 and a set of states @xmath38 , @xmath39 , @xmath40,@xmath41 , @xmath42 . the algorithm consists in updating those configurations by making use of the green operator . assuming that the green operator is acting at time @xmath43 , it can `` create '' a @xmath2 operator ( that is to say a @xmath2 operator can be inserted in the operator string ) at the same time , thus introducing a new intermediate state , then it can be shifted to a different time . while shifting , any @xmath2 operator encountered by the green operator is `` destroyed '' ( that is to say removed from the operator string ) . assuming a left ( or right ) move , creating an operator will update the state @xmath44 ( or @xmath41 ) , while destroying will update the state @xmath41 ( or @xmath44 ) . when a diagonal configuration of the green operator occurs , @xmath45 , such a configuration associated to the extended partition function ( [ extendedpartitionfunction ] ) is also a configuration associated to the partition function ( [ partitionfunction ] ) . measurements can be done when this occurs ( see ref.@xcite for details on measurements ) . next section presents a simple update scheme that meets the requirements of ergodicity and detailed balance . before introducing the directed update , we start by simplifying the update scheme used in the original sgf algorithm . we will assume in the following that a left move of the green operator is chosen . in the original version , the green operator @xmath26 can choose to create or not on its right a @xmath2 operator at time @xmath43 . then a time shift @xmath46 to the left is chosen for the green operator with an exponential distribution in the range @xmath47 . if an operator is encountered while shifting the green operator , then the operator is destroyed and the move stops there . as a result , four possible situations can occur during one move : 1 . no creation , shift , no destruction . 2 . creation , shift , no destruction . 3 . no creation , shift , destruction . 4 . creation , shift , destruction . it appears that the first possibility `` no creation , no destruction '' is actually useless , since no change is performed in the operator string . the idea is to get rid of this possibility by forcing the green operator to destroy an operator if no creation is chosen a further simplification can be done by noticing that the last possibility `` creation , destruction '' is not necessary for the ergodicity of the algorithm , and can be avoided by restricting the range of the time shift after having created an operator . therefore we replace the original update scheme by the following : we assume that the green operator is acting at time @xmath43 and that the operator on its left is acting at time @xmath35 . the green operator @xmath26 chooses to create or not an operator on its right at time @xmath43 . if creation is chosen , then a time shift @xmath46 of the green operator is chosen to the left in the range @xmath48 , with the probability distribution defined below . if no creation is chosen , then the green operator is directly shifted to the operator on its left at time @xmath35 , and the operator is destroyed . as a result only two possibilities have to be considered : 1 . creation , shift . 2 . shift , destruction . figure [ simplfiedupdatescheme ] shows the associated organigram . section iii.b explains how detailed balance can be satisfied with this simplified update scheme . when updating the configurations according to the chosen update scheme , we need to generate different transitions from initial to final states with probabilities that satisfy detailed balance . in this section we propose a choice for these probabilities , and determine the corresponding acceptance factors . we denote the probability of the initial ( final ) configuration by @xmath49 ( @xmath50 ) . we denote by @xmath51 the probability of the transition from configuration @xmath17 to configuration @xmath52 , and by @xmath53 the probability of the reverse transition . finally we denote by @xmath54 the acceptance rate of the transition from @xmath17 to @xmath52 , and by @xmath55 the acceptance rate of the reverse transition . the detailed balance can be written as @xmath56 we will make use of the metropolis solution @xcite , @xmath57 with @xmath58 we will use primed ( non - primed ) labels for states and time indices to denote final ( initial ) configurations . we consider here the case where a left move is chosen , an operator is created on the right of the green operator at time @xmath43 , and a new state is chosen . then a time shift to the left is chosen for the green operator in the range @xmath59 . it is important to note that @xmath60 and @xmath61 correspond to the time indices of the operators appearing on the left and the right of the green operator after the new operator has been inserted , that is to say at the moment where the time shift needs to be performed . thus we have @xmath62 and @xmath63 . the probability of the initial configuration is the boltzmann weight appearing in the extended partition function ( [ extendedpartitionfunction ] ) : @xmath64 the probability of the final configuration takes the form : @xmath65 it is important here to realize that the green operator only inserted on its right the operator @xmath66 , before being shifted from @xmath61 to @xmath67 . therefore we have the equalities @xmath68 , @xmath69 , @xmath70 , and @xmath71 . the probability @xmath51 of the transition from the initial configuration to the final configuration is the probability @xmath72 of a left move , times the probability @xmath73 of a creation , times the probability @xmath74 to choose the new state @xmath75 , times the probability @xmath76 to shift the green operator by @xmath77 , knowing that the states on the left and the right of the green operator at the moment of the shift are @xmath78 and @xmath79 : @xmath80 the probability of the reverse transition is simply the probability @xmath81 of a right move , times the probability of no creation , @xmath82 : @xmath83\ ] ] from the original version of the sgf algorithm , we know that choosing the time shift with an exponential distribution is a good choice , because it cancels the exponentials appearing in the probabilities of the initial ( [ initial ] ) and final ( [ final ] ) configurations , avoiding exponentially small acceptance factors . however a different normalization must be used here , since the time shift is chosen in the range @xmath84 instead of @xmath47 . the suitable solution is : @xmath85 it is straightforward to check that the above probability is correctly normalized and well - defined for any real value of @xmath86 , the particular case @xmath87 reducing to the uniform distribution @xmath88 ( note that @xmath89 is always a positive number ) . for the probability @xmath74 to choose the new state @xmath75 , the convenient solution is the same as in the original version : @xmath90 putting everything together , the acceptance factor ( [ metropolis2 ] ) becomes @xmath91\big[1-e^{-(\tau_l^\prime-\tau_r^\prime)(v_r^\prime - v_l^\prime)}\big]}{v_r^\prime - v_l^\prime},\end{aligned}\ ] ] where we have used the notation @xmath92 to emphasize that this acceptance factor corresponds to a creation . it is also important for the remaining of this paper to note that @xmath92 is written as a quantity that depends on the initial configuration , times a quantity that depends on the final configuration . we consider here the case where a left move is chosen , and the operator on the left of the green operator is destroyed . this move corresponds to the inverse of the above `` creation , shift '' move . thus , the corresponding acceptance factor @xmath93 is obtained by inverting the acceptance factor @xmath92 , exchanging the initial time @xmath43 and final time @xmath67 , and switching the direction . however @xmath94 represents an absolute time shift , so @xmath35 and @xmath36 do not have to be exchanged . we get @xmath95\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big ] } \\ & \times & \frac{\big\langle\psi_l^\prime\big|\hat\mathcal g\big|\psi_r^\prime\big\rangle p(\rightarrow^\prime)p_\rightarrow^\dagger(\tau^\prime)}{\big\langle\psi_l^\prime\big|\hat\mathcal t\hat\mathcal g\big|\psi_r^\prime\big\rangle},\end{aligned}\ ] ] which is written as a quantity that depends on the initial configuration , times a quantity that depends on the final configuration . we will use here the short notation @xmath96 , @xmath97 , and @xmath98 to denote respectively the quantities @xmath99 , @xmath100 , and @xmath101 . as in ref . @xcite , we have some freedom for the choice of the probabilities of choosing a left or right move , @xmath72 and @xmath102 , and the probabilities of creation @xmath73 and @xmath103 . a suitable choice for those probabilities can be done in order to accept all moves , resulting in an appreciable simplification of the algorithm . for this purpose , we impose the acceptance factor @xmath92 ( or @xmath104 ) to be equal to the acceptance factor @xmath93 ( or @xmath105 ) . this allows to determine the probabilities @xmath73 and @xmath103 , @xmath106 and the acceptance factors @xmath107 and @xmath108 take the form @xmath109 with @xmath110 finally we can impose the acceptance factors @xmath111 and @xmath112 to be equal . this implies @xmath113 defining @xmath114 , we are left with a single acceptance factor , @xmath115 which is independent of the chosen direction , and independent of the nature of the move ( creation or destruction ) . thus all moves can be accepted by making use of a proper reweighting , as explained in ref . the appendix shows how to generate random numbers with the appropriate exponential distribution ( [ exponentialdistribution ] ) . although the above simplified update scheme works , it turns out to have a poor efficiency . this is because of a lack of `` directionality '' : the green operator has , in average , a probability of @xmath116 to choose a left move or a right move . therefore the green operator propagates along the operator string like a `` drunk man '' , with a diffusion - like law . the basic creation and destruction processes correspond to the steps of the random walk . this suggests that the efficiency of the update scheme can be improved if one can force the green operator to move in the same direction for several iterations . next section presents a modified version of the simplified update scheme , which allows to control the mean length of the steps of the random walk , that is to say the mean number of creations and destructions in a given direction . the proposed directed update scheme can be considered analogous to the `` directed loop update '' used in the stochastic series expansion algorithm @xcite , which prevents a worm from going backwards . however the connection should not be pushed too far . indeed the picture of a worm whose head is evolving both in space and imaginary time accross vertices is obvious in a loop algorithm . in such algorithm , a creation ( or an annihilation ) operator which is represented by the head of a worm is propagated both in space and imaginary time , while an annihilation ( or a creation ) operator represented by the tail of the worm remains at rest . the loop ends when the head of the worm bites the tail . such a worm picture is not obvious in the sgf algorithm : instead of single creation or annihilation operators , it is the full green operator over the whole space that is propagated only in imaginary time . this creates open worldlines , thus introducing discontinuities . these discontinuities increase or decrease while propagating in imaginary time . all open ends of the worldlines are localized at the same imaginary time index . therefore it is actually not possible to draw step by step a worm whose head is evolving in space and imaginary time until it bites its tail . we present in this section a directed update scheme which is obtained by modifying slightly the simplified update scheme , thus keeping the simplicity and generality of the algorithm . assuming that a left move is chosen , the green operator chooses between starting the move by a creation or a destruction . after having created ( or destroyed ) an operator , the green operator can choose to keep moving in the same direction and destroy ( or create ) with a probability @xmath117 ( or @xmath118 ) , or to stop . if it keeps moving , then a destruction ( or creation ) occurs , and the green operator can choose to keep moving and create ( or destroy ) with a probability @xmath118 ( or @xmath117 ) ... and so on , until it decides to stop . if the last action of the move is a creation , then a time shift is chosen . the organigram is represented in figure [ directedupdatescheme ] . in order to satisfy detailed balance , in addition to the acceptance factors @xmath92 and @xmath93 , we need to determine new acceptance factors of the form @xmath119 and @xmath120 . we first determine the new expressions of @xmath92 and @xmath93 resulting from the directed update scheme . for @xmath92 , the previous probability @xmath51 has to be multiplied by the probability to stop the move after having created , @xmath121 . the previous probability @xmath53 has to be multiplied by the probability to stop the move after having destroyed , @xmath122 . we get for @xmath92 and @xmath93 the new expressions : @xmath123}{\big\langle\psi_l\big|\hat\mathcal g\big|\psi_r\big\rangle p(\leftarrow)p_\leftarrow^\dagger(\tau ) } \\ & \times & \frac{p(\rightarrow^\prime)\big[1-p_\rightarrow^\dagger(\tau^\prime)\big]\big[1-e^{-(\tau_l^\prime-\tau_r^\prime)(v_r^\prime - v_l^\prime)}\big]}{\big[1-p_\leftarrow^{kd}(\tau^\prime)\big]\big(v_r^\prime - v_l^\prime\big ) } \\ \nonumber q_\leftarrow^d & = & \frac{\big[1-p_\rightarrow^{kd}(\tau)\big]\big(v_l - v_r\big)}{p(\leftarrow)\big[1-p_\leftarrow^\dagger(\tau)\big]\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big ] } \\ & \times & \frac{\big\langle\psi_l^\prime\big|\hat\mathcal g\big|\psi_r^\prime\big\rangle p(\rightarrow^\prime)p_\rightarrow^\dagger(\tau^\prime)}{\big\langle\psi_l^\prime\big|\hat\mathcal t\hat\mathcal g\big|\psi_r^\prime\big\rangle\big[1-p_\leftarrow^{kc}(\tau^\prime)\big]},\end{aligned}\ ] ] we consider here the case where a left move is chosen , an operator is created on the right of the green operator , and a new state is chosen . then the operator on the left of the green operator is destroyed . using the superscripts @xmath124 to denote intermediate configurations between initial and final configurations , the sequence is the following 1 . @xmath125 2 . @xmath126 3 . @xmath127 , where we have @xmath128 , @xmath129 , @xmath130 , and @xmath131 . the probability of the transition from the initial configuration to the final configuration is the probability @xmath72 to choose a left move , times the probability @xmath73 to create an operator at time @xmath43 , times the probability @xmath132 to choose the new state @xmath133 , times the probability @xmath134 to keep moving and destroy , times the probability @xmath135 to stop the move after having destroyed : @xmath136\ ] ] the probability of the reverse move is exactly symmetric : @xmath137\ ] ] it is important to notice that , when in the intermediate configuration @xmath7 , the time @xmath138 of the operator to the left of the green operator is equal to @xmath35 , and the time @xmath139 of the operator to the right of the green operator is equal to @xmath43 . thus the acceptance factor takes the form @xmath140}{\big\langle\psi_l\big|\hat\mathcal g\big|\psi_r\big\rangle p(\leftarrow)p_\leftarrow^\dagger(\tau ) } \\ \nonumber & \times & \frac{e^{-\big(\tau_l^a-\tau_r^a\big)v_r^a}p_\rightarrow^{kd}(a)}{e^{-\big(\tau_l^a-\tau_r^a\big)v_l^a}p_\leftarrow^{kd}(a ) } \\ & \times & \frac{\big\langle\psi_l^\prime\big|\hat\mathcal g\big|\psi_r^\prime\big\rangle p(\rightarrow^\prime)p_\rightarrow^\dagger(\tau^\prime)}{\big\langle\psi_l^\prime\big|\hat\mathcal t\hat\mathcal g\big|\psi_r^\prime\big\rangle\big[1-p_\leftarrow^{kc}(\tau^\prime)\big]},\end{aligned}\ ] ] and is written as a quantity that depends on the initial configuration , times a quantity that depends on the intermediate configuration @xmath7 , times a quantity that depends on the final configuration . it is useful for the remaining of the paper to define the intermediate acceptance factor , @xmath141 we consider here the case where a left move is chosen , the operator on the left of the green operator is destroyed , then an operator is created on its right , and a new state is chosen . finally a time shift is chosen . the sequence of configurations is the following 1 . @xmath125 2 . @xmath142 3 . @xmath127 , where we have @xmath143 , and @xmath144 . the probability of the transition from the initial configuration to the final configuration is the probability @xmath72 to choose a left move , times the probability @xmath145 of no creation , times the probability @xmath146 to keep moving and create , times the probability @xmath74 to choose the new state @xmath75 , times the probability @xmath121 to stop the move after having destroyed , times the probability @xmath76 to shift the green operator by @xmath77 : @xmath147p_\leftarrow^{kc}(a)p_\leftarrow(\psi_r^\prime ) \\ & \times & \big[1-p_\leftarrow^{kd}(\tau^\prime)\big]p_\leftarrow^{l^\prime r^\prime}(\tau^\prime-\tau_r^\prime)\end{aligned}\ ] ] the probability of the reverse move is exactly symmetric : @xmath148p_\rightarrow^{kc}(a)p_\rightarrow(\psi_l ) \\ & \times & \big[1-p_\rightarrow^{kd}(\tau)\big]p_\rightarrow^{lr}(\tau_l-\tau)\end{aligned}\ ] ] the acceptance factor takes the form @xmath149\big(v_l - v_r\big)}{p(\leftarrow)\big[1-p_\leftarrow^\dagger(\tau)\big]\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big ] } \\ \nonumber & \times & \frac{\big\langle\psi_l^a\big|\hat\mathcal g\hat\mathcal t\big|\psi_r^a\big\rangle p_\rightarrow^{kc}(a)}{\big\langle\psi_l^a\big|\hat\mathcal t\hat\mathcal g\big|\psi_r^a\big\rangle p_\leftarrow^{kc}(a ) } \\ & \times & \frac{p(\rightarrow^\prime)\big[1-p_\rightarrow^\dagger(\tau^\prime)\big]\big[1-e^{-(\tau_l^\prime-\tau_r^\prime)(v_r^\prime - v_l^\prime)}\big]}{\big[1-p_\leftarrow^{kd}(\tau^\prime)\big]\big(v_r^\prime - v_l^\prime\big)},\end{aligned}\ ] ] and is written as a quantity that depends on the initial configuration , times a quantity that depends on the intermediate configuration @xmath7 , times a quantity that depends on the final configuration . it is useful for the remaining of the paper to define the intermediate acceptance factor , @xmath150 we consider here the case where a left move is chosen , an operator is created on the right of the green operator , then the operator on its left is destroyed , then a second operator is created on its right . finally , a time shift of the green operator is performed . the sequence of configurations is the following 1 . @xmath125 2 . @xmath126 3 . @xmath151 4 . @xmath152 , considering the intermediate configurations @xmath7 and @xmath153 between the intial and final configurations , it is easy to show that the corresponding acceptance factor can be written @xmath154 we consider here the case where a left move is chosen , the operator on the left of the green operator is destroyed , then an operator is created on its right . finally a second operator on the left of green operator is destroyed . the sequence of configurations is the following 1 . @xmath155 2 . @xmath156 3 . @xmath157 4 . @xmath127 , considering the intermediate configurations @xmath7 and @xmath153 between the intial and final configurations , it is easy to show that the corresponding acceptance factor can be written @xmath158 it is straighforward to show that the acceptance factors of the form @xmath159 , @xmath160 , @xmath161 ( or @xmath162 , @xmath163 , @xmath164 ) can be expressed as products of the acceptance factor @xmath92 ( or @xmath93 ) and the intermediate factors @xmath165 and @xmath166 . in the same manner , the acceptance factors of the form @xmath167 , @xmath168 , @xmath169 ( or @xmath170 , @xmath171 , @xmath172 ) can be expressed as products of the acceptance factor @xmath173 ( or @xmath174 ) and the intermediate factors @xmath165 and @xmath166 . here again it is possible to take advantage of the freedom that we have for the choice of the probabilities @xmath72 , @xmath175 , @xmath118 , and @xmath117 ( or @xmath102 , @xmath176 , @xmath177 , and @xmath178 ) . a proper choice of these probabilities can be done in order to allow us to accept all moves , simplicity and generality being the leitmotiv of the sgf algorithm . for this purpose , we impose to all acceptance factors corresponding to left ( or right ) moves to be equal . this requires the intermediate acceptance factors @xmath165 and @xmath166 ( or @xmath179 and @xmath180 ) to be equal to 1 . this is realized if @xmath181 where @xmath182 and @xmath183 are optimization parameters belonging to @xmath184 . by tuning these parameters , the mean length of the steps of the green operator can be controlled . note that we have explicitly excluded @xmath185 from the allowed values for these optimization parameters . this is necessary for the green operator to have a chance to end in a diagonal configuration , @xmath45 . indeed , the choice @xmath186 would systematically lead to values of @xmath185 for the probabilities @xmath187 and @xmath188 for diagonal configurations . therefore the green operator would never stop in a diagonal configution , and no measurement could be done . it is important here to note that the quantities @xmath96 , @xmath97 , and @xmath98 are evaluated between the states on the left and the right of the green operator that are present at the moment where those quantities are needed , as well as for the times indices @xmath189 and @xmath190 and the potentials @xmath191 and @xmath192 . all acceptance factors corresponding to a given direction of propagation become equal if we choose for the creation probabilities : @xmath193(v_l - v_r)}{\big[1-p_\rightarrow^{kc}\big]\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big ] } } \\ & & p_\rightarrow^\dagger(\tau)=\frac{\big\langle\hat\mathcal t\hat\mathcal g\big\rangle}{\big\langle\hat\mathcal t\hat\mathcal g\big\rangle+\big\langle\hat\mathcal g\big\rangle\frac{\big[1-p_\leftarrow^{kd}\big](v_r - v_l)}{\big[1-p_\leftarrow^{kc}\big]\big[1-e^{-(\tau_l-\tau_r)(v_r - v_l)}\big]}},\end{aligned}\ ] ] finally , all acceptances factors become independant of the direction of propagation if we choose @xmath194 and @xmath195 with @xmath196\frac{\big\langle\hat\mathcal g\hat\mathcal t\big\rangle}{\big\langle\hat\mathcal g\big\rangle}+\frac{\big[1-p_\rightarrow^{kd}\big](v_l - v_r)}{\big[1-e^{-(\tau_l-\tau_r)(v_l - v_r)}\big ] } \\ r_\rightarrow(\tau)=\big[1-p_\leftarrow^{kc}\big]\frac{\big\langle\hat\mathcal t\hat\mathcal g\big\rangle}{\big\langle\hat\mathcal g\big\rangle}+\frac{\big[1-p_\leftarrow^{kd}\big](v_r - v_l)}{\big[1-e^{-(\tau_l-\tau_r)(v_r - v_l)}\big]}.\end{aligned}\ ] ] as a result all moves can be accepted again , ensuring the maximum of simplicity of the algorithm . we still have some freedom for the choice of the optimization parameters @xmath182 and @xmath183 . this is discussed in next section . from the central limit theorem , we know that the errorbar associated to any measured quantity must decrease as the square root of the number of measurements , or equivalently , the square root of the time of the simulation . therefore it makes sense to define the efficiency @xmath197 of a qmc algorithm by @xmath198 where @xmath199 represents the set of all optimization parameters of the algorithm , @xmath200 is the measured quantity of interest , @xmath201 is the time of the simulation , and @xmath202 is the errorbar associated to the measured quantity @xmath200 . this definition ensures that @xmath197 is independent of the time of the simulation . as a result , the larger @xmath197 the more efficient the algorithm . in the present case we have @xmath203 , while @xmath204 for the original sgf algorithm . it is useful here to realize that , by symmetry , the mean values of @xmath118 and @xmath177 ( and @xmath117 and @xmath178 ) must be equal . therefore we define @xmath205 and @xmath206 . it seems reasonable to impose a condition of uniform sampling , @xmath207 . this condition can be satisfied by adjusting dynamically the values of @xmath182 and @xmath183 during the thermalization process . for this purpose we introduce a new optimization parameter @xmath208 and apply the following algorithm from time to time while thermalizing ( we start with @xmath209 ) : @xmath210 thus we are left with the optimization parameter @xmath211 . in order to determine the optimal value , we have considered 2 different hamiltonians @xmath212 and @xmath213 , and evaluated the efficiency of the algorithm while scanning @xmath211 . the first hamiltonian we have considered describes free hardcore bosons and is exactly solvable , @xmath214 where the sum runs over pairs of first neighboring sites and @xmath215 is the hopping parameter . the second hamiltonian is highly non - trivial and describes a mixture of atoms and diatomic molecules , with a special term allowing conversions between the two species @xcite , @xmath216 where @xmath217 and @xmath218 ( @xmath219 and @xmath220 ) are the creation and annihilation operators of atoms ( molecules ) , @xmath221 , @xmath222 , @xmath223 , @xmath224 , and @xmath225 are respectively the hopping parameter of atoms , the hopping parameter of molecules , the atomic onsite interaction parameter , the molecular onsite interaction parameter , and the inter - species interaction parameter . the conversion term is tunable via the parameter @xmath226 and does not conserve the number @xmath227 of atoms or the number @xmath228 of molecules . however the total number of particles @xmath229 is conserved and is the canonical constraint . the parameter @xmath230 allows to control the ratio between the number of atoms and molecules . the application of the sgf algorithm to the hamiltonian ( [ twospecies ] ) is described in details in ref.@xcite . the changes coming with the directed update scheme are completely independent of the chosen hamiltonian . the following table shows the mean number of creations and destructions in one step , @xmath231 , and the relative efficiency @xmath232 of the algorithm applied to @xmath212 at half filling , for which we have measured the energy @xmath233 , the superfluid density @xmath234 , and the number of particles in the zero momentum state @xmath235 : . relative efficiency of the algorithm applied to @xmath212 at half filling for the energy , the superfluid density , and the number of particles in the zero momentum state . [ cols="^,^,^,^,^",options="header " , ] while the best value of @xmath211 depends on the hamiltonian which is considered and the measured quantity , it appears that a good compromise is to choose @xmath211 between @xmath236 and @xmath237 . the improvment of the efficiency is remarkable . in the following , we illustrate the applicability of the algorithm to problems with non - uniform potentials , by adding a parabolic trap to the hamiltonian ( [ twospecies ] ) : @xmath238 the parameters @xmath239 and @xmath240 allow to control the curvature of the trap associated to atoms and molecules , respectively , and @xmath31 is the number of lattice sites . the inclusion of this term in the algorithm is trivial since only the values of the diagonal energies @xmath241 and @xmath242 are changed . figures ( [ density ] ) and ( [ momentum ] ) show the density profiles and momentum distribution functions obtained for a system with @xmath243 lattice sites initially loaded with @xmath244 atoms and no molecules , and the parameters @xmath245 , @xmath246 , @xmath247 , @xmath248 , @xmath249 , @xmath250 , @xmath251 , @xmath252 , @xmath253 , and @xmath254 . the presented results have been obtained by performing @xmath255 updates for thermalization , and @xmath256 updates with measurements ( an update is to be understood as the occurence of a diagonal configuration ) . the time of the simulation is about 8 hours on a cheap 32 bits laptop with 1ghz processor , with an implementation of the algorithm involving dynamical structures with pointers ( see ref.@xcite ) . ) to the hamiltonian ( [ twospecies ] ) . the errorbars are smaller than the symbol sizes , and are the biggest in the neighborhood of site indices 23 and 47 where they equal the size of the symbols . , scaledwidth=45.0% ] ) to the hamiltonian ( [ twospecies ] ) . the errorbars are smaller than the symbol sizes , and are the biggest for @xmath257 where they equal the size of the symbols . , scaledwidth=45.0% ] we have presented a directed update scheme for the sgf algorithm , which has the properties of keeping the simplicity and generality of the original algorithm , and improves significantly its efficiency . i would like to express special thanks to peter denteneer for useful suggestions . this work is part of the research program of the `` stichting voor fundamenteel onderzoek der materie ( fom ) , '' which is financially supported by the `` nederlandse organisatie voor wetenschappelijk onderzoek ( nwo ) . '' we describe here how to generate numbers with the appropriate exponential distribution ( [ exponentialdistribution ] ) . assuming that we have at our disposal a uniform random number generator that generates a random variable @xmath258 with the distribution @xmath259 for @xmath260 , we would like to find a function @xmath52 such that the random variable @xmath261 is generated with the distribution @xmath262 where @xmath46 and @xmath263 are the parameters of the exponential distribution . because of the relation @xmath261 , the probability to find @xmath264 in the range @xmath265 must be equal to the probability to find @xmath258 in the range @xmath266 . this implies the condition @xmath267 with @xmath268 . thus we have @xmath269 taking the anti - derivative with respect to @xmath270 on both sides of the equation , we get @xmath271 where @xmath272 is a constant . this constant and the correct sign are determined by imposing the conditions @xmath273 and @xmath274 . as a result , if @xmath270 is a realization of @xmath258 , then a realization of @xmath264 is given by @xmath275.\ ] ] 10 nicholas metropolis and s. ulam , journal of the american statistical association , number 247 , volume 44 ( 1949 ) . handscomb , proc . 58 , 594 ( 1962 ) . kalos , phys . 128 , 1791 ( 1962 ) . r. blankenbecler , d.j . scalapino and r.l . sugar , phys . d 24 , 2278 ( 1981 ) . g.g . batrouni and r.t . scalettar , phys . b * 46 * , 9051 ( 1992 ) . w. von der linden , phys . rep . 220 , 53 ( 1992 ) . evertz , g. lana and m. marcu , phys . 70 , 875 - 879 ( 1993 ) . ceperley , rev . 67 , 279 ( 1995 ) . beard and u .- j . wiese , phys . 77 5130 ( 1996 ) . `` quantum monte carlo methods in physics and chemistry '' , ed . m.p . nightingale and c.j . umrigar , nato science series c 525 , kluwer academic publishers , dordrecht , ( 1999 ) . sandvik , j. phys . a * 25 * , 3667 ( 1992 ) ; phys . rev . b * 59 * , 14157 ( 1999 ) . n.v . prokofev , b.v . svistunov , and i.s . tupitsyn , jetp lett . * 87 * , 310 ( 1998 ) . m. rigol , a. muramatsu , g.g . batrouni , and r.t . scalettar , phys . lett . * 91 * , 130403 ( 2003 ) . k. van houcke , s.m.a . rombouts , and l. pollet , phys . e * 73*,056703 ( 2006 ) . rousseau , phys . e * 77 * , 056705 ( 2008 ) . sandvik , s. daul , r.r.p . singh , and d.j . lett . * 89 * , 247201 ( 2002 ) . rousseau , r.t . scalettar , and g.g . batrouni , phys . b * 72 * , 054524 ( 2005 ) . n. metropolis , a.w . rosenbluth , m.n . metropolis , a.h . teller , and e. teller , j. chem . phys . * 21 * , 1087 ( 1953 ) . olav f. syljuasen , anders w. sandvik , phys . e * 66 * , 046701 ( 2002 ) . rousseau and p.j.h . denteneer , phys . a * 77 * , 013609 ( 2008 ) .

in a recent publication we have presented the stochastic green function ( sgf ) algorithm , which has the properties of being general and easy to apply to any lattice hamiltonian of the form @xmath0 , where @xmath1 is diagonal in the chosen occupation number basis and @xmath2 has only positive matrix elements . we propose here a modified version of the update scheme that keeps the simplicity and generality of the original sgf algorithm , and enhances significantly its efficiency .

0806.1410

model selection is an important problem in many areas including machine learning . if a proper model is not selected , any effort for parameter estimation or prediction of the algorithm s outcome is hopeless . given a set of candidate models , the goal of model selection is to select the model that best approximates the observed data and captures its underlying regularities . model selection criteria are defined such that they strike a balance between the _ goodness - of - fit ( gof ) _ , and the _ generalizability _ or _ complexity _ of the models . goodness - of - fit measures how well a model capture the regularity in the data . generalizability / complexity is the assessment of the performance of the model on unseen data or how accurately the model fits / predicts the future data . models with higher complexity than necessary can suffer from overfitting and poor generalization , while models that are too simple will underfit and have low gof @xcite . cross - validation @xcite , bootstrapping @xcite , akaike information criterion ( aic ) @xcite , and bayesian information criterion ( bic ) @xcite , are well known examples of traditional model selection . in re - sampling methods such as cross - validation and bootstraping , the generalization error of the model is estimated using monte carlo simulation . in contrast with re - sampling methods , the model selection methods like aic and bic do not require validation to compute the model error , and are computationally efficient . in these procedures an _ information criterion _ is defined such that the generalization error is estimated by penalizing the model s error on observed data . a large number of information criteria have been introduced with different motivations that lead to different theoretical properties . for instance , the tighter penalization parameter in bic favors simpler models , while aic works better when the dataset has a very large sample size . kernel methods are strong , computationally efficient analytical tools that are capable of working on high dimensional data with arbitrarily complex structure . they have been successfully applied in wide range of applications such as classification , and regression . in kernel methods , the data are mapped from their original space to a higher dimensional feature space , the reproducing kernel hilbert space ( rkhs ) . the idea behind this mapping is to transform the nonlinear relationships between data points in the original space into an easy - to - compute linear learning problem in the feature space . for example , in kernel regression the response variable is described as a linear combination of the embedded data . any algorithm that can be represented through dot products has a kernel evaluation . this operation , called kernelization , makes it possible to transform traditional , already proven , model selection methods into stronger , corresponding kernel methods . the literature on kernel methods has , however , mostly focused on kernel selection and on tuning the kernel parameters , but only limited work being done on kernel - based model selection @xcite . in this study , we investigate a kernel - based information criterion for ridge regression models . in kernel ridge regression ( krr ) , tuning the ridge parameters to find the most predictive subspace with respect to the data at hand and the unseen data is the goal of the kernel model selection criterion . in classical model selection methods the performance of the model selection criterion is evaluated theoretically by providing a consistency proof where the sample size tends to infinity and empirically through simulated studies for finite sample sizes . other methods investigate a probabilistic upper bound of the generalization error @xcite . proving the consistency properties of the model selection in _ kernel model selection _ is challenging . the proof procedure of the classical methods does not work here . some reasons for that are : the size of the model to evaluate problems such as under / overfitting @xcite is not apparent ( for @xmath1 data points of dimension @xmath2 , the kernel is @xmath3 , which is independent of @xmath2 ) and asymptotic probabilities of generalization error or estimators are hard to compute in rkhs . researchers have kernelized the traditional model selection criteria and shown the success of their kernel model selection empirically . kobayashi and komaki @xcite extracted the kernel - based regularization information criterion ( kric ) using an eigenvalue equation to set the regularization parameters in kernel logistic regression and support vector machines ( svm ) . rosipal et al . @xcite developed covariance information criterion ( cic ) for model selection in kernel principal component analysis , because of its outperformed results compared to aic and bic in orthogonal linear regression . demyanov et al . @xcite , provided alternative way of calculating the likelihood function in akaike information criterion ( aic , @xcite and bayesian information criterion ( bic , @xcite ) , and used it for parameter selection in svms using the gaussian kernel . as pointed out by van emden @xcite , a desirable model is the one with the fewest dependent variables . thus defining a complexity term that measures the interdependency of model parameters enables one to select the most desirable model . in this study , we define a novel variable - wise variance and obtain a complexity measure as the additive combination of kernels defined on model parameters . formalizing the complexity term in this way effectively captures the interdependency of each parameter of the model . we call this novel method _ kernel - based information criterion ( kic)_. model selection criterion in gaussian process regression ( gpr ; @xcite ) , and kernel - based information complexity ( icomp ; @xcite ) resemble kic in using a covariance - based complexity measure . however , the methods differ because these complexity measures capture the interdependency between the data points rather than the model parameters . although we can not establish the consistency properties of kic theoretically , we empirically evaluate the efficiency of kic both on synthetic and real datasets obtaining state - of - the - art results compared to leave - one - out - cross - validation ( loocv ) , kernel - based icomp , and maximum log marginal likelihood in gpr . the paper is organized as follows . in section [ sec : krr ] , we give an overview of kernel ridge regression . kic is described in detail in section [ sec : kic ] . section [ sec : om ] is provides a brief explanation of the methods to which kic is compared , and in section [ sec : exp ] we evaluate the performance of kic through sets of experiments . in regression analysis , the regression model of the form : @xmath4 where @xmath5 can be either a linear or non - linear function . in linear regression we have , @xmath6 , where @xmath7 is an observation vector ( response variable ) of size @xmath8 , @xmath9 is a full rank data matrix of independent variables of size @xmath10 , and @xmath11 , is an unknown vector of regression parameters , where @xmath12 denotes the transposition . we also assume that the error ( noise ) vector @xmath13 is an @xmath1-dimensional vector whose elements are drawn i.i.d , @xmath14 , where @xmath15 is an @xmath1-dimensional identity matrix and @xmath16 is an unknown variance . the regression coefficients minimize the squared errors , @xmath17 , between estimated function @xmath18 , and target function @xmath5 . when @xmath19 , the problem is ill - posed , so that some kind of regularization , such as tikhanov regularization ( ridge regression ) is required , and the coefficients minimize the following optimization problem @xmath20 where @xmath21 is the regularization parameter . the estimated regression coefficients in ridge regression @xmath22 are : @xmath23 in _ kernel _ ridge regression ( krr ) , the data matrix @xmath9 is non - linearly transformed in rkhs using a feature map @xmath24 . the estimated regression coefficients based on @xmath25 are : @xmath26 where @xmath27 is the kernel matrix . equation [ eq : theta ] does not obtain an explicit expression for @xmath28 because of @xmath24 ( the kernel trick enables one to avoid explicitly defining @xmath25 that could be numerically intractable if computed in rkhs , if known ) , thus a ridge estimator is used ( e.g. @xcite ) that excludes @xmath24 : @xmath29 using @xmath30 in the calculation of krr is similar to regularizing the regression function instead of the regression coefficients , where the objective function is : @xmath31 and @xmath32 denotes the relevant rkhs . for @xmath33 , and @xmath34 we have : @xmath35 where @xmath36 is the kernel function , and @xmath37 . the main contribution of this study is to introduce a new kernel - based information criterion ( kic ) for the model selection in kernel - based regression . according to equation kic balances between the goodness - of - fit and the complexity of the model . gof is defined using a log - likelihood - based function ( we maximize penalized log likelihood ) and the complexity measure is a function based on the covariance function of the parameters of the model . in the next subsections we elaborate on these terms . the definition of van emden @xcite for the complexity measure of a random vector is based on the interactions among random variables in the corresponding covariance matrix . a desirable model is the one with the fewest dependent variables . this reduces the information entropy and yields lower complexity . in this paper we focus on this definition of the complexity measures . considering a @xmath2-variate normal distribution @xmath38 , the complexity of a covariance matrix , @xmath39 , is given by the shannon s entropy @xcite , @xmath40 where @xmath41 , @xmath42 are the marginal and the joint entropy , and @xmath43 is the @xmath44 diagonal element of @xmath39 . @xmath45 if and only if the covariates are independent . the complexity measure in equation changes with orthonormal transformations because it is dependent on the coordinates of the random variable vectors @xmath46 @xcite . to overcome these drawbacks , bozodgan and haughton @xcite introduced icomp information criterion with a complexity measure based on the maximal covariance complexity , which is an upper bound on the complexity measure in equation : @xmath47 this complexity measure is proportional to the estimated arithmetic ( @xmath48 ) and geometric mean ( @xmath49 ) of the eigenvalues of the covariance matrix . larger values of @xmath50 , indicates higher dependency between random variables , and vice versa . zhang @xcite introduced a kernel form of this complexity measure @xmath50 , that is computed on kernel - based covariance of the ridge estimator : @xmath51 the complexity measure in gaussian process regression ( gpr ; @xcite ) is defined as @xmath52 , a concept from the joint entropy @xmath42 ( as shown in equation [ eq : complexity ] ) . in contrast to icomp and gpr , the complexity measure in kic is defined using the hilbert - schmidt ( hs ) norm of the covariance matrix , @xmath53 . minimizing this complexity measure obtains a model with more independent variables . in the next sections , we explain in detail how to define the needed variable - wise variance in the complexity measure , and the computation of the complexity measure . + in kernel - based model selection methods such as icomp , and gpr , the complexity measure is defined on a covariance matrix that is of size @xmath54 for @xmath9 of size @xmath10 . the idea behind this measure is to compute the interdependency between the model parameters , which independent of the number of the model parameters @xmath2 . in the other words , the concept of the size of the model is hidden because of the definition of a kernel . to have a complexity measure that depends on @xmath2 , we introduce variable - wise variance using an additive combination of kernels for each parameter of the model . let @xmath55 be the parameter vector of the kernel ridge regression : @xmath56 where @xmath57 and @xmath58 , and @xmath59 the solution of krr is given by @xmath60 . the quantity @xmath61 = \sigma^2 \operatorname{tr}[k(k+\alpha i)^{-2 } ] $ ] can be interpreted as the sum of variances for the component - wise parameter vectors , if the following sum of component - wise kernel is introduced : @xmath62 where @xmath63 and @xmath64 denote the j - th component of vectors @xmath65 and @xmath66 . with this sum kernel , the function @xmath67 can be written as : @xmath68 where @xmath69 is a function in @xmath70 , the rkhs defined by @xmath71 . the parameter @xmath28 in this case is given by @xmath72 where @xmath73 , and thus @xmath69 in equation [ eq : g ] is equal to @xmath74 . let @xmath75 be the conditional covariance of @xmath74 or @xmath69 given @xmath76 . we have @xmath77,\end{aligned}\ ] ] where @xmath78 be the gram matrix with @xmath71 . since @xmath79 , we have @xmath80 = \operatorname{tr}[\sigma_{\theta}].\end{aligned}\ ] ] formalizing the complexity term with variable - wise variance effectively captures the interdependency of each parameter of the model ( measures the significance of the contribution by the variables ) explicitly . + gretton et al . @xcite introduced a kernel - based independence measure , namely the hilbert - schmidt independence criterion ( hsic ) , which is explained here . suppose @xmath81 , and @xmath82 are random vectors with feature maps @xmath83 , and @xmath84 , where @xmath85 , and @xmath86 are rkhss . the cross - covariance operator corresponding to the joint probability distribution @xmath87 is a linear operator , @xmath88 such that : @xmath89,\end{aligned}\ ] ] where @xmath90 denotes the tensor product , @xmath91= e[k(\cdot , x)]$ ] , and @xmath92=e[k(\cdot , y)]$ ] , for @xmath93 , and associated kernel function @xmath36 . the hsic measure for separable rkhs @xmath85 , and @xmath86 is the squared hs - norm of the cross - covariance operator and is denoted as : @xmath94\end{aligned}\ ] ] * theorem 1 . * assume @xmath95 , and @xmath96 are compact , for all @xmath97 , and @xmath98 , @xmath99 , and @xmath100 , @xmath101 if and only if @xmath102 , and @xmath7 are independent ( theorem 4 in @xcite ) . + by computing the hsic on covariance matrix associated with model s parameters @xmath103 we can measure the independence between the parameters . since @xmath104 is a symmetric positive semi - definite matrix , @xmath105 , and the trace of the hs norm of the covariance matrix is equal to : @xmath106 = \sum_{j=1}^p v_j^2\nonumber\\ ~~&= \sigma^4 \operatorname{tr}[k(k+\alpha i)^{-2 } k(k+\alpha i)^{-2}]\end{aligned}\ ] ] kic is defined as : @xmath107 where @xmath108 is the complexity term based on equation [ eq : hs ] . the normalization by @xmath109 obtains a complexity measure that is robust to changes in variance ( similar to icomp criterion ) . the minimum kic defines the best model . ] . the penalized log - likelihood ( pll ) in krr for normally distribution data is defined by : @xmath110 the unknown parameters @xmath22 , and @xmath111 are calculated by minimizing the kic objective function . @xmath112 we also investigated the effect of using @xmath113 $ ] , and @xmath61 $ ] as complexity terms . the empirical results reported in subsection [ subsec : realdata ] on real datasets , and compared with kic . we denote these information criteria as : @xmath114,\end{aligned}\ ] ] @xmath115.\end{aligned}\ ] ] in both kic_1 , and kic_2 , similar to kic , @xmath116 , while because the complexity term is dependent on @xmath16 , @xmath111 for kic_1 is : @xmath117=0.\end{aligned}\ ] ] if we denote @xmath118 , @xmath111 is the solution of a quadratic optimization problem , @xmath119 , where @xmath120 . in the case of kic_2 , the @xmath111 is the real root of the following cubic problem : @xmath121 where @xmath122 $ ] . we compared kic with loocv @xcite , kernel - based icomp @xcite , and maximum log of marginal likelihood in gpr ( abbreviated as gpr ) @xcite to find the optimal ridge regressors . the reason to compare kic with icomp and gpr is that in all of these methods the complexity measure computes the interdependency of model parameters as a function of covariance matrix in different ways . loocv is a standard and commonly used methods for model selection . * loocv : * re - sampling model selection methods like cross - validation is time consuming @xcite . for instance , the leave - one - out - cross - validation ( loocv ) has the computational cost of @xmath123 the number of parameter combinations ( @xmath124 is the processing time of the model selection algorithm @xmath125 ) for @xmath126 training samples . to have cross - validation methods with faster processing time , the closed form formula for the risk estimators of the algorithm under special conditions are provided . we consider the kernel - based closed form of loocv for linear regression introduced by @xcite : @xmath127^{-1}[i - h]y\|_2 ^ 2}{n}\end{aligned}\ ] ] where @xmath128 is the hat matrix . * maximizing the log of marginal likelihood ( gpr ) * is a kernel - based regression method . for a given training set @xmath129 , and @xmath130 , a multivariate gaussian distribution is defined on any function @xmath5 such that , @xmath131 , where @xmath39 is a kernel . marginal likelihood is used as the model selection criterion in gpr , since it balances between the lack - of - fit and complexity of a model . maximizing the log of marginal likelihood obtains the optimal parameters for model selection . the log of marginal likelihood is denoted as : @xmath132 where @xmath133 denotes the model s fit , @xmath134 , denotes the complexity , and @xmath135 is a normalization constant . without loss of generality in this paper gpr means the model selection criterion is used in gpr . * icomp : * the kernel - based icomp introduced in @xcite is an information criterion to select the models and is defined as @xmath136 , where @xmath50 , and @xmath39 elaborated in equations [ eq : cicomp ] , and [ eq : sigmaicomp ] . in this section we evaluate the performance of kic on synthetic , and real datasets , and compare with competing model selection methods . kic was first evaluated on the problem of approximating @xmath137 from a set of 100 points sampled at regular intervals in @xmath138 $ ] . to evaluate robustness to noise , normal random noise was added to the @xmath139 function at two noise - to - signal ( nsr ) ratios : @xmath140 , and @xmath141 . figure [ sinc ] shows the sinc function and the perturbed datasets . the following experiments were conducted : ( 1 ) shows how kic balances between gof and complexity , ( 2 ) shows how kic and mse on training sets change when the sample size and the level of noise in the data change ( 3 ) investigates the effect of using different kernels , and ( 4 ) evaluates the consistency of kic in parameter selection . all experiments were run 100 times using randomly generated datasets , and corresponding test sets of size 1000 . * experiment 1 . * the effect of @xmath21 on complexity , lack - of - fit and kic values was measured by setting @xmath142 , with krr models being generated using a gaussian kernel with different standard deviations , @xmath143 , computed over the 100 data points . the results are shown in figure [ co_la_kic ] . the model generated with @xmath144 overfits , because it is overly complex , while @xmath145 gives a simpler model that underfits . as the ridge parameter @xmath21 increases , the model complexity decreases while the goodness - of - fit is adversely affected . kic balances between these two terms , which yields a criterion to select a model that has good generalization , as well as goodness of fit to the data . * experiment 2 . * the influence of training sample size was investigated by comparing sample sizes , @xmath1 , of 50 , and 100 , for a total of four sets of experiments : ( @xmath146 ) : ( @xmath147 ) , ( @xmath148 ) , ( @xmath149 ) , ( @xmath150 ) . the gaussian kernel was used with @xmath151 . the kic value and mean squared error ( mse , @xmath152 ) , for different @xmath153 @xmath154 is shown in figure [ kic - mse ] . the data with nsr=@xmath141 has larger mse values , and larger error bars , and consequently larger kic values compared to data with nsr=@xmath140 . in both cases , kic and mse change with similar profiles with respect to @xmath21 . the noise and the sample size have no effect on kic for selecting the best model ( parameter @xmath21 ) . * experiment 3 . * the effect of using a gaussian kernel , @xmath155 , versus the cauchy kernel , @xmath156 , was investigated , where @xmath157 , and @xmath158 in the computation of the kernel - based model selection criteria icomp , kic , gpr , and loocv . the results are reported in figures [ gaussian kernel ] and [ cauchy kernel ] . the graphs show box plots with markers at @xmath159 , and @xmath160 of the empirical distributions of mse values . as expected , the mse of all methods is larger when nsr is high , @xmath161 , and smaller for the larger of the two training sets ( 100 samples ) . loocv , icomp , and kic performed comparably , and better than gpr using a gaussian kernel for data with nsr @xmath162 . in the other cases , the best results ( smallest mse ) was achieved by kic . all methods have smaller mse values using the gaussian kernel versus the cauchy kernel . gpr with the cauchy kernel obtains results comparable with kic , but with a standard deviation close to zero . * experiment 4 . * we assessed the consistency of selecting / tuning the parameters of the models in comparison with loocv . we considered four experiment of sample size , @xmath163 , and nsr @xmath164 . the parameters to tune or select are @xmath165 @xmath166 , and @xmath167 for the gaussian kernel . the frequency of selecting the parameters are shown in figure [ loocv ] for loocv , and in figure [ kic_frequency ] for kic . the more concentrated frequency shows the more consistent selecting criterion . the diagrams show that kic is more consistent in selecting the parameters rather than loocv . loocv is also sensitive to sample size . it provides a more consistent result for benchmarks with @xmath168 samples . + we used three benchmarks selected from the delve datasets ( www.cs.toronto.edu/~delve/data ) : ( 1 ) abalone dataset ( 4177 instances , 7 dimensions ) , ( 2 ) kin - family of datasets ( 4 datasets ; 8192 instances , 8 dimensions ) , and ( 3 ) puma - family of datasets ( 4 datasets ; 8192 instances , 8 dimensions ) . for the abalone dataset , the task is to estimate the age of abalones . we used normalized attributes in range [ 0,1 ] . the experiment is repeated 100 times to obtain the confidence interval . in each trial 100 samples were selected randomly as the training set and the remaining 4077 samples as the test set . the kin - family and puma - family datasets are realistic simulations of a robot arm taking into consideration combinations of attributes such as whether the arm movement is nonlinear ( n ) or fairly linear ( f ) , and whether the level of noise ( unpredictability ) in the data is : medium ( m ) , or high ( h ) . the kin - family includes : kin-8fm , kin-8fh , kin-8 nm , kin-8nh datasets , and the puma - family contains : puma-8fm , puma-8fh , puma-8 nm , and puma-8nh datasets . in the kin - family of datasets , having the angular positions of an 8-link robot arm , the distance of the end effector of the robot arm from a starting position is predicted . the angular position of a link of the robot arm is predicted given the angular positions , angular velocities , and the torques of the links . we compared kic_1 ( [ eq : kic1 ] ) , kic_2 ( [ eq : kic2 ] ) , and kic with loocv , icomp , and gpr on the three datasets . the results are shown as box - plots in figures [ abalone ] , [ kin - family ] , and [ puma - family ] for abalone , kin - family , and puma - family datasets , respectively . the best results across all three datasets were achieved using kic , and the second best results were for loocv . for the abalone dataset , comparable results were achieved for kic and loocv , that are better than icomp , and the smallest mse value obtained by sgpr . kic_1 , and kic_2 had similar mse values , which are larger than for the other methods . for the kin - family datasets , except for kin-8fm , kic gets better results than gpr , icomp , and loocv . kic_1 , and kic_2 obtain better results than gpr , and loocv for kin-8fm , and kin-8 nm , which are datasets with medium level of noise , but larger mse value for datasets with high noise ( kin-8fh , and kin-8nh ) . for the puma - family datasets , kic got the best results on all datasets except for on puma-8 nm , where the smallest mse was achieved by loocv . the result of kic is comparable to icomp and better than gpr for puma-8 nm dataset . for puma-8fm , puma-8fh , and puma-8nh , although the median of mse for loocv and gpr are comparable to kic , kic has a more significant mse ( smaller interquartile in the box bots ) . the median mse value for kic_1 , and kic_2 are closer to the median mse values of the other methods on puma-8fm , and puma-8 nm , where the noise level is moderate compared to puma-8fh , and puma-8nh , where the noise level is high . the sensitivity of kic_1 , and kic_2 to noise is due to the existence of variance in their formula . kic_2 has a larger interquartile of mse than kic_1 in datasets with high noise , which highlights the effect of @xmath109 in its formula ( equation [ eq : kic2 ] ) rather than @xmath16 in equation . we introduced a novel kernel - based information criterion ( kic ) for model selection in regression analysis . the complexity measure in kic is defined on a variable - wise variance which explicitly computes the interdependency of each parameter involved in the model ; whereas in methods such as kernel - based icomp and gpr , this interdependency is defined on a covariance matrix , which obscures the true contribution of the model parameters . we provided empirical evidence showing how kic outperforms loocv ( with kernel - based closed form formula of the estimator ) , kernel - based icomp , and gpr , on both artificial data and real benchmark datasets : abalon , kin family , and puma family . in these experiments , kic efficiently balances the goodness of fit and complexity of the model , is robust to noise ( although for higher noise we have larger confidence interval as expected ) and sample size , is consistent in tuning / selecting the ridge and kernel parameters , and has significantly smaller or comparable mean squared values with respect to competing methods , while yielding stronger regressors . the effect of using different kernels was also investigated since the definition of a proper kernel plays an important role in kernel methods . kic had superior performance using different kernels and for the proper one obtains smaller mse . this work was funded by fnsnf grants ( p1tip2_148352 , pbtip2_140015 ) . we want to thank arthur gretton , and zoltn szab for the fruitful discussions .

this paper introduces kernel - based information criterion ( kic ) for model selection in regression analysis . the novel kernel - based complexity measure in kic efficiently computes the interdependency between parameters of the model using a variable - wise variance and yields selection of better , more robust regressors . experimental results show superior performance on both simulated and real data sets compared to leave - one - out cross - validation ( loocv ) , kernel - based information complexity ( icomp ) , and maximum log of marginal likelihood in gaussian process regression ( gpr ) .

1408.5810

this work on pricing american options under proportional transaction costs goes back to the seminal discovery by @xcite that to hedge against a buyer who can exercise the option at any ( ordinary ) stopping time , the seller must in effect be protected against all mixed ( randomised ) stopping times . this was followed by @xcite , who established a non - constructive dual representation for the set of strategies superhedging the seller s ( though not the buyer s ) position in an american option under transaction costs . efficient iterative algorithms for computing the upper and lower hedging prices of the option , the hedging strategies , optimal stopping times as well as dual representations for both the seller and the buyer of an american option under transaction costs were developed by @xcite in a model with two assets , and @xcite in a multi - asset model . all these approaches take it for granted that the buyer can only exercise the option instantly , at an ordinary stopping time of his choosing . by contrast , in the present paper we allow the buyer the flexibility to exercise an american option gradually , rather than all at a single time instance . though it would be difficult in practice to exercise a fraction of an option contract and to hold on to the reminder to exercise it later , the holder of a large portfolio of options may well choose to exercise the individual contracts on different dates if that proves beneficial . does this ability to exercise gradually affect the pricing bounds , hedging strategies and optimal stopping times for the buyer and/or seller ? perhaps surprisingly , the answer to this question is yes , it does in the presence of transaction costs . gradual exercise turns out to be linked to another feature , referred to as deferred solvency , which will also be studied here . if a temporary loss of liquidity occurs in the market , as reflected by unusually large bid - ask spreads , agents may become insolvent . being allowed to defer closing their positions until liquidity is restored might enable them to become solvent once again . this gives more leeway when constructing hedging strategies than the usual requirement that agents should remain solvent at all times . @xcite was the first to explore the consequences of gradual exercise and deferred solvency using a model with a single risky asset as a testing ground . in the present paper these ideas are developed in a systematic manner and extended to the much more general setting of the multi - asset market model with transaction costs due to @xcite ; see also @xcite and @xcite . pricing and hedging for the seller of an american option under transaction costs is a convex optimisation problem irrespective of whether instant or gradual exercise is permitted . however , this is not so for the buyer . in this case one has to tackle a non - convex optimisation problem for options that can only be exercised instantly . a very interesting consequence of gradual exercise is that pricing and hedging becomes a convex optimisation problem also for the buyer of an american option , making it possible to deploy convex duality methods . the convexity of the problem also makes it much easier to implement the pricing and hedging algorithms numerically . we will make use of this new opportunity in this paper . the paper is organised as follows . section [ sect - multi - curr - mod ] recalls the general setting of kabanov s multi - asset model with transaction costs . in section [ sect : inst - versus - grad - exe ] the hedging strategies for the buyer and seller and the corresponding option prices under gradual exercise are introduced and compared with the same notions under instant exercise . a toy example is set up to demonstrate that it is easier to hedge an option and that the bid - ask spread of the option prices can be narrower under gradual exercise as compared to instant exercise . in section [ sect : seller ] the seller s case is studied in detail . the notion of deferred solvency is first discussed and linked in proposition [ prop : am : seller : immediate - ultimate ] with the hedging problem for the seller of an american option with gradual exercise . the sets of seller s hedging portfolios are then constructed and related to the ask price of the option under gradual exercise and to a construction of a seller s hedging strategy realising the ask price ; see theorem [ prop : seller : zau0=initial - endowments ] . a dual representation of the seller s price is established in theorem [ thm : ask - price - representation ] . the toy example is revisited to illustrate the various constructions and results for the seller . section [ sect : buyer ] is devoted to the buyer s case . buyer s hedging portfolios and strategies are constructed and used to compute the bid price of the option ; see theorem [ prop:2012 - 07 - 26:hedging - construct ] . finally , the dual representation for the buyer is explored in theorem [ th : bu - buyer ] . once again , the toy example serves to illustrate the results . a numerical example with three assets can be found in section [ sec : num - example ] . some conclusions and possible further developments and ramifications are touched upon in section [ sect : conclusions ] . technical information and proofs are collected in the appendix . let @xmath0 be a filtered probability space . we assume that @xmath1 is finite , @xmath2 , @xmath3 and @xmath4 for all @xmath5 . for each @xmath6 let @xmath7 be the collection of atoms of @xmath8 , called the _ nodes _ of the associated tree model . a node @xmath9 is said to be a _ successor _ of a node @xmath10 if @xmath11 . for each @xmath12 we denote the collection of successors of any given node @xmath10 by @xmath13 . for each @xmath6 let @xmath14 be the collection of @xmath8-measurable @xmath15-valued random variables . we identify elements of @xmath16 with functions on @xmath7 whenever convenient . we consider the discrete - time currency model introduced by @xcite and developed further by @xcite and @xcite among others . the model contains @xmath17 assets or currencies . at each trading date @xmath18 and for each @xmath19 one unit of asset @xmath20 can be obtained by exchanging @xmath21 units of asset @xmath22 . we assume that the exchange rates @xmath23 are @xmath8-measurable and @xmath24 for all @xmath25 and @xmath26 . we say that a portfolio @xmath27 is can be _ exchanged _ into a portfolio @xmath28 at time @xmath25 whenever there are @xmath8-measurable random variables @xmath29 , @xmath30 such that for all @xmath31 @xmath32 where @xmath33 represents the number of units of asset @xmath20 received as a result of exchanging some units of asset @xmath22 . the _ solvency cone _ @xmath34 is the set of portfolios that are _ solvent _ at time @xmath25 , i.e. the portfolios at time @xmath25 that can be exchanged into portfolios with non - negative holdings in all @xmath17 assets . it is straightforward to show that @xmath35 is the convex cone generated by the canonical basis @xmath36 of @xmath15 and the vectors @xmath37 for @xmath30 , and so @xmath35 is a polyhedral cone , hence closed . note that @xmath35 contains all the non - negative elements of @xmath16 . a _ trading strategy _ @xmath38 is a predictable @xmath15-valued process with final value @xmath39 and initial endowment @xmath40 . for each @xmath41 the portfolio @xmath42 is held from time @xmath43 to time @xmath25 . let @xmath44 be the set of trading strategies . we say that @xmath45 is a _ self - financing _ strategy whenever @xmath46 for all @xmath12 . note that no implicitly assumed self - financing condition is included in the definition of @xmath44 . a trading strategy @xmath45 is an _ arbitrage opportunity _ if it is self - financing , @xmath47 and there is a portfolio @xmath48 with non - negative holdings in all @xmath17 assets such that @xmath49 . this notion of arbitrage was considered by @xcite , and its absence is formally different but equivalent to the weak no - arbitrage condition introduced by @xcite . [ th:2012 - 10 - 03:ftap ] the model admits no arbitrage opportunity if and only if there exists a probability measure @xmath50 equivalent to @xmath51 and an @xmath15-valued @xmath50-martingale @xmath52 such that @xmath53 where @xmath54 is the polar of @xmath55 ; see ( [ eq:2012 - 09 - 20:aast ] ) in the appendix . we denote by @xmath56 the set of pairs @xmath57 satisfying the conditions in theorem [ th:2012 - 10 - 03:ftap ] , and by @xmath58 the set of pairs @xmath57 satisfying the conditions in theorem [ th:2012 - 10 - 03:ftap ] but with @xmath50 absolutely continuous with respect to ( and not necessarily equivalent to ) @xmath51 . we assume for the remainder of this paper that the model admits no arbitrage opportunities , i.e. @xmath59 . in place of a pair @xmath60 one can equivalently use the so - called _ consistent price process _ @xmath61 ; see @xcite . we also define for any @xmath62 @xmath63 in the absence of arbitrage @xmath54 is a non - empty compactly @xmath64-generated polyhedral cone for all @xmath25 ( * ? ? ? * remark 2.2 ) , which means that @xmath65 . ( for the definition of a compactly @xmath64-generated cone , see appendix [ subsect : comp - gen - cones ] . ) the payoff of an american option in the model with @xmath17 underlying currencies is , in general , an @xmath15-valued adapted process @xmath66 . the seller of the american option is obliged to deliver , and the buyer is entitled to receive the portfolio of currencies @xmath67 at a stopping time @xmath68 chosen by the buyer . here @xmath69 denotes the family of stopping times with values in @xmath70 . this is the usual setup in which the option is exercised _ instantly _ at a stopping time @xmath68 . american options with the provision for instant exercise in the multi - currency model under proportional transaction costs have been studied by @xcite , who established a non - constructive characterisation of the superhedging strategies for the option seller only , and by @xcite , who provided computationally efficient iterative constructions of the ask and bid option prices and the superhedging strategies for both the option seller and buyer . in the present paper we relax the requirement that the option needs to be exercised instantly at a stopping time @xmath68 . instead , we allow the buyer to exercise _ gradually _ at a mixed stopping time @xmath71 . ( for the definition of mixed stopping times , see appendix [ sect : mixed - stop - times ] . ) if the buyer chooses to exercise the option gradually according to a mixed stopping time @xmath71 , then the seller of the american option will be obliged to deliver , and the buyer will be entitled to receive the fraction @xmath72 of the portfolio of currencies @xmath73 at each time @xmath6 . the question then arises whether or not it would be more beneficial for the buyer to exercise the option gradually rather than instantly ? what will be the optimal mixed stopping time @xmath71 for the buyer ? how should the seller hedge against gradual exercise ? are the ask ( seller s ) and bid ( buyer s ) option prices and hedging strategies affected by gradual exercise as compared to instant exercise ? in the case of instant exercise the seller of an american option @xmath74 needs to hedge by means of a trading strategy @xmath45 against all ordinary stopping times @xmath68 chosen by the buyer . the trading strategy @xmath75 needs to be self - financing up to time @xmath76 and to allow the seller to remain solvent on delivering the portfolio @xmath67 at time @xmath76 , for any @xmath68 . hence the family of seller s superhedging strategies is defined as @xmath77 and the _ ask price _ ( _ seller s price _ ) of the option in currency @xmath62 is @xmath78 this is the smallest amount in currency @xmath64 needed to superhedge a short position in @xmath74 . on the other hand , the buyer of an american option @xmath74 can select both a stopping time @xmath68 and a trading strategy @xmath45 . the trading strategy @xmath75 needs to be self - financing up to time @xmath76 and to allow the buyer to remain solvent on receiving the portfolio @xmath67 at time @xmath76 . thus , the family of buyer s superhedging strategies is defined as @xmath79 and the _ bid price _ ( _ buyer s price _ ) of the option in currency @xmath62 is @xmath80 this is the largest amount in currency @xmath64 that the buyer can raise using the option @xmath74 as surety . for american options with instant exercise , iterative constructions of the ask and bid option prices @xmath81 and @xmath82 and the corresponding seller s and buyer s superhedging strategies from @xmath83 and @xmath84 were established by @xcite . when the buyer is allowed to exercise gradually , the seller needs to follow a suitable trading strategy to hedge his exposure . since the seller can react to the buyer s actions , this strategy may in general depend on the mixed stopping time @xmath71 followed by the buyer , and will be denoted by @xmath85 . in other words , we consider a function @xmath86 . at each time @xmath25 the seller will be holding a portfolio @xmath87 and will be obliged to deliver a fraction @xmath72 of the payoff @xmath73 . he can then rebalance the remaining portfolio @xmath88 into @xmath89 in a self - financing manner , so that @xmath90the self - financing and superhedging conditions have merged into one . we call ( [ eq : seller - self - fin - superhedge ] ) the _ rebalancing _ condition . when creating the portfolio @xmath87 at time @xmath43 , the seller can only use information available at that time . this includes @xmath91 , but the seller has no way of knowing the future values @xmath92 that will be chosen by the buyer . the trading strategies @xmath85 that can be adopted by the seller are therefore restricted to those satisfying the _ non - anticipation _ condition @xmath93 in particular , the initial endowment @xmath94 of the trading strategy @xmath95 is the same for all @xmath71 . we denote this common value by @xmath96 . we define the family of seller s superhedging strategies against gradual exercise by @xmath97 and the corresponding _ ask price _ ( _ seller s price _ ) of the option in currency @xmath98 by @xmath99 this is the smallest amount in currency @xmath22 that the seller needs to superhedge a short position in the american option @xmath74 when the buyer is allowed to exercise gradually . on the other hand , the buyer is able to select both a mixed stopping time @xmath71 and a trading strategy @xmath45 , and will be taking delivery of a fraction @xmath72 of the payoff @xmath73 at each time @xmath25 . because the choice of the mixed stopping time @xmath100 is up to the buyer , the trading strategy @xmath75 needs to be good just for the one chosen stopping time , and does not need to be considered as a function of @xmath100 , in contrast to the seller s case . the _ rebalancing _ condition @xmath101 needs to be satisfied . hence , the family of superhedging strategies for the buyer of an american option @xmath74 with gradual exercise is defined as @xmath102 and the corresponding _ bid price _ ( _ buyer s price _ ) of the option in currency @xmath98 is @xmath103 this is the largest amount in currency @xmath22 that can be raised using the option as surety by a buyer who is able to exercise gradually . [ exl : new]we consider a toy example with two assets , a foreign currency ( asset 1 ) and domestic currency ( asset 2 ) in a two - step binomial tree model with the following bid / ask foreign currency prices @xmath104 in each of the four scenarios in @xmath105:@xmath106{|c|cc|cc|cc|}\hline $ \rule[-0.2cm]{0pt}{0.6cm}$ & $ s_{0}^{\mathrm{b}}$ & $ s_{0}^{\mathrm{a}}$ & $ s_{1}^{\mathrm{b}}$ & $ s_{1}^{\mathrm{a}}$ & $ s_{2}^{\mathrm{b}}$ & $ s_{2}^{\mathrm{a}}$\\\hline $ \omega_{1}$ & $ 5 $ & $ 5 $ & $ 3 $ & $ 9 $ & $ 4 $ & $ 8$\\\cline{6 - 7}$\omega_{2}$ & $ 5 $ & $ 5 $ & $ 3 $ & $ 9 $ & $ 4 $ & $ 4$\\\cline{4 - 7}$\omega_{3}$ & $ 5 $ & $ 5 $ & $ 2 $ & $ 2 $ & $ 3 $ & $ 3$\\\cline{6 - 7}$\omega_{4}$ & $ 5 $ & $ 5 $ & $ 2 $ & $ 2 $ & $ 1 $ & $ 1$\\\hline \end{tabular}\ ] ] note there are only two nodes with a non - trivial bid / ask spread , namely the ` up ' node @xmath107 and the ` up - up ' node @xmath108 . the corresponding exchange rates are @xmath109{cc}\pi_{t}^{11 } & \pi_{t}^{12}\\ \pi_{t}^{21 } & \pi_{t}^{22}\end{array } \right ] = \left [ \begin{array } [ c]{cc}1 & 1/s_{t}^{\mathrm{b}}\\ s_{t}^{\mathrm{a } } & 1 \end{array } \right ] .\ ] ] in this model we consider an american option with the following payoff process @xmath110:@xmath106{|c|c|c|c|}\hline $ \rule[-0.15cm]{0pt}{0.5cm}$ & $ \xi_{0}$ & $ \xi_{1}$ & $ \xi_{2}$\\\hline $ \omega_{1}$ & $ \left ( 0,0\right ) $ & $ \left ( 0,4\right ) $ & $ \left ( 2,-8\right ) $ \\\cline{4 - 4}$\omega_{2}$ & $ \left ( 0,0\right ) $ & $ \left ( 0,4\right ) $ & $ \left ( 0,0\right ) $ \\\cline{3 - 4}$\omega_{3}$ & $ \left ( 0,0\right ) $ & $ \left ( 0,0\right ) $ & $ \left ( 0,0\right ) $ \\\cline{4 - 4}$\omega_{4}$ & $ \left ( 0,0\right ) $ & $ \left ( 0,0\right ) $ & $ \left ( 0,0\right ) $ \\\hline \end{tabular}\ ] ] in the case when the option can only be exercised instantly , using the algorithms of @xcite we can compute the bid and ask prices of the option in the domestic currency to be@xmath111 now consider @xmath112 given by@xmath106{|c|c|c|c|}\hline $ \rule[-0.2cm]{0pt}{0.6cm}$ & $ y_{0}^{\chi}$ & $ y_{1}^{\chi}$ & $ y_{2}^{\chi}$\\\hline $ \omega_{1}\rule[-0.2cm]{0pt}{0.6cm}$ & $ \left ( 0,5\right ) $ & $ \left ( 1,0\right ) $ & $ \left ( 1,-4\chi_{1}^{\omega_{1}}\right ) $ \\ $ \omega_{2}\rule[-0.2cm]{0pt}{0.6cm}$ & $ \left ( 0,5\right ) $ & $ \left ( 1,0\right ) $ & $ \left ( 1,-4\chi_{1}^{\omega_{2}}\right ) $ \\\cline{4 - 4}$\omega_{3}\rule[-0.2cm]{0pt}{0.6cm}$ & $ \left ( 0,5\right ) $ & $ \left ( 1,0\right ) $ & $ \left ( 0,0\right ) $ \\ $ \omega_{4}\rule[-0.2cm]{0pt}{0.6cm}$ & $ \left ( 0,5\right ) $ & $ \left ( 1,0\right ) $ & $ \left ( 0,0\right ) $ \\\hline \end{tabular}\ ] ] for any @xmath71 . also consider @xmath45 and @xmath113 such that@xmath106{|c|c|c|c|c|c|c|}\hline $ \rule[-0.15cm]{0pt}{0.5cm}$ & $ y_{0}$ & $ y_{1}$ & $ y_{2}$ & $ \chi_{0}$ & $ \chi_{1}$ & $ \chi_{2}$\\\hline $ \omega_{1}\rule[-0.2cm]{0pt}{0.6cm}$ & $ \left ( 0,-3\right ) $ & $ \left ( -1,2\right ) $ & $ \left ( -1,4\right ) $ & $ 0 $ & $ \frac{1}{2}$ & $ \frac{1}{2}$\\ $ \omega_{2}\rule[-0.2cm]{0pt}{0.6cm}$ & $ \left ( 0,-3\right ) $ & $ \left ( -1,2\right ) $ & $ \left ( -1,4\right ) $ & $ 0 $ & $ \frac{1}{2}$ & $ \frac{1}{2}$\\\cline{4 - 4}\cline{6 - 7}$\omega_{3}\rule[-0.2cm]{0pt}{0.6cm}$ & $ \left ( 0,-3\right ) $ & $ \left ( -1,2\right ) $ & $ \left ( 0,0\right ) $ & $ 0 $ & $ 0 $ & $ 1$\\ $ \omega_{4}\rule[-0.2cm]{0pt}{0.6cm}$ & $ \left ( 0,-3\right ) $ & $ \left ( -1,2\right ) $ & $ \left ( 0,0\right ) $ & $ 0 $ & $ 0 $ & $ 1$\\\hline \end{tabular}\ ] ] we can verify that @xmath114 and @xmath115 . the existence of these strategies means that@xmath116 this example demonstrates that the seller s and buyer s prices @xmath117 under gradual exercise may differ from their respective counterparts @xmath118 under instant exercise . it demonstrates the need to revisit and investigate the pricing and superhedging results in the case when the instant exercise provision is relaxed and replaced by gradual exercise . we have seen in example [ exl : new ] that the seller s price @xmath119 may be higher than @xmath120 . the reason is that an option seller who follows a hedging strategy @xmath121 is required to be instantly solvent upon delivering the payoff at the stopping time @xmath68 when the buyer has chosen to exercise the option . meanwhile , a seller who follows a strategy @xmath122 will be able to continue rebalancing the strategy up to the time horizon @xmath123 as long as a solvent position can be reached eventually . being able to defer solvency in this fashion allows more flexibility for the seller , resulting in a lower seller s price . on the other hand , it might appear that a seller who hedges against gradual exercise ( against mixed stopping times ) would have a harder task to accomplish than someone who only needs to hedge against instant exercise ( ordinary stopping times ) . however , this turns out not to be a factor affecting the seller s price , as we shall see in proposition [ prop : am : seller : immediate - ultimate ] . these considerations indicate that the notion of solvency needs to be relaxed . we say that a portfolio @xmath124 satisfies the _ deferred solvency _ condition at time @xmath25 if it can be exchanged into a solvent portfolio by time @xmath123 without any additional investment , i.e. if there is a sequence @xmath125 such that @xmath126 for all @xmath127 and @xmath128 we call such a sequence @xmath125 a _ liquidation strategy _ starting from @xmath129 at time @xmath25 . the set of portfolios satisfying the deferred solvency condition at time @xmath25 is a cone . we call it the _ deferred solvency cone _ and denote by @xmath130 . in example [ exl : new ] the portfolio with @xmath131 in the domestic currency and @xmath132 in the foreign currency is insolvent at the ` up ' node @xmath133 at time @xmath134 , that is , @xmath135 . it does , however , satisfy the deferred solvency condition at that node , i.e. @xmath136 . the large bid - ask spread @xmath137=[3,9]$ ] at node @xmath138 indicates a temporary loss of liquidity . although the portfolio is insolvent at that node , waiting until the market recovers from the loss of liquidity can restore solvency . the liquidation strategy is to hold the portfolio until time @xmath139 and to buy the foreign currency then . the following result shows that the deferred solvency cones @xmath130 can be regarded as the sets of time @xmath25 superhedging portfolios for the seller of a european option with expiry time @xmath123 and zero payoff ; see @xcite . [ prop : constr : ultimate - solvent ] the deferred solvency cones can be constructed by backward induction as follows : @xmath140 the proof of proposition [ prop : constr : ultimate - solvent ] can be found in appendix [ sect : appendix : defer - solv ] . from ( [ eq : qt - recursive ] ) we can see that for any @xmath12 and for any @xmath10 @xmath141 by backward induction , @xmath142 is given as an intersection and algebraic sum of a finite number of polyhedral cones , so it is a polyhedral cone . this also means the solvency cones can readily be computed using standard operations on polyhedral convex sets . the next result shows that theorem [ th:2012 - 10 - 03:ftap ] can be formulated equivalently in terms of the deferred solvency cones @xmath130 instead of the solvency cones @xmath35 . [ prop : mart - ito - q ] if @xmath50 is a probability measure and @xmath52 is an @xmath15-valued @xmath50-martingale , then @xmath143 satisfies if and only if @xmath144 where @xmath145 is the polar of @xmath146 . the proof of proposition [ prop : mart - ito - q ] is in appendix [ sect : appendix : defer - solv ] . we extend the family @xmath83 of seller s superhedging strategies by allowing for deferred solvency : @xmath147 the following proposition shows that the set of initial endowments that allow the seller to hedge against gradual exercise is the same as that allowing to hedge against instant exercise with deferred solvency . [ prop : am : seller : immediate - ultimate ] for any american option @xmath74 @xmath148 for the proof of proposition [ prop : am : seller : immediate - ultimate ] , see appendix [ sect : appendix : technical : seller ] . we now present an iterative construction of the set of initial endowments that allow superhedging for the seller under deferred solvency . by proposition [ prop : am : seller : immediate - ultimate ] , this also gives the set of initial endowments that allow superhedging for the seller under gradual exercise . [ const : am : seller ] construct adapted sequences @xmath149 , @xmath150 , @xmath151 , @xmath152 for @xmath6 by @xmath153 it follows by backward induction that the sets @xmath154 are convex and polyhedral for each @xmath12 and @xmath10 because the algebraic sum and the intersection of a finite number of convex polyhedral sets are convex and polyhedral , and @xmath155 are convex polyhedral sets for each @xmath156 . moreover , @xmath149 , @xmath150 , @xmath151 , @xmath152 are non - empty for each @xmath6 because the portfolio @xmath157 belongs to all of them when @xmath158 is large enough . [ prop : seller : zau0=initial - endowments ] the set of initial endowments that superhedge the seller s position in the american option @xmath74 under gradual exercise is equal to @xmath159 and the ask ( seller s ) price of the option in currency @xmath98 can be computed as @xmath160 moreover , a strategy @xmath122 can be constructed such that @xmath161 the proof of theorem [ prop : seller : zau0=initial - endowments ] can be found in appendix [ sect : appendix : technical : seller ] . we can conclude that the set of initial endowments @xmath162 superhedging the seller s position , the option ask price @xmath163 , and a superhedging strategy @xmath164 realising the ask price can be computed by means of standard operations on convex polyhedral sets . working within the setting of example [ exl : new ] , we can now apply the constructions described in the current section to compute the sets @xmath165 of superhedging portfolios for the seller . these are sets of portfolios @xmath166 satisfying the inequalities@xmath106{|c|c|c|c|}\hline $ \rule[-0.2cm]{0pt}{0.6cm}$ & $ \mathcal{z}_{0}^{\mathrm{ad}}$ & $ \mathcal{z}_{1}^{\mathrm{ad}}$ & $ \mathcal{z}_{2}^{\mathrm{ad}}$\\\hline $ \omega_{1}\rule[-0.4cm]{0cm}{1.05cm}$ & $ 5x^{1}+x^{2}\geq5 $ & $ \begin{array } [ c]{l}8x^{1}+x^{2}\geq8\\ 4x^{1}+x^{2}\geq0 \end{array } $ & $ \begin{array } [ c]{l}8x^{1}+x^{2}\geq8\\ 4x^{1}+x^{2}\geq0 \end{array } $ \\\cline{4 - 4}$\omega_{2}\rule[-0.4cm]{0cm}{1.05cm}$ & $ 5x^{1}+x^{2}\geq5 $ & $ \begin{array } [ c]{l}8x^{1}+x^{2}\geq8\\ 4x^{1}+x^{2}\geq0 \end{array } $ & $ 4x^{1}+x^{2}\geq0$\\\cline{3 - 4}$\omega_{3}\rule[-0.4cm]{0cm}{1.05cm}$ & $ 5x^{1}+x^{2}\geq5 $ & $ 2x^{1}+x^{2}\geq0 $ & $ 3x^{1}+x^{2}\geq0$\\\cline{4 - 4}$\omega_{4}\rule[-0.4cm]{0cm}{1.05cm}$ & $ 5x^{1}+x^{2}\geq5 $ & $ 2x^{1}+x^{2}\geq0 $ & $ x^{1}+x^{2}\geq0$\\\hline \end{tabular}\ ] ] from @xmath167 we obtain the ask price @xmath168 we can also construct a superhedging strategy @xmath114 such that @xmath169 it is the strategy @xmath164 specified in example [ exl : new ] . a dual representation of the seller s price @xmath163 can be obtained with the aid of the support function @xmath170 of @xmath171 . for the definition of the support function of a convex set , see appendix [ subsect : conv - anal ] . more generally , let @xmath172 , @xmath173 , @xmath174 , @xmath175 be the support functions of the sets @xmath176 , @xmath177 , @xmath178 , @xmath179 of construction [ const : am : seller ] . the functions @xmath172 , @xmath173 @xmath174 , @xmath175 are polyhedral ( * ? ? ? * corollary 19.2.1 ) , hence continuous . proposition [ prop : seller : dual ] in appendix [ sect : appendix : technical : seller ] lists a number of properties of support functions , which will prove useful in what follows . [ prop:20130727:pi - ag - dual ] the seller s price of an american option @xmath74 with gradual exercise can be written as @xmath180 for some mixed stopping time @xmath71 , a probability measure @xmath50 and an @xmath15-valued adapted process @xmath143 such that @xmath181 and @xmath182 for all @xmath6 . such @xmath100 , @xmath50 and @xmath143 can be constructed by a recursive procedure . the notation @xmath183 , @xmath184 and @xmath185 used in proposition [ prop:20130727:pi - ag - dual ] is defined by ( [ eq:2012 - 10 - 03:sigmai ] ) , ( [ eq:20130726-x - chi - star ] ) and ( [ eq:20130726-x - stopped - at - chi ] ) . the proof is provided in appendix [ sect : appendix : technical : seller ] . for any @xmath71 denote by @xmath186 the set of pairs @xmath57 such that @xmath50 is a probability measure and @xmath143 is an @xmath15-valued adapted process satisfying ( [ eq : chi - approx - u ] ) . also define for @xmath62 @xmath187 the lack of arbitrage opportunities and proposition [ prop : mart - ito - q ] ensure that @xmath188 for all @xmath71 . the superscript @xmath189 indicating deferred solvency distinguishes @xmath186 and @xmath190 from the collections @xmath191 and @xmath192 defined by @xcite in a similar way as above , but with the weaker condition @xmath193 in place of . the following result provides a representation of @xmath163 dual to the representation ( [ eq:20130728:def - pi - ag ] ) in terms of superhedging strategies . [ thm : ask - price - representation ] the ask price in currency @xmath98 of an american option @xmath74 with gradual exercise can be written as @xmath194 moreover , we can algorithmically construct @xmath195 , and @xmath196 such that @xmath197 this theorem is proved in appendix [ sect : appendix : technical : seller ] . we continue working in the setting of example [ exl : new ] . the mixed stopping time @xmath195 and a pair @xmath198 such that @xmath199 are@xmath106{|c|c|c|c|c|c|c|c|}\hline $ \rule[-0.2cm]{0pt}{0.6cm}$ & $ \mathbb{\hat{q}}$ & $ \hat{s}_{0}$ & $ \hat { s}_{1}$ & $ \hat{s}_{2}$ & $ \hat{\chi}_{0}$ & $ \hat{\chi}_{1}$ & $ \hat{\chi } _ { 2}$\\\hline $ \omega_{1}\rule[-0.2cm]{0pt}{0.6cm}$ & $ 1 $ & $ ( 5,1)$ & $ ( 4,1)$ & $ ( 8,1)$ & $ 0 $ & $ \frac{3}{4}$ & $ \frac{1}{4}$\\\cline{2 - 2}\cline{5 - 5}$\omega_{2}\rule[-0.2cm]{0pt}{0.6cm}$ & $ 0 $ & $ ( 5,1)$ & $ ( 4,1)$ & $ ( 4,1)$ & $ 0 $ & $ \frac{3}{4}$ & $ \frac{1}{4}$\\\cline{0 - 2}\cline{4 - 5}\cline{7 - 8}$\omega_{3}\rule[-0.2cm]{0pt}{0.6cm}$ & $ 0 $ & $ ( 5,1)$ & $ ( 2,1)$ & $ ( 3,1)$ & $ 0 $ & $ 0 $ & $ 1$\\\cline{2 - 2}\cline{5 - 5}$\omega_{4}\rule[-0.2cm]{0pt}{0.6cm}$ & $ 0 $ & $ ( 5,1)$ & $ ( 2,1)$ & $ ( 1,1)$ & $ 0 $ & $ 0 $ & $ 1$\\\hline \end{tabular}\ ] ] the buyer of an american option @xmath74 is entitled to receive the payoff according to a mixed stopping time @xmath71 of his choosing . in other words , the buyer receives @xmath200 at each time @xmath6 . the family @xmath201 of superhedging strategies for the buyer and the bid price ( buyer s price ) @xmath202 under gradual exercise are defined in section [ sect : grad - exe ] . we turn to the task of computing the bid price and an optimal superhedging strategy for the buyer . we start by computing the set if initial endowments that allow superhedging for the buyer . [ const:2012 - 07 - 26.1 ] construct adapted sequences @xmath203 , @xmath204 , @xmath205 , @xmath206 for @xmath6 by @xmath207 for each @xmath25 the convex hull in is taken on each atom of @xmath8 , i.e. for all @xmath10 @xmath208 the index @xmath189 indicates that the deferred solvency cones @xmath130 are used in this construction . the sets @xmath203 , @xmath204 , @xmath205 , @xmath206 are non - empty for each @xmath6 because the portfolio @xmath157 belongs to all of them when @xmath158 is large enough . in contrast with construction 4.6 of @xcite , which was used the case of instant exercise at an ordinary stopping time , we have the convex hull of @xmath209 in ( [ eq : mathcal - zt ] ) rather than the union of sets . this means that @xmath203 , @xmath204 , @xmath205 , @xmath206 are convex sets , unlike their counterparts in construction 4.6 of @xcite . this is important because , once it is established in the next proposition that the @xmath206 are polyhedral , it becomes possible to implement techniques from convex analysis to compute them . [ prop:2012 - 09 - 19:zt - closed ] the set @xmath206 in construction [ const:2012 - 07 - 26.1 ] is polyhedral with recession cone @xmath130 for each @xmath6 . the proof of proposition [ prop:2012 - 09 - 19:zt - closed ] can be found in appendix [ sect : appendix : technical : buyer ] . the next result shows that construction [ const:2012 - 07 - 26.1 ] produces the set of initial endowments that superhedges @xmath74 for the buyer , which in turn makes it possible to compute the option bid price and also to construct a strategy that realises this price . this is similar to theorem [ prop : seller : zau0=initial - endowments ] for the seller . [ prop:2012 - 07 - 26:hedging - construct ] the set of initial endowments that superhedge the buyer s position in the american option @xmath74 with gradual exercise is equal to @xmath210 and the bid ( buyer s ) price of the option in currency @xmath98 can be computed as @xmath211 moreover , a strategy @xmath212 can be constructed such that @xmath213 the proof of this theorem is also in appendix [ sect : appendix : technical : buyer ] . still within the setting of example [ exl : new ] , we apply the constructions described in the current section to compute the sets @xmath214 of superhedging portfolios for the buyer . these are sets of portfolios @xmath166 satisfying the inequalities@xmath106{|c|c|c|c|}\hline $ \rule[-0.2cm]{0pt}{0.6cm}$ & $ \mathcal{z}_{0}^{\mathrm{bd}}$ & $ \mathcal{z}_{1}^{\mathrm{bd}}$ & $ \mathcal{z}_{2}^{\mathrm{bd}}$\\\hline $ \omega_{1}\rule[-0.6cm]{0cm}{1.45cm}$ & $ 5x^{1}+x^{2}\geq-3 $ & $ \begin{array } [ c]{c}8x^{1}+x^{2}\geq-8\\ 6x^{1}+x^{2}\geq-4\\ 4x^{1}+x^{2}\geq-4 \end{array } $ & $ \begin{array } [ c]{l}8x^{1}+x^{2}\geq-8\\ 4x^{1}+x^{2}\geq0 \end{array } $ \\\cline{1 - 1}\cline{4 - 4}$\omega_{2}\rule[-0.6cm]{0cm}{1.45cm}$ & $ 5x^{1}+x^{2}\geq-3 $ & $ \begin{array } [ c]{c}8x^{1}+x^{2}\geq-8\\ 6x^{1}+x^{2}\geq-4\\ 4x^{1}+x^{2}\geq-4 \end{array } $ & $ 4x^{1}+x^{2}\geq0$\\\cline{1 - 1}\cline{3 - 4}$\omega_{3}\rule[-0.6cm]{0cm}{1.45cm}$ & $ 5x^{1}+x^{2}\geq-3 $ & $ 2x^{1}+x^{2}\geq0 $ & $ 3x^{1}+x^{2}\geq0$\\\cline{1 - 1}\cline{4 - 4}$\omega_{4}\rule[-0.6cm]{0cm}{1.45cm}$ & $ 5x^{1}+x^{2}\geq-3 $ & $ 2x^{1}+x^{2}\geq0 $ & $ x^{1}+x^{2}\geq0$\\\hline \end{tabular}\ ] ] from @xmath215 we obtain the ask price @xmath216 we can also construct a superhedging strategy @xmath217 such that @xmath218 it is the strategy @xmath219 specified in example [ exl : new ] . since the @xmath220 , @xmath221 , @xmath222 , @xmath214 are convex , it becomes possible to apply convex duality methods not just in the seller s case but also in the buyer s case . ( this was impossible to do in @xcite for american options with instant exercise because of the lack of convexity in the buyer s case . ) in particular , in a similar way as in the proof of proposition [ prop:20130727:pi - ag - dual ] , we can show that the bid price of an american option with payoff @xmath74 under gradual exercise can be expressed as@xmath223 in terms of the support function @xmath224 of @xmath225 . however , we follow a different approach to obtain a representation of the bid price @xmath226 dual to the representation ( [ eq : buyer - bid - price - gradual ] ) of @xmath226 by means of superhedging strategies . in theorem [ prop:2012 - 07 - 26:hedging - construct ] a mixed stopping time @xmath71 has already been constructed as part of a superhedging strategy @xmath217 such that @xmath227 . as a result , the bid price given by ( [ eq : buyer - bid - price - gradual ] ) can be written as@xmath228 for this mixed stopping time @xmath100 . it turns out that the set on the right - hand side can be expressed by means of the family @xmath229 of superhedging strategies for the seller of a european option with expiry time @xmath123 and payoff @xmath230 as described in appendix [ sect : eur - opt ] , where @xmath231 is defined by ( [ eq:20130726-x - stopped - at - chi ] ) . [ prop : am - eur ] for any american option @xmath74 and any mixed stopping time @xmath232 we have@xmath233 this proposition is proved in appendix [ sect : appendix : technical : buyer ] . we are now in a position to state a representation of the bid price dual to ( [ eq : buyer - bid - price - gradual ] ) , and to prove it with the aid of proposition [ prop : am - eur ] . [ th : bu - buyer ] the buyer s ( bid ) price of an american option @xmath74 in currency @xmath98 can be represented as @xmath234 where @xmath184 is defined by ( [ eq:20130726-x - stopped - at - chi ] ) . moreover , we can algorithmically construct @xmath195 and @xmath235 such that @xmath236 the proof of theorem [ th : bu - buyer ] is in appendix [ sect : appendix : technical : buyer ] . we revisit example [ exl : new ] one more time to construct a mixed stopping time @xmath195 and a pair @xmath237 such that @xmath238 they are@xmath106{|c|c|c|c|c|c|c|c|}\hline $ \rule[-0.2cm]{0pt}{0.6cm}$ & $ \mathbb{\hat{q}}$ & $ \hat{s}_{0}$ & $ \hat { s}_{1}$ & $ \hat{s}_{2}$ & $ \hat{\chi}_{0}$ & $ \hat{\chi}_{1}$ & $ \hat{\chi } _ { 2}$\\\hline $ \omega_{1}\rule[-0.2cm]{0pt}{0.6cm}$ & $ 1 $ & $ ( 5,1)$ & $ ( 5,1)$ & $ ( 5,1)$ & $ 0 $ & $ \frac{1}{2}$ & $ \frac{1}{2}$\\\cline{2 - 2}\cline{5 - 5}$\omega_{2}\rule[-0.2cm]{0pt}{0.6cm}$ & $ 0 $ & $ ( 5,1)$ & $ ( 5,1)$ & $ ( 4,1)$ & $ 0 $ & $ \frac{1}{2}$ & $ \frac{1}{2}$\\\cline{0 - 2}\cline{4 - 5}\cline{7 - 8}$\omega_{3}\rule[-0.2cm]{0pt}{0.6cm}$ & $ 0 $ & $ ( 5,1)$ & $ ( 2,1)$ & $ ( 3,1)$ & $ 0 $ & $ 0 $ & $ 1$\\\cline{2 - 2}\cline{5 - 5}$\omega_{4}\rule[-0.2cm]{0pt}{0.6cm}$ & $ 0 $ & $ ( 5,1)$ & $ ( 2,1)$ & $ ( 1,1)$ & $ 0 $ & $ 0 $ & $ 1$\\\hline \end{tabular}\ ] ] in this section we present a three - dimensional numerical example with a realistic flavour to illustrate constructions [ const : am : seller ] and [ const:2012 - 07 - 26.1 ] . the numerical procedures below were implemented in _ maple _ with the aid of the _ convex _ package @xcite . consider a model involving a domestic currency and two foreign currencies , with time horizon @xmath239 and with * @xmath240 * time steps . the friction - free nominal exchange rates @xmath241 between the domestic currency and the two foreign currencies follow the two - asset recombinant @xcite model with cholesky decomposition . that is , there are @xmath242 possibilities for the exchange rates at each time step @xmath6 , indexed by pairs @xmath243 with @xmath244 , and each non - terminal node with exchange rates @xmath245 has four successors , associated with exchange rates @xmath246 , @xmath247 , @xmath248 and @xmath249 . with @xmath250 defined for convenience , the exchange rates are given by @xmath251 for @xmath6 and @xmath252 , where @xmath253 and @xmath254 are the initial exchange rates , @xmath255 and @xmath256 are the volatilities and @xmath257 is the correlation between the logarithmic growth of the exchange rates . assume that proportional transaction costs of @xmath258 are payable on all currency exchanges , except at time step @xmath134 , when @xmath259 is payable , modelling a temporary loss of liquidity . in other words , the matrix of exchange rates between each pair among the three currencies at each time step @xmath25 is @xmath260 where @xmath261 consider an american put option with physical delivery and strike @xmath262 on a basket containing one unit of each of the foreign currencies . it offers the payoff @xmath263 we allow for the possibility that the option may never be exercised by adding an extra time step @xmath264 to the model and setting the payoff to be @xmath265 at that time step . constructions [ const : am : seller ] and [ const:2012 - 07 - 26.1 ] give @xmath266 where @xmath267 is the convex cone generated by the vectors @xmath268 the sets @xmath269 and @xmath270 , which appear in figure [ fig:2 ] , yield the ask and bid prices @xmath271 in each of the three currencies . [ c]cc & + @xmath269 & @xmath270 in this paper we have explored american options with gradual exercise within kabanov s model @xcite of many assets under transaction costs , along with the related notion of deferred solvency , which helps to deal with a temporary loss of liquidity ( large bid - ask spreads ) in the market . we have demonstrated that gradual exercise ( at a mixed stopping time chosen by the buyer ) can reduce the ask ( seller s ) price and increase the bid ( buyer s ) price of the option compared with the case when the option can only be exercised instantly ( at an ordinary stopping time ) . in this context we have constructed and implemented algorithms to compute the ask and bid option prices , the buyer s and seller s optimal hedging portfolios and strategies , and their optimal mixed stopping times . we have studied dual representations for both the buyer and the seller of an american option with gradual exercise . the results have been illustrated by numerical examples . compared to options with instant exercise , a novel feature is that pricing and hedging an american option is a convex optimisation problem not just for the seller but also for the buyer of the option , making it possible to use convex duality in both cases . ramifications to be explored further may include an extension of bouchard and temam s representation of the strategies hedging the seller s ( short ) position @xcite to the case of hedging the buyer s ( long ) position in the option . we also conjecture that it should be possible to adapt the constructions presented here so that linear vector optimisation methods can be used to price and hedge both the seller s and buyer s positions in an american option with gradual exercise , along similar lines as was done by @xcite for european options under transaction costs . for any non - empty convex cone @xmath272 , denote by @xmath273 the _ polar _ of @xmath274 , i.e. @xmath275 for any set @xmath272 define the _ cone generated by @xmath276 _ as @xmath277 the _ recession cone _ of a non - empty convex set @xmath272 is defined as @xmath278 it is a convex cone containing the origin ( * ? ? ? * theorem 8.1 ) . if @xmath276 is a polyhedral cone , then @xmath279 ( * ? ? * corollary 8.3.2 ) . the _ convex hull _ of sets @xmath280 in @xmath15 is the smallest convex set in @xmath15 that contains @xmath280 , and is denoted by @xmath281 . the _ convex hull _ of convex functions @xmath282 is the function @xmath283 defined by @xmath284 the _ effective domain _ of a convex function @xmath285 is defined as @xmath286 the _ support function _ @xmath287 of a convex set @xmath272 is defined as @xmath288 equality follows directly from . a vector @xmath295 is an element of @xmath296 if and only if @xmath297 and @xmath298 , if and only if @xmath299 and @xmath300 and @xmath298 , if and only if @xmath301 and @xmath302 , if and only if @xmath303 . the set @xmath296 is compact since it is the intersection of two compact sets @xmath291 and @xmath304 . it remains to show that @xmath296 is non - empty and generates @xmath293 . to this end , fix any @xmath305 . as @xmath276 and @xmath306 are generated , respectively , by @xmath291 and @xmath304 , there exist @xmath307 , @xmath308 , @xmath309 and @xmath310 such that @xmath311 as @xmath312 , we must have @xmath313 and @xmath314 . moreover , since @xmath315 , we have @xmath316 which in turn implies @xmath317 , completing the proof . [ lem : roux_zastawniak2011 ] fix any @xmath62 , and suppose that @xmath280 are non - empty closed convex sets in @xmath15 such that @xmath318 and @xmath319 is compactly @xmath64-generated for all @xmath20 . then @xmath320 the cone @xmath321 is compactly @xmath64-generated and @xmath322 and for each @xmath323 there exist @xmath324 and @xmath325 with @xmath326 for all @xmath327 such that @xmath328 for any @xmath71 we put @xmath333moreover , for any adapted process @xmath334 and for any @xmath71 we put @xmath335 we also define _ @xmath334 evaluated at @xmath100 _ by @xmath336 with each ordinary stopping time @xmath68 we associate the mixed stopping time @xmath337 defined as @xmath338{cc}1 & \text{on } \{\tau = t\}\\ 0 & \text{on } \{\tau\neq t\ } \end{array } \right . \quad\text{for each } t=0,\ldots , t.\ ] ] equality ( [ eq : qt = kt ] ) is obvious . by the definition of the deferred solvency cones , for any @xmath12 the following conditions are equivalent : @xmath339 if and only if there is a @xmath340 such that @xmath341 if and only if there is a sequence @xmath125 such that @xmath126 for each @xmath342 and @xmath343 if and only if @xmath344 . this proves ( [ eq : qt - recursive ] ) . conversely , suppose that @xmath143 is an @xmath15-valued @xmath50-martingale that satisfies . to show that it satisfies ( [ eq:2012 - 10 - 03:qastt - not - empty ] ) we proceed by backward induction . by , we have @xmath347 . for any @xmath12 suppose that @xmath348 . as @xmath143 is a @xmath50-martingale , we have for all @xmath10 that @xmath349 for every @xmath12 and @xmath10 , observe from that @xmath350^\ast.\ ] ] successive application of corollaries 16.4.2 and 16.5.2 in @xcite then gives @xmath351^\ast\cap\mathcal{k}^{\mu\ast}_t = \operatorname{conv}\{\mathcal{q}^{\nu\ast}_{t+1}{{\,|\,}}\nu\in\operatorname{succ}\mu\}\cap\mathcal{k}^{\mu\ast}_t.\ ] ] since @xmath352 by , it follows that @xmath353 , which concludes the inductive step . we show first that for any @xmath354 there exists @xmath122 such that @xmath355 . if @xmath354 , then for each @xmath12 we have @xmath356 , i.e. there exists a liquidation strategy @xmath357 starting from @xmath358 at time @xmath25 . we also put @xmath359 for notational convenience . moreover , for each @xmath6 we have @xmath360 , i.e. there exists a liquidation strategy @xmath361 starting from @xmath362 at time @xmath25 . for each @xmath71 define @xmath363 where @xmath364 is defined by ( [ eq:2013 - 07 - 13-chi - star ] ) . the process @xmath95 belongs to @xmath44 and satisfies the non - anticipation condition ( [ eq : non - anticipate ] ) . moreover , for each @xmath6 @xmath365 because @xmath366 and @xmath35 is a convex cone . hence @xmath164 satisfies ( [ eq : seller - self - fin - superhedge ] ) in addition to ( [ eq : non - anticipate ] ) , and so @xmath122 . conversely , fix any @xmath122 and put @xmath367 , where @xmath368 is defined by ( [ eq:20130812-chi - tau ] ) . then for all @xmath369 we have @xmath370 and @xmath371 fix any @xmath6 . then @xmath372 for each @xmath373 , and the non - anticipation property ( [ eq : non - anticipate ] ) of @xmath164 gives @xmath374 . since @xmath375 , it means that @xmath376 moreover , for each @xmath342 we have @xmath377 , and so @xmath378 we verify by backward induction that @xmath379 for each @xmath380 . clearly , @xmath381 . now suppose that @xmath379 for some @xmath342 . from ( [ eq:2012 - 12 - 29_1 ] ) we can see that @xmath382 . because @xmath383 is predictable , we have @xmath384 by ( [ eq : qt - recursive ] ) , completing the backward induction argument . in particular , this means that @xmath385 . together with ( [ eq:2012 - 12 - 29_2 ] ) it gives @xmath386 for any @xmath6 . as a result , we have constructed @xmath354 such that @xmath387 . suppose that @xmath388 . we construct a sequence @xmath38 of random variables by induction . first take @xmath389 . now suppose that we have already constructed @xmath390 such that @xmath391 for some @xmath12 . from we obtain @xmath392 , whence @xmath393 by . we also obtain @xmath394 , and by ( [ eq : seller : mathcal - vt ] ) there exists a random variable @xmath395 such that @xmath356 . from we have @xmath396 , which concludes the inductive step . finally , we put @xmath397 . it follows that @xmath354 with @xmath389 . by proposition [ prop : am : seller : immediate - ultimate ] , a strategy @xmath122 can be constructed such that @xmath398 . suppose now that @xmath122 . by proposition [ prop : am : seller : immediate - ultimate ] , there is a @xmath354 such that @xmath355 . clearly , @xmath399 for all @xmath6 , and in particular @xmath400 . we now show by backward induction that @xmath390 for all @xmath6 . suppose that @xmath401 for some @xmath12 . since @xmath340 , this means by that @xmath395 . the condition @xmath356 implies that @xmath402 . property then gives @xmath390 by , which completes the inductive step . we conclude that @xmath403 . we have proved ( [ eq:20130727-zad0 ] ) . it follows that @xmath404 we know that @xmath162 is polyhedral , hence closed , so @xmath405 is also a closed set . it is non - empty and bounded below because @xmath406 for any @xmath407 large enough , and @xmath408 for any @xmath407 small enough . as a result , the infimum is attained . it means , in particular , that @xmath409 , so we know that a strategy @xmath122 can be constructed such that @xmath410 . 1 . [ item : prop : seller : dual:0 ] for each @xmath6 the set @xmath145 is compactly @xmath64-generated . item : prop : seller : dual:1 ] for all @xmath6 and @xmath28 we have @xmath411 3 . [ item : prop : seller : dual:2 ] we have @xmath412 for all @xmath6 . for all @xmath12 and @xmath10 we have @xmath413 and for each @xmath414 there exist @xmath415 $ ] , @xmath416 and @xmath417 such that @xmath418 4 . [ item : prop : seller : dual:3]for every @xmath12 and @xmath10 we have @xmath419 and for each @xmath420 there exist @xmath421 $ ] and @xmath422 for all @xmath423 such that @xmath424 we first consider claim . as @xmath130 is a cone , @xmath425 note in particular that @xmath426 . for all @xmath25 and @xmath28 we have @xcite @xmath427 which leads to . equation follows similarly from @xmath428 now fix any @xmath12 and @xmath10 , and suppose that @xmath431 and that this set is compactly @xmath64-generated . since @xmath432 lemma [ lem : roux_zastawniak2011 ] can be applied to the sets @xmath433 for all @xmath423 . this justifies claim for this @xmath25 and also that @xmath434 is compactly @xmath64-generated . by theorem [ th:2012 - 10 - 03:ftap ] and proposition [ prop : mart - ito - q ] , the lack of arbitrage opportunities implies that there is a pair @xmath60 such that @xmath435 and @xmath436 for each @xmath423 . since @xmath143 is a martingale under @xmath50 , it follows that @xmath437 and so @xmath438 . as @xmath439 and @xmath440 are compactly @xmath64-generated , it follows from lemma [ lem:2012 - 10 - 03:sigmai - intersect ] and that @xmath441 is compactly @xmath64-generated , which justifies claim for this value of @xmath25 . in view of and , lemma [ lem:2012 - 10 - 03:sigmai - intersect ] consequently shows that @xmath442 is also compactly @xmath64-generated . we may apply lemma [ lem : roux_zastawniak2011 ] to the sets @xmath443 and @xmath444 since @xmath445 claim for this value of @xmath25 follows upon observing that @xmath446 this concludes the inductive step . by proposition [ prop : seller : dual ] , @xmath447 is compactly @xmath64-generated . since @xmath448 is polyhedral , it is continuous on its effective domain and therefore attains a maximum on the non - empty compact set @xmath449 . from theorem [ prop : seller : zau0=initial - endowments ] it follows ( * ? ? ? * theorem 13.1 ) that@xmath450 the following construction produces adapted processes @xmath451 , @xmath452 , @xmath453 and @xmath454 for @xmath6 , and @xmath455 for @xmath456 . we already know that the maximum of @xmath448 over the set @xmath183 is attained , i.e. there exists some @xmath457 such that @xmath458 for any @xmath12 , suppose that @xmath459 is given , and fix any @xmath10 . then by proposition [ prop : seller : dual ] , there exist @xmath460 $ ] , @xmath461 and @xmath462 such that @xmath463 by and proposition [ prop : seller : dual ] , there exist @xmath464 $ ] and @xmath465 for all @xmath423 such that @xmath466 this completes the inductive step . also define for all @xmath156 @xmath467 then , are also satisfied when @xmath468 . the mixed stopping time @xmath100 is defined by setting @xmath474 and @xmath475 it is straightforward to show by induction that @xmath476 for all @xmath25 . moreover , since @xmath477 , we have @xmath331 observe also that @xmath478 for all @xmath6 , where @xmath364 is defined by ( [ eq:2013 - 07 - 13-chi - star ] ) . it then follows from , and that for all @xmath6 @xmath479 we now show by backward induction that for all @xmath6 @xmath480 at time @xmath123 the result is trivial because @xmath481 . suppose now that ( [ eq : seller : dual - opt:5 ] ) holds for some @xmath456 . then , by the tower property of conditional expectation , @xmath482 and , by , the predictability of @xmath364 , and , @xmath483 this concludes the inductive step . we also show by backward induction that for all @xmath6 @xmath484 at time @xmath123 @xmath485 suppose now that ( [ eq : seller : dual - opt:10 ] ) holds for some @xmath456 . then by , and the tower property of conditional expectation , we have @xmath486 this concludes the inductive step . by proposition [ prop:20130727:pi - ag - dual ] , a stopping time @xmath195 and a pair @xmath196 can be constructed such that @xmath491 to establish the reverse inequality we prove by backward induction that for any @xmath354 , @xmath71 and @xmath492 @xmath493 when @xmath468 , @xmath494 since @xmath495 and @xmath496 . now fix any @xmath12 , and suppose that @xmath497 then , by the tower property of conditional expectation , and since @xmath356 and @xmath498 , it follows that @xmath499 which proves ( [ eq : reverse - ineq - dual repr - seller ] ) . the construction in the proof of theorem [ prop : seller : zau0=initial - endowments ] with initial portfolio @xmath500 yields a strategy @xmath501 . for any @xmath71 and @xmath502 we have @xmath503 , and therefore ( [ eq : reverse - ineq - dual repr - seller ] ) with @xmath504 yields @xmath505 it follows that @xmath506 the set @xmath510 is clearly polyhedral with recession cone @xmath511 . for @xmath12 we proceed by induction . suppose that @xmath512 is polyhedral and its recession cone is @xmath513 . then @xmath514 is polyhedral and its recession cone is @xmath515 ( * ? ? ? * corollary 8.3.3 ) . being polyhedral , @xmath205 is the convex hull of a finite set of points and directions , and its recession cone @xmath515 is the convex hull of the origin and the directions in @xmath205 . the set @xmath516 is polyhedral ( * ? ? ? * corollary 19.3.2 ) and hence it is the convex hull of a finite set of points and directions . since the cone @xmath130 can be written as the convex hull of the origin and a finite number of directions , it is possible to write @xmath204 as the convex hull of a finite set of points , all in @xmath205 , and a finite set of directions . these directions are exactly the directions in @xmath205 and @xmath130 , i.e. the directions in @xmath515 and @xmath130 . thus the recession cone of @xmath204 is @xmath517 since @xmath518 by ( [ eq : qt - recursive ] ) . this means that the set @xmath519 is closed and its recession cone is @xmath130 ( * ? ? ? * corollary 9.8.1 ) . moreover , since @xmath204 and @xmath203 are polyhedral , it follows that @xmath520 is polyhedral ( * ? ? ? * theorem 19.6 ) , which means that @xmath521 is polyhedral , concluding the inductive step . the proof is by backward induction . since @xmath523 , from we have @xmath524 it immediately follows that @xmath525 on the set @xmath526 . on the set @xmath527 we have @xmath528 because @xmath511 is a cone , and therefore @xmath529 * on the set @xmath535 we have @xmath536 and therefore @xmath537 so that @xmath538 since @xmath539 it follows that @xmath540 on @xmath535 . * on the set @xmath541 we have @xmath542 because @xmath543 by . there are two further possibilities . * * on @xmath544 we have @xmath545 and therefore @xmath546 * * on @xmath547 we have @xmath548 and therefore @xmath549 as claimed . in view of proposition [ prop:2012 - 07 - 26:hedging - construct - converse ] , to verify ( [ eq:2012 - 07 - 26:constr - equivalence ] ) it is sufficient to show that for every @xmath550 there exists a pair @xmath217 such that @xmath551 . to this end , define @xmath552 and @xmath553 . suppose by induction that for some @xmath12 we have constructed predictable sequences @xmath554 and @xmath555 such that @xmath556 and @xmath557 because of , there exists an @xmath558-measurable random variable @xmath559 such that @xmath560 and @xmath561 equations and then give @xmath562 where @xmath563 follows from the fact that @xmath564 is a convex cone . this means there exists a random variable @xmath565 such that @xmath566 put @xmath567 . then @xmath568 , which concludes the inductive step . now define the mixed stopping time @xmath569 by @xmath570 we also put @xmath571 . we have constructed @xmath572 and @xmath71 such that @xmath573 and@xmath574 finally , we construct @xmath45 such that @xmath217 and @xmath551 . by the definition of the deferred solvency cones @xmath564 , for each @xmath6 there is a liquidation strategy @xmath575 starting from @xmath576 at time @xmath25 . we put@xmath577 which means that@xmath578 for each @xmath6 , with @xmath579 , completing the proof of ( [ eq:2012 - 07 - 26:constr - equivalence ] ) . next , if follows from ( [ eq:2012 - 07 - 26:constr - equivalence ] ) that @xmath580 by proposition [ prop:2012 - 09 - 19:zt - closed ] , @xmath215 is polyhedral , hence closed . as a result , the set @xmath581 is also closed . it is non - empty and bounded above because @xmath582 for any @xmath407 large enough , and @xmath583 for any @xmath407 small enough . this means that the supremum is attained . it follows that @xmath584 , so we know that a strategy @xmath212 can be constructed such that @xmath585 . theorem [ prop:2012 - 07 - 26:hedging - construct ] gives @xmath596 the maximum is attained , so @xmath597 . the strategy @xmath598 constructed by the method in the proof of theorem [ prop:2012 - 07 - 26:hedging - construct ] from the initial portfolio @xmath599 therefore realises the supremum in ( [ eq : buyer - bid - price - gradual ] ) . we write this supremum as a maximum , @xmath600,\end{aligned}\ ] ] and apply proposition [ prop : am - eur ] , which gives @xmath601\\ & = \max_{\chi\in\mathcal{x } } \left[-p^\mathrm{a}_{j}(-\xi_\chi)\right],\end{aligned}\ ] ] where @xmath602 is the ask ( seller s ) price in currency @xmath22 of a european option with expiry time @xmath123 and payoff @xmath603 as defined in appendix [ sect : eur - opt ] . we can now apply lemma [ lem : eur - ask - price - dual - repr ] to write @xmath604 for any @xmath605 , since @xmath143 is a martingale under @xmath50 , we have @xmath606 this means that @xmath607 proving ( [ eq : pi - bu ] ) . we know that @xmath608 realises the supremum in ( [ eq : buyer - bid - price - gradual ] ) , and therefore the above maxima over @xmath71 are attained at @xmath609 . a pair @xmath610 such that @xmath611 can be constructed by the method of ( * ? ? ? * proposition 5.3 ) for the european option with payoff @xmath612 , completing the proof . we recall a result for european options in the market model with @xmath17 assets under transaction costs . this is needed in the proof of the dual representation for the bid price of an american option . a european option obliges the seller ( writer ) to deliver a portfolio @xmath613 at time @xmath123 . the set of strategies superhedging the seller s position is given as@xmath614 and the _ ask price _ ( _ seller s price _ ) of such an option in currency @xmath98 is@xmath615 the following result can be found in ( * ? ? ? * section 4.3.1 ) . [ lem : eur - ask - price - dual - repr ] the ask price in currency @xmath98 of a european option @xmath616 can be represented as@xmath617 moreover , a pair @xmath235 such that @xmath618 can be constructed algorithmically . roux , a. zastawniak , t. 2009 , american options under proportional transaction costs : pricing , hedging and stopping algorithms for long and short positions , _ acta applicandae mathematicae _ * 106 * , 199228 .

american options in a multi - asset market model with proportional transaction costs are studied in the case when the holder of an option is able to exercise it gradually at a so - called mixed ( randomised ) stopping time . the introduction of gradual exercise leads to tighter bounds on the option price when compared to the case studied in the existing literature , where the standard assumption is that the option can only be exercised instantly at an ordinary stopping time . algorithmic constructions for the bid and ask prices and the associated superhedging strategies and optimal mixed stoping times for an american option with gradual exercise are developed and implemented , and dual representations are established .

1308.2688

wimps captured by the gravitational field of the sun that are slowed through collisions with solar matter can accumulate in the solar core . there , wimp annihilation may produce neutrinos with much larger energy than solar neutrinos . the signal would be an excess of high - energy ( @xmath1gev ) neutrino events pointing back to the sun @xcite . the cleanest signature at no@xmath0a will be from @xmath2 cc events producing upward - going muons that can be reconstructed in the no@xmath0a detector . the large and unique no@xmath0a far detector , with its excellent granularity and energy resolution , and relatively low - energy neutrino thresholds , is an ideal tool for these indirect dark matter searches . at no@xmath0a , the neutrino analyses simply store events synchronous with the numi beam . for non - beam exotic physics searches , so - called data - driven triggers @xcite are required to select events of interest . only the upward - going flux will be considered in order to suppress the cosmic - ray background . the downward - going muon rate in the no@xmath0a far detector is approximately 100,000 hz . we expect to keep the upward - going muon trigger rate to about 10 hz or less , so a rejection of at least four orders of magnitude is required by the trigger . of course , this rejection must be accomplished while keeping the acceptance for upward - going muons relatively high . the neutrino flux from dark matter annihilation is model dependent ; however , energies from @xmath30.5gev to many tev should be detected with high acceptance . for high - mass signal hypothesis , no@xmath0a will not be able to compete with the high acceptance of the icecube detector @xcite . for lower - mass scenarios ( below @xmath320 gev ) the super - kamiokande experiment currently has the best sensitivity @xcite . if an efficient upward - going muon trigger and sufficient cosmic ray background rejection can be achieved , no@xmath0a will be competitive with super kamiokande for wimp mass hypotheses below 20 gev / c@xmath4 . one advantage that no@xmath0a has compared to past experiments that performed similar searches for dark matter annihilation is the relatively low energy threshold for muons . a 1 gev muon track travels approximately 5 meters in the no@xmath0a detector resulting in an energy threshold well below 1 gev . the challenge for the dark matter search is triggering efficiently on these low - energy muons . for shorter track lengths , the timing information will not be as powerful for rejecting downward - going backgrounds . using stopping or fully - contained events and using the top and sides of the detector to veto downward - going events can provide an additional two orders of magnitude rejection . in this note we focus on using the timing information from all of the hits on a track to reject the downward - going muon background and efficiently select upward - going events . a trigger for upward - going muons based on timing information required a minor upgrade to the readout of the no@xmath0a far detector . this upgrade to the so - called `` multipoint '' readout occurred on september 11 , 2014 , and resulted immediately in a single - hit timing resolution of about 25 ns ( note that the timing resolution with the previous algorithm was about 125 ns , so this is a significant improvement ) . with dozens of hits per track , it is possible to reject downward - going muons by many orders of magnitude using hit timing information alone . to resolve the directionality of the muon track , the upward - going muon trigger takes advantage of the timing information from each individual hit in the reconstructed tracks . the tracks are reconstructed using the hough transform algorithm , and are required to match in both xz and yz views . we start from the hit with lowest @xmath5 cell value , @xmath6 , in the track in the yz view . the measured time of the corresponding hit is defined as @xmath7 . the observed and expected time of each hit on the track in the yz view is therefore : @xmath8 similarly , for the xz view : @xmath9 where @xmath10 and @xmath11 are the cell numbers in xz and yz view , and @xmath12 is the time measurement in tdc units , which is converted to ns using the factor of 15.625 ns / tdc . @xmath13 is the time - of - flight of the muon track defined as : @xmath14 where @xmath15 is track length in cm , and 29.97cm / ns is the expected speed assuming that the muon is relativistic . since we require that each track is reconstructed and matched in both views , ( @xmath16 ; @xmath6 ) and ( @xmath17 ; @xmath18 ) must correspond to the lowest and highest points of the track respectively . in addition , we can estimate the missing coordinate for a particular hit in either view using 3d requirement . for the yz view , track coordinates can be calculated as such : @xmath19 similarly , for the xz view : @xmath20 where @xmath21 cm and @xmath22 cm are the widths of detector cells and planes . the cell and plane with id=0 have coordinates @xmath23 cm and @xmath24 cm . since for each hit in each view we can estimate ( x ; y ; z ) coordinates , we can calculate the distance from the hit to the apd readout end . the further the hit is located from the readout the longer it takes for the light to propagate and be detected by the apd . we are interested in the hit time of the muon passing through the extrusion , so we have to correct for the light propagation time in the fiber . the speed of light in the fiber is measured to be 15.3 cm / ns . the light level in each channel in the no@xmath0a detector is independently sampled every 500 ns . the electronic response to an incident particle depositing energy in a cell can be parameterized in terms of two intrinsic timing values ( @xmath25 and @xmath26 ) , the number of photoelectrons ( @xmath27 ) , and a timing `` offset '' ( @xmath28 ) , or the elapsed time between a read - out and the time of incidence of the particle : @xmath29 here , @xmath30 is a proportionality factor that does not affect the timing fit . the parameters @xmath25 and @xmath26 correspond to the intrinsic falling and rising time of the response curve , respectively . as such , they are approximately known . for the purpose of determining hit timing , the parameter of note is @xmath28 . by performing a simple @xmath31 minimization , the data - preferred value of @xmath28 can be extracted from multiple readouts on a single channel . for the purposes of the trigger , where hit processing time must be minimized , fit results were pre - calculated and tabulated such that the computationally expensive minimization need not be repeated for each individual hit . an example of fitting the electronics response curve to multiple readouts from a single cell hit . the time coordinate of the inflection point where the curve begins to rise is the fitted parameter @xmath28.,width=192 ] single - hit timing resolution as observed in no@xmath0a far detector data with four - point readout , before ( left ) and after ( right ) fine timing implementation . see ref . @xcite for more details.,width=576 ] each time measurement has an uncertainty , which varies with the amount of energy deposited . the time uncertainty on a given hit from a reconstructed muon track affects the determination of track directionality , so a parameterization of uncertainty in terms of energy deposition is necessary for the timing - based trigger . single - hit time resolution is plotted against energy deposition in fig . [ fig : deltat ] . for high energy hits the @xmath32 is measured to be approximately 10 ns in the data using the four - point readout scheme , which is consistent with that observed in simulation @xcite . we can use equations [ eq : ytimes ] and [ eq : xtimes ] to produce the distribution of the expected v / s observed time for each track . the expected versus observed time distribution for an upward - going muon track reconstructed in the no@xmath0a far detector , using fine timing . the linear unconstrained fit ( red solid line ) has slope value close to `` 1 '' . the fit with the upward - going track hypothesis ( slope = 1 ) is shown as the blue - dashed line . the fit with the downward - going hypothesis is shown in the green dashed line and has a very poor probability . , width=384 ] an example of expected v / s observed time distribution is shown in fig . [ fig : track_example ] . the distribution is produced using a reconstructed upward - going muon track simulated with wimpsim @xcite . as can be seen , the points follow a rising trend with a slope value consistent with the upward - going track hypothesis . it is clear from the figure that the fitted slope value can be used to estimate the muon direction ( up or down ) . as shown in fig . [ fig : disc_slope ] , the slope values for cosmics and wimpsim mc samples are consistent with the downward- and upward - going hypothesis , respectively . in the relativistic limit , it is safe to assume that there are only two options for the slope values . therefore , we can fit the time distribution on fig . [ fig : track_example ] with fixed values of slopes . for the upward - going track the fit with the slope constrained to `` 1 '' results in a good @xmath31 probability value of the fit , @xmath33 . however the fit with slope of `` -1 '' yields a low probability value , @xmath34 . using the probability values from the fits with the fixed slope value , we can form a log - likelihood ratio ( llr ) : @xmath35 the llr distributions for the cosmic and wimpsim mc samples are shown in fig . [ fig : disc_llr ] . from this distribution , it is clear that a cut on llr slightly above zero will reduce the cosmic background by the desired amount while preserving a high signal acceptance . note that the wimpsim sample used is for dark matter with a 20 gev mass annihilating through the @xmath36 channel . as such the neutrinos from the b - meson decay produce muons which , on average , have a much lower energy compared to the cosmic ray muons . this explains why the llr for the signal has a larger component close to zero than the cosmic sample . the llr yields better performance for cosmic background rejection for the same signal acceptance in the regime where the cosmic rejection is sufficient ( at least four orders of magnitude ) , compared to a cut on the best - fit slope . for example , for a signal acceptance of 0.7 the background rejection is about a factor of three better for the llr . at this point the mc predicts background rejection of close to five orders of magnitude . in addition to being a more powerful discriminator as observed in the mc studies , the llr estimator is more robust to mis - reconstructed tracks which will be an important feature in real data . since mis - reconstructions will result in time distributions that follow neither the upward- nor the downward - going hypothesis , the result of mis - reconstruction will yield llr values close to `` 0 '' , and not values consistent with a high - probability for being upward - going . the slope distributions for cosmics ( red ) and wimpsim ( blue ) mc samples.,width=576 ] the llr distributions for cosmics ( red ) and wimpsim ( blue ) mc samples . note that only tracks longer then 5 m and with more than 50 hits are included.,width=384 ] a timing - based upward - going muon trigger was implemented for the no@xmath0a far detector and was deployed in november 2014 . triggering at @xmath37hz , the algorithm suppresses cosmic ray muons by five orders of magnitude . on the left is a display of a triggered event that is a strong candidate , based on its topology , for an upward - going muon . the activity at the bottom right indicates a cc scattering interaction . the curving at the other end probably indicates that the muon ranged out . there is also evidence for a michel electron , based on the timing information on the right . this event confirms that the llr algorithm is successfully selecting upward - going muons in the data.,title="fig:",width=384 ] on the left is a display of a triggered event that is a strong candidate , based on its topology , for an upward - going muon . the activity at the bottom right indicates a cc scattering interaction . the curving at the other end probably indicates that the muon ranged out . there is also evidence for a michel electron , based on the timing information on the right . this event confirms that the llr algorithm is successfully selecting upward - going muons in the data.,title="fig:",width=192 ] as can be seen in fig . [ fig : conc_example ] , events with probable michel electrons and contained vertices have been used to confirm upward - going muons in the triggered sample . atmospheric neutrinos generated on the other side of the earth are also capable of producing upward - going muons in the detector . these events represent an irreducible background in this search . the only method of discriminating atmospheric neutrino events from wimp events is to reconstruct the directionality of the incident neutrinos , which has yet to be attempted in no@xmath0a . the accumulating data sample opens the door to a program to study atmospheric neutrinos and sets the stage for a competitive dark matter search by the no@xmath0a experiment . this conference presentation was made possible by a grant from the university of virginia college of arts and sciences . additional financial support was provided by the jefferson trust , the uva physics department , and the fermilab particle physics division . the authors also acknowledge that support for this research was carried out by the fermilab scientific and technical staff . fermilab is operated by fermi research alliance , llc under contract no . de - ac02 - 07ch11359 with the united states department of energy . the university of virginia particle physics group is supported by de - sc0007838 . 7 j. s. hagelin , k. w. ng and k. a. olive , phys . b * 180 * , 375 ( 1986 ) . j. buckley , d. f. cowen , s. profumo , a. archer , m. cahill - rowley , r. cotta , s. digel and a. drlica - wagner _ _ , arxiv:1310.7040 . m. fischler , c. green , j. kowalkowski , a. norman , m. paterno and r. rechenmacher , j. phys . ser . * 396 * , 012020 ( 2012 ) .

the no@xmath0a collaboration has constructed a 14,000 ton , fine - grained , low - z , total absorption tracking calorimeter at an off - axis angle to an upgraded numi neutrino beam . this detector , with its excellent granularity and energy resolution and relatively low - energy neutrino thresholds , was designed to observe electron neutrino appearance in a muon neutrino beam , but it also has unique capabilities suitable for more exotic efforts . in fact , if an efficient upward - going muon trigger with sufficient cosmic ray background rejection can be demonstrated , no@xmath0a will be capable of a competitive indirect dark matter search for low - mass wimps . the cosmic ray muon rate at the no@xmath0a far detector is about 100 khz and provides the primary challenge for triggering and optimizing such a search analysis . the status of the no@xmath0a upward - going muon trigger is presented .

1510.07571

shape memory alloys ( sma ) have attracted a great deal of attention due to their important technological applications , including mechanical actuator devices and medical stents . the shape memory effect also gives rise to superelasticity , which finds applications in stents and spectacle frames . the shape memory effect is related to a reversible martensitic ( diffusionless ) phase transformation . it has been shown that the martensitic transformation can be induced by applied fields , temperature or both , and the mechanical properties of materials , therefore , can be controlled accordingly . in many systems , including those discussed in the present work , alloying can dramatically change the properties and transition temperatures of the materials , reflecting the importance of electronic features , specifically fermi surface effects , in the structural energetics of sma . there are several complementary approaches to modelling of the shape memory effect . continuum modelling allows investigation of the microstructural behavior , specifically martensitic twins , at the relevant long length scales . material - specific behavior is incorporated through an empirical functional for the free energy in terms of strain and a twin boundary energy to set the length scale . in atomistic models , the individual atoms are considered explicitly and their interaction given by an interatomic potential , which may be determined empirically , from first - principles density - functional - theory ( dft ) calculations , or a combination of the two . crystal symmetry and defect energies emerge from this approach , which gives microstructures with both natural length scales ( from defect energies ) and time scales ( since the atoms have definite newtonian forces and masses ) . however , in atomistic models , the electronic degrees of freedom do not appear explicitly . first principles dft methods are so computationally intensive that direct studies of microstructural behavior are impossible , but they are valuable both for obtaining quantitative atomic - level information regarding energies , forces and stresses independent of empirical input , and for understanding the electronic origin of this behavior . thus , first - principles investigation of the energetic instability of the high - temperature structure towards the low - symmetry martensitic structure is in itself quite illuminating . the resulting information can then also be used as inputs to atomistic@xcite and continuum modelling of shape memory behavior . typically , martensitic transformations are described using the strain as an order parameter , the classic example being the bain bcc - fcc transformation of iron . however , there is an alternative approach appropriate for cases where the strain degrees of freedom are coupled to atomic degrees of freedom ( phonons ) . following the soft - mode theory of structural transitions,@xcite we start from a high - symmetry reference structure ( here @xmath2 ) and freeze in unstable phonons of this structure , with corresponding lattice relaxation , to produce the ground - state structure . the symmetry of the phonons determines the symmetry of the low temperature structure . this approach has been successfully used in the study of minerals@xcite and ferroelectric materials @xcite and has been extended to shape memory alloys in our previous study of niti@xcite . closely related to niti , pdti and ptti are shape memory materials with reasonable structural simplicity and extraordinary mechanical behavior . they undergo a martensitic transformation at tunable temperatures : pdti transforms at 810k , but this can be reduced to 410k with 8% substitution of cr for pd@xcite . the high - temperature `` austenitic '' phase has a simple cubic @xmath2 structure ( space group @xmath3 ) , while the ambient temperature `` martensitic '' phase has been reported as the orthorhombic @xmath0 structure@xcite ( space group : @xmath4 ) . previous first - principles studies in pdti and ptti@xcite have shown that the observed electronic and elastic properties of the @xmath0 structure are well reproduced by density - functional theory calculations assuming the experimentally determined structure . in this paper , we investigate the structural energetics of pdti and ptti from first - principles calculations of phonon frequencies as well as total energies . this allows us to examine local as well as global stability and to investigate transition mechanisms , drawing on the strong analogy between the @xmath2-@xmath0 and bcc - hcp transformations and showing that coupling of unstable modes to the strain is a crucial aspect of the structural energetics . in sec . ii , we describe the first - principles calculations . in sec . iii , we present and discuss the results for the phonon dispersion of pdti and ptti in the @xmath2 structure and for the relaxed structures in which unstable modes and strains are coupled , yielding a low - symmetry ground state . in addition , we present results of calculations of the electronic structure , identifying and discussing features that affect the relative stability of the phases . v concludes the paper . first - principles total energy calculations were carried out within density - functional theory with a plane - wave pseudopotential approach . the calculations were performed with the vienna _ ab - initio _ simulations package@xcite , using the perdew - zunger@xcite parametrization of the local - density approximation ( lda ) . vanderbilt ultrasoft pseudopotentials @xcite were used . our pseudopotentials include nonlinear core corrections and for ti , we treated the occupied @xmath5 levels as valence . the electronic wave functions were represented in a plane - wave basis set with a kinetic energy cutoff of 278ev . the brillouin zone ( bz ) integrations were carried out by the hermite - gaussian smearing technique @xcite with the smearing parameter of 0.1ev . the unit cells contain two atoms in the cubic @xmath2 structure and four atoms in the orthorhombic @xmath0 and monoclinic @xmath1 structures . the calculations were performed with a @xmath6 monkhorst - pack ( mp ) @xmath7point mesh for the cubic @xmath2 structure and a @xmath8 mp @xmath7point mesh for both orthorhombic @xmath0 and monoclinic @xmath1 structures ( space group : @xmath9 ) , corresponding to 120 @xmath10 points in the @xmath11 irreducible bz of the simple cubic cell , 288 @xmath10 points in the @xmath12 irreducible bz of the orthorhombic cell and 576 @xmath10 points in the @xmath13 irreducible bz of the monoclinic cell . this choice of parameters converges the total energy to within 1 mev / atom . the density of states ( dos ) for the @xmath0 and @xmath1 structures were calculated using the tetrahedron method with blchl corrections@xcite . the bz s for the orthorhombic @xmath0 and monoclinic @xmath1 structures are different . to compare the band structure of the two structures , we label the @xmath0 band structure by regarding the @xmath0 structure as a special case of @xmath1 . the phonon dispersion relations were obtained with the pwscf and phonon codes @xcite , using the perdew - zunger@xcite parametrization of the lda , as above . ultrasoft pseudopotentials @xcite for pd , pt and ti were generated according to a modified rappe - rabe - kaxiras - joannopoulos ( rrkj ) scheme @xcite with three bessel functions @xcite . the electronic wave functions were represented in a plane - wave basis set with a kinetic energy cutoff of 408ev . the augmentation charges were expanded up to 9000ev . the brillouin zone ( bz ) integrations were carried out by the hermite - gaussian smearing technique @xcite using a 56 _ k_-point mesh ( corresponding to @xmath14 regular divisions along the @xmath15 , @xmath16 and @xmath17 axes ) in the @xmath11 irreducible wedge . the value of the smearing parameter was @xmath18=0.2ev . these parameters yield phonon frequencies converged within 5 @xmath19 . the dynamical matrix was computed on a @xmath20 _ q_-point mesh commensurate with the _ k_-point mesh . the complete phonon dispersion relation was obtained through the computation of real - space interatomic force constants within the corresponding box@xcite . the choice to use two different first - principles codes was dictated by the individual strengths of each . vasp has a highly efficient scheme for calculating total energies , forces , and stresses , and relaxing to the minimum energy structure , but does not have the density - functional perturbation theory capabilities of pwscf / phonon . even with slightly different pseudopotentials and _ k_-point sampling , the results of the two codes are quite compatible . for example , the difference between the computed lattice parameters for the @xmath2 structure of pdti is less than 0.2% , and for ptti the difference is less than 0.1% . comparisons of normalized eigenvector components computed by vasp using the frozen phonon method and by pwscf / phonon also show good agreement , generally within 5% . for the cubic @xmath2 structure , our calculations yield the equilibrium lattice parameters of 3.112 and 3.125 for pdti and ptti respectively . for comparison , we also performed full - potential linearized - augmented - plane - wave calculations ( flapw ) within the lda@xcite . the results are in excellent agreement with flapw results of 3.113 ( pdti ) and 3.127 ( ptti ) and in in good agreement with experimental values of 3.18 and 3.17 . the phonon dispersion relations along high symmetry lines , computed at the theoretical lattice parameters , are shown in figure [ phonons ] . the frequencies are obtained by taking the square root of the eigenvalues of the dynamical matrix@xcite . imaginary frequencies , as plotted as negative , are obtained from negative eigenvalues of the dynamical matrix . thus , the structure is dynamically unstable against distortions following the corresponding eigenvector . it should be noted that the phonon frequency @xmath21 is _ not _ the reciprocal of the period of oscillation of this mode ( as measured in molecular dynamics ) nor is it the energy difference between adjacent quantum levels ( as measured in neutron scattering experiments ) . these three quantities are equal only for a stable harmonic crystal . in the materials studied here the unstable modes may be related to a soft mode@xcite ( as defined by md or neutron scattering ) at high temperature , and even the stable modes are expected to be significantly renormalized as a function of temperature by anharmonic effects . phonon dispersion relations for pdti ( upper ) and ptti ( lower ) in the @xmath2 structure calculated at the lda equilibrium parameters 3.112 and 3.125 respectively . the negative slope of the acoustic @xmath22 branch corresponds to a pure elastic instability ( @xmath23 ) . symmetry labels are assigned according to the conventions of ref . with pd / pt at the origin . the imaginary frequencies of the unstable modes are plotted as negative values.,title="fig : " ] + phonon dispersion relations for pdti ( upper ) and ptti ( lower ) in the @xmath2 structure calculated at the lda equilibrium parameters 3.112 and 3.125 respectively . the negative slope of the acoustic @xmath22 branch corresponds to a pure elastic instability ( @xmath23 ) . symmetry labels are assigned according to the conventions of ref . with pd / pt at the origin . the imaginary frequencies of the unstable modes are plotted as negative values.,title="fig : " ] the dynamical matrices are related by mass factors to the force constant matrix : the second derivatives of the internal energy with respect to atomic displacements . the eigenmodes of the force constant matrix describe the potential energy landscape , and a negative eigenvalue indicates a static instability against a distortion following the corresponding eigenvector . while the actual normalized displacements of these eigenmodes are in general slightly different they carry the same symmetry labels as the eigenmodes of the dynamical matrix . either choice is expected to serve as a useful pointer to a lower energy structure if the distorted structure obtained by freezing in " an unstable mode is relaxed using first - principles forces and stresses , as we describe below . the phonon dispersion relations shown in figure [ phonons ] show instability of the @xmath2 structure similar to and even stronger than that of niti@xcite . there are large regions of reciprocal space where one , two or even three modes are unstable , with dominant instabilities at @xmath24 and along @xmath25-@xmath26 . the phonon instability shows that the observed high - temperature @xmath2 phases of pdti and ptti are dynamically stabilized by anharmonic phonons , and should be characterized by large fluctuating local distortions . the calculated phonon dispersions are also reminiscent of those of unstable bcc materials such as zr and ti , which undergo martensitic transformations to hcp or @xmath21@xcite ( via the @xmath24 and @xmath25-@xmath26 bcc - phonon equivalents respectively ) phases . the analogy based on the view of @xmath0 phases of pdti and ptti as chemically ordered hcp will be further strengthened below . in the soft mode approach , we search for local energy minima by choosing an unstable mode of the high symmetry structure , freezing in the distortion with varying amplitude , and relaxing the resulting structure . in many cases , the mode with the largest negative eigenvalue will generate the lowest energy structure . however , this is by no means generally true , as the energy gain is determined not only by the curvature of the energy surface but by higher order terms as well as the strength of coupling to strain and other modes , both unstable and stable , of appropriate symmetry . indeed , in pdti this `` most unstable '' mode ( i.e. largest negative eigenvalue ) lies in the @xmath25-@xmath26 branch rather than at @xmath24 . this mode is typically unstable in dynamically - stabilized bcc materials such as titanium and zirconium where it is associated with a phase transition to the ideal @xmath21 ( @xmath27 ) phase . the fact that in the chemically ordered analog ( pd / pt)ti , the observed phase transition is to @xmath0 rather than @xmath27 illustrates the importance of anharmonic effects and strain coupling in the energetics of these materials . eigenmode of the doubly - degenerate @xmath28 unstable phonon in the @xmath2 structure . this mode generates the @xmath0 structure in pdti and ptti . the structure is shown projected along the @xmath29 direction , with pd / pt represented by filled circles and ti by open circles . ] [ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ] in the soft - mode approach , there is no guarantee that the energy - minimizing freezing - in of one unstable mode will stablize the other unstable modes of the high - symmetry structure . in the present case , the undistorted doubled unit cell contains two sets of ( 110)@xmath30 planes each of which is unstable to strain - coupled shuffling at all _ q-_points . the simplest such mode in the @xmath0 structure is @xmath31 , which lowers the symmetry to monoclinic @xmath32 . the resulting @xmath1 structure has three additional free parameters : the monoclinic angle @xmath33 and two internal parameters @xmath34 and @xmath35 . these values are given for the relaxed @xmath1 structure in table [ table : struct ] , and compared with the normalized eigenvector displacements in table [ table : ratio ] . however , it is important to note that the computed @xmath25 phonon modes in the @xmath0 structures are in fact all stable ( table [ table : phonon - gamma ] ) . the lowering of energy by distortion to @xmath1 can not be obtained by a pure @xmath31 phonon distortion , but only if the strain is allowed to relax simulataneously ( table [ table : struct ] ) . this may be the reason that in a previous calculation @xcite , @xmath0 was reported to be the minimum energy structure . the relative energies of the various relaxed structures are given in table [ table : struct ] . the @xmath0 total energy is lower than @xmath2 . a simple estimate of the transition temperature is given by @xmath36 which suggest @xmath37 of 1050k ( pdti ) and 1755k ( ptti ) . these rough values are significantly larger than the experimental data for the hysteretical transition region@xcite ( approximately 800k and 1400k respectively ) , but show the correct material trend . for neither system has a @xmath1 phase yet been observed experimentally . the small computed energy differences between @xmath1 and @xmath1 , translated into temperature , are 28k and 39k for pdti and ptti , respectively . this suggests that the transition to the @xmath1 phase should occur at temperatures well below those at which the experiments were performed , so that our results are fully consistent with the available experimental work . + + the binary - hcp phase interpretation also suggests that we should examine the binary - fcc equivalent , which is the @xmath38 phase . @xmath38 is accessible from @xmath2 by a simple ( 001 ) shear and has lower energy@xcite , however we find that @xmath38 has slightly higher energy than @xmath0 . we consider one further structure : at the special @xmath1 values @xmath39 and @xmath40 , orthorhombic symmetry is restored . the side of the conventional cell of this body centered orthorhombic ( bco ) ( space group @xmath41 ) structure is doubled in the * b * direction , though the primitive cell still contains four atoms . although at relatively high energy ( [ table : struct ] ) , this provides us a reference point for structures with large @xmath33@xcite . note that a further shear to @xmath42 would give the @xmath0 structure once more . the potential energies of continuous paths between the structures identified above are important for understanding the transformation mechanism . we compute the energies of three paths : @xmath2-@xmath0 , @xmath0-@xmath1 and @xmath1-bco . in keeping with the timescale separation between bulk strain and atomic motion , we define intermediate configurations by relaxing the atoms to their minimum energy configuration consistent with the applied symmetry and strain on the cell . the remaining four strain degrees of freedom are reduced to a single parameter by taking interpolations between the strains of the endpoint structures . for the @xmath2 structure , we minimize the energy assuming the @xmath0 space group , which gives a smooth evolution of the structure along the path . > from figure [ tote ] , it is clear that there is no total energy barrier along the @xmath2-@xmath0-@xmath1 path , and that @xmath0 represents the total energy barrier between @xmath1 martensitic variants . the @xmath0 phase can be viewed as the binary equivalent of hcp , and the transformation path as the binary equivalent of the nishiyama - wassermann ( nw ) path . using the analogy with the nw mechanism for the bcc - hcp transition , we can attribute the transition to a shuffling of ( 110)@xmath30 planes@xcite . structural instabilities in metals are typically related to details of the fermi surface , and we have calculated the band structures of @xmath2 , @xmath0 and @xmath1 to investigate this . in pdti / ptti the band structure is dominated by the _ d_-bands , with the pd / pt bands lying below the fermi level and almost fully occupied , and the ti @xmath43-bands lying above the fermi level ( figure [ bands - pdti ] and [ bands - ptti ] ) , the band centers being offset by some 6ev . the free - electron like @xmath44 bands are very broad , and play little role in the bonding except to donate some electrons to the ti - d band . the large strain involved in the @xmath45 transition means that the fermi surfaces are quite different ( @xmath0 having the lowest dos at @xmath46@xcite ) and this transition can not be related directly to the band structure . by contrast , the @xmath0-@xmath1 transition is accompanied by the opening of a pseudogap at the fermi level , a typical signature of increased stability . the band structures are very similar , the small difference which stabilises @xmath1 being traceable to the shifting above the fermi level of a pocket of electrons around @xmath47 . the distortion to @xmath1 is just enough to complete the topological phase transition which eliminates this pocket of electrons in both materials . in conclusion , we have performed _ ab initio _ calculations of the structural energetics of pdti and ptti . in each case we predict that the low temperature ground state structure will be @xmath1 , with the ( observed ) @xmath2 and @xmath0 phases being dynamically stabilized . there are no total energy barriers between the structures , meaning that the phase space microstates that belong to the @xmath1 structure _ also _ belong to the @xmath0 and @xmath2 structures . in contrast to niti@xcite , the entire @xmath22 phonon branch is unstable . we showed that the @xmath0 structure can be obtained by a `` freezing in '' of phonons of the @xmath2 structure coupled to the shear associated with the @xmath48 elastic constant , but that no single dynamical - matrix or force - constant - matrix eigenvector leads to the low - symmetry phase . the @xmath1 then corresponds to a further strain coupled to a @xmath31 phonon of the @xmath0 phase . tracing the atomic motions of these instabilities shows that they are both related to shears of alternate @xmath49 phases , and hence that the transition mechanism is the binary equivalent of the nishiyama - wassermann bcc - hcp mechanism . we thank r. d. james , i. i. naumov , and k. bhattacharya for valuable discussions . this work was supported by afosr / muri f49620 - 98 - 1 - 0433 . the calculations were performed on the sgi origin 3000 and ibm sp3 at arl msrc . p. blaha , k. schwarz , and j. luitz , wien97 , vienna university of technology , vienna 1997 . ( improved and updated unix version of the original copyrighted wien - code , published by p. blaha , k. schwarz , p. sorantin , and s. b. trickey , in comput . phys . commun . * 59 , * 399 1990 ) . our flapw calculations were performed using a 120 _ k_-point mesh in the @xmath11 irreducible wedge . no shape approximations were made to the density or potential . the muffin - tin radii for pd / pt and ti were chosen to be 1.343and 1.278 , repectively . a @xmath50 of 10 were used . the tetrahedron method was used in the brillouin zone integrations .

the structural energetics of pdti and ptti have been studied using first - principles density - functional theory with pseudopotentials and a plane - wave basis . we predict that in both materials , the experimentally reported orthorhombic @xmath0 phase will undergo a low - temperature phase transition to a monoclinic @xmath1 ground state . within a soft - mode framework , we relate the @xmath0 structure to the cubic @xmath2 structure , observed at high temperature , and the @xmath1 structure to @xmath0 via phonon modes strongly coupled to strain . in contrast to niti , the @xmath0 structure is extremely close to hcp . we draw on the analogy to the bcc - hcp transition to suggest likely transition mechanisms in the present case .

cond-mat0207090

consider a network which evolves under the removal and addition of vertices . in each unit of time we add @xmath1 vertex and remove @xmath2 vertices . removal of a vertex also implies that all the edges incident on that vertex vanish and consequently the degree of vertices at the end of those edges decrease . here @xmath2 can be interpreted as the ratio of vertices removed to those added , so @xmath3 represents a growing network , @xmath4 a shrinking one , while @xmath5 implies vertex turnover but fixed network size . the equations to follow represent the completely general case . however , for the purposes of this paper we will specialize to networks of constant size as we assume that the network already exists and we would like to preserve its original structure , by balancing the rate of attack against the rate of repair . let @xmath6 be the fraction of nodes in the network that at a given time have degree @xmath7 . by definition then it has the normalization : @xmath8 in addition to this we would like to have freedom over the degree of the incoming vertex . let @xmath9 be the probability distribution governing this , with the constraint @xmath10 . we also have to consider how a newly arriving vertex chooses to attach to other vertices extant in the network and how a vertex is removed from the same . let @xmath11 be the probability that a given edge from a new node is connected to a node of degree @xmath7 , multiplied by the total number of nodes @xmath0 . then @xmath12 is the probability that an edge from a new node is connected to some node of degree @xmath7 . similarly , let @xmath13 be the probability that a given node with degree @xmath7 fails or is attacked during one node removal also multiplied by @xmath0 . then @xmath14 is the total probability to remove a node with degree @xmath7 during one node removal . note that the introduction of the deletion kernel @xmath13 is what sets our model apart from previous models describing the network evolution process . since each newly attached edge goes to some vertex with degree @xmath7 , we have the following normalization conditions : @xmath15 armed with the given definitions and building on the work done previously by @xcite , we are now in a position to write down a rate equation governing the evolution of the degree distribution . for a network of @xmath0 nodes at a given unit of time , the total number of nodes with degree @xmath7 is @xmath16 . after one unit of time we add one vertex and take away @xmath2 vertices , so the number is @xmath17 , where @xmath18 is the new value of @xmath6 . therefore we have , @xmath19 where @xmath20 is the conditional probability of following an edge from a node of degree @xmath21 and reaching a node of degree @xmath7 . alternatively , it is the degree distribution of nodes at the end of an edge emanating from a node of degree @xmath21 . note that @xmath22 and @xmath23 are always zero , and for an uncorrelated network , @xmath24 . the terms involving @xmath11 describe the flow of vertices with degree @xmath25 to @xmath7 and @xmath7 to @xmath26 as a consequence of edges gained due to the addition of new vertices . the first two terms involving @xmath27 describes the flow of vertices with degree @xmath28 to @xmath7 and @xmath7 to @xmath25 as vertices lose edges as a result of losing neighbors . the term @xmath29 represents the direct removal of a node of degree @xmath7 at rate @xmath2 . finally @xmath9 represents the addition of a vertex with degree @xmath7 . processes where vertices gain or lose two or more edges vanish in the limit of large @xmath0 and are not included in eq . . the rate equation described above presents a formidable challenge due to the appearance of @xmath20 from the terms representing deleted edges from lost neighbors . rate equations for recovery schemes based on edge rewiring are slightly easier to deal with . upon failure , all edges connected to that node are rewired so that the degrees of the deleted node s neighbors do not change , and this term does not appear . the specific case of preferential failure in power - law networks was considered previously in this context by @xcite . however , this recovery protocol can only be used on strictly growing networks , because a network of constant size would become dense under its application . moreover , it is dependent on the power - law structure of the network . the methods described here are general and are applicable to arbitrary degree distributions . apart from edge rewiring , the special case of random deletion also leads to a significant simplification . uniform deletion amounts to setting @xmath30 . doing so , then leads to the following , @xmath31 which renders eq . independent of @xmath32 and thus independent of any degree - degree correlations . random deletion hence closes equation for @xmath33 , enabling us to seek a solution for the degree distribution for a given @xmath34 and @xmath11 . with non - uniform deletion , the degree distribution depends on a two - point probability distribution , and as we shall see in section [ sec : correlations ] , the two - point probability distribution will depend on the three - point probability distribution and so on . this hierarchy of distributions , where the @xmath0-point distribution depends on the @xmath35-point distribution , is not closed under non - uniform failure and hence it is difficult to seek an exact solution for the degree distribution . nevertheless , in the following , we demonstrate a method that allows us to navigate our way around this problem . as mentioned before , for the purposes of this paper we will be interested in a network of constant size , where the rate of attack is compensated by the rate of repair . assuming that the network reaches ( or already is ) a stationary distribution and does not possess degree - degree correlations , we set @xmath5 and can further simplify eq . . let @xmath36 be the mean degree of nodes removed from the network ( i.e. @xmath37 ) , and @xmath38 the mean degree of the original degree distribution @xmath33 . then we have , @xmath39 the evolution process , specifically non - uniform removal of nodes , can and in many cases will introduce degree - degree correlations into our networks . in order to confront this issue , we will first find choices for @xmath9 and @xmath40 that satisfy the solutions to the rate equation , for a given @xmath6 , in a network that is uncorrelated . we will then demonstrate that a special subset of those solutions for @xmath34 and @xmath11 is an uncorrelated fixed point of the rate equation for the degree - degree correlations . this opens up the possibility , that a network that initially has no degree - degree correlations will not develop correlations from the evolution process . although the rate equation described in eq . is fairly complicated , it is a relatively straightforward exercise to determine the relation between edges added to those removed . multiplying eq . by @xmath7 , summing over @xmath7 and rearranging yields @xmath41 . this equation is simple to interpret . since the network has a constant fixed - point degree distribution , the average degree of the network remains constant , and therefore edges are removed and added at the the same rate . in this section we describe our method under which networks can recover from various forms of attack . the types of attack we consider are those studied generally by most authors ( though in static networks ) , namely preferential and targeted attacks . random failures are the most generally studied schemes in both static and evolving networks , in view of the fact that they lend themselves to relatively simple analysis . these types of failures may be representative , say , of disruption of power lines or transformers in a power grid owing to extraneous factors such as weather . however , the functionality of most networks often depends on the performance of higher degree nodes , consequently non - uniform attack schemes focus on these . for example , in a peer - to - peer network , a high degree node could be a central user with large amounts of data . high degree could also be indicative of the amount of load on a node during its operation , or on the public visibility of a person in a social network . it is reasonable to assume that a malicious entity such as a computer virus is more likely to strike these important nodes . et al _ @xcite have employed this removal strategy ( among others ) on a variety of simulated and real networks and have found it to be highly effective in disrupting the structure of the attacked network . nodes ) with mean @xmath42 , under preferential attack @xmath43 and uniform attachment @xmath44 using @xmath45.,width=302 ] we simulate these kinds of attacks using preferential failure @xmath43 , that sample nodes in proportion to their number of connections , and through an outright attack on the highest degree nodes represented by @xmath46 , where @xmath47 is the heaviside step function . our method of compensation will involve control over two processes : the first where our newly incoming / repaired vertex chooses a degree for itself drawn from some distribution @xmath9 , and second , the process by which this vertex decides to attach to any other vertex in the network , governed by the attachment kernel @xmath40 . our goal here is to solve for the attachment kernel @xmath11 , that will preserve the original probability distribution @xmath33 , subject to a deletion kernel @xmath13 for some choice of @xmath34 . we will assume that the final network is uncorrelated and work with eq . , keeping in mind that any arbitrary choice of @xmath34 and @xmath11 is probably not consistent with that assumption . introducing the cumulative distribution for the attacked and newly added vertices , @xmath48 and @xmath49 respectively , @xmath50 we sum eq . from @xmath51 to @xmath52 , noting that @xmath41 for our steady state network . this leads to the following relation , @xmath53 dividing both sides by @xmath33 gives us an expression for the attachment kernel , @xmath54 . \nonumber\\ \label{eq : genattachment}\end{aligned}\ ] ] nodes ) with @xmath42 , under high degree attack @xmath55 and uniform attachment @xmath44 using @xmath45.,width=302 ] equation represents the set of possible solutions for the attachment kernel that will lead to the desired degree distribution , given that the final network is uncorrelated . the correct choice of solution from the above set , must obey the consistency condition , that when inserted into the rate equation for the degree - degree correlations , the correlations vanish . in section [ sec : correlations ] , we will show that the following _ ansatz _ chosen from the above set is such a choice : @xmath56 equation was previously derived by @xcite for the case of random deletion . here we posit that it works more generally for the case of non - uniform attack when our initial network is uncorrelated ( with some caveats that will be explained shortly ) . the choice of @xmath11 makes intuitive sense because the quantity @xmath57 is the probability distribution governing the number of edges belonging to a node , reached by following a randomly chosen edge to one of its ends , _ not including _ the edge that was followed . this is one less than the total degree of the node and is also referred to as the _ excess _ degree distribution . note that in our model we specify the degree of incoming nodes . therefore the appearance of the excess degree distribution is a signature of an uncorrelated network , implying the newly arriving edges are being introduced in an uncorrelated fashion . there are basically two conditions for the existence of a solution given by eq . ; @xmath58 must be a valid probability distribution , and @xmath38 must be finite . these are not very stringent conditions and are typically satisfied by most degree distributions . in other words , barring some pathological cases , it is always possible to find a solution of the form of eq . . there is an additional consideration , the deletion process may lead to nodes of degree zero in a network that originally did not have any such nodes . while the fraction of such nodes is vanishingly small for networks with say , poisson degree distributions , they may be non - trivial for power - law networks . as such , it is important to set @xmath59 ( the probability to attach to a node of degree zero ) to a generous value in order to reconnect these nodes to the network . we are now in a position to effect our repair on the network . given the original degree distribution @xmath6 and the form of the attack @xmath13 , eq . gives us the precise recipe for recovering the degree distribution . we need to sample the degrees of the newly introduced nodes in proportion to the product of the deletion kernel _ and _ the degree distribution , and then attach these edges in proportion to the excess degree distribution of the network . to test our repair method , we provide four examples for initially uncorrelated networks with @xmath60 nodes generated using the _ configuration model _ @xcite . in the configuration model , only the degrees of vertices are specified , apart from this sole constraint the connections between vertices are made at random . nodes ) with @xmath61 under targeted attack @xmath62 using @xmath11 from eq . after setting @xmath45.,width=302 ] the simulation results show the initial degree distribution and the compensated one subject to two types of attacks on poissonian networks with degree distribution given by , @xmath63 in fig . [ fig : poissonpref ] we show the resulting degree distribution where nodes were attacked preferentially , i.e. @xmath43 , while in fig . [ fig : poissontheta ] we show the case for targeted attack only on high degree nodes represented by @xmath64 where @xmath65 is the _ minimum _ degree of the node attacked . the degrees of newly added nodes were chosen from the distribution @xmath58 with the attachment kernel @xmath11 set to one , corresponding to the solution of equation after substituting in the appropriate @xmath6 . the data points in all the figures are averaged over multiple realizations of the network each subject to @xmath66 iterations of addition and deletion . the points along with corresponding error bars represent the final degree distribution , whereas the solid line represents the initial network . as the figures show , the final networks are in excellent agreement with the initial degree distribution . nodes ) with exponent @xmath67 and exponential cutoff @xmath68 , under preferential attack @xmath43 using @xmath11 from eq . after setting @xmath45.,width=302 ] we employ the same attack kernels , @xmath43 and a targeted attack only on high degree nodes represented by @xmath64 on two other examples . our first example network has links distributed according to a power - law with an exponential cutoff , @xmath69 @xmath70 is a normalization constant which in this case is @xmath71 , where the function @xmath72 is the poly - logarithm function defined as : @xmath73 the exponential cut - off has been introduced for three reasons . first , many real world networks appear to show this cutoff @xcite and second , it renders the distribution normalizable for ranges of the exponent @xmath74 . finally , for a pure power - law network it is in principle possible to assign a degree to a node that is greater than the system size . the exponential cutoff ensures that the probability for this to happen is vanishingly small . in the other examples that we consider , the functional form of the distribution already ensures this property . the second network has an exponential distribution given by , @xmath75 fig . [ fig : exptheta ] shows the results for the exponentially distributed network ( @xmath61 ) undergoing targeted attack . in fig . [ fig : powerpref ] we show the resulting degree distribution for the power - law network ( @xmath67 and @xmath68 ) where nodes were attacked preferentially . both figures indicate the initial and final networks are in excellent agreement . at this point , aside from the technical details , it is worth reminding ourselves of the big picture . we have demonstrated above that if a network with a certain degree structure is subjected to an attack that aims to destabilize that structure , one can recover the same , by manipulating the rules by which vertices are introduced to the network . the rules that we employ in our repair method are dependent on the types of attacks that our networks are subject to . in the following section we give a detailed justification of the employment of our method . in order for our results from the previous sections to be valid , we must demonstrate that our initially uncorrelated networks remain uncorrelated under our repair scheme . to accomplish this , we will define a rate equation for the degree - degree correlations and demonstrate that the uncorrelated network is a fixed point of this equation . our rate equation will describe the evolution of the expected number of edges in the network with ends of degree @xmath7 and @xmath76 . let the expected number of such edges in the network be , @xmath77 where @xmath78 , and @xmath79 is the probability that a randomly selected edge has degree @xmath7 at one end and degree @xmath76 in the other . the expected number of edges after one time step where we add @xmath80 and take away @xmath81 edges is then , @xmath82 e'_{l , k } = m e_{l , k } + \delta , \label{eq : edgerate1}\ ] ] where @xmath83 represents all other edge addition and removal processes . we have already established that in the steady state case , @xmath84 irrespective of the degree distribution , so our goal is equivalent to showing that @xmath83 is equal to zero for an uncorrelated network generated / repaired with our special choices of @xmath11 and @xmath34 . as a result @xmath85 , implying that the degree - degree correlations ( if any ) remain constant over time . we will assume that our network is locally tree - like , something which holds true for most random graphs . in addition we will only consider processes out to second nearest - neighbors of a node . these assumptions allows us to avoid including terms in the rate equation representing removal of nodes with neighbors that are connected to each other . nevertheless , there are a large number of remaining processes that we will need to consider . to start things off , note that the rate equation is symmetric in the indices @xmath76 and @xmath7 . any process that contributes to changing @xmath7 while holding @xmath76 constant also contributes to changing @xmath76 while holding @xmath7 constant . we can therefore consider contributions to @xmath83 from @xmath86 , @xmath87 and @xmath88 and add on the corresponding symmetric terms at the end . the first process we need to take into account is a direct addition of a node of degree @xmath76 . this contributes two flows to the rate equation , @xmath89 and @xmath90 . similarly , the direct deletion of a node of degree @xmath76 contributes @xmath91 and @xmath92 . next , we will have to take into account second nearest - neighbor processes . we can be certain that these terms are of the same order by merely counting the number of unsummed probability distributions that go into each process . there will be two terms for the attachment process representing the situation where a new node of any degree attaches to a node of degree @xmath7 or @xmath25 , that was previously attached to a node of degree @xmath76 . these terms are @xmath93 and @xmath94 . similarly there are two removal processes , where a node of any degree that is removed from the network was previously attached to a node of degree @xmath7 or @xmath28 that has neighbor(s ) of degree @xmath76 . unfortunately these terms introduce three - point correlations into the rate equation . analogous to methods employed in similar hierarchy problems , we use a moment - closure approximation to represent these processes as a product of two two - point correlations in the following manner , @xmath95 adding all of these terms together our final equation for @xmath83 is , @xmath96 in addition to terms where @xmath76 and @xmath7 are interchanged . after inserting the appropriate @xmath11 and @xmath34 from eq . along with the uncorrelated solution @xmath97 , it can be shown that , @xmath98 according to eq . , there exist a set of solutions such that an initially uncorrelated network will not develop any degree - degree correlations as a consequence of the evolution process . the attachment kernel that was employed in the network evolution process , described in section [ sec : designattachment ] , was a subset of these solutions . this allowed the repair method to be employed by maintaining negligible correlations in the network . one must point out , that we have not explicitly demonstrated the stability of the uncorrelated solution to perturbations . for example fluctuations in @xmath87 or in the number of edges may drive the network away from the uncorrelated steady - state . an analytical approach to determine this , say using linear stability analysis is difficult , due to the numerous related probability distributions involved . so instead we resort to a numerical approach . we measured the pearson correlation coefficient between the degrees of nodes at both ends of an edge for all our model networks . for the poisson and exponential cases , the correlations remained negligible during the evolution process . on the other hand , the power - law network developed non - trivial correlations . we have not been able to determine whether the appearance of these correlations was due to finite - size effects , or instability in the uncorrelated solution , or to some other cause . the results show that the agreement between the initial and final degree distributions is very good , and it seems that in this particular case , the correlations did not demonstrate a significant effect on the final state of the network . in this paper , we have shown how to preserve a network s degree distribution from various forms of attack or failures by allowing it to adapt via the simple manipulation of rules that govern the introduction of nodes and edges . we based our analysis on a rate equation describing the evolution of the network under arbitrary schemes of addition and deletion . in addition to choosing the degree of incoming nodes , we allow ourselves to choose how nodes attach to the existing network . to deal with the special case of non - uniform deletion we have introduced a rate equation for the evolution of degree - degree correlations and have used that in combination with the equation for the degree distribution to come to our solution . we have provided examples of the applicability of this method using a combination of analytical techniques and numerical simulations on a variety of degree distributions , yielding excellent results in each case . the structure of many networks in the real world is crucially related to their performance . many authors have seized on the fact that technological networks such as the internet and peer - to - peer networks are power - law in nature , and have used this to design efficient search schemes among other things . loss of structural properties of these networks then lead to severe constraints on their performance . recent empirical studies @xcite have suggested that node removal , for example , in the world wide web , is typically non - uniform in nature . in view of this , it is crucial for researchers to come up with effective solutions to try and manage these types of disruptions . to the best of our knowledge , there is a considerable gap in understanding the non - uniform deletion process of nodes and edges and corresponding methods to deal with them . this paper begins to address this gap . it must be pointed out that the methods we have described depends crucially on the assumption of negligible correlations as the network evolves . curiously enough , in our example power - law network , we were able to get very good agreement between the initial and final degree distributions , in spite of the appearance of non - trivial correlations . it will certainly be interesting to see if our methods can be extended to the case of networks with strong correlations , and other metrics describing network structure . perhaps it is possible to directly confront the rate equation for the degree - degree correlations , although this seems a difficult prospect at the moment . the idea of preserving the structure of networks from attacks by allowing it to react in real - time is a relatively nascent one and the authors look forward to more developments in this area .

there has been a considerable amount of interest in recent years on the robustness of networks to failures . many previous studies have concentrated on the effects of node and edge removals on the connectivity structure of a _ static _ network ; the networks are considered to be static in the sense that no compensatory measures are allowed for recovery of the original structure . real world networks such as the world wide web , however , are not static and experience a considerable amount of turnover , where nodes and edges are both added and deleted . considering degree - based node removals , we examine the possibility of preserving networks from these types of disruptions . we recover the original degree distribution by allowing the network to react to the attack by introducing new nodes and attaching their edges via specially tailored schemes . we focus particularly on the case of non - uniform failures , a subject that has received little attention in the context of evolving networks . using a combination of analytical techniques and numerical simulations , we demonstrate how to preserve the _ exact _ degree distribution of the studied networks from various forms of attack . recent years have witnessed a substantial amount of interest within the physics community in the properties of networks @xcite . techniques from statistical physics coupled with the widespread availability of computing resources have facilitated studies ranging from large scale empirical analysis of the worldwide web , social networks , biological systems , to the development of theoretical models and tools to explore the various properties of these systems @xcite . a relatively large body of work has been devoted to the study of degree distributions of networks , focusing both on their measurement , and formulation of theories to explain their emergence and their effects on various properties such as resilience and percolation . these studies are mostly aimed at networks in the real world that evolve naturally , in the sense that they are driven by dynamical processes not under our control . representative examples being social , biological networks and information networks like the world wide web , which though manmade , grows in a distributed fashion . there are however different classes of infrastructure related networks such as the transportation and power grids , communication networks such as the telephone and internet , that evolve under the direction of a centrally controlled authority . in addition to these is a relatively new class of networks which fall in between these two types , the classic example being peer - to - peer file - sharing networks . these networks grow in a collaborative , distributed fashion , so that we have no direct influence over their structure . however , we can manipulate some of the rules by which these form , giving us a limited but potentially useful influence over their properties . it is a well established fact , that the structure of such networks is directly related to their performance . in view of this , a certain degree of effort has been made to tailor these _ designer _ networks towards structures that optimize certain properties such as robustness to removal of nodes and efficient information transfer among other things @xcite . these networks typically experience a significant amount of vertex / edge turnover , with users joining and leaving the network voluntarily , possible failures of key components and resources , or intentional attacks such as denial of service . these factors can lead to severe disruption of the network structure and as a result , loss of its key properties . in the face of this , it is natural to extend our analysis to the effects of these failures / attacks and use our limited control to attempt to adaptively restore the original structure of these networks . previous work has focused on the effects of disruption on static networks , where authors have studied the connectivity structure under the random / targeted removal of nodes and edges @xcite . the network is considered static in that no compensatory measures , such as the introduction of new edges or nodes , are permitted . the effect of these removals have been measured against the existence of the _ giant component _ : the largest set of vertices in the network of o(@xmath0 ) , where @xmath0 is the number of nodes , that are connected to each other by at least one path . a representative example can be found in the paper by albert _ et al _ @xcite , where they studied the size of the giant component of scale free networks such as the internet , under simulated random failures and targeted attacks on high degree nodes . one of the interesting things they found was that , while these networks were remarkably robust to random failures , they were extremely fragile to targeted attacks . this emphasizes the importance of non - uniform removal strategies . unlike in the static case , the networks considered in this paper evolve in time with sustained node and edge removals . the network is allowed to react to these disruptions via the introduction of new nodes and edges , chosen to be attached in a manner such that the network retains it original form , at least in terms of the degree distribution . such models , conventionally referred to in the literature as _ reactive networks _ have been discussed before , see @xcite for instance . here we assume that the designers of the network are only aware of the statistical properties of the removed nodes and have no ability to influence the existing network beyond the introduction of new nodes or reattachment of those removed . consequently they have two processes under their control to compensate for the attack . the first is the degree of the introduced vertices and the second is the process by which a newly introduced vertex chooses to attach to a previously extant vertex on the network . failure is thus compensated by adding nodes and edges chosen from an appropriate degree distribution and attaching them to the network via specially tailored schemes . note that in our model , one can re - introduce nodes that have been removed or introduce completely new sets of nodes . the former case could be indicative of say a computer in a peer - to - peer network that loses its connection , and would like to reconnect . the latter could represent the permanent loss of web - pages from the world wide web and the introduction of a new web - page . we use the attachment kernel of krapivsky and redner @xcite , to simulate the introduction of nodes and edges , and via the introduction of a deletion kernel we analyze the interesting and neglected case of non - uniform deletion . a variety of models have been proposed to simulate network evolution and growth where vertices are both added and deleted @xcite , but these have concentrated on the relatively simple case of uniform deletion . we will show that under uniform failures , the appearance of degree - degree correlations , that typically arise as a result of growth processes , as discussed in @xcite , can be neglected . previous models have taken advantage of precisely this fact to circumvent the difficulty of dealing with degree - degree correlations . for the case of non - uniform deletion , correlations can not be ignored . in this paper we confront this issue by demonstrating how to preserve an initially uncorrelated network throughout the evolution process with the introduction of an additional rate equation for the degree - degree correlations . we give analytical results and numerical simulations for a variety of degree distributions under various forms of attack . in all the cases that we study , we recover the _ exact _ degree distributions .

0711.1602

at a continuous transition , the expression @xmath13 for the leading critical behavior of a thermodynamic observable @xmath14 has the well known form @xmath15 where @xmath0 and @xmath16 are the transition temperature and the critical exponent respectively . for the concrete analysis of numerical data , a normalization factor with non - critical behavior at @xmath0 must be introduced . the simplest and most traditional convention , which will be referred to below as @xmath17 scaling , is to normalize each @xmath18 by a temperature independent constant . for obvious reasons this constant is chosen to be @xmath19 for each observable ; one then writes the normalized leading term as the familiar text - book expression : @xmath20^{-\rho_f } = { \cal c}_{f}t^{-\rho_f } , \label{t_scaling}\ ] ] where @xmath21 and @xmath22 is the critical amplitude ( see [ ] for a detailed review ) . an alternative and _ a priori _ equally valid choice is to write @xmath23^{-\rho_f } = { \cal c}_{f}[(t-\tc)/t]^{-\rho_f}\nonumber \\ & = & { \cal c}_{f}\left[1-\frac{\beta}{\bec}\right]^{-\rho_f } = { \cal c}_{f}\tau^{-\rho_f } , \label{beta_scaling}\end{aligned}\ ] ] where @xmath24 is the inverse temperature @xmath25 and @xmath26 . note that the temperature dependence of the normalization is now different for each observable . this `` @xmath24 scaling '' form has become the standard normalization for theoretical work on the critical properties of ferromagnets and analogous systems , see for instance @xcite , although more complex normalizations have been used in special cases . at higher order , confluent and analytic correction terms ( such as temperature independent constants ) are introduced . thus including the confluent correction terms , the critical behavior , @xmath27 , is written in terms of the @xmath24 scaling as @xmath28 where @xmath29 with @xmath30 being the [ universal ] confluent correction exponent , and @xmath31 is the confluent correction amplitude . in the @xmath17 scaling form , @xmath5 in the above equation is replaced by @xmath32 . this critical scaling form is firmly established by field theory in the limit of temperatures very close to @xmath33 @xcite . ratios of the @xmath31 for different observables are universal @xcite . the exponent @xmath34 is common in both scaling forms so long as @xmath35 . however , no general argument seems to have been given which would show that either the @xmath17 or the @xmath24 scaling is optimal for all ( or any ) observables when a much wider temperature range is considered . recently we have proposed an extended scaling scheme for normalizing observables such that the leading critical expressions remain good approximations right up to the trivial fixed point at infinite temperature @xcite . our extended scaling scheme is based on a consideration of high - temperature series expansions ( htse ) , and so is naturally formulated in terms of the @xmath24 scaling . the most important ingredient of the scheme is the introduction of non - critical prefactors @xmath36 in the normalizations , where each exponent @xmath37 is uniquely chosen such that the normalized @xmath38 tends to the correct asymptotic form in the limit @xmath39 . in the present work our aim is to further develop our extended scaling scheme to include explicitly the confluent and analytical correction terms . we then validate our scheme by analyzing data for three canonical ferromagnets : the @xmath40 ising , xy and heisenberg models on simple cubic lattices in dimension three . these models have been intensively studied over many years and their main critical parameters : @xmath0 , the critical exponents @xmath16 , @xmath34 , and certain critical amplitudes are known to high precision . careful accounts of studies using different complementary approaches are given for instance in refs . accurate simulation and htse results have been published in the form of tabulated data . the present analyses show that the appropriately normalized leading terms are good approximations over the entire temperature range , with small but identifiable corrections due to confluent and non - critical terms . we obtain estimates of non - universal critical parameters like critical amplitudes @xmath41 and confluent correction amplitudes @xmath42 from the high precision numerical data . our extended scaling analyses are in each case entirely consistent with field theoretical and htse estimates of the critical parameters . an important result of the present analysis is to demonstrate that the prefactors @xmath36 which have been introduced play a crucial role in extracting accurate values of the critical exponents from simulation data even in a temperature range close to @xmath0 , such as @xmath43 . in the standard scalings without the prefactors the estimates of the leading critical term and of the confluent term from analyses of numerical data turn out to be modified to order @xmath44 ( note @xmath45 ) . the same approach based on the htse should be directly applicable to a wide class of systems having the same intrinsic htse structure as the simple ferromagnets . extensions to more complex systems such as spin glasses are in principle straightforward @xcite . the paper is organized as follows . in sec . [ sec : basics ] we explain our extended scaling scheme for various thermodynamic observables , and discuss confluent corrections to scaling terms in our scheme . in sec . [ sec : analysis ] we give methods of analysis for numerical data using our extended scaling scheme . we show how they work in practice for ising , xy and heisenberg ferromagnets in sec . [ sec:3dising ] , [ sec:3dxy ] and [ sec:3dh ] , respectively . in sec . [ sec : conclusion ] we make concluding remarks and discuss related problems . let us suppose htse of an observable @xmath46 is given by @xmath47 the most important ingredient of our extended scaling scheme is then to write @xmath27 as @xmath48 where @xmath49 with @xmath50 . in particular , the leading contribution without the confluent correction is represented as @xmath51 the idea here is to let @xmath27 not only represent the correct power - law divergence @xmath52 with the critical amplitude @xmath53 ( and with certain confluent correction terms ) at temperatures close to @xmath0 but also have an asymptotic form consistent with the htse in the high temperature limit . the observable @xmath46 is then approximated as @xmath54 here the second term represents the analytic ( non - critical ) correction term in the present scheme . its coefficients @xmath55 and @xmath56 are determined in such a way that eq . ( [ f_expression ] ) , combined with eqs . ( [ fc_expression ] ) and ( [ rfc_expression ] ) , coincides with eq . ( [ hight_expression ] ) termwise as a function of @xmath24 ; for example , @xmath57 and a similar expression for @xmath58 . the above set of equations with the minimum number of the confluent and analytic correction terms is an optimized expression we propose for the function @xmath46 which is analytic in the range @xmath59 and is singular at @xmath60 . an important quantity for analyzing our extended scaling scheme is @xmath61 defined by @xmath62 it is the ratio of the measured values of observable @xmath14 to its leading critical term including the @xmath36 prefactor but without the critical amplitude @xmath41 . explicitly , in the vicinity of @xmath0 where @xmath63 , it behaves as @xmath64 the plot @xmath61 versus @xmath65 near @xmath66 thus becomes a straight line with intercept @xmath67 and slope @xmath68 , where the values of @xmath0 , @xmath16 and @xmath34 are assumed to be known ( @xmath37 and @xmath69 are given by htse analysis ) . in the limit @xmath70 , on the other hand , it becomes @xmath71 between these limits the form of @xmath61 will depend on the entire collection of unspecified higher order corrections to scaling . the `` true '' susceptibility , naturally defined through the magnetization response to an infinitesimal applied field , is given by the fluctuation - dissipation theorem as @xmath72 the reduced susceptibility @xmath73 is ( confusingly ) almost always referred to in the literature as `` the susceptibility '' . for consistency we will follow this convention and write the reduced susceptibility as @xmath74 , but we will refer systematically in the text to `` reduced susceptibility '' . the htse for the reduced susceptibility @xmath75 in @xmath40 ferromagnets is of the form with @xmath76 and @xmath77 , or with abbreviation of @xmath78 , @xmath79 then the leading divergent expression , eq . ( [ fc_leading ] ) , is written as @xmath80 with @xmath81 . the ratio @xmath82 of eq . ( [ rf_expression ] ) is reduced to @xmath83 where @xmath84 is eq . ( [ fc_expression ] ) for @xmath8 . note that @xmath85 at @xmath0 , @xmath86 near @xmath33 , and @xmath87 near infinite temperature . if @xmath82 remains close to 1 over the whole temperature range ( which is the case for the systems we consider as we will see below ) , the leading critical contribution without the correction terms , @xmath88 , is a good approximation for the reduced susceptibility , @xmath75 . furthermore , the small difference @xmath89 of the ising and xy ferromagnets in the whole temperature range @xmath59 turns out to be reproduced surprisingly well by our optimized expression , @xmath90 of eq . ( [ f_expression ] ) , only with one confluent and two non - critical correction terms . there are different alternative definitions for the correlation length , but any correlation length diverges at criticality as @xmath91 . the second moment correlation length @xmath92 is defined through the second moment @xmath93 with @xmath94 the space dimensionality @xcite . from now on we will refer to @xmath95 simply as @xmath96 . the htse results show that for @xmath97-vector @xmath40 spins , the series for @xmath98 is of the form @xmath99 and is well behaved with @xmath100 , where @xmath101 is the number of nearest neighbors . this yields @xmath102 and @xmath103 . we then reduce eq . ( [ fc_leading ] ) to @xmath104 where @xmath105 with @xmath106 being the standard critical amplitude in eq . ( [ beta_scaling ] ) for @xmath107 . the non - standard normalization prefactor @xmath108 for @xmath109 is our main result . the mean - field calculation @xcite of the correlation length through the fluctuation - dissipation theorem provides an example confirming the extended scaling form of eq . ( [ cri_xi ] ) . see also the analysis of fisher and burford @xcite , particularly their temperature dependent `` effective interaction range '' parameter @xmath110 . the critically divergent part of @xmath96 with the confluent correction terms is represented by @xmath111 and is written as @xmath112 the ratio @xmath113 becomes @xmath114 again , because of the confluent correction , it becomes @xmath115 near @xmath33 and @xmath116\beta + \cdots$ ] with @xmath117 being the coefficient in eq . ( [ htse_chi ] ) near infinite temperature . the usual analysis of the specific heat ( defined as the derivative of the internal energy at fixed volume @xmath118 ) near criticality assumes the form @xmath119^{-\alpha } + k \label{standard_c}\ ] ] where @xmath120 and @xmath121 is the critical amplitude of @xmath122 , and it is standard practice to introduce a large non - critical ( in fact temperature independent ) contribution @xmath123 ( see e.g. ref [ ] ) . while the series for the reduced susceptibility and the second moment @xmath124 are polynomial functions of @xmath24 with both odd and even terms , for bipartite ( such as bcc and simple cubic ) lattices the htse expression for @xmath125 consists of even powers of @xmath24 only @xcite , and can be written as @xmath126 one can carry through the same type of argument @xcite as in the case of @xmath127 , except that as all the terms in the series are even in @xmath128 , the critical behavior must be re - written in terms of the scaling variable @xmath129 $ ] replacing @xmath130 $ ] in the equivalent expressions for the correlation length . thus , with @xmath131 , one can write the leading critical term , which corresponds to eq . ( [ fc_leading ] ) , as @xmath132 where @xmath133 . if the confluent correction terms are included , we obtain the expression corresponding to eq . ( [ fc_expression ] ) as @xmath134 . \label{c_cri}\ ] ] where @xmath135 is the confluent correction amplitude . it is noted that , since the two critical amplitudes @xmath121 and @xmath136 are introduced in the standard way ( as represented by eq . ( [ standard_c ] ) for @xmath121 ) , the factors @xmath137 and @xmath138 appear in the definition of @xmath139 and in eq . ( [ c_cri ] ) , respectively . in fact there is a hyper - universal relationship linking this @xmath140 to @xmath106 @xcite : @xmath141 where @xmath142 is a constant whose value is known rather accurately @xcite . equation ( [ cri_c ] ) is not standard , but it can be seen to tend to the appropriate limit , @xmath143 , as @xmath17 approaches @xmath0 . in practice @xmath139 is much larger than unity ( as will be seen later in the case of the @xmath144 ising model ) which is the reason for the large non - critical contribution to @xmath125 . the non - critical contribution is in fact not a parameter to be adjusted freely , but it has to be determined through the high temperature limit of an equation which corresponds to eq . ( [ f_expression ] ) . ignoring the confluent correction so as to clarify the discussion , we know the exact high temperature limits for @xmath145 from eq . ( [ cri_c ] ) and for @xmath125 by eq . ( [ htse_c ] ) . then @xmath125 truncated to two leading non - critical correction terms is explicitly written as @xmath146 where the non - critical parameters @xmath147 are given by @xmath148 and @xmath149 . the coefficients @xmath150 and @xmath151 are known from htse . so if @xmath152 and @xmath153 have been measured independently , we can evaluate all the parameters which one needs to fix the functional form of @xmath125 , such as @xmath154 and @xmath139 determined through eq . ( [ eqn : hyper ] ) . this we assume to be @xmath155 , an optimized expression for @xmath125 , in the whole @xmath24 range @xmath156 . the thus calculated curve @xmath155 can be tested by comparing with simulation and htse data . though we will discuss thermodynamic limit behavior only and will not analyze finite - size - scaling ( fss ) data explicitly in the present paper , we note for reference that the extended scaling normalization modifies the fss expressions . the canonical fss ansatz @xcite is @xmath157 , \label{fss}\ ] ] where @xmath158 is a universal scaling function . the frequently used fss expression derived from eq . ( [ fss ] ) , @xmath159\ ] ] contains the implicit assumption of @xmath17 scaling for the correlation length . it is thus only appropriate if restricted to a very narrow range of temperature around @xmath33 . with the extended scaling and the finite size correlation length @xmath160 , the fss ansatz can be rewritten @xcite @xmath161 , \label{fss_ferro}\ ] ] or @xmath162,\ ] ] where the scaling functions behave as @xmath163 and @xmath164 at @xmath165 . for the reduced susceptibility with @xmath76 , the fss form is written as @xmath166 , \nonumber \\ & = & \tau^{-\gamma } \tilde{\mathcal f}_{\chi } \left[\left(\frac{l}{\beta^{1/2}}\right)^{1/\nu}\tau\right ] , \label{fss_chi}\end{aligned}\ ] ] where @xmath167 and @xmath168 constant . at @xmath165 . in a similar manner , the fss form for the correlation length @xmath160 , for which @xmath169 and @xmath102 , is written as @xmath170 , \label{fss_xi}\ ] ] where @xmath171 at @xmath165 . while the extended fss scheme for the susceptibility is modified from the standard one only by the @xmath24-prefactor in the argument of @xmath172 , the scaling plot is significantly improved for @xmath173 ising ferromagnetic and @xmath144 ising spin glass models @xcite . in order to make a stringent test of the extended scaling scheme , we study the three canonical ferromagnets : ising , xy and heisenberg , on three dimensional simple cubic lattices . high precision numerical data have been obtained for each of these systems for the temperature domain ranging from close to @xmath0 to about @xmath174 and the authors have generously published their data in tabulated form @xcite . the data have been taken on systems large enough for the data points to be representative of the thermodynamic limit . long htses have also been published for @xmath74 and @xmath124 and for @xmath175 for all three systems @xcite , and relatively longer series for the free - energy and the specific heat have been calculated for the ising model @xcite ; these series can be used to calculate @xmath8 , @xmath9 and @xmath122 explicitly for the region @xmath17 well above @xmath0 . below we call these htse and mc data as the _ real _ data . thanks to a combination of results from field theory and htse the values of the critical temperatures , the critical exponents and the critical amplitudes are known to a high degree of accuracy , and the confluent correction exponents are also well known . the [ non - universal ] confluent correction amplitudes are small for these three systems and the estimates are much less accurate ( see butera and comi @xcite for a detailed account ) . in each case we will plot the ratios @xmath82 and @xmath113 respectively defined by eqs . ( [ rchi_def ] ) and ( [ rxi_def ] ) with respect to @xmath65 , where we have used @xmath176 for simple cubic lattices . the plots near @xmath66 give us the critical and the confluent amplitudes as explained at the end of sec . [ sec : optimal ] . the ratios are defined in a way that they approach unity at infinite temperature . we will see that the two ratios defined in the extended scaling scheme are in fact close to unity ( within several hundredth deviation from unity at most ) in the whole range of @xmath5 . in addition , a simple scaling relation links the observables @xmath75 and @xmath96 through @xmath177 to leading order . this equation has the advantage that it can in principle be used to determine the exponent @xmath178 directly from a log - log plot of @xmath75 against @xmath96 near @xmath179 without any explicit knowledge of @xmath179 . for the extended scaling scheme ( @xmath24 scaling with the @xmath36 factors ) , the relation can be rewritten to leading order @xmath180 we will analyze the ratio defined by @xmath181 including the leading confluent correction factors , it behaves near @xmath33 as @xmath182 where @xmath183 and @xmath184 . we also note that the ratios @xmath185 are universal and are known to be about @xmath186 @xcite . this means that @xmath187 . for the @xmath144 simple cubic ising case @xmath188 , together with the high precision mc data at temperatures close to @xmath0 by kim et al@xcite and the htse estimates at relatively high temperatures by butera and comi@xcite , our own mc data are also used in order to interpolate them and to see overall temperature dependences of @xmath189 and @xmath175 . in our simulation we used the exchange mc method in combination with 64 bit multi - spin coding technique for making equilibration fast . the 64 different temperatures simulated are distributed in the range of @xmath190 . the amount of total mc steps for @xmath191 is @xmath192 and the last @xmath193 mc steps are used for taking thermal averages . figure [ fig:3di - chixi - e ] shows the parameter free log - log plot in the extended scaling form of the reduced susceptibility @xmath74 against @xmath194 data . without allowing for corrections , the slope of the line fitted to the data points ( ignoring our mc data when they are polluted by finite - size effects ) gives a first estimate @xmath195 . figure [ fig:3di - chixi - c ] is the equivalent standard ( @xmath17 or @xmath24 scaling ) log - log plot of @xmath74 against @xmath107 with the slope fixed to the one obtained from fig . [ fig:3di - chixi - e ] . it can be seen that in the standard scaling form the linear relationship breaks down rather quickly while in the extended scaling form with the same input data , the linearity persists to a good approximation up to an infinite temperature and down to temperatures near @xmath33 until limited by finite - size effects . we examine the leading correction of the extended scaling formula given by eq . ( [ ratio_normalized ] ) . to higher precision , fig . [ fig:3di - chixi - e2 ] shows a plot of @xmath196 against @xmath197 , assuming @xmath198 , @xmath199 and @xmath200 @xcite . the line is obtained by fitting the data points at @xmath201 to eq . ( [ ratio_normalized ] ) . the intercept at @xmath66 , @xmath202 , is in good agreement with the value @xmath203 assuming the critical amplitudes from htse @xcite . from the initial slope , @xmath204 , which we will comment on below . figures [ fig:3di - rchi ] and [ fig:3di - rxi ] show the ratios @xmath82 and @xmath113 of eqs . ( [ rchi_def ] ) and ( [ rxi_def ] ) , respectively . the numerical data are taken from kim et al @xcite , and the higher temperature values are calculated using the tabulated series of butera and comi@xcite . the htse terms were simply summed , and the points quoted correspond to the temperature range where the contributions from further terms can be considered negligible on the scale of the plots . by using appropriate extrapolation techniques , like differential approximations , the range over which the published htse data@xcite could be used to evaluate the temperature dependence of the observables to high precision could be considerably extended . the assumed critical parameters are @xmath205 , @xmath206 and @xmath207@xcite . from the initial intercepts and slopes of the fitted line at small @xmath5 , we obtain @xmath208 , @xmath209 , @xmath210 and @xmath211 . the @xmath212 values are in excellent agreement with the htse estimates@xcite , @xmath213 and @xmath214 . the @xmath42 values are in qualitative agreement with the htse estimates @xmath215 and @xmath216 @xcite . an overall conclusion on the extended scaling analysis of the @xmath144 simple cubic ising data , which will be confirmed by the analyses of the two other systems as well , is that this form of scaling is entirely consistent with the high precision values of critical parameters from extensive htse and field theoretical ( ft ) work . it is remarkable that over the entire temperature range from @xmath0 to infinity , the maximum deviations from the leading critical expressions of eqs . ( [ cri_chi ] ) and ( [ cri_xi ] ) are of the order of a few percent . let us go into further discussions about the @xmath217 behavior . in the inset of fig . [ fig:3di - rchi ] , we show the corresponding @xmath17 scaling ratio @xmath218 plotted against @xmath219 . the latter is calculated using the same values of the critical parameters @xmath220 and @xmath221 as those for @xmath217 , and so by construction in the low @xmath222 limit the intercepts and slopes of both ratios must coincide . it can be seen that in fact the @xmath17 scaling curve only approaches the @xmath24 scaling curve closely in the range of @xmath222 extremely close to zero . this result for @xmath74 with @xmath223 strongly suggests the superiority of the @xmath24 scaling , and hence our extended scaling , over the @xmath17 scaling . the full curve in the main frame of fig . [ fig:3di - rchi ] is the optimized expression , @xmath224 , which is evaluated through @xmath225 , with one confluent correction term discussed above and the two non - critical terms . the first term of the latter is a constant , @xmath226 , which yields simply @xmath227 at @xmath228 , or at an infinite temperature . its second term @xmath229 , which is also calculated via the parameters already fixed , specifies the slope of @xmath224 at @xmath228 . by taking into account only these three correction terms to the leading critical term , @xmath230 of eq . ( [ cri_chi ] ) , we obtain @xmath224 which reproduces surprisingly well the _ real _ data in the whole temperature range @xmath231 . notice that @xmath232 corresponds to @xmath233 . this result also indicates the superiority of our extended scaling with the @xmath24 scaling : @xmath230 not only represents the critical behavior of @xmath75 close to @xmath0 but also @xmath75 in the whole temperature range up to infinity . in this context we note again that @xmath234 for the `` true '' susceptibility and that the reduced susceptibility @xmath75 is derived through our extended scaling scheme . we also note that the similarity between the @xmath217 plot in fig . [ fig:3di - rchi ] and the @xmath235 plot in fig . [ fig:3di - rxi ] over the entire range of temperature is striking . lastly , fig . [ fig:3di - spec ] shows @xmath236 as a function of @xmath237 . the data points are calculated from the htse of arisue and fujiwara which extends to powers up to @xmath238 @xcite , mc energy data at @xmath239 and @xmath240 @xcite , and our numerical simulations for different sizes up to @xmath191 . we examine the extended scaling with non - critical contributions to @xmath125 given by eq . ( [ c_beta ] ) . by using the hyper - universal relation with the value of @xmath142 equal to @xmath241 @xcite for the @xmath144 ising model and our @xmath107 analysis , we obtain @xmath242 . then the non - critical parameters @xmath243 and @xmath244 are determined by @xmath150 and @xmath151 of htse and with putting @xmath245 . the solid curve represents the no - free parameter plot of eq . ( [ c_beta ] ) with the @xmath246 values cited or estimated above . the agreement over the whole temperature range is very satisfactory ; the non - critical correction is so strong that the bare leading power law is a poor approximation until very much closer to @xmath0 than the range covered by the figure . we consider this result as an indication that the extended scaling scheme combined with the optimized introduction of correction terms is an effective method for analyzing critically - divergent quantities in general . the same analysis has been carried out for the @xmath144 xy model ( @xmath247 ) . high precision numerical data were published by gottlob and hasenbusch@xcite , and are supplemented here by unpublished data kindly provided by m. hasebusch@xcite . the higher temperature data are calculated using the tabulated series of butera and comi@xcite . the critical point is @xmath248 and the exponents @xmath178 , @xmath34 , @xmath249 and @xmath250 are close to @xmath251 and @xmath252 , respectively@xcite . figure [ fig:3dxy - chixi - e ] shows the @xmath253 log - log plot . the leading scaling scheme works well up to very high temperatures , as in the ising case . the slope in fig . [ fig:3dxy - chixi - e ] gives us the value of @xmath178 which is in agreement with the previously reported values@xcite . figure [ fig:3dxy - chixi - e2 ] shows the plot of @xmath254^{2-\eta}$ ] against @xmath65 assuming the central values for the exponents @xmath178 and @xmath34 as mentioned above . figures [ fig:3dxy - rchi ] and [ fig:3dxy - rxi ] show @xmath82 and @xmath113 respectively against @xmath197 . from the @xmath66 intercept and the initial slope one can estimate @xmath255 , @xmath256 , @xmath257 and @xmath258 . these are all reasonably close to the quite independent htse values@xcite @xmath259 , @xmath260 , @xmath261 and @xmath262 , but are probably more reliable as they are consistent with the independent ft estimate of the universal ratio @xmath263 , see comments in ref . [ ] . also , the values of @xmath264 and @xmath265 in eq . ( [ ratio_normalized ] ) calculated from thus obtained set of the parameters reproduce well the data as shown in fig . [ fig:3dxy - chixi - e2 ] . this agreement again validates the extended scaling protocol and demonstrates that a combination of information from ft , htse , and simulations analyzed using this protocol can lead to consistent high precision critical parameter measurements . for comparison , we plot the standard @xmath17 scaling ratio @xmath266 introduced in sec . [ sec:3dising ] also in fig . [ fig:3dxy - rchi ] . its coincidence with @xmath217 will only hold for @xmath267 . as is the case for the ising system , the slope of @xmath268 is opposite to that of @xmath82 and the magnitude of @xmath269 is much larger than the corresponding magnitude of the extended ratio already at @xmath270 , or @xmath271 . in fig . [ fig:3dxy - rxi ] , we also show the @xmath17 scaling @xmath272 and the @xmath273 by @xmath24-scaling . the true leading term plus confluent correction holds with the extended scaling form , @xmath235 of eq . ( [ rxi_def ] ) with @xmath247 up to @xmath274 while with the other forms of scaling the correct limit will hold only for @xmath267 . in particular , the comparison of @xmath24 scaling @xmath275 with extended scaling @xmath235 demonstrates the importance of the @xmath108 prefactor in eq . ( [ rxi_def ] ) of the extended scaling scheme . these results imply that even close to @xmath0 the extended scaling is a considerable improvement over the standard scaling analysis for estimating critical parameters including the correction terms . the curve in fig . [ fig:3dxy - rchi ] represents our optimized estimates @xmath224 up to the second order of non - critical corrections . it reproduces about 5 percents change in @xmath217 , from about 1.05 at @xmath66 to 1 at @xmath228 , to a very good approximation . the corresponding relative change in @xmath235 is only less than 2 percents as seen in fig . [ fig:3dxy - rxi ] . to reproduce this change by @xmath276 to an approximation as good as @xmath224 in fig . [ fig:3dxy - rchi ] , however , more than third order non - critical correction terms are required . the same analysis has been carried out for the @xmath144 heisenberg model ( @xmath277 ) . high precision numerical data were published by holm and janke@xcite , and are supplemented here by higher temperature data calculated using the tabulated series of butera and comi@xcite . the critical point is @xmath278 and the exponents @xmath178 and @xmath34 are close to @xmath279 and @xmath280 @xcite . a recent exponent set @xcite gives @xmath281 , @xmath282 and @xmath283 . figure [ fig:3dh - chixi - e ] shows the @xmath284 log - log plot , which gives an estimate of @xmath178 consistent with that of ref . [ ] . figure [ fig:3dh - chixi - e2 ] shows the plot of @xmath285^{2-\eta}$ ] against @xmath197 assuming the exponent values as @xmath286 and @xmath287 . from this plot it appears that the initial slope is very small , corresponding to almost zero values for @xmath288 and @xmath289 . figures [ fig:3dh - rchi ] and [ fig:3dh - rxi ] show respectively @xmath82 and @xmath113 against @xmath197 , assuming the values of @xmath249 and @xmath250 in ref . [ ] . the mc and htse points may not appear to connect smoothly in these figures , because the manner in which the plots are presented enhances small deviations from the leading term form . however , the change in the values of both @xmath82 in fig . [ fig:3dh - rchi ] and @xmath113 in fig . [ fig:3dh - rxi ] are limited to within a few percent of their absolute magnitude in a whole range of @xmath5 as is the case for the other two ferromagnets studied . from the straight line fit of the mc data at small @xmath197 , one can estimate @xmath290 , @xmath291 , @xmath292 and @xmath293 . in this case the parameters are slightly less consistent with the htse estimates@xcite , @xmath294 , @xmath295 , @xmath296 and @xmath297 , but it should be noted that the estimates for these [ non - universal ] parameters depend very sensitively on the precise values taken for the critical exponents . we certainly need more precise data near @xmath0 to fix the values of these critical parameters for the heisenberg ferromagnet . we have outlined a systematic rule for the scaling and normalization of thermodynamic observables having critical behavior at continuous phase transitions . this `` extended scaling '' rule corresponds for ferromagnets to scaling of the leading term of the reduced susceptibility above @xmath0 as @xmath298 in agreement with standard practice , for the leading term of the second moment correlation length as @xmath299 with @xmath300 and for the leading term of the specific heat in bipartite lattices @xmath301 with @xmath133 plus strong non - critical correction terms which we explicitly evaluate by linking to the htse . analyses are made of high precision numerical data on three canonical ferromagnets using these expressions allowing for confluent scaling correction terms , plus non - critical corrections for the specific heat . near @xmath0 the results are entirely consistent with the critical parameter sets ( including the confluent corrections ) which have been obtained independently using sophisticated ft , htse and simulation techniques@xcite . the most important result found in the present work is that , for @xmath8 and @xmath9 the leading critical expressions with the extended scaling normalizations @xmath302 of eq . ( [ fc_leading ] ) agree to a very good approximation with the true @xmath46 up to infinite temperature . to demonstrate this fact more in details we have introduced the ratio @xmath61 defined by eq . ( [ rf_expression ] ) . for @xmath74 of the ising ferromagnet , for example , it is equal to the critical amplitude @xmath303 at @xmath0 ( @xmath66 ) and to unity at infinite temperature ( @xmath228 ) by definition . @xmath217 evaluated from the true data are represented by the data points in fig . [ fig:3di - rchi ] , while @xmath217 evaluated through the leading expression @xmath304 is independent of @xmath5 and equal to @xmath303 . the difference between the two is , however , at most 13 percent in this case . the corresponding differences for @xmath217 s of the two other ferromagnets as well as for @xmath235 s of the all three ferromagnets are less than several percent . this is our first result mentioned just above . we have next demonstrated that our extended scaling scheme , in terms of the @xmath24 scaling and with the temperature dependent prefactor @xmath36 , is of crucial importance in precisely extracting the small amplitude @xmath42 of the leading confluent correction term . the result is represented by the solid line in fig . [ fig:3di - rchi ] as well as those in figs . [ fig:3di - rxi ] , [ fig:3dxy - rchi ] , and [ fig:3dxy - rxi ] . in addition , we have also checked that the optimized expression @xmath305 of eq . ( [ f_expression ] ) , consisting of @xmath306 and one confluent and two non - critical correction , yield @xmath307 which reproduces the true @xmath61 surprisingly well as shown the curves in figs . [ fig:3di - rchi ] and [ fig:3dxy - rchi ] , though more than third non - critical correction terms would be required for equally good agreement in other observables . the large non - critical terms in the specific heat @xmath122 are also incorporated explicitly within our extended scaling scheme with no further adjustable input parameters . for the ising ferromagnet on the simple cubic lattice @xmath122 is calculated to a good approximation over the entire temperature range ( see eq . ( [ c_beta ] ) ) . although the non - critical correction terms are large for @xmath122 , the principle of the analysis is the same as the one applied above to @xmath8 and @xmath9 , for which the corrections to scaling are quite small . namely , each critically - divergent observable @xmath46 is represented by @xmath305 of eq . ( [ f_expression ] ) over the whole range of @xmath24 to a good approximation . the input consists of @xmath308 , a confluent correction term and a very limited numbers of non - critical correction terms derived from htse . together these results can be taken as validating the `` extended scaling '' approach . the approach could be systematically implemented in numerical work so as to improve yet further the accuracy of critical parameter sets derived for standard systems , possibly incorporating where necessary further higher order correction terms . perhaps a more fruitful application would concern the analyses of numerical data in more complex systems , where the present accuracy of the critical parameter sets is much poorer . for instance , it has been pointed out that for the analysis of data on spin glasses with symmetric interaction distributions @xmath24 should be replaced by @xmath309 in all expressions@xcite as all terms in the htse in these spin glasses are strictly even in @xmath24 . the extended scaling protocol allowing for this and with appropriate @xmath310 normalization factors has indeed been shown to significantly improve the consistency of critical exponent values derived from numerical simulations on ising spin glasses@xcite . we would like to thank p. butera for all his careful and patient advice , h. arisue for providing extensive tabulated series data , m. hasenbusch for allowing us to use his unpublished high - precision numerical data , and w. janke for helpful discussions . this work was supported by the grants - in - aid for scientific research ( no . 17540348 and no . 18079004 ) and naregi nanoscience project , both from mext of japan . the numerical calculations were mainly performed on the sgi origin 2800/384 at the supercomputer center , issp , the university at tokyo . 99 a. pelissetto and e. vicari , phys . rept.*368 * 549 ( 2002 ) . s. gartenhaus and w. s. mccullough , phys . b * 38 * , 11688 ( 1988 ) . butera and m. comi , phys . rev . b * 65 * , 144431 ( 2002 ) . wegner , phys . b * 5 * 4529 ( 1972 ) . c. bagnuls and c. bervillier , phys . b * 24 * , 1226 ( 1981 ) . i. a. campbell , k. hukushima and h. takayama , phys . lett . * 97 * , 117202 ( 2006 ) . r. guida and j. zinn - justin , j. phys . a * 31 * , 8103 ( 1998 ) . m. campostrini , m. hasenbusch , a. pelissetto , and e. vicari , phys . b * 74 * , 144506 ( 2006 ) . g. parisi , in _ statistical field theory _ , ( perseus books , 1988 ) . m. e. fisher and r. j. burford , phys . rev.*156 * , 583 ( 1967 ) . m. hasenbusch and k. pinn , j. phys . a * 31 * , 6185 ( 1998 ) . p. butera and m. comi , j. stat . phys . * 109 * , 311 ( 2002 ) , hep - lat/0204007 . h. arisue and t. fujiwara , phys.rev . e * 67 * , 066109 ( 2003 ) . d. stauffer , m. ferer and m. wortis , phys . * 29 * , 345 ( 1972 ) . m. e. fisher and m. n. barber , phys . * 28 * , 1516 ( 1972 ) . kim , a.j.f . de souza and d.p . landau , phys . e * 54 * , 2291 ( 1996 ) . a.p . gottlob and a. hasenbusch , physica a * 201 * , 593 ( 1993 ) . c. holm and w. janke , phys . b * 48 * , 936 ( 1993 ) . p. butera and m. comi , phys . b * 58 * , 11552 ( 1998 ) . y. deng and h.w.j . blte , phys . e * 68 * , 036125 ( 2003 ) . m. hasenbusch , private communication . m. campostrini , m. hasenbusch , a. pelissetto , p. rossi , and e. vicari , phys . b * 65 * , 144520 ( 2002 ) . d. daboul , i. chang and a. aharony , eur . j. b * 41 * , 231 ( 2004 ) . katzgraber , m . krner , and a. p. young , phys rev b * 73 * , 224432 ( 2006 ) .

a simple systematic rule , inspired by high - temperature series expansion ( htse ) results , is proposed for optimizing the expression for thermodynamic observables of ferromagnets exhibiting critical behavior at @xmath0 . this `` extended scaling '' scheme leads to a protocol for the choice of scaling variables , @xmath1 or @xmath2 depending on the observable instead of @xmath3 , and more importantly to temperature dependent non - critical prefactors for each observable . the rule corresponds to scaling of the leading of the reduced susceptibility above @xmath0 as @xmath4 in agreement with standard practice with scaling variable @xmath5 , and for the leading term of the second - moment correlation length as @xmath6 . for the specific heat in bipartite lattices the rule gives @xmath7^{-\alpha}$ ] . the latter two expressions are not standard . the scheme can allow for confluent and non - critical correction terms . a stringent test of the extended scaling is made through analyses of high precision numerical and htse data , or _ real _ data , on the three - dimensional canonical ising , xy , and heisenberg ferromagnets . for the susceptibility @xmath8 and the correlation length @xmath9 of the three ferromagnets , their optimized expression , which consists of the leading term ( respectively @xmath10 and @xmath11 ) and a quite limited number of confluent and non - critical correction terms , represents _ real _ data to surprisingly good approximations over the entire temperature range from @xmath0 to infinity . the temperature dependent prefactors introduced are of crucial importance not only in fixing the optimized expression at relatively high temperatures but also in determining appropriately the small amplitude correction terms . for the specific heat of the ising ferromagnet , @xmath12 combined with two non - critical correction terms which are calculated with no free parameters once the correlation length critical parameters are known , reproduces _ real _ data nicely also over the whole temperature range .

cond-mat0612665

laser - cooling @xcite allows to cool ions and atoms to very low temperatures . for this purpose , the full knowledge of the effects of the various physical parameters determining the cooling process is very important . among the various schemes , raman sideband cooling has been demonstrated to be a very successful technique for preparing atoms in the ground state of a harmonic potential @xcite . this cooling method exploits two stable or metastable atomic internal levels , which we call @xmath0 and @xmath1 , connected by dipole transitions to a common excited state @xmath2 . the transitions are usually driven by alternating pulses . a typical sequence alternates a coherent pulse , in which the atom is coherently transferred from @xmath0 to @xmath1 via a properly designed raman pulse , with a re - pumping pulse , in which the atom is incoherently re - scattered to @xmath0 by means of a laser resonant with @xmath3 . a change of the motional state during the repumping is a process of higher order in the ratio @xmath4 of the recoil frequency @xmath5 and the trap frequency @xmath6 , with @xmath7 being the mass of the atom and @xmath8 the wave vector of the one - photon transition . in the lamb - dicke regime , where @xmath9 , the probability for a change of the motional state is negligible and therefore , on the average , the system is cooled at a rate of one phonon of energy @xmath10 per cooling cycle . since there is a finite probability for the atom to be returned to the state @xmath1 instead of being repumped , a number of incoherent scattering events may be required before the atom is finally scattered into @xmath0 , which significantly increases the motional energy at the end of the optical pumping , reducing the cooling efficiency . furthermore , since two and three level schemes are realized using zeeman or hyperfine substates , decays from @xmath2 into other electronic substates can occur , leading to additional heating . + in this work we quantify the effect of a finite branching ratio in pulsed raman sideband cooling by calculating the average shift and diffusion of the vibrational energy distribution at the end of an incoherent pumping pulse . it should be pointed out that theoretical studies on laser - cooling for multilevel ions exist , which systematically include the branching ratio in their treatments @xcite . those studies have focussed on the lamb - dicke regime and on certain cooling schemes . here , we single out the effect of the branching ratio on cooling for an arbitrary ratio @xmath4 by applying sum rules . hence , we infer the cooling efficiency in the lamb - dicke regime and we discuss the result outside the lamb - dicke regime in connection with the proposal in @xcite . in particular , we show that in some parameter ranges the average effect of the multiple photon scattering can be described with an effective wave vector @xmath11 for the `` effective '' two - level transition @xmath12 @xcite . + this article is organized as follows . in section 2 we introduce the model for the evolution of a trapped ion during the repumping pulse in a raman transition , and we evaluate the average shift and variance of the ion energy at the end of the pulse . in section 3 we extend our analysis to cases where the channels of decay are multiple . in section 4 we draw some conclusions , and in the appendix we report the details of our calculations . we consider a three level atom as in fig . [ fig : scheme0 ] , whose internal levels are a ground state @xmath13 , stable or metastable state @xmath14 and excited state @xmath15 of radiative width @xmath16 ; @xmath17 , @xmath18 are dipole transitions , with respective probabilities of decay @xmath19 , @xmath20 , where @xmath21 . a laser resonantly drives the transition @xmath18 with rabi frequency @xmath22 . in the following we assume the wave vectors for both transitions to be equal to @xmath8 , which is a good approximation if , e.g. , @xmath14 and @xmath13 are hyperfine components of the ground state . we study the ion motion in one - dimension . + the master equation for the atomic density matrix @xmath23 is written as ( @xmath24 ) : @xmath25+l\rho_3 , \label{master0}\ ] ] where @xmath26 has the form : @xmath27 here , @xmath28 is the detuning of the laser on the @xmath29 transition , which we take to be zero , and @xmath6 is the frequency of the harmonic oscillator which traps the ion along the @xmath30-direction , with @xmath31 annihilation and creation operator , respectively . the interaction of the ion with the laser light is described in the dipole approximation by the operator @xmath32 : @xmath33 with @xmath34 ( with @xmath35 ) dipole raising operator , @xmath36 its adjoint , and @xmath30 the position of the atom . in writing ( [ coherent1 ] ) , ( [ master0 ] ) we have applied the rotating wave approximation and we have moved to the inertial frame rotating at the laser frequency . finally , the relaxation super operator has the form @xmath37 where @xmath38 is the dipole pattern of the spontaneous emission , which we take @xmath39 . + in the limit @xmath40 we can eliminate the excited state @xmath2 in second order perturbation theory @xcite , and reduce the three - level scheme to a two level one , with excited state @xmath14 and linewidth @xmath41 @xcite . in the limit @xmath42 the master equation for the density matrix @xmath43 , projection of @xmath23 on the subspace @xmath44 , can be rewritten as @xcite : @xmath45+\gamma_e\left[j_e\rho + j_g\rho\right ] , \label{master1}\ ] ] with @xmath46 effective hamiltonian @xmath47 , @xmath48 jump operators , defined as : where @xmath50 and where @xmath51 the solution of eq . ( [ master1 ] ) can be written as follows @xcite : @xmath52 with @xmath53 , and @xmath54 is the propagator for the effective hamiltonian : @xmath55 in eq . ( [ sol ] ) the successive contributions to the multiple scattering event are singled out : the first term on the rhs corresponds to the case in which at time @xmath56 no spontaneous decay has occurred . the second term describes a single scattering event , and the @xmath57-th term @xmath58 scattering events . the trace of each term corresponds to the probability associated with each event , and we can thus interpret eq . ( [ sol ] ) as the sum over all the possible paths of the scattering event weighted by their respective probabilities . at @xmath59 , @xmath60 , the atom is in @xmath0 and @xmath61 . for a pulse of duration @xmath62 we can replace @xmath56 by @xmath63 in the integrals of eq . ( [ sol ] ) and assume that the atom has been scattered into @xmath0 at the end of the pulse . now , each term on the rhs of eq . ( [ sol ] ) corresponds to the path associated with a certain number of scattering events into @xmath1 before the atom is finally scattered into @xmath0 . through ( [ sol ] ) we can evaluate the shift and the variance of the energy distribution at the end of the repumping pulse , which are defined as : @xmath64 where @xmath65 and @xmath66 is the initial motional energy of the atom . for simplifying the form of the discussion presented below , we rewrite the operator @xmath67 as follows : @xmath68 where @xmath69 , @xmath70 are defined as : @xmath71|l\rangle\langle l|,&\\ & \hat{j}\rho=\sum_l\sum_{l_1,l_1\neq l } and where @xmath72 is the basis of eigenstates of the harmonic oscillator . for @xmath73 , with @xmath74 initial distribution over the motional states , and according to eq . ( [ sol ] ) the steady state distribution has the form : @xmath75 where @xmath76 is the final distribution over the motional states . the first term in the rhs of ( [ sol1 ] ) is the sum over all paths from @xmath1 into @xmath0 , where after each jump the density operator is diagonal in the basis @xmath72 , whereas the second term contains all other paths . these latter terms can be neglected @xcite , and for @xmath77 the following relation holds : @xmath78|s\rangle = d_n(s ) . \label{distri}\ ] ] here , @xmath79 is the probability for the atom to be found in the state @xmath80 at @xmath59 , given the initial state @xmath81 at @xmath82 . using the explicit form ( [ tilde ] ) of @xmath69 in ( [ distri ] ) , @xmath79 has the form : @xmath83 where we have used the relation @xmath84 , with @xmath85 size of the ground state of the harmonic oscillator . substituting ( [ explicit ] ) into eqs . ( [ shift0 ] ) , ( [ diff0 ] ) , and applying the commutation properties of @xmath31 [ see the appendix ] , we find : @xmath86 , \label{width1 } \end{aligned}\ ] ] where @xmath87 is the lamb - dicke parameter . equation ( [ center1 ] ) represents the average shift to the vibrational energy at the end of the repumping pulse . for @xmath88 it corresponds to the average recoil energy @xmath89 associated with one incoherent raman scattering into @xmath0 . in this case , the second term in the rhs of eq . ( [ width1 ] ) vanishes , and eqs . ( [ center1 ] ) , ( [ width1 ] ) describe the scattering of one photon of wave vector @xmath90 on the effective two - level transition @xmath91 . similarly for @xmath92 an effective wave vector @xmath11 can be defined for the incoherent scattering on the two - level transition @xmath91 , which has the form @xmath93 thus , @xmath94 describes the average mechanical effect on the ion resulting from the multiple scattering of photons during the repumping pulse in a raman transition with branching ratio @xmath95 : this description is valid in the limit in which we may neglect the second term in the rhs of ( [ width1 ] ) , _ i.e. _ for @xmath20 and/or @xmath96 sufficiently small . in fig . 2 the first term of rhs of eq . ( [ width1 ] ) is compared with the complete expression for @xmath97 , for different values of the lamb - dicke parameter and as a function of @xmath20 . here , we see that @xmath11 characterizes the scattering process for almost any branching ratio in the lamb - dicke regime , whereas for @xmath98 an appreciable difference is already visible at @xmath99 . + from ( [ eta_eff ] ) we can define the effective lamb - dicke parameter @xmath100 describing an incoherent scattering into the state @xmath0 . this parameter provides an immediate estimate of the effect of the branching ratio on cooling . for @xmath101 , if @xmath102 the system is still in the lamb - dicke regime once it has been finally scattered into @xmath0 . furthermore , the coarse - grained dynamics of the system can be described by a rate equation for the motional states @xmath103 projected onto @xmath0 , where the rate of cooling ( heating ) is the real part of the sum of two terms : one corresponding to the component of the fluctuation spectrum of the dipole force at frequency @xmath6 ( @xmath104 ) , the other to the diffusion coefficient due to spontaneous emission from the excited state @xcite . this latter term is proportional to the squared lamb - dicke parameter for the incoherent scattering , and thus in our case to @xmath105 . from the well - known solution of the rate equation @xcite , the diffusion term affects the steady state average vibrational number @xmath106 , which is proportional to the diffusion coefficient . + outside the lamb - dicke regime , when @xmath107 is comparable to , or larger than , @xmath6 , there are no estabilished ground - state laser - cooling techniques for trapped atoms . here , we discuss our result in connection to the proposal in @xcite . there , a cooling scheme similar to raman sideband cooling has been presented , where pulses which pump the atoms to the ground state alternate with pulses confining the atoms to a limited region of motional energy . these confinement pulses have two - photon detuning @xmath108 to the red of the two - photon resonance frequency , where @xmath109 . then , the presence of a branching ratio must be taken into account by choosing @xmath110 . in this regime , pulses which efficiently counteract the average kick @xmath111 can be designed , provided that the following condition is fulfilled : @xmath112 where @xmath113 is the projection on @xmath30 of the two - photon wave vector of the coherent pulse . for two counterpropagating beams parallel to @xmath30 , @xmath114 and ( [ valid ] ) is fulfilled for @xmath115 , i.e. up to branching ratios @xmath116 . finally , outside the lamb - dicke regime the second term in the rhs of eq . ( [ width1 ] ) can not be neglected . hence , the diffusion is larger , and the efficiency of cooling may decrease dramatically as @xmath20 increases . in the following , we show that the average heating associated with the repumping pulse in multilevel - schemes can be described in the same way as discussed in the previous sections . + let us consider the level - scheme of fig . 3(a ) , where we have added to the scheme of fig . 1 a further channel of decay from @xmath2 into the stable or metastable state @xmath117 , with probability of decay @xmath118 such that @xmath119 , where @xmath120 , @xmath121 are the probability of decay onto @xmath122 , respectively . a laser resonantly drives the transition @xmath123 with rabi frequency @xmath124 . for @xmath125 the state @xmath2 can be adiabatically eliminated from the equations of motion . in this limit the master equation aquires the form @xmath126\\ & & + p_e'\gamma^{\prime}j_e\rho + p_1\gamma^{\prime}j_1\rho + p_g'\gamma^{\prime}j_g\rho,\nonumber\end{aligned}\ ] ] where @xmath127 , with @xmath128 . the effective hamiltonian is now : @xmath129 with @xmath130 . the solution at @xmath59 can be written as : @xmath131 hence , the shift and variance have the form evaluated in eqs . ( [ center1 ] ) , ( [ width1 ] ) where now the probability @xmath19 , @xmath20 are defined as @xmath132 , @xmath133 ( @xmath21 ) . in a similar way we have evaluated these quantities for schemes like the one shown in fig . 3(b ) , where a second excited state @xmath134 is coupled to @xmath1 via the same recycling laser tuned on the transition @xmath135 . for simplifying the treatment , we assume that a fourth laser resonantly drives the transition @xmath136 with rabi frequency @xmath137 ( grey arrow in fig . 3(b ) ) . thus , for low saturation eq . ( [ mastermany ] ) describes the dynamics , where now @xmath138 ( @xmath139 ) , with @xmath140 being the rate of scattering through the excited state @xmath141 ( @xmath142 ) . assuming that @xmath137 is such that @xmath143 , the solution in eqs . ( [ center1 ] ) , ( [ width1 ] ) applies to this case too , where now @xmath20 is defined as : @xmath144 and the probability @xmath19 of decaying into @xmath0 is @xmath145 . + the result ( [ result_prob ] ) shows that the total heating is minimum for @xmath146 , which can be obtained by choosing properly the laser intensity of the repumping lasers , or simply by removing degeneracies in the zeeman multiplet , for example with the help of a magnetic field . we have studied the motional heating associated with a finite branching ratio and in the presence of multiple decay and excitation channels at the end of a repumping pulse in raman sideband cooling . the first and second moments of the final energy distribution has been evaluated analytically , and the effect of the branching ratio has been singled out . we have shown that in a certain range of parameters the diffusion can be described with an effective wave vector @xmath11 , corresponding to an effective lamb dicke parameter @xmath147 for the incoherent scattering on the two - level transition @xmath12 . finally , on the basis of this result we have discussed the efficiency of raman sideband cooling and of a recent proposal of ground - state cooling outside the lamb - dicke regime @xcite . + analogous sum rules and considerations can be applied to raman cooling for free atoms @xcite . in that case the calculations are much simpler , since the total momentum of radiation and atom is a conserved quantity in the scattering event . + in general , these results can be applied to cooling schemes in multilevel atoms . the authors acknowledge many stimulating discussions with s. khler and v. ludsteck . g.m . thanks j.i . cirac , j. eschner and p. lambropoulos for many stimulating discussions . this work is supported in parts by the european commission within the tmr - networks erb - fmrx - ct96 - 0087 and erb - fmrx - ct96 - 0077 . using ( [ distri ] ) , we rewrite ( [ shift0 ] ) and ( [ diff0 ] ) as : the sum over @xmath151 can be contracted by observing that @xmath152 . then , using the commutation properties of the bosonic operators @xmath153 and the closure relation for the eigenstates of the harmonic oscillator , eq . ( [ b ] ) takes the form : c. monroe , d.m . meekhof , b.e . king , s.r . jefferts , w.m . itano , d.j . wineland , and p. gould , phys . lett . * 75 * , 4011 ( 1995 ) ; h. perrin , a. kuhn , i. bouchoule , c. salomon , europhys 42 * , 395 ( 1998 ) ; s.e . hamann , d.l . haycock , g. klose , p.h . pax , i.h . deutsch , and p.s . jessen , phys . lett . * 80 * , 4149 ( 1998 ) ; v. vuletic , c. chin , a.j . kerman , and s. chu , phys . 81 * , 5768 ( 1998 ) . r. dum , p. zoller , and h. ritsch , phys . rev . a * 45 * , 4879 ( 1992 ) . it can be shown that the terms in eq . ( [ sol ] ) containing at some time @xmath159 coherences between vibrational states are of order @xmath160 with respect to the terms that contain the populations only . this condition alone would not be sufficient , as the number of terms of @xmath161 corresponding to @xmath57 scattering events increases with @xmath57 . however , these term are oscillating functions of the intermediate vibrational states , and thus their sum is much smaller than the first term of ( [ distri ] ) . m. kasevich and s. chu , phys . * 69 * , 1741 ( 1992 ) ; n. davidson , h.j . less , m. kasevich , and s. chu , phys . 72 * , 3158 ( 1994 ) ; j. reichel , o. morice , g.m . tino , and c. salomon , europhys . lett . * 28 * , 477 ( 1994 ) . ( a ) level scheme with @xmath0 , @xmath1 , @xmath117 , stable or metastable states , @xmath2 excited state of radiative width @xmath16 and probability of decaying in the three ground states @xmath121 , @xmath120 and @xmath118 , respectively . two lasers couple @xmath1 and @xmath117 to @xmath2 ; ( b ) level scheme as in ( a ) with the addition of the excited state @xmath134 with decay probability on @xmath117 , @xmath1 equal to @xmath162 , @xmath163 , respectively , @xmath164 . two lasers couple @xmath1 and @xmath117 to @xmath134 .

we have investigated the efficiency of pulsed raman sideband cooling in the presence of multiple decay and excitation channels . by applying sum rules we identify parameter regimes in which multiple scattering of photons can be described by an effective wave vector . using this method we determine the rate of heating caused by optical pumping inside and outside the lamb - dicke regime . on this basis we discuss also the efficiency of a recently proposed scheme for ground - state cooling outside the lamb - dicke regime [ g. morigi , j.i . cirac , m. lewenstein , and p. zoller , europhys . lett . * 39 * , 13 ( 1997 ) ] .

quant-ph0011024

infrared spectroscopy provides a crucial means in the identification of interstellar dust compositions . recent infrared satellite observations by the _ infrared telescope in space _ ( _ irts _ ; * ? ? ? * ) and the _ infrared space observatory _ ( _ iso _ ; * ? ? ? * ) have revealed several new dust features in the diffuse emission , indicating the presence of new dust components in the interstellar medium @xcite . observations by _ iso _ also clearly show the presence of crystalline silicates around young and evolved stars for the first time @xcite , while it is not yet certain whether crystalline silicates exist commonly in interstellar space . in the present paper we report the observations of active star - forming regions , the carina nebula and the sharpless 171 ( s171 ) region with the long - wavelength spectrometer ( lws ; * ? ? ? * ) on board _ iso _ and the detection of far - infrared features around 65@xmath0 m and 100@xmath0 m in the diffuse emission . the carina nebula is one of the most active regions on the galactic plane and known to contain a number of early - type stars @xcite . the s171 region is a typical region and molecular cloud complex @xcite . both regions are supposed to represent the characteristics of active regions in the galaxy . possible carriers of the two detected features are discussed and we investigate the possibility that carbon onion grains of curved graphitic shells are the carrier for the broad interstellar 100@xmath0 m feature . the central @xmath1 portion of the carina nebula was observed by two - dimensional raster scans with the lws full grating scan mode and the far - infrared spectra for 43197@xmath0 m were obtained for 132 positions ( for details of the observations , see * ? ? ? a one - dimensional scan was made for 24 positions on a line from the heating source to the molecular cloud region in s171 with the same lws observing mode @xcite . the observed area of the both objects includes ionized regions and molecular clouds and the spectra sample the diffuse emission from the interstellar matter rather than the emission from point - like sources . the off - line processing data of version 10.1 ( olp 10.1 ) provided by the iso archival data center were used in the present study . the spectra were defringed , converted into the surface brightness , and the extended source correction was applied by the iso spectral analysis package ( isap ) software . the beam size and the correction factors were taken from the latest lws handbook @xcite . there are gaps in the spectra between the detector channels , which can be ascribed to the uncertainties either in the responsivity , in the dark current , or in the spatial brightness distribution in the beam . figure [ fig1 ] shows examples of the obtained original spectra of the two regions , while figure [ fig2 ] indicates their stitched spectra to correct the gaps . both spectra were taken at the interface regions between the molecular cloud and ionized gas , where the far - infrared intensity is sufficiently large to investigate dust features . the stitched spectra are made by scaling each detector signal because the observed regions are bright enough that the uncertainty in the dark current should be less significant than those in the responsivity or in the spatial brightness distribution . as can be seen in figure [ fig1 ] , the amount of the gaps is small ( @xmath2 5% ) except for the three longest channels ( @xmath3@xmath0 m ) , where 1020% scaling is necessary to correct the gaps . the presence of a relatively narrow band feature at 65@xmath0 m is seen even in the unstitched spectra , particularly in s171 . in the spectrum of the carina nebula , the appearance of this feature is slightly disturbed by the higher levels of the adjacent channel spectra ( sw2 and sw4 ) relative to the level of the sw3 channel , but it can still be seen in the individual spectrum of the sw3 channel . a broad feature centered around 100@xmath0 m is also noticeable in the unstitched spectrum of s171 . the slope of the continuum starts to become flatter around the boundary between the sw4 and sw5 channels , indicating a feature starting around 80@xmath0 m . in the unstitched spectrum of the carina nebula , the gap between the sw4 and sw5 channels makes the feature less obvious , but the change in the slope in the sw5 channel can still be seen . the stitched spectrum clearly indicates the presence of the feature . however there is no appreciable abrupt change in the slope at longer wavelengths and the longer wavelength end of the feature is difficult to estimate from these spectra . neither spurious features have been reported nor the relative spectral response functions have the corresponding features in these spectral ranges @xcite . we will discuss possible underlying continua to confirm the presence of the feature and estimate the feature profile in next section . similar features are seen at about a half of the observed positions both in the carina and s171 regions . since these features are seen in a wide area of the interstellar medium , the band carriers must be ubiquitous species in interstellar space . @xcite reported the presence of 65@xmath0 m and 90@xmath0 m features in the spectra of evolved stars . figure [ fig3]a shows a spectrum of ngc6302 taken from the iso archival data center for comparison ( cf . * ? ? ? the continuum emission indicates a much higher temperature than those in figure [ fig2 ] and the features are weakly seen on the steep continuum . to see the features more clearly , the flux is multiplied by the square of the wavelength ( @xmath4 ) and plotted in figure [ fig3]b . the interstellar 65@xmath0 m feature seems very similar to that detected in evolved stars . the peak of the 65@xmath0 m feature is located obviously longer than [ ] 63@xmath0 m line ( fig . [ fig4 ] ; see also fig . [ fig6 ] ) and thus is not compatible with the crystalline ice band at 62@xmath0 m @xcite . @xcite have proposed a ca - bearing crystalline silicate , diopside ( camgsi@xmath5o@xmath6 ) , as a possible carrier of the 65@xmath0 m feature in evolved stars . cryogenic measurements of the optical properties of diopside support the identification @xcite . figure [ fig4 ] shows a comparison of the observed spectra with the laboratory data . the best fit continua described in next subsection are also plotted . the laboratory spectrum shows a narrower profile than those observed and other species , such as water ice and dolomite ( camg(co@xmath7)@xmath5 ) , have been suggested to contribute also to the 65@xmath0 m band emission @xcite . diopside has a weak feature also at 44.5@xmath0 m . the lws detector in this spectral range ( sw1 ) is less sensitive and known to have strong hysteresis . in the present spectra a band feature is seen around 45@xmath0 m both in the upward and downward scans of both spectra , suggesting the presence of the 45@xmath0 m feature . however large noises in this spectral range preclude the firm detection and further observations are needed to confirm the feature . band features of other crystalline minerals , such as the 69@xmath0 m band of forsterite seen in ngc6302 , are not seen in the present lws spectra . diopside also has strong features in 3040@xmath0 m . the carina nebula was observed by short - wavelength spectrometer ( sws ; * ? ? ? * ) and the spectra of 2.345@xmath0 m have been obtained . however the sws spectra were not taken at the same positions as the lws spectrum and thus the direct examination is difficult . the sws spectra are dominated by strong continuum and do not clearly show any solid bands except for the broad 22@xmath0 m feature @xcite . if the identification of the interstellar 65@xmath0 m with diopside is correct , this is the first detection of a crystalline silicate in the diffuse interstellar medium . efficient destruction of dust grains by interstellar shocks suggests that a large fraction of interstellar dust must be formed in interstellar space in addition to those supplied from stars @xcite . diopside is a high - temperature condensate and may survive harsh conditions . calcium is a less abundant element than magnesium or silicon , but it is highly depleted in the gas phase of the interstellar medium @xcite . therefore the presence of ca - containing dust should not be surprising in interstellar space . based on the measured band strength of diopside @xcite and the observed strength relative to the continuum , we roughly estimate that 510% of solar abundance calcium in diopside grains is sufficient to account for the observed band emission if we take the commonly used mass absorption coefficient of 50 @xmath8g@xmath9 for the continuum emission at 100@xmath0 m @xcite . @xcite have suggested that the crystallinity of the silicates is less than 12% in the interstellar medium based on the observations of protostars . since the feature seen around 100@xmath0 m is quite broad and weak , we investigate several cases for the underlying continuum to examine the presence of the broad 100@xmath0 m feature in detail and to make a rough estimate of the 100@xmath0 m feature profile . in the estimate of the continuum we assume the baseline positions to be at 5560@xmath0 m , 7080@xmath0 m , and 140190@xmath0 m or 120190@xmath0 m ( see below ) and we try to fit the observed spectra in these ranges with the model continuum as much as possible . because the shortest spectral range ( 5560@xmath0 m ) has higher noises , less weight is put on this range in the fitting . we first adopt the dust model with the power - law emissivity ( @xmath10 , where @xmath11 is a constant ) for the continuum emission . we found that the single - temperature graybody model can not fit the entire baseline positions satisfactorily . particularly the model always gives a higher flux at long wavelengths than the observed spectra . this discrepancy can not be solved by increasing @xmath11 because then the model would provide unnecessarily large fluxes at shorter wavelengths . introducing a second component with a low temperature improves the fit drastically . the model of @xmath12 both for warm and cold grains gives reasonable fits , but still has slightly larger fluxes at longest wavelengths ( @xmath13@xmath0 m ) than the observed spectra . increase of @xmath11 from 2 to unrealistically large 3 for the cold grains does not improve the fit appreciably . the power - law emissivity model has a spectral dependence of @xmath14 in the rayleigh - jeans regime . the discrepancy in the fit at longest wavelengths comes from the fact that the observed spectra have a gradually changing power - law index . the brightness distribution within the lws beam affects the global shape of the spectrum . as shown in figure [ fig1 ] the unstitched spectra have relatively large gaps in longer wavelengths ( @xmath15@xmath0 m ) , suggesting an uncertainty associated with the slope in this spectral range . it also suggests a difficulty in defining the assumed baseline in the longer wavelengths . in the following we present two cases for the baseline ; one with 140190@xmath0 m ( case a ) and 120190@xmath0 m ( case b ) to examine the effect of the assumed baseline and as a more realistic model we examine the astronomical silicate and graphite grain model @xcite the silicate and graphite grains both have approximately a power - law emissivity of @xmath16 in the far - infrared and this model provides slightly better fits than the power - law emissivity model of @xmath17 . we present the results of the silicate - graphite model in the following . we assume different single temperatures for each of the astronomical silicate and graphite grains and search for the best fit temperatures . the observed regions may contain various temperature components of various dust grains and thus these fits are a simple approximation for the underlying continuum . in figure [ fig2 ] the best fit results are plotted together with the observed spectra . the dotted lines indicate the results for the case a baseline , which fit the observed spectra reasonably well even in the longest wavelengths . they overlap mostly with the observed spectra for @xmath18@xmath0 m in the plot . the dashed lines show those for the case b baseline , which have obviously higher fluxes at longest wavelengths ( @xmath13@xmath0 m ) than the observed spectra . both cases clearly indicate the presence of an excess feature starting around 80@xmath0 m . the slope change around 80@xmath0 m is steep and can not be accounted for by extra graybodies . the similarity of the excess profile in two different regions supports the presence of the feature and suggests the common origin . @xcite have attributed the 90@xmath0 m feature in evolved stars to calcite ( caco@xmath7 ) , a carbonate mineral . in figure [ fig3]b we also plot a single - temperature graybody as a simple reference continuum . comparison with figure [ fig2 ] indicates that the 90@xmath0 m feature in ngc6302 is narrower than the interstellar 100@xmath0 m feature . the spectrum of ngc6302 shows a clear slope change around 100@xmath0 m , which indicates the longer wavelength edge of the feature . in contrast , the interstellar spectra do not show the clear change in the slope and suggest that the feature is extended to longer wavelengths than the 90@xmath0 m feature . the longer wavelength edge of the interstellar feature can not be well determined . although the exact peak position and width of the feature depend on the assumed continuum and the location of the baseline , the interstellar 100@xmath0 m feature seems to be shifted to longer wavelengths and have a wider width than the 90@xmath0 m feature seen in evolved stars . while carbonate grains are a likely candidate for the 90@xmath0 m emission around evolved stars and may partly contribute to the interstellar 100@xmath0 m feature , we examine the possibility of alternative species which has a broad feature around 100@xmath0 m for the band carrier in the diffuse emission . in the following we investigate whether carbon onion grains consisting of concentric curved graphitic sheets @xcite can account for the observed broad 100@xmath0 m feature or not . graphite is an anisotropic material and has different optical properties in the directions parallel and perpendicular to the c - axis ( the c - axis is perpendicular to the graphitic plane ) . it has an interband transition around 80@xmath0 m in the direction perpendicular to the c - axis @xcite . the emission efficiency of graphite spheres can be calculated by the so - called approximation @xcite , in which the efficiencies in the perpendicular and parallel to the c - axis are averaged with the weight of and , respectively . this approximation is valid in the small particle limit if the sphere consists of layered graphitic sheets and the optical properties in both directions are independent . in the graphite sphere , the emission efficiency in the direction parallel to the c - axis is much larger than that in the direction perpendicular to the axis in the far - infrared region . therefore the interband transition feature mentioned above is not visible in the averaged efficiency @xcite . in carbon onions , on the other hand , the graphitic layer is curved and approximately constitutes closed shells . thus the optical properties in the both directions should be mutually coupled and the interband feature can become visible in the emission efficiency of carbon onion grains . figure [ fig5 ] shows the emission efficiency factors divided by the grain radius for a graphite sphere and a carbon onion grain under the assumption that the grain radius is much smaller than the wavelengths in question . here the dielectric constants of graphite in the directions parallel and perpendicular to the c - axis at room temperature measurements are adopted in the calculations ( * ? ? ? * ; * ? ? ? * see below for discussion ) . the efficiency for the carbon onion is calculated by the formulation by henrard et al . ( 1993 ) and is assumed to have a central cavity of 0.7 in radius relative to the particle size . the appearance of the feature is insensitive to the size of the cavity . a broad feature around 100@xmath0 m is seen in the emission efficiency of the graphite sphere in the direction perpendicular to the c - axis , but it is hardly seen in the averaged efficiency . on the other hand , the far - infrared feature is evident in the emission efficiency of carbon onion particles . figure [ fig6 ] shows a comparison of the observed feature with that of carbon onion grains . to make the comparison easy the observed spectra are divided by the assumed continuum , while the efficiency of the carbon onion is divided by @xmath19 . two lines in the upper two panels indicate the effect of the assumed continuum . carbon onion grains show a similar broad feature to that observed in the diffuse interstellar emission , but details of the profile do not match perfectly . taking account of the uncertainties in the shape of the underlying continuum and the optical properties of carbon onions ( see below ) , the similarity of the band feature suggests that carbon onions are a possible carrier of the interstellar 100@xmath0 m feature . the 100@xmath0 m feature of carbon onions results from the surface resonance of small particles @xcite and appears near the wavelength where the real part of the dielectric constants in the perpendicular direction just becomes below zero . the exact position and profile of the feature thus depend on the adopted dielectric constants . while the electronic structure of carbon onions has been suggested to not differ significantly from that of graphite @xcite , the contribution of free electrons , which dominates in the far - infrared regions , may be different . in fact , measurements of electron spin resonance and electron energy - loss spectroscopy suggest that @xmath20 electrons in carbon onions are mostly localized in small domains @xcite . the localization of @xmath20 electron will decrease the contribution of free electrons , shifting the surface mode to wavelengths longer than 100@xmath0 m . we surmise that the shift can be more than 10@xmath0 m . but it is difficult to estimate the possible range of the shift because the behavior of free electrons depends also on the temperature and the strength of the interband transition in carbon onions could also be affected by the localization . the @xmath20 electron localization , the temperature dependence , and the possible change in the interband transition strength should affect the optical properties of carbon onions in the far - infrared . the match seen in figure [ fig6 ] may be just a coincidence in this sense . the present calculation suggests that the observed band feature can be accounted for if carbon onion grains contribute to 2030% of the far - infrared emission . carbon is an abundant element , but the exact form of carbon dust in the interstellar medium is not yet clear ( e.g. * ? ? ? carbon onions are a likely form other than graphite or amorphous carbon in addition to small aromatic particles or large molecules whose presence has been confirmed by the infrared emission bands in the diffuse interstellar radiation @xcite . carbon onions have recently attracted attentions as a new form of carbon material following the discovery of fullerens and their family . in astronomy , they are suggested to be formed in interstellar processes @xcite and the harsh conditions accompanying interstellar dust formation are favorable for the formation of onions @xcite . they have been proposed as a likely candidate for the interstellar 220 nm extinction hump @xcite . the quenched carbonaceous composite ( qcc ) , which shows a feature similar to the interstellar 220 nm hump @xcite , has also been shown to contain graphitic shell structures @xcite . it is not unexpected that the band features of carbon onions , if exist , also appear in the infrared region . in the present paper we simply propose the possibility that the interband feature of graphite in the far - infrared could appear in the emissivity spectrum of particles consisting of curved graphitic sheets and the observed broad interstellar feature around 100@xmath0 m may be accounted for by carbon onion grains . experimental work is definitely needed for further investigations . in the present paper we reported the detection of two far - infrared features at 65@xmath0 m and 100@xmath0 m in the diffuse infrared emission . the 65@xmath0 m band can reasonably be attributed to the ca - rich silicate , diopside . if this identification is correct , this is the first detection of a crystalline silicate feature in the interstellar diffuse emission . the interstellar 100@xmath0 m feature seems to be broader and peaked at longer wavelengths than the calcite feature seen in evolved stars although the precise estimate of the band profile is difficult . as a possible band carrier we investigate the possibility that the feature originates from carbon onion grains . while the observed feature may be accounted for by carbon onion grains if the assumed optical properties are adequate , the appearance of the feature is sensitive to the electronic structure of carbon onions . the origin of the interstellar 100@xmath0 m feature must be investigated in further experimental studies . the authors thank k. kawara , y. satoh , t. tanab , h. okuda , t. tsuji , h. shibai , and other members of the japanese iso group for their continuous help and support . we also thank s. tomita and s. hayashi for stimulating discussions on the optical properties of carbon onions and h. chihara and c. koike for providing us the far - infrared data of diopside and calcite . this work was supported in part by grant - in - aids for scientific research from the japan society of promotion of science ( jsps ) . 1982 , absorption and scattering of light by small particles ( new york : wiley ) chan , k .- w . , & onaka , t. 2000 , , 533 , l33 chihara , h. , koike , c. , & tsuchiyama , a. 2001 , , 53 , 243 clegg , p. , et al . 1996 , , 315 , l38 cohen , m. , barlow , m. j. , sylvester , r. j. , liu , x .- w . , cox , p. , lim , t. , schmitt , b. , & speck , a. k. 1999 , , 513 , l135 de graauw , t. , et al . 1996 , , 315 , l49 demyk , k. , jones , a. p. , dartois , e. , cox , p. , & dhendecourt , l. 1999 , , 349 , 267 draine , b. t. , & lee , h .- m . 1984 , 285 , 89 draine , b. t. , & malhotra , s. 1993 , , 414 , 632 feinstein , a. 1995 , rev . astrofis . , 2 , 57 gry , c. , et al . 2002 , iso handbook ; volume iv : lws the long - wavelength spectrometer , version 2.0 ( esa ) henrard , l. , lucas , a. a. , & lambin , ph . 1993 , , 406 , 92 henrard , l. , lambin , ph . , & lucas , a. a. 1997 , , 487 , 719 hildebrand , r. h. 1983 , , 24 , 267 jones , a. p. , tielens , a. g. g. m. , & hollenbach , d. j. 1996 , , 469 , 740 kemper , f. , jger , c. , waters , l. b. f. m. , henning , th . , molster , f. j. , barlow , m. j. , lim , t. , & de koter , a. 2002 , , 415 , 295 kessler , m. f. , et al . 1996 , , 315 , l27 koike , c. , et al . 2000 , , 363 , 1115 mattila , k , lemke , d. , haikala , l. k. , laureijs , r. j. , leger , a. , lehtinen , k. , leinert , c. , & mezger , p. g. 1996 , , 315 , l353 mizutani , m. , onaka , t. , & shibai , h. 2002 , , 382 , 610 molinari , s. , ceccarelli , c. , white , g. j. , saraceno , p. , nisini , b. , giannini , t. , & caux , e. 1999 , , 521 , l71 molster , f. j. , lim , t. l. , sylvester , r. j. , waters , l. b. f. m. , barlow , m. j. , beintema , d. a. , cohen , m. , cox , p. , & schmitt , b. 2001 , , 372 , 165 molster , f. j. , waters , l. b. f. m. , & tielens , a. g. g. m. 2002 , , 382 , 222 murakami , h. , et al . 1996 , , 48 , l41 nuth , j. 1985 , , 318 , 166 okada , y. , onaka , t. , shibai , h. & doi , y. 2002 , , submitted onaka , t. , yamamura , i. , tanab , t. , roellig , t. l. , & yuen , l. 1996 , , 48 , l59 philipp , h. r. 1977 , , 16 , 2896 pichler , t. , knupfer , m. , golden , m. s. , fink , j. , & cabioch , t. 2001 , , 63 , 155415 sakata , a. , wada , s. , okutsu , y. , shintani , h. , & nakada , y. 1983 , , 301 , 493 savage , b. d. , & sembach , k. r. 1996 , , 34 , 279 smith , r. g. , robinson , g. , hyland , a. r. , & carpenter , g. l. 1994 , , 271 , 481 tomita , s. , sakurai , m. , ohta , h. , & hayashi , s. 2001 , , 114 , 7477 ugarte , d. 1992 , , 359 , 707 ugarte , d. 1995 , , 443 , l85 venghaus , h. 1977 , phys . solidi ( b ) , 81 , 221 wada , s. , kaito , c. , kitamura , s. , ono , h. , & tokunaga , t. a. 1999 , , 345 , 259 waelkens , c. , et al . 1996 , , 315 , l245 walborn , n. r. 1995 , rev . , 2 , 51 waters , l. b. f. m. , et al . 1996 , , 315 , l361 yang , j. , & fukui , y. 1992 , , 386 , 618

we report the detection of a feature at 65@xmath0 m and a broad feature around 100@xmath0 m in the far - infrared spectra of the diffuse emission from two active star - forming regions , and . the features are seen in the spectra over a wide area of the observed regions , indicating that the carriers are fairly ubiquitous species in the interstellar medium . a similar 65@xmath0 m feature has been detected in evolved stars and attributed to diopside , a ca - bearing crystalline silicate . the present observations indicate the first detection of a crystalline silicate in the interstellar medium if this identification holds true also for the interstellar feature . a similar broad feature around 90@xmath0 m reported in the spectra of evolved stars has been attributed to calcite , a ca - bearing carbonate mineral . the interstellar feature seems to be shifted to longer wavelengths and have a broader width although the precise estimate of the feature profile is difficult . as a carrier for the interstellar 100@xmath0 m feature , we investigate the possibility that the feature originates from carbon onions , grains consisting of curved graphitic shells . because of the curved graphitic sheet structure , the optical properties in the direction parallel to the graphitic plane interacts with those in the vertical direction in carbon onion grains . this effect enhances the interband transition feature in the direction parallel to the graphitic plane in carbon onions , which is suppressed in graphite particles . simple calculations suggest that carbon onion grains are a likely candidate for the observed 100@xmath0 m feature carrier , but the appearance of the feature is sensitive to the assumed optical properties .

astro-ph0211378

processes of meson electroproduction have played an important role in studying the structure and properties of matter ( see , _ e.g. _ review @xcite ) . in the past few years , however , reactions with production of dileptons in hadron - hadron and hadron - nucleus collisions have drawn much attention @xcite . in these reactions virtual photons , which materialize as dileptons ( _ e.g. _ , the @xmath2 pair ) , carry unique information on properties of matter because the processes in which the particle structure is formed proceed in the timelike region of the `` mass '' ( @xmath3 ) of the virtual photon . therefore , further investigation of these reactions is necessary and promising in acquisition of new and perhaps unexpected information about the form factors of hadrons and nuclei . the inverse pion electroproduction ( ipe ) , @xmath1 , ( or dileptonproduction ) , being for a long time the only source of information on the nucleon electromagnetic structure in the timelike region , has been investigated both theoretically @xcite and experimentally @xcite since the beginning of the 1960s . in refs . @xcite , we worked out the method of extracting the pion and nucleon electromagnetic form factors from ipe at low energies . this method has been successfully applied in the analysis of experimental data on the nucleon and @xmath4c and @xmath5li nuclei @xcite and values of the form factors were obtained for the first time in the timelike region of @xmath3 ranging from 0.05 to 0.22 ( gev / c)@xmath6 . in refs . @xcite , the authors proposed to use ipe at intermediate ( above @xmath7 resonances ) energies and small @xmath8 to study the nucleon electromagnetic structure and justified it up to @xmath9 . though experimental data @xcite on the @xmath10 process are now available , there still remains a wide range of @xmath3 ( up to @xmath11 ) , where the form factors can not be measured directly in these experiments . on the other hand , the intense pion beams available now enable one to perform more detailed experiments on ipe aimed at both extracting the hadron structure and carrying out a multipole analysis similar to those for photoproduction and electroproduction ( see,_e.g . _ , @xcite ) . such experiments can address interesting topics . for example , in the @xmath12 region it is challenging to verify the @xmath3 dependence of the color - magnetic - force contribution found in the constituent quark model @xcite . it is , therefore , worth recalling the earlier discovered properties of the photoproduction , electroproduction , and ipe to consistently substantiate methods of studying the electromagnetic and weak structure of the nucleon on the basis of the ipe data in the first resonance region , and to provide new results of this analysis . additional motive for studying ipe in the first resonance region is the possibility of investigating the nucleon weak structure by utilizing the same data as for the electromagnetic structure . this possibility is based on the current algebra ( ca ) description and on the remarkable property of ipe . in the ipe process the creation of the @xmath2 pairs of maximal mass ( at the `` quasithreshold '' ) is dominated by the born mechanism , whereas the rescattering - effect contributions are at the level of radiative corrections up to the total @xmath7 energy @xmath13 gev ( the `` quasithreshold theorem '' ) @xcite . due to this property , the threshold ca theorems for the pion electroproduction and photoproduction can be justified in the case of ipe up to the indicated energy @xcite . this allows one to avoid threshold difficulties when using the ipe data ( unlike the electroproduction one ) for extracting the weak form factors of the nucleon . furthermore , in the case of ipe there is no strong kinematic restriction inherent to the @xmath14 capture and no kinematic suppression of contributions of the induced pseudoscalar nucleon form factor to the cross sections of `` straight '' processes , such as @xmath15 , present because of multiplying by the lepton masses . information on the pseudoscalar nucleon form factor @xmath16 , which is practically absent for the above reasons , is important because @xmath16 contains contributions of states with the pion quantum numbers and , therefore , it is related to the chiral symmetry breaking . this would enable us , _ e.g. _ , to test the goldberger treiman relation . another aim of this paper is to draw attention of experimenters to the process @xmath1 as a natural and unique laboratory for investigating the hadron structure . one could use these processes for determining the baryon resonance dynamics based on the study of the @xmath17 interference pattern @xcite . on the other hand , investigation of the exclusive reactions @xmath18 in experiments with high - energy pions at large invariant mass of the dilepton and small squared momentum transfer to the nucleon could provide access to generalized parton distributions as suggested in @xcite . this paper is organized as follows . in sec . ii we give the basic formalism for the unified treatment of the reactions @xmath19 , @xmath20 , and @xmath21 . in sec . iii we present our dispersion - relation model for unified description of these three reactions and compare the calculated results with experimental data . to clearly explain our method , we choose a simple version of the model , which satisfactorily describes the data on the photoproduction and electroproduction . in sec . iv , we outline the method of determining the nucleon electromagnetic form factors from low - energy ipe and discuss some results of its application to analysis of the ipe data on the nucleon . section v is devoted to extracting the pseudoscalar nucleon form factor from the same ipe data , and interpretation of the results is given . concluding remarks are presented in sec . appendices present the relations between the amplitudes , derivation of the quasithreshold constraints for the multipole amplitudes on the basis of the principle of the first - class maximum analyticity , and explanation of the compensation effect . we consider the reactions @xmath19 , @xmath20 , and @xmath21 in the framework of the unified model . this approach is natural because in the one - photon approximation , due to the @xmath22-invariance , research into these three processes is related to a study of the process @xmath23 with the hadron current @xmath24 , where @xmath25 , @xmath26 , and @xmath27 correspond to the pion photoproduction , electroproduction , and ipe , respectively . this allows us to predict peculiarities of the ipe dynamics on the basis of a rich experimental material on the electroproduction and photoproduction and to test the reliability of the unified model for these three processes . the @xmath28-matrix element for the electroproduction in the one - photon approximation is @xmath29 where @xmath30 is the matrix element of the lepton electromagnetic current , @xmath31 ( @xmath32 ) , @xmath33 ( @xmath34 ) , and @xmath35 are the four - momenta of the initial ( final ) electron , nucleon , and final pion , respectively . momentum of the virtual photon is @xmath36 ( @xmath37 for ipe ) , @xmath38 , and @xmath39 and @xmath40 are mandelstam variables . conservation of the lepton and hadron electromagnetic currents implies @xmath41 . assuming the @xmath22-invariance , for the ipe process one must use the spinor @xmath42 instead of @xmath43 in the lepton current ( [ lept.current ] ) . then @xmath44 is timelike , and @xmath45 is the range of @xmath3 values for fixed @xmath46 . the hadron current @xmath24 can be expanded using the six independent covariant gauge - invariant structures @xmath47 @xcite : @xmath48 where @xmath49 with @xmath50 and @xmath51 . the invariant amplitudes @xmath52 are free from kinematic constraints , but @xmath53 and @xmath54 have a kinematic pole at @xmath55 . the amplitudes @xmath54 and @xmath56 are absent in the photoproduction . the matrix element ( [ j : covar.expantion ] ) can be expressed through the scalar c.m . amplitudes @xmath57 where @xmath58 and @xmath59 are the pauli spinors , and @xmath60f_2 + i{\vec\sigma}\cdot{\tilde{\vec k}}({\tilde{\vec q}}\cdot{\vec \varepsilon } - { \tilde{\vec q}}\cdot{\tilde{\vec k}}{\tilde{\vec k}}\cdot{\vec \varepsilon})f_3+\nonumber\\ & & + i{\vec\sigma}\cdot{\tilde{\vec q}}\,({\tilde{\vec q}}\cdot{\vec \varepsilon } - { \tilde{\vec q}}\cdot{\tilde{\vec k}}{\tilde{\vec k}}\cdot{\vec \varepsilon}\,)f_4 + i{k^2\over k_0}{\vec \sigma}\cdot{\tilde{\vec k } } { \tilde{\vec k}}\cdot{\vec \varepsilon}\,f_5 + i{k^2\over k_0}{\vec \sigma}\cdot{\tilde{\vec q}}\,{\tilde{\vec k}}\cdot{\vec \varepsilon}\,f_6\;.\end{aligned}\ ] ] here @xmath61 and @xmath62 are unit vectors . the amplitudes @xmath63 describe the process with the transversal photons and @xmath64 with the longitudinal ones . the relations between the amplitudes @xmath65 and @xmath66 are listed in appendix a. in the isotopic space @xmath65 ( and @xmath66 ) are matrices @xmath67f_i^{(-)}\ ; , \ ] ] where the upper sign corresponds to @xmath68 and the lower one to @xmath69 process . for the physical processes , we have @xmath70 the differential cross section for the electroproduction is written down in the following form taking into account a possibility of the longitudinal polarization of the electron @xmath71\;,\end{aligned}\ ] ] where @xmath72^{-1}$ ] , @xmath73 and * q * are magnitudes of the photon and pion c.m . momenta , respectively ; @xmath74 is the azimuthal angle between planes of the electron scattering and the reaction @xmath68 ; @xmath75 , @xmath76 ( @xmath77 ) and @xmath78 are the photon - momentum magnitude , the initial ( final ) energy and the electron scattering angle in lab frame , respectively ; and @xmath79 and @xmath80 are the solid angles of the scattered electron in lab and pion in c.m . of the @xmath7 system , respectively . degree of the longitudinal polarization of the electron is @xmath81 where @xmath82 is the polarization vector of the electron in its rest system . the ipe differential cross section reads @xmath83\;,\end{aligned}\ ] ] where @xmath84 is the angle between the momenta of the final nucleon and electron in the @xmath2 c.m . system , @xmath74 is the angle between planes of the reaction @xmath85 and the @xmath2 pair production , @xmath86 is the solid angle of the virtual photon production in the c.m . of the @xmath7 system , and @xmath87 is the solid angle of electron in the c.m . of the @xmath2 pair . @xmath88 in formulas ( [ cr.section:el-pr-n ] ) and ( [ cr.section:ipe ] ) describes the processes @xmath89 with the unpolarized transversal photons , @xmath90 characterizes the asymmetry of contributions of the transversal virtual photons linearly polarized in the plane of the @xmath89 and normally to it , @xmath91 is the contribution of the longitudinal photons , and @xmath92 and @xmath93 are the real and imaginary parts of the interference contribution of transversal and longitudinal photon in the helicity basis . it is seen that @xmath93 is related with the contribution of longitudinal polarization of electron to the cross section . the differential cross sections ( [ cr.section:el-pr-n ] ) and ( [ cr.section:ipe ] ) , measured for the processes with electron polarized in one direction , @xmath94 , and in the opposite one , @xmath95 , generally should display the asymmetry @xmath96 given by the contribution of @xmath93 @xmath97 the quantities @xmath98 are related to the amplitudes @xmath65 in the c.m . frame as @xmath99+t_2,\nonumber\\ & & t_2=\frac{\alpha m^2}{2\pi s}\sin^2\theta^\gamma\left[|f_3|^2 + |f_4|^2 - 2\mbox{re}(f_1f_4^*+f_2f_3^*+\cos\theta^\gamma f_3f_4^ * ) \right],\nonumber\\ & & t_3=\frac{\alpha m^2}{\pi s}\sin\theta^\gamma\mbox{re}\left[(f_2 + f_3+\cos\theta^\gamma f_4)f_5^*+(f_1+\cos\theta^\gamma f_3 + f_4)f_6^*\right],\\ & & t_4=\frac{\alpha m^2}{\pi s}\left[|f_5|^2+|f_6|^2 + 2\cos\theta^\gamma\mbox{re}(f_5f_6^*)\right],\nonumber\\ & & t_5=2\frac{\alpha m^2}{\pi s}\sin\theta^\gamma\mbox{im}\left[(f_2^ * + f_3^*+\cos\theta^\gamma f_4^*)f_5+(f_1^*+\cos\theta^\gamma f_3^ * + f_4^*)f_6\right].\nonumber\end{aligned}\ ] ] to obtain reliable information on the nucleon structure , it is important to find kinematic conditions under which the ipe dynamics is determined mainly by a model - independent part of interactions , the born one . to this end , we utilize such general principles , as analyticity , unitarity , and lorentz invariance and the phenomenology of the processes @xmath100 and @xmath101 , considered in the framework of the unified ( including ipe ) model . we use a simple version of the model , which describes the experimental data satisfactorily with a minimal number of parameters ( electromagnetic form factors ) . this allows one to carry out a simple analytic continuation from the spacelike to timelike values of @xmath3 . for isovector amplitudes , the fixed-@xmath102 dispersion relations without subtractions at finite energy are used , with the spectral functions describing the magnetic excitation of the @xmath103 resonance . the isoscalar amplitudes are described only by the born terms since the @xmath103 resonance does not contribute to them due to the isospin conservation . in refs . @xcite , it was shown that this model is successful in combined description of experimental data on the pion electroproduction , photoproduction , and ipe in the total - energy region from the threshold up to @xmath104 gev . moreover , this model is adequate for our purpose , namely to formulate the method for determination of the form factors in the timelike region , for the following reasons . first , in the dispersion - relation approach , the spectral functions in the first resonance region are expressed through the nucleon electromagnetic form factors and the phase shift of the @xmath7-scattering @xmath105 . this reduces considerably the number of fitted parameters , which is especially important in the ipe analysis . second , in the @xmath103 region , the electric @xmath106 and scalar @xmath107 quadrupoles , which ( as the magnetic dipole @xmath108 ) describe excitation of the resonance @xmath103 , amount to not more than 15% of @xmath108 . for example , the photoproduction data @xcite give the value @xmath109 for the @xmath110 ratio . phenomenological results for the @xmath3 dependence of the @xmath110 and @xmath111 ratios are not stable yet and depend upon the method of their extraction from electroproduction data ( see discussion of this point in ref . @xcite , p. 698 ) . however , these ratios do not exceed respectively , 7 and 15% up to @xmath112 ( gev / c)@xmath6 . based on the quark model calculations ( see , _ e.g. _ , refs . @xcite ) , we suppose that an analogous situation takes place also for @xmath113 , at least up to @xmath114 0.3 ( gev / c)@xmath6 . therefore , at the first stage of our analysis we neglect the quadrupole excitations of @xmath103 , which we expect to be a good approximation for processes with unpolarised nucleons . the conventional procedure of reggeization provides us with behaviour of the invariant amplitudes for @xmath115 and small @xmath8 @xcite @xmath116 consequently , in a complete @xmath46-channel description , we should write a fixed-@xmath102 dispersion relation with one subtraction at finite energy for the isovector amplitude @xmath117 and without subtractions for the remaining amplitudes , taking into account their crossing properties . however , the dispersion integrals with the spectral functions which describe the magnetic excitation of the @xmath103 resonance converge very well already at @xmath118 2 gev for all the amplitudes @xmath66 . therefore , we use the fixed-@xmath102 dispersion relations without subtraction at the finite energy @xcite for all isovector amplitudes @xmath119 and we take the born approximation for the isoscalar ones @xmath120 where @xmath121 , @xmath122\ ; , \nonumber\end{aligned}\ ] ] and @xmath123\;,\ ] ] with the @xmath7 coupling constant @xmath124 and the normalisation of the form factors : @xmath125 . the terms @xmath126 and @xmath127 belong only to the amplitude @xmath117 . note that @xmath53 and @xmath54 have the kinematic pole at @xmath55 . however , these singularities are cancelled out kinematically because these amplitudes enter into the matrix element through the combination @xmath128 which is equal to @xmath129 , where @xmath130 and @xmath131 are the ball amplitudes which have been proved to have no kinematic singularities @xcite . in specific model calculations , the condition @xmath132=0\ ; , \ ] ] which is ensured by the form of the term @xmath127 ( [ eq : c5 ] ) , guarantees absence of the singularity at @xmath55 . the spectral functions @xmath133 are supposed to describe the magnetic excitation of the @xmath103 resonance @xmath134}~ a_i(w , t,\lambda^2),\ ] ] where @xmath135 , @xmath105 is the corresponding phase shift of the @xmath7-scattering amplitude , for which we utilize the prescription from ref . @xcite , and @xmath136 with the coefficients @xmath137 @xmath138 , & ~~\beta_1=\frac{1}{2}(w+m - q_0 ) , \\ \alpha_2=\frac{3}{2}(w+m)(m_\pi^2-t ) , & ~~\beta_2=\frac{1}{2}(w+m)+q_0 , \\ \alpha_3=-\frac{1}{2}(w+m)(w+m - q_0)-\frac{3}{4}(m_\pi^2-t ) , & ~~\beta_3=-\frac{3}{4}\ ; , \\ \alpha_4=(w+m)(w+m+\frac{1}{2}q_0)-\frac{3}{4}(m_\pi^2-t ) , & ~~\beta_4=\frac{3}{4}\;,\\ \alpha_5=2(s - m^2)(w+m+\frac{1}{2}q_0)-\frac{3}{2}(w - m)(m_\pi^2-t ) , & ~~ \beta_5=\frac{3}{2}(w - m),\\ \alpha_6=-\frac{1}{2}(w+m)q_0-\frac{1}{4}(m_\pi^2 + 3 t ) , & ~~\beta_6=-\frac{3}{4}\;. \end{array } \right.\end{aligned}\ ] ] furthermore , the results of the photoproduction multipole analyses @xcite allow us to set @xmath139 above the @xmath103 energy . prescriptions for the pion and nucleon electromagnetic form factors are taken from refs . the model described above is the first ( simple ) reliable version of the more complex model for unified treatment of contemporary experimental data on the pion photoproduction , electroproduction , and ipe in the energy region which spans from the threshold up to the second @xmath7 resonance . in fig . [ fig : ph - pr - n ] we compare results of our model for the differential cross section with @xmath140-photoproduction data . in fig . [ fig : e - pr - n1 ] we show comparison with the electroproduction data on the proton for the transversal @xmath141 and longitudinal @xmath142 parts of the cross section @xmath143\;.\ ] ] the differential cross sections for the forward electroproduction of pions by transversal and longitudinal virtual photons are presented in fig . [ fig : e - pr - n2 ] . in general , we obtain quite a good agreement with the pion photoproduction and electroproduction data on unpolarized nucleons , especially in the region of the @xmath103 resonance ( @xmath144 gev ) . in the case of the @xmath140 photoproduction , the results are very good up to @xmath13 gev ( @xmath145 gev ) , see fig . [ fig : ph - pr - n ] . it is obvious that our model can be further improved by including the quadrupole excitation of the @xmath103 resonance ( @xmath146 and @xmath147 ) in the spectral functions . however , a still more elaborate model should include , in addition to the quadrupole excitation , contributions of other nucleon isobars and high - energy `` tails '' to the absorption parts of the amplitudes to ensure a balanced consideration of small corrections . furthermore , analytic continuation of the corrected absorption parts of the amplitudes into the unobservable region in the dispersion integrals , @xmath148 for @xmath149 , requires use of the quasithreshold relations ( following from causality analyticity ) between the electric and longitudinal multipoles @xcite in which `` toroid '' multipoles appear @xcite . on the contrary , the analytic continuation with the approximation ( [ eq : ima ] ) is immediate . however , having in mind the quality of contemporary experimental data , the above - stated simple model seems to be quite sufficient ( see figs . [ fig : ph - pr - n]-[fig : e - pr - n2 ] ) . application of the model to the calculations for ipe shows an interesting growth of the relative contribution of the born terms with @xmath3 and their dominance in the neighbourhood of the value @xmath150 @xcite . this approximate dominance of the born terms has a model - independent explanation . it is related to the quasithreshold theorem @xcite , which means that at the quasithreshold , @xmath151 , the ipe amplitude becomes the born one in the energy region from the threshold up to @xmath13 gev . this remarkable dynamics of ipe distinguishes it essentially from the photoproduction and electroproduction , where rescattering effects amount @xmath152 . let us explain the quasithreshold behaviour of the ipe amplitude . as @xmath153 the multipole amplitudes behave in the following way @xmath154 therefore , at @xmath155 only the electric ( @xmath156 and @xmath157 ) and longitudinal ( @xmath158 and @xmath159 ) dipoles survive . in addition , the number of independent dipole transitions diminishes to the two ones at the quasithreshold due to the quasithreshold constraints @xmath160 which arise from the causality ( analyticity ) . the selection rules which follow from the parity conservation and the value of the angular momentum of the stopped virtual photon , @xmath161 , restrict possible s - channel resonances at the quasithreshold to the following sets : @xmath162 , _ etc . _ ] and @xmath163 , _ etc . behaviour of the multipole amplitudes as @xmath164 and @xmath153 and the quasithreshold constraints among them are derived in appendix b. since the @xmath46- and @xmath165-wave @xmath7 resonances are excited at energies above 1.5 gev , one can expect that the dipoles @xmath156 and @xmath157 are dominated by the born terms below this energy . this is in agreement with the multipole analyses of charged pion photoproduction and confirmed by the dispersion - relation calculations at @xmath166 . we can , therefore , conclude that in the quasithreshold region , the ipe amplitude is given by the born terms with accuracy better than @xmath167 . at energies below @xmath168 gev , we can write for the quasithreshold ipe @xmath169.\end{aligned}\ ] ] in the real experiment , however , one can not realize strictly the quasithreshold conditions and , therefore , the realistic model ( presented above ) is needed . in choosing the optimal geometry of the experiment for deriving the form factors , the `` compensation curves '' @xcite can help . the curves are defined as curves in the @xmath170 plane along which the differential cross section is given only by the born terms . these curves can be constructed by comparing photoproduction experimental data with the born cross section and employing the existence theorem for implicit functions ( more details on the compensation effect is given in appendix c ) . the method of determining the electromagnetic form factors from low - energy ipe is , therefore , based on utilizing the quasithreshold theorem , the realistic dispersion - relation model , and the compensation curve . this method has been already used in the experiments on the nucleon and nuclei @xmath4c and @xmath5li @xcite . .[tab1]electromagnetic form factors obtained in experiments on the nucleon . the virtual photon momentum @xmath3 is given in units of the pion mass . [ cols="^,^,^,^,^,^,^,^,^,^,^",options="header " , ] in table [ tab1 ] , we present values of the electromagnetic form factors obtained in experiments on the nucleon . in tab . [ tab1 ] , the same experimental errors are given for @xmath171 and @xmath172 because in this @xmath3-range these form factors can be connected with each other via the spectral function by the relation @xmath173 . the quantity @xmath174 , taken from the dispersion calculations @xcite , possesses a significantly smaller theoretical uncertainty than the calculated quantities @xmath171 and @xmath172 . this is caused by cancellation of terms with the large uncertainties in the spectral function of @xmath174 which , therefore , is dominated by the one - nucleon exchange contribution in the region @xmath175 . one can see that this result is rather model - independent . then the same experimental errors can be given for @xmath171 and @xmath172 . of course , having high - statistics data for ipe one need not use the relation of @xmath171 with @xmath172 in extracting these quantities . the values of @xmath171 are quite consistent with the calculations of the nucleon electromagnetic structure in framework of the unitary and analytic vector - meson dominance model @xcite . let us look at the possibility of investigating the form factor @xmath176 of the @xmath177 vertex at @xmath178 . whereas a measurement of the differential cross section for the electroproduction with unpolarised electrons in the @xmath12 region at @xmath179 out of the compensation curve allows one to extract information about the form factors of the @xmath177 vertex , this approach is not sufficiently effective for ipe because the dominance of the born mechanism which reaches @xmath180 at the quasithreshold is also considerable at lower values of @xmath3 . it turns out that the analysis of asymmetry @xmath181 ( [ p_e ] ) in the dilepton production near the quasithreshold gives us that chance . we consider the quantity @xmath93 , connected to the asymmetry @xmath181 . it can be expanded to series in @xmath182 near the quasithreshold with taking into account formulas ( [ t_i - f_i ] ) , ( [ m - pole - expantion ] ) and ( [ gk - behavior ] ) @xmath183 the lowest term of the formal series in eq . ( [ t_5-expand ] ) @xmath184_{{\bf k}=0}\ ; , \ ] ] equals zero at the quasithreshold as it follows from the constraints ( [ q - thr - constr ] ) . behaviour of @xmath93 for @xmath153 then is @xmath185 where @xmath186_{\bf k=0 } .\ ] ] in the @xmath103 region , the amplitudes @xmath156 and @xmath157 are dominated by the born terms , and the imaginary parts of form factors are negligible . contributions of @xmath106 and @xmath187 amount to less than 15% of that of @xmath108 . multipole amplitudes @xmath188 and @xmath187 , related to the excitation of the @xmath189 resonance , generally have to be very small , as it is seen in analyses of photoproduction ( especially on the neutron ) . therefore , with a good accuracy , we obtain @xmath190 since the quasithreshold relations ( [ gk - behavior ] ) seem to be approximately realized in a rather wide interval in @xmath3 , the asymmetry @xmath181 has to be sensitive to @xmath191 in the @xmath103 region . the measurement of @xmath181 would , therefore , allow one to study quantitatively the assumption about the dominance of the magnetic dipole transition and to extract information on the form factor @xmath176 at @xmath113 . this is possible because the contribution of the background part of the amplitude to @xmath93 is considerably suppressed in the quasithreshold region , the background part reducing as @xmath192 for @xmath182 becoming zero . now let us discuss another interesting possibility of investigating the weak nucleon structure related to the nucleon gamow teller transition described by the matrix element @xmath193\gamma_5u(p_1),\ ] ] where @xmath194 is the axial - vector current , and @xmath195 and @xmath196 are the axial and induced pseudoscalar form factors , respectively . an alternative description of ipe which utilizes the current commutators , pcac , and completeness allows one to derive a low - energy theorem at the threshold , @xmath197 and @xmath198 , related to the approximate chiral symmetry and @xmath199 corrections . minimization of the continuum contribution at the quasithreshold justifies this approach up to @xmath13 gev @xcite with the continuum corrections being practically the same as in the dispersion - relation description . then , one obtains for the longitudinal part of the @xmath200 amplitude at the quasithreshold retaining only the leading terms in @xmath201 @xcite , such that @xmath202\bigr\},\end{aligned}\ ] ] where the constant of the @xmath203 decay @xmath204 is defined by @xmath205 , @xmath206 , and the quasithreshold values of the variables are @xmath207 @xmath208 was measured in various experiments , first of all in @xmath209 and @xmath210 . it is reasonable to use first this result @xmath211 however , it is difficult to obtain reliable information on @xmath16 in these experiments since its contribution to cross sections is kinematically suppressed ( it is multiplied by the lepton masses ) . in the @xmath14-capture and @xmath212-decay experiments , there is a kinematically restricted small range of @xmath8 , @xmath213 , in which the weak form factor can be determined , though with a large error . for example , its value extracted from measurements of the @xmath14-capture in hydrogen @xcite is @xmath214 . @xmath16 has also been measured in the capture of polarized muons by @xmath215si nuclei @xcite . equation ( [ long.part:ca ] ) shows that the kinematic suppression of the @xmath16 contribution is absent when the ipe data at the quasithreshold are used for extracting @xmath16 . in this way , the pseudoscalar form factor @xmath216 can be determined in the range up to @xmath217 ( which corresponds to @xmath13 gev ) . adopting the quasithreshold kinematics , one can avoid the threshold difficulties that are present when applying an analogous method to the analysis of electroproduction data . next we shall follow the method of ref . first , using the @xmath218 and @xmath219 values obtained in the analysis of the ipe data on the nucleon @xcite , we obtain 10 data points for the longitudinal part of the @xmath220 amplitude at the quasithreshold . these data points , which can be considered as experimental ones , are depicted in fig . [ fig : long - exp ] . for @xmath216 we chose the dispersion relation without subtractions : @xmath221 the residue at the pole @xmath222 is determined by the pcac relation . when only the @xmath223-pole term is considered in @xmath16 , the result is inconsistent with the experimental data as demonstrated in fig . [ fig : long - exp ] ( dashed line ) . therefore , the dispersion integral in eq . ( [ g_p : dr ] ) should be considered . it could be approximated by the contributions of possible intermediate three - pion and resonance states with the pion quantum numbers . however , since the contributions of nonresonant three - particle states must be suppressed by the phase volume , it is , therefore , reasonable to approximate the integral in ( [ g_p : dr ] ) by a resonance - pole term . a satisfactory description is obtained if one takes the following expression for @xmath216 with the indicated values of parameters : @xmath224 where @xmath225 , the @xmath226 weak - decay constant @xmath227 is defined by @xmath228 , and @xmath229 and @xmath230 are the coupling constants of the @xmath223 and @xmath226 states with the nucleon , respectively . as seen from the definitions of the weak - decay constants , one must expect that @xmath231 to reflect a tendency of another way ( in addition to the goldstone one ) in which the axial current is conserved for vanishing quark masses . this behaviour is demonstrated in various models with some nonlocality which describe chiral symmetry breaking @xcite . notice that the pole at @xmath232 in eq . ( [ g_p ] ) , situated considerably lower than the poles of the known contributing states @xmath233 and @xmath234 , is indispensable for describing the obtained experimental data on ipe . in fig . [ fig : gp - ratio ] we show the ratio @xmath235 in comparison with the values obtained in @xmath14-capture on hydrogen @xcite and in analysis of data on the @xmath236 electroproduction off the proton near the threshold @xcite . the error bars on the solid line indicate that the values of @xmath216 determined by this method are of high accuracy . one can see that the results of refs . @xcite agree with the pion - pole dominance hypothesis in a large range of momentum transfer unlike our result in which the hypothesis is valid only in a narrow @xmath102 range , whereas outside the range the contribution of continuum is considerable . note that the contributions of the pion radial excitations , @xmath233 and @xmath234 , which are rather distant from this region , are suppressed and their consideration would only slightly increase the mass of @xmath237 . the parameters of this pole term in ( [ g_p ] ) might be changed more considerably if one assumes the channel @xmath238 with the recently discovered scalar @xmath239 @xcite , due to the possible multichannel nature of this state . in any case , the conclusion will remain valid that the state with @xmath240 in the range 500 - 800 mev is needed for explaining the obtained ipe data . let us add that a possible signal of the charged state of this isotriplet was observed in the @xmath241 system @xcite and interpreted as the first radial excitation of the pion in the framework of the relativistic quark model @xcite based on the covariant formalism for two - particle equations . accepting this designation for @xmath242 and taking the estimate for the @xmath226 weak - decay constant in the nambu jona - lasinio ( njl ) model generalized by using effective quark interactions with a finite range , @xmath243 mev , we obtain @xmath244 . there are no suitable theoretical calculations for this coupling constant now . in the njl model , the consideration of radial excitations of states requires introducing some nonlocality . since a successful calculation of the @xmath7 coupling constant in that model enforces one to go beyond the framework of the tree approximation and take loop corrections into account @xcite , it seems that a satisfactory evaluation of @xmath245 in that approximation can not avoid assuming some nonlocality . of course , a more reliable interpretation of @xmath226 requires investigation of other processes with @xmath226 . existence of this state would also raise the question on its su(3 ) partners . a careful ( re)analysis of the corresponding processes is , therefore , desirable in this energy region . we have demonstrated that a subsequent investigation of ipe is necessary for extracting both unique information about the electromagnetic structure of particles in the sub-@xmath0 threshold region of the timelike values of @xmath3 and the nucleon weak structure in the spacelike region . the former is interesting especially now , _ e.g. _ , in connection with discussion about the hidden strangeness of the nucleon ( see , _ e.g. _ , @xcite ) and quasinuclear bound @xmath246 state @xcite . analyses of the experimental ipe data in the first @xmath7 resonance region allow one to obtain the values of the form factor @xmath171 at timelike values of @xmath3 which are quite consistent with the calculations in the framework of the unitary analytic vector - meson dominance model @xcite . an inevitable step that is necessary for the study of the electromagnetic structure of nucleon - isobar systems in the timelike @xmath3-region is a multipole analysis of ipe similar to that for the photoproduction and electroproduction , _ e.g. _ @xcite . at present , with the intense pion beams being available , it is possible to perform experiments aiming at carrying out that analysis . in a construction of the dispersion - quark model in the second and third @xmath7 resonance region , the multichannel character of the nucleon isobars must be taken into account , _ e.g. _ , by the method of ref . @xcite or utilizing the proper uniforming variables @xcite . as we already mentioned , our method was used in the analysis of the pion - induced dilepton production on the nucleon and light nuclei . it is worth making some remarks about the analysis of the experiment on the @xmath5li nucleus with a @xmath236 beam at 500 mev / c @xcite . the missing mass analysis of the data has shown that about half of the events are related to disintegration processes of the nucleus which are dominated by the reaction + @xmath247 . + on analyzing this process it was assumed that the pion - nucleus amplitude is determined by the neutron - pole mechanism , and the nuclear part ( the vertex function of the @xmath248 ) was calculated in the nucleon cluster model @xcite . the remaining events belong to the process + @xmath249 . + when all the events were analyzed ( with and without disintegration of the nucleus ) , the cross section on the nucleus was supposed to be additively connected with the cross section on the individual nucleon and nuclear effects were taken into account via screening . in both cases , our model was used for describing ipe on the individual nucleon . the obtained values for @xmath171 are again quite consistent with the calculations in ref . @xcite . in the case of the reaction without disintegration of the nucleus , one observed there the electromagnetic form factor of the nucleus in the timelike @xmath3 region for the first time . unfortunately , in the indicated analysis , unique information on the electromagnetic structure of the nucleus in the timelike region was lost . generally , it seems at present that there is no satisfactory concept of the electromagnetic form factors of the nucleus in the timelike region . a satisfactory description must take into account both a constituent character of the nucleus ( and the corresponding analytic properties ) and more subtle ( than the screening ) collective nuclear effects . finally , it should be noticed that a more reliable interpretation of the observed state @xmath250 requires solving a number of questions , both theoretical and experimental . in the pseudoscalar sector , states of various nature are possible , except for @xmath251 , the @xmath252 and @xmath253 glueballs , @xmath254 hybrids , and multiquark states . however , all the models and lattice calculations give masses of those unusual states considerably greater than 1 gev . therefore , the most probable interpretation of @xmath250 seems to be the first radial pion excitation . the authors are grateful to s.b . gerasimov , v.a . meshcheryakov , and g.b . pontecorvo for useful discussions and interest in this work . yu.s . acknowledges also support provided by the votruba - blokhintsev program for theoretical physics of the committee for cooperation of the czech republic with jinr , dubna . thanks the grant agency of the czech republic , grant no . 202/05/2142 , and the institutional research plan avoz10480505 . m.n . acknowledges the slovak scientific grant agency , grant vega no . 2/3105/23 . the invariant amplitudes @xmath255 relate to the scalar c.m . amplitudes @xmath256 as follows @xmath257\left(\frac{f_3}{p_{10}-m}+ \frac{p_{20}+m}{\bf qk}f_4\right)-2m\lambda^2\left(\frac{f_5}{p_{10}-m}+ \frac{p_{20}+m}{\bf qk}f_6\right),\nonumber\\ { k}{a}_2 & = & \frac{2}{t - m_{\pi}^2-\lambda^2}\left\{\lambda^2\left(\frac{f_1}{p_{10}-m}- \frac{p_{20}+m}{\bf qk}f_2\right)+\frac{1}{2\bf qk}[(t - m_{\pi}^2-\lambda^2)k_0 + \right.\nonumber\\ & & \left.2q_0\lambda^2]\left[(w - m)\frac{f_3}{p_{10}-m}-\frac{p_{20}+m } { \bf qk}(w+m)f_4\right]-\right.\nonumber\\ & & \left.\lambda^2\left[(w - m)\frac{f_5}{p_{10}-m}- \frac{p_{20}+m}{\bf qk}(w+m)f_6\right]\right\},\nonumber\\ { k}{a}_3 & = & { k}{a}_4 + 2w\frac{\bf k}{\bf q}\left(\frac{f_3}{p_{10}-m}+ \frac{p_{20}+m}{\bf qk}f_4\right),\\ { k}{a}_4 & = & ( w - m)\frac{f_1}{p_{10}-m}+(w+m)\frac{p_{20}+m}{\bf qk}f_2 + \frac{1}{2\bf qk}[(t - m_{\pi}^2-\lambda^2)k_0+\nonumber\\ & & 2q_0\lambda^2]\left(\frac{f_3}{p_{10}-m}+\frac{p_{20}+m}{\bf qk}f_4\right)- \lambda^2\left[\frac{f_5}{p_{10}-m}+\frac{p_{20}+m}{\bf qk}f_6\right],\nonumber\\ { k}{a}_5 & = & \frac{2}{t - m_{\pi}^2-\lambda^2}\left\{(s - m^2)\left(\frac{-f_1}{p_{10}-m}+ \frac{p_{20}+m}{\bf qk}f_2\right)+\frac{1}{2\bf qk}[(t - m_{\pi}^2-\right.\nonumber\\ & & \left.\lambda^2)(p_{10}+w)+2q_0(s - m^2)]\left[(w - m)\frac{-f_3}{p_{10}-m}+ \frac{p_{20}+m}{\bf qk}(w+m)f_4\right]+\right.\nonumber\\ & & \left.(s - m^2)\left[(w - m)\frac{f_5}{p_{10}-m}-\frac{p_{20}+m}{\bf qk}(w+m)f_6 \right]\right\},\nonumber\\ { k}{a}_6 & = & ( w+m)\frac{f_1}{p_{10}-m}+(w - m)\frac{p_{20}+m}{\bf qk}f_2- \frac{1}{2\bf qk}[(t - m_{\pi}^2-\lambda^2)(p_{10}+w)+\nonumber\\ & & 2q_0(s - m^2)]\left(\frac{f_3}{p_{10}-m}+\frac{p_{20}+m}{\bf qk}f_4\right)- ( s - m^2)\left(\frac{f_5}{p_{10}-m}+\frac{p_{20}+m}{\bf qk}f_6\right),\nonumber\end{aligned}\ ] ] where @xmath258 . the inverse relations reads @xmath259 , \nonumber\\ f_3 & = & \sqrt{((w - m)^2-\lambda^2)((w - m)^2-m_{\pi}^2)}\frac{(w+m)^2-m_{\pi}^2}{8ms } [ ( s - m^2){a}_2+\nonumber\\ & & \lambda^2{a}_5+(w+m)({a}_3-{a}_4)],\\ f_5 & = & \frac{1}{4mw}\sqrt{\frac{(w+m)^2-m_{\pi}^2}{(w+m)^2-\lambda^2 } } \left\{\frac{(t - m_{\pi}^2-\lambda^2)}{2}[(3s+m^2-\lambda^2){a}_2 + 2w({a}_3-{a}_4)+\right.\nonumber\\ & & \left.(s - m^2+\lambda^2){a}_5]+[(w+m)^2-\lambda^2][{a}_1+(w - m)({a}_4- { a}_6)]+\right.\nonumber\\ & & \left.(s - m^2+m_{\pi}^2)[(w+m)({a}_3-{a}_4)+(s - m^2){a}_2 + \lambda^2{a}_5]\right\}\,.\nonumber\end{aligned}\ ] ] formulas for the remaining amplitudes can be obtained from eqs . ( [ f : a ] ) using the formal substitution @xmath260 and symmetry properties of the amplitudes @xmath65 under the substitution @xmath261 here we demonstrate a derivation @xcite of the behaviour of the multipole amplitudes as @xmath164 and @xmath153 on the basis of the first - class maximum analyticity principle @xcite . according to this principle , the total amplitude possesses only singularities that are related to the dynamic processes and whose positions depend on masses of intermediate ( and external ) states which are involved in these processes . then at the known arrangement of poles ( and , therefore , of the threshold branch points ) all the other singularities are determined by the systematic consideration of formulas for discontinuities in all channels . notice that the limit @xmath153 is fulfilled in two cases : @xmath262 and @xmath263 . in the former case , the limit is fulfilled in ipe at the quasithreshold when the virtual photon has a maximal mass , whereas in the latter , the limit is fulfilled in the @xmath264-channel , @xmath265 , at the stopped antinucleon @xmath266 . the amplitudes @xmath65 can be decomposed in terms of the multipole amplitudes : the magnetic @xmath267 , electric @xmath268 , and longitudinal @xmath269 ( or scalar @xmath270 but due to the current conservation , @xmath271 ) : @xmath272p^\prime_{l+1 } ( \cos{\theta^{\gamma}})+ \sum_{l=2}^{\infty}[(l+1)m_{l-}+e_{l-}]p^\prime_{l-1 } ( \cos{\theta^{\gamma}}),\nonumber\\ f_2&=&\sum_{l=1}^{\infty}[(l+1)m_{l+}+lm_{l-}]p^\prime_l ( \cos{\theta^{\gamma}}),\nonumber\\ f_3&=&\sum_{l=1}^{\infty}[-m_{l+}+e_{l+}]p^{\prime\prime}_{l+1 } ( \cos{\theta^{\gamma}})+ \sum_{l=3}^{\infty}[m_{l-}+e_{l-}]p^{\prime\prime}_{l-1 } ( \cos{\theta^{\gamma}}),\\ f_4&=&\sum_{l=2}^{\infty}[m_{l+}-m_{l-}-e_{l+}-e_{l-}]p^{\prime\prime}_l ( \cos{\theta^{\gamma}}),\nonumber\\ k_0f_5&=&\sum_{l=0}^{\infty}l_{l+}p^{\prime}_{l+1}(\cos{\theta^{\gamma}})- \sum_{l=2}^{\infty}l_{l-}p^{\prime}_{l-1}(\cos{\theta^{\gamma}}),\nonumber\\ k_0f_6&=&\sum_{l=1}^{\infty}[l_{l-}-l_{l+}]p^{\prime}_{l+1 } ( \cos{\theta^{\gamma}}).\nonumber\end{aligned}\ ] ] notice the unphysical multipoles : @xmath273 . the multipole amplitudes are functions of @xmath274 and @xmath3 only . the properties of the amplitudes @xmath275 with respect to the substitution @xmath260 ( [ f : w - w ] ) result in the corresponding symmetry relations for the multipole amplitudes @xmath276 in the following , it is convenient to introduce new functions @xmath277 instead of the legendre polynomial which have singularities ( due to @xmath278 ) @xmath279 taking into account the relation between @xmath280 and @xmath281 , we can conclude that the quantities @xmath282^{-1/2}f_1,~~~ [ ( p_{10}-m)(p_{20}-m)]^{-1/2}f_2,~~~ \frac{[(p_{10}-m)(p_{20}-m)]^{-1/2}}{p_{20}+m}f_3,\nonumber\\ & & \frac{[(p_{10}+m)(p_{20}+m)]^{-1/2}}{p_{20}-m}f_4,~~~ \left(\frac{p_{10}+m}{p_{20}+m}\right)^{1/2}f_5,~~~ \left(\frac{p_{10}-m}{p_{20}-m}\right)^{1/2}f_6\end{aligned}\ ] ] possess no kinematic singularities . to guarantee this property of the quantities ( [ nksf ] ) the multipole amplitudes in ( [ m - pole - expantion ] ) must have the form @xmath283^{-1/2}({\bf qk})^l\tilde{m}_{l-},\nonumber\\ e_{l+}&=&\left(\frac{p_{20}+m}{p_{10}+m}\right)^{1/2}({\bf qk})^l\tilde{e}_{l+},~~~ e_{l-}=\left(\frac{p_{10}+m}{p_{20}+m}\right)^{1/2}({\bf q})^l{\bf k}^{l-2 } \tilde{e}_{l-},\\ l_{l+}&=&k_0\left(\frac{p_{20}+m}{p_{10}+m}\right)^{1/2}({\bf qk})^l \tilde{l}_{l+},~~~ l_{l-}=k_0\left(\frac{p_{10}+m}{p_{20}+m}\right)^{1/2}({\bf q})^l{\bf k}^{l-2 } \tilde{l}_{l-}.\nonumber\end{aligned}\ ] ] furthermore , we must ensure that the expression @xmath284{\tilde p}^\prime_{l+1}\over p_{10}+m\ ] ] in @xmath285 has no kinematic singularities at @xmath286 , _ i.e. _ , we must demand the following constraint : @xmath287 in the exceptional case @xmath288 , ( @xmath289 ) we obtain @xmath290 _ i.e. _ , @xmath291^{1/2}\mu_0.\ ] ] it is clear that a question about the kinematic zeros in @xmath65 is left open . since we are interested mainly in the @xmath3 dependence , we give only the kinematic @xmath3 behaviour of the amplitudes @xmath65 as @xmath262 and @xmath292 , not writing down explicitly the coefficients that depend only on @xmath293 at @xmath294 . from ( [ m - pole - expantion ] ) and ( [ tildep])-([e0 + ] ) , we have + a ) as @xmath262 @xmath295 b ) as @xmath292 @xmath296 taking into account formulas ( [ a : f ] ) , ( [ fbehavior.m ] ) and ( [ fbehavior .- m ] ) , and the fact that the amplitudes @xmath66 have no kinematic singularities ( except for the pole at @xmath297 in @xmath53 and @xmath54 ) , we conclude that the following constraints must be fulfilled to satisfy the maximal analyticity : + a ) as @xmath262 @xmath298 b ) as @xmath292 @xmath299 constraints ( [ c : f1,f3,f5.m])-([c : f3,f5.-m ] ) and the @xmath3 behaviour of the amplitudes @xmath65 , ( [ fbehavior.m ] ) and ( [ fbehavior .- m ] ) , ensure the absence of kinematic singularities in the amplitude @xmath300 , eq . ( [ f : scal.expantion ] ) , as @xmath301 and @xmath302 . taking into account the constraints ( [ c : f1,f3,f5.m])-([c : f3,f5.-m ] ) and eq . ( [ gk - behavior ] ) and using the recurrent formulas for the legendre polynomials , we obtain kinematic relations between the multipole amplitudes at the quasithreshold and for @xmath292 . the constraint ( [ c : f4,f6.m ] ) gives as @xmath262 @xmath303 and @xmath304^{1/2}\kappa_1\;.\ ] ] the constraint ( [ c : f1,f3,f5.m ] ) gives @xmath305 and ( [ c : f3,f5.-m ] ) , as @xmath292 , results in @xmath306 or @xmath307\;.\ ] ] the multipole amplitude @xmath158 , behaves as @xmath308^{1/2}\nu_0\;.\ ] ] finally , the constraint ( [ c : f2,f4,f6.-m ] ) gives the following relations between the multipole amplitudes as @xmath292 @xmath309 so , we derived the constraints for the multipole amplitudes at the quasithreshold and as @xmath292 . the relations of the first type ( [ c : el-,ll-.m])-([c : el+,ll+.m ] ) were also derived in refs . @xcite by a different method that shades their kinematic nature . the relations for @xmath292 , analogous to ( [ c : m+e+.-m ] ) , ( [ c : el+,ll+.-m])-([c : m1-,l1-.-m ] ) , were also obtained in a different way in ref . note that since kinematic conditions as @xmath292 ( @xmath310 ) are obtained from the quasithreshold kinematic configuration @xmath262 ( @xmath311 ) through the substitution @xmath312 , constraints for the multipole amplitudes as @xmath292 can be derived from the ones at the quasithreshold with the help of the symmetry relations for multipole amplitudes ( [ mac - dowell ] ) taking into account the complete @xmath3 behaviour of the latter as @xmath262 ( [ gk - behavior ] ) , ( [ c : el-,ll-.m])-([c : el+,ll+.m ] ) . the relations for multipole amplitudes as @xmath292 are , of course , not fulfilled in the reaction under consideration . a region of their applicability is the c.m . system of pion and nucleon in the @xmath3 channel @xmath313 at the stopped antinucleon ( @xmath314 ) . the relations between multipole amplitudes at the quasithreshold are very important for our considerations because the condition @xmath262 is realized ( at @xmath149 ) in the investigated reaction @xmath315 unlike the electroproduction and photoproduction . the introduced quantities @xmath316 ( [ c : el-,ll-.m ] ) and @xmath317 ( [ c : el+,ll+.m ] ) are not formal coefficients of expansion in a series but they can be related to `` toroid '' multipole amplitudes @xmath318 @xcite that have a definite physical meaning . the relations ( [ c : el-,ll-.m ] ) and ( [ c : el+,ll+.m ] ) must be used with necessity when carrying out analytic continuations of spectral functions in the dispersion relations for ipe into the unphysical region @xmath319 . therefore , it seems that application ( from the beginning ) of toroid multipoles @xmath318 instead of @xmath320 is rather relevant in the timelike region . here we shall find a region of variables @xmath321 , and @xmath3 where the cross sections of the processes of pion photoproduction , electroproduction , and ipe are described only by the born terms . let us write the differential cross section of virtual photoproduction as a sum of two terms @xmath322 where the first term is the born cross section and @xmath323 takes into account the final - state interaction and its interference with the born part of the amplitude . to establish the conditions under which @xmath324 we use the existence theorem for implicit functions . by this theorem , if equation ( [ phi - cond . ] ) allows the solution @xmath325 , @xmath326 , @xmath327 , and the function @xmath323 and its partial derivatives of the first order are continuous in the vicinity of the point @xmath328 and the derivative @xmath329 at this point is different from zero , that is , @xmath330 then there exists only one function @xmath331 which satisfies equation ( [ phi - cond . ] ) in some vicinity of the point @xmath332 and takes the value @xmath326 at @xmath325 , @xmath333 . this function and its partial derivatives are continuous in the vicinity of the point @xmath332 . according to this theorem , if we find at least one point in the space of @xmath334 , where the effects of rescattering and their interference with the born terms compensate each other , then , since the cross section is continuous in the physical region , there is a surface of `` compensation '' in this space on which the cross section is the born one . the intersections of this surface with every plane @xmath335 define some curves in the plane @xmath170 each of them being characterized by its own value of @xmath3 and the cross section ( [ dif.cr.sec . ] ) being the born one along them . thus , we obtain a one - parameter set of the compensation curves with @xmath3 as a parameter . consider the curves @xmath25 in more detail . we shall use @xmath336 instead of variable @xmath102 , where @xmath84 is a scattering angle in the c.m . system . if the compensation takes place , then in the plane @xmath337 we ought to have the compensation curve @xmath338 continuous in the physical region . generally speaking , there may be several curves of this kind , because the function @xmath339 can always be represented as a polynomial in powers of @xmath340 : @xmath341 and the equation @xmath342 can have several real roots @xmath343 . however , due to the theorem mentioned above , the curves do not intersect , since only one curve can pass through the given point . in the case of the pion photoproduction , available experimental data are sufficient to construct the compensation curves @xcite . to this end , it is necessary to find out the intersection points of the experimental differential cross section with the calculated born cross section and plot these points in the plane @xmath337 or @xmath344 . in fig . [ fig : comp - curv ] the compensation curves for the charged pion photoproduction are shown . calculating the born cross sections , we take into account the interaction of the photon only with the charge . therefore , along these curves the experimental cross sections are completely described by the electric born terms . one can see that at high energies , the compensation curves pass through small angles in accordance with the electric born model ( ebm ) @xcite . the compensation curves show also that an applicability of ebm to the cross section extends to the whole energy range , but at the corresponding angles , giving a more complete and transparent picture . of course , there are also compensation curves corresponding to the total born amplitude . now , if one assumes that for processes of the pion electroproduction and ipe , when @xmath166 , the compensation curves are not much different from the corresponding curves for the photoproduction @xmath345 , then along these curves the model dependence of description of the processes should be minimal . in the resonance region , this assumption would be reasonable if the change of @xmath3 does not lead to the essential reconstruction of the helicity ( or multipole ) structure of the excitation of the corresponding resonance . in the first resonance region , calculations , _ e.g. _ , in the dispersion theory , which agree with the predictions of the quark models ( _ e.g. _ @xcite ) and with the result of the phenomenological analysis of the electroproduction data ( _ e.g. _ @xcite ) , confirm a dominance preservation of the magnetic dipole excitation of the @xmath103 resonance with the change of @xmath346 . quark models expand this situation to values of @xmath178 . in our dispersion model , it was verified that the compensation angle @xmath347 is invariable with @xmath3 for the cross sections of the processes @xmath348 with the transverse virtual photons at @xmath349 gev @xmath350 . this prediction was confirmed by the experiment @xcite for the timelike values of @xmath3 : @xmath351 . in the second and third resonance regions , the analysis of experimental data for electroproduction ( _ e.g. _ @xcite ) gives a considerable change ( with @xmath352 ) of the helicity structure for the excitation of leading resonances , which is especially essential approximately at @xmath353 . the compensation curves exist also for the asymmetry in the charged pion production by the polarized @xmath354-quanta @xmath355/ [ d{\sigma}_{\perp}/dt + d{\sigma}_{\parallel}/dt]\;.\ ] ] for example , at @xmath356 gev the compensation in @xmath357 ( [ ph - asym ] ) takes place at @xmath358 for the @xmath359 photoproduction . since at high energies and small @xmath360 ebm ( as any reasonable model explaining the sharp forward peak in @xmath361 for the @xmath359 photoproduction ) gives the sign and size of @xmath362 in the forward direction , the compensation curves for @xmath362 above the resonance region go through approximately the same values of the angles @xmath363 as for @xmath364 . this fact is explained practically model independently . the plausibility of assumption that the compensation curves at high energies and small angles and at @xmath166 are not much different from the corresponding curves for @xmath25 is justified by successful application of ebm to the description of the processes of electroproduction and @xmath365 at high energies and @xmath366 @xcite . in any case , the compensation curves help one to reveal the optimal experimental conditions for studying the form factors @xmath367 and @xmath368 in processes of the pion electroproduction and ipe . e. amaldi , s. fubini , g. furlan , _ electroproduction at low energy and hadron form factors _ , springer tracts in modern physics * 83 * ( springer , berlin , 1979 ) . surovtsev , f.g . tkebuchava , yad . fiz . * 55 * , 2138 ( 1992 ) . blokhintseva , yu.s . surovtsev , m. nagy , acta physica slovaca * 49 * , 351 ( 1999 ) . titov , b. kmpfer , eur . j. a * 12 * , 217 ( 2001 ) . berger , m. diehl , b. pire , phys . b * 523 * , 265 ( 2001 ) . lutz , b. friman , m. soyeur , nucl . a * 713 * , 97 ( 2003 ) . bratkovskaya , w. cassing , u. mosel , nucl . a * 686 * , 568 ( 2001 ) . geffen , phys . rev . * 125 * , 1745 ( 1962 ) ; j. loubaton , j. tran thanh van , nucl . b * 2 * , 342 ( 1967 ) . surovtsev , f.g . tkebuchava , jinr communication p2 - 4561 ( dubna , 1969 ) ; t.d . blokhintseva , yu.s . surovtsev , f.g . tkebuchava , yad . fiz . * 21 * , 850 ( 1975 ) . a.m. baldin , v.a . suleymanov , phys . b * 37 * , 305 ( 1971 ) ; jinr communication p2 - 7096 ( dubna , 1973 ) . surovtsev , f.g . tkebuchava , yad . fiz . * 16 * , 1294 ( 1972 ) . kulish , yad . fiz . * 16 * , 1102 ( 1972 ) . g. furlan , n. paver , g. verzegnassi , nuovo cimento a * 32 * , 75 ( 1976 ) . n. dombey , b.s . read , j. phys . g * 3 * , 1659 ( 1977 ) . tkebuchava , nuovo cimento a * 47 * , 415 ( 1978 ) . smirnov , n.m . shumeiko , yad . fiz . * 17 * , 1266 ( 1973 ) . a. bietti , s. petrarca , nuovo cimento a * 22 * , 595 ( 1974 ) ; lett . nuovo cimento * 13 * , 539 ( 1975 ) . surovtsev , f.g . tkebuchava , yad . fiz . * 21 * , 753 ( 1975 ) . samios , phys . rev . * 121 * , 275 ( 1961 ) ; h. kobrak , nuovo cimento * 20 * , 1115 ( 1961 ) ; s. devons _ et al . _ , phys . rev . * 184 * , 1356 ( 1969 ) ; m.n . et al . _ , b * 24 * , 349 ( 1967 ) ; c.m . et al . _ , d * 28 * , 660 ( 1983 ) ; h. fonvieille _ et al . _ , b * 233 * , 60 , 65 ( 1989 ) . yu.k . akimov _ et al . _ , yad . fiz . * 13 * , 748 ( 1971 ) ; s.f . * 16 * , 185 ( 1972 ) ; * 18 * , 102 ( 1973 ) ; * 24 * , 1127 ( 1976 ) ; g.d . et al . _ , ibid . * 36 * , 322 ( 1982 ) ; * 46 * , 1360 ( 1987 ) ; v.v . et al . _ , ibid . * 33 * , 357 ( 1981 ) . v.n . baturin _ et al . _ , yad . fiz . * 47 * , 708 ( 1988 ) . g. bardin , phys . b * 255 * , 149 ( 1991 ) ; ibid . * 257 * , 514 ( 1991 ) . b * 52 * , 227 ( 1974 ) ; r.c.e . et al . _ , nuovo cimento a * 1 * , 475 ( 1971 ) . goghilidze , yu.s . surovtsev , f.g . tkebuchava , yad . fiz . * 45 * , 1085 ( 1987 ) . adler , ann . ( ny ) * 50 * , 189 ( 1968 ) . c * 61 * , 035204 ( 2000 ) . particle data group , d.e . groom _ et al . j. c * 15 * , 1 ( 2000 ) . s. capstick , phys . d * 46 * , 2864 ( 1992 ) . yu.s . surovtsev , author s summary , 2 - 11047 , dubna , 1977 ( unpublished ) . aznauryan , phys . c * 67 * , 015209 ( 2003 ) . j. volmer ( jlab f@xmath223 collaboration ) , phys . rev . lett . * 86 * , 1713 ( 2001 ) . gari and w. kruempelmann , phys . d * 45 * , 1817 ( 1992 ) . et al . _ , * 26 * , 1672 ( 1971 ) ; nucl . b * 120 * , 395 ( 1977 ) ; p. benz _ et al . _ , nucl . b * 65 * , 158 ( 1973 ) ; v. rossi _ et al . _ , nuovo cimento a * 13 * , 59 ( 1973 ) ; m. beneventano _ et al . _ , nuovo cimento a * 19 * , 529 ( 1974 ) ; g.von holtey , g. knop , h. stein , j. stmpfig , and h. wahlen , nucl . b * 70 * , 379 ( 1974 ) ; p.e . scheffler and p.l . walden , nucl . b * 75 * , 125 ( 1974 ) ; a.j . b * 101 * , 1 ( 1975 ) ; p.e . et al . _ , a * 296 * , 373 ( 1978 ) ; m.t . et al . _ , a * 324 * , 301 ( 1979 ) ; j.c . comiso , d.j . blasberg , r.p . haddock , b.m.k . nefkens , p. truoel , and l.j . verhey , phys . d * 12 * , 719 ( 1979 ) ; a. bagheri , k.a . aniol , f. entezami , m.d . hasinoff , d.f . measday j - m . poutissou , m. salomon , and b.c . robertson , phys . c * 38 * , 875 ( 1988 ) ; g.j . kima , j. arendsa , w.j . briscoec , j. engelagea , b.m.k . nefkensa , m.e . sadlerd , and h.j . ziocka , phys . d * 40 * , 244 ( 1989 ) . k. btzner _ et al . _ , b * 39 * , 575 ( 1972 ) . g. fischer _ et al . _ , z. phys . * 245 * , 225 ( 1971 ) . g. bardin , j. duclos , a. magnon , b. michel and j.c . montret , nucl . b * 120 * , 45 ( 1977 ) . h. breuker _ et al . _ , nucl . b * 146 * , 285 ( 1978 ) . dubovik , l.a . tosunyan , fiz . elem . chastits . at . 14*(5 ) , 1193 ( 1983 ) . g. hhler and e. pietarinen , nucl . b * 95 * , 210 ( 1975 ) . s. dubnika _ et al . _ , nuovo cimento a * 106 * , 1253 ( 1993 ) . g. bardin _ et al . _ , a * 352 * , 365 ( 1981 ) ; phys . b * 104 * , 320 ( 1981 ) . v. brudanin _ et al . _ , a * 587 * , 577 ( 1995 ) . volkov and c. weiss , phys . d * 56 * , 221 ( 1997 ) . yu.l . kalinovsky and c. weiss , z. phys . c * 63 * , 275 ( 1994 ) ; i.v . et al . _ , modelling * 6 * , 57 ( 1994 ) . s. choi _ et al . _ , lett . * 71 * , 3927 ( 1993 ) . k. hagiwara _ ( particle data group ) , phys . d * 66 * , 010001 ( 2002 ) . yu.i . ivanshin _ et al . _ , nuovo cimento a * 107 * , 2855 ( 1994 ) . ivanshin , n.b . skachkov , nuovo cimento a * 108 * , 1263 ( 1995 ) . a.n . ivanov , m. nagy , and n.i . troitskaya , phys . c * 59 * , 451 ( 1999 ) . gerasimov , chinese j. phys . * 34 * , 848 ( 1996 ) . v.a . meshcheryakov and g.v . meshcheryakov , yad . fiz . * 60 * , 1400 ( 1997 ) . guan - yeu chen , s.s . kamalov , shin nan yang , d. drechsel , l. tiator , nucl . a * 723 * , 447 ( 2003 ) . d. krupa , v.a . meshcheryakov , yu.s . surovtsev , nuovo cimento a * 109 * , 281 ( 1996 ) . et al . _ , yad . fiz . * 44 * , 1471 ( 1986 ) . @xmath28-matrix theory of strong interactions _ ( benjamin , new york , 1962 ) . j.d . bjorken and j.d . walecka , ann . ( ny ) * 38 * , 35 ( 1966 ) . devenish and d.h . lyth , nucl . b * 93 * , 109 ( 1975 ) . n.n . achasov and l.m . samkov , yad . fiz . * 8 * , 1201 ( 1968 ) . cho and j.j . sakurai , phys . b * 30 * , 119 ( 1969 ) . f.e . close and f.j . gilmal , phys . b * 38 * , 541 ( 1972 ) . s.f . berezhnev _ _ , yad . fiz . * 26 * , 547 ( 1977 ) .

methods are set forth for determining the hadron electromagnetic structure in the sub-@xmath0-threshold timelike region of the virtual - photon `` mass '' and for investigating the nucleon weak structure in the spacelike region from experimental data on the process @xmath1 at low energies . these methods are formulated using the unified description of photoproduction , electroproduction , and inverse electroproduction of pions in the first resonance region in the framework of the dispersion - relation model and on the basis of the model - independent properties of inverse electroproduction . applications of these methods are also shown .

nucl-th0401033

nuclei in interaction with external fields display a wide variety of collective vibrations known as giant resonances , associated with various degrees of freedom and multipolarities . the giant isovector dipole resonance and the giant isoscalar quadrupole resonance are the most studied examples in this class of phenomena . a particular mode , that is associated with vibrations in the number of particles , has been predicted in the 70s@xcite and discussed , under the name of giant pairing resonance , in the middle of the 80 s in a number of papers@xcite . this phenomenon , despite some early efforts aimed to resolve some broad bump in the high - lying spectrum in ( p , t ) reactions@xcite , is still without any conclusive experimental confirmation . for a discussion , in particluar in connection with two - particle transfer reactions , on many aspects of pairing correlations in nuclei we refer to a recent review@xcite . we have studied the problem of collective pairing modes at high excitation energy in two neutron transfer reactions with the aim to prove the advantage of using unstable beam as a new tool to enhance the excitation of such modes @xcite . the main point is that with standard available beams one is faced with a large energy mismatch that strongly hinders the excitation of high - lying states and favours the transition to the ground state of the final system . instead the optimum q - value condition in the ( @xmath3he,@xmath4he ) stripping reaction suppresses the ground state and should allow the transition to 10 - 15 mev energy region . we have performed particle - particle rpa calculations on lead and bcs+rpa on tin , as paradigmatic examples of normal and superfluid systems , evaluating the response to the pairing operator . subsequently the two - neutron transfer form factors have been constructed in the framework of the macroscopic model@xcite and used in dwba computer codes . we have estimated cross - sections of the order of some millibarns , dominating over the mismatched transition to the ground state . recently we added similar calculations on other much studied targets to give some guide for experimental work . the formal analogy between particle - hole and particle - particle excitations is very well established both from the theoretical side@xcite and from the experimental side for what concern low - lying pairing vibrations around closed shell nuclei and pairing rotations in open shells . the predicted concentration of strength of a @xmath5 character in the high - energy region ( 8 - 15 mev for most nuclei ) is understood microscopically as the coherent superposition of 2p ( or 2h ) states in the next major shell above the fermi level . we have roughly depicted the situation in fig . ( [ fig1 ] ) . in closed shell nuclei the addition of a pair of particles ( or holes ) to the next major shell , with a total energy @xmath6 , is expected to have a high degree of collectivity . also in the case of open shell nuclei the same is expected for the excitation of a pair of particles with @xmath7 energies . for normal nuclei the hamiltonian with a monopole strength interaction reads : @xmath8 where @xmath9 annihilates a pair of particles coupled to @xmath10 total angular momentum . getting rid of all the technicalities of the solution of the pp - rpa equations ( that may be found in the already cited work by the author ) we merely state that the pairing phonon may be expressed as a superposition of 2p ( or 2h ) states with proper forward and backward amplitudes ( @xmath11 and @xmath12 ) . the pair transfer strength , that is a measure of the amount of collectivity of a each state @xmath13 , is given by : @xmath14 . \label{p5}\ ] ] this quantity is plotted in the first column of fig . ( [ fig2 ] ) for the removal ( upper panel ) and addition mode ( lower panel ) . in the same figure are reported the pairing strength parameters for the states of @xmath1sn . to obtain these last quantities for superfluid spherical nuclei one has to rewrite the hamiltonian according to the bcs transformation and has to solve more complex rpa equations . in this case the pairing strength for the addition of two particles is given , for each state @xmath13 , by : @xmath15_{00}|0\rangle = \sum_{j } \sqrt{2j+1 } [ u^{2}_{j } x_{n}(j ) + v^{2}_{j}y_{n}(j)]\ ] ] where the @xmath16 and @xmath17 are the usual occupation probabilities . the amount of collectivity is a clear signal of the structural existence of giant pairing vibrations in the high - lying energy region . we also report here a number of analogous results for other commonly studied targets = 9.4pc = 9.4pc = 9.4pc with the aim of giving some indications to experimentalists on the reasons why we think that lead and tin are some of the most promising candidates . we have studied two isotopes of calcium with closed shells . even if the absolute magnitudes of the @xmath18 is lower , it is worthwhile to notice that some enhancement is seen in the more neutron - rich @xmath19ca with respect to @xmath20ca . an important role in this change is certainly due to the different shell structure of the two nuclei as well as to the scheme that we implemented to obtain the set of single particle levels . the latter is responsible for the collectivity of the removal modes in both ca isotopes and also for the difficulty in finding out a collective state in the addition modes . we display also results for @xmath21zr where the strength is much more fragmented and the identification of the gpv is more difficult . in the work of broglia and bes estimates for the energy of the pairing resonance are given as @xmath22 mev and @xmath23 mev for normal and superfluid systems respectively . our figures follow roughly these prescriptions based on simple arguments ( and much more grounded in the case of normal nuclei ) as evident from table [ ta1 ] . .comparison of position of gpv between our calculation and the broglia and bes estimate . [ cols="^,^,^",options="header " , ] = 13.8pc = 13.8pc these cross - sections have been derived for sharp states , and we refer to the numbers in the last table when speaking of order of magnitude estimates . obviously cross - section in the high - lying energy region have a finite ( and large ) width that should be inserted for a more realistic description of the spectrum . we have chosen a simple scheme that gives a lorentzian distribution with a width that grows quadratically with the excitation energy , @xmath24 , with @xmath25 adjusted to give a width of 4 mev for the gpv . this could seem rather arbitrary since there is no reason for an _ a priori _ assignment of this quantity . we have been brought to this simple prescription because other collective states ( of different nature ) lying in the same energy region display similar values for their width , and it is reasonable to assume some rule to narrow the low - energy states and to broaden the high - energy ones . the final achievements for the four reactions studied in detail are presented in figure [ fig4 ] where the areas corresponding to the cross - sections given above have been shaded to give a feeling of the relative magnitudes of the transition to the ground states and to the gpv s . it is worthwhile to note that in the case of pb there is a considerable gain in using unstable beams , while in sn is much less evident . one sees the need for unstable helium when compares the magnitude for the pairing resonance in the right a ) and b ) panels with the peak at zero energy : in the first panel the transition to the ground state is extremely hindered . a @xmath3he beam is currently available ( or it will be available in the very near future ) in many radioactive ion beams facilities around the world and the calculations that we have presented could allow a planning for future experiments aimed to study the not yet completely unraveled role of pairing interaction in common nuclei , using exotic weakly bound nuclei as useful tools . the author wishes to gratefully acknowledge discussions with andrea vitturi , hugo sofia and wolfram von oertzen on various aspects of theoretical and experimental nuclear physics . the participation at the _ vii international school - seminar on heavy ion physics , dubna , russia _ 2002 has been supported by the infn . xxxx r.a.broglia and d.r.bes , plb691291977 . m.w.herzog , r.j.liotta and l.j.sibanda , _ phys . _ c * 31 * , 259 , ( 1985 ) . et al _ , prl 3914511977 . w.von oertzen and a.vitturi , _ rep . phys . _ * 64 * , 1247 - 1337 , ( 2001 ) . l.fortunato , w.von oertzen , h.m.sofia and a. vitturi , _ eur . _ a * 14 * , ( 2002 ) , in press . c.h.dasso and a.vitturi ( editors ) , _ collective aspects in pair transfer phenomena _ , sif proc . 18 , ( editrice compositori bologna , 1987 ) . d.r.bes and r.a.broglia , _ phys . _ c * 3 * , 2349 , ( 1971 ) . c.h.dasso and g.pollarolo , plb 1552231985 . c.h.dasso and a.vitturi , prl 596341987 .

we investigate the possible signature of the presence of giant pairing states at excitation energy of about 10 mev via two - particle transfer reactions induced by neutron - rich weakly - bound projectiles . performing particle - particle rpa calculations on @xmath0pb and bcs+rpa calculations on @xmath1sn , we obtain the pairing strength distribution for two particles addition and removal modes . estimates of two - particle transfer cross sections can be obtained in the framework of the macroscopic model. the weak - binding nature of the projectile kinematically favours transitions to high - lying states . in the case of @xmath2 reaction we predict a population of the giant pairing vibration with cross sections of the order of a millibarn , dominating over the mismatched transition to the ground state .

nucl-th0206008

the detection of gravitational waves ( gws ) will rank as one of the major scientific achievements of this century . their detection will open up a new observational window to the universe , revealing dynamic sources of strong field relativistic gravity previously inaccessible through conventional astronomical instruments . our understanding of space - time and matter under the most extreme conditions will be transformed . although there has been no direct detection of gws to date , indirect evidence for their existence comes from high precision , nobel - prize winning measurements of the pulsar psr 1913 + 16 and its companion neutron star ( ns ; * ? ? ? * ; * ? ? ? the gw emission that drives the system s orbital decay is in agreement with the predictions of general relativity to better than 1% @xcite . when such binary neutron star ( bns ) systems eventually coalesce , they are predicted to emit copious amounts of gws @xcite . these sources will be prime targets for the new generation of gw detectors , led by advanced ligo ( aligo ; * ? ? ? * ) which is set to begin observing during the second half of 2015 and advanced virgo a year later @xcite . at final sensitivity , these advanced detectors are expected to detect bns mergers at a rate within the range 0.4400 yr@xmath0 @xcite . compact binary coalescences ( cbcs ) consisting of at least one black hole ( bh ) are also targets for gw detectors ; although there is compelling evidence for their existence @xcite , the event rates of these sources for aligo detection is not well known . one realisation in the last decade is that coalescing systems of ns / ns or ns / bh events could be the progenitors of short - hard gamma ray bursts ( sgrbs ) ; transient events routinely observed throughout the electromagnetic ( em ) spectrum @xcite . there exist other types of em , neutrino and cosmic ray emissions that may also be associated with gw events . these include long - duration gamma ray bursts ( lgrbs ; * ? ? ? * ) , short gamma ray repeaters @xcite , supernovae @xcite , fast radio bursts @xcite as well as others . history has already shown that multi - wavelength astronomy can play an important role in unveiling new phenomena . in the last decade , x - ray , optical and radio follow - ups have all transformed and revealed new processes in our understanding of gamma ray bursts ( grbs ) ; combining em observations with those in the gw domain will too provide new insight into the internal engines and mechanisms at play in a multitude of different sources . a new generation of sensitive , wide - field telescopes , advancements in time domain astronomy and upgrades to neutrino and cosmic ray detectors can provide a coordinated network for discovery . the possible simultaneous detection of photons , neutrinos or high energy particles with gws would be a landmark moment for astrophysics , initiating a new era of _ _ multi - messenger _ _ astronomy , for the first time including gw . maximising the potential offered by gw observations involves the development of a worldwide , multi - messenger network . australian facilities are ideally placed to foster scientific exchanges in this new era and agreements have already been established . to conduct em follow - up of gw triggers , memorandums of understanding ( mous ) have been signed between the ligo / virgo gw collaboration and a number of facilities either based in australia or with strong australian involvement ; these include : the anglo - australian telescope , the australian square kilometer array pathfinder ( askap ; * ? ? ? * ) , the cherenkov telescope array ( cta ; * ? ? ? * ) , the high energy stereoscopic system ( h.e.s.s ; * ? ? ? * ) , icecube @xcite , the murchison widefield array ( mwa ; * ? ? ? * ) , and the skymapper @xcite , the gw optical transient observer ( goto ) and zadko @xcite optical telescopes . in this paper , we focus on the most probable multi - messenger observations from the advanced detector era ; those associated with grbs . whilst doing so , we consider the contribution that the australian facilities can make to the worldwide multi - messenger effort . the structure of this paper is as follows : section 2 describes gw astronomy . sections 3 and 4 introduce sgrbs and lgrbs and describe how co - ordinated gw and multiwavelength observations of these events can provide breakthrough science . section 5 acts as a primer for those unfamiliar with the concepts and terminologies of detection and data analysis often used in the gw domain ; this section is not designed to be exhaustive but to present some of the most important concepts in gw detection and data analysis . section 6 discusses the expected rates and detection ranges for gw sources . the next two sections describe two of the strategies that form the basis for coordinated gw and em observations in the gw era . section 7 discusses em triggered gw searches ; these could likely yield the first coincident gw - em event through archival gw data . section 8 discusses the em follow - up of gw triggers ; this strategy is highly challenging due to the large positional uncertainties of gw observations but the potential rewards for success are without doubt highly significant . section 10 discusses the australian facilities involved in the co - ordinated science programs with aligo / adv and we highlight the areas in which they could contribute in this new frontier . finally , in section 11 we discuss the role neutrino follow - up plays in gw detection . gravitational waves are produced by regions of space - time that are distorted by high velocity bulk motions of matter . the timescale of the motions determine the frequency of the gw emission ; ground based detectors will target systems with masses in the range 110@xmath1 @xmath2 , which emit in the 1 hz10 khz band . this frequency range , covering the audio band , has motivated the characterisation of interferometric gw astronomy as `` listening to the universe '' . instruments capable of achieving detections will begin observations in the second half of 2015 . advanced ligo , a pair of us based interferometric detectors at hanford and livingston ( usa * ? ? ? * ) will have its first observational science run ( o1 ) in late-2015 ; a year later it will be joined by the italian advanced virgo ( adv ; * ? ? ? * ; * ? ? ? * ) for a second observing run ( o2 ) . the `` advanced '' network of interferometric gw detectors will eventually have 10 times the sensitivity of the first generation instruments . the increased sensitivity translates into a factor @xmath3 increase in observed volume , making detections expected rather than plausible . additional instruments are expected to eventually join the network . kagra , a japanese detector , is envisioned to begin operation in 2018 - 19 @xcite and ligo - india is expected to be operational from 2020 , reaching a design sensitivity at the same level as aligo by around 2022 @xcite . the gw observations made by these instruments will differ from most conventional em observations in several ways : 1 . gws are not scattered or obscured by intervening material like dust so provide a window into the densest regions of the universe . as gw detectors observe an amplitude rather than a flux , the measure of detectability follows an inverse relationship with distance rather than the conventional inverse square law . therefore , number counts of a homogeneous distribution of standard - candle sources increases with distance , @xmath4 , as , @xmath5 , rather than , @xmath6 ; 3 . as gws couple weakly to the detectors , even very local astronomical sources of gws have to be highly energetic emitters of gravitational radiation ; 4 . gw detectors are nearly omnidirectional , with a nearly @xmath7 steradian sensitivity to astrophysical events with a greater than average response over more than 40% of the sky . the first point implies that gw observations can allow us to view astrophysical phenomena inaccessible by other means . the gravitational window can therefore enable frontier explorations in the low to intermediate redshift universe ( @xmath8 ) of sources that are electromagnetically invisible for much , or all , of their lives . the second point means that a factor 2 improvement in the sensitivity of a gw detector results in a factor 8 increase in the volume of the universe being probed . the third point emphasises a detection bias for detecting the most highly energetic astrophysical events . the typical fluxes of gw sources are of order @xmath9 jy , far greater than equivalent fluxes typically observed in the radio domain ( @xmath10jy jy ) . the final point means that gw detectors are naturally survey instruments over a wide band of frequencies ( 105000 hz ) . there are a number of types of em counterparts that may be associated with gw emissions @xcite . as some of these counterparts are quite speculative , this paper focuses on gw signals associated with grbs . other sources of simultaneous em and gw emission include supernovae as well as multiple emission mechanisms from nss ; for a review of the latter , see the accompanying article in this series @xcite . in the next few sections we provide a summary of both sgrbs and lgrbs and the type of gw / em associations that could be targeted in the gw era . some of these predictions are based on solid foundations whilst some are more speculative . in considering the latter , we note that when a new window of observation has been opened in the past , the discoveries that transform our understanding of the universe have often been the least expected . + gravitational waves from the merger of coalescing binary systems of ns / ns and ns / bhs are confidently predicted to have observable em counterparts . this expectation is a result of the connection between these events and sgrbs ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? the evidence stems from a number of different channels . firstly , the dynamic timescales of discs predicted from the merger of cbcs are consistent with the durations of sgrbs . secondly , em follow - ups of sgrbs have never provided an associated supernova . thirdly , sgrb afterglows have been localised to galaxies harbouring older stellar populations with offsets of order tens of kpc from their galactic centers ; this is consistent with post - natal kick velocities of 100s of km s@xmath0 , and also with the fainter and shorter lived afterglows expected from an ambient interstellar medium at a large offset . finally , as is discussed in [ kilonova ] , the discovery of a faint em transient called a _ kilonova _ has provided the strongest observational evidence to date of the sgrb / cbc association . conclusive proof of the cbc / sgrb association will be provided through gw observations . coincident em and gw observations of sgrbs could also provide a fascinating insight to the dominant mechanisms at the heart of grbs . low - latency gw pipelines could enable multi - wavelength follow - up measurements of the prompt emission , constraining both the underlying central engines and the emission mechanisms at work . later - time multi - wavelength follow - ups can provide insight through extensive coverage of the sgrb afterglow . a number of em counterparts have been predicted to accompany the inspiral and merger of ns / ns and ns / bh systems . in figure [ fig_mm_scenarios ] we show the likely outcomes of these mergers and in the following sections we will briefly review the most likely em counterparts that could accompany cbcs . during the final stages of the merger of a compact binary , the system is expected to launch a highly relativistic jet that interacts with itself and the surrounding medium ( the fireball model for grbs ; e.g. * ? ? ? collisions of material moving at different velocities within the jet will lead to internal shocks , giving short - lived bursts of gamma - rays that we detect as the sgrb prompt emission . as the accretion timescale is expected to be @xmath112 seconds @xcite , the grbs associated with compact binary mergers are typically shorter in duration than those associated with core - collapse supernovae ( explaining the observed distribution of grbs ; * ? ? ? . however , the division between these two populations is not easily identifiable from the prompt gamma - ray emission alone ( e.g. * ? ? ? * ; * ? ? ? * ) . a number of _ fermi_-lat grbs have shown @xmath12gev emission ( even at redshifts as distant as @xmath13 ) . two bursts were observed with gamma - ray photons reaching energies up to 94gev ( grb 130427a ) and 62gev ( grb 131231a ) this supports the suggestion that the photon energies may extend higher than previously assumed @xcite . significantly , these discoveries have not been limited to lgrbs , with sgrbs also showing high energy photons and gev emission often continuing for 10 s of seconds beyond the initial burst . the fact that _ fermi_-lat discovered a photon of energy 31gev during the prompt phase of grb 090510 @xcite is promising for co - ordinated observations between gw detectors and ground - based cherenkov telescope arrays @xcite operating at @xmath12 10 gev . additionally , sgrbs with time - extended emission have recently been cited as promising targets for cherenkov telescope arrays @xcite . one exciting possibility is the observation of prompt optical flashes . so far , these emissions have only been observed in lgrbs @xcite . an early optical emission correlated with the prompt gamma - rays could indicate a common origin related to the internal shocks @xcite . a number of studies have suggested that compact binary mergers could generate prompt coherent radio emission ( e.g. , * ? ? ? such mechanisms include excitation of the plasma surrounding a compact binary merger by gws @xcite , from a dynamically - generated magnetic field after the merger @xcite , or from the onset of the collision of the forward shock with the surrounding medium @xcite . however , the detectability of emission from these processes will be dependent upon the scattering by the surrounding environment @xcite . nonetheless , these studies suggest compact binary mergers are an interesting contender for the progenitors of fast radio bursts ( frbs ; * ? ? ? * ; * ? ? ? * ) , which are currently unknown . plateaus and flares in x - ray light curves following grbs are signatures of ongoing energy injection . this could be caused by late - time accretion onto a central black hole ( unlikely in the compact binary scenario ; see discussion in * ? ? ? * ) , or from ongoing energy injection from the spindown of a newly born neutron star . indeed , recent studies ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) have shown that the merger of two nss could result in a supramassive ns ; a star with a mass greater than the non - rotating maximum mass but supported from further collapse through rotation @xcite . around 60% of x - ray afterglow light curves of sgrbs observed by the _ swift _ satellite @xcite have shown plateaus lasting 10010000 s after the burst ; these have been attributed to electromagnetic spin - down emissions from protomagnetars @xcite formed via the merger of two neutron stars @xcite . observations of the plateau phase can also be used to constrain the ns equation of state , with gw observations of the inspiral phase significantly aiding this endeavour @xcite if the post - merger remnant is an ns , early optical afterglow as bright as 17th magnitude in r band ( assuming a distance of @xmath14 mpc ; see [ section_cbc_ranges ] ) could be produced from dissipation of a wide - beamed protomagnetar wind @xcite . this magnetar wind could launch ejecta at relativistic speeds which would interact with the surrounding medium and produce a bright broadband afterglow from synchrotron radiation @xcite . gravitational wave emission may also accompany an afterglow plateau if a millisecond magnetar is born from the collision . multiple mechanisms for generating such gws exist in nascent neutron stars , including secular bar modes ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , @xmath15-modes @xcite , and magnetic - field induced stellar deformations @xcite . such emission could be observable by aligo out to @xmath16 mpc @xcite . in fact , the x - ray light curve itself can be used to constrain the total gw emission from these systems @xcite . some plateaus following sgrbs exhibit an extremely steep decay phase , commonly interpreted as the collapse of the nascent neutron star to a black hole @xcite . such collapse could potentially produce an frb when the magnetic field lines snap as they cross the bh horizon @xcite , which is expected to occur @xmath17 s after the merger @xcite . a low latency gw trigger could enable prompt follow - ups to test this connection @xcite . as the relativistic jet propagates , it collides with the medium surrounding the progenitor resulting in a forward shock travelling into the surrounding medium , and a reverse shock propagating back up the jet ( e.g. * ? ? ? * ; * ? ? ? these shock fronts produce multi - wavelength synchrotron emission , initially peaking in the x - ray and moving through the different wavelengths to radio as it fades . the typical afterglow of grbs is attributed to the forward shock emission and the brightness of this afterglow is dependent upon a number of parameters , including the density of the surrounding medium . therefore , in a low density environment , the forward shock component is expected to be relatively faint . the multi - wavelength afterglows of sgrbs have been observed and are typically fainter than those of lgrbs @xcite . this is consistent with sgrbs being less energetic than lgrbs and with their locations in lower density environments . the reverse shock has also been observed for sgrb 051221a ( e.g. * ? ? ? * ) . a ` kilonova ' is been predicted to form after the merger of two nss . this faint optical transient is powered by the radioactive decay of the ejected neutron rich matter @xcite and could reach around 21-23 mag in the optical and 21-24 mag in the nir for a source at 200 mpc @xcite . recent optical and near - infrared follow - up observations of grb 130603b have provided the most conclusive evidence to date of this scenario , reinforcing the theory that compact object mergers are the progenitors of sgrbs @xcite . these observations have added significantly to other observational evidence in support of this scenario @xcite . coincident em and gw observations could confirm that sgrbs are indeed the result of coalescing compact binaries . an additional prompt em emission related to the kilonova mechanism has also recently been suggested by @xcite . this has been inspired by studies that suggest a small fraction of the ejected neutron rich matter can expand rapidly enough to avoid r - process capture @xcite . the suggestion is that @xmath18-decay from free neutrons in the outermost layers of this ejecta could power optical emission on a timescale of hours after the merger , peaking at around magnitude 22 in the u - band for a source . for a source at 200 mpc this signal would peak at around magnitude 22 in the u - band and would act as a precursor to a kilonova . lgrbs are amongst the most luminous transient events in the universe in terms of em radiation per unit solid angle . these beamed emissions have been observed to last up to @xmath19 s @xcite and can radiate a total energy equivalent to that of the sun in its entire 10 gyr lifetime . the extreme luminosities allow lgrbs to be seen out to cosmological volumes , making them a probe of the high redshift universe ( @xmath20 ) . the favoured scenario for these bursts is described by the collapsar model @xcite in which the inner part of a wolf - rayet star progenitor collapses to form a rapidly rotating black hole . high angular momentum enables the infalling matter to form an accretion disk , which in turn provides the energy reservoir to power an ultra - relativistic jet that blasts its way through the stellar envelope . the observed radiation is explained through synchrotron and/or inverse compton emission from the accelerated electrons in internal and external shocks . some authors have suggested instead that the central engines may consist of magnetars @xcite . there is observational evidence to support this scenario for at least a proportion of lgrbs @xcite . the connection between lgrbs with the collapse of massive stars @xcite has been supported by afterglow observations in or near dense regions of active star - formation ; predominantly dwarf starburst field galaxies @xcite . as mentioned earlier , their denser environments , as well as their higher emission energies , mean that the multiwavelength afterglows of lgrbs are typically brighter than those that occur from sgrbs @xcite . in terms of gw emissions from these events , a number of lgrbs have been associated with core - collapse supernova @xcite . modelling the gw emission from these supernovae is very complex , requiring a combination of general relativistic hydrodynamics , magnetic fields , rotation , neutrino transport and nuclear physics @xcite . simulations have so far provided a picture of a very complex and chaotic behavior that includes shock formation and turbulence that create highly complex waveforms with multiple sharp bursts over ms durations . however , most scenarios suggest an event may have to be within tens of kpc for detection . as most lgrbs occur at cosmological distances , the vast majority of their gw signals will be out of reach for advanced detectors . the requirement for rapid rotation to produce the disc in a grb @xcite allows for alternative emission mechanisms that could produce detectable gws out to 10s of mpc @xcite . fragmentation instabilities could be produced in the core or in the disc @xcite . rapid rotation could also give rise to rotational instabilities in the protoneutron star remnant @xcite a number of studies have suggested there exists a sub - population of lgrbs known as low - luminosity grbs ( _ _ ll__grbs ) . these events have isotropic equivalent gamma - ray luminosities 2 - 3 orders of magnitude below classical lgrbs @xcite and have only been detected at low-@xmath21 due to their lower energy emissions ( the closest was grb980425 at z = 0.0085 or 36mpc ) . as such their rates have been predicted to be 23 orders of magnitude greater than lgrbs . observations have confirmed that both lgrbs and _ _ ll__grbs produce supernovae , suggesting that the _ _ ll__grbs may just be lower - energy events from the tail of the distribution . this has been a long going debate and attempts to address it have used statistical arguments @xcite , fits to the peak flux distribution @xcite , and simulation @xcite . the suggestion that _ _ ll__grbs could be just normal lgrbs viewed off - axis was discounted based on statistical arguments , as it would produce a far higher local rate density than expected from lgrbs and would require narrower opening angles for lgrbs than determined from the breaks in afterglow lightcurves @xcite . recently , an analysis of _ _ ll__grb 060218 has suggested that the main difference in the two bursts arises from an extended low - mass envelope in _ _ the existence of such an envelope can smother the jet and drive a mildly relativistic shock resulting in a much lower luminosity than that produced by an ultra - relativistic jet that is able to penetrate through the bare progenitor star . interestingly , the statistical arguments suggesting separate populations put forward by @xcite can also support these two different scenarios . it is therefore possible that gw emission mechanisms could be driven by the same type of engine for both these classes . the output from a single gw detector consists of a time series data stream , @xmath22 , composed of the detector response to a gw signal , @xmath23 , and the detector noise @xmath24 : @xmath25 in general , @xmath23 will be a linear combination of the two orthogonal transverse polarizations , @xmath26 , weighted by the dimensionless detector antenna pattern functions for the two polarizations @xmath27 : @xmath28 which describe the detector sensitivity to radiation of different polarizations , incident from different directions @xcite . the angles , @xmath29 and @xmath30 , represent the direction to the source and @xmath31 is the polarization angle of the wave . a gw detector can follow the phase of a gw signal , so the time series is generally represented in the frequency domain by the strain amplitude spectral density , @xmath32 . this quantity is defined through the power spectral density @xmath33 , with @xmath34 the fourier transform of the time series . similarly , one can define a signal power spectral density , @xmath35 , and a noise power spectral density , @xmath36 . the strain amplitude spectral density is given by : @xmath37 with dimensions of @xmath38 @xcite . this quantity is often used in plots to display the sensitivity of gw interferometers . a single gw detector can not determine the polarization state or source direction of a transient signal . to obtain source localization , a widely separated network of gw detectors is essential . such a network may employ techniques such as _ coincidence analysis _ , in which individual events from different detectors are correlated in time @xcite , or _ coherent analysis _ , in which synchronized detector outputs are merged before searching for a common pattern @xcite . by effectively resolving the different times of arrival of gw events between members of a network , coherent network analysis enables a detector array to become an all - sky monitor with good angular resolution over all source directions . achieving good directional sensitivity is of paramount importance for gw / em associations . for the sources considered in this review , directional sensitivity is determined through triangulation of arrival times . to maximize the time delays , and hence improve directionality , it is advantageous that a network be as geographically widely separated as possible @xcite and as such , a number of detectors are planned to join the aligo / adv network throughout the next decade . the japanese observatory kagra , should begin operations by around 2018 - 19 @xcite ; at design sensitivity this detector could improve the directional precision of a aligo / adv network by a factor of 1.5 - 2 and the detection rate by a similar factor @xcite . ligo - india operating at aligo sensitivity will be added to the aligo / adv network by 2022 by then bnss will be detectable out to @xmath39200 mpc and up to 400 events are possible per year @xcite . an indian detector will improve the angular resolution sufficiently to increase the percentage of gw sources detected within 5 deg@xmath40 from 3 - 7% to 17% @xcite . it has long been recognized that a gw detector in australia would add the longest baseline to the proposed advanced detector network ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? for example , adding an australian detector to an aligo / adv three detector network can reduce the error in solid angle to tens of arc - minutes for high signal - to - noise ratio ( snr ) signals @xcite , dramatically improving the ability to localise gw sources for multi - wavelength follow - up observations . this scenario could be realised when third generation observatories such as the ` einstein gravitational wave telescope ' ( et ) become a reality in the next decade @xcite . the optimal site for a detector in the southern hemisphere been shown to be western australia @xcite , the current home to an 80-m baseline prototype gw detector . the false alarm rate ( far ) is the rate that false positives appear above a given snr threshold , and is dependent on the number of glitches ( non - stationary transients ) in the gw data stream . it is a critical measure as it determines whether a candidate should be considered for follow - up . for well modeled sources , the background of false alarms is at a level close to that of gaussian noise . for un - modeled sources typically short duration transients the data quality has a greater effect on detection confidence . one therefore sets the threshold high enough so that noise generated false alarms are negligible . given that the probability , @xmath41 , of observing an event with an amplitude in the range @xmath42 to @xmath43 is given by a gaussian distribution of standard deviation @xmath44 , the probability of obtaining a far greater than a given threshold , @xmath45 , is : @xmath46 to be 99% confident that a gw has been detected , one can set an snr @xmath47 which is equivalent to a far of 1 in 100 years of observation . to see this one can approximate number of noise instances during that period . if the detector output sampling rate is 1 khz and the output is processed through @xmath48 filters , in 100 years we get @xmath49 , yielding @xmath50 which is our required snr . for a network of three equivalent detectors combined snr , @xmath51 is given as : @xmath52 where @xmath53 represents the snr in the ith detector @xcite . this shows that for a network of 3 equivalent detectors , to dismiss false alarms at a level @xmath54hz requires @xmath55 . + the expected gw signals from cbcs takes on the well modelled chirp form shown in figure [ fig_chirp ] . the figure shows how the signal increases in both amplitude and frequency towards merger ; as it does so it sweeps across the sensitive bandwidth of advanced gw interferometric detectors . for such well - modeled signals , the most efficient signal detection method to extract signals from noisy detector data is _ matched filtering _ , in which a _ template _ , representing the predicted waveform as a function of time is correlated with the output of a detector @xcite . a matched signal will produce an output much greater than that expected for pure noise with an optimal snr given as : @xmath56^{1/2}. \label{eq : snr}\ ] ] for well - modelled sources , matched filtering enhances the value of the signal by a factor @xmath57 , where @xmath58 is the number of cycles used in the integration . as inspiralling systems approach merger , even though the rest frame gw amplitude will increase , the number of cycles in each frequency bin , @xmath59 , gets smaller ; therefore the detected signal will decrease . this means that for inspiraling systems , rather than solely base the predicted amplitude of the radiation as a true indicator of the detectability , we include a measure of the observed cycles . the value of @xmath58 increases with the compactness of the system as it approaches merger and if observed from a frequency of 10 hz until merger , could produce @xmath60 cycles effectively improving the detectability by a factor of 100 . however , to achieve such gains , a gw data - stream would have to be filtered by a large number of templates ( of order @xmath61 in near real time the significant challenges in both theoretical modeling and computational efficiency to achieve this can not be underestimated . one important aspect of well - modeled inspiralling systems is that a detection can be made 10s of seconds before the merger if enough cycles can be detected to boost the snr @xcite . figure 2 illustrates this concept showing a chirp signal 40s before the merger phase . this scenario could allow a low - latency alert to be sent out to em facilities as near real - time as possible to catch a prompt em signature ; the combination of em and gw data in this regime would provide valuable insight into the inner workings of such cataclysmic events . it is also worth noting that gws can provide an independent measure of luminosity distance , @xmath62 @xcite . during the inspiral phase , the gw strain , and the rate of change of gw frequency are given as @xmath63 where @xmath64 is the redshifted chirp mass , @xmath65 , and @xmath66 , @xmath67 are the component masses of the binary . therefore , if one can determine the redshift through , for example , a galaxy association , one can measure the redshift - luminosity distance relation independent of the cosmic distance ladder . a recent series of papers has reinvigorated this topic by introducing novel methods for breaking the redshift - chirp mass degeneracy with future gw observations @xcite . although matched filtering is the optimal strategy for gaussian , stationary noise , high amplitude transients due to instrumental and environmental artifacts can render gw data to be non - stationary and non - gaussian . therefore , one must employ robust methods that can reject instrumental artifacts and retain the true gw events . one such method is the @xmath68 veto that is a powerful consistency test used to reject false alarms @xcite . this method uses the fact that the quantity @xmath45 is an integral over all frequencies and therefore not sensitive to the contributions from different frequency regions of the broadband signal . one can split the signal spectrum into @xmath58 bins of equal snr contribution , and draw a comparison with the expected value in each bin ( based on the model template ) . a true gw event will have power accumulated approximately equally in each of @xmath58 bins ; a noise glitch will have power unevenly distributed and will yield a large @xmath68 value . , b. l. , abbott , t. m. c. , angstadt , r. , et al . 2012 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 8446 , society of photo - optical instrumentation engineers ( spie ) conference series , 11 , d. w. , price , p. a. , soderberg , a. m. , berger , e. , kulkarni , s. r. , sari , r. , frail , d. a. , harrison , f. a. , yost , s. a. , matthews , k. , peterson , b. a. , tanaka , i. , christiansen , j. , & moriarty - schieven , g. h. 2003 , apj , 586 , l5 , r. m. , dekany , r. g. , bebek , c. , et al . 2014 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 9147 , society of photo - optical instrumentation engineers ( spie ) conference series , 79 , r. d. , barthelmy , s. d. , denny , r. b. , graham , m. j. , & swinbank , j. 2012 , in society of photo - optical instrumentation engineers ( spie ) conference series , vol . 8448 , society of photo - optical instrumentation engineers ( spie ) conference series , 0

the first observations by a worldwide network of advanced interferometric gravitational wave detectors offer a unique opportunity for the astronomical community . at design sensitivity , these facilities will be able to detect coalescing binary neutron stars to distances approaching , and neutron star - black hole systems to . both of these sources are associated with gamma ray bursts which are known to emit across the entire electromagnetic spectrum . gravitational wave detections provide the opportunity for `` multi - messenger '' observations , combining gravitational wave with electromagnetic , cosmic ray or neutrino observations . this review provides an overview of how australian astronomical facilities and collaborations with the gravitational wave community can contribute to this new era of discovery , via contemporaneous follow - up observations from the radio to the optical and high energy . we discuss some of the frontier discoveries that will be made possible when this new window to the universe is opened . binaries : close gravitational waves gamma - ray burst : general methods : observational supernovae : general stars : neutron

1511.02959

the physical content of general relativity is contained in einstein s equation , which has a well - posed initial value formulation ( see , e.g. , @xcite ) . in principle , therefore , to determine the motion of bodies in general relativity such as binary neutron stars or black holes one simply needs to provide appropriate initial data ( satisfying the constraint equations ) on a spacelike slice and then evolve this data via einstein s equation . however , in practice , it is generally impossible to find exact solutions of physical interest describing the motion of bodies by analytic methods . although it now is possible to find solutions numerically in many cases of interest , it is difficult and cumbersome to do so , and one may overlook subtle effects and/or remain unenlightened about some basic general features of the solutions . therefore , it is of considerable interest to develop methods that yield approximate descriptions of motion in some cases of interest . in general , the motion of a body of finite size will depend on the details of its composition as well as the details of its internal states of motion . therefore , one can expect to get a simple description of motion only in some kind of `` point particle limit '' . however , einstein s equation is nonlinear , and a straightforward analysis @xcite shows that it does not make any mathematical sense to consider solutions of einstein s equation with a distributional stress - energy tensor supported on a worldline . physically , if one tried to shrink a body down to zero radius at fixed mass , collapse to a black hole would occur before the point particle limit could be reached . distributional stress - energy tensors supported on a world - line _ do _ make mathematical sense in the context of the linearized einstein equation . therefore , one might begin a treatment of gravitational self - force by considering a metric perturbation , @xmath3 , in a background metric , @xmath4 , sourced by the stress - energy tensor of a `` point particle '' of mass m , given in coordinates @xmath5 by @xmath6(t , x^i ) = 8 \pi m u_a(t ) u_b(t ) \frac{\delta^{(3)}(x^i - z^i(t))}{\sqrt{-g } } \frac{d\tau}{dt}\,\ , .\ ] ] where @xmath7 is the unit tangent ( i.e. , 4-velocity ) of the worldline @xmath8 defined by @xmath9 , and @xmath10 is the proper time along @xmath8 . ( here @xmath11 is the `` coordinate delta function '' , i.e. , @xmath12 . the right side also could be written covariantly as @xmath13 where @xmath14 is the covariant 4-dimensional delta - function and @xmath10 denotes the proper time along @xmath8 . ) however , this approach presents two major difficulties . first , the linearized bianchi identity implies that the point particle stress - energy must be conserved . however , as we shall see explicitly in section [ sec : geodesic ] below , this requires that the worldline @xmath8 of the particle be a geodesic of the background spacetime . therefore , there are no solutions to equation for non - geodesic source curves , making it hopeless to use this equation to derive corrections to geodesic motion . this difficulty has been circumvented in @xcite and other references by modifying as follows . choose the lorenz gauge condition , so that equation takes the form @xmath15 where @xmath16 with @xmath17 . equation by itself has solutions for any source curve @xmath8 ; it is only when the lorenz gauge condition is adjoined that the equations are equivalent to the linearized einstein equation and geodesic motion is enforced . therefore , if one solves the lorenz - gauge form of the linearized einstein equation while simply _ ignoring _ the lorenz gauge condition ) truly is ignored in the sense that it is not even pointed out that one has modified the linearized einstein equation , and no attempt is made to justify this modification . ] that was used to derive , one allows for the possibility non - geodesic motion . of course , this `` gauge relaxation '' of the linearized einstein equation produces an equation inequivalent to the original . however , because deviations from geodesic motion are expected to be small , the lorenz gauge violation should likewise be small , and it thus has been argued @xcite that solutions to the two systems should agree to sufficient accuracy . the second difficulty is that the solutions to eq . ( [ eq : wave ] ) are singular on the worldine of the particle . therefore , naive attempts to compute corrections to the motion due to @xmath3such as demanding that the particle move on a geodesic of @xmath18are virtually certain to encounter severe mathematical difficulties , analogous to the difficulties encountered in treatments of the electromagnetic self - force problem . despite these difficulties , there is a general consensus that in the approximation that spin and higher multipole moments may be neglected , the motion of a sufficiently small body ( with no `` incoming radiation '' ) should be described by self consistently solving eq . ( [ eq : wave ] ) via the retarded solution together with @xmath19 where @xmath20 with @xmath21 the retarded green s function for eq . ( [ eq : wave ] ) , normalized with a factor of @xmath22 , following @xcite . the symbol @xmath23 indicates that the range of the integral extends just short of the retarded time @xmath24 , so that only the `` tail '' ( i.e. , interior of the light cone ) portion of the green s function is used ( see , e.g. , reference @xcite for details ) . equations ( [ eq : wave ] ) and ( [ eq : misataquwa - intro ] ) are known as the misataquwa equations , and have been derived by a variety of approaches . however , there are difficulties with all of these approaches . one approach @xcite that has been taken is to parallel the analysis of @xcite in the electromagnetic case and use conservation of effective gravitational stress - energy to determine the motion . however , this use of distributional sources at second - order in perturbation theory results in infinities that must be `` regularized '' . although these regularization procedures are relatively natural - looking , the mathematical status of such a derivation is unclear . another approach @xcite is to postulate certain properties that equations of gravitational self - force should satisfy . this yields a mathematically clean derivation of the self - force corrected equations of motion . however , as the authors of @xcite emphasized , the motion of bodies in general relativity is fully described by einstein s equation together with the field equations of the matter sources , so no additional postulates should be needed to obtain an equation of motion , beyond the `` small body '' assumption and other such approximations . the analysis given by @xcite shows that equation follows from certain plausible assumptions . however , their derivation is thus only a plausibility argument for equation . similar remarks apply to a later derivation by poisson @xcite that uses a green s function decomposition developed by detweiler and whiting @xcite . a third approach , taken by mino , sasaki , and tanaka @xcite and later poisson @xcite building on previous work of burke @xcite , death @xcite , kates @xcite , thorne and hartle @xcite , and others involves the use of matched astymptotic expansions . here one assumes a metric form in the `` near zone''where the metric is assumed to be that of the body , with a small correction due to the background spacetime and in the `` far zone''where the metric is assumed to be that of the background spacetime , with a small correction due to the body . one then assumes that there is an overlap region of the body where both metric forms apply , and matches the expressions . the equations of motion of the body then arise from the matching conditions . however , as we shall indicate below , in addition to the `` lorenz gauge relaxation '' , there are a number of assumptions and steps in these derivations that have not been adequately justified . a more rigorous approach to deriving gravitational self - force is suggested by the work of geroch and jang @xcite and later geroch and ehlers @xcite on geodesic motion of small bodies ( see also @xcite ) . in @xcite , one considers a fixed spacetime background metric @xmath4 and considers a smooth one - parameter family of stress - energy smooth tensors @xmath25 that satisfy the dominant energy condition and have support on a world tube . as the parameter goes to zero , the world tube shrinks to a timelike curve . it is then proven that this timelike curve must be a geodesic . this result was generalized in @xcite to allow @xmath4 to also vary with @xmath2 so that einstein s equation is satisfied . within the framework of @xcite , it therefore should be possible to derive perturbative corrections to geodesic motion , including gravitational self - force . however , the conditions imposed in @xcite in effect require the mass of the body to go to zero faster than @xmath26 . consequently , in this approach , a self - force correction like to the motion of the body would arise at higher order than finite size effects and possibly other effects that would depend on the composition of the body . thus , while the work of @xcite provides a rigorous derivation of geodesic motion of a `` small body '' to lowest order , it is not a promising approach to derive gravitatational self - force corrections to geodesic motion . in this paper , we shall take an approach similar in spirit to that of @xcite , but we will consider a different smooth , one - parameter family of metrics @xmath0 , wherein , in effect , we have a body ( or black hole ) present that scales to zero size in a self - similar manner , with both the size and the mass of the body being proportional to @xmath2 . in the limit as @xmath27 , the body ( or black hole ) shrinks down to a worldline , @xmath8 . as in @xcite and @xcite , we prove that @xmath8 must be a geodesic of the `` background spacetime '' @xmath28although our method of proving this differs significantly from @xcite and @xcite . to first order in @xmath2 , the correction to the motion is described by a vector field , @xmath29 , on @xmath8 , which gives the `` infinitesimal displacement '' to the new worldline . we will show that , for any such one parameter family @xmath0 , in the lorenz gauge @xmath30 satisfies @xmath31 here @xmath32 and @xmath33 are , respectively , the mass and spin of the body . the terms in parentheses on the right side of this equation correspond exactly to the gravitational self - force term in eq . ; the first term is the papapetrou spin - force @xcite ; the second term is simply the usual right hand side of the geodesic deviation equation . equation is `` universal '' in the sense that it holds for any one - parameter family satisfying our assumptions , so it holds equally well for a ( sufficiently small ) blob of ordinary matter or a ( sufficiently small ) black hole . our derivation of is closely related to the matched asymptotic expansions analyses . however , our derivation is a rigorous , perturbative result . in addition , we eliminate a number of ad hoc , unjustified , and/or unnecssary assumptions made in previous approaches , including assumptions about the form of the `` body metric '' and its perturbations , assumptions about rate of change of these quantities with time , the imposition of certain gauge conditions , the imposition of boundary conditions at the body , and , most importantly , the step of lorenz gauge relaxation . furthermore , in our approach , the notion of a `` point particle '' is a concept that is _ derived _ rather than assumed . it also will be manifest in our approach that the results hold for all bodies ( or black holes ) and that the physical results do not depend on a choice of gauge ( although @xmath30 itself is a gauge dependent quantity , i.e. , the description of the corrections to particle motion depend on how the spacetimes at different @xmath2 are identified see section [ sec : dipole ] and appendix a ) . in particular , because the lorenz gauge plays no preferred role in our derivation ( aside from being a computationally convenient choice ) , our gauge transformation law is not , as in previous work @xcite , restricted to gauges continuously related to the lorenz gauge . our approach also holds out the possibility of being extended so as to systematically take higher order corrections into account . however , we shall not attempt to undertake such an extension in this paper . although holds rigorously as a first - order perturbative correction to geodesic motion , this equation would not be expected to provide a good global - in - time approximation to motion , since the small local corrections to the motion given by may accumulate with time , and eq . would not be expected to provide a good description of the perturbed motion when @xmath29 becomes large . we will argue in this paper that the misataquwa equations , eqs . and , should provide a much better global - in - time approximation to motion than eq . , and they therefore should be used for self - consistent calculations of the motion of a small body , such as for calculations of extreme - mass - ratio inspiral . we note in passing that , in contrast to einstein s equation , maxwell s equations are linear , and it makes perfectly good mathematical sense to consider distributional solutions to maxwell s equations with point particle sources . however , if the charge - current sources are not specified in advance but rather are determined by solving the matter equations of motion which are assumed to be such that the total stress - energy of the matter and electromagnetic field is conserved then the full , coupled system of maxwell s equations together with the equations of motion of the sources becomes nonlinear in the the electromagnetic field . point particle sources do not make mathematical sense in this context either . it is for this reason that despite more than a century of work on this problem there is no mathematically legitimate derivation of the electromagnetic self - force on a charged particle . the methods of this paper can be used to rigorously derive a perturbative expression for electromagnetic self - force by considering suitable one - parameter families of coupled electromagnetic - matter systems , and we shall carry out this analysis elsewhere @xcite . however , we shall restrict consideration in this paper to the gravitational case . this paper is organized as follows . in section [ sec : example ] , we motivate the kind of one - parameter family of metrics , @xmath34 , that we will consider by examining the one - parameter family of schwarzschild - desitter spacetimes with black hole mass equal to @xmath2 . one way of taking the limit as @xmath35 yields desitter spacetime . we refer to this way of taking the limit as the `` ordinary limit '' . but we show that if we take the limit as @xmath35 in another way and also rescale the metric by @xmath36 , we obtain schwarzschild spacetime . we refer to this second way of taking the limit as @xmath35 as the `` scaled limit '' . the scaled limit can be taken at any time @xmath37 on the worldline @xmath8 . the simultaneous existence of both types of limits defines the kind of one parameter family of metrics we seek , wherein a body ( or black hole ) is shrinking down to a world - line @xmath8 in an asymptotically self - similar manner . the precise , general conditions we impose on @xmath0 are formulated in section [ sec : example ] . some basic properties of @xmath0 that follow directly from our assumptions are given in section [ sec : assumptions ] . in particular , we obtain general `` far zone '' and `` near zone '' expansions and we show that , at any @xmath37 , the scaled limit always yields a stationary , asymptotically flat spacetime at @xmath38 . in section [ sec : geodesic ] , we prove that @xmath8 must be a geodesic of the `` background spacetime '' ( i.e. , the spacetime at @xmath38 resulting from taking the ordinary limit ) . in other words , to zeroth order in @xmath2 a small body or black hole always moves on a geodesic . we also show that , to first order in @xmath2 , the metric perturbation associated with such a body or black hole corresponds to that of a structureless `` point particle '' . in section [ sec : dipole ] , we define the motion of the body ( or black hole ) to first order in @xmath2 by finding a coordinate displacement that makes the mass dipole moment of the stationary , asymptotically flat spacetime appearing in the scaled limit vanish . ( this can be interpreted as a displacement to the `` center of mass '' of the body . ) in section [ sec : calculation ] , we then derive eq . as the first order in @xmath2 correction to @xmath8 in the lorenz gauge . ( an appendix provides the transformation to other gauges . ) finally , in section [ sec : beyond ] we explain the status of the misataquwa equation . our spacetime conventions are those of wald @xcite , and we work in units where @xmath39 . lower case latin indices early in the alphabet ( @xmath40 ) will denote abstract spacetime indices ; greek indices will denote coordinate components of these tensors ; latin indices from mid - alphabet ( @xmath41 ) will denote spatial coordinate components . as discussed in the introduction , we seek a one - parameter family of metrics @xmath42 that describes a material body or black hole that `` shrinks down to zero size '' in an asymptotically self - similar manner . in order to motivate the general conditions on @xmath42 that we shall impose , we consider here an extremely simple example of the type of one - parameter family that we seek . this example will provide an illustration of the two types of limits that we shall use to characterize @xmath42 . our example is built from schwarzschild - desitter spacetime , @xmath43 ( this metric , of course , is a solution to the vacuum einstein s equation with a cosmological constant @xmath44 rather than a solution with @xmath45 , but the field equations will not play any role in the considerations of this section ; we prefer to use this example because of its simplicity and familiarity . ) the desired one - parameter family is @xmath46 where we consider only the portion of the spacetime with @xmath47 for some @xmath48 . for each @xmath2 , this spacetime describes the exterior gravitational field of a spherical body or black hole of mass @xmath49 in an asymptotically desitter spacetime . as @xmath35 , the body / black hole shrinks to zero size and mass . for @xmath38 , the spacetime is desitter spacetime , which extends smoothly to the worldline @xmath50 , corresponding to where the body / black hole was just before it `` disappeared '' . as explained clearly in @xcite , the limit of a one - parameter family of metrics @xmath42 depends on how the spacetime manifolds at different values of @xmath2 are identified . this identification of spacetime manifolds at different @xmath2 can be specified by choosing coordinates @xmath51 for each @xmath2 and identifying points labeled by the same value of the coordinates @xmath51 . if we use the coordinates @xmath52 in which the one - parameter family of metrics was presented to do the identification , then it is obvious that the limit as @xmath35 of @xmath42 is the desitter metric @xmath53 this corresponds to the view that the body / black hole shrinks to zero size and mass and `` disappears '' as @xmath35 . however , there is another way of taking the limit of @xmath42 as @xmath35 ; the existence of this second limit is one of the key ideas in this paper . choose an arbitrary time @xmath37 and , for @xmath54 , introduce new time and radial coordinates by @xmath55 and @xmath56 . in the new coordinates , the one - parameter family of metrics becomes @xmath57 we now consider the limit as @xmath35 by identifying the spacetimes with different @xmath2 at the same values of the barred coordinates . it is clear by inspection of eq.([gbar ] ) that the the limit of @xmath42 as @xmath35 at fixed @xmath58 exists , but is zero . in essence , the spacetime interval between any two events labeled by @xmath59 and @xmath60 goes to zero as @xmath35 because these events are converging to the same point on @xmath8 . thus , this limit of @xmath42 exists but is not very interesting . however , an interesting limit can be taken by considering a new one - parameter family of metrics @xmath61 by @xmath62 so that @xmath63 by inspection , the limit of this family of metrics is simply , @xmath64 which is just the schwarzschild metric with mass @xmath65 . the meaning of this second limit can be understood as follows . as already discussed above , the one - parameter family of metrics ( [ eq : family ] ) describes the exterior gravitational field of a spherical body or black hole that shrinks to zero size and mass as @xmath35 . the second limit corresponds to how this family of spacetimes looks to the family of observers placed at the events labeled by fixed values of @xmath58 . in going from , say , the @xmath66 to the @xmath67 spacetime , each observer will see that the body / black hole has shrunk in size and mass by a factor of @xmath68 and each observer also will find himself `` closer to the origin '' by this same factor of @xmath68 . suppose now that this family of observers also `` changes units '' by this same factor of @xmath68 , i.e. , they use centimeters rather than meters to measure distances . then , except for small effects due to the desitter background , the family of observers for the @xmath67 spacetime will report the same results ( expressed in centimeters ) as the observers for the @xmath66 spacetime had reported ( in meters ) . in the limit as @xmath35 , these observers simply see a schwarzschild black hole of mass @xmath65 , since the effects of the desitter background on what these observers will report disappear entirely in this limit . we will refer to the first type of limit ( i.e. , the limit as @xmath35 of @xmath42 taken at fixed @xmath51 ) as the _ ordinary limit _ of @xmath42 , and we will refer to the second limit ( i.e. , the limit as @xmath35 of @xmath69 taken at fixed @xmath58 ) as the _ scaled limit _ of @xmath42 . the simultaneous existence of both types of the above limits contains a great deal of relevant information about the one - parameter family of spacetimes ( [ eq : family ] ) . in essence , the existence of the first type of limit is telling us that the body / black hole is shrinking down to a worldline @xmath8 , with its mass going to zero ( at least ) as rapidly as its radius . the existence of the second type of limit is telling us that the body / black hole is shrinking to zero size in an asymptotically self - similar manner : in particular , its mass is proportional to its size , its shape is not changing , and it is not undergoing any ( high frequency ) oscillations in time . figure [ fig:2limits - spacetime ] illustrates the two limits we consider . , the body shrinks and finally disappears , leaving behind a smooth background spacetime with a preferred world - line , @xmath8 , picked out . the solid lines illustrate this `` ordinary limit '' of @xmath35 at fixed @xmath70 , which is taken along paths that terminate away from @xmath8 ( i.e. , @xmath71 ) . by contrast , the `` scaled limit '' as @xmath35 , shown in dashed lines , is taken along paths at fixed @xmath72 that converge to a point on @xmath8.,width=288 ] we wish now to abstract from the above example the general conditions to impose upon one - parameter families that express in a simple and precise way the condition that we have an arbitrary body ( or black hole ) that is shrinking to zero size in an asymptotically self - similar way in an arbitrary background spacetime . most of the remainder of this paper will be devoted to obtaining `` equations of motion '' for these bodies that are accurate enough to include gravitational self - force effects . a first attempt at specifying the type of one - parameter families @xmath42 that we seek is as follows : * \(i ) _ existence of the `` ordinary limit '' _ : @xmath42 is such that there exists coordinates @xmath73 such that @xmath74 is jointly smooth in @xmath75 , at least for @xmath76 for some constant @xmath77 , where @xmath78 ( @xmath79 ) . for all @xmath2 and for @xmath76 , @xmath42 is a vacuum solution of einstein s equation . furthermore , @xmath80 is smooth in @xmath73 , including at @xmath81 , and , for @xmath82 , the curve @xmath8 defined by @xmath81 is timelike . * \(ii ) _ existence of the `` scaled limit '' _ : for each @xmath37 , we define @xmath55 , @xmath83 . then the metric @xmath84 is jointly smooth in @xmath85 for @xmath86 . here we have used the notation @xmath87 to denote the components of @xmath4 in the @xmath88 coordinates , whereas @xmath89 denotes the components of @xmath90 in the @xmath88 coordinates . it should be noted that , since the barred coordinates differ only by scale ( and shift of time origin ) from the corresponding unbarred coordinates , we have @xmath91 consequently , we have @xmath92 since there is a cancelation of the factors of @xmath2 resulting from the definition of @xmath90 and the coordinate rescalings . it also should be noted that there is a redundancy in our description of the one - parameter family of metrics when taking the scaled limit : we define a scaled limit for all values of @xmath37 , but these arise from a single one - parameter family of metrics @xmath42 . indeed , it is not difficult to see that we have @xmath93 in fact , our requirements on @xmath4 of the existence of both an `` ordinary limit '' and a `` scaled limit '' are not quite strong enough to properly specify the one - parameter families we seek . to explain this and obtain the desired strengthened condition , it is convenient to define the new variables @xmath94 where the range of @xmath95 is @xmath96 . let @xmath97 denote a component of @xmath42 in the coordinates @xmath73 . however , instead of viewing @xmath97 as a function of @xmath98 , we view @xmath97 as a function of @xmath99 , where @xmath100 and @xmath101 are defined in terms of @xmath102 by the usual formula for spherical polar angles . in terms of these new variables , taking the `` ordinary limit '' corresponds to letting @xmath103 at any fixed @xmath104 , whereas taking the `` scaled limit '' corresponds to letting @xmath105 at any fixed @xmath106 ( see figure [ fig:2limits - ab ] ) . now , our assumptions concerning the ordinary limit imply that , at fixed @xmath107 and at fixed @xmath108 , @xmath97 depends smoothly on @xmath95 , including at @xmath109 . on the other hand , our assumptions concerning the scaled limit imply that at fixed @xmath107 and at fixed @xmath106 , @xmath97 is smooth in @xmath110 . furthermore , the last condition in the ordinary limit implies that for @xmath109 and fixed @xmath107 , @xmath97 is smooth in @xmath110 , including at @xmath111 . thus , at fixed @xmath107 , our previously stated assumptions imply that @xmath97 is well defined at the `` origin '' @xmath112 , and is smooth in @xmath110 along the @xmath110-axis ( i.e. , @xmath109 ) . however , our previously stated assumptions do not say anything about the continuity or smoothness of @xmath97 as @xmath113 from directions other than along the @xmath110-axis . such limiting behavior would correspond to letting @xmath114 as @xmath35 but at a rate _ slower _ than @xmath2 , i.e. , such that @xmath115 . to see the meaning and relevance of this limiting behavior , let us return to our original motivating example , eq . ( [ eq : family ] ) and take @xmath97 to be the time - time component of this metric to `` cartesian - like '' coordinates @xmath102 that are well behaved at the origin @xmath116 when @xmath38 ] . in terms of our new variables ( [ alphabeta ] ) , we have @xmath118 which is jointly smooth in @xmath119 at @xmath120 . by contrast , suppose we had a one - parameter family of metrics @xmath121 that satisfies our above assumptions about the ordinary and scaled limits , but fails to be jointly smooth in @xmath119 at @xmath120 . for example , suppose the time - time component of such a one - parameter family varied as @xmath122 in terms of the original variables @xmath123 , this corresponds to behavior of the form @xmath124 if we take the `` ordinary limit '' ( @xmath35 at fixed @xmath71 ) of @xmath125 , we find that @xmath125 smoothly goes to @xmath126 . similarly , if we take the `` scaled limit '' ( @xmath127 at fixed @xmath128 ) , we also find that @xmath125 smoothly goes to @xmath126 . however , suppose we let @xmath35 but let @xmath70 go to zero as @xmath129 . then @xmath125 will approach a different limit , namely @xmath130 . in essence , @xmath131 corresponds to a one - parameter family in which there is a `` bump of curvature '' at @xmath132 . although this `` bump of curvature '' does not register when one takes the ordinary or scaled limits , it is present in the one - parameter family of spacetimes and represents unacceptable limiting behavior as @xmath35 of this one - parameter family . in order to eliminate this kind of non - uniform behavior in @xmath2 and @xmath70 , we now impose the following addtional condition : * \(iii ) _ uniformity condition _ : each component of @xmath42 in the coordinates @xmath51 is a jointly smooth function of the variables @xmath133 at @xmath120 ( at fixed @xmath134 ) , where @xmath110 and @xmath95 are defined by eq.([alphabeta ] ) . assumptions ( i)-(iii ) constitute all of the conditions that we shall impose on @xmath42 . no additional assumptions will be made in this paper . we note that the coordinate freedom allowed by our conditions are precisely all coordinate transformations @xmath135 such that @xmath136 is jointly smooth in @xmath137 for all @xmath138 for some constant @xmath139 , and such that the jacobian matrix @xmath140 is jointly smooth in @xmath119 at @xmath120 at fixed @xmath134 . it should be emphasized that our assumtions place absolutely no restrictions on the one - parameter family of spacetimes for @xmath141 , i.e. , this portion of these spacetimes could equally well be `` filled in '' with ordinary matter or a black hole satisfying our assumptions . ] . it also should be noted that the `` large @xmath70 '' region of the spacetime will not be relevant to any of our considerations , so it is only necessary that our conditions hold for @xmath142 for some constant @xmath143 . finally , although it may not be obvious upon first reading , we note that our assumptions concerning @xmath0 are closely related to the assumptions made in matched asymptotic expansion analyses . as we shall see in the next section , in essence , our assumption about the existence of an ordinary limit of @xmath0 corresponds to assuming the existence of a `` far zone '' expansion ; our assumption about the existence of a scaled limit of @xmath0 corresponds to assuming the existence of a `` near zone '' expansion ; and our uniformity assumption corresponds closely to the assumption of the existence of a `` buffer zone '' where both expansions are valid . in this section , we derive some immediate consequences of the assumptions of the previous section that will play a key role in our analysis . these results will follow directly from the `` uniformity condition '' and the consistency relation ( [ consist ] ) . since , by the uniformity assumption , the coordinate components of the one - parameter family of metrics @xmath42 are jointly smooth in the variables @xmath119 at @xmath120 , we may approximate @xmath144 by a joint taylor series in @xmath110 and @xmath95 to any finite orders @xmath145 and @xmath32 by @xmath146 substituting for @xmath110 and @xmath95 from eq.([alphabeta ] ) , we have @xmath147 where here and in the following , we drop the error term . we can rewrite this equation as a perturbation expansion in @xmath2 : @xmath148 we will refer to eq.([eq : rseries ] ) as the _ far zone expansion _ of @xmath42 . it should be noted that the @xmath149th - order term in @xmath2 in the far zone perturbation expansion has leading order behavior of @xmath150 at small @xmath70 . however , arbitrarily high positive powers of @xmath70 may occur at each order in @xmath2 . finally , we note that the angular dependence of @xmath151 is further restricted by the requirement that the metric components @xmath152 must be smooth at @xmath50 when re - expressed as functions of @xmath102 . in particular , this implies that @xmath153 can not have any angular dependence . equivalently , in view of eq.([barg ] ) , we can expand @xmath89 as @xmath154 by further expanding @xmath155 in @xmath156 about @xmath157 , we obtain @xmath158 where @xmath159 we can rewrite this as a perturbation series expansion in @xmath2 : @xmath160 we will refer to eq.([eq : fullnearzone ] ) as the _ near zone expansion _ of @xmath42 . we see from this formula that the scaled metric , viewed as a perturbation series in @xmath2 , follows the rule that the combined powers of @xmath156 and @xmath72 are allowed to be only as positive as the order in perturbation theory . by contrast inverse powers of @xmath72 of arbitrarily high order are always allowed . of course , only non - negative powers of @xmath156 can occur . by inspection of eq.([eq : fullnearzone ] ) , we see that the `` background '' ( @xmath82 ) scaled metric is given by @xmath161 where we have used the fact that @xmath162 . thus , we see that there is no dependence of @xmath163 on @xmath156 and only non - positive powers of @xmath72 occur . _ thus , we see that @xmath164 is a stationary , asymptotically flat spacetime . _ however , the limiting , stationary , asymtotically flat spacetime that we obtain may depend on the choice of the time , @xmath37 , on the worldline , @xmath8 , about which the scaling is done . our `` far zone expansion '' , eq.([eq : rseries ] ) , appears to correspond closely to the far zone expansion used in matched asymptotic expansion analyses @xcite . however , our `` near zone expansion '' differs in that we define a separate expansion for each time @xmath37 rather than attempting a uniform in time approximation with a single expansion . such expansions require an additional `` quasi - static '' or slow - time variation assumption for the evolution of the metric perturbations . a further difference is that the conclusion that the background ( @xmath38 ) metric is stationary and asymptotically flat has been derived here rather than assumed . indeed , in other analyses , a particular form of the background metric ( such as the schwarzschild metric ) is assumed , and the possibility that this metric form might change with time ( i.e. , depend upon @xmath37 ) is not considered . in addition , in other analyses boundary conditions at small @xmath72 ( such as regularity on the schwarzschild horizon ) are imposed . in our analysis , we make no assumptions other than the assumptions ( i)-(iii ) stated in the previous section . in particular , since we make no assumptions about the form of the metric for @xmath165 , we do not impose any boundary conditions at small @xmath72 . finally , it is also useful to express the consistency relation ( [ consist ] ) in a simple , differential form . we define @xmath166 then , a short calculation shows that @xmath167 as well as @xmath168 setting @xmath169 , we see that the last relation implies that @xmath170 is stationary , as we have already noted . in this section , we will prove that the worldline @xmath8 appearing in assumption ( i ) of section [ sec : example ] must , in fact , be a geodesic of the background metric @xmath171 . this can be interpreted as establishing that , to zeroth order in @xmath2 , any body ( or black hole ) moves on a geodesic of the background spacetime . in fact , we will show considerably more than this : we will show that , to first order in @xmath2 , the far zone description of @xmath42 is that of a `` point particle '' . as previously mentioned in the introduction , our derivation of geodesic motion is similar in spirit to that of @xcite in that we consider one - parameter families of solutions with a worldtube that shrinks down to a curve @xmath8 , but the nature of the one - parameter families that we consider here are very different from those considered by @xcite , and our proof of geodesic motion is very different as well . our derivation of geodesic motion also appears to differ significantly from pervious derivations using matched asymptotic expansions @xcite . we begin by writing the lowest order terms in the far zone expansion , eq . ( [ eq : rseries ] ) , as follows : @xmath172 + o(\lambda^2)\ , , r>0 \,\ , , \end{aligned}\ ] ] where we have used the fact that @xmath173 can depend only on @xmath174 , as noted in the previous section . since the worldline @xmath8 , given by @xmath116 , was assumed to be timelike however , if , instead , we had assumed that the `` interior region '' @xmath175 , were `` filled in '' with matter satisfying the dominant energy condition , then it should be possible to prove that @xmath8 must be timelike . ] in the spacetime @xmath176 , without loss of generality , we may choose our coordinates @xmath73 so that @xmath177 . ( one such possible choice of coordinates would be fermi normal coordinates with respect to @xmath8 in the metric @xmath176 . we emphasize that we make the coordinate choice @xmath178 purely for convenience so that , e.g. , coordinate time coincides with proper time on @xmath8but it plays no essential role in our arguments . ) choosing these coordinates , and letting @xmath179 denote the @xmath180 piece of the metric , we see that @xmath181 takes the form @xmath182 where @xmath183 where in eq.([eq : perturbations - series ] ) , the term `` @xmath184 '' denotes a term that , when multiplied by @xmath185 for any @xmath186 , vanishes as @xmath187 . now , by assumption ( i ) of section [ sec : example ] , for each @xmath2 , @xmath0 is a vacuum solution of einstein s equation for @xmath188 and is jointly smooth in @xmath98 in this coordinate range . it follows that for all @xmath189 , @xmath3 is a solution the linearized einstein equation off of @xmath1 , i.e. , @xmath190 = -\frac{1}{2 } \nabla_a \nabla_b h^c_{\ c } - \frac{1}{2}\nabla^c\nabla_c h_{ab } + \nabla^c \nabla_{(b}h_{a)c } = 0 , \,\ , r>0 \,\ , , \label{lee}\ ] ] where here and in the following , @xmath191 denotes the derivative operator associated with @xmath1 , and indices are raised and lowered with @xmath1 . equation ( [ lee ] ) holds only for @xmath189 , and , indeed , if @xmath192 , @xmath3 is singular at @xmath193 . however , even if @xmath192 , the singularity of each component of @xmath3 is locally integrable with respect to the volume element associated with @xmath171 , i.e. , each component , @xmath179 , is a locally @xmath194 function on the entire spacetime manifold , including @xmath50 . _ thus , @xmath3 is well defined as a distribution on all of spacetime . _ the quantity @xmath195/8 \pi$ ] is therefore automatically well defined as a distribution . this quantity has the interpretation of being the `` source '' for the metric perturbation ( [ eq : perturbations - series])even though all of our spacetimes @xmath42 for @xmath54 have excluded the `` source region '' @xmath196 . it follows immediately from eq.([lee ] ) that , as a distribution , @xmath197 must have support on @xmath8 in the sense that it must vanish when acting on any test tensor field @xmath198 whose support does not intersect @xmath8 . we now compute @xmath197 . by definition , @xmath195/8 \pi$ ] is the distribution on spacetime whose action on an arbitrary smooth , compact support , symmetric tensor field @xmath199 is given by @xmath200 h^{ab } \sqrt{-g } d^4x = 0 \,\ , , \ ] ] where @xmath201 denotes the volume element associated with @xmath171 and we have used the fact that the operator @xmath202 is self - adjoint were not self - adjoint , then the adjoint of @xmath202 would have appeared in eq . ( [ eq : distributional - equation ] ) . ] . note that the right side of this equation is well defined since @xmath203 $ ] is a smooth tensor field of compact support and @xmath204 is locally @xmath194 . we can evaluate the right side of eq . ( [ eq : distributional - equation ] ) by integrating over the region @xmath205 and then taking the limit as @xmath206 . in the region @xmath207 , @xmath204 is smooth , and a straightforward `` integration by parts '' calculation shows that @xmath208 h^{ab } - g^{(1)}_{ab } [ h_{cd } ] f^{ab } = \nabla_c s^c \,\ , , \ ] ] where @xmath209 where @xmath210 . since @xmath211 = 0 $ ] for @xmath189 , it follows immediately that @xmath212 h^{ab } = \frac{1}{8 \pi } \lim_{\epsilon \rightarrow 0 } \int_{r=\epsilon } s^a n_a ds \,\ , .\ ] ] using eqs . ( [ eq : perturbations - series ] ) and ( [ eq : adjoint - current ] ) , we find that @xmath213 takes the form @xmath214 where @xmath215 is a smooth , symmetric ( @xmath216 ) tensor field on @xmath8 whose components are given in terms of suitable angular averages of @xmath217 and its first angular derivatives . in other words , the distribution @xmath197 is given by on @xmath8 and @xmath218 , but we prefer to leave in these factors so that this formula holds for an arbitrary choice of coordinates . ] @xmath219 where @xmath220 is the `` coordinate delta - function '' ( i.e. , @xmath221 ) . we now use the fact that , since the differential operator @xmath202 satisfies the linearized bianchi identity @xmath222 , the distribution @xmath197 must satisfy @xmath223 in the distributional sense . this means that the action of @xmath197 must vanish on any test tensor field of the form @xmath224 where @xmath225 is smooth and of compact support . in other words , by eq . ( [ tn ] ) , the tensor field @xmath226 on @xmath8 must be such that for an arbitrary smooth vector field @xmath227 on spacetime , we have @xmath228 now for any @xmath79 , choose @xmath227 to have components of the form @xmath229 , where each @xmath230 ( @xmath231 ) is an arbitrary smooth function of compact support in @xmath174 and @xmath232 is a smooth function of compact spatial support , with @xmath233 in a neighborhood of @xmath8 . then eq . ( [ nf ] ) yields @xmath234 for all @xmath235 , which immediately implies that @xmath236 for all @xmath79 and all @xmath231 . in other words , we have shown that @xmath215 must take the form @xmath237 where @xmath7 denotes the unit tangent to @xmath8 , i.e. , @xmath7 is the 4-velocity of @xmath8 . now choose @xmath227 to be an arbitrary smooth vector field of compact support . then eqs . ( [ nf ] ) and ( [ n ] ) yield @xmath238 where we integrated by parts in @xmath174 to obtain the last equality . since @xmath227 is arbitrary , this immediately implies that @xmath239 this , in turn , implies that @xmath240 i.e. , @xmath32 is a constant along @xmath8 , and , if @xmath241 , @xmath242 i.e. , in the case where @xmath241 , @xmath8 is a geodesic of @xmath176 , as we desired to show .. it is critical that this assumption be used in any valid derivation of geodesic motion , since a derivation that holds for @xmath243 effectively would show that all curves are geodesics . ] in summary , we have shown that for any one - parameter family of metrics @xmath42 satisfying assumptions ( i)-(iii ) of section [ sec : example ] , to first order in @xmath2 , the far zone metric perturbation @xmath3 corresponds to a solution to the linearized einstein equation with a point particle source @xmath244 where @xmath32 is a constant and @xmath7 is the 4-velocity of @xmath8 , which must be a geodesic if @xmath241 . we refer to @xmath32 as the _ mass _ of the particle . it is rather remarkable that the point particle source ( [ ptparticle ] ) is an _ output _ of our analysis rather than an input . indeed , we maintain that the result we have just derived is what provides the justification for the notion of `` point particles''a notion that has played a dominant role in classical physics for more than three centuries . in fact , the notion of point particles makes no mathematical sense in the context of nonlinear field theories like general relativity . nevertheless , we have just shown that the notion of a ( structureless ) `` point particle '' arises naturally as an _ approximate _ description of sufficiently small bodies namely , a description that is valid to first order in @xmath2 in the far zone for arbitrary one - parameter families of metrics @xmath42 satisfying the assumptions of section [ sec : example ] . this description is valid independently of the nature of the `` body '' , e.g. , it holds with equal validity for a small black hole as for a small blob of ordinary matter . in the previous section , we established that , to zeroth order in @xmath2 , any body ( or black hole ) of nonvanishing mass moves on a geodesic of the background spacetime . much of the remainder of this paper will be devoted to finding the corrections to this motion , valid to first order in @xmath2 in the far zone . in this section , we address the issue of what is meant by the `` motion of the body '' to first order in @xmath2 . the first point that should be clearly recognized is that it is far from obvious how to describe `` motion '' in terms of a worldline for @xmath54 . indeed , the metric @xmath42 is defined only for @xmath188 , so at finite @xmath2 the spacetime metric may not even be defined in a neighborhood of @xmath8 . if we were to assume that @xmath245 and that the region @xmath246 were `` filled in '' with sufficiently `` weak field matter''so that @xmath247 , where @xmath248 denotes the supremum of the components of the riemann curvature tensor of @xmath61 in the `` filled in '' region then it should be possible to define a `` center of mass '' worldline at finite @xmath2 , and we could use this worldline to characterize the motion of the body @xcite . however , we do not wish to make any `` weak field '' assumptions here , since we wish to describe to motion of small black holes and other `` strong field '' objects . since it is not clear how to associate a worldline to the body at finite @xmath2and , indeed , the `` body '' is excluded from the spacetime region we consider at finite @xmath2it is not clear what one would mean by a `` perturbative correction '' to @xmath8 to first or higher order in @xmath2 . a second point that should be understood is that if we have succeeded in defining the worldlines describing the motion the body at finite @xmath2 , @xmath249 then the `` first order in @xmath2 perturbative correction '' , @xmath29 , to the zeroth order motion @xmath8 ( given by @xmath250 ) is most properly viewed as the spatial components of a vector field , @xmath251 , defined along @xmath8 . this vector field describes the `` infinitesimal displacement '' to the corrected motion to first order in @xmath2 . the time component , @xmath252 , of @xmath251 depends on on how we identify the time parameter of the worldlines at different values of @xmath2 and is not physically relevant ; we will set @xmath253 so that @xmath251 is orthogonal to the tangent , @xmath7 , to @xmath8 in the background metric @xmath176 . thus , when we seek equations of motion describing the first order perturbative corrections to geodesic motion , we are seeking equations satisfied by the vector field @xmath254 on @xmath8 . a third point that should be clearly recognized is that @xmath251 and any equations of motion satisfied by @xmath251 will depend on our choice of gauge for @xmath3 . to see this explicitly , suppose that we perform a smooth is not smooth at @xmath116 are also permitted under the coordinate freedom stated at the end of section [ sec : example ] . however , it suffices to consider smooth @xmath255 for our consderations here . the change in the description of motion under non - smooth gauge transformations will be treated in appendix a. ] gauge transformation of the form @xmath256 under this gauge transformation , we have @xmath257 however , clearly , the new description of motion will be of the form @xcite @xmath258 where @xmath259 thus , we see that @xmath251 transforms as @xmath260 in order that it describe the same perturbed motion . thus , the first order correction , @xmath254 , to the background geodesic motion contains no meaningful information by itself and , indeed , it can always be set to zero by a smooth gauge transformation . only the pair @xmath261 has gauge invariant meaning . we turn now to the definition of the first order perturbed motion . our definition relies on the fact , proven in section [ sec : assumptions ] above , that for each @xmath37 , @xmath262 is a stationary , asymptotically flat spacetime . therefore , @xmath164 has well defined sets of mass ( `` electric parity '' ) and angular momentum ( `` magnetic parity '' ) multipole moments @xcite and , indeed , the spacetime is characterized by the values of these two sets of multipole moments @xcite . the multipole moments ( other than the lowest nonvanishing multipoles of each type ) depend upon a choice of conformal factor @xcite , which , rougly speaking , corresponds to a choice of `` origin '' . we choose the conformal factor @xmath263 to define all of the multipoles , corresponding to choosing the origin at @xmath264 . for a metric of the form eq . ( [ eq : fullnearzone2])with @xmath265 by our coordinate choice imposed in the previous section that @xmath266 on @xmath8the mass will be simply the @xmath267 part of the coefficient of @xmath268 in the large @xmath72 expansion of @xmath269 . similarly , the mass dipole moment will be the coefficient of the @xmath270 part of this quantity at order @xmath271 . it is well known that if the mass is nonzero , the mass dipole moment is `` pure gauge '' and can be set to zero by choice of conformal factor/``origin '' . we now explicitly show that , with our choice of conformal factor @xmath263 , the mass dipole moment can be set to zero by a smooth gauge transformation of the form ( [ eq : transform ] ) . it follows from the linearized einstein equation ( [ lee ] ) with source ( [ ptparticle ] ) applied to @xmath3 , eq . ( [ eq : perturbations - series ] ) , that the time - time component of @xmath3 takes the form @xmath272 i.e. , in the notation of eq . ( [ eq : rseries ] ) , we have @xmath273 . comparing with eq . ( [ eq : fullnearzone ] ) ( and also using the fact that @xmath274 ) , we see that at each @xmath37 @xmath275 from this equation , we see that the `` particle mass '' , @xmath32 , of the `` source '' of the far zone metric perturbation ( see eq . ( [ ptparticle ] ) ) is also the komar / adm mass of the stationary , asymptotically flat spacetime @xmath276 . we now calculate the effect of the coordinate transformation ( [ eq : transform ] ) on @xmath277 . the transformation ( [ eq : transform ] ) corresponds to changing the barred coordinates by @xmath278 i.e. , to zeroth order in @xmath2 , it corresponds to a `` constant displacement '' of coordinates . since @xmath279 it can be seen that this transformation have the effect of changing the mass dipole moment by @xmath280 . in particular , we can always choose @xmath281 so as to set the mass dipole moment to zero . now , the `` near zone '' coordinates @xmath282 for which the mass dipole moment vanishes have the interpretation of being `` body centered '' coordinates to zeroth order in @xmath2 . the origin @xmath283 of the corresponding `` far zone '' coordinates @xmath284 therefore has the interpretation of representing the `` position '' of the center of mass of the body to first order in @xmath2 . we shall use this to define the correction to geodesic motion to first order in @xmath2 by proceeding as follows : first , we shall choose our coordinates , @xmath51 , to zeroth order in @xmath2 by choosing convenient coordinates for the `` background spacetime '' @xmath176 . ( we will use fermi normal coordinates based on @xmath8 . ) next , we will define our coordinates , @xmath51 , to first order in @xmath2 by choosing a convenient gauge for @xmath3 , eq . ( [ eq : perturbations - series ] ) . ( we will choose the lorenz gauge @xmath285 . ) then we will introduce the smooth coordinate transformation ( [ eq : transform ] ) , and impose the requirement that @xmath286 be such that the mass dipole moment of @xmath276 vanish for all @xmath37 . since the `` location '' of the body in the new coordinates is @xmath287 , the first order perturbative correction @xmath254 to the motion of the body in our original coordinates @xmath51 will be given by @xmath288 of course , the particular @xmath254 that we obtain in any given case will depend upon the particular one - parameter family @xmath42 that we consider . what is of interest is any `` universal relations '' satisfied by @xmath254 that are independent of the choice of one - parameter family satisfying assumptions ( i)-(iii ) of section [ sec : example ] . such universal relations would provide us with `` laws of motion '' for point particles that take self - force effects into account . in the next section , we will show ( via a lengthy calculation ) that such a universal relation exists for @xmath289 , thus providing us with general `` equations of motion '' for all `` point particles '' , valid to first order in @xmath2 . finally , we note that if we wish to describe motion beyond first order in @xmath2 , it will be necessary to define a `` representative world line '' in the far zone to at least second order in @xmath2 . we shall not attempt to do so in this paper . the definition of a suitable representative worldline is probably the greatest obstacle to extending the results of this paper to higher order in perturbation theory . in the section [ sec : geodesic ] we found that first - order far zone perturbations of the background spacetime @xmath176 are sourced by a point particle stress - energy , eq . ( [ ptparticle ] ) . for the remainder of this paper , we will assume that @xmath241 , so that , as shown in section [ sec : geodesic ] , the lowest order motion is described by a geodesic , @xmath8 , of the background spacetime . we will need expressions for the components of the far zone metric , @xmath290 , its first order perturbation , @xmath291 , and its second order perturbation @xmath292 . it is convenient to choose our coordinates @xmath51 to zeroth order in @xmath2 to be fermi normal coordinates with respect to the background geodesic @xmath8 , and choose these coordinates to first order in @xmath2 so that @xmath293 satisfies the lorenz gauge condition @xmath294 , where @xmath295 with @xmath296 . then the linearized einstein equation reads @xmath297 this system of equations can be solved using the hadamard expansion techniques of dewitt and brehme @xcite . since this technology has been used in all previous derivations of gravitational self - force , we do not review it here but simply present results . equation ( 2.27 ) of mino , sasaki , and tanaka @xcite provides a covariant expression for the perturbations in terms of parallel propagators and synge s world function on the background metric ( see , e.g. , reference @xcite for definitions of these quantities ) . the fermi normal coordinate components of these tensors are easily calculated with the aid of expressions from section 8 of poisson @xcite . combining this with the form of the background metric in fermi normal coordinates , we obtain @xmath298 where the quantities @xmath299 and @xmath300 are defined by the following expressions in terms of the fermi normal coordinate components of the riemann tensor of the background metric @xmath301 and @xmath302 and @xmath303 are given by @xmath304 in these expressions , @xmath305 is the lorenz gauge retarded green s function , normalized with a factor of @xmath306 , following @xcite . as previously mentioned , the symbol @xmath23 indicates that the range of the integral extends just short of the retarded time @xmath24 , so that only the `` tail '' ( i.e. , interior of the light cone ) portion of the green s function is used ( see , e.g. , reference @xcite for details ) . we define @xmath303 , rather than working with derivatives of @xmath302 , because @xmath302 is not differentiable on the worldline . ( however , this non - differentiability is limited only to spatial derivatives of spatial components of @xmath302 on the worldline @xmath8 yields @xmath303 plus the coincidence limit of the integrand of , which is proportional to @xmath307 times the gradient of @xmath24 . ] , so that expressions expressions like are well - defined . ) a choice of retarded solution ( corresponding to `` no incoming radiation '' ) was made in writing these equations . this choice is not necessary , and one could add an arbitrary smooth solution @xmath179 of the linearized einstein equation to the first order in @xmath2 term on the right side of eq . ( [ eq : metric ] ) , which could then be carried through all of our calculations straightforwardly . however , for simplicity , we will not consider the addition of such a term . our derivation of gravitational self - force to first order in @xmath2 will require consideration of second - order metric perturbations , so we will have to carry the expansion of @xmath42 somewhat beyond eq . ( [ eq : metric ] ) . ( this should not be surprising in view of fact that our above derivation in section [ sec : geodesic ] of geodesic motion at zeroth order in @xmath2 , required consideration of first - order metric perturbations . ) in particular , we will need an explicit expression for the quantity @xmath308 appearing in the far zone expansion eq . ( [ eq : rseries ] ) , i.e. , the term of order @xmath26 that has the most singular behavior in @xmath309 ( namely , @xmath310 ) . the second - order perturbation @xmath311 satisfies the second - order einstein equation , which takes the form @xmath312 = - g^{(2)}_{ab}[h , h]\ ] ] where @xmath313 denotes the second order einstein tensor about the background metric @xmath314 . since the @xmath315 part of @xmath3 corresponds to the linearized schwarzschild metric in isotropic coordinates ( see eq . ( [ eq : metric ] ) ) , it is clear that there is a particular solution to eq . ( [ eq : e2 ] ) of the form @xmath316 as @xmath187 , where @xmath317 and @xmath318 , whereas @xmath319 . ( the explicitly written term on the right side of eq . ( [ ji ] ) is just the @xmath320 part of the schwarzschild metric in isotropic coordinates . ) the general solution to eq . ( [ eq : e2 ] ) can then be written as @xmath321 where @xmath322 is a homogeneous solution of the linearized einstein equation . we wish to compute the @xmath323 part of @xmath322 , i.e. , writing @xmath324 we wish to compute @xmath325 . now , although the equations of motion to first order in @xmath2 depend upon a choice of gauge to first order in @xmath2 ( see section [ sec : dipole ] ) , they can not depend upon a choice of gauge to second order in @xmath2 , since a second order gauge transformation can not affect the mass dipole moment of the background scaled metric @xmath326 . ( we have also verified by a direct , lengthy computation that second order gauge transformations do not produce changes in the equations of motion to first order in @xmath2 . ) therefore , we are free to impose any ( admissible ) second order gauge condition on @xmath322 . it will be convenient to require that the lorenz gauge condition @xmath327 be satisfied to order @xmath328 . the @xmath329 part of the linearized einstein equation together with the @xmath330 part of the lorenz gauge condition then yields @xmath331 where @xmath332 . this system of equations for @xmath333 is the same system of equations as is satisfied by stationary solutions of the flat spacetime linearized einstein equation ( but our @xmath334 may depend on time ) . the general solution of these equations is @xmath335 , @xmath336 , and @xmath337 , where @xmath33 is antisymmetric , @xmath338 , where @xmath232 , @xmath33 , and @xmath339 have no spatial dependence , and where @xmath340 was defined by eq . ( [ ni ] ) . by a further second order gauge transformation ( of the form @xmath341 ) , we can set @xmath342 . we thus obtain @xmath343 which is of the same form as the general stationary @xmath270 perturbation of minkowski spacetime ( see , e.g. , @xcite ) , except that time dependence is allowed for @xmath33 and @xmath339 . as we shall see shortly , @xmath33 and @xmath339 correspond , respectively , to the spin and mass dipole moment of the body . we now may write for the metric through @xmath344 , @xmath345 where we have introduced the unknown tensors @xmath143 and @xmath346 , and @xmath347 is the antisymmetric tensor whose spatial components are @xmath33 and whose time components vanish , i.e. , @xmath348 we now follow the strategy outlined in section [ sec : dipole ] . we consider a _ smooth _ coordinate shift of the form , @xmath349 and choose @xmath286 so as to make the mass dipole moment of @xmath350 vanish for all @xmath37 . a straightforward application of the coordinate transformation ( [ eq : transform2 ] ) to the metric ( [ eq : metric2 ] ) yields @xmath351 n^i \delta_{\alpha \beta } + \frac{1}{\hat{r}^2 } t_{(\alpha } s_{\beta ) j}(\hat{t } ) n^j \\ & \qquad \quad + \frac{1}{\hat{r } } k_{\alpha \beta}(\hat{t},\theta,\phi ) + h_{\alpha \beta}(\hat{t},\theta,\phi ) + o(r ) \biggr ) + o(\lambda^3 ) , \end{split}\ ] ] where we have `` absorbed '' the effects of the gauge transformation at orders @xmath352 and @xmath353 into the unknown tensors @xmath354 . the corresponding `` near zone expansion '' ( see eqs . ( [ eq : rseries ] ) , ( [ bnmp ] ) , and ( [ eq : fullnearzone ] ) ) is @xmath355 n^i \delta_{\alpha \beta } + o \left ( \frac{1}{\hat{\bar{r}}^3 } \right ) \\ & \quad + \lambda \left [ h^{\textrm{\tiny tail}}_{\alpha \beta } + 2 a_{(\alpha , \beta ) } + \frac{1}{\hat{\bar{r } } } k_{\alpha \beta}+ \frac{\hat{\bar{t}}}{\hat{\bar{r}}^2 } \left ( t_{(\alpha } s_{\beta ) j,0 } n^j + 2 \left[p_{i,0}-m a_{i,0}\right ] n^i \delta_{\alpha \beta } \right ) + o \left ( \frac{1}{\hat{\bar{r}}^2 } \right ) + \hat{\bar{t } } o\left ( \frac{1}{\hat{\bar{r}}^3 } \right ) \right ] \\ & \quad + \lambda^2 \biggl [ b_{\alpha i \beta j } \hat{\bar{x}}^i \hat{\bar{x}}^j + h^{\textrm{\tiny tail}}_{\alpha \beta \gamma}\hat{\bar{x}}^\gamma + m \mathcal{r}_{\alpha \beta}(\hat{\bar{x}}^i ) + 2 b_{\alpha i \beta j}a^i \hat{\bar{x}}^j + 2 a_{(\alpha , \beta ) \gamma } \hat{\bar{x}}^\gamma \\ & \qquad \qquad + h_{\alpha \beta } + \frac{\hat{\bar{t}}}{\hat{\bar{r } } } k_{\alpha \beta,0 } + \frac{\hat{\bar{t}}^2}{\hat{\bar{r}}^2 } \left ( t_{(\alpha } s_{\beta ) j,00 } n^j + 2 \left[p_{i,00}-m a_{i,00}\right ] n^i \delta_{\alpha \beta } \right ) \\ & \qquad \qquad + o\left ( \frac{1}{\hat{\bar{r } } } \right ) + \hat{\bar{t } } \ o \left ( \frac{1}{\hat{\bar{r}}^3 } \right ) + \hat{\bar{t}}^2 \ o \left ( \frac{1}{\hat{\bar{r}}^3 } \right ) \biggr ] + o(\lambda^3 ) \,\ , . \end{split}\ ] ] notice that the indices on the left side of this equation have both a `` hat '' and `` bar '' on them to denote that they are components of @xmath90 in the scaled coordinates associated with our new coordinates @xmath356 . by contrast , the indices on the right side have neither a `` hat '' nor a `` bar '' , since they denote the corresponding components in the unscaled , original coordinates @xmath51 . thus , for example , @xmath357 denotes the matrix of first partial derivatives of the @xmath51-components of @xmath358 with respect to the @xmath51 coordinates arises from taylor expanding @xmath359 with respect to the @xmath356 coordinates , so , in principle , the second partial derivative in this expression should be with respect to @xmath360 rather than @xmath361 . however , since @xmath360 coincides with @xmath361 at zeroth order in @xmath2 and the @xmath362 appears at second order in @xmath2 , we may replace the partial derivative with respect to @xmath360 by the partial derivative with respect to @xmath361 . ] . it also should be understood that all tensor components appearing on the right side of eq . ( [ eq : scaled - metric ] ) are evaluated at time @xmath363 , and that @xmath364 and its derivatives , as well as @xmath365 and @xmath366 , are evaluated at @xmath367 ( i.e. , on the worldline @xmath8 ) . finally , the `` reversals '' in the roles of various terms in going from the far zone expansion of the metric eq . ( [ eq : metric2 ] ) to the near zone expansion eq . ( [ eq : scaled - metric ] ) should be noted . for example , the spin term @xmath368 originated as a second order perturbation in the far zone , but it now appears as part of the background scaled metric in the near zone expansion . by contrast , the term @xmath369 originated as part of the background metric in the far zone , but it now appears as a second order perturbation in the near zone expansion . it is easy to see from eq . ( [ eq : scaled - metric ] ) that @xmath370 is the mass dipole moment of @xmath371 at time @xmath37 . we therefore set @xmath372 for all @xmath174 . consequently , no mass dipole term will appear in our expressions below . although we have `` solved '' for @xmath281 in eq . ( [ ap ] ) , we have not learned anything useful about the motion . is equivalent to simply determining the mass dipole moment @xmath373 in the original coordinates . the computations of this section may therefore be recast as simply solving enough of the second - order perturbation equations for the mass dipole moment and hence the motion to be determined . ] all useful information about @xmath281 will come from demanding that the metrics @xmath42or , equivalently , @xmath61be solutions of einstein s equation . we may apply einstein s equation perturbatively either via the far zone expansion or the near zone expansion . the resulting systems of equations are entirely equivalent , but the terms are organized very differently . we find it more convenient to work with the near zone expansion , and will do so below . we emphasize , however , that we could equally well have used the far zone perturbation expansion . we also emphasize that no new information whatsoever can be generated by matching the near and far zone expansions , since these expansions have already been fully `` matched '' via eqs . ( [ eq : rseries ] ) , ( [ bnmp ] ) , and ( [ eq : fullnearzone ] ) . in the following , in order to make the notation less cumbersome , we will drop the `` hat '' on the near - zone coordinates @xmath374 and on the components @xmath375 . no confusion should arise from this , since we will not have occassion to use the original scaled coordinates @xmath58 below . on the other hand , we will maintain the `` hat '' on the coordinates @xmath356 , since we will have occassion to use both @xmath356 and @xmath51 below . using this notation and setting the mass dipole terms to zero , eq . ( [ eq : scaled - metric ] ) becomes @xmath376 \\ & \quad + \lambda^2 \biggl [ b_{\alpha i \beta j } \bar{x}^i \bar{x}^j + h^{\textrm{\tiny tail}}_{\alpha \beta \gamma}\bar{x}^\gamma + m \mathcal{r}_{\alpha \beta}(\bar{x}^i ) + 2 b_{\alpha i \beta j}a^i \bar{x}^j + 2 a_{(\alpha , \beta ) \gamma } \bar{x}^\gamma \\ & \qquad \qquad + h_{\alpha \beta } + \frac{\bar{t}}{\bar{r } } \dot{k}_{\alpha \beta } + \frac{\bar{t}^2}{\bar{r}^2 } t_{(\alpha } \ddot{s}_{\beta ) j } n^j + o\left ( \frac{1}{\bar{r } } \right ) + \bar{t } \ o \left ( \frac{1}{\bar{r}^3 } \right ) + \bar{t}^2 \ o \left ( \frac{1}{\bar{r}^3 } \right ) \biggr ] + o(\lambda^3 ) \,\ , , \end{split}\ ] ] where the `` dots '' denote derivatives with respect to @xmath174 . we now apply the vacuum linearized einstein equation up to leading order , @xmath377 , in @xmath268 as @xmath378to the first order term in @xmath2 appearing in eq . ( [ eq : scaled - metric2 ] ) , namely @xmath379 it is clear that the terms of order @xmath271 and @xmath380 in @xmath381 can not contribute to the linearized ricci tensor to order @xmath377 . similarly , it is clear that the terms of order @xmath271 and higher in the background scaled metric can not contribute to the linearized ricci tensor to order @xmath377 , so , to order @xmath377 , we see that @xmath382 satisfies the linearized einstein equation about the schwarzschild metric . it is therefore useful to expand @xmath382 in tensor spherical harmonics . we obtain one very useful piece of information by extracting the @xmath383 , magnetic parity part of the linearized ricci tensor that is even under time reversal . on account of the symmetries of the background schwarzschild metric , only the @xmath383 , magnetic parity , even under time reversal part of the metric perturbation can contribute . now , a general @xmath383 , symmetric ( but not necessarily trace free ) tensor field @xmath384 can be expanded in tensor spherical harmonics as ( see , e.g. , @xcite or @xcite equations ( a16 - 18 ) ) @xmath385 where the expansion coefficients @xmath386 are functions of @xmath387 . the three - vector index on these coefficients corresponds to the three different `` @xmath149-values '' for each @xmath383 harmonic . thus , we see that there are a grand total of eight types of @xmath383 tensor spherical harmonics . the six harmonics associated with labeling indices @xmath388 are of electric parity , whereas the two harmonics associated with @xmath389 are of magnetic parity . for the metric perturbation ( [ eq : scaled-1 ] ) the `` constant tensors '' @xmath365 and @xmath390 are purely electric parity and can not contribute . it turns out that @xmath391 also does not contribute to the @xmath383 , magnetic parity part of the linearized ricci tensor that is even under time reversal : since @xmath392 is independent of @xmath156 the `` @xmath32 '' part of @xmath392 is odd under time reversal , whereas the `` @xmath145 '' part of @xmath391 is pure gauge . thus , the only term that contributes to order @xmath377 to the @xmath383 , magnetic parity part of of the linearized ricci tensor that is even under time reversal is @xmath393 . satisfaction of vacuum linearized einstein equation at order @xmath377 requires that this term vanish . we thereby learn that @xmath394 i.e. , to lowest order , the spin is parallelly propagated with respect to the background metric along the worldline @xmath8 . having set the spin term to zero in eq.([eq : scaled-1 ] ) , we may now substitute the remaining terms in eq . ( [ eq : scaled-1 ] ) into the linearized einstein equation and set the @xmath377 part equal to zero . it is clear that we will thereby obtain relations between @xmath395 , @xmath359 , and @xmath396 . however , these relations will not be of direct interest for obtaining `` equations of motion''i.e . , equations relating @xmath281 and its time derivatives to known quantities because the quantity of interest @xmath397 always appears in combination with the quantity @xmath398 , which is unrelated to the motion . therefore , we shall not explicitly compute the relations arising from the linearized einstein equation here . we now consider the information on @xmath281 that can be obtained from the near zone second - order einstein equation @xmath399 = - g^{(2)}_{ab}[\bar{g}^{(1 ) } , \bar{g}^{(1)}],\ ] ] where , from eq . ( [ eq : scaled - metric2 ] ) , we see that @xmath400 where we have defined @xmath401 we wish to impose the second order einstein equation to orders @xmath271 and @xmath380 , which , as we shall see below , are the lowest nontrivial orders in @xmath268 as @xmath378 that occur . first , we consider @xmath402 $ ] . the terms appearing in this quantity can be organized into terms of the following general forms ( i ) @xmath403 ; ( ii ) @xmath404 ; ( iii ) @xmath405 where @xmath406 denotes a christoffel symbol of the background scaled metric ; ( iv ) @xmath407 ; and ( v ) @xmath408 . from the form of @xmath409 together with the fact that @xmath410 and @xmath411 , it is clear that none of these terms can contribute to @xmath412 $ ] to order @xmath271 or @xmath380 . therefore , we may treat @xmath413 as satisfying the homogeneous , vacuum linearized einstein equation . we now consider the linearized ricci tensor of the perturbation @xmath413 . by inspection of eq . ( [ eq : scaled-2 ] ) , it might appear that terms that are @xmath184 ( from two partial derivatives acting on the `` b '' term ) and @xmath414 ( from various terms ) will arise . however , it is not difficult to show that the total contribution to the @xmath184 and @xmath414 terms will vanish by virtue of the fact that the metric @xmath171 is a solution to einstein s equation and the term proportional to @xmath2 in eq . ( [ eq : metric2 ] ) satisfies the far zone linearized einstein equation ( which has already been imposed ) . it also is clear that there is no contribution of @xmath413 to the linearized ricci tensor that is of order @xmath415 . thus , the lowest nontrivial orders that arise in the second order einstein equation are indeed @xmath271 and @xmath380 , as claimed above . the computation of the linearized ricci tensor to orders @xmath271 and @xmath380 for the metric perturbation @xmath413 is quite complicated , so we will save considerable labor by focusing on the relevant part of the linearized einstein equation to these orders . our hope / expectation ( which will be borne out by our calculation ) is to obtain an equation for @xmath416 . since this quantity is of @xmath383 , electric parity type and is even under time reversal , we shall focus on the @xmath383 , electric parity , even under time reversal part of the linearized ricci tensor of @xmath413 at orders @xmath271 and @xmath380 . from eq . ( [ eq : harmonics ] ) , we see that the @xmath383 electric parity part of the ricci tensor that is @xmath417 and even under time reversal can be written as @xmath418 whereas the @xmath383 part of the ricci tensor that is @xmath419 and even under time reversal can be written as @xmath420 here , in contrast to the usage of , @xmath421 are `` constants '' , i.e , they have no dependence on @xmath422 . we now consider the terms in @xmath413 that can contribute to these ricci terms . the term @xmath423 has no @xmath424 part . nevertheless , the @xmath425 magnetic parity part of this term can , in effect , combine with the @xmath383 magnetic parity `` spin term '' @xmath426 in the background scaled metric to produce a contribution to the linearized ricci tensor of the correct type . this contribution will be proportional to @xmath427 for the remaining terms in @xmath413 , the `` spin term '' @xmath426 in the background scaled metric will not contribute to the relevant parts of the linearized ricci tensor , so we may treat the remaining terms in @xmath413 as though they were perturbations of schwarzschild . thus , only the @xmath383 , electric parity , even under time reversal part of these terms can contribute . the remaining contributors to @xmath428 , and @xmath429 are @xmath430 and @xmath431 where , in eq . ( [ d ] ) , we have @xmath432(the curvature term @xmath300 has not appeared in the above equations because it has no @xmath383 part . ) the @xmath433 are also constants " in these expressions . a lengthy calculation now yields @xmath434 using the vacuum linearized einstein equation @xmath435 , we thus obtain 6 linear equations for our 11 unknowns . however , in order to find `` universial '' behavior , we are interested in relations that do not involve @xmath436 and @xmath437 . it can be shown that there are two such relations in near - zone perturbation theory the vanishing of the value , time derivative , and second time derivative of the mass dipole at time @xmath37 . the third condition should follow from the first - order near - zone einstein equation , which we did not fully use . in fact , it should only be necessary to impose that the mass dipole have no second time derivative in order to define the motion . ] , namely @xmath438 and @xmath439 the first equation involves @xmath440 and spatial derivatives of @xmath281 , and does not yield restrictions on the motion . however , the second equation gives desired equations of motion . plugging in the definitions of the quantities appearing in eq . ( [ motion ] ) , we obtain @xmath441 where we have taken advantage of the fact ( noted above ) that for @xmath442 and @xmath443 components we have @xmath444 . using the equality of mixed partials @xmath445 , we obtain @xmath446 thus , according to the interpretation provided in section [ sec : dipole ] above , the first order perturbative correction , @xmath447 , to the geodesic @xmath8 of the background spacetime satisfies @xmath448 in addition , we have previously found that @xmath32 and @xmath33 are constant along @xmath8 . taking account of the fact that this equation is written in fermi normal coordinates of @xmath8 and that @xmath253 , we can rewrite this equation in a more manifestly covariant looking form as @xmath449 where @xmath7 is the tangent to @xmath8 and @xmath450 . however , it should be emphasized that this equation describes the perturbed motion only when the metric perturbation is in the lorenz gauge ( see appendix a ) . the first term in eq . ( [ eq : eom ] ) ( or , equivalently , in eq . ( [ eq : eomcov ] ) ) is the `` spin force '' first obtained by papapetrou @xcite . contributions from higher multipole moments do not appear in our equation because they scale to zero faster than the spin dipole moment , and thus would arise at higher order in @xmath2 in our perturbation scheme . the second term corresponds to the right side of the geodesic deviation equation , and appears because the perturbed worldline is not ( except at special points ) coincident with the background worldline . in the limit @xmath451 , the body then moves on a family of geodesics of the background metric parameterized by @xmath2 , and the perturbative description of motion should indeed be the geodesic deviation equation . ] . the final term is the `` gravitational self - force '' , which is seen to take the form of a ( regularized ) gravitational force from the particle s own field . our derivation has thus provided a rigorous justification of the regularization schemes that have been proposed elsewhere . finally , we note that , although our analysis has many points of contact with previous analyses using matched asymptotic expansions , there are a number of significant differences . we have already noted in section [ sec : geodesic ] that our derivation of geodesic motion at zeroth order in @xmath2 appears to differ from some other derivations @xcite , which do not appear to impose the requirement that @xmath241 . we also have already noted that in other approaches to self - force @xcite , what corresponds to our scaled metric at @xmath38 is _ assumed _ to be of schwarzschild form . in these other approaches , first order perturbations in the near zone expansion are treated as time independent , and are required to be regular on the schwarzschild horizon . by contrast we make no assumptions about the time - dependence of the perturbations of the scaled metric beyond those that follow from our fundamental assumptions ( i)-(iii ) of section [ sec : example ] . thus , our first order perturbations are allowed to have linear dependence on @xmath156 , and our second order perturbations can depend quadratically on @xmath156 . we also make no assumptions about the spacetime at @xmath165 and therefore impose no boundary conditions at small @xmath72 . finally , there is a significant difference in the manner in which the gauge conditions used to define the motion are imposed . in @xcite , the entire @xmath424 electric parity part of what corresponds to our second order near zone perturbation is set to zero without proper justification electric parity perturbation that is relevant for obtaining equations of motion in @xcite is of `` acceleration type '' ( with linear growth in @xmath72 ) and does not have an obvious interpretation in terms of a shift in the center of mass . ] . by contrast , our `` no mass dipole '' condition applies to the _ background _ near - zone metric and has been justified as providing `` center of mass '' coordinates . as already mentioned near the beginning of section [ sec : dipole ] , the quantity @xmath29 in eq . is a `` deviation vector '' defined on the background geodesic @xmath8 that describes the first order in @xmath2 perturbation to the motion . for any one parameter family of spacetimes @xmath42 satisfying the assumptions stated in section [ sec : example ] , eq . is therefore guaranteed to give a good approximation to the deviation from the background geodesic motion @xmath8 as @xmath35 . in other words , if @xmath8 is described by @xmath250 , then the new worldline obtained defined by @xmath452 is the correct description of motion to first order in @xmath2 ( when the metric perturbation is in lorenz gauge ) and is therefore guaranteed to be accurate at small @xmath2 . however , this guarantee is of the form that if one wants to describe the motion accurately up to time @xmath174 , then it always will be possible to choose @xmath2 sufficiently small that @xmath452 is a good approximation up to time @xmath174 . the guarantee is _ not _ of the form that if @xmath2 is chosen to be sufficiently small , then @xmath452 will accurately describe the motion for all time . indeed , for any fixed @xmath453 , it is to be expected that @xmath447 will grow large at sufficiently late times , and it is clear that the approximate description of motion @xmath452 can not be expected to be good when @xmath447 is large , since by the time the motion has deviated significantly from the original background geodesic @xmath8 , the motion clearly can not be accurately described in the framework of being a `` small correction '' to @xmath8 . however , the main intended application of the first order corrected equations of motion is to compute motion in cases , such as inspiral , where the deviations from the original geodesic motion become large at late times . it is therefore clear that eq . , as it stands , is useless for computing long term effects , such as inspiral . one possible response to the above difficulty would be to go to higher order in perturbation theory . however , it seems clear that this will not help . although the equations of motion obtained from @xmath454th order perturbation theory will be more accurate than the first order equations , they will not have a domain of validity that is significantly larger than the first order equations . the perturbative description at any finite order will continue to treat the motion as a `` small deviation '' from @xmath8 , and can not be expected to describe motion accurately when the deviations are , in fact , large . in essence , by the time that the deviation from @xmath8 has become sufficiently large to invalidate first order perturbation theory so that , e.g. , the second order corrections are comparable in magnitude to the first order corrections then one would expect that the @xmath455th order corrections will also be comparable to the @xmath454th order corrections , so @xmath454th order perturbation theory will not be accurate either . only by going to all orders in perturbation theory can one expect to get an accurate , global in time , description of motion via perturbation theory . of course , if one goes to all orders in perturbation theory , then there is little point in having done perturbation theory at all . nevertheless , for a sufficiently small body of suffciently small mass , it seems clear that the corrections to geodesic motion should be _ locally _ small and should be locally described by eq . . by the time these small corrections have built up and the body has deviated significantly from the original geodesic approximating its motion , it should then be close to a _ geodesic , perturbing off of which should give a better approximation to the motion for that portion of time . one could then attempt to `` patch together '' such solutions to construct a world - line that accurately describes the motion of the particle for a longer time . in the limit of many such patches with small times between them , one expects the resulting worldline to be described by a single `` self - consistent '' differential equation , which should then well - approximate the motion as long as it remains _ locally _ close to geodesic motion . a simple , familiar example will help illustrate all of the above points . consider the cooling of a `` black body '' . to choose a definite problem that can be put in a framework similar to that considered in this paper , let us consider a body ( such as a lump of hot coal ) that is put in a box with perfect reflecting walls , but a hole of area @xmath456 is cut in the this wall . we are interested in determining how the energy , @xmath457 , of the body changes with time . at finite @xmath456 , this is a very difficult problem , since the body will not remain in exact thermal equilibrium as it radiates energy through the hole . however , let us consider a one - parameter family of cavities where @xmath458 smoothly goes to zero as @xmath35 . when @xmath82 , we find that the energy , @xmath459 , does not change with time , and the body will remain in thermal equilibrium at temperature @xmath460 for all time . when we do first order perturbation theory in @xmath2 , we will find that the first order in @xmath2 correction , @xmath461 , to the energy satisfies becomes small compared to the typical wavelengths of the radiation ( as it must as we let @xmath462 ) , we enter a physical optics regime where our formulas are no longer valid . we ignore such effects here , just as in our above analysis of the motion of bodies in general relativity we ignored quantum gravity effects even though they should be important when the size of the body is smaller than the planck scale . ] @xmath463 where @xmath464 is the stefan - boltzmann constant and @xmath465 . note that only the zeroth order temperature , @xmath460 , enters the right side of this equation because the quantity @xmath466 is already first order in @xmath2 , so the effect of any changes in temperature would appear only to higher order in @xmath2 . since @xmath460 is a constant , it is easy to integrate eq . ( [ de ] ) to obtain , @xmath467 thus , first order perturbation theory approximates the behavior of @xmath468 as @xmath469 although this is a good approximation at early times , it is a horrible approximation at late times , as it predicts that the energy will go negative . if one went to second order in perturbation theory , one would obtain corrections to eq . ( [ de ] ) that would take into account the first order energy loss as well as various non - equilibrium effects . however , one would still be perturbing off of the non - radiating background , and the late time predictions using second ( or any finite higher order ) perturbation theory would still be very poor . however , there is an obvious major improvement that can be obtained by noting that if @xmath456 is sufficiently small , then the body should remain nearly in thermal equilibrium as it loses energy . therefore , although perturbation theory off of the zeroth order solution may give poor results at late times , first order perturbation theory off of _ some _ thermal equilibrium solution should give locally accurate results at all times . this suggests that if @xmath456 is sufficiently small , the cooling of the body should be described by @xmath470 when supplemented with the formula , @xmath471 , that relates energy to temperature when the body is in thermal equilibrium , this equation should provide an excellent description of the cooling of the body that is valid at all times . in effect , eq . ( [ denew ] ) takes into account the higher order perturbative effects ( to all orders in @xmath2 ) associated with the cooling of the body , but it neglects various perturbative effects associated with the body failing to remain in thermal equilibrium as it cools . equation ( [ denew ] ) is _ not _ an exact equation ( since it does not take various non - equilibrium effects into account ) and it is _ not _ an equation that arises directly from perturbation theory . rather , it is an equation that corresponds to applying first order perturbation theory to a background that itself undergoes changes resulting from the perturbation . we will refer to such an equation as a `` self - consistent perturbative equation '' . such equations are commonly written down for systems that can be described _ locally in time _ by a small deviation from a simple solution . how does one find a `` self - consistent perturbative equation '' for a given system for which one has derived first order perturbative equations ? we do not believe that there is any general method for deriving a self - consistent perturbative equation . however , the following appear to be appropriate criteria to impose on a self - consistent perturbative equation : ( 1 ) it should have a well posed initial value formulation . ( 2 ) it should have the same number of degrees of freedom as the first order perturbative system , so that a correspondence can be made between initial data for the self - consistent perturbative equation and the first order perturbative system . ( 3 ) for corresponding initial data , the solutions to the self - consistent perturbative equation should be close to the corresponding solutions of the first order perturbative system over the time interval for which the first order perturbative description should be accurate . we do not know of any reason why , for any given system , there need exist a self - consistent perturbative equation satisfying these criteria . in cases where a self - consistent perturbative equation satisfying these criteria does exist , we would not expect it to be unique . for example , we could modify eq . ( [ denew ] ) by adding suitable terms proportional to @xmath472 to the right side of this equation . the first order perturbative equations for the motion of a small body are that the first order metric perturbation satisfies @xmath473 where @xmath116 corresponds to a geodesic , @xmath8 of the background spacetime , and @xmath7 is the tangent to @xmath8 . if we consider the retarded solution to this equation ( which automatically satisfies the lorenz gauge condition ) , we have proven rigorously in this paper that the first order in @xmath2 deviation of the motion from @xmath8 satisfies @xmath474 with @xmath475 where , for simplicity , we have dropped the spin term . the misataquwa equations @xmath476 @xmath477 @xmath20 ( where one chooses the retarded solution to eq . ( [ misa2 ] ) ) are an excellent candidate for self - consistent perturbative equations corresponding to the above first order perturbative system ) , since in the self - consistent perturbative equation , the deviation from the self - consistent worldline should vanish . ] . here , @xmath478 ( normalized in the background metric ) refers to the self - consistent motion @xmath479 , rather than to a background geodesic as before . although a proper mathematical analysis of this integro - differential system has not been carried out , it appears plausible that our above criteria ( 1)-(3 ) will be satisfied by the misataquwa equations . if so , they should provide a good , global in time , description of motion for problems like extreme mass ratio inspiral . we wish to thank abraham harte and eric poisson for helpful discussions . this research was supported in part by nsf grant phy04 - 56619 to the university of chicago and a national science foundation graduate research fellowship to sg . as discussed in section [ sec : dipole ] , the description of motion will change under first - order changes of gauge . indeed , in that section , we noted that under a smooth gauge transformation , the description of motion changes by eq . ( [ smoothgauge ] ) . however , as previously stated near the end of section [ sec : example ] above ( see equation ) , the allowed coordinate freedom includes transformations that are not smooth at @xmath50 . since such gauges may arise in practice singularity along a string @xcite , and therefore do not satisfy our assumptions . ] , we provide here the expression for the first order perturbative equation of motion in an arbitrary gauge allowed by our assumptions . we also present the corresponding self - consistent perturbative equations of motion . as previously noted in section [ sec : calculation ] ( see the remark below eq . ( [ jh ] ) ) , the equations of motion to first order in @xmath2 depend only upon the first order gauge transformation @xmath480 . as we have seen , the mass dipole moment appears at second - order in ( far zone ) perturbation theory , so we must consider the effects of first - order gauge transformations on second - order perturbations . this is given by @xmath481 , with @xmath482 where @xmath483 denotes the lie derivative . equivalently , we have @xmath484 where @xmath191 is the derivative operator associated with the background metric @xmath171 . in order to satisfy the criteria on allowed gauge transformations ( see equation ) , the components of @xmath480 must be of the form @xmath485 i.e. , @xmath480 can not `` blow up '' at @xmath50 but it can be singular in the sense that its components can have direction - dependent limits . the mass dipole moment , @xmath373 , is one - half of the coefficient of the @xmath383 part of the leading order , @xmath310 , part of the second order metric perturbation , @xmath486 . therefore , @xmath373 may be extracted from the formula , @xmath487 where ds is the area element on the sphere of radius @xmath488 . under the gauge transformation generated by @xmath480 , we have @xmath489 as previously noted , for an arbitrary first - order perturbation satisfying our assumptions , we have @xmath490 where @xmath32 is the mass of the body . from eqs . ( [ xiform ] ) , ( [ dg200 ] ) and ( [ g100 ] ) , we see that the change in @xmath486 induced by our gauge transformation is @xmath491 therefore , by eq . ( [ eq : app - dip ] ) , the induced change in the mass dipole moment is @xmath492 where @xmath493 is the area element on the unit sphere . equation ( [ dp ] ) gives the change in the mass dipole moment induced by the possibly non - smooth gauge transformation generated by @xmath480 . the corresponding change in the first order perturbative equation of motion is determined by the change in the _ smooth _ vector field @xmath494 required to eliminate the mass dipole . writing @xmath495 , this change is given by @xmath496 ( see eq . ) . thus , the change @xmath497 induced in the deviation vector describing the perturbed worldline is @xmath498 with @xmath499 given by eq . . in the case where our original gauge was the lorenz gauge , it follows immediately from eq . ( [ eq : eom ] ) that the new equation of motion for @xmath500 is @xmath501 where @xmath502 is given by eq . ( [ eq : gauge - law ] ) , and where , for simplicity , we have dropped the spin term . we may rewrite eq . ( [ eq : eom - app2 ] ) as @xmath503 note that although eq . ( [ eq : eom - app6 ] ) provides us with the desired equation of motion in an arbitrary allowed gauge , the terms involving components of @xmath504 must still be computed in the lorenz gauge . now suppose one wishes to pass to a self - consistent perturbative equation associated with the new choice of gauge . it is not obvious how one might wish to modify the evolution equations for the metric perturbations in the new gauge . ( one possibility would be to simply use eq . ( [ misa2 ] ) and then modify the result by the addition of @xmath505 but it might be preferable to find a new equation based on a suitable `` relaxed '' version of the linearized einstein equation for the new gauge . ) however , it appears that a natural choice of self - consistent perturbative equation associated to eq . ( [ eq : eom - app6 ] ) would be @xmath506 in the case where @xmath480 is smooth ( so that , by eq . ( [ eq : gauge - law ] ) , we have @xmath507 ) this agrees with the proposal of barack and ori @xcite . wald 1984 _ general relativity _ ( chicago , il : university of chicago press ) r. geroch and j. traschen , _ phys . rev . d _ * 36 * , 1017 ( 1987 ) s. weinberg 1972 _ gravitation and cosmology _ ( new york : wiley ) w. israel , _ nuovo cim . b _ * 44 * 1 ( 1966 ) y. mino , m. sasaki , and t. tanaka , _ phys d _ , * 55 * , 3457 - 3476 , ( 1997 ) t.c . quinn and r.m . wald , _ phys . d _ , * 56 * , 3381 - 3394 , ( 1997 ) s. detweiler and b.f . whiting , _ phys . * 67 * , 024025 , ( 2003 ) e. poisson , _ liv . rel . _ * 7 * 6 ( 2004 ) b.s . dewitt and r.w . brehme , _ ann . _ , * 9 * , 220 - 259 , ( 1960 ) p. a. m. dirac , _ proc . lond . a _ * 167 * 148 ( 1938 ) w. l. burke , _ j. math . ( n.y . ) * 12 * , 401 ( 1971 ) p. d. death , _ phys . d _ * 11 * 1387 ( 1975 ) r. kates , _ phys . * 22 * 1853 ( 1980 ) k. s. thorne and j. b. hartle , _ phys . d _ * 31 * 1815 ( 1985 ) r. geroch and p.s . * 16 * ( 1975 ) , pp . 65 - 67 j. ehlers and r. geroch , _ ann . phys . _ * 309 * , 232 ( 2004 ) d.a.m . stuart , _ j. math . pures appl . _ * 83 * , 541 ( 2004 ) a. papapetrou , _ proc . . london _ * a209 * , pp . 248 - 258 ( 1951 ) l. barack and a. ori , _ phys . d _ , * 64 * , 124003 , 1 - 13 , ( 2001 ) s. gralla , a. harte , and r. wald , in preparation r. geroch , _ commun . * 13 * 180 ( 1969 ) t. futamase and y. itoh , _ living rev . relativity _ * 10 * , 2 ( 2007 ) r.m . wald , _ phys . lett . _ * 41 * 203 ( 1978 ) w. beiglbock , _ commun . _ * 5 * , 106 ( 1967 ) r. geroch , _ j. math * 11 * 2580 ( 1970 ) r. o. hansen , _ j. math . phys . _ * 15 * 46 ( 1974 ) r. beig and w. simon , _ gen . * 12 * 1003 ( 1980 ) p. kundu , _ _ , 22 , 1236 ( 1981 ) x.h . zhang , _ phys . d _ * 34 * 991 ( 1986 ) k. s. thorne , _ rev . _ _ 52 _ 299 ( 1980 ) l. blanchet and t. damour , _ philos . . london _ * a320 * , 379 ( 1986 )

there is general agreement that the misataquwa equations should describe the motion of a small body " in general relativity , taking into account the leading order self - force effects . however , previous derivations of these equations have made a number of ad hoc assumptions and/or contain a number of unsatisfactory features . for example , all previous derivations have invoked , without proper justification , the step of `` lorenz gauge relaxation '' , wherein the linearized einstein equation is written down in the form appropriate to the lorenz gauge , but the lorenz gauge condition is then not imposed thereby making the resulting equations for the metric perturbation inequivalent to the linearized einstein equations . ( such a `` relaxation '' of the linearized einstein equations is essential in order to avoid the conclusion that `` point particles '' move on geodesics . ) in this paper , we analyze the issue of `` particle motion '' in general relativity in a systematic and rigorous way by considering a one - parameter family of metrics , @xmath0 , corresponding to having a body ( or black hole ) that is `` scaled down '' to zero size and mass in an appropriate manner . we prove that the limiting worldline of such a one - parameter family must be a geodesic of the background metric , @xmath1 . gravitational self - force as well as the force due to coupling of the spin of the body to curvature then arises as a first - order perturbative correction in @xmath2 to this worldline . no assumptions are made in our analysis apart from the smoothness and limit properties of the one - parameter family of metrics , @xmath0 . our approach should provide a framework for systematically calculating higher order corrections to gravitational self - force , including higher multipole effects , although we do not attempt to go beyond first order calculations here . the status of the misataquwa equations is explained .

0806.3293

the schwinger - dyson ( sd ) equation is one of the most popular approaches to investigate the non - perturbative features of quantum field theory . the analyses by making use of the sd equation for quark propagator are well - known . recently , the coupled sd equations for the gluon and ghost propagators in yang - mills theory have been studied mainly in the lorentz ( landau ) gauge.@xcite in this paper , we derive the sd equations for the @xmath0 yang - mills theory in the maximal abelian ( ma ) gauge and solve them analytically in the infrared ( ir ) asymptotic region . the ma gauge is useful to investigate the yang - mills theory from the view point of the dual superconductivity . in the ma gauge , in contrast to the ordinary lorentz gauge , we must explicitly distinguish the diagonal components of the fields from the off - diagonal components . this is indeed the case even in the perturbative analysis in the uv region.@xcite therefore , we must take account of the four propagators for the diagonal gluon , off - diagonal gluon , diagonal ghost and off - diagonal ghost . numerical behaviors of gluon propagators in the ma gauge are also investigated on a lattice simulation.@xcite first , we derive the sd equations from the @xmath0 yang - mills action in the ma gauge@xcite . the graphical representation of sd equations are shown in figure [ fig : sde ] . = .001 in ( 6000,1800 ) ( 0,-200)(0,500)(0,150)(450,300)(600,160)(800,200)(1250,300)(1400,160)(1600,0)(2000,350)(2200,160)(2400,160)(3600,160)(3800,160)(0,1000)(0,150)(450,300)(600,160)(800,200)(1250,300)(1400,160)(1600,100)(2000,350)(2200,160)(2400,160)(3600,160)(3800,160)(0,1500)(0,150)(0,250)(450,300)(600,160)(800,200)(1000,250)(1250,300)(1400,160)(1600,0)(1570,230)(2200,160)(2400,0)(2370,230)(3000,160)(3200,160)(4400,160)(4600,160)(0,0)(0,150)(0,250)(450,300)(600,160)(800,200)(1000,250)(1250,300 ) for the diagonal gluon propagator , we adopt the landau gauge so that the diagonal gluon propagator @xmath1 has only the transverse part @xmath2 where we defined the form factor @xmath3 . while , the off - diagonal gluon propagator @xmath4 has both the transverse and longitudinal parts @xmath5\delta^{ab},\ ] ] where we defined the form factors @xmath6 and @xmath7 . the form factor @xmath8 for the off - diagonal ghost propagator @xmath9 is defined @xmath10 the diagonal ghost propagator is decoupled from the other fields so that we omit it hereafter . now , we write down the sd equations : @xmath11 @xmath12 and @xmath13 here the contributions from the two - loop graphs have been omitted . the full form of sd equations will be given in a separate paper@xcite . @xmath14 is the full vertex function for the diagonal gluon , off - diagonal ghost and off - diagonal antighost interaction , while @xmath15 is the full vertex function for an interaction of the diagonal gluon and two off - diagonal gluons , and the superscript `` @xmath16 '' means a _ bare _ propagator or vertex function . in the ma gauge , we obtain the slavnov - taylor ( st ) identities @xmath17 @xmath18 in order to solve the sd equations analytically , we employ the following approximations . @xmath19 we neglect the two - loop contributions . instead of the full vertex functions , we adopt modified vertex functions which are compatible with the st identities . we adopt approximations for vertex functions as @xmath20 and @xmath21 here , we adopt the feynman gauge for the off - diagonal gluon for simplicity , that is , @xmath22 and @xmath23 . substituting the bare form factors , which are @xmath24 , into the right hand side of the ansatz ( [ eq : acc ] ) and ( [ eq : aaa ] ) , we obtain the bare vertex functions . moreover , these ansatz are compatible with the st identities ( [ eq : sti - c ] ) and ( [ eq : sti - a ] ) in the limit of @xmath25 . in the momentum integration , we use the higashijima - miransky approximation@xcite as @xmath26 now we adopt the ansatz for the form factors in the ir region : @xmath27 g(p^2 ) = b(p^2)^v+\cdots,\\[1 mm ] f_{\rm t}(p^2 ) = c(p^2)^w+\cdots . \end{array } \label{eq : ir solutions}\ ] ] substituting the ansatz ( [ eq : ir solutions ] ) for the form factors , and the ansatz ( [ eq : acc ] ) and ( [ eq : aaa ] ) for vertex functions into the sd equations ( [ eq : diagonal gluon ] ) , ( [ eq : off - diagonal ghost ] ) and ( [ eq : off - diagonal gluon ] ) , and comparing the leading term in the both sides of each equation , we obtain the following results for @xmath22 . from eqs . ( [ eq : off - diagonal ghost ] ) and ( [ eq : off - diagonal gluon ] ) , we obtain the relations @xmath28 and @xmath29 . in the case of @xmath30 and @xmath31 , from the eq . ( [ eq : diagonal gluon ] ) , we obtain the relation @xmath32 so that @xmath33 is less than @xmath34 . in the case of @xmath35 and @xmath31 , we need redefine the form factor @xmath8 as @xmath36 with @xmath37 since contributions from the leading term of @xmath8 are canceled each other in the ansatz ( [ eq : acc ] ) . therefore we need the information of next leading term of the form factor @xmath8 . in this case we obtain the relation @xmath38 from the eq . ( [ eq : diagonal gluon ] ) so that @xmath33 is also less than @xmath34 . next , we consider the case of @xmath30 and @xmath39 . as well as the above case , we need redefine the form factor @xmath6 as @xmath40 with @xmath41 and we obtain the relation @xmath42 ( @xmath43 ) . similarly , in the case of @xmath44 , we obtain the relation @xmath45 ( @xmath43 ) . the results are summarized in table [ tbl : feynman gauge ] . @xmath32 & @xmath42 @xmath35 & @xmath38 & @xmath45 [ tbl : feynman gauge ] in the gauge other than the feynman gauge , that is , @xmath46 , the calculation and discussion are very tedious . however , the qualitative results are identical to the above case except for the following one point . in this case , even if @xmath39 , there occurs no cancellation as in the above two cases 2c and 2d . this is because the off - diagonal gluon propagator has the momentum dependent tensor structure for @xmath46 , while it is proportional to @xmath47 for @xmath22 . therefore , we obtain the relation @xmath48 in the case of @xmath39 . ( see table [ tbl : not feynman gauge ] . ) @xmath30 & @xmath32 & @xmath48 @xmath35 & @xmath38 & @xmath48 [ tbl : not feynman gauge ] in the ir limit , the form factors of each propagator behave as @xmath49 @xmath50 @xmath51 therefore the solution shows that the diagonal gluon propagator is enhanced in the ir limit , while the off - diagonal gluon and off - diagonal ghost propagators are suppressed in the ir region . our results are compatible with a hypothesis of abelian dominance@xcite .

we derive the schwinger - dyson equations for the @xmath0 yang - mills theory in the maximal abelian gauge and solve them in the infrared asymptotic region . we find that the infrared asymptotic solutions for the gluon and ghost propagators are consistent with the hypothesis of abelian dominance .

hep-lat0309164

information - theoretic research on capacity and coding for write - limited memory originates in @xcite , @xcite , @xcite and @xcite . in @xcite , the authors consider a model of write - once memory ( wom ) . in particular , each memory cell can be in state either 0 or 1 . the state of a cell can go from 0 to 1 , but not from 1 back to 0 later . these write - once bits are called _ wits_. it is shown that , the efficiency of storing information in a wom can be improved if one allows multiple rewrites and designs the storage / rewrite scheme carefully . multilevel flash memory is a storage technology where the charge level of any cell can be easily increased , but is difficult to decrease . recent multilevel cell technology allows many charge levels to be stored in a cell . cells are organized into blocks that contain roughly @xmath2 cells . the only way to decrease the charge level of a cell is to erase the whole block ( i.e. , set the charge on all cells to zero ) and reprogram each cell . this takes time , consumes energy , and reduces the lifetime of the memory . therefore , it is important to design efficient rewriting schemes that maximize the number of rewrites between two erasures @xcite , @xcite , @xcite , @xcite . the rewriting schemes increase some cell charge levels based on the current cell state and message to be stored . in this paper , we call a rewriting scheme a _ modulation code_. two different objective functions for modulation codes are primarily considered in previous work : ( i ) maximizing the number of rewrites for the worst case @xcite and ( ii ) maximizing for the average case @xcite . as finucane et al . @xcite mentioned , the reason for considering average performance is the averaging effect caused by the large number of erasures during the lifetime of a flash memory device . our analysis shows that the worst - case objective and the average case objective are two extreme cases of our optimization objective . we also discuss under what conditions each optimality measure makes sense . in previous work ( e.g. , @xcite ) , many modulation codes are shown to be asymptotically optimal as the number of cell - levels @xmath0 goes to infinity . but the condition that @xmath1 can not be satisfied in practical systems . therefore , we also analyze asymptotically optimal modulation codes when @xmath0 is only moderately large using the results from load - balancing theory @xcite . this suggests an enhanced algorithm that improves the performance of practical system significantly . theoretical analysis and simulation results show that this algorithm performs better than other asymptotically optimal algorithms when @xmath0 is moderately large . the structure of the paper is as follows . the system model and performance evaluation metrics are discussed in section [ sec : optimality - measure ] . an asymptotically optimal modulation code , which is universal over arbitrary i.i.d . input distributions , is proposed in section [ sub : another - rewriting - algorithm ] . the storage efficiency of this asymptotically optimal modulation code is analyzed in section [ sec : an - enhanced - algorithm ] . an enhanced modulation code is also presented in section [ sec : an - enhanced - algorithm ] . the storage efficiency of the enhanced algorithm is also analyzed in section [ sec : an - enhanced - algorithm ] . simulation results and comparisons are presented in section [ sec : simulation - results ] . the paper is concluded in section [ sec : conclusion ] . flash memory devices usually rely on error detecting / correcting codes to ensure a low error rate . so far , practical systems tend to use bose - chaudhuri - hocquenghem ( bch ) and reed - solomon ( rs ) codes . the error - correcting codes ( ecc s ) are used as the outer codes while the modulation codes are the inner codes . in this paper , we focus on the modulation codes and ignore the noise and the design of ecc for now . let us assume that a block contains @xmath3 @xmath0-level cells and that @xmath4 cells ( called an @xmath4-cell ) are used together to store @xmath5 @xmath6-ary variables ( called a @xmath5-variable ) . a block contains @xmath7 @xmath4-cells and the @xmath7 @xmath5-variables are assumed to be i.i.d . random variables . we assume that all the @xmath5-variables are updated together randomly at the same time and the new values are stored in the corresponding @xmath4-cells . this is a reasonable assumption in a system with an outer ecc . we use the subscript @xmath8 to denote the time index and each rewrite increases @xmath8 by 1 . when we discuss a modulation code , we focus on a single @xmath4-cell . ( the encoder of the modulation code increases some of the cell - levels based on the current cell - levels and the new value of the @xmath5-variable . ) remember that cell - levels can only be increased during a rewrite . so , when any cell - level must be increased beyond the maximum value @xmath9 , the whole block is erased and all the cell levels are reset to zero . we let the maximal allowable number of block - erasures be @xmath10 and assume that after @xmath10 block erasures , the device becomes unreliable . assume the @xmath5-variable written at time @xmath8 is a random variable @xmath11 sampled from the set @xmath12 with distribution @xmath13 . for convenience , we also represent the @xmath5-variable at time @xmath8 in the vector form as @xmath14 where @xmath15 denotes the set of integers modulo @xmath6 . the cell - state vector at time @xmath8 is denoted as @xmath16 and @xmath17 denotes the charge level of the @xmath18-th cell at time @xmath19 when we say @xmath20 we mean @xmath21 for @xmath22 since the charge level of a cell can only be increased , continuous use of the memory implies that an erasure of the whole block will be required at some point . although writes , reads and erasures can all introduce noise into the memory , we neglect this and assume that the writes , reads and erasures are noise - free . consider writing information to a flash memory when encoder knows the previous cell state @xmath23 the current @xmath5-variable @xmath24 , and an encoding function @xmath25 that maps @xmath24 and @xmath26 to a new cell - state vector @xmath27 . the decoder only knows the current cell state @xmath27 and the decoding function @xmath28 that maps the cell state @xmath27 back to the variable vector @xmath29 . of course , the encoding and decoding functions could change over time to improve performance , but we only consider time - invariant encoding / decoding functions for simplicity . the idea of designing efficient modulation codes jointly to store multiple variables in multiple cells was introduced by jiang @xcite . in previous work on modulation codes design for flash memory ( e.g. @xcite , @xcite , @xcite , @xcite ) , the lifetime of the memory ( either worst - case or average ) is maximized given fixed amount of information per rewrite . improving storage density and extending the lifetime of the device are two conflicting objectives . one can either fix one and optimize the other or optimize over these two jointly . most previous work ( e.g. , @xcite ) takes the first approach by fixing the amount of information for each rewrite and maximizing the number of rewrites between two erasures . in this paper , we consider the latter approach and our objective is to maximize the total amount of information stored in the device until the device dies . this is equivalent to maximizing the average ( over the @xmath5-variable distribution @xmath13 ) amount of information stored per cell - level , @xmath30 where @xmath31 is the amount of information stored at the @xmath18-th rewrite , @xmath32 is the number of rewrites between two erasures , and the expectation is over the @xmath5-variable distribution . we also call @xmath33 as _ storage efficiency_. in previous work on modulation codes for flash memory , the number of rewrites of an @xmath4-cell has been maximized in two different ways . the authors in @xcite consider the worst case number of rewrites and the authors in @xcite consider the average number of rewrites . as mentioned in @xcite , the reason for considering the average case is due to the large number of erasures in the lifetime of a flash memory device . interestingly , these two considerations can be seen as two extreme cases of the optimization objective in ( [ eq : opt ] ) . let the @xmath5-variables be a sequence of i.i.d . random variables over time and all the @xmath4-cells . the objective of optimization is to maximize the amount of information stored until the device dies . the total amount of information stored in the device - cell changes to the same value , should it count as stored information ? should this count as a rewrite ? this formula assumes that it counts as a rewrite , so that @xmath34 values ( rather than @xmath35 ) can be stored during each rewrite . ] can be upper - bounded by @xmath36 where @xmath37 is the number of rewrites between the @xmath38-th and the @xmath18-th erasures . note that the upper bound in ( [ eq : total_info_ub ] ) is achievable by uniform input distribution , i.e. , when the input @xmath5-variable is uniformly distributed over @xmath39 , each rewrite stores @xmath40 bits of information . due to the i.i.d . property of the input variables over time , @xmath37 s are i.i.d . random variables over time . since @xmath37 s are i.i.d . over time , we can drop the subscript @xmath18 . since @xmath10 , which is the maximum number of erasures allowed , is approximately on the order of @xmath41 , by the law of large numbers ( lln ) , we have @xmath42k\log_{2}(l).\ ] ] let the set of all valid encoder / decoder pairs be @xmath43 where @xmath44 implies the charge levels are element - wise non - decreasing . this allows us to treat the problem @xmath45 as the following equivalent problem @xmath46k\log_{2}(l).\label{eq : opt2 - 1}\ ] ] denote the maximal charge level of the @xmath18-th @xmath4-cell at time @xmath8 as @xmath47 . note that time index @xmath8 is reset to zero when a block erasure occurs and increased by one at each rewrite otherwise . denote the maximal charge level in a block at time @xmath8 as @xmath48 which can be calculated as @xmath49 define @xmath50 as the time when the @xmath18-th @xmath4-cell reaches its maximal allowed value , i.e. , @xmath51 . we assume , perhaps naively , that a block - erasure is required when any cell within a block reaches its maximum allowed value . the time when a block erasure is required is defined as @xmath52 it is easy to see that @xmath53=ne\left[t\right],$ ] where the expectations are over the @xmath5-variable distribution . so maximizing @xmath54 $ ] is equivalent to maximizing @xmath55 . so the optimization problem ( [ eq : opt2 - 1 ] ) can be written as the following optimization problem @xmath56.\label{eq : opt3}\ ] ] under the assumption that the input is i.i.d . over all the @xmath4-cells and time indices , one finds that the @xmath50 s are i.i.d . random variables . let their common probability density function ( pdf ) be @xmath57 it is easy to see that @xmath58 is the minimum of @xmath7 i.i.d . random variables with pdf @xmath57 therefore , we have @xmath59 where @xmath60 is the cumulative distribution function ( cdf ) of @xmath61 so , the optimization problem ( [ eq : opt3 ] ) becomes @xmath62=\max_{f , g\in\mathcal{q}}\int nf_{t}(x)\left(1-f_{t}(x)\right)^{n-1}x\mbox{d}x.\label{eq : opt}\ ] ] note that when @xmath63 the optimization problem in ( [ eq : opt ] ) simplifies to @xmath64.\label{eq : opt2}\ ] ] this is essentially the case that the authors in @xcite consider . when the whole block is used as one @xmath4-cell and the number of erasures allowed is large , optimizing the average ( over all input sequences ) number of rewrites of an @xmath4-cell is equivalent to maximizing the total amount of information stored @xmath65 the analysis also shows that the reason we consider average performance is not only due to the averaging effect caused by the large number of erasures . one other important assumption is that there is only one @xmath4-cell per block . the other extreme is when @xmath66 in this case , the pdf @xmath67 tends to a point mass at the minimum of @xmath8 and the integral @xmath68 approaches the minimum of @xmath8 . this gives the worst case stopping time for the programming process of an @xmath4-cell . this case is considered by @xcite . our analysis shows that we should consider the worst case when @xmath69 even though the device experiences a large number of erasures . so the optimality measure is not determined only by @xmath10 , but also by @xmath70 when @xmath7 and @xmath10 are large , it makes more sense to consider the worst case performance . when @xmath71 , it is better to consider the average performance . when @xmath7 is moderately large , we should maximize the number of rewrites using ( [ eq : opt ] ) which balances the worst case and the average case . when @xmath7 is moderately large , one should probably focus on optimizing the function in ( [ eq : opt ] ) , but it is not clear how to do this directly . so , this remains an open problem for future research . instead , we will consider a load - balancing approach to improve practical systems where @xmath0 is moderately large . if we assume that there is only one variable changed each time , the average amount of information per cell - level can be bounded by @xmath72 because there are @xmath73 possible new values . since the number of rewrites can be bounded by @xmath74 we have @xmath75 if we allow arbitrary change on the @xmath5-variables , there are totally @xmath76 possible new values . it can be shown that @xmath77 for fixed @xmath6 and @xmath0 , the bound in ( [ eq : storage_efficiency_bound ] ) suggests using a large @xmath5 can improve the storage efficiency . this is also the reason jointly coding over multiple cells can improve the storage efficiency @xcite . since optimal rewriting schemes only allow a single cell - level to increase by one during each rewrite , decodability implies that @xmath78 for the first case and @xmath79 for the second case . therefore , the bounds in ( [ eq : storage_efficiency_bound2 ] ) and ( [ eq : storage_efficiency_bound ] ) also require large @xmath4 to improve storage efficiency . the upper bound in ( [ eq : storage_efficiency_bound ] ) grows linearly with @xmath5 while the upper bound in ( [ eq : storage_efficiency_bound2 ] ) grows logarithmically with @xmath5 . therefore , in the remainder of this paper , we assume an arbitrary change in the @xmath5-variable per rewrite and @xmath71 , i.e. , the whole block is used as an @xmath4-cell , to improve the storage efficiency . this approach implicitly trades instantaneous capacity for future storage capacity because more cells are used to store the same number of bits , but the cells can also be reused many more times . note that the assumption of @xmath71 might be difficult for real implementation , but its analysis gives an upper bound on the storage efficiency . from the analysis above with @xmath71 , we also know that maximizing @xmath33 is equivalent to maximize the average number of rewrites . in @xcite , modulation codes are proposed that are asymptotically optimal ( as @xmath0 goes to infinity ) in the average sense when @xmath80 . in this section , we introduce a modulation code that is asymptotically optimal for arbitrary input distributions and arbitrary @xmath5 and @xmath6 . this rewriting algorithm can be seen as an extension of the one in @xcite . the goal is , to increase the cell - levels uniformly on average for an arbitrary input distribution . of course , decodability must be maintained . the solution is to use common information , known to both the encoder ( to encode the input value ) and the decoder ( to ensure the decodability ) , to randomize the cell index over time for each particular input value . let us assume the @xmath5-variable is an i.i.d . random variable over time with arbitrary distribution @xmath13 and the @xmath5-variable at time @xmath8 is denoted as @xmath81 the output of the decoder is denoted as @xmath82 we choose @xmath83 and let the cell state vector at time @xmath8 be @xmath84 , where @xmath17 is the charge level of the @xmath18-th cell at time @xmath19 at @xmath85 , the variables are initialized to @xmath86 , @xmath87 and @xmath88 . the decoding algorithm @xmath89 is described as follows . * step 1 : read cell state vector @xmath27 and calculate the @xmath90 norm @xmath91 . * step 2 : calculate @xmath92 and @xmath93 the encoding algorithm @xmath94 is described as follows . * step 1 : read cell state @xmath26 and calculate @xmath95 and @xmath96 as above . if @xmath97 then do nothing . * step 2 : calculate @xmath98 and @xmath99 * step 3 : increase the charge level of the @xmath100-th cell by 1 . for convenience , in the rest of the paper , we refer the above rewriting algorithm as `` self - randomized modulation code '' . the self - randomized modulation code achieves at least @xmath101 rewrites with high probability , as @xmath102 for arbitrary @xmath103 @xmath104 and i.i.d . input distribution @xmath13 . therefore , it is asymptotically optimal for random inputs as @xmath1 . [ sketch of proof ] the proof is similar to the proof in @xcite . since exactly one cell has its level increased by 1 during each rewrite , @xmath105 is an integer sequence that increases by 1 at each rewrite . the cell index to be written @xmath100 is randomized by adding the value @xmath106 . this causes each consecutive sequence of @xmath76 rewrites to have a uniform affect on all cell levels . as @xmath1 , an unbounded number of rewrites is possible and we can assume @xmath107 . consider the first @xmath108 steps , the value @xmath109 is as even as possible over @xmath110 for convenience , we say there are @xmath111 @xmath112 s at each value , as the rounding difference by 1 is absorbed in the @xmath113 term . assuming the input distribution is @xmath114 . for the case that @xmath115 , the probability that @xmath116 is @xmath117 for @xmath118 . therefore , @xmath119 has a uniform distribution over @xmath12 . since inputs are independent over time , by applying the same chernoff bound argument as @xcite , it follows that the number of times @xmath116 is at most @xmath120 with high probability ( larger than @xmath121 ) for all @xmath122 . summing over @xmath122 , we finish the proof . notice that the randomizing term @xmath105 a deterministic term which makes @xmath100 look _ random _ over time in the sense that there are equally many terms for each value . moreover , @xmath105 is known to both the encoder and the decoder such that the encoder can generate `` uniform '' cell indices over time and the decoder knows the accumulated value of @xmath105 , it can subtract it out and recover the data correctly . although this algorithm is asymptotically optimal as @xmath1 , the maximum number of rewrites @xmath123 can not be achieved for moderate @xmath0 . this motivates the analysis and the design of an enhanced version of this algorithm for practical systems in next section . a self - randomized modulation code uses @xmath83 cells to store a @xmath5-variable . this is much larger than the @xmath124 used by previous asymptotically optimal algorithms because we allow the @xmath5-variable to change arbitrarily . although this seems to be a waste of cells , the average amount of information stored per cell - level is actually maximized ( see ( [ eq : storage_efficiency_bound2 ] ) and ( [ eq : storage_efficiency_bound ] ) ) . in fact , the definition of asymptotic optimality requires @xmath79 if we allow arbitrary changes to the @xmath5-variable . we note that the optimality of the self - randomized modulation codes is similar to the weak robust codes presented in @xcite . we use @xmath83 cells to store one of @xmath125 possible messages . this is slightly worse than the simple method of using @xmath126 . is it possible to have self - randomization using only @xmath126 cells ? a preliminary analysis of this question based on group theory indicates that it is not . thus , the extra cell provides the possibility to randomize the mappings between message values and the cell indices over time . while asymptotically optimal modulation codes ( e.g. , codes in @xcite , @xcite , @xcite , @xcite and the self - randomized modulation codes described in section [ sec : another - rewriting - algorithm ] ) require @xmath1 , practical systems use @xmath0 values between @xmath127 and @xmath128 . compared to the number of cells @xmath4 , the size of @xmath0 is not quite large enough for asymptotic optimality to suffice . in other words , codes that are asymptotically optimal may have significantly suboptimal performance when the system parameters are not large enough . moreover , different asymptotically optimal codes may perform differently when @xmath0 is not large enough . therefore , asymptotic optimality can be misleading in this case . in this section , we first analyze the storage efficiency of self - randomized modulation codes when @xmath0 is not large enough and then propose an enhanced algorithm which improves the storage efficiency significantly . before we analyze the storage efficiency of asymptotically optimal modulation codes for moderately large @xmath0 , we first show the connection between rewriting process and the load - balancing problem ( aka the balls - into - bins or balls - and - bins problem ) which is well studied in mathematics and computer science @xcite . basically , the load - balancing problem considers how to distribute objects among a set of locations as evenly as possible . specifically , the balls - and - bins model considers the following problem . if @xmath129 balls are thrown into @xmath4 bins , with each ball being placed into a bin chosen independently and uniformly at random , define the _ load _ as the number of balls in a bin , what is the maximal load over all the bins ? based on the results in theorem 1 in @xcite , we take a simpler and less accurate approach to the balls - into - bins problem and arrive at the following theorem . [ thm : random_loading]suppose that @xmath129 balls are sequentially placed into @xmath4 bins . each time a bin is chosen independently and uniformly at random . the maximal load over all the bins is @xmath130 and : ( @xmath18 ) if @xmath131 the maximally loaded bin has @xmath132 balls , @xmath133 and @xmath134 , with high probability ( @xmath135 ) as @xmath136 ( @xmath137 ) if @xmath138 , the maximally loaded bin has @xmath139 balls , @xmath140 , with high probability ( @xmath135 ) as @xmath136 ( @xmath141 ) if @xmath142 the maximally loaded bin has @xmath143 , @xmath144 , @xmath145 and @xmath146 , with high probability ( @xmath135 ) as @xmath136 denote the event that there are at least @xmath5 balls in a particular bin as @xmath147 . using the union bound over all subsets of size @xmath103 it is easy to show that the probability that @xmath147 occurs is upper bounded by @xmath148 using stirling s formula , we have @xmath149 . then @xmath150 can be further bounded by @xmath151 if @xmath152 , substitute @xmath153 to the rhs of ( [ eq : maxload_ub ] ) , we have @xmath154 denote the event that all bins have at most @xmath5 balls as @xmath155 . by applying the union bound , it is shown that @xmath156 since @xmath157 we finish the proof for the case of @xmath158 if @xmath138 , substitute @xmath159 to the rhs of ( [ eq : maxload_ub ] ) , we have @xmath160 by applying the union bound , we finish the proof for the case of @xmath161 if @xmath142 substitute @xmath162 to the rhs of ( [ eq : maxload_ub ] ) , we have @xmath163 where @xmath164 by applying the union bound , it is shown that @xmath165 since @xmath166 we finish the proof for the case of @xmath167 note that theorem [ thm : random_loading ] only shows an upper bound on the maximum load @xmath130 with a simple proof . more precise results can be found in theorem 1 of @xcite , where the exact order of @xmath130 is given for different cases . it is worth mentioning that the results in theorem 1 of @xcite are different from theorem [ thm : random_loading ] because theorem 1 of @xcite holds with probability @xmath168 while theorem [ thm : random_loading ] holds with probability ( @xmath135 ) . the asymptotic optimality in the rewriting process implies that each rewrite only increases the cell - level of a cell by 1 and all the cell - levels are fully used when an erasure occurs . this actually implies @xmath169 . since @xmath4 is usually a large number and @xmath0 is not large enough in practice , the theorem shows that , when @xmath0 is not large enough , asymptotic optimality is not achievable . for example , in practical systems , the number of cell - levels @xmath0 does not depend on the number of cells in a block . therefore , rather than @xmath74 only roughly @xmath170 charge levels can be used as @xmath171 if @xmath0 is a small constant which is independent of @xmath4 . in practice , this loss could be mitigated by using writes that increase the charge level in multiple cells simultaneously ( instead of erasing the block ) . [ thm : gamma1]the self - randomized modulation code has storage efficiency @xmath172 when @xmath173 and @xmath174 when @xmath175 as @xmath4 goes to infinity with high probability ( i.e. , @xmath168 ) . consider the problem of throwing @xmath129 balls into @xmath4 bins and let the r.v . @xmath10 be the number of balls thrown into @xmath4 bins until some bin has more than @xmath9 balls in it . while we would like to calculate @xmath176 $ ] exactly , we still settle for an approximation based on the following result . if @xmath177 , then there is a constant @xmath178 such that maximum number of balls @xmath130 in any bin satisfies@xmath179 with probability @xmath168 as @xmath171 @xcite . the constant @xmath178 is given by the largest @xmath180-root of@xmath181 and solving this equation for @xmath182 gives the implicit expression @xmath183 . since the lower bound matches the expected maximum value better , we define @xmath184 and apply it to our problem using the equation @xmath185 or @xmath186 . therefore , the storage efficiency is @xmath187 if @xmath188 , the maximum load is approximately @xmath189 with probability @xmath168 for large @xmath4 @xcite . by definition , therefore , the storage efficiency is @xmath191 the results in theorem [ thm : gamma1 ] show that when @xmath0 is on the order of @xmath192 , the storage efficiency is on the order of @xmath193 . taking the limit as @xmath194 with @xmath195 , we have @xmath196 when @xmath0 is a constant independent of @xmath4 , the storage efficiency is on the order of @xmath197 taking the limit as @xmath171 with @xmath173 , we have @xmath198 . in this regime , the self - randomized modulation codes actually perform very poorly even though they are asymptotically optimal as @xmath1 . considering the bins - and - balls problem , can we distribute balls more evenly when @xmath199 is on the order of @xmath200 fortunately , when @xmath201 , the maximal load can be reduced by a factor of roughly @xmath202 by using _ the power of two random choices _ @xcite . in detail , the strategy is , every time we pick two bins independently and uniformly at random and throw a ball into the less loaded bin . by doing this , the maximally loaded bin has roughly @xmath203 balls with high probability . theorem 1 in @xcite gives the answer in a general form when we consider @xmath204 random choices . the theorem shows there is a large gain when the number of random choice is increased from 1 to 2 . beyond that , the gain is on the same order and only the constant can be improved . based on the idea of 2 random choices , we define the following load - balanced modulation code . again , we let the cell state vector at time @xmath8 be @xmath84 , where @xmath17 is the charge level of the @xmath18-th cell at time @xmath19 this time , we use @xmath205 cells to store a @xmath5-variable @xmath206 ( i.e. , we write @xmath207 bits to store @xmath208 bits of information ) . the information loss provides @xmath6 ways to write the same value . this flexibility allows us to avoid sequences of writes that increase one cell level too much . we are primarily interested in binary variables with 2 random choices or @xmath209 . for the power of @xmath6 choices to be effective , we must try to randomize ( over time ) , the @xmath6 possible choices over the set of all @xmath210 possibilities . the value @xmath91 is used to do this . let @xmath211 be the galois field with @xmath212 elements and @xmath213 be a bijection that satisfies @xmath214 ( i.e. , the galois field element 0 is associated with the integer 0 ) . the decoding algorithm calculates @xmath215 from @xmath27 and operates as follows : * step 1 : read cell state vector @xmath27 and calculate the @xmath90 norm @xmath91 . * step 2 : calculate @xmath216 and @xmath217 * step 3 : calculate @xmath218 and @xmath219 * step 4 : calculate @xmath220 . the encoding algorithm stores @xmath11 and operates as follows . * step 1 : read cell state @xmath26 and decode to @xmath221 and @xmath96 . if @xmath222 then do nothing . * step 2 : calculate @xmath223 , @xmath218 , and @xmath219 * step 3 : calculate @xmath224 and @xmath225 for @xmath226 . * step 4 : calculate @xmath227 . increase the charge level by 1 of cell @xmath228 . note that the state vector at @xmath85 is initialized to @xmath229 and therefore @xmath87 . the first arbitrary value that can be stored is @xmath230 . the following conjecture suggests that the ball - loading performance of the above algorithm is identical to the random loading algorithm with @xmath209 random choices . [ thm : gamma2]if @xmath209 and @xmath175 , then the load - balancing modulation code has storage efficiency @xmath231 with probability 1-@xmath232 as @xmath171 . if @xmath233 the storage efficiency @xmath234 with probability 1-@xmath232 . [ sketch of proof ] consider the affine permutation @xmath235 for @xmath236 and @xmath237 . as @xmath238 vary , this permutation maps the two elements @xmath11 and @xmath239 uniformly over all pairs of cell indices . after @xmath240 steps , we see that all pairs of @xmath238 occur equally often . therefore , by picking the less charged cell , the modulation code is almost identical to the random loading algorithm with two random choices . unfortunately , we are interested in the case where @xmath241 so the analysis is somewhat more delicate . if @xmath177 , the highest charge level is @xmath242 with probability @xmath168 @xcite . since @xmath175 in this case , the storage efficiency is @xmath243 . if @xmath188 , then @xmath173 and the maximum load is @xmath244 . by definition , we have @xmath245 therefore , we have @xmath246 if @xmath209 and @xmath0 is on the order of @xmath247 conjecture [ thm : gamma2 ] shows that the bound ( [ eq : storage_efficiency_bound ] ) is achievable by load - balancing modulation codes as @xmath4 goes to infinity . in this regime , the load - balancing modulation codes provide a better constant than self - randomized modulation codes by using twice many cells . [ rem : if @xmath209 and @xmath0 is a constant independent of @xmath4 , the storage efficiency is @xmath248 for the self - randomized modulation code and @xmath249 for the load - balancing modulation code . but , the self - randomized modulation code uses @xmath250 cells and the load - balancing modulation code uses @xmath251 cells . to make fair comparison on the storage efficiency between them , we let @xmath251 for both codes . then we have @xmath252 and @xmath249 . so , as @xmath171 , we see that @xmath253 . therefore , the load - balancing modulation code outperforms the self - randomized code when @xmath4 is sufficiently large . in this section , we present the simulation results for the modulation codes described in sections [ sub : another - rewriting - algorithm ] and [ sub : an - enhanced - algorithm ] . in the figures , the first modulation code is called the `` self - randomized modulation code '' while the second is called the `` load - balancing modulation code '' . let the `` loss factor '' @xmath254 be the fraction of cell - levels which are not used when a block erasure is required : @xmath255}{n(q-1)}.$ ] we show the loss factor for random loading with 1 and 2 random choices as comparison . note that @xmath254 does not take the amount of information per cell - level into account . results in fig . [ flo : fig2 ] show that the self - randomized modulation code has the same @xmath254 with random loading with 1 random choice and the load - balancing modulation code has the same @xmath254 with random loading with 2 random choices . this shows the optimality of these two modulation codes in terms of ball loading . , @xmath209 and 1000 erasures.[flo : fig2 ] ] with @xmath80 , @xmath256 @xmath257 and 1000 erasures.[flo : fig4 ] ] .[fig : fig5 ] ] .[fig : fig6 ] ] we also provide the simulation results for random loading with 1 random choice and the codes designed in @xcite , which we denote as flm-(@xmath258 ) algorithm , in fig . [ flo : fig4 ] . from results shown in fig . [ flo : fig4 ] , we see that the flm-(@xmath258 ) algorithm has the same loss factor as random loading with 1 random choice . this can be actually seen from the proof of asymptotic optimality in @xcite as the algorithm transforms an arbitrary input distribution into an uniform distribution on the cell - level increment . note that flm algorithm is only proved to be optimal when 1 bit of information is stored . so we just compare the flm algorithm with random loading algorithm in this case . [ fig : fig5 ] and fig . [ fig : fig6 ] show the storage efficiency @xmath33 for these two modulation codes . [ fig : fig5 ] and fig . [ fig : fig6 ] show that the load - balancing modulation code performs better than self - randomized modulation code when @xmath4 is large . this is also shown by the theoretical analysis in remark [ rem : if is ] . in this paper , we consider modulation code design problem for practical flash memory storage systems . the storage efficiency , or average ( over the distribution of input variables ) amount of information per cell - level is maximized . under this framework , we show the maximization of the number of rewrites for the the worst - case criterion @xcite and the average - case criterion @xcite are two extreme cases of our optimization objective . the self - randomized modulation code is proposed which is asymptotically optimal for arbitrary input distribution and arbitrary @xmath5 and @xmath6 , as the number of cell - levels @xmath1 . we further consider performance of practical systems where @xmath0 is not large enough for asymptotic results to dominate . then we analyze the storage efficiency of the self - randomized modulation code when @xmath0 is only moderately large . then the load - balancing modulation codes are proposed based on the power of two random choices @xcite @xcite . analysis and numerical simulations show that the load - balancing scheme outperforms previously proposed algorithms .

in this paper , we consider modulation codes for practical multilevel flash memory storage systems with @xmath0 cell levels . instead of maximizing the lifetime of the device @xcite , we maximize the average amount of information stored per cell - level , which is defined as storage efficiency . using this framework , we show that the worst - case criterion @xcite and the average - case criterion @xcite are two extreme cases of our objective function . a self - randomized modulation code is proposed which is asymptotically optimal , as @xmath1 , for an arbitrary input alphabet and i.i.d . input distribution . in practical flash memory systems , the number of cell - levels @xmath0 is only moderately large . so the asymptotic performance as @xmath1 may not tell the whole story . using the tools from load - balancing theory , we analyze the storage efficiency of the self - randomized modulation code . the result shows that only a fraction of the cells are utilized when the number of cell - levels @xmath0 is only moderately large . we also propose a load - balancing modulation code , based on a phenomenon known as `` the power of two random choices '' @xcite , to improve the storage efficiency of practical systems . theoretical analysis and simulation results show that our load - balancing modulation codes can provide significant gain to practical flash memory storage systems . though pseudo - random , our approach achieves the same load - balancing performance , for i.i.d . inputs , as a purely random approach based on the power of two random choices .

0910.2005

stephan s quintet ( hereafter sq ) is a strongly interacting compact group which has produced a highly disturbed intragroup medium ( igm ) @xcite through a complex sequence of interactions and harrassment @xcite . this interplay has produced a large - scale intergalactic shock - wave , first observed as a narrow filament in the radio continuum @xcite , and subsequently detected in the x - ray @xcite . the high - velocity ( @xmath61000 km s@xmath13 ) collision of an intruder galaxy , ngc 7318b , with the intergalactic medium of the group @xcite is believed to be responsible for the shock - heating of the x - ray emitting gas . optical emission line ratios and observed broad linewidths provide evidence that the region is powered by strong shocks and not star formation ( xu et al . 2003 ; duc et al . 2010 , in preparation ) . the main elements of stephan s quintet are shown in figure 1 . central to the system is the primary shock , as defined by the 20 cm radio continuum emission . ngc 7318b , the intruder galaxy , lies to the west of the shock , while the large seyfert 2 galaxy @xcite , lies to the east . other members of the group are also indicated . the peculiar extranuclear star formation region , named sq - a @xcite , lies at the extreme northern end of the main shock wave . ngc 7318a ( west of ngc 7318b ) is also a strongly interacting group member , with ngc 7317 further away from the core . the unexpected discovery of extremely powerful , pure - rotational h@xmath9 line emission from the center of the shock @xcite , using the _ spitzer space telescope _ @xcite , has sparked intense interest in the sq system . warm molecular hydrogen emission was found with a line luminosity exceeding the x - ray luminosity from the shock . the mid - infrared ( mir ) h@xmath9 linewidth was resolved ( @xmath14 870 km s@xmath13 ) suggesting that the h@xmath9-emitting clouds carry a large bulk - kinetic energy , tapping a large percentage of the energy available in the shock . a recent model of sq , involving the collision between two inhomogeneous gas flows , describes h@xmath9 formation out of the multiphase , shocked gas , and an efficient cooling channel for high - speed shocks as an alternative to x - ray emission @xcite since the sq detection , several other systems exhibiting similarly strong h@xmath9 emission have been discovered . @xcite find that a large - subset of the local 3cr radio galaxies have extremely dominant mir rotational h@xmath9 lines , often seen against a very weak thermal continuum . the low agn and star formation power are insufficient to drive the mir h@xmath5 emission . mechanical heating driven by the radio jet interaction with the host galaxy ism is the favoured mechanism . in addition , total h@xmath9 luminosities in the range @xmath15 ergs s@xmath13 have been detected in some central cluster galaxies out to @xmath16 ( * ? ? ? * g. de messieres - university of virginia , private communication ) and in filaments in clusters @xcite . the study of nearby prototypes may provide valuable insight into the nature of these more distant systems . the large scale ( @xmath6 30 kpc ) of the sq shock is well - suited for such a study . in this paper i , we extend the single pointing observations of @xcite to full spectral maps of sq using the irs instrument on the _ spitzer _ space telescope , hereafter _ spitzer_. in paper ii , we will present detailed 2-dimensional excitation maps of the h@xmath5 emission across the face of the x - ray emitting shock , and compare them with models . in addition , several other papers are being prepared by our team which will discuss the relationship between the uv / x - ray emission and emission from dust . in section 2 , we present our observations and data reduction methods . in section 3 we discuss the mapping results for various detected lines . in sections 4 , 5 and 6 we present our discussion on the properties of the system and in section 7 the implication for galaxy formation . in section 8 we present our conclusions . additional material is included as appendices , with a discussion on ngc 7319 in appendix a , and a reanalysis of the high resolution mir spectrum of the shock , as well as renanalysis of the x - ray data presented in appendix b and c , respectively . throughout this paper , we assume a systemic velocity of @xmath17 for the group , corresponding to a distance of 94 mpc with @xmath18 . mid - infrared spectroscopy of the shock region in sq was obtained using the irs instrument @xcite onboard _ spitzer_. observations were done in low - resolution mapping mode , using the short - low ( @xmath19 ; @xmath20 ) and long - low ( @xmath21 ; @xmath22 ) modules and taken on january 11 2008 and december 10 2007 , respectively . figure 1 displays the outline of the areas observed superimposed on a composite image of the group . the sl spectral mapping consists of two separate , partially overlapping maps , centered north and south on the x - ray emission associated with the shock . the map was constructed with 23 steps of 2.8(0.75 @xmath2 slit width ) perpendicular to the slit and one parallel step of 7.2 . observations consisted of 60s integrations with 5 cycles per step . the ll module was used to map an area of @xmath23 using 21 steps of 8.0 ( 0.75 @xmath2 slit width ) perpendicular to the slit and a parallel step of 24.0 . an integration time of 120s was used with 3 cycles per step . primary data reductions were done by the _ spitzer _ science center ( ssc ) pipeline , version s17.0.4 and s17.2.0 for sl and ll respectively , which performs standard reductions such as ramp fitting , dark current subtraction and flat - fielding . background subtraction for ll data was performed by subtracting dedicated off - source observations , with the same observing mode , taken shortly after the mapping sequence . in the case of sl , where for scheduling reasons the dedicated `` off '' observations were too far away in time to be optimal , backgrounds were generated from observations at the periphery of the map that contain no spectral line signatures . examination of the pipeline products showed that the stray - light - corrected images ( that account for potential spillover from the peak - up arrays onto the sl1/2 spectral apertures ) contained wavelength - dependent over - corrections in some of the images , seemingly due to a high background of cosmic rays during the sl portion of the mapping . to negate this effect , the alternative flat - fielded images ( also available from the science pipeline ) that are uncorrected for crosstalk and straylight removal were used . it was determined that after background subtraction , any uncorrected stray light in the sl spectral extraction areas was unmeasurable at the @xmath24% level . thus stray - light correction was unneccesary , and the resulting basic calibrated data ( bcd ) images used were of better quality than the standard bcds . for all modules , individual frames for each pointing were median - combined and obvious `` bad '' pixels were replaced using customised software that allows for manual `` average '' replacement . the spectra were assembled into spectral cubes for each module using the software tool , cubism @xcite . further bad pixel removal was performed within cubism . spectral maps were generated by making continuum maps on either side of a feature and subtracting the averaged continuum map from the relevant emission line map . one dimensional spectra were further extracted from the data cubes using matched apertures . broad - band images at 16 and 24@xmath25 m were obtained with the _ spitzer _ irs blue peak - up imager ( pui ) and the mips instrument . the pui was obtained in a 5 x 5 map with 3 cycles of 30s duration each on 2007 december 10 . the mips instrument @xcite on _ spitzer _ obtained 24 imaging of sq on 2008 july 29 , achieving a spatial resolution of @xmath66 . primary data reduction was done by the _ spitzer _ science center ( ssc ) science pipeline ( version s18.0.2 ) run through the mopex software , and for the mips image a smooth 2d polynomial background was further removed to correct for a large - scale background gradient . _ spitzer _ irac 3.6 , 4.5 , 5.8 and 8.0 data of sq ( p.i . houck ) were obtained from the ssc archive ; these were reduced using science pipeline version s18.0.2 . the final mosaics have a pixel scale of 0.61 . the pure rotational transitions of molecular hydrogen can be excited by several mechanisms . these include fuv ( far ultraviolet ) induced pumping , and possible additional collisional heating , of the h@xmath5 in photodissociation regions associated with star formation ( e. g. black & van dishoeck 1987 , hollenbach & tielens 1997 ) , hard x - rays penetrating and heating regions within molecular clouds , which in turn excite h@xmath5 via collisions with electrons or hydrogen atoms ( lepp & mccray 1983 ; draine & woods 1992 ) and finally collisional excitation of h@xmath5 due to acceleration produced by shocks ( e.g. shull & hollenbach 1978 ) . the pure rotational mir line ratios are not especially good diagnostics for distinguishing between these mechanisms since all three mechanisms discussed can lead to well thermalized level distributions of lower - level rotational states . the rotational h@xmath5 emission lines do , however , allow us to trace gas at different temperatures and compare with model predictions ( this will be the main emphasis of paper ii ) . higher level transitions 0 - 0 s(3)-s(5 ) tend to trace warmer gas , whereas the s(0 ) and s(1 ) lines are sensitive to the `` coolest '' warm h@xmath5 . although the line ratios themselves can not be used directly as shock diagnostics , in stephan s quintet the distribution of large - scale x - ray and radio emission , plus optical emission line diagnostics , provide strong evidence that the giant filament seen in figure 1 is the result of a strong shock . in appleton et al . ( 2006 ) this fact was used to reveal the surprising association of detected h@xmath5 emission with the shock . however , in this paper we can make a more definitive association of the emission with the shock by means of spectral maps . the spectral cubes were used to extract maps of all the pure rotational emission lines of molecular hydrogen that were detected , namely the 0 - 0 s(0)28.22 , s(1)17.03 , s(2)12.28 , s(3)9.66 , s(4)8.03 and s(5)6.91 lines . specific intensity contour maps of these lines are presented in figure 2 overlaid on a @xmath26-band image of the region from @xcite . the s(0 ) and s(1 ) lines were mapped by the ll modules , while s(2 ) - s(5 ) transitions were mapped by the sl modules of the irs . as indicated in figure 1 , the sl observations were concentrated on the main shock to provide high signal to noise ( s / n ) measurements there . as a result , these maps do not fully cover sq - a or ngc 7319 . we note that the s(4 ) line at 8.03 ( fig . 2e ) is faint , and also suffers from contamination from the pah bands at 7.7 and 8.6 . the contours indicate powerful , widespread emission running north - south along the shock ridge ( see fig . 1 ) . in addition we detect strong emission from the star forming region , sq - a , as well as associated with ngc 7319 . we discuss further details of the agn - like mir emission lines from ngc 7319 in appendix a. figures 2a - d reveal a new h@xmath5 structure running eastward from the `` main '' shock ridge . in what follows , we refer to this feature as the `` bridge '' . this structure is observed faintly in the _ chandra _ @xcite and xmm @xcite x - ray images and detected as faint h@xmath27 emission by @xcite , but is not strong in radio continuum images . as is evident in figure 2 , there is distinct variation in the distribution of the warm h@xmath5 emission . the brightest 0 - 0 s(0 ) emission ( fig . 2a ) appears to be concentrated towards the north of the shock , whereas the 0 - 0 s(1 ) transition emission ( fig . 2b ) appears more concentrated towards the center . the s(0 ) and s(1 ) maps demonstrate that the h@xmath5 in the bridge terminates in a large clump a few arcsecs west of the nucleus of the seyfert 2 galaxy ngc 7319 , and in a small detour to the north ( especially in the s(1 ) map which has the highest s / n ) . the s(2 ) through s(5 ) lines clearly show that the warm h@xmath5 emission breaks into clumps in the shock . despite the limited coverage compared to the ll mapping , the base of the `` bridge '' is visible and sq - a is partially covered . sq - a is fully covered by the sl2 module ( because of fortuitous `` off observation '' coverage ) and hence the s(5 ) emission line reveals that in sq - a the h@xmath5 emission is also clumpy . the molecular - line maps provide considerable information about the excitation of the h@xmath5 along and across the x - ray shock , but these discussions will be deferred to a full 2-d modelling of the h@xmath5 excitation in paper ii ( appleton et al . , in preparation ) . instead we shall limit ourselves to global properties of the h@xmath5 here . in section [ spec ] we shall present spectra of some selected regions of the emission and discuss a global excitation diagram for the shock . to further demonstrate the close connection between the h@xmath5 emission and the main global shock - wave in sq , we now consider the distribution of warm h@xmath5 in relation to the x - ray and radio emission . figure [ fig : xray_rad]a and b show the s(0 ) and s(3 ) contours overlaid on an xmm - newton x - ray image @xcite of stephan s quintet . the warm molecular hydrogen is distributed along the length of the main north - south ( ns ) x - ray shock ridge and along the `` bridge '' , demonstrating the remarkable projected coexistence of of hot x - ray plasma ( 10@xmath28 @xmath29 t @xmath29 10@xmath30 k ) and warm h@xmath5 ( 10@xmath31 @xmath29 t@xmath29 10@xmath32 k ) . although the h@xmath5 appears to follow the x - ray , there are subtle differences . the cooler s(0 ) line has emission concentrated to the north and follows the x - ray less closely compared to the warmer s(3 ) line . the s(3 ) line exhibits a clear correspondence to the x - ray , notably at the center of the shock , where we find peaks at both wavelengths . thus the region of greatest shock heating , as traced by the stronger x - ray emission , appears to correspond to the higher - j h@xmath5 transitions , perhaps implying a causal connection . the intergalactic star formation region sq - a , is essentially absent in x - ray emission , as seen in figure [ fig : xray_rad]a , but is strongly detected in h@xmath9 . we find a similar picture in the radio continuum ( fig . [ fig : xray_rad]c & d ) with the s(0 ) line demonstrating correspondence with the main shock , but dominated by emission in the north where we observe less powerful radio emission . the s(3 ) ( and s(2)see fig . 2c ) line presents a much tighter correlation with regions of the shock that are more radio luminous than the lower - j transitions . the radio emission is quite likely sensitive to the most compressed regions of the shock where cosmic ray particles are accelerated more strongly @xcite , whereas the brightest x - ray patches are likely due to the fastest regions of the shock @xcite . as already mentioned the `` bridge '' emission is detected only faintly in the x - ray and is weak or absent at radio continuum wavelengths . this noticeable difference compared to the main shock likely implies that the conditions that give rise to strong synchrotron emission in the main shock are absent in the bridge . emission from fine - structure lines provide key diagnostics that trace the interplay between the various constituents of the shocked interstellar medium . in @xcite the spectra were limited to the very core of the shock and only weak emission was detected from all but two metal lines , namely @xmath10neii@xmath11$]12.81and @xmath10siii@xmath11$]34.82 ( this data has been reanalysed and is presented in appendix b ) . in this section we discuss the spatial distribution of emission from the @xmath10feii@xmath11$]25.99 , @xmath10oiv@xmath11$]25.89 , @xmath10siii@xmath11$]33.48 , @xmath10siii@xmath11$]34.82 , @xmath10neii@xmath11$]12.81 and @xmath10neiii@xmath11$]15.56 lines . in figure [ fig : fs ] we present the specific intensity contours of the @xmath10feii@xmath11$]25.99 and @xmath10oiv@xmath11$]25.89 blend , @xmath10siii@xmath11$]33.48 and @xmath10siii@xmath11$]34.82 emission lines . given the low spectral resolution of the sl and ll modules of _ spitzer _ , we can not distinguish between emission from @xmath10fe ii@xmath11$]25.99 and @xmath10oiv@xmath11$]25.89 . however , except in the direction of the seyfert ii galaxy ngc 7319 , the emission near 26 seen in figure [ fig : fs]a is likely to be pure @xmath10feii@xmath11 $ ] with little contamination from @xmath10oiv@xmath11 $ ] as there is no evidence from the spectra of high - excitation emission from the intragroup medium in sq . for example , @xmath10oiv@xmath11 $ ] has an excitation potential of 56 ev ( compared to 7.9 ev of @xmath10feii@xmath11 $ ] ) , and yet @xcite have shown that the @xmath10neiii@xmath11$]/@xmath10neii@xmath11 $ ] ratio is low suggesting low - excitation conditions for the ions in the shock , further supported in section [ em ] . assuming that , apart from towards ngc 7319 , @xmath10feii@xmath11 $ ] dominates the @xmath10feii@xmath11$]+@xmath10oiv@xmath11 $ ] complex , we detect faint emission from @xmath10feii@xmath11$]25.99 at the location of the center of the shock ( as defined by the x - ray `` hotspot '' in fig . [ fig : xray_rad]a ) . the energetic requirements for shocks to produce strong @xmath10feii@xmath11 $ ] emission are usually present in @xmath33 shocks while the ion abundance in @xmath34 shocks are low in comparison @xcite . we will discuss the production of @xmath10feii@xmath11 $ ] in the shock in section 6.2 . the @xmath10siii@xmath11$]33.48 distribution is presented in figure [ fig : fs]b . this fine structure line acts as a strong tracer of hii regions @xcite and we observe emission from sq - a and from other regions of star formation in the south ( see section [ sf ] ) . in the primary shock region the distributions of @xmath10feii@xmath11 $ ] and @xmath10siii@xmath11 $ ] are anti - correlated . in strong contrast to the weak @xmath10feii@xmath11 $ ] emission , there is copious @xmath10siii@xmath11$]34.82 emission ( fig . [ fig : fs]c ) , which follows the s(1 ) distribution closely with respect to the primary shock , as mapped by x - ray emission ( fig . [ fig : fs]d ) . although @xmath10siii@xmath11 $ ] is commonly found in normal hii regions , we will demonstrate below that , apart from in sq - a , the strong silicon emission does not correlate with regions of strong pah emission ( tracing star formation ) in sq , but instead closely follows the h@xmath5 and x - ray ( shock ) distributions . @xmath10siii@xmath11 $ ] acts as an efficient coolant of x - ray - irradiated gas and is predicted to be one of the top four cooling lines under these circumstances @xcite . we will discuss the excitation of @xmath10siii@xmath11 $ ] in the shock in section 6.2 . @xmath10neii@xmath11$]12.81 ( with an ionisation potential of 21.6 ev ) is also represented in the shock , as shown in figure [ fig : ne]a , although it is also emitted from some hii regions associated with star formation , such as sq - a and the star - forming region south of ngc 7318b ( see section [ sf ] ) . the @xmath10neiii@xmath11$]15.56 contours ( with an ionisation potential of 41 ev ) are shown in figure [ fig : ne]b and are associated with excitation in the shock , as well as from regions of star formation ( see section [ sf ] ) . the higher ionisation line of @xmath10neiii@xmath11 $ ] is much weaker in the shock ridge , but regions of emission correspond closely to peaks seen for the h@xmath5 s(3 ) line . the @xmath10neii@xmath11 $ ] emission , however , exhibits clearly extended emission with regions of greatest luminosity matching those seen for the h@xmath5 s(3 ) line . the @xmath10neii@xmath11 $ ] emission suggests excitation from the shock , with @xmath10neiii@xmath11 $ ] found at the location of the center of the shock , similar to what is observed for @xmath10feii@xmath11 $ ] . previous observations and spectroscopy of sq @xcite have determined that there are some regions of star formation associated with the spiral arms of the intruder galaxy ngc 7318b . we discuss in this section how these regions , which have a different spatial distribution from the shocked gas , are correlated with the pah emission we detect in the irs spectra . in figure [ fig : dust]a there is a strong correlation between the 11.3 pah distribution ( from the irs cube ) superimposed on a near - uv image from _ galex _ @xcite , which maps the uv emission from hot stars associated with weak star formation from the system . this correlation suggests that the pah molecules are excited by star formation . a similar close correlation is shown in figure [ fig : dust]b where we overlay the 11.3 contours on the irac 8 band , which is dominated by the 7.7 and 8.6 @xmath25 m feature . it is noticeable , however , that regions with strong 11.3 emission in the shock , do not appear similarly strong at 8 . this point will be addressed in section 6.4 . we compare the 11.3 pah map to the distribution of warm dust in figure [ fig : dust]c , using the mips 24@xmath25 m map of sq . again , it is clear that there is a good correlation between the pah emission and the thermal dust , most of which seems only poorly correlated with the shock ridge . the lack of conspicuous star formation in the ridge was also observed by @xcite . the main point we emphasize here is that the dust , pah and uv emission appears to be associated with previously known star formation regions and no additional star formation is observed in the shock ; this can be seen in figure [ fig : dust]d showing the h@xmath5 0 - 0 s(1 ) emission overlaid on an irac 8 image . there is little correspondence between the h@xmath5 emission in the shock and 8@xmath25 m ( hot dust plus pah ) image . this is important because it implies that there is very little triggered star formation in the molecular gas associated with the shock - excited h@xmath5 . figure [ fig : dust]c demonstrates that there is only faint dust emission at 24@xmath25 m from the shock ridge . the presence of dust in the shock is required in the model of guillard et al . ( 2009 ) to explain the formation of h@xmath5 behind the shock and we do observe evidence of depletion onto dust grains ( see section 6.2 ) . thus the faint 24 emission could be the result of destruction of very small grains ( vsgs ) , with only larger grains surviving , or indicate that the grains are cold and radiating more strongly at longer wavelengths , where _ spitzer _ has the least spatial resolution . a more detailed description of the dust and faint pah emission in the sq group ( including results from mips 70 imaging ) is discussed in separate papers ( guillard et al . 2010 . ; natale et al . 2009 , in preparation ) . guillard et al . ( 2010 ) show that the ir emission in the shock is faint due to dust being heated by a relatively low intensity uv radiation field and determine a galactic pah / vsg abundance ratio in this region . a more complete understanding of the likely existence of cool dust in the shock will require higher angular resolution and a broader wavelength coverage than that achieved by _ spitzer_. despite the faintness of emission from the main shock , the 24@xmath25 m map presents a new result , which was less obvious in previous studies , namely that the dominant regions of star formation in sq lie not in the galaxies themselves , but in two strikingly powerful , almost symmetrically disposed regions at either end of the shock . the region to the north is the well studied sq - a , but the region to the south ( which we refer to as 7318b - south ) is also very powerful and both regions lie at the ends of the shock , as defined by the h@xmath5 distribution . this may not be a coincidence , and we will discuss this further in section 6.1 . the @xmath10neii@xmath11$]12.81 , @xmath10neiii@xmath11$]15.56 and @xmath10siii@xmath11$]33.48 fine - structure lines , as mentioned above , are also tracers of star formation in sq as these lines are often associated with hii regions . the @xmath10neii@xmath11$]12.81 emission appears to follow _ both _ the h@xmath5 and the star formation regions ( see fig . [ fig : ne]a ) , appearing more extended in the south than the corresponding h@xmath5 emission and flaring out where star formation regions , especially 7318b - south , are observed optically , and through pah emission ( see fig . [ fig : dust ] ) . the warm molecular hydrogen in stephan s quintet follows the x - ray distribution in the main shock and in the `` bridge '' structure . this might suggest that the molecular hydrogen is excited directly by the x - ray heating . however , we will show that the h@xmath5 emission exceeds by at least a factor of 3 the x - ray luminosity from the various shocked filaments , thus ruling out direct x - ray excitation from the shock . to measure the strength of the h@xmath5 emission , we extract spectra from various rectangular sub - areas of sq which are defined in figure [ fig : ext ] . the spectra were extracted from cubism cubes built from each irs module , and joined to make a continous spectrum no scaling was necessary to join the spectra . figures [ fig : ext ] a , b and c indicate the spectral extraction regions of the main ns shock , a sub - region of the shock and a characteristic part of the `` bridge '' . the shock sub - region is chosen to be just north of the center of the shock , avoiding regions contaminated by star formation in the intruder . spectra for these extractions are shown in figure [ fig : spectra ] a , b and c. all three spectra , share the common property that they are dominated by molecular hydrogen emission . the shock sub - region ( fig . [ fig : spectra]b ) , unlike figure [ fig : spectra]a , is less contaminated by the star forming regions discussed in the previous section . the mir continuum of the main shock appears stronger than the shock sub - region indicating stronger emission from warm dust ; this is likely the result of contamination from star - forming regions in the main shock extraction . the bridge exhibits a similarly weak continuum emission compared to the shock sub - region . figure [ fig : spectra]b includes photometry from the irac bands , 16 peak - up image ( pui ) and mips-24 image superimposed on the irs spectrum . these are useful to probe conditions in the shock , in particular star formation ( see section 6.4 ) . stellar light in the extraction area , from an extended spiral arm of ngc 7318b , produces contamination of the shock spectrum , visible as continuum emission shortwards of 6 , in both figure 8a and b. there exists a striking similarity between the mid - ir spectrum of all three regions , showing powerful h@xmath5 lines and low excitation weak emission from fine structure lines . also the pah emission observed in the spectra of the shock regions , and in the region of the `` bridge '' in the irac 8 image ( fig . [ fig : dust]b ) , appears weak . this confirms that these properties , observed in the @xcite observations of the shock core , extend to both the full extent of the main shock and the `` bridge '' . this , and the fact that the `` bridge '' has similar x - ray properties to the main shock ( see later ) suggests that the bridge is a `` scaled - down '' version of the main shock . the weaker radio continuum emission at this location is significant . one possibility , that the `` bridge '' is older than the main shock , and the cosmic rays compressed in it have diffused away , will be discussed further in paper ii.thus the new irs observations seem to suggest that more than one large - scale group - wide shock is present in the group . this could be the result of previous tidal interactions and imply multiple shock heating events have taken place in sq , consistent with what is seen in the x - ray @xcite . figures [ fig : ext]d and [ fig : spectra]d present the extraction region and spectrum of sq - a , the extragalactic star - forming region . in this case , although h@xmath5 lines are still strong , a rising continuum and an increase in the strength of the metal lines relative to the h@xmath5 is consistent with a spectrum that is increasingly dominated by star formation a result which is already known from previous optical observations @xcite . line fluxes for all the h@xmath5 and metal lines in the spectra discussed above are presented in table [ tableh2fluxes ] and table [ tablemetalfluxes ] , respectively . we estimate the luminosity emitted from the h@xmath5 lines in the main aperture shown in figure [ fig : spectra]a . the emission from the 0 - 0 s(1 ) line alone can be calculated from table [ tableh2fluxes ] ( for d = 94 mpc ) to be 2.3 @xmath210@xmath35 ergs@xmath13 . summing the emission measured in the observed lines for the main shock ( 0 - 0 s(0 ) through s(5 ) lines ) and including an extra 28% emission from unobserved lines ( see model fit to excitation diagram below ) , yields a total h@xmath5 line luminosity from the main shock of 9.7 @xmath210@xmath35 ergs@xmath13 . this phenomenal power in the molecular hydrogen lines dwarfs by a factor of ten the next brightest mid - ir line , which is @xmath10siii@xmath11$]34.82 with a line luminosity of l@xmath36 = 0.85 @xmath210@xmath35 ergs@xmath13 . figure [ fig : excite ] presents the excitation diagram of the low - j 0 - 0 h@xmath5 transitions for the main shock extraction . the points are well fit by a model including three temperature components ( t@xmath37 = 158@xmath38k , t@xmath9 = 412@xmath39k , and t@xmath40 = 1500@xmath41k ) . it is likely that in reality , many different temperature components are present in the shock , and the three - temperature fit is only an approximation . however , it does allow us to provide an estimate of the total mass of warm h@xmath5 of 5.0@xmath4210@xmath43m@xmath44 . temperature t@xmath9 is more uncertain than formally represented by the fit because it depends on the value of the s(4 ) flux , which may be systematically too low due to pah contamination ( see fig . [ fig : excite ] ) . in paper ii we will present a more complete two - dimensional map of the excitation of the h@xmath5 in sq and explore variations in the shape of the excitation diagram along and across the shock in more detail . our observations have shown that h@xmath5 is the dominant line coolant in the mir . however , how does it compare with the most important coolant in high - speed shocks namely the x - ray emission ? @xcite suggested that the h@xmath5 emission was stronger than the x - ray emission at the shock center . this can now be evaluated over much of the inner sq group . we present a complete reanalysis of the xmm - newton observations of sq using the latest calibrations ( see appendix c for full details ) in order to determine the fluxes and luminosities of the x - ray emission to match our spectral extractions . the results indicate the striking dominance of the h@xmath5 line luminosities compared with the x - ray emission from the same regions . for the main shock , the x - ray `` bolometric '' flux of l@xmath45 = 2.8 @xmath2 10@xmath46erg s@xmath13@xmath47 corresponds to 2.95 @xmath2 10@xmath35 erg s@xmath13 , or l(h@xmath5)/l@xmath45 = 2.9 . this is a lower limit since we have not attempted to remove the contribution to the main shock aperture of an extended group - wide x - ray component upon which the emission from the shock lies . therefore it is likely that the h@xmath5 line luminosity dominates over the main shock x - ray gas by a factor @xmath48 3 . similar calculations can be done for the other regions for which h@xmath5 spectra have been extracted . for example , in the `` bridge '' region , which we have already indicated has many of the same characteristics as the main shock , we find l(h@xmath5)/l@xmath45 = 2.5 . these values demonstrate that throughout the extended regions of sq , the molecular hydrogen cooling pathway dominates over the x - ray in this shocked system . this is a very significant result , upturning the traditional view that x - ray emission always dominates cooling in the later stages of evolution in compact groups of galaxies . the fine - structure flux ratios ( see table [ tablemetalfluxes ] ) can be used to probe the conditions within the extracted regions of sq . the @xmath10siii@xmath11$]34.82/@xmath10siii@xmath11$]33.48 ratio provides an indication of the sources of excitation within the system . as mentioned previously , @xmath10siii@xmath11 $ ] is mainly a tracer of hii regions , whereas enhanced @xmath10siii@xmath11 $ ] emission can be generated via several mechanisms , including thermal excitation by x - rays ( xdrs ) , or in shocks . in the main shock , the @xmath10siii@xmath11$]34.82/@xmath10siii@xmath11$]33.48 ratio of @xmath6 4.59 is high compared to , for example , both normal galaxies ( @xmath6 1.2 ) and agn ( @xmath6 2.9 ) in the sings sample @xcite . however , this large aperture is contaminated by star formation emission from sq - a and the intruder galaxy . a better measure is given by the smaller shock sub - region , where the @xmath10siii@xmath11$]34.82/@xmath10siii@xmath11$]33.48 ratio is @xmath49 . thus it is clear that the @xmath10siii@xmath11 $ ] emission is well outside the normal range of values , even for local well - studied agn . using the upper limit found for @xmath10siii@xmath11$]33.48 in the `` bridge '' structure , we find a ratio of @xmath50 again values well outside the range of normal galaxy disk emission . indeed , these high values are typical of galactic supernova remnants where shock excitation is well determined ( e.g. * ? ? ? * ; * ? ? ? we will argue in the next section that silicon is being ionised in regions experiencing fast shocks @xmath51km@xmath0 and depleted onto dust grains . the @xmath10feii@xmath11$]26.0/@xmath10siii@xmath11$]34.82 ratio for the main shock and sub - region ( @xmath6 0.12 ) is in agreement with values found by @xcite for their sample of snr . for the extragalactic star forming region sq - a , @xmath10siii@xmath11$]34.82/@xmath10siii@xmath11$]33.48 is @xmath61.49 , only slightly higher than the average of @xmath6 1.2 found for star - forming regions in the sings sample @xcite ; another indication that sq - a is dominated by star formation . in star - forming galaxies the @xmath10neiii@xmath11$]15.56/@xmath10neii@xmath11$]12.81 ratio can be used as a measure of the hardness of the radiation field as it is sensitive to the effective temperature of the ionising sources . in the main shock , we find a value of @xmath52 which would be considered typical compared to those found in starburst systems , which range from @xmath53 @xcite , and in supernovae remnants ranging from @xmath54 @xcite . the shock subregion has a ratio of only @xmath55 indicating a lower intensity radiation field north of the shock center . however , it is clear from its spatial distribution relative to the 8 and h@xmath5 emission , that most of the neon is not originating from star formation , and so shocks are an obvious source of excitation . the @xmath10neii@xmath11$]12.81/@xmath10neiii@xmath11$]15.56 ratio can be used to estimate the shock velocity using the mappings shock model library of @xcite . in the main shock this ratio corresponds to shock speeds of between 100 and 300kms@xmath13 ( using preshock densities of @xmath56 @xmath57 and magnetic parameter @xmath58 = 1 and 3.23 the nominal equipartition value ) . we do , however , have contamination from starforming regions in the main shock and can not disentangle this emission from that produced in the shock . to address this we use the @xmath10neii@xmath11$]12.81 and @xmath10neiii@xmath11$]15.56 maps to mask areas associated with star formation ( see section 3.3 ) and determine a lower limit for the @xmath10neii@xmath11$]12.81/@xmath10neiii@xmath11$]15.56 ratio in the shock of @xmath6 4.54 . this value corresponds to a shock velocity of @xmath6 150kms@xmath13 . the average electron density is determined from the @xmath10siii@xmath11$]18.71/@xmath10siii@xmath11$]33.48 ( two lines of the same ionisation state ) ratio . for sq - a we find a ratio of @xmath60.56 , and in the main shock @xmath60.41 . this corresponds to an electron density of @xmath59 @xmath57 @xcite for both , i.e. in the low - density limit for this diagnostic @xcite . our observations have shown that the molecular hydrogen and x - ray emitting plasma appear to follow a similar distribution , and we have ruled out the possibility that this is a consequence of x - rays heating the h@xmath5 , since the h@xmath5 has the dominant luminosity . how then can we explain the similar distributions ? are these results consistent with the hypothesis that the shock is formed where the intruding galaxy ngc 7318b collides with a pre - existing tidal filament of hi drawn out of ngc 7319 in a previous interaction with another group member @xcite ? this basic mechanical picture appears plausible as can be seen in figure [ fig : hi+h2 ] which shows that the h@xmath5 distribution `` fills in '' the gap in the hi tidal tail as observed by the vla @xcite . the implication is that the hi has been converted into both a hot x - ray component and a warm h@xmath5 component by the collision of the intruder with the now missing hi . part of the puzzle of how this high - speed ( @xmath60 @xmath61 kms@xmath13 ) shock can lead to both x - ray and very strong molecular line emission is presented in a model by @xcite . the high - speed collision of ngc 7318b with the hi filament ( assumed to be composed of a multiphase medium ) leads to multiple shocks passing through and compressing denser clumps ( which become dusty nucleation sites for h@xmath5 formation ) as opposed to the lower - density gas , which is shock - heated to x - ray tempertures . the h@xmath5 therefore forms in denser clouds experiencing slower shocks . thus the coexistence of both hot x - ray gas , and cooler molecular material is a natural consequence of the multiphase medium of the pre - shocked material . modeling of the h@xmath5 excitation by @xcite demonstrates that the emission can be reproduced by low velocity ( @xmath62 ) magnetohydrodynamic shocks within the dense ( @xmath63 ) h@xmath5 gas . the denser clouds survive long enough to be heated by turbulence in the hot - gas component , tapping into the large available kinetic energy of the shock . this picture is consistent with both the broad h@xmath5 linewidth ( 870kms@xmath13 ) measured in irs high - resolution spectrometer observations of @xcite , and the velocity center of the warm h@xmath5 ( based on new irs spectral calibrations - see appendix b ) which places the gas at intermediate velocities between the intruder and the group igm . both these measurements are consistent with h@xmath5 being accelerated in a turbulent post - shocked layer . intermediate pre - shock densities and post - shock temperatures result in regions of hi and hii that have cooled , but where the dust content has been destroyed @xcite . pre - existing giant molecular clouds ( gmcs ) which may have been embedded in the hi gas , would be rapidly compressed and collapse quickly , thus forming stars.this mechanism , proposed for sq - a by @xcite , might also apply to 7318b - south . however , if this was the case , it would have to explain why two such gmcs happened to be positioned at the extreme ends of the current shock an unlikely coincidence . more probable , however , is that the geometry of the shock somehow favors the collapse of clouds at the ends of the shock perhaps in regions where the turbulent heating is less efficient . as outlined above , the combination of emission detected in the shock region of sq can be understood in terms of a spectrum of shock velocities . the fastest shock velocities ( @xmath64 km@xmath0 ) are associated with the lowest density pre - shock regions and the post - shock x - ray emitting plasma . these are fast @xmath33 shocks and represent a discontinous change of hydrodynamic variables and are often dissociative @xcite . @xmath34 shocks have a broad transition region such that the transition from pre - shock to post - shock is continuous and are usually non - dissociative @xcite . the lowest velocity shocks associated with the turbulent h@xmath5 emission are @xmath62 @xmath34 shocks @xcite . @xcite find optical emission line ratios consistent with shock models that do not include a radiative precursor @xcite . the @xmath10feii@xmath11$]25.99 emission associated with the shock region in sq is relatively weak , but coincides with the most energetic part of the shock as traced by the x - rays ( fig . [ fig : fs ] ) . we also detect abundant @xmath10siii@xmath11$]34.82 emission associated with the main shock . silicon and iron have very similar first and second ionization potentials . their first ionization potentials ( 7.9 and 8.15 ev for fe and si , respectively ) are lower than that of hydrogen but their second ionization potential is higher ( 16.19 and 16.35 ev ) . the mir @xmath10feii@xmath11 $ ] and @xmath10siii@xmath11 $ ] line emission observed from the sq shock could thus arise from predominantly neutral , as well as ionized gas . we discuss the contribution from the ionized gas using the @xmath10feii@xmath11$](@xmath65m)/@xmath10neii@xmath11$](@xmath66 m ) and @xmath10siii@xmath11$](@xmath67m)/@xmath10neii@xmath11$](@xmath66 m ) line ratios . the high @xmath10neii@xmath11$](@xmath66m)/@xmath10neiii@xmath11$](@xmath68 m ) mid - ir line ratio ( see section 5 ) implies that @xmath10neii@xmath11 $ ] is the dominant ionization state of ne in the sq shock . unlike fe and si , ne is not much depleted on dust @xcite . because the @xmath10neii@xmath11$]@xmath66 m line has a high critical density ( @xmath69@xmath57 , ho and keto 2007 ) , the neon line strength scales with the emission measure of the ionised gas . the optical line emission from the sq shock is discussed in detail in xu et al . the high values of the @xmath10oi@xmath11$](6300 ) and @xmath10nii@xmath11$](6584 ) lines to h@xmath27 line ratios are evidence of shock ionization . the optical @xmath10sii@xmath11$](6716/6731 ) line ratio as well as the mir @xmath10siii@xmath11 $ ] line ratio correspond to the low density limit ( see section 5 ) and comparison with the shock models of allen et al . ( 2008 ) constrain the pre - shock gas density to be about @xmath70@xmath57 or smaller . in figure [ fig : sil_ne ] we indicate the region ( in grey ) corresponding to the observed mir ratios ( using the upper and lower limits determined for the @[email protected]\mu$]m line emission as discussed in section 5 ) . for comparison the expected emission from the shock models of @xcite are displayed for a pre - shock gas density of @xmath70@xmath57 and two values of the magnetic parameter , @xmath58 = 0.5 and 3 . for clarity of the figure , only shock velocities from 100 to 300kms@xmath13 are used and higher velocities shocks which do not match the observed @xmath10neii@xmath11$]/@xmath10neiii@xmath11 $ ] line emission ratio ( see section 5 ) are discarded . the observed @xmath10feii@xmath11$](@xmath65m)/@xmath10neii@xmath11$](@xmath66 m ) and @xmath10siii@xmath11$](@xmath67m)/@xmath10neii@xmath11$](@xmath66 m ) ratios are both smaller than the shock values . the iron and silicon lines are not dominant cooling lines of ionizing shocks . the gas abundances of these elements do not impact the thermal structure of the shock and the line intensities roughly scale with the gas phase abundances . we thus interpret the offset between the irs observation and model values as evidence for fe and si depletion . we consider the magnitude of the depletions indicated by the arrow ( @xmath650% and @xmath660% for fe and si , respectively ) in figure [ fig : sil_ne ] as lower limits since there could be a contribution to the @xmath10feii@xmath11 $ ] and @xmath10siii@xmath11 $ ] line emission from non - ionizing @xmath33-shocks into molecular gas @xcite . such shocks could also contribute to the @xmath10oi@xmath11$](6300 ) line emission as discussed in guillard et al . forthcoming observations of the h@xmath9 ro - vibrational line emission in the near - ir should allow us to estimate whether they may be significant . a contribution from non - ionizing shocks to the @xmath10feii@xmath11 $ ] and @xmath10siii@xmath11 $ ] line emission will raise the depletion of both fe and si as well as the fe / si depletion ratio because non - ionizing shocks do not produce @xmath10neii@xmath11 $ ] line emission and the @xmath10feii@xmath11$](@xmath71m)/@xmath10siii@xmath11$](@xmath72 m ) line emission ratio in @xmath33-shocks within dense gas is larger than for shocks plotted in figure [ fig : sil_ne ] . a key aspect to understanding the emission we observe in sq , as well as a test of the proposed model of h@xmath5 excitation , is the amount and distribution of cold h@xmath5 , as measured by co. a reservoir of cold h@xmath5 associated with the warm gas observed in sq would provide key insight into conditions within the intragroup medium . in figure [ fig : co ] we show the bima co ( 1 - 0 ) integrated intensity contours from @xcite , shown with the h@xmath5 s(0 ) and s(1 ) emission contours , overlaid on an optical image of sq . these interferometric observations use the large primary beam of 110 , centered on the shock , to determine areas of high column densities in the group . the co traces areas of known star formation seen at 24 and in the nuv ( see section [ sf ] ) , notably sq - a and 7318b - south . as observed by @xcite , the co around ngc 7319 is concentrated in two regions . the dominant complex is north of the nucleus residing in a dusty tidal feature . the nuclear co is elongated perpendicular to the stellar disk suggesting a deficiency of ongoing star formation in the disk . the co distribution does not correlate with the location of warm h@xmath5 emission , particularly around ngc 7319 . the cold h@xmath5 complexes are clearly offset from the concentrations of warm h@xmath9 . even the s(0 ) emission line ( see fig . [ fig : co]a ) , which follows the coldest warm h@xmath9 , does not have peaks corresponding to the strongest co detections . in a forthcoming paper ( guillard et al . 2010 , in preparation ) , we will report the recent detection of @xmath73co(1 - 0 ) and ( 2 - 1 ) emission , associated with the warm h@xmath9 in the sq shock , using the single - dish 30 m iram telescope . these observations suggest that most of the co emission in the shock has been missed by interferometers ( because of the broad linewidth ) and show that the co emission is both present in the shock , and extends along the h@xmath9-emitting bridge and towards ngc 7319 . the kinematics of the co gas lying outside of star - forming regions , in the new observations , appears to be highly disturbed with a broad linewidth in agreement with the appleton et al . ( 2006 ) interpretation that the mir h@xmath5 lines were intrinsically very broad and resolved by the high resolution module of irs . the co data also agrees with our re - analysis of the appleton et al . ( 2006 ) data ( see appendix b ) using more recent and reliable wavelength calibration , which places the bulk of the h@xmath5 gas at velocities intermediate to that of the intruder and the group supporting the idea that the h@xmath5 gas is accelerated in the shock . section [ sf ] discussed the star forming regions observed in sq and noted that there was very little evidence for star formation in the shock associated with the warm h@xmath5 emission . the main shock contains a large quantity of warm molecular hydrogen ( 5.0 @xmath210@xmath43m@xmath44 ) providing a reservoir of fuel for star formation once it cools ( see section [ spec ] ) . we investigate star formation in the warm h@xmath9-dominated medium by considering the shock sub - region , chosen to avoid star forming regions in the intruder galaxy , but likely still subject to some contamination from these regions . pah emission is a classical tracer of star formation , but the molecules are fragile and easily destroyed in hard radiation fields . in the spectra of the main shock and subregion ( fig . [ fig : spectra]a and b ) the pah emission bands at 6.2 , 7.7 and 8.6 are far weaker compared to the 11.3 bands , which when strong are predominantly produced by neutral pah molecules @xcite . in the shock subregion we find an upper limit flux for the 6.2 pah of @xmath74 wm@xmath75 and fluxes of @xmath76 wm@xmath75 and @xmath77 wm@xmath75 for the 7.7 and 11.3 bands respectively . this corresponds to a 7.7/11.3 pah ratio of 0.35 , very low compared to the median value found for the sings sample of 3.6 @xcite . the suppression of the 7.7/11.3 pah ratio has been observed in agn environments ( e.g. low - luminosity agn in the sings sample of * ? ? ? * ) and is favored to be the result of selective destruction of pah molecules small enough to emit at 7.7 . pah processing in the shock due to larger molecules being less fragile than smaller ones is discussed further in guillard et al . an alternative explanation is that the pah molecules are chiefly large and neutral in the shock , producing enhanced 11.3 emission in comparison to the smaller pah molecules . a detailed comparison of the dust and pah emission properties can be found in guillard et al . ( 2010 ) . dust emission in the mid- and far - infrared can be used to infer the amount of star formation taking place @xcite . we measure a 24 flux in this region of 0.408 mjy , corresponding to a spectral luminosity ( @xmath78 ) of @xmath79 . this low luminosity is consistent with the weak mir continuum ( fig . [ fig : spectra]b ) , arising from emission from vsgs heated by the uv radiation field . we can combine this with a measurement of the h@xmath27 emission in this region to obtain a star formation rate ( sfr ) , given that they are complimentary ( h@xmath27 tracing the young stellar population and 24 as a measure of dust - absorbed stellar light ) . we find @xmath80 , but caution that h@xmath27 emission in sq is also the result of shock - excitation @xcite and must be considered an upper limit for measuring star formation . when we combine @xmath81 with @xmath82 , using the relation of @xcite , we find a sfr of @xmath290.05@xmath83 . a further measure of star formation can be obtained from the pah strength . using the relation of @xcite , derived from the starburst sample of @xcite , we can use the 7.7 flux density as a measure of star formation . the shock subregion has a 7.7 flux density of 0.73 mjy which corresponds to a sfr of @xmath84 , in good agreement with our previous calculation , but also an upper limit as some pah emission is contamination from known star formation regions in the group . a point of caution , however , is that this star formation indicator may be biased given that we detect suppressed 7.7 emission compared to the 11.3 pah . comparing the @xmath85(7.7)/@xmath82 ratio to the galaxies in sings , we find that the value of @xmath86 is typical of star - forming galaxies @xcite , which suggests that both measures of star formation are low , but self - consistent . the low _ upper limits _ for the sfr in the shock suggests that star formation is depressed in the shock apart from in the igm starburst , sq - a , which has a sfr of @xmath87 at the velocity of the group @xcite . this would be consistent with a picture of molecular hydrogen being reheated by mhd shocks in the turbulent medium . since the jeans mass increases with gas temperature and turbulence and shearing morions will prevent collapse , the cold molecular gas clouds may be too short - lived or undersized to facilitate collapse and produce significant star formation @xcite . the extent of a cold reservoir of molecular gas in the shock is a key consideration in this scenario , as discussed in section [ co ] . the present observations suggest that molecular line cooling in dense clumps dissipates a significant fraction of the kinetic energy available in high - speed shocks . for reference , and to provide a glimpse of what this group might look like at high redshift when filling a single beam , we present in figure [ fig : full ] the spectrum extracted for the entire group ( i.e. including the three neighboring galaxies and the shock ) . the total h@xmath5 line emission from the whole group exceeds @xmath8 and is still the most dominant mir coolant . indeed the luminosity of the rotational h@xmath5 lines is sufficient that it could be detected at high redshift with future far - infrared or sub - mm instrumentation like spica or safir ( see appleton et al . how likely is it , however , that high - speed shocks play a role in the assembly of galaxies ? there is growing evidence that galaxies at high redshift are turbulent @xcite and increasingly clumpy ( e.g. conselice et al . 2005 , elmegreen & elmegreen 2005 ) . indeed bournaud & elmegreen ( 2009 ) discuss the importance of the growth instabilities in massive gas clumps in forming disks at z @xmath88 1 , and favor at least a large fraction of the clump systems being formed in smooth flows , perhaps similar to those discussed by dekel et al . to what extent the build - up of these disks is truly `` smooth '' is not yet clear since the medium is likely to be a multiphase one . in a more standard picture , gas flowing into the more massive dark halos will experience strong shocks , most likely in an inhomogenious medium ( e. g. greif et al . 2008 ) thus it begs the question of how important h@xmath9 cooling may be in these different cases . models of the collapse of the first structures predict that strong metal lines soon dominate the cooling over molecular hydrogen when the first stars pollute the environment . it is therefore usually assumed that , except at very early stages , molecular hydrogen is a minority coolant in gas that forms the first major structures ( e.g. bromm et al . 2001 , santoro and shull 2006 ) . however , our observations show that , under the right conditions , even in high metallicity environments , molecular hydrogen can be extremely powerful in this case dominating by a factor of ten over the usually powerful @[email protected] \mu$]m line in strong shocks . if there are situations at high redshift where strong shocks propagate into a clumpy , multiphase medium , then our observations imply that molecular hydrogen cooling can not be assumed to be negligible . on the other hand , this will not be a trivial problem . our best model of stephan s quintet @xcite involves the formation of h@xmath5 in a complex multiphase turbulent medium in which shocks destroy dust in some places , but allow survival in others - thus encouraging h@xmath5 formation . in the early universe , this enhanced cooling , which has so far been neglected , will depend on the distribution and nature of the first dust grains , in concert with the formation , temperature and abundance of gas , and the feedback effects from the first stars and agn . in this paper we have presented the results of the mid - infrared spectral mapping of the stephan s quintet system using the _ spitzer space telescope_. we highlight here our five main conclusions : * the powerful h@xmath5 emission detected by @xcite surprisingly represents only a small fraction of the group - wide warm h@xmath5 ( with a lower limit luminosity of @xmath89 spread over @xmath90 kpc@xmath7 ) that dominates the mid - infrared emission of the system . there is evidence for another shock - excited feature , the so - called h@xmath5 bridge between the main shock and ngc 7319 , which is likely a remnant of past tidal interactions within the group . the spatial variation in the distribution of the h@xmath5 0 - 0 line ratios implies differences in temperature and excitation in the shocked system - this will be explored fully in paper ii . * the global l(h@xmath5)/l@xmath91 ratio in the main shock is @xmath92 , and @xmath62.5 in the new `` bridge '' feature . the results confirm that mir h@xmath9 lines are a stronger coolant than x - ray emission over the shock structures , indicating a new cooling pathway seen on a large scale in sq . this modifies the traditional view that x - rays dominate cooling at all times in the later stages of compact group evolution . since h@xmath5 forms on the surfaces of dust grains , we expect dust emission associated with these regions , but a low intensity radiation field produces only weak emission at 24 . * following earlier interpretations of nebular line ratios in the optical , we interpret infrared ionic lines within the framework of fast ( @xmath93 km@xmath0 ) ionizing shocks . comparison between the @xmath10neii@xmath11 $ ] , @xmath10neiii@xmath11 $ ] , @xmath10siii@xmath11 $ ] and @xmath10feii@xmath11 $ ] line intensities implies that both silicon and iron are depleted onto dust . this result implies that dust is not destroyed in the shock . * star formation in sq is dominated by sq - a and 7318b - south , located at the extreme ends of the shock ridge seen at radio wavelengths , suggesting they are both shock triggered starbursts . however , regions dominated by warm h@xmath5 emission exhibit very low star formation rates , consistent with a turbulent model where h@xmath5 is significantly reheated and cool clouds are too short - lived or undersized to collapse . * in sq we observe the projected coexistence of @xmath10siii@xmath11 $ ] and h@xmath5 being produced by @xmath6200km@xmath0 and @xmath620km@xmath0 velocity shocks , respectively . our observational results are consistent with a model of a multiphase postshock medium produced by a galaxy - wide collision @xcite . the cooling pathway of warm h@xmath5 emission we observe group - wide in sq is clearly a significant , albeit surprising , mechanism in shock systems . to determine the overall dominant cooling mechanism in sq , we require an inventory of lines and continuum processes at all wavelengths . early shock models @xcite predict that , apart from the rotational emission from h@xmath5 , contributions from lines such as @xmath10oi@xmath11$]63.2 , @xmath10cii@xmath11$]157.7 and the thz spectrum of h@xmath50 @xcite could be significant . we hope to explore this chemistry more fully , and the detailed distribution of cool dust , using the capabilities of the _ herschel _ space observatory . in addition , we can not rule out strong uv line - cooling . stephan s quintet provides the ideal laboratory for probing a mechanism potentially crucial in systems ranging from ulirgs to radio galaxies to supernova remnants . mec is supported by nasa through an award issued by jpl / caltech under program 40142 . we thank tom jarrett for use of his irs pixel cleaning software and irac / mips photometry software . aaaaa allen , m.g . , groves , b.a . , dopita , m.a . , sutherland , r.s . , & kewley , l.j . 2008 , , 178 , 20 allen , r.j . , & hartsuiker , j.w . 1972 , nature , 239 , 324 allen , r.j . , & sullivan , w.t.,iii . 1980 , , 84 , 181 appleton , p. , et al . 2009 , astro2010 : the astronomy and astrophysics decadal survey , science white papers , 2010 , 2 appleton , p.n . et al . 2006 , , 639 , l51 appleton , p.n . , charmandaris , v. , horrelou , c. , mirabel , i.f . , ghigo , f. , higdon , j.l . , & lord , s. 1999 , , 527 , 143 bernard - salas , j. et al . 2009 , , in press ( arxiv : astro - ph/0908.2821 ) black , j. h. , & van dishoeck , e. f. 1987 , , 322 , 412 bournaud , f. , & elmegreen , b. g. 2009 , , 694 , l158 brandl , b.r . 2006 , , 653 , 1129 bromm , v. , ferrara , a. , coppi , p. s. , & larson , r. b. 2001 , , 328 , 969 buchanan , c. l. , gallimore , j. f. , odea , c. p. , baum , s. a. , axon , d. j. , robinson , a. , elitzur , m. , & elvis , m. 2006 , , 132 , 401 calzetti , d. , et al . , 2007 , , 666 , 870 conselice , c. j. , et al . 2005 , , 633 , 29 dale , d.a . , et al . 2006 , , 646 , 161 dekel , a. , sari , r. , & ceverino , d. 2009 , , 703 , 785 deo , r.p . , et al . 2007 , , , 671 , 124 dopita , m. a. , & sutherland , r. s. 1995 , , 455 , 468 draine , b.t . , roberge , w.g . , & dalgarno , a. 1983 , , 264 , 485 draine , b. t. , & woods , d. t. 1992 , , 387 , 732 draine , b.t . , & li a. 2001 , , 551 , 807 durret , f. 1994 , s , 105 , 57 egami , e. , rieke , g.h . , fadda , d. & hines , d.c . 2006 , , 652 , l21 elmegreen , b. g. , & elmegreen , d. m. 2005 , , 627 , 632 frster schreiber , n. m. , et al . 2006 , , 645 , 1062 gao , y. & xu , c. 2000 , , 542 , l83 genzel , r. , et al . 2008 , , 687 , 59 greif , t. h. , johnson , j. l. , klessen , r. s. , & bromm , v. 2008 , , 387 , 1021 guillard , p. , boulanger , f. , cluver , m. , appleton , p.n . , pineau des forts , g. , & ogle , p. 2010 , , submitted guillard , p. , boulanger , f. , pineau des forts , g. , & appleton , p.n . 2009 , , 502 , 515 hatch , n.a . , crawford , c.s , fabian , a.c . , & johnstone , r.m . 2005 , , 358 , 765 helou , g. , & bicay , m.d . 1993 , , 415 , 93 hewitt , j.w . , rho , j. , andersen , m. , & reach , w.t . 2009 , , 694 , 1266 ho , l.c . , & keto , e. 2007 , , 658 , 314 hollenbach , d. , & mckee c.f . 1989 , , 342 , 306 hollenbach , d. j. , & tielens , a. g. g. m. 1997 , , 35 , 179 houck , j.r . , weedman , d.w . , le floch , & hao , l. 2007 , , 671 , 323 houck , j.r . 2004 , , 154 , 18 kennicutt jr . , 1998 , , 36 , 189 lepp , s. , & mccray , r. 1983 , , 269 , 560 maloney , p.r . , hollenbach , d.j . , & tielens , a.g.g.m . 1996 , , 466 , 561 martn - hernndez , n.l . , vermeij , r. , tielens , a.g.g.m . , van der hulst , j.m . , & peeters , e. 2002 , , 389 , 286 moles , m. , sulentic , j.w . , & mrquez , i. 1997 , , 485 , l69 neufeld , d. a. , hollenbach , d. j. , kaufman , m. j. , snell , r. l. , melnick , g. j. , bergin , e. a. , & sonnentrucker , p. 2007 , , 664 , 890 pietsch w. , trinchieri , g. , arp , h. , & sulentic , j.w . 1997 , , 322 , 89 pearson j. c. , de lucia f. c. , anderson , t. , herbst , e. , & helminger p. 1991 , , 379 , l41 ogle , p. , antonucci , r. , appleton , p. , & whysong , d. 2007 , , 668 , 699 ogle , p. et al . 2009 , , submitted osullivan , e.o . , giacintucci , s. , vrtilek , j.m . , raychaudhury , s. , & david , l.p . 2009 , , 701 , 1560 rieke , g. et al . , 2004 , , 154 , 204 rigopoulou , d. , kunze , d. , lutz , d. , genzel , r. , & moorwood , a.f.m . 2002 , , 389 , 374 santoro , f. , & shull , j. m. 2006 , , 643 , 26 shull , j. m. , & hollenbach , d. j. 1978 , , 220 , 525 simpson , j. p. , witteborn , f. c. , price , s. d. , & cohen , m. 1998 , , 508 , 268 smith , j.d.t . et al . 2007 , , 119 , 1133 smith , j.d.t . 2009 , , 693 , 713 sulentic , j.w . , rosado , m. , dultzin - hacyan , d. , verdes - montenegro , l. , trincheri , g. , xu , c. , & pietsch , w. 2001 , , 122 , 2993 trinchieri , g. , sulentic , j. , breitschwerdt , d. , & pietsch , w. , 2003 , , 401 , 173 trinchieri , g. , sulentic , j. , pietsch , w. , & breitschwerdt , d. 2005 , , 444 , 697 verma , a. , lutz , d. , sturm , e. , sternberg , a. , genzel , r. , & vacca , w. 2003 , , 403 , 829 weedman , d. , et al . 2005 , , 633 , 706 werner , m. w. , et al . 2004 , , 154 , 1 williams , b.a . , yun , m.s . , & verdes - montenegro , l. 2002 , , 123 , 2417 xanthopoulos , e. , muxlow , t.w.b . , thomasson , p. , & garrington , s.t . , 2004 , , 353 , 1117 xu , c. , sulentic , j.w . , & tuffs , r. 1999 , , 512 , 178 xu , c.k . , lu , n. , condon , j.j . , dopita , m. , & tuffs , r.j . 2003 , , 595 , 665 xu , c.k . , et al . 2005 , , 619 , l95 yun , m.s . , verdes - montenegro , l. , del olmo , a. , & perea , j. 1997 , , 475 , l21 in this section we discuss the results pertaining to ngc 7319 , a seyfert 2 galaxy @xcite lying to the east in the sq group ( see fig . the specific intensity contours for the h@xmath5 s(0 ) and h@xmath5 s(1 ) lines ( fig . 2a and b ) show emission associated with the galaxy , seemingly connected to the rest of the group by the h@xmath5 `` bridge '' discussed in section [ molhy ] . figure [ fig : xray_rad ] shows that the nucleus of ngc 7319 produces strong x - ray emission @xcite and is prominent at radio wavelengths @xcite . however , we observe an offset between the peak of the h@xmath5 emission near ngc 7319 and the seyfert nucleus , which suggests that the `` bridge '' is a separate structure and not being excited by the agn . the agn in ngc 7319 does not have a well - collimated jet , but two extended lobes with compact hotspots , asymetrically distributed along the minor axis of the galaxy @xcite . this structure runs ne / sw and its orientation compared to the h@xmath5 filament ( which runs ew ) is not consistent with causing the excitation of the h@xmath5 bridge . this is also evident from the relatively weak power in the agn , as inferred by the emission line diagnostics discussed below , and in the x - ray where it is only a factor of @xmath62 greater than the emission associated with the main shock and `` bridge '' . in figure [ fig : fs ] we present the specific intensity contours of the @xmath10feii@xmath11$]25.99 ( blended with @xmath10oiv@xmath11$]25.89@xmath25 m ) , @xmath10siii@xmath11$]33.48 and @xmath10siii@xmath11$]34.82 emission lines . given the low spectral resolution of the sl and ll modules of _ spitzer _ , we can not distinguish between emission from @xmath10fe ii@xmath11 $ ] and @xmath10oiv@xmath11 $ ] . the agn in ngc 7319 is likely to produce both , with @xmath10feii@xmath11 $ ] emission likely originating in x - ray dissociation regions ( xdrs ) surrounding the agn @xcite . prominent emission from @xmath10siii@xmath11$]33.48 and @xmath10siii@xmath11$]34.82 are due to the high excitation conditions associated with the agn . dense pdrs and x - ray dominated regions , powered by agn , show strong @xmath10siii@xmath11 $ ] emission at 34.82 , while @xmath10siii@xmath11 $ ] 33.48 emission acts as a tracer of hii regions . we now focus on the emission line properties of ngc 7319 . figure [ fig : agn ] shows the spectrum extracted from the galaxy and the measured properties of the line ratios are listed in table [ tableh2fluxes ] and [ tablemetalfluxes ] . for the first time in the sq group , we observe a spectrum which is no longer dominated by h@xmath5 emission , but instead the brightest lines are the high - excitation @xmath10oiv@xmath11$]25.91@xmath25 m and @xmath10feii@xmath11$]25.98 blended lines , as well as the @xmath10siii@xmath11$]34.81@xmath25 m line . @xmath10nev@xmath11 $ ] is prominent at both 14.32 and 24.30@xmath25 m a line typically seen in agn . also , unlike the majority of the extended shocked regions , there is a rising thermal continuum more typical of a starforming galaxy than a classical seyfert galaxy , although as various studies have shown ( buchanan et al . 2006 ; deo et al . 2007 ) , seyfert ii galaxies exhibit a variety of mir spectral characteristics at long wavelengths . ngc 7319 s rising continuum is similar to that seen in the seyfert ii galaxy ngc 3079 ( deo et al . 2007 ) , and likely represents a dominant starburst component in the far - ir . spitzer imaging reveals not only a bright nucleus , but also extended emission regions in the galaxy . using the @xmath10siii@xmath11$]34.81@xmath25m/@xmath10siii@xmath11$]33.48 ratio as a probe of excitation sources , we find a ratio of @xmath61.85 which is low compared to the average value found for agn galaxies ( @xmath94 ) in the _ spitzer _ infrared nearby galaxy sample ( sings ) , but high compared to star - forming regions ( @xmath95 ) in the same sample @xcite . this suggests a relatively weak agn . the @xmath10neiii@xmath11$]15.56/@xmath10neii@xmath11$]12.81 ratio is a measure of radiation field strength and the value of 0.97 indicates a typical radiation field strength compared to other agn ( the sample of weedman et al . 2005 shows a range of @xmath60.17 to 1.9 ) . the average electron density , estimated from the @xmath10siii@xmath11$]18.71/@xmath10siii@xmath11$]33.48 ratio of @xmath60.56 , is @xmath59 @xmath57 @xcite , in the low - density limit for this diagnostic @xcite . the properties of the h@xmath5 emission of ngc 7319 are limited to the two long wavelength lines observed by irs - ll ( the galaxy lies outside the region mapped by sl ) . as a result , the excitation diagram contains only two points allowing only an approximate idea of the h@xmath5 mass , since without the sl wavelength coverage to provide information on possible warmer components , we are likely to overestimate the temperature of the h@xmath5 by fitting a straight line to the points . any warmer component would contribute to the flux of the 0 - 0 s(1 ) line thus leading to a reduction of the temperature ( and increase in h@xmath5 mass ) for any cooler component . however , to provide a guide , we estimate the temperature of h@xmath5 @xmath96171@xmath97 k ( assuming the gas is in thermal equilibrium and thus an ortho - para ratio of 2.65 ) and a total h@xmath5 mass of 3.0@xmath9810@xmath3m@xmath4 . we consider this a lower limit to the total warm mass if ( as is likely ) more than one component is present . gao & xu ( 2000 ) estimated the _ cold _ molecular hydrogen mass of ngc 7319 based on @xmath73co ( 1 - 0 ) observations as @xmath88 3.6 @xmath210@xmath99m@xmath4 , a factor of roughly ten greater than the warm h@xmath5 mass . such a ratio is not atypical of large spiral galaxies @xcite . the high resolution spectrum at the center of the shock in sq , obtained by @xcite , has been reanalysed using the latest calibrations available for the irs instrument ( ssc pipeline version s17 ) and is included as figure [ fig : hr ] . this has shown that the h@xmath5 gas lies at a velocity of 6360(@xmath100100 ) km s@xmath13 , between the velocity of the group ( 6600 kms@xmath13 ) and the velocity of the intruder ( 5700 kms@xmath13 ) . this is consistent with a model of gas being accelerated by the shock , as well as the turbulence demonstrated by the broad linewidth of the h@xmath5 in the shock ( 860 kms@xmath13 ) . the new sh spectrum shows that with the improved calibrations , the 11.3 pah feature is detected , although faint . this is consistent with what is found in the larger extractions that show enhanced 11.3 pah emission compared to ionised pah features emitting at 6.2 and 7.7 . in this section we present the x - ray fluxes in the extraction regions shown in figure [ fig : ext ] . the reanalysis of archival data is necessary to obtain accurate fluxes and luminosities for the x - ray emission over the various apertures matched to our spectral extractions . we use the xmm - newton epic - pn data ( see * ? ? ? * for observational details ) to obtain the most sensitive measurements . a calibrated event file was generated and filtered using standard quality flags , and subsequently cleaned of background flares . a 0.32 kev image was then extracted and corrected for instrumental response . all point - like sources were masked out to @xmath101 in the analysis of diffuse emission , and the local background level was estimated within a @xmath102 arcmin@xmath7 rectangular region away from the group core . for each region in fig . 7 , resulting background - subtracted photon count rates were converted to 0.32 kev and bolometric " ( 0.00110 kev ) x - ray fluxes assuming an absorbed thermal plasma model of metallicity 0.4 solar , with temperature as estimated from the map of osullivan et al . ( 2009 ) , and an absorbing galactic hi column density of @xmath103 @xmath47 . for ngc7319 , which harbors an agn , an absorbed power - law spectrum of photon index @xmath104 was assumed instead . results are listed in table [ x - ray ] . lcccc + region & count rate ( 0.3 - 2 kev ) & flux ( 0.3 - 2 kev ) & bolometric flux ( 0.001 - 10 kev ) & adopted spectral model + & ( photonss@xmath105 ) & ergs@xmath13@xmath47 & ergs@xmath13@xmath47 & + + main shock & 0.102 & 1.8 @xmath106 & 2.8 @xmath106 & t = 0.7 kev + shock sub - region & 0.017 & 3.3 @xmath107 & 5.1 @xmath107 & t = 0.8 kev + bridge & 0.017 & 3.0 @xmath107 & 4.5 @xmath107 & t = 0.6 kev + ngc 7319 & 0.051 & 1.1 @xmath106 & 3.7 @xmath106 & @xmath108 = 1.7 + sq - a & 0.013 & 2.4 @xmath107 & 3.7 @xmath107 & t = 0.6 kev + we note that for the main shock ( with luminosity 1.9 @xmath210@xmath35ergs@xmath13 in the 0 - 3 - 2 kev band ) we are within a factor of 2 of the 0.5 - 2 kev luminosity @xmath63.1 @xmath210@xmath35ergs@xmath13 obtained for a similar , but larger extraction of the shock by @xcite . @xcite obtain a 0.5 - 2 kev surface brightness of 0.07 l@xmath4pc@xmath75 in the main shock compared to our value of 0.1 l@xmath4pc@xmath75 for the 0.3 - 2 kev surface brightness . lccccccc + target region & aperture & h@xmath5 0 - 0 s(0 ) & h@xmath5 0 - 0 s(1 ) & h@xmath5 0 - 0 s(2 ) & h@xmath5 0 - 0 s(3 ) & h@xmath5 0 - 0 s(4 ) & h@xmath5 0 - 0 s(5 ) + & ( arcsec@xmath7 ) & @xmath10928.22@xmath25 m & @xmath10917.03@xmath25 m & @xmath10912.28@xmath25 m & @xmath1099.66@xmath25 m & @xmath1098.03@xmath25 m & @xmath1096.91@xmath25 m + + main shock & 2307 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + shock sub - region & 242 & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] + bridge & 413 & [email protected] & [email protected] & & & & + ngc 7319 & 1302 & [email protected] & [email protected] & & & & + sq - a & 671 & [email protected] & [email protected] & & & & [email protected] + lcccccccc + target region & @xmath10neii@xmath11 $ ] & @xmath10nev@xmath11 $ ] & @xmath10neiii@xmath11 $ ] & @xmath10siii@xmath11 $ ] & @xmath10nev@xmath11 $ ] & @xmath10feii@xmath11$]+@xmath10oiv@xmath11 $ ] & @xmath10siii@xmath11 $ ] & @xmath10siii@xmath11 $ ] + & @xmath10912.81@xmath25 m & @xmath10914.32@xmath25 m & @xmath10915.56@xmath25 m & @xmath10918.71@xmath25 m & @xmath10924.32@xmath25 m & @xmath10925.99 + 25.89@xmath25 m & @xmath10933.48@xmath25 m & @xmath10934.82@xmath25 m + + main shock & [email protected] & @xmath290.9 & [email protected] & [email protected] & @xmath290.14 & [email protected] & [email protected] & [email protected] + shock sub - region & [email protected] & @xmath290.18 & [email protected] & @xmath290.12 & @xmath290.14 & [email protected] & [email protected] & [email protected] + bridge & & & [email protected] & @xmath290.1 & @xmath290.15 & [email protected] & @xmath290.22 & [email protected] + ngc 7319 & & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & [email protected] & 9.24 @xmath1000.49 + sq - a & & @xmath290.8 & [email protected] & [email protected] & @xmath290.18 & @xmath290.17 & [email protected] & [email protected] +

we present results from the mid - infrared spectral mapping of stephan s quintet using the _ spitzer space telescope_. a 1000km@xmath0 collision ( @xmath1yr ) has produced a group - wide shock and for the first time the large - scale distribution of warm molecular hydrogen emission is revealed , as well as its close association with known shock structures . in the main shock region alone we find 5.0 @xmath210@xmath3m@xmath4 of warm h@xmath5 spread over @xmath6480kpc@xmath7 and additionally report the discovery of a second major shock - excited h@xmath5 feature , likely a remnant of previous tidal interactions . this brings the total h@xmath5 line luminosity of the group in excess of @xmath8 . in the main shock , the h@xmath5 line luminosity exceeds , by a factor of three , the x - ray luminosity from the hot shocked gas , confirming that the h@xmath9-cooling pathway dominates over the x - ray . @xmath10siii@xmath11$]34.82emission , detected at a luminosity of 1/10th of that of the h@xmath9 , appears to trace the group - wide shock closely and in addition , we detect weak @xmath10feii@xmath11$]25.99 emission from the most x - ray luminous part of the shock . comparison with shock models reveals that this emission is consistent with regions of fast shocks ( @xmath12km@xmath0 ) experiencing depletion of iron and silicon onto dust grains . star formation in the shock ( as traced via ionic lines , pah and dust emission ) appears in the intruder galaxy , but most strikingly at either end of the radio shock . the shock ridge itself shows little star formation , consistent with a model in which the tremendous h@xmath5 power is driven by turbulent energy transfer from motions in a post - shocked layer which suppresses star formation . the significance of the molecular hydrogen lines over other measured sources of cooling in fast galaxy - scale shocks may have crucial implications for the cooling of gas in the assembly of the first galaxies .

0912.0282

if a word @xmath0 can be written as @xmath10 , then words @xmath11 , @xmath12 , and @xmath13 are called the prefix , factor , and suffix of @xmath0 , respectively . a word @xmath0 over @xmath14 is called a de bruijn sequence of order @xmath1 , if each word in @xmath15 appears exactly once in @xmath0 as a factor . for example , @xmath16 is a binary de bruijn sequence of order @xmath17 since each binary word of length two appears in it exactly once as a factor : @xmath18 . the de bruijn sequence can be understood by the following game . suppose there are infinite supplies of balls , each of which is labeled by a letter in @xmath14 , and there is a glass pipe that can hold balls in a vertical line . on the top of that pipe is an opening , through which one can drop balls into that pipe , and on the bottom is a trap - door , which can support the weight of at most @xmath1 balls . when there are more than @xmath1 balls in the pipe , the trap - door opens and those balls at the bottom drop off until only @xmath1 balls remain . if we put balls as numbered as in a de bruijn sequence on the alphabet @xmath14 of order @xmath1 , then every @xmath1 ball sequence will appear exactly once in the pipe . it is easy to see that a de - bruijn sequence of order @xmath1 , if exists , is of length @xmath19 and its suffix of length @xmath20 is identical to its prefix of length @xmath20 . so , sometimes a de - bruijn sequence is written in a circular form by omitting the last @xmath20 letters , which can be viewed as the equivalence class of words under the conjugate relation . the de bruijn sequence is also called the de bruijn - good sequence , named after de bruijn @xcite and good @xcite who independently studied the existence of such words over binary alphabet ; the former also provided a formula @xmath21 for the total number of those words of order @xmath1 . the study of the de bruijn sequence , however , dates back at least to 1894 , when flye sainte - marie @xcite studied the words and provided the same formula @xmath21 . for an arbitrary alphabet @xmath14 , van aardenne - ehrenfest and de bruijn provided the formula @xmath22 for the total number of de bruijn sequences of order @xmath1 . besides the total number of de bruijn sequences , another interesting topic is how to generate a de bruijn sequence ( arbitrary one , lexicographically least one , lexicographically largest one ) . for generating de bruijn sequences , see the surveys @xcite . the de bruijn sequence is some times called the full cycle @xcite , and has connections to the following concepts : feedback shift registers @xcite , normal words @xcite , generating random binary sequences @xcite , primitive polynomials over a galois field @xcite , lyndon words and necklaces , euler tours and spanning trees . in this paper , we consider a generalization of the de bruijn sequence . to understand the concept , let us return to the glass pipe game presented at the beginning . now the trap - door can support more weight . when there are @xmath23 or more balls in the pipe , the trap - door opens and the balls drop off until there are only @xmath1 balls in the pipe . is there an arrangement of putting the balls such that every @xmath1 ball sequence appears exactly once in the pipe ? the answer is `` yes '' for arbitrary positive integers @xmath24 . the solution represents a multi - shift de bruijn sequence . we will discuss the existence of the multi - shift de bruijn sequence , the total number of multi - shift de bruijn sequences , generating a multi - shift de bruijn sequence , and the application of the multi - shift de bruijn sequence in the frobenius problem in a free monoid . let @xmath25 be the _ alphabet _ and let @xmath26 be a word over @xmath14 . the _ length _ of @xmath0 is denoted by @xmath27 and the _ factor _ @xmath28 of @xmath0 is denoted by @xmath29 . if @xmath30 for some non - negative integer @xmath31 , we say factor @xmath32 appears in @xmath0 at _ a modulo @xmath2 position_. the set of all words of length @xmath1 is denoted by @xmath15 and the set of all finite words is denoted by @xmath33 , where @xmath34 is the _ empty word_. the concatenation of two words @xmath35 is denoted by @xmath36 , or simply @xmath37 . a word @xmath0 over @xmath14 is called a _ multi - shift de bruijn sequence _ of shift @xmath2 and order @xmath1 , if each word in @xmath15 appears exactly once in @xmath0 as a factor at a modulo @xmath2 position . for example , one of the @xmath17-shift de bruijn sequence of order @xmath38 is @xmath39 , which can be verified as follows : @xmath40 the multi - shift de bruijn sequence generalizes the de bruijn sequence in the sense de bruijn sequences are exactly @xmath41-shift de bruijn sequences of the same order . it is easy to see that the length of each @xmath2-shift de bruijn sequence of order @xmath1 , if exists , is equal to @xmath42 . by the definition of multi - shift de bruijn sequence , the following proposition holds . [ lemma : circular ] let @xmath0 be one @xmath2-shift de bruijn sequence @xmath0 of order @xmath1 , @xmath43 . then the suffix of length @xmath44 of @xmath0 is identical to the prefix of length @xmath44 of @xmath0 . let @xmath0 be one @xmath2-shift de bruijn sequence @xmath0 of order @xmath1 over @xmath14 and let @xmath45 . write @xmath46 such that @xmath47 . if @xmath48 , then we compare the set of all factors @xmath49 and the set of all factors @xmath50 for @xmath51 . the former covers factors @xmath52 and the latter covers factors @xmath53 for every @xmath54 . since the two are identical , we have @xmath55 . now we assume @xmath56 . consider the set of all factors @xmath57 and the set of all factors @xmath58 for @xmath51 and @xmath59 . by the same argument , we have @xmath60 for @xmath61 . finally , comparing the set of all @xmath62 and the set of all @xmath63 for @xmath51 , we have the equality @xmath64 . therefore , we have the equality @xmath65 . from proposition [ lemma : circular ] , we know that when @xmath43 , every multi - shift de bruijn sequence can be written as a circular word and the discussion on multi - shift de bruijn sequences of the two different forms are equivalent . in this paper , we discuss the multi - shift de bruijn sequence in the form of ordinary words . a _ ( non - strict ) directed graph _ , or _ digraph _ for short , is a triple @xmath66 consisting of a set @xmath67 of _ vertices _ , a set @xmath68 of _ arcs _ , and an _ incidence function _ @xmath69 . here we do not take the convention @xmath70 , since we allow a digraph contains self - loops and multiple arcs regarding the same pair of vertices . when @xmath71 , we say the arc @xmath9 joins @xmath32 to @xmath72 , where vertex @xmath73 and vertex @xmath74 are called _ tail _ and _ head _ , respectively . the indegree @xmath75 ( outdegree @xmath76 , respectively ) of a vertex @xmath72 is the number of arcs with @xmath72 being the head ( the tail , respectively ) . a _ walk _ in @xmath77 is a sequence @xmath78 such that @xmath79 for each @xmath80 . the walk is _ closed _ , if @xmath81 . two closed walks are regarded as identical if one is the circular shift of the other . an _ euler tour _ is a closed walk that traverses each arc exactly once . hamilton cycle _ is a closed walk that traverses each vertex exactly once . an _ ( spanning ) arborescence _ is a digraph with a particular vertex , called the _ root _ , such that it contains every vertices of @xmath77 , its number of arcs is exactly one less than the number of vertices , and there is exactly one walk from the root to any other vertex . we denote the total number of euler tours , hamilton cycles , and arborescence of @xmath77 by @xmath82 , @xmath83 , and @xmath84 , respectively . an _ ( undirected ) graph _ is defined as a digraph such that for any pair of vertices @xmath85 , there is an arc @xmath9 , @xmath86 , if and only if there is a corresponding arc @xmath87 , @xmath88 . in this case , we write @xmath89 and a spanning arborescence is just a _ spanning tree_. the arc - graph @xmath90 of @xmath66 is defined as @xmath91 such that for every pair of arcs @xmath92 , @xmath93 , there is an arc @xmath94 , @xmath95 and those arcs are the only arcs in @xmath96 . euler tours exist in a graph @xmath77 if and only if hamilton cycles exist in the arc - graph @xmath90 . we define the word graph @xmath97 by @xmath98 , where @xmath99 for @xmath100 . then by definition , the following lemmas are straightforward . [ lemma : arcgraph ] the digraph @xmath101 is the digraph @xmath102 . by definition , @xmath103 where @xmath104 , and @xmath105 so for every pair of arcs @xmath106 of @xmath97 with @xmath107 , there is an arc @xmath108 of @xmath102 ; and for every arc @xmath109 of @xmath102 , @xmath110 hence , by definition , @xmath102 is the arc - graph of @xmath97 . [ lemma : equivalent ] suppose @xmath111 . ( 1 ) there is a @xmath112-to-@xmath41 mapping from the set of @xmath2-shift de bruijn sequences of order @xmath1 onto the set of hamilton cycles in @xmath97 . ( 2 ) there is a @xmath112-to-@xmath41 mapping from the set of @xmath2-shift de bruijn sequences of order @xmath1 onto the set of euler tours in @xmath113 . let @xmath114 . ( 1 ) notice that any hamilton cycle @xmath115 together with a starting arc @xmath116 uniquely determines one @xmath2-shift de bruijn sequences of order @xmath1 specified by @xmath117 and vice versa . so the @xmath118-to-@xmath41 mapping exists . ( 2 ) applying lemma [ lemma : arcgraph ] , this part follows from ( 1 ) . [ theorem : existence ] for any alphabet @xmath14 , positive integers @xmath24 , the @xmath2-shift de bruijn sequences of order @xmath1 over @xmath14 exist . first we assume @xmath119 . let @xmath120 be any permutation of the words in @xmath15 for @xmath114 . then the word @xmath121 is one @xmath2-shift de bruijn sequence of order @xmath1 over @xmath14 . now we assume @xmath122 and prove there exists an euler tour in @xmath113 . then by lemma [ lemma : equivalent ] , the existence of @xmath2-shift de bruijn sequences of order @xmath1 over @xmath14 is ensured . to show the existence of an euler tour , we only need to verify that @xmath113 is connected and that @xmath123 for every vertex @xmath72 , both of which are straightforward : for every vertex @xmath72 in @xmath113 , @xmath72 is connected to the vertex @xmath124 in both directions and @xmath125 . since @xmath2-shift de bruijn sequence of order @xmath1 exists , in this section we discuss the total number of different @xmath2-shift de bruijn sequence of order @xmath1 , and we denote the number by @xmath126 . first , we study the degenerated case . [ lemma : countnlesm ] for @xmath6 , @xmath127 , where @xmath45 . let @xmath45 . by the definition of the multi - shift de bruijn sequence , in the case @xmath6 , @xmath2-shift de bruijn sequences of order @xmath1 are exactly those of the form @xmath128 , where @xmath129 and @xmath120 is a permutation of all words in @xmath15 . therefore , the total number of such words is @xmath5 . to study the case @xmath8 , we need a theorem by van aardenne - ehrenfest and de bruijn , which describes the relation between the number of euler tours in a particular type of digraph and the number of euler tours in its arc - graph . [ theorem : ehrenfestbruijn ] let @xmath66 be a digraph such that @xmath130 for every @xmath131 . then @xmath132 . the digraph @xmath97 satisfies the conditions in theorem [ theorem : ehrenfestbruijn ] with @xmath133 . so , by the relation between the multi - shift de bruijn sequences and the euler tours in the word graph @xmath97 , we have the following recursive expression on @xmath126 . [ lemma : recursion ] for @xmath134 , @xmath135 , where @xmath45 , @xmath136 . let @xmath45 , @xmath136 . by lemma [ lemma : equivalent ] , @xmath137 to finish the last step of obtaining @xmath126 for @xmath8 , we again need two theorems , which are often used in the literature to count the number of euler tours in various types of digraphs . [ theorem : best ] in a digraph @xmath66 , @xmath138 . [ theorem : kirchhoff ] in a graph @xmath66 , the number of spanning trees is equal to any cofactor of the laplacian matrix of @xmath77 , which is the diagonal matrix of degrees minus the adjacency matrix . [ lemma : countnles2 m ] for @xmath139 , @xmath140 , where @xmath45 . let @xmath141 and @xmath45 . then @xmath142 . by definition , so from any vertex to any vertex , there are @xmath144-many arcs in @xmath77 . we convert @xmath77 into a undirected graph @xmath145 by omitting all self - loops ; there are @xmath144-many of them for each vertex . since for every pair of vertices @xmath85 there are @xmath144-many arcs joins @xmath146 to @xmath147 and correspondingly there are @xmath144-many arcs joins @xmath147 to @xmath146 , the graph @xmath145 is indeed an undirected graph by our definition . each vertex in @xmath145 is of degree @xmath148 . then the laplacian matrix of @xmath145 is @xmath149 by theorem [ theorem : kirchhoff ] , the number of arborescence @xmath150 is equal to the cofactor of @xmath151 , which is @xmath152 . then by theorem [ theorem : best ] , the number of euler tours in digraph @xmath77 is @xmath153 . finally , by lemma [ lemma : equivalent ] , the number of @xmath2-shift de bruijn sequence of order @xmath1 is @xmath154 . [ theorem : count ] for @xmath6 , @xmath127 , and for @xmath155 , @xmath140 , where @xmath45 . for @xmath6 , the equality @xmath127 is shown in lemma [ lemma : countnlesm ] . now we assume @xmath156 . let @xmath136 . then by lemmas [ lemma : recursion],[lemma : countnles2 m ] , we have @xmath157 . in this section , we study the problem of generating one @xmath2-shift de bruijn sequence of order @xmath1 for arbitrary alphabet and positive integers @xmath24 . when @xmath6 , a @xmath2-shift de bruijn sequence of order @xmath1 is easy to construct as given in theorem [ theorem : existence ] . now we consider the case @xmath158 . we will present two algorithms for generating a @xmath2-shift de bruijn sequence of order @xmath1 . we claim that @xmath2-shift de bruijn sequences of order @xmath159 can be generated using the ordinary de bruijn sequence generating algorithm , such as described by fredricksen @xcite . to do this , we first generate a de bruijn sequence @xmath0 of order @xmath160 over the alphabet @xmath161 . then we replace each letter of @xmath0 in @xmath162 by the corresponding word of length @xmath2 over @xmath14 . it is easy to see that the new word is a @xmath2-shift de bruijn sequence of order @xmath159 . the first algorithm of generating multi - shift de bruijn sequence is to generate @xmath163-shift de bruijn sequences of order @xmath164 for @xmath165 before rearranging the words to obtain an arbitrary @xmath2-shift de bruijn sequence of order @xmath1 . let @xmath166 be two integers , and @xmath46 , where @xmath136 . the case @xmath167 is already discussed and the case @xmath168 is trivial . so we assume @xmath169 and @xmath170 . we define @xmath171 , @xmath172 and generate @xmath173 such that @xmath174 is a @xmath175-shift de bruijn sequence of order @xmath176 and @xmath177 ; and define @xmath178 , @xmath179 and generate @xmath180 such that @xmath181 is a @xmath182-shift de bruijn sequence of order @xmath183 and @xmath184 . let @xmath45 , @xmath185 , @xmath186 . we define @xmath187 , @xmath188 , @xmath189 , @xmath190 . then the following word @xmath191 is one @xmath2-shift de bruijn sequence of order @xmath1 , where @xmath192 and @xmath193 . to show the correctness , we claim that every word in @xmath194 appears in @xmath195 as a factor at a modulo @xmath2 position exactly once . furthermore , since @xmath196 , every word in @xmath197 appears in @xmath198 as a factor at a modulo @xmath2 position exactly once . therefore , the generated word is indeed a @xmath2-shift de bruijn sequence of order @xmath1 . now , we will see an example . consider generating a @xmath17-shift de bruijn sequence of order @xmath199 . then @xmath200 and we can obtain two words @xmath201 , which is @xmath202 , and @xmath203 , which is @xmath204 . so one @xmath17-shift de bruijn sequence of order @xmath199 is as follows @xmath205 where the subscripts @xmath41 and @xmath17 denote whether the letter is from the word @xmath206 ( words @xmath207 ) or from the word @xmath208 ( words @xmath209 ) . now we present the second algorithm , which uses the same idea of `` prefer one '' algorithm @xcite for generating ordinary de bruijn sequences . let @xmath24 be two positive integers . the following algorithm generates a @xmath2-shift de bruijn sequence of order @xmath1 : 1 . start the sequence @xmath0 with @xmath1 zeros ; 2 . append to the end of current sequence @xmath0 the lexicographically largest word of length @xmath2 such that the suffix of length @xmath1 of new sequence has not yet appeared as factor at a modulo @xmath2 position ; 3 . repeat the last step until no word can be added . to show the correctness , first we claim that when the algorithm stops , the suffix @xmath32 of length @xmath44 of @xmath0 contains only zeros . to see this , suppose @xmath32 is not @xmath124 . since no word can be added , all @xmath210 words of length @xmath1 with prefix @xmath32 appear in @xmath0 and thus @xmath32 appears in @xmath0 as a factor at a modulo @xmath2 position @xmath211 times . so there are @xmath211 words of length @xmath1 with suffix @xmath32 that appear in @xmath0 at a modulo @xmath2 position , which contradicts the definition of the multi - shift de bruijn sequence . therefore , @xmath212 furthermore , word @xmath124 appears in @xmath0 as a factor at a modulo @xmath2 position @xmath211 times and thus all words in @xmath213 appear in @xmath0 as a factor at a modulo @xmath2 position . by the algorithm , no word of length @xmath1 can appear twice in @xmath0 at a modulo position . so , in order to prove the correctness of the algorithm , it remains to show every word of length @xmath1 appears in @xmath0 as a factor at a modulo @xmath2 position . suppose a word @xmath72 does not appear in @xmath0 at a modulo @xmath2 position . then @xmath214 and the word @xmath215 does not appear in @xmath0 as a factor at a modulo @xmath2 position as well ; otherwise , there are @xmath216 appearance of @xmath217 in @xmath0 at a modulo @xmath2 position , which means @xmath72 appears in @xmath0 as a factor at a modulo @xmath2 position . repeat this procedure , none of the words @xmath215 , @xmath218 , @xmath219 , @xmath220 appears in @xmath0 as a factor at a modulo @xmath2 position . but for @xmath221 , we proved that @xmath222 appears in @xmath0 as a factor at a modulo @xmath2 position , a contradiction . therefore , every word of length @xmath1 appears at a modulo @xmath2 position . now , we use the algorithm to generate one @xmath17-shift de bruijn sequence of order @xmath199 . starting from @xmath223 , since @xmath224 does not appear as a factor at a modulo @xmath17 position , we append @xmath225 to the current sequence @xmath223 . repeating this procedure and appending words @xmath225 , @xmath225 , @xmath226 , @xmath225 , , finally we obtain the word : @xmath227 if we circularly move the prefix @xmath228 to the end , the sequence generated by the second algorithm is the lexicographically largest @xmath2-shift de bruijn sequence of order @xmath1 . the study of multi - shift de bruijn sequences is inspired by a problems of words , called the frobenius problem in a free monoid . given @xmath160 integers @xmath229 , such that @xmath230 , then there are only finitely many positive integers that _ can not _ be written as a non - negative integer linear combination of @xmath229 . the integer _ frobenius problem _ is to find the largest such integer , which is denoted by @xmath231 . for example , @xmath232 . if words @xmath229 , instead of integers , are given such that there are only finitely many words that _ can not _ be written as concatenation of words from the set @xmath233 , the _ frobenius problem in a free monoid _ @xcite is to find the longest such words . if all @xmath229 are of length either @xmath2 or @xmath1 , @xmath234 , there is an upper bound : the length of the longest word that can not be written as concatenation of words from the set @xmath233 is less than or equal to @xmath235 , where @xmath236 . @xcite furthermore , the upper bound is tight and the construction is based on the multi - shift de bruijn sequences . we denote the set of all words that can be written as concatenation of words in @xmath237 , including the empty word , by @xmath238 . @xcite there exists @xmath239 , @xmath234 , such that @xmath240 is finite and the longest words in @xmath240 constitute exactly the language @xmath241 , where @xmath242 is a @xmath2-shift de bruijn sequence of order @xmath44 . for example , for any set of words @xmath243 such that @xmath244 is finite , the longest words in @xmath244 are of length less than or equal to @xmath245 . to construct @xmath237 to reach the upper bound , we first choose an anbitrary @xmath38-shift de bruijn sequence of order @xmath246 as @xmath247 . then based on @xmath242 , we construct the set @xmath248 @xmath249 , @xmath250 , @xmath251 , @xmath252 , @xmath253 , @xmath254 , @xmath255 , @xmath256 , @xmath257 , @xmath258 , @xmath259 , @xmath260 , @xmath261 , @xmath262 , @xmath263 @xmath264 . we have @xmath265 and one of the longest words in @xmath151 of length exactly @xmath266 is given below : @xmath267 in this paper , we generalized the classic de bruijn sequence to a new multi - shift setting . a word @xmath0 is a @xmath2-shift de bruijn sequence @xmath268 of order @xmath1 , if each word of length @xmath1 appears exactly once as a factor at a modulo @xmath2 position . an ordinary de bruijn sequence is a @xmath41-shift de bruijn sequence . we showed the total number of distinct @xmath2-shift de bruijn sequences of order @xmath1 is @xmath269 for @xmath6 and is @xmath140 for @xmath8 , where @xmath45 . this result generalizes the formula @xmath270 for the number of ordinary de bruijn sequences . here we use an ordinary word form ; if counting the sequences in a circular form , then the number is to be divided by @xmath271 . we provided two algorithms for generating a @xmath2-shift de bruijn sequence of order @xmath1 . the first algorithm is to rearrange factors from two simpler multi - shift de bruijn sequences , where the order is a multiple of the shift . the second is the analogue of the `` prefer one '' algorithm ( for example , see ) for generating ordinary de bruijn sequence . the multi - shift de bruijn sequence has application in the frobenius problem in a free monoid by providing constructions of examples . it will be interesting to see that this generalized concept of the de bruijn sequence can help in other fields of theoretical computer science and discrete mathematics .

a ( non - circular ) de bruijn sequence @xmath0 of order @xmath1 is a word such that every word of length @xmath1 appears exactly once in @xmath0 as a factor . in this paper , we generalize the concept to a multi - shift setting : a @xmath2-shift de bruijn sequence of order @xmath1 is a word such that every word of length @xmath1 appears exactly once in @xmath0 as a factor that starts at an index @xmath3 for some integer @xmath4 . we show the number of the @xmath2-shift de bruijn sequences of order @xmath1 is @xmath5 for @xmath6 and is @xmath7 for @xmath8 , where @xmath9 is the size of the alphabet . we provide two algorithms for generating a multi - shift de bruijn sequence . the multi - shift de bruijn sequence is important in solving the frobenius problem in a free monoid .

1004.1216

quantum rings and quantum dots are prototype systems for the observation of mesoscopic interference on the one hand and for spectroscopic investigations of discrete level spectra of interacting systems on the other hand . ring shaped structures give rise to aharonov bohm ( ab ) interference @xcite which can be tuned by applying a magnetic flux through the area enclosed by the ring . open ring geometries have been used as interferometers , e.g. , to study the transmission phase of quantum dots in the coulomb blockade ( cb ) regime.@xcite the discrete level spectrum of coulomb blockaded quantum dots has been extensively studied using tunneling spectroscopy to probe interaction and spin effects when a gate voltage is used to successively add electrons to such artificial atoms.@xcite and @xmath0 simultaneously and under symmetric bias conditions . @xmath1 reflects the transmission from the ring to the dot . ( c ) aharonov bohm oscillations in the two terminal ring conductance @xmath2 . ( d ) coulomb - blockade oscillations in the two - terminal dot conductance @xmath3 as a function of the in - plane gate voltage @xmath4 . , width=302 ] interference of a resonant state , e.g. a discrete level of a quantum dot , and a continuum of states , e.g. , in a quantum wire or an open ring , typically gives rise to asymmetric line shapes characteristic of the fano effect.@xcite a theoretical treatment of the fano effect was originally developed for scattering cross - sections of nuclei @xcite and in optical spectroscopy@xcite , but recently fano resonnances were also addressed in a multitude of mesoscopic systems @xcite . in mesoscopic transport the energy dependent conductance of a fano resonance can be written as @xmath5 here @xmath6 is a dimensionless energy parameter with @xmath7 the energy of an electron , @xmath8 the energy of the resonance and @xmath9 the width of the resonance . the quantity @xmath10 is known as the _ fano parameter _ and determines the shape of the resonance line.@xcite for @xmath11 resonant transmission dominates and the fano resonance becomes equivalent to a symmetric breit wigner resonance . for @xmath12 a breit wigner shaped anti - resonance is observed and for @xmath13 the asymmetry is strongest . theoretically the fano effect in mesoscopic systems has been discussed for both resonant and open cavities,@xcite and in conjunction with rings.@xcite it has been proposed that fano resonances in quantum dots could be used as a measure of phase coherence@xcite or a means to produce spin polarized currents.@xcite experimentally fano resonances were observed in the tunneling current through a single cobalt atom on a gold surface.@xcite in coulomb blockaded quantum dots it was found that the interference of a broad resonance ( quasi - continuum ) with a narrow resonance can lead to fano peak shapes.@xcite a recent experiment investigated a coulomb blockaded quantum dot side - coupled to a quantum wire and discussed the implications of coulomb interactions between the dot and the wire in the fano regime@xcite . kobayashi et al . further studied the fano effect in an ab - ring with a quantum dot embedded in one arm of the ring.@xcite in these experiments the magnetic field allowed them to tune the relative phase between the non - resonant ( ring ) and the resonant ( dot ) path , periodically changing the asymmetry of the fano line shape . their interpretation required the introduction of a complex @xmath10-parameter to account for the ab - phase . similar results were also found in crossed carbon nanotubes with a resonant state at the point where the tubes are touching.@xcite in these ring systems the fano effect arises due to the interference of a breit - wigner type resonance in one arm of the ring ( containing a strongly coupled quantum dot ) with a continuum channel in the other arm of the ring . here we present transport studies on a structure where a quantum dot in cb - regime is side - coupled to a ring [ see fig.[fig1](a ) ] . our structure has a tunable channel between the ring and the dot which permits us to couple coherently the two subsystems while keeping them spatially separated and therefore minimize capacitive effects of the quantum dot on the interference in the ring , as investigated in ref and ref . in contrast to previous ring systems , our experiment constitutes a tunable fano scatterer consisting of one arm of the ring and the side coupled dot , which is made to interfere with alternative paths along the second arm of the ring . this allows us to study the interplay between continuous ab - interference in the ring and interference involving a resonant level in the dot . the paper is organized as follows : in section ii , the experimental realization of the coupled ring - dot structure is discussed and low temperature transport measurements are presented . in section iii we give a model for the ring - dot system within the scattering matrix formalism and link it to the fano formula in eq.[eqn1 ] . in section iv model and experimental results are compared and we follow ref . to model decoherence in the dot due to finite temperatures and coupling to the leads . in section v we discuss limitations of the model . as a function of @xmath4 when the channel between the dot and the ring is pinched off . the dashed white lines show the cb - peak positions as determined from the dot conductance @xmath3 . this indicates that electrostatic coupling is negligibly small . ( b ) three simultaneously measured currents @xmath0 ( dotted lines ) , @xmath14 ( dashed lines ) , @xmath1 ( solid lines ) , when @xmath15= -50mv,-46mv,-42mv is raised to increase the coupling between the ring and the dot . the thickest line indicates strongest coupling . ( c ) @xmath0 and @xmath14 when @xmath16 is used to tune the electron number of the dot . while @xmath14 shows the expected coulomb oscillations the current through the ring exhibits the typical asymmetric line shapes characteristic of the fano effect . , width=326 ] the coupled ring dot structure was realized in a ga[al]as heterostructure with a high - quality two - dimensional electron gas ( 2deg ) 34 nm below the surface . the electron density in the 2deg was @xmath17m@xmath18 and the mobility was @xmath19m@xmath20/vs at 1.7k . fig.[fig1](a ) shows an afm image of the device defined using afm - lithography . the 2deg below the oxide lines is depleted ( white lines ) and the 2deg regions at the side of the ring dot structure are used as in - plane gates ( _ igt , igr , igb , igd _ ) . in a subsequent step an 8 nm thin titanium film was evaporated on top of the structure and the afm was again used to cut this top gate into individual split gates ( _ tq4,tgd , tq3,tgc , tq1,tgr , tq2 _ ) as indicated by the gray lines . these top gates allow for a unique tunability of the electron number in and the coupling between the two subsystems as well as to the leads . more specifically a narrow top gate stripe ( _ tgc _ ) can be used to completely isolate the ring electronically from the dot . details about the fabrication process may be found in ref.and ref .. the measurements were performed in a dilution refrigerator with an electron temperature @xmath21mk . the measurement setup is shown schematically in fig.[fig1](b ) . a symmetric ac bias voltage @xmath22v was applied to the two leads of the ring at a frequency of 21hz and in the same way @xmath23v was applied to the two leads of the dot with a frequency of 14.4hz . using lock - in techniques this allows us to simultaneously detect the following three currents related to the transmission matrix of the four terminal structure ( assuming spin degeneracy ) : @xmath24\\%=t_{12}+\frac{1}{2}t_{1\leftarrow dot},\\ \frac{i_\mathrm{dot}}{v_\mathrm{dot}}&=&\frac{2 e^2 } { h}\left[t_{34}+\frac{1}{2 } ( t_{31}+t_{32})\right]\\%=t_{34}+\frac{1}{2}t_{3\leftarrow ring } , \\ \frac{i_\mathrm{cross}}{v_\mathrm{ring}}&=&\frac{2 e^2 } { h}(t_{31}-t_{32}).\end{aligned}\ ] ] here t@xmath25 is the transmission from lead @xmath26 to @xmath27 following the numbering of the gates _ tqi _ in fig . [ fig1](a ) . for negligible coupling between the structures the first two values become equivalent to the two terminal conductance @xmath2 of the ring and @xmath3 of the dot respectively since the transmission through the connecting channel vanishes . a measurement of @xmath2 and @xmath3 when the central gate pinches off the channel between the structures is shown in fig.[fig1](c ) and ( d ) . the ring conductance exhibits pronounced ab - oscillations as a function of a magnetic field @xmath28 applied perpendicular to the sample and the period @xmath29mt is in agreement with adding single flux quanta to the area enclosed by the ring with an average radius @xmath30135 nm . the ab - oscillations persist up to @xmath31k . this is slightly lower than in previous measurements of similar two - terminal ring structures where a phase coherence length @xmath32 of more than a micron was found at 4.2k.@xcite the conductance @xmath3 shows cb - resonances as a function of @xmath4 applied to the in - plane gate electrode _ closest to the dot . from measurements of coulomb diamonds we find an average addition energy @xmath33ev for the dot . our set - up allows us to simultaneously detect the resonance position in the dot and the interference pattern in the ring current , without changing the coupling or any additional gate voltages . furthermore , the magnetic field is used to tune the interference between the phase sensitive ring and the dot which , when side coupled to one arm of the ring , will act as a fano scatterer ( see below ) . before coupling the ring and the dot coherently we assess the magnitude of the crosstalk between the structures due to the cross - coupling of the gate electrodes and direct coulomb interaction . to this end @xmath15 was set to -75mv in order to deplete the channel between the two structures . figure[fig2](a ) shows a measurement of ab - oscillations in @xmath2 when @xmath4 is used to tune the electron number @xmath34 in the dot . the white dotted lines show the cb - peak positions in the dot . the unperturbed ab - oscillations in fig.[fig2 ] indicate that unlike recently reported results @xcite we are in a regime where the direct coulomb interaction between the two structures is negligibly small . furthermore , the gate on the dot changes neither the characteristic amplitude nor the phase of ab - oscillations in the ring over a range @xmath35mv . we find that the same is true when changing @xmath36 applied to the top gate of the dot . the top gate electrodes _ tgr _ and _ tgd _ respectively allow us to tune the electron density in the ring or electron number in the dot over a large range without changing the coupling to source and drain strongly . however , where possible we restrict ourselves to tuning the in - plane gates , since in this case we find the structure to be more stable and the gate voltages lead to fewer charge rearrangements . in order to couple the two structures we increase @xmath15 . figure[fig2](b ) shows the three currents @xmath0(dotted line ) , @xmath14(dashed line ) and @xmath1(solid line ) where thicker lines correspond to less negative @xmath15 . with stronger coupling , @xmath1 increases from negligible current at @xmath37mv to a value around 4pa with clear ab - oscillations at @xmath38mv . in this regime @xmath0 exhibits larger deviations from the perfect symmetry as a function of magnetic field @xmath28 , an effect that is expected for the transition from a two - terminal to multi - terminal system.@xcite the current @xmath14 is not significantly modified by the increase in @xmath15 except that the source - drain coupling of the dot is influenced slightly which leads to a larger current for the strongest coupling . of the currents @xmath0 ( dotted lines ) , @xmath14(dashed lines ) and @xmath1 ( solid lines ) at @xmath39mt ( top panel ) and @xmath4030 mt ( bottom panel ) . the vertical dashed lines mark the peak positions . curves are slightly offset at @xmath4=-25mv to correct for a parametric charge rearrangement at that gate value . ( b ) - ( d ) gray scale plots of the currents @xmath0 ( a ) , @xmath14(b ) and @xmath1(c ) as a function of @xmath28 and @xmath4 . the white vertical lines are fits to the cb - peak positions in ( c ) . , width=326 ] starting from this strongly coupled situation , we have optimized the amplitude of the ab - oscillations in the ring and their coupling to the cb - resonances in the dot by changing several gate voltages . note that specifically @xmath41 and @xmath42 were increased from 0mv to 200mv in order to optimize the interconnecting channel from the ring to the dot . in the following , more negative @xmath15 values are therefore required to close this channel . figure[fig2](c ) shows a simultaneous measurement of @xmath0 and @xmath14 when @xmath36 is tuned over a large range . while @xmath14 shows clear cb - oscillations the ring current depends strongly on the level alignment in the dot . when the dot is tuned through a resonance peak with increasing @xmath36 , @xmath0 is reduced by up to a factor of two . this is a manifestation of the fano effect . it has been shown previously that a dot which is side - coupled to a wire is a typical fano system.@xcite here we probe the properties of such a fano scatterer by embedding it in a two terminal ring structure . figure[fig3](a ) shows curve traces of @xmath0 and @xmath14 as a function of @xmath4 for @xmath43mt and after half an ab - period ( @xmath44mt ) . the vertical lines mark the positions of the cb - resonances as determined from peak fits to @xmath14 with a lorentzian line shape . increasing the flux through the ring by half a flux quantum changes @xmath0 from a behavior where a dip occurs at each gate voltage before the dot resonance to a situation where a similar asymmetric dip occurs after the dot resonance . figures[fig3](b)-(d ) show @xmath0 , @xmath14 and @xmath1 in gray - scale plots as a function of @xmath4 and @xmath28 . the dips in @xmath0 [ bright areas in fig.[fig3](b ) ] continue to oscillate back and forth for each additional flux quantum through the ring until at @xmath45mt normal ab - oscillations are recovered . the vertical white lines are again the peak positions determined from fits to @xmath14 shown in fig.[fig3](c ) . the slight peak shifts as a function of magnetic field are due to the influence of the magnetic field on individual orbital levels of the dot . from the evolution of the peak amplitude and position , we find that neighboring peaks show highly correlated behavior over large magnetic field ranges indicating spin pairing in the dot for the six peaks that are shown . as expected @xmath1 [ fig.[fig3](d ) ] is different from zero only when @xmath14 is finite . @xmath1 which flows through both structures oscillates around zero with the ab - period as a function of @xmath28 . this is in contrast to @xmath0 which has a significant @xmath28-independent background . we explain this by the fact that @xmath1 measures the difference between the two transmissions @xmath46 and @xmath47 and the @xmath48-contribution therefore cancels out if the structure geometry and the couplings are ideally symmetric . due to the three terminals the different ab - geometry for @xmath46 and @xmath47 retains a net ab - effect . we model the coherent quantum dot in the single particle , single level approximation as a closed - end resonator consisting of a 1d wire with a barrier and a perfect reflector as shown in fig . [ fig4](a ) . the unitary scattering matrix of a general barrier is given by @xmath49 where @xmath50 is the reflection probability and @xmath51 are free phase parameters . the reflection amplitude of such a dot is found from a sum over all feynman paths @xmath52 t = r+\frac{t\lambda r_\mathrm{t}\lambda t}{1-r'\lambda r_\mathrm{t}\lambda}.\ ] ] here @xmath53 describes the propagation between the barrier and the reflector and @xmath54 is the reflection amplitude at the reflector . by introducing the phase @xmath55 which an electron accumulates during a round trip in the dot , the total reflection amplitude becomes @xmath56 this can be rewritten as @xmath57 with @xmath58 and @xmath59 . for this closed - end structure , the amplitude is restricted to the unit circle and resonances arise when @xmath60 , @xmath61 . . ( d ) model calculation for -2 to + 1 flux quanta passing through the ring and over a change of @xmath62 in the dot phase @xmath63 . the parameters used were @xmath64 , @xmath65 , @xmath66 and all leads to the ring are assumed to be ideal beam splitters . ( e ) section of @xmath0 for two spin paired states . the white dashed lines are the peak positions extracted from fits to @xmath14 . , width=326 ] the dot resonator is then side - coupled to a wire through a three terminal junction as seen in fig . [ fig4](b ) . the transmission and reflection amplitudes of the wire are @xmath67 where the amplitudes @xmath68 and @xmath69 describe the direct transmission from the entrance to the exit of the wire and the direct reflection at the entrance of the wire respectively . @xmath70 and @xmath71 are the amplitudes for transmission from the entrance into the dot and from the dot to the entrance respectively , and @xmath72 is the amplitude for transmission from the dot into the exit of the wire . the side - coupled dot structure can hence be mapped onto a @xmath73 transfer matrix @xmath74 as indicated in fig . [ fig4](b ) . the transmission probability of the wire with the side - coupled dot can then be shown to have the fano form @xmath75 where @xmath76 is the fano parameter . for @xmath77 @xmath78 produces fano line shapes . this becomes clear when @xmath79 is approximated close to a resonance at energy @xmath80 , neglecting any energy dependence of @xmath81 . it follows that @xmath82 with @xmath83 giving the link to the fano formula in eq.[eqn1 ] . note , that the fano parameter @xmath10 in eq.[tw ] , and through it the line shape , is uniquely determined by the direct reflection phase @xmath84 of the barrier in the side - coupled dot structure , i.e. , @xmath85 produces a symmetric resonance peak ( @xmath86 ) , @xmath87 produces a symmetric dip in the transmission ( @xmath88 ) , and @xmath89 produce maximally asymmetric line shapes ( @xmath90 ) . in the following we assume the three terminal junction to be a symmetric beam splitter which is described by the unitary scattering matrix @xcite @xmath91 using this for the coupling between the dot and the wire we get @xmath92 resulting in a vanishing constant term in the transmission probability in eq . it follows that @xmath93 a _ fano scatterer _ is therefore characterized by the transfer matrix @xmath94 in eq.[ft ] , with the two parameters @xmath10 and @xmath79 . the ring - dot structure in the experiment can now be modeled by placing the transfer matrix @xmath95 in one of the arms of the ring as shown in fig . [ fig4](c ) . the ring is attached to two ideal semi - infinite leads at the junctions 1 and 2 . these junctions together with the transfer matrix @xmath94 partitions the ring into three sections of equal length . when an electron wave traverses each section it acquires the fermi phase @xmath96 , where @xmath97 is the fermi wave - number , as well as a magnetic phase @xmath98 ( opposite signs for clockwise and anticlockwise propagation ) where @xmath99 is the magnetic flux enclosed by the ring and @xmath100 is the magnetic flux quantum . these phases enter the transfer matrix for each ring section @xmath101 as indicated in fig.[fig4](c ) . with the boundary conditions on the incoming amplitudes in the source and drain leads @xmath102 and @xmath103 , one finds the two - terminal conductance from the landauer formula @xmath104 with @xmath105 and @xmath106 for comparison with the experimental data we set @xmath107 , a value which gives pronounced ab - oscillations in the ring while suppressing higher harmonics when the dot is decoupled ( r=1 ) . figure[fig4](d ) shows the calculated conductance @xmath108 as a function of @xmath109 and the number of flux quanta penetrating the ring with @xmath110 and @xmath66 . for comparison , a similar section of the measured @xmath0 as a function of b and @xmath111 is shown in fig.[fig4](e ) . very good agreement is found , indicating that our simple model quite accurately describes the experimental situation of the coupled ring - dot structure . this agreement is robust against variations in @xmath50 but sensitive to changes in @xmath84 , since @xmath84 is linked to the fano factor @xmath10 . we determine @xmath112 in agreement with the experimental procedure where we optimized the coupling between the ring and the dot to result in most asymmetric fano line shapes . the sign of the asymmetric line shape in @xmath0 is periodically tuned by the flux penetrating the ring . this behavior is qualitatively similar to recently reported results in ref . where a complex q was introduced in order to accommodate for the magnetic field dependence of the fano line shapes . however , we have not been able to transform eq.[tw ] into a form where the magnetic flux through the ring @xmath113 enters as a phase @xmath114 in a complex @xmath10-parameter and we therefore restrict ourselves to a real @xmath10-parameter reflecting only the fano effect of one arm of the ring with the side coupled dot . the interference properties of this fano scatterer are then probed by ab - interference in the ring . this is different to ref . where the whole ring structure with a dot embedded in one arm is considered as a fano scatterer . the @xmath10-parameter is indicative of the coupling between the dot and the ring and to further strengthen the agreement between model and experiment we tune @xmath115 successively more negative to reduce the coupling and thereby tune the @xmath10-parameter . figure[fig5](a ) shows @xmath0 as a function of @xmath28 and @xmath111 for decreasing values of @xmath115 from left to right . as the coupling is decreased @xmath0 becomes more strongly suppressed at @xmath116mt while the dips ( white areas ) at @xmath43mt seem to disappear . within our model , this can be understood assuming that a decrease in @xmath115 reduces the phase pick - up @xmath84 of the electrons traversing the interconnecting channel and therefore decreases @xmath10 from its initial value of 1 . in addition , the reflection probability r is likely to increase . figure[fig5](b ) shows @xmath108 for the same parameters as in fig.[fig4](a ) while varying @xmath10 from 1 to 0 . again we find good qualitative agreement with the experiment except for very small @xmath10-values where it is difficult to determine which q matches the experimental situation best . the last plot on the right shows @xmath108 for @xmath1170 where ab - like oscillations are found except very close to the resonance of the dot . this transition is more complete in the experimental data , where for @xmath118mv only small dips in @xmath0 are found . in fig.[fig6](a ) traces of @xmath0 and @xmath14 as a function of @xmath4 at @xmath43mt ( solid lines ) and @xmath44mt ( dashed lines ) are plotted for @xmath119 - 68.3,-68.7,-69.1,-69.5 , and -69.9mv . the traces have been taken across the coulomb resonance marked with arrows in fig[fig5](a ) and for each value of @xmath15 they were shifted in @xmath4 such that the peak maximum in @xmath14 ( only @xmath43mt shown ) comes to lie at @xmath120mv . the topmost solid ( lowermost dashed ) line respectively mark an ab - maximum(minimum ) for @xmath119 - 69.9mv which is the most weakly coupled situation and corresponds to a @xmath121 . here the dip(peak)in @xmath0 close to the resonance position are more clearly visible . the behavior in fig.[fig5](b ) is qualitatively different in the calculated @xmath108 where even for @xmath122 the dip close to resonance at @xmath43 yields a sharp transmission zero for the parameters that were chosen . we will argue in the following that this discrepancy between model and experiment can be understood when taking into account decoherence in the dot . in contrast to the model , the experimental quantum dot is connected to two contacts which will lead to partial phase randomization of the electron paths visiting the quantum dot.@xcite at @xmath123=0.1k for decreasing gate coupling tuned by @xmath119 - 68.3,-68.7,-69.1,-69.5 , and @xmath124mv . the curves are measured over the peak marked with an arrow in fig.[fig5 ] and for @xmath39mt ( solid lines ) , @xmath125mt ( dashed lines ) . for each value of @xmath15 we subtract an offset in @xmath4 in order for the peak in @xmath14 to be at @xmath126mv . ( b ) @xmath127 for @xmath128 and 0 including decoherence in the dot with a decoherence rate @xmath129 . ( c ) @xmath127 with strong decoherence in the dot using @xmath130 . ( d ) @xmath0 as a function of @xmath4 and @xmath28 at @xmath123=0.65k.,width=326 ] since the observed fano resonances rely on coherent transport through the whole structure , decoherence at finite temperatures due to coupling of the structure to the environment , e.g. the source and drain leads of the dot , is important . we have measured the temperature dependence of the fano line shapes in @xmath0 and found that at @xmath131k the ab - oscillations in the ring persist while the fano behavior is strongly suppressed . this is in agreement with a picture where the paths that pass through the dot are most strongly affected by decoherence due to the fact that such a path is longer than the direct paths in the ring which do nt traverse the dot . furthermore , the dot paths will be influenced by decoherence induced by the two contacts attached to it . we follow ref . in order to include decoherence in the dot by the modification of the phase @xmath132 where @xmath133 is the length of the path through the dot and @xmath32 is the phase coherence length . this introduces a damping of the coherent channel passing through the dot . the `` absorbed amplitude '' is re - injected into the incoherent channel by introducing an additional term in the two - terminal conductance @xmath134 where @xmath135 is the transmission(reflection ) probability for injection in lead @xmath26 and detection in lead @xmath27 and is given by @xmath136 with @xmath137 and @xmath138 the re - injection asserts probability conservation in the structure and hence the symmetries required by the two - terminal onsager relation . figure[fig6](b ) shows @xmath127 for @xmath139 , @xmath140 , @xmath141 , @xmath142 , @xmath143 , and @xmath144 taking into account a finite phase coherence length @xmath145 . the solid lines are again for @xmath146 and the dashed lines for @xmath147 . the gray solid line shows the line shape for @xmath122 in the fully coherent case for comparison . decoherence leads , as expected , to broadening of the fano line shapes and to strong amplitude reduction , a fact which has been suggested to be used as a measure for decoherence in a quantum dot in the fano regime@xcite . the close resemblance to the measured data at @xmath148mk [ fig.[fig6](a ) ] suggests that even at these low temperatures the contacts to the quantum dot lead to decoherence of paths entering the quantum dot from the ring , hence , a modification of the fano effect which we measure in @xmath0 . however , the deviation from the fully coherent limit is strongest in the regime of weak coupling between the two structures close to @xmath122 . the experimental traces show a reduction in oscillation - amplitude ( separation between solid and dashed curve ) of about 20% close to resonance , while @xmath127 exhibits a reduction which is more than twice as large . we believe that in the real structure both the reflection probability @xmath50 and the decoherence rate increase as the coupling between the ring and the dot is decreased . for weaker coupling , the time an electron spends in the dot becomes larger and this means that close to the dot resonance its probability for tunneling into one of the contacts of the dot becomes larger . we have adjusted the parameters in the model accordingly and find good agreement for @xmath149 and @xmath150 ( not shown ) . note , that the cross - coupling of @xmath15 has the opposite effect and the coulomb peak width in @xmath14 decreases for decreasing @xmath115 . for @xmath64 the effect of decoherence due to the dot contacts is small since we expect strong coherent coupling between the two structures . this can be seen from the good agreement between @xmath127 and @xmath0 for this situation . when the temperature is increased decoherence becomes important also for the @xmath64 case and the fano effect is almost completely lost at @xmath151 . figure[fig6](c ) shows @xmath127 for @xmath149 , @xmath152 , @xmath64 and @xmath153 . these parameters were chosen such as to resemble the data in fig.[fig6](d ) measured at @xmath151 . using the dimensions of the ring dot structure we estimate a phase coherence length @xmath154 nm at this temperature . the decoherence of paths through the dot strongly suppresses the asymmetric line shapes , leading to an ab pattern in the conductance . this clearly links the observed asymmetric line shapes to the interaction between discrete states formed in the dot and continuous states of the ring and further strengthens our explanation in terms of the fano effect . the scattering model clearly reproduces all the characteristic features of the experimental data but has a few obvious limitations . while we restrict decoherence to the dot , a more accurate model would include both thermal averaging as well as decoherence in the dot , the ring and the interconnecting arm . using thermal averaging alone we were not able to reproduce the experimental data at finite temperature and the inclusion of decoherence in the ring and the interconnecting arm did not improve the model . we therefore restricted ourselves to decoherence as discussed above and speculate that the additional leads on the dot are the main source for decoherence of the fano effect . the model only treats a two - terminal geometry which neglects effects of the two contacts on the dot in terms of the symmetry of the current as a function of @xmath28 . this means that the model also assumes that there are no net currents flowing from the dot to the ring ( @xmath155 ) . from the simultaneous measurements of @xmath1 we estimate this contribution to be about two times smaller than the dips ( peaks ) in @xmath0 for @xmath122 and we are therefore led to believe that it has little influence on @xmath0 . however , where the current through the dot is suppressed at larger magnetic fields [ e.g. , in the lower left corner in fig.[fig3](c ) ] ab - periodic features are found in @xmath14 indicating that either the coherent coupling between the two systems also influences @xmath14 or that there is a considerable ab - modulated net current flowing from the ring leads into the dot . in a quantum dot the addition energy between two resonances is not only determined by space quantization effects but is enhanced by coulomb interactions between the electrons on the dot . this is not included in our calculation and could explain the slightly stronger modulation as a function of gate voltage in the experimental data when comparing with the model at @xmath156k in fig.[fig4](c)-(d ) . while we think that our model accurately describes the regime where the fano effect is strong , we find that in the experimental data at @xmath156k the dips in @xmath0 are closer to the actual peak positions of the dot than in the model . we have shown that the influence of cross - coupling from the dot is negligible in the completely decoupled system ( see also ref ) . however for @xmath119 - 69.9 mv the channel between the two structures is not pinched off and coulomb interactions could be stronger and possibly lead to modified fano line shapes close to the resonance condition of the dot.@xcite we find that spin paired peaks involving subsequent filling of the same orbital do not only show similar behavior in @xmath14 but also lead to a similar interference pattern in @xmath0 as evident , e.g. , in fig.[fig4](b ) . we conclude , that spin is not important for the observed interference effect and kondo correlations@xcite are therefore negligible . in conclusion , we have demonstrated that the current through a ring with a coherently side coupled dot leads to asymmetric line shapes in the current through the ring as a function of a gate voltage which tunes the discrete dot levels . this is indicative of the fano effect and we have shown that the symmetry of the observed fano line shapes can be tuned by applying a magnetic field . the @xmath10-parameter was adjusted with a gate voltage tuning the coupling between the ring and the dot . good agreement between the data and a single - channel scattering matrix model allows us to identify our structure with a fano scatterer embedded in a two - terminal ab - ring . comparison with this model also lets us to identify the additional leads on the dot as the dominant source of decoherence for the fano effect . raising the temperature to @xmath156k the fano effect almost completely disappears while ab - interference in the ring persists to above @xmath157k .

transport measurements are presented on a quantum ring that is tunnel - coupled to a quantum dot . when the dot is in the coulomb blockade regime , but strongly coupled to the open ring , fano line shapes are observed in the current through the ring , when the electron number in the dot changes by one . the symmetry of the fano resonances is found to depend on the magnetic flux penetrating the area of the ring and on the strength of the ring dot coupling . at temperatures above t=0.65k the fano effect disappears while the aharonov bohm interference in the ring persists up to t=4.2k . good agreement is found between these experimental observations and a single - channel scattering matrix model including decoherence in the dot .

cond-mat0602246

information encoded in qubits can be used for reliable quantum communication or efficient quantum computing @xcite . this information is encoded in a quantum state @xmath4 which in the case of a qubit can be parameterized as |(,)= |0 + e^i |1 ; [ 1 ] where @xmath5 and @xmath6 are basis vectors of the 2-dimensional space of the qubit and @xmath7 ; @xmath8 . qubits are very fragile , that is the state of a qubit can easily be changed by the influence of the environment or a random error . one ( very inefficient ) way to protect the quantum information encoded in a qubit is to measure it . with the help of an optimal measurement one can estimate the state of a qubit , with an average fidelity equal to 2/3 ( see below ) . in this way a quantum information is transformed into a classical information which can be stored , copied , and processed according the laws of classical physics with arbitrarily high precision . however , in order to utilize the full potential of quantum information processing we have to keep the information in states of quantum systems , but then we are forced to face the problem of decoherence . recently it has been proposed that quantum information and quantum information processing can be stabilized via symmetrization @xcite . in particular , the qubit in an unknown state is entangled with a set of @xmath1 ( ancilla ) qubits in a specific reference state ( let us say @xmath5 ) so the symmetric state @xmath9 of @xmath10 qubits , |(|,0 ,0+|0,, ,0+ + |0,0 , ) , [ 2 ] is generated . if we introduce a notation for completely symmetric states @xmath11 of @xmath10 qubits with @xmath12 of them being in the state @xmath6 and @xmath13 of them in the state @xmath5 , then the state ( [ 2 ] ) can be expressed in the simple form |(,|)= |n;0 + e^i| |n;1[3 ] where the parameters @xmath14 and @xmath15 are specified by the relations = ; [ 4 ] and @xmath16 , while @xmath17 . we see that symmetric @xmath10 qubit state @xmath18 is isomorphic to a single qubit state . but in this case the information is spread among @xmath10 entangled qubits - the original quantum information is `` diluted '' . each of the qubits of the @xmath10-qubit state ( [ 3 ] ) is in the state @xmath19 . we define the average fidelity between the single state @xmath20 and the original qubit @xmath21 as = d(;)|_j(,| ) |(;)[5 ] where @xmath22 is the invariant measure on the state space of the original qubit ( i.e. we assume no _ prior _ knowledge about the pure state @xmath21 ) . for this fidelity we find the expression _ 0 = . [ 5a ] we see that for @xmath23 the fidelity @xmath24 is equal to unity ( as it should , because in this case @xmath25 ) while in the limit @xmath26 we find @xmath27 . in fact in this limit density operators of individual qubits are approximately equal to @xmath28 . in other words , individually the qubits of the symmetric state @xmath18 in the large @xmath10 limit do not carry any information about the original single - qubit state @xmath29 . so how can we extract the information from the @xmath10-qubit symmetric state ( [ 3 ] ) ? the ideal possibility would be to have have a perfect _ universal _ disentangler which would perform a unitary transformation type of disentangling transformation @xcite . while the perfect transformation is impossible , there are a number of things we can do to concentrate the information from the @xmath10-qubit state @xmath18 back into a single qubit . in principle , we have the following possibilities : * i ) * we can either optimally measure the @xmath10 qubit state and based on the information obtained prepare a single - qubit state . * ii ) * we can design a quantum disentangler which would perform a transformation as close as possible to the ideal disentangling ( [ 7 ] ) . in this quantum scenario we have several options - the process of disentanglement can be input - state dependent . this means that states ( [ 3 ] ) for some values of the parameters @xmath14 and @xmath30 will be disentangled better than for other values of these parameters . alternatively , we can construct a quantum device which disentangles all the state with the same fidelity . * iii ) * finally , we propose a probabilistic disentangler , such that when a specific projective measurement over an ancilla is performed at the output , the desired single - qubit state is generated . the probability of the outcome of the measurement in this case is state - dependent . in what follows we shall investigate all these possibilities . before proceeding we note that a different type of disentangler has been considered by terno and mor @xcite - @xcite . they considered two different operations . the first would take the state of a bipartite quantum system and transform it into a state that is just the product of the reduced density matrixes of the two subsystems . the second , which is a generalization of the first , would again start with a state of a bipartate quantum system , and map it into a separable state which has the same reduced density matrixes as the original state . they showed that while both of these processes are impossible in general , they can be realized for particular sets of input states . an approximate disentangler of the first type has been considered by bandyopadhyay , et . al . @xcite . the disentanglers we are considering extract , to some degree of approximation , an unknown state from an entangled state formed from that state and a known state . here we first describe a measurement scenario utilizing a set of specific projection operators . then we present the optimal measurement - based approach to quantum disentanglement and we derive an upper bound on the fidelity of the measurement - based disentangler . we utilize the fact that the @xmath10 qubit system prepared in the state @xmath18 is isomorphic to a single qubit . therefore we first consider a strategy based on a a projective measurement with two projectors @xcite @xmath31 ( @xmath32 ) with |_0(,)&= & @xmath33 and @xmath34 , where the angles @xmath35 and @xmath36 are chosen randomly if no _ prior _ information about the measured @xmath10-qubit state is available . we can use the result of the measurement to manufacture a a single - qubit state . specifically , if the result of the measurement is positive for @xmath37 then the single qubit is prepared in the state @xmath38 then the single qubit is prepared in the orthogonal state @xmath39 . for a particular orientation of the measurement apparatus ( i.e. the angles @xmath40 ) this measurement - based scenario gives us a single qubit prepared in the state described by the density operator ^(meas ) ( , |;,)= _ j=0 ^ 1 ||_j|^2 |_j_j| [ 9 ] after we average over all possible orientations of the measurement apparatus we obtain on average a single qubit prepared in the state ^(est ) ( , |)= |(,|)(,|)| + . [ 10 ] to find the average fidelity of this measurement - based disentangling procedure we have to evaluate the mean fidelity @xmath41 , that is the overlap between the state ( [ 10 ] ) and the original input state @xmath42 averaged over all possible orientations of the input qubit : _ 1 = d (,)|^(est ) ( , |)| ( , ) . [ 11 ] taking into account the relation ( [ 4 ] ) we perform the integration in eq.([11 ] ) and we find _ 1 = ( 1+f_n ) [ 12 ] where the function @xmath43 reads f_n= . [ 6 ] for @xmath23 : @xmath44 which is the optimal fidelity of estimation of the state of a single qubit . from fig . [ fig1 ] we see that the fidelity ( [ 12 ] ) is a decreasing function of @xmath10 and in the limit @xmath45 we find @xmath46 , which is equal to the fidelity of a _ random _ guess associated with a binary system such as the two projectors under consideration . in other words , when the original qubit is diluted into an infinite qubit state of the form ( [ 3 ] ) no relevant information can be gained from the measurement . the estimated density operator ( [ 10 ] ) in this case is simply equal to @xmath47 , which is understandable , because as we have shown earlier in this limit the @xmath10-qubit state is approximately in the state @xmath48 , so information about the original is `` almost '' totally lost . we now want to find an upper bound @xmath49 for the average fidelity which can be achieved by a wide class of measurement - based disentanglement procedures . we assume that it is _ a priori _ known that our @xmath10-qubit is prepared in the symmetric state ( [ 2 ] ) with unknown parameters @xmath50 and @xmath51 associated with a single - qubit state ( [ 1 ] ) . the integration measure on the state space of the single qubit is @xmath52 and the corresponding prior probability density distribution on this state space is constant . our strategy is to measure the input state @xmath9 along the vector @xmath53 [ see eq . ( [ 8 ] ) ] , where the angles @xmath54 and @xmath55 are chosen according to the distribution @xmath56 , which will be left unspecified for the moment . if the answer is positive , we produce the output density matrix @xmath57 , and if it is negative we produce @xmath58 , where @xmath59 with @xmath32 and @xmath60 given by eq . ( [ 8a ] ) . we shall also leave the conditional probabilities , @xmath61 unspecified , as this allows us to consider a wide range of strategies . for a fixed @xmath53 , the probability of the output being @xmath57 is @xmath62 and the probability of it being @xmath58 is @xmath63 . averaging over all vectors , @xmath64 gives us ^(out)(,| ) & = & d^[|_0| & + & |_1 ||^2_1(^,^ ) ] q(^ , ^ ) . in order to find the average fidelity of the output produced by this procedure , we compute the fidelity for a particular input state and average over the input ensemble @xmath65 where @xmath14 is a function of @xmath50 [ see eq.([4 ] ) ] . this can be expressed as @xmath66 where @xmath67 is a normalized joint probability distribution , and @xmath68 where @xmath69 and the supremum is taken over the variables @xmath70 . we then have that @xmath71 , \ ] ] where the supremum is now taken over @xmath72 and @xmath73 . in order to calculate this upper bound we must find explicit expressions for @xmath74 and @xmath75 . after performing the necessary calculations we find for @xmath49 the expression ^max= . this fidelity for @xmath23 is equal to 2/3 while in the limit @xmath26 is equal to 1/2 . for any other @xmath10 is larger than the fidelity @xmath41 of the measurement given by eq.([12 ] ) as discussed in our previous example . nevertheless , as we will show later it is alway smaller than the fidelity of the universal quantum device . in what follows we show that a quantum disentangler which preserves quantum coherences can distill the information back to a single qubit more efficiently than can the measurement - based method . as we have already said in the introduction quantum mechanics does not allow one to construct a perfect disentangler which would perform transformation ( [ 7 ] ) for an arbitrary ( unknown ) state @xmath42 diluted in the @xmath10 qubit symmetric state ( [ 3 ] ) . nevertheless , we can try to design optimal disentanglers which perform best under given constraints . so let us assume our quantum disentangler , @xmath76 , is a quantum system with a @xmath77-dimensional hilbert space spanned by basis vectors @xmath78 ( @xmath79 ) . the disentangler is always initially prepared in the state @xmath80 , and then it interacts with the @xmath10-qubit system in the state ( [ 3 ] ) . at the output we want to disentangle the @xmath1 ancilla qubits from the original qubit , so we expect to have |(,|)|d_0|n-1;0_k=1^k _ j=0 ^ 1 c_j(,| ) |j|d_k . [ 13 ] as seen from eq.([13 ] ) during the disentanglement process the entanglement between the @xmath1 ancilla qubits and the original qubit is transferred ( swapped ) into the entanglement between the original qubit and the disentangler itself . by tracing over the disentangler we then expect to obtain the best possible disentangled qubit in the state @xmath81 . now we impose several constraints which would specify what we mean by the optimal covariant ( universal ) disentangler : the fidelity between the output of the disentangler and the original state @xmath42 has to be invariant with respect to rotations of the original qubit , so the fidelity has to be input - state independent . this universality of the disentangler would then guarantee that the information from the symmetric state ( [ 3 ] ) is extracted for all states equally well . we are looking for the _ optimal _ disentangler which would disentangle the information with the highest fidelity . imposing these two conditions we have found the unitary transformation which realizes the _ optimal covariant _ disentangler , i.e. which disentangle the qubit - state @xmath29 from the @xmath10-qubit state @xmath9 in the optimal and the @xmath29-state independent way ( see appendix ) . this disentangler is described by the transformation : |n;0|d_0 & & |n-1;0 ; + basis vectors of the disentangler . the amplitudes @xmath82 and @xmath83 given by the relation _ n=()^1/2 ; _ n=. [ 15 ] we can directly verify , that the fidelity @xmath84 is input - state independent and equal to @xmath85 . moreover , it can be shown that the transformation ( [ 15 ] ) is optimal , i.e. among all unitary transformations satisfying the given conditions the transformation ( [ 15 ] ) has the largest fidelity . we see that for @xmath23 the fidelity @xmath86 , which is obvious , because the original qubit has not been entangled with ancilla qubits . we plot @xmath87 in fig . we see , that it is _ always _ larger than the fidelity of the disentanglement via measurement . in the limit @xmath26 even the quantum disentangler gives us a totally random outcome . so in this limit , even optimal quantum entangler on which we impose the universality condition , is not able to extract information from the state ( [ 3 ] ) . this is one of the main results of our paper - the optimal covariant quantum disentangler operates better than if the information is extracted ( disentangled , distilled ) from the symmetrized state ( [ 3 ] ) with the help the of optimal measurement . this is due to the fact that @xmath88 . one can also ask the opposite question , how can we generate out of a qubit in an unknown state @xmath29 the symmetric state of the form ( [ 3 ] ) . it can be shown that within quantum mechanics perfect universal entanglers , which would realize the inverse of the relation ( [ 7 ] ) do not exist . if one wants to create a state ( [ 3 ] ) from a qubit in an unknown state and @xmath1 ancilla qubits in the known state @xmath5 again two scenarios are possible , the measurement - based and quantum scenarios . it is not surprising that the quantum scenario works better . we have found the optimal universal ( covariant with respect to rotations of the input qubit ) quantum entangler given by the transformations : |0|n-1;0|e_0 & & ; + basis states of the quantum entangler , @xmath89 is its initial state and the parameters @xmath90 and @xmath83 are given by eq.([15 ] ) . one can check that the fidelity between the output of this entangler described by the density operator @xmath91 and the ideally entangled state ( [ 3 ] ) is input - state independent ( i.e. does not depend on the parameters @xmath92 ) and is equal to @xmath93 . this is the best possible universal ( covariant ) entangler . the universal disentangler gives a higher fidelity than does the best measurement - based procedure , but it is not obvious that this is the best that one can do . in the case of quantum cloning , the universal cloners are the ones which maximize the average fidelity @xcite . as we shall see , however , in the case of disentanglers this is no longer the case ; there are state - dependent devices which are better . consider the general disentangler transformation @xmath94 , are states of the disentangler itself and need not be orthogonal . they must , however , satisfy the constraints imposed by the unitarity of the above transformation . the input state for the device is assumed to be @xmath95 , and the ideal output state , to which the actual output should be compared , is @xmath96 . the output state is calculated by starting with the input state , using the above transformation , and then tracing over the disentangler to obtain an output density matrix , @xmath97 . one then finds the average fidelity for this process , which we shall call @xmath98 , from @xmath99 note that we are assuming a specific ensemble of input states ; the probability of the one - qubit state @xmath100 is assumed to be constant on the bloch sphere . our result for the average fidelity for a state - dependent device depends on our choice of input ensemble , while for a state - independent device the average fidelity is independent of this ensemble . the calculation of the average fidelity is given in the appendix , and will not be given in detail . we find that @xmath101 and @xmath102 . this implies that the final state is just a product of the state of the @xmath10 particles and the entangler state , which means that the entangler states can be dropped from the problem . therefore , the transformation which maximizes the average fidelity is just a kind of state swapping transformation . the average fidelity itself is given by @xmath103 , where the coefficient @xmath43 is given by eq.([6 ] ) . this average fidelity is larger than the fidelity of the optimal _ universal _ disentangler ( see fig . [ fig1 ] ) . in this case , the fact that the universality condition forces us to use an additional quantum device , the disentangler , with which the qubit at the output becomes partially entangled , results in a net loss of information . as a result the fidelity of the universal ( covariant ) entangler is smaller . analogously , we find that quantum state - dependent entanglement can also be performed by a kind of state swapping transformation , i.e. |(,)|n;0|0 @xmath104 . nevertheless , when averaged over all values of @xmath105 we find the mean fidelity of this state - dependent entangler to be equal to @xmath43 which on average is larger than the fidelity of the state - independent entangler . let us examine a simple quantum network which takes as an input the @xmath10-qubit state ( [ 3 ] ) . the network is composed of a sequence of @xmath1 c - not gates @xmath106 where @xmath107 is the c - not with @xmath108 being the control bit and @xmath12 being the target bit . this sequence of the c - not gates acts on the two vectors @xmath109 and @xmath110 as [ 20 ] p_n|n;0 & & |n-1;0|0 + p_n |n;1 & & ( |n-1;1+|n-1;0 ) |1from which it follows that the input vector ( [ 3 ] ) is transformed as |(,| ) & & ( |v_+| ( , ) + & & + |v_-|0 ) [ 21 ] where @xmath111 is the normalization constant . in eq.([21 ] ) we have introduced two orthogonal vectors of @xmath1 qubits @xmath112 . measurement on the first @xmath1 qubits is performed in order to determine whether they are in the state @xmath113 or @xmath114 . if the result @xmath115 is obtained , then the @xmath10th qubit is in the desired state @xmath42 . the probability of this outcome is given by p_|v_+= . [ 23 ] this probability is input - state - dependent , and it decreases with @xmath10 . there is a difference between this probabilistic process and those considered previously , such as probabilistic cloning @xcite . those only work for set of input states which is finite . the process considered above , however , works for a continuous , and hence infinite , set of input states . it , in fact , works for all input states of the type we are considering . therefore , we can conclude that the range of applicability of probabilistic devices depends on the process being considered . we have considered a number of different methods of extracting an unknown state from an entangled state formed from that state and a known state . measuring the state is , as expected , the least effective method . in the case of quantum devices , the universal device was not best one , at least if average fidelity is used as the criterion . probabilistic quantum devices were seen to work very well for this operation in that they can be used for the entire set of input states . this work was supported by the national science foundation under grant phy-9970507 , by the ist project equip under the contract ist-1999 - 11053 and by the crest , research team for interacting career electronics . let us consider the optimal quantum disentangler which acts as close as possible to the ideal transformation ( [ 7 ] ) . the disentangler maps the space spanned by the vectors @xmath116 and @xmath117 , into the space spanned by @xmath118 and @xmath119 . this suggests that we consider a transformation of the following form @xmath120 where @xmath80 is the initial state of the disentangler which is supposed to be the same for all inputs and @xmath121 ( @xmath122 ) are some unnormalized disentangler state - vectors . our task is to determine these vectors . unitarity immediately implies that @xmath123 we shall now use our disentangler transformations ( a.1 ) to calculate the fidelity of the actual output to the ideal output ( [ 7 ] ) the input of the disentangler is given by eq . ( [ 3 ] ) . if we introduce a notation @xmath124 and @xmath125 we can write the result of the transformation ( a.1 ) @xmath126 we now use this expression to find the output density matrix and trace out the disentangler itself . we define the @xmath10-qubit output density matrix to be @xmath127 the output fidelity is given by @xmath128 where @xmath129 is given by eq . ( [ 7 ] ) . if we denote @xmath130 and @xmath131 we can express this fidelity as @xmath132 \nonumber \\ & + & \alpha^{\ast}\beta \langle d_{3}|d_{4}\rangle ) \nonumber \\ & + & ( \alpha^{\ast})^{2}\beta^{2}\sqrt{n}\langle d_{2}| d_{3}\rangle + \alpha^{2}(\beta^{\ast})^{2}\sqrt{n } \langle d_{3}|d_{2}\rangle \}. \label{a.5 } \eqnum{a.5}\end{aligned}\ ] ] demanding that the fidelity be independent of phases of @xmath133 and @xmath134 we find that @xmath135 assuming these conditions to be satisfied the fidelity becomes @xmath136 \ } . \nonumber\end{aligned}\ ] ] in order for this to be independent of @xmath133 and @xmath134 , the term in brackets must be proportional to @xmath137 comparing eqs . ( a.7 ) and ( a.8 ) we find that @xmath138 combining these requirements with those imposed by unitarity we conclude that @xmath139 and @xmath140 . this means that in order to maximize @xmath141 , we must maximize @xmath142 . our first step in accomplishing this is to note that by combining the results of eqs . ( a.9 ) and ( a.10 ) we have that @xmath143 where @xmath144 and @xmath145 . solving for @xmath142 we find that @xmath146 which , assuming @xmath147 , is greatest when @xmath148 . this implies that @xmath149 and that @xmath150 imposing now the conditions on inner products we find that @xmath151 we can summarize our results in the following way . let @xmath152 be a set of three orthonormal vectors and define two parameters @xmath82 and @xmath83 given by eq . ( [ 15 ] ) we then have that @xmath153 and the universal _ optimal _ disentangler transformation is given explicitly by eq . ( [ 14 ] ) . in order to find the optimal input - state dependent disentangler we find the explicit form of the transformation ( a.1 ) such that the _ averaged _ fidelity @xmath154 ( with @xmath141 given by eq . ( a.5 ) ) is maximized . here , as usually , the integration measure is @xmath155 . therefore after the integral over the phase @xmath51 is performed we can write the average fidelity as @xmath156\ } \nonumber\end{aligned}\ ] ] with [ a.18 ] _ 1&=&_0^ ^4 + _ 2 & = & _ 0^ ^4 + _ 3&=&_0^ ^2 ^2 after the integration over the parameter @xmath50 we find [ a.19 ] _ 1&= & + _ 2&= & + _ 3&= & from the unitarity of the disentangling transformation it follows that @xmath157 and @xmath158 . when we introduce the notation @xmath159 where @xmath160 , and @xmath161 ; @xmath162 we can rewrite the average fidelity ( a.17 ) as & = & [ _ 1 n ( _ 1-_3 ) + _ 4(_2-_3 ) + & + & 2_3 u + _ 3(1+n ) ] . [ a.21 ] taking into account that @xmath163 and @xmath164 we easily find that the maximum of the mean fidelity ( a.21 ) is achieved for @xmath165 and @xmath166 . in this case we rewrite ( a.21 ) as = [ _ 1 n + _ 2 + 2 _ 3 ] . [ a.22 ] when we substitute into eq . ( a.22 ) the explicit expression for the parameters @xmath167 given by eq . ( a.19 ) we find that the mean fidelity is equal to the function @xmath43 given by eq . ( [ 6 ] ) . this exactly is equal to the mean fidelity of the input - state disentanglement performed via the state swapping transformation described by eq . ( [ 18a ] ) . in fact , from our conditions @xmath166 it directly follows that @xmath168 while @xmath169 . in addition , from @xmath165 it follows that @xmath102 , so that the optimal state - dependent disentangling transformation is indeed equal to eq . ( [ 18a ] ) , which we wanted to prove .

it is not possible to disentangle a qubit in an _ unknown _ state @xmath0 from a set of @xmath1 ancilla qubits prepared in a specific reference state @xmath2 . that is , it is not possible to _ perfectly _ perform the transformation @xmath3 . the question is then how well we can do ? we consider a number of different methods of extracting an unknown state from an entangled state formed from that qubit and a set of ancilla qubits in an known state . measuring the whole system is , as expected , the least effective method . we present various quantum `` devices '' which disentangle the unknown qubit from the set of ancilla qubits . in particular , we present the _ optimal universal _ disentangler which disentangles the unknown qubit with the fidelity which does not depend on the state of the qubit , and a probabilistic disentangler which performs the perfect disentangling transformation , but with a probability less than one . * pacs number : 03.67.-a , 03.65.bz * 2

quant-ph0006047

in the past decade many neutrino oscillation results have been presented by different collaborations @xcite , and a phenomenological extension of the standard model has been proposed that involves three neutrino mass states , over which the three flavors of neutrinos are distributed . despite its successful predictions , this can be considered as an extension of the standard model that does not address fundamental questions , e.g. , small masses and large mixing angles compared to quark sector , and has raised a large debate over other possible unexpected properties of neutrinos that could lead to a more complete understanding of neutrino physics @xcite . to gain a deeper understanding of neutrino phenomenology the reduction of uncertainties in baseline neutrino oscillation experiments is mandatory . because of the interest in oscillation measurements , in recent years various neutrino - nucleus differential cross sections have been presented @xcite and are planned in the near future @xcite . differential cross sections are important to obtain a complete kinematical determination of neutrino - nucleus scattering and a clear understanding of neutrino - nucleus reactions is crucial for the analysis of experimental measurements . the argoneut collaboration has recently reported @xcite a measurement of the muon neutrino charged - current ( cc ) flux - averaged differential cross section on @xmath0ar in an energy range up to 50 gev . a liquid argon detector is very interesting because it has excellent potentialities to make precise measurements of a very large class of neutrino interactions from the mev energy scale to multi - gev events @xcite . the argoneut measurement has proven the validity of this experimental technique and , hoperfully , new data will be available in the future . for instance , a calculation of neutrino capture cross sections for solar neutrinos that could be addressed by this new generation of detectors is presented in @xcite . the energy region considered in the argoneut experiment , with average neutrino energy of @xmath1 gev , requires the use of a relativistic model , where not only relativistic kinematics is considered , but also nuclear dynamics and current operators are described within a relativistic framework . the first measurement of the charged - current quasielastic ( ccqe ) flux - averaged double - differential muon neutrino cross section on @xmath2c by the miniboone collaboration @xcite has raised extensive discussions . in particular , the experimental cross section is usually underestimated by the relativistic fermi gas model and by other more sophisticated models based on the impulse approximation @xcite , unless the nucleon axial mass @xmath3 is significantly enlarged with respect to the world average value of 1.03 gev/@xmath4 . it is reasonable to assume that the larger axial mass obtained from the miniboone data on @xmath2c can be interpreted as an effective way to include medium effects that are not taken into account by the models ; this is another indication that a precise knowledge of lepton - nucleus cross sections , where uncertainties on nuclear effects are reduced as much as possible , is necessary . moreover , any model aimed to describe neutrino - nucles scattering should first be tested against electron scattering data in the same kinematic region . at intermediate energy , quasielastic ( qe ) electron scattering calculations @xcite , which were able to successfully describe a wide number of experimental data , can provide a useful tool to study neutrino - induced processes . there are , however , indications that the reaction can have significant contributions from effects beyond the impulse approximation ( ia ) in some energy regions where the neutrino flux has still significant strength . for instance , in the models of @xcite the contribution of multinucleon excitations to ccqe scattering has been found sizable and able to bring the theory in agreement with the experimental miniboone cross sections without increasing the value of @xmath3 . the role of processes involving two - body currents compared to the ia models has been discussed in @xcite . a careful evaluation of all nuclear effects and of the relevance of multinucleon emission and of some non - nucleonic contributions @xcite would be , without a doubt , useful for a deeper understanding of the reaction dynamics . the relevance of final state interactions ( fsi ) has been clearly stated for the exclusive @xmath5 reaction , where the use of a complex optical potential ( op ) in the distorted - wave impulse approximation ( dwia ) is required @xcite . the imaginary part of the op produces an absorption that reduces the cross section and accounts for the loss of part of the incident flux in the elastically scattered beam to the inelastic channels which are open . in the inclusive scattering only the emitted lepton is detected , the final nuclear state is not determined and all elastic and inelastic channels contribute . thus , a different treatment of fsi is required , where all final - state channels are retained and the total flux , although redistributed among all possible channels , is conserved . different approaches have been used to describe fsi in relativistic calculations for the inclusive qe electron- and neutrino - nucleus scattering @xcite . in the relativistic plane - wave impulse approximation ( rpwia ) , fsi are simply neglected . in other approaches fsi are included in dwia calculations where the final nucleon state is evaluated with real potentials , either retaining only the real part of the relativistic energy - dependent complex optical potential ( rrop ) , or using the same relativistic mean field potential considered in describing the initial nucleon state ( rmf ) . although conserving the flux , the rrop is unsatisfactory from a theoretical point of view . on the contrary , the rmf , where the same strong energy - independent real potential is used for both bound and scattering states , fulfills the dispersion relations @xcite and also the continuity equation . in a different description of fsi relativistic green s function ( rgf ) techniques @xcite are used . in the rgf model , under suitable approximations , which are basically related to the ia , the components of the nuclear response are written in terms of the single particle optical model green s function ; its spectral representation , that is based on a biorthogonal expansion in terms of a non - hermitian optical potential @xmath6 and of its hermitian conjugate @xmath7 , can be exploited to avoid the explicit calculation of the single particle green s function and obtain the components of the hadron tensor @xcite . calculations require matrix elements of the same type as the dwia ones of the exclusive process in @xcite , but involve eigenfunctions of both @xmath6 and @xmath7 , where the imaginary part has an opposite sign and gives in one case a loss and in the other case a gain of strength . the rgf formalism allows us to reconstruct the flux lost into nonelastic channels in the case of the inclusive response starting from the complex optical potential which describes elastic nucleon - nucleus scattering data and to include contributions which are not included in the rmf and in other models based on the ia . moreover , with the use of the same complex optical potential , it provides a consistent treatment of fsi in the exclusive and in the inclusive scattering . in addition , because of the analyticity properties of the optical potential , it fulfills the coulomb sum rule @xcite . these different descriptions of fsi have been compared in @xcite for the inclusive qe electron scattering , in @xcite for the ccqe neutrino scattering , and in @xcite with the ccqe and nce miniboone data . both rmf and rgf are able to describe the shape of the ccqe experimental data , only the rgf gives cross sections of the same magnitude as the experimental ones without the need to increase the value of @xmath3 @xcite . similar results are obtained in @xcite , where the rgf results and their interpretation in comparison with the nce data from miniboone are discussed . in this paper the results of different relativistic descriptions of fsi for cc @xmath8-nucleus scattering are presented and discussed for the differential cross section averaged over the @xmath9 argoneut flux . we are aware of the interpretative questions that may be connected to the use of models developed for the qe regime in a kinematic situation , with the @xmath9 argoneut flux up to @xmath10 gev , where other processes beyond the ia , which are not included in the models considered here , can give significant contributions . nevertheless we believe that our calculations can give useful information about the role of nuclear effects in the analysis of @xmath11ar scattering and about the uncertainties which are related to their evaluation . for the reaction @xmath0ar@xmath12 as a function of the muon scattering angle @xmath13 . the data are from argoneut @xcite . [ csteta ] , title="fig : " ] -0.2 cm for the reaction @xmath0ar@xmath12 as a function of the muon momentum @xmath14 . the data are from argoneut @xcite . [ cspmu ] , title="fig : " ] -0.2 cm and @xmath15 on @xmath0ar calculated in rpwia . the data are from argoneut @xcite . [ csrpwia - area ] , title="fig : " ] + and @xmath15 on @xmath0ar calculated in rpwia . the data are from argoneut @xcite . [ csrpwia - area ] , title="fig : " ] -0.2 cm but in rmf . [ csrmf - area ] , title="fig : " ] + but in rmf . [ csrmf - area ] , title="fig : " ] -0.2 cm in all the calculations presented in this work the bound nucleon states are taken as self - consistent dirac - hartree solutions derived within a relativistic mean field approach using a lagrangian containing @xmath16 , @xmath17 , and @xmath18 mesons @xcite , with medium dependent parameterizations of the meson - nucleon vertices that can be more directly related to the underlying microscopic description of nuclear interactions @xcite . the same relativistic mean field approach has been used to calculate the bound state wave functions in @xcite , where the cross sections of the exclusive quasi - free @xmath5 and inclusive qe @xmath19 reactions have been presented and discussed for oxygen and calcium isotopic chains . in the rgf calculations we have used three parameterizations for the relativistic op of @xmath0ar : the energy - dependent and a - dependent edad1 ( where the @xmath20 represents the energy and the @xmath21 the atomic number ) op of @xcite and the more recent democratic ( dem ) and the undemocratic ( undem ) phenomenological ops of @xcite . we note that all these three parameterizations are global ones , since they are obtained through a fit to elastic proton - scattering data on a wide range of nuclei and , as such , they depend on the atomic number @xmath21 and are not constructed to reproduce the @xmath0ar phenomenology . in fig . [ csteta ] the cc differential cross section @xmath22 integrated over the argoneut flux is shown as a function of the muon scattering angle @xmath13 . all the calculations give results in reasonable agreement with the experimental shape but generally underpredict the magnitude of the experimental cross section . we note that in the rpwia fsi are completely neglected , while in the rmf the same strong energy - independent real potential is used for bound and scattering states . the comparison between the rgf results obtained with the edad1 , dem , and undem potentials can give an idea of how the predictions of the model are affected by uncertainties in the determination of the phenomenological op . the differences depend on the energy and momentum transfer and are essentially due to the different imaginary part of the three potentials , which accounts for the overall effects of inelastic channels and is not univocally determined from the elastic phenomenology . in contrast , the real term is similar for different parameterizations and gives similar results . we observe that the dem and undem potentials give in general close results ; in fact , even if they are obtained using very different fitting procedures @xcite , they are based on the same dataset of elastic - scattering data . in constrast , the edad1 potential produces somewhat smaller results and larger differences in comparison with the results of the dem and undem potentials : for instance , in the peak region with @xmath23 deg , the rgf - dem and rgf - edad1 cross sections differ by about @xmath24 . the rgf cross sections are generally larger than the rpwia and the rmf ones , but they are in general significantly lower than the data , but for @xmath25 deg and @xmath26 deg . in the rgf the imaginary part of the optical potential redistributes the flux in all the final - state channels and , in each channel , the flux lost towards other channels is compensated by the flux gained from the other channels . the larger cross sections in the rgf arise from the translation to the inclusive strength of the overall effects of inelastic channels which are not included in the other models such as , for instance , rescattering processes of the nucleon in its way out of the nucleus , non - nucleonic @xmath27 excitations which may arise during nucleon propagation , or also some multinucleon processes . these contributions are not included explicitly in the rgf , but they can be recovered , to some extent , by the imaginary part of the phenomenological op . we note that in all the calculations presented in this work we have used the standard value of the axial mass @xmath28 gev . a larger value of @xmath3 would increase the cross section and improve the agreement with data . in fig . [ cspmu ] the cc differential cross section @xmath29 integrated over the argoneut flux is shown as a function of the muon momentum @xmath30 . also in this case the rpwia and the rmf results are lower than the experimental data , while the rgf produces larger cross sections which are in better agreement with the data . anyway , the first two measurements in the low energy bins of @xmath30 are underestimated by all the results by a factor of @xmath31 . all the models which have been adopted for the present calculations are based on the ia , where the cross section is given by an incoherent sum of interactions between the incident neutrino and all the single nucleons of the nucleus . also the rgf , which is a more complex model and , with the use of the complex op , is able to recover contributions of channels which are not included in the other models , is essentially based on the ia . models based on ia have been successful in the description of qe exclusive and inclusive electron scattering @xcite . moreover , the rgf is able to give a reasonable description of ccqe and nce miniboone cross sections @xcite . in the inclusive cc neutrino - nucleus scattering at energies larger than a few gevs , however , all these models may neglect important contributions of reaction processes which can be included in the experimental cross sections . it is therefore not surprising that the calculated cross sections in figs . [ csteta ] and [ cspmu ] are generally lower than the experimental data . but in rgf - dem . [ csrgf - area - dem ] , title="fig : " ] + but in rgf - dem . [ csrgf - area - dem ] , title="fig : " ] -0.2 cm with the aim to give a more quantitative information we have tried to estimate the uncertainties of our calculations . the most obvious source of uncertainty is the neutrino flux : it is usually known with sufficient precision but its errors are not negligible @xcite . for energies up to @xmath32 gev the argoneut flux is given in bins of resolution of @xmath33 or @xmath31 gev and , for each bin , we compute an averagecross section starting from five or more calculations at different energies . it is straightforward to assume that this procedure introduces additional uncertainties in our results . in figs . [ csrpwia - area ] and [ csrmf - area ] we show our results for the @xmath22 and @xmath29 differential cross section in rpwia and in rmf . the bands in the figures represent these two errors added in quadrature . in the interval of @xmath13 and @xmath30 covered by the argoneut kinematics the two errors are generally small and neither the rpwia nor the rmf cross sections can reproduce the experimental data within the error bands . this is a reasonable result since the rpwia and rmf cross sections in each bin are stable and the uncertainties on the neutrino flux are generally small . in the case of the rgf we consider two additional sources of errors . the rgf results presented here contain the contribution of both terms of the hadron tensor in eq . ( 25 ) of @xcite . the calculation of the second term , which is entirely due to the imaginary part of the op , is a hard and time consuming numerical task which requires the integration over all the eigenfunctions of the continuum spectrum of the optical potential . numerical uncertainties on this term are anyhow under control and , from many calculations in different kinematics , have been estimated at most within @xmath34 . the fact that in actual rgf calculations we have to use a phenomenological energy dependent op introduces additional complications . the ops in @xcite have been tested for energies up to @xmath35 gev and to provide results up to @xmath36 gev we have to extend the range of validity of these parameterizations . this has been done multiplying each term of the op by a realistic function of the energy that has been chosen to carefully reproduce the behavior of the op around @xmath33 gev . we have checked that our results do not depend significantly on the multiplying function . the rgf - dem @xmath22 and @xmath29 differential cross sections are shown in fig . [ csrgf - area - dem ] , where the error bands represent all the uncertainties that we have considered added in quadrature . similar results with similar uncertainties are obtained in the case of rgf - undem and rgf - edad1 . the error bands for the rgf results are larger than for rpwia and rmf and , as a consequence , the upper limits of the rgf cross sections are closer to the data . this outcome can be ascribed to the moderately large uncertainties on the cross sections in each experimental bin of neutrino energy , in particular for neutrino energies of @xmath37 gev and small energy transferred to the nucleus . the large error bands in fig . [ csrgf - area - dem ] do not allow us to draw any definite conclusion . however , the results presented in this work indicate that the rgf , as well as the other models based on the ia , generally underpredict argoneut data , but they are able to reproduce the shape and the correct order of magnitude of the experimental cross section . in this paper we have compared the predictions of different relativistic descriptions of fsi for cc neutrino - nucleus scattering in the argoneut kinematics . in the rpwia fsi are neglected ; in the rmf they are described using the same relativistic mean field potential considered in describing the initial nucleon state ; in the rgf the full complex op , with its real and imaginary parts , is used to account for fsi . all final - state channels are included in the rgf , the flux lost in each channel is recovered in the other channels by the imaginary part of the op making use of the dispersion relations and the total flux is conserved . the rgf gives a good description of the @xmath19 data in the qe region and it is also able to describe both ccqe and nce miniboone data . in the rgf cross sections the contribution of reaction channels that are neglected in the other models , e.g. , rescattering processes of the nucleon in its way out of the nucleus , non - nucleonic @xmath27 excitations , which may arise during nucleon propagation , with or without real pion production , or also multinucleon processes , is translated , to some extent , into the inclusive strength by the imaginary part of the phenomenological op . however , the role of the various reaction channels included in the phenomenological op , can not be disentangled and the enhancement of the cross section can not be attributed to a specific process . in order to clarify this point , a careful evaluation of all nuclear effects and of the relevance of multinucleon emission and of some non - nucleonic contributions , as well as a better determination of the relativistic op , which closely fulfills the dispersion relations , would reduce the theoretical uncertainties on the rgf . our results give a clear indication that ia - based models are able to reproduce the correct order of magnitude and the shape of the argoneut data but they generally underpredict the experimental cross sections , in particular for lower values of @xmath30 and for values of @xmath13 between @xmath38 deg and @xmath39 deg . a careful evaluation of all nuclear effects is required to recover some important contributions to the cc inclusive strenght . in particular , a careful study of medium effects in the few - gev energy region that takes into account quasielastic , inelastic , as well as deep - inelastic processes , is highly desirable . 89ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1''''@noop [ 0]secondoftwosanitize@url [ 0 ] + 12$12 & 12#1212_12%12@startlink[1]@endlink[0]@bib@innerbibempty link:\doibase 10.1103/physrevd.83.052010 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.107.241801 [ * * , ( ) ] link:\doibase 10.1140/epjc / s10052 - 013 - 2345 - 6 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop * * , ( ) link:\doibase 10.1103/physrevlett.110.161801 [ * * , ( ) ] link:\doibase 10.1103/physrevd.85.032007 [ * * , ( ) ] link:\doibase 10.1103/physrevd.86.052009 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.131801 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.171803 [ * * , ( ) ] link:\doibase 10.1088/1674 - 1137/37/1/011001 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.191802 [ * * , ( ) ] link:\doibase 10.1103/physrevd.74.072003 [ * * , ( ) ] link:\doibase 10.1103/physrevd.64.112007 [ * * , ( ) ] link:\doibase 10.1155/2013/163897 [ * * , ( ) ] link:\doibase 10.1155/2013/852987 [ * * , ( ) ] link:\doibase 10.1103/physrevd.85.083522 [ * * , ( ) ] link:\doibase 10.1103/physrevd.83.073006 [ * * , ( ) ] http://stacks.iop.org/1475-7516/2011/i=09/a=034 [ * * , ( ) ] link:\doibase 10.1103/physrevd.70.073004 [ * * , ( ) ] @noop * * , ( ) @noop * * , ( ) @noop ( ) , link:\doibase 10.1103/physrevd.83.012005 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.161802 [ * * , ( ) ] miniboone , @noop miner@xmath8a , @noop t2k , @noop @noop ( ) , link:\doibase 10.1103/physrevc.87.014607 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.105.132301 [ * * , ( ) ] link:\doibase 10.1016/j.physletb.2011.07.032 [ * * , ( ) ] link:\doibase 10.1103/physrevc.82.055501 [ * * , ( ) ] link:\doibase 10.1103/physrevc.84.015501 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1016/0370 - 1573(93)90132-w [ * * , ( ) ] @noop _ _ , , vol . ( , , ) link:\doibase 10.1103/physrevc.80.065501 [ * * , ( ) ] link:\doibase 10.1103/physrevc.81.045502 [ * * , ( ) ] link:\doibase 10.1103/physrevc.84.055502 [ * * , ( ) ] link:\doibase 10.1103/physrevc.83.045501 [ * * , ( ) ] link:\doibase 10.1016/j.physletb.2011.11.061 [ * * , ( ) ] link:\doibase 10.1016/j.physletb.2013.03.002 [ * * , ( ) ] link:\doibase 10.1016/j.physletb.2010.12.007 [ * * , ( ) ] link:\doibase 10.1103/physrevd.84.033004 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.108.152501 [ * * , ( ) ] link:\doibase 10.1140/epjc / s10052 - 011 - 1726-y [ * * , ( ) ] @noop ( ) , link:\doibase 10.1103/physrevc.79.034601 [ * * , ( ) ] link:\doibase 10.1103/physrevc.81.064614 [ * * , ( ) ] link:\doibase 10.1103/physrevc.83.054616 [ * * , ( ) ] link:\doibase 10.1016/j.physletb.2011.02.043 [ * * , ( ) ] link:\doibase 10.1103/physrevc.48.2731 [ * * , ( ) ] link:\doibase 10.1103/physrevc.64.014604 [ * * , ( ) ] link:\doibase 10.1103/physrevc.64.064615 [ * * , ( ) ] link:\doibase 10.1103/physrevc.65.044601 [ * * , ( ) ] link:\doibase 10.1140/epja / i2002 - 10137 - 2 [ * * , ( ) ] link:\doibase 10.1103/physrevc.84.024615 [ * * , ( ) ] link:\doibase 10.1103/physrevc.68.048501 [ * * , ( ) ] link:\doibase 10.1103/physrevc.74.015502 [ * * , ( ) ] link:\doibase 10.1016/j.physletb.2010.03.078 [ * * , ( ) ] link:\doibase 10.1016/j.nuclphysa.2004.04.108 [ * * , ( ) ] link:\doibase 10.1103/physrevc.67.054601 [ * * , ( ) ] link:\doibase 10.1016/j.nuclphysa.2004.08.023 [ * * , ( ) ] http://th-www.if.uj.edu.pl/acta/vol37/pdf/v37p2279.pdf [ * * , ( ) ] link:\doibase 10.1016/j.nuclphysa.2006.05.005 [ * * , ( ) ] link:\doibase 10.1103/physrevc.77.034606 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1103/physrevc.80.024605 [ * * , ( ) ] link:\doibase 10.1103/physrevc.83.064614 [ * * , ( ) ] link:\doibase 10.1051/epjconf/20123814004 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1016/0375 - 9474(91)90269-c [ * * , ( ) ] link:\doibase 10.1016/j.aop.2004.12.005 [ * * , ( ) ] link:\doibase 10.1016/j.nuclphysa.2005.04.007 [ * * , ( ) ] @noop * * , ( ) @noop ( ) , link:\doibase 10.1103/physrevlett.107.172501 [ * * , ( ) ] link:\doibase 10.1103/physrevd.85.093002 [ * * , ( ) ] link:\doibase 10.1103/physrevd.84.113003 [ * * , ( ) ] @noop * * , ( ) link:\doibase 10.1016/j.nuclphysa.2004.02.001 [ * * , ( ) ] `` , '' in link:\doibase 10.1142/9789812701985_0005 [ _ _ ] , ( , , ) chap . , link:\doibase 10.1016/j.nuclphysa.2006.02.007 [ * * , ( ) ] link:\doibase 10.1103/physrevc.85.024322 [ * * , ( ) ] link:\doibase 10.1088/1742 - 6596/366/1/012019 [ * * , ( ) ] link:\doibase 10.1103/physrevc.47.297 [ * * , ( ) ] link:\doibase 10.1103/physrevc.80.034605 [ * * , ( ) ]

the analysis of the recent charged - current neutrino - nucleus scattering cross sections measured by the argoneut collaboration requires relativistic theoretical descriptions also accounting for the role of final - state interactions . in this work , we evaluate differential neutrino - nucleus cross sections with the relativistic green s function model , where final - state interactions are described in the inclusive scattering consistently with the exclusive scattering using a complex optical potential . the sensitivity to the parameterization adopted for the phenomenological optical potential is discussed . the predictions of the relativistic green s function model are compared with the results of different descriptions of final - state interactions .

1305.5466

the short - lived radioisotope ( slri ) @xmath4al was alive during the formation of the first refractory solids in the solar nebula , the ca- , al - rich inclusions ( cais ) found in primitive chondritic meteorites . this means that at least some of the solar system s slris may have been injected into either the presolar cloud ( e.g. , boss & keiser 2012 ; boss 2012 ) or the solar nebula ( ouellette et al . 2007 , 2010 ; dauphas & chaussidon 2011 ) by a supernova or agb star shock wave . in either case , injection occurred as a single event that was spatially heterogeneous , which would potentially reduce the usefulness of @xmath4al as a spatially homogeneous chronometer ( dauphas & chaussidon 2011 ) for precise studies of the earliest phases of planet formation ( macpherson et al . 2012 ; cf . krot et al . previous models ( e.g. , boss 2011 , 2012 ) have shown how such initial spatial isotopic heterogeneity can be substantially reduced in a marginally gravitationally unstable ( mgu ) disk , as a result of the large - scale inward and outward transport and mixing of gas and particles small enough to move with the gas ( e.g. , boss et al . 2012 ) . other elements and their isotopes suggest a similarly well - mixed solar nebula ( e.g. , os : walker 2012 ; fe : wang et al . 2013 ) . the stable oxygen isotopes , on the other hand , appear to have been spatially heterogeneous in the solar nebula during the early phases of planet formation ; e.g. , small refractory particles from comet 81p / wild 2 have normalized @xmath5o/@xmath6o ratios that span the entire solar system range of @xmath0 6% variations ( nakashima et al . 2012 ) . the leading explanation for generating these oxygen anomalies is uv photodissociation of co molecules at the surface of the outer solar nebula ( e.g. , podio et al . 2013 ) , where self - shielding could lead to isotopic fractionation between gas - phase and solid - phase oxygen atoms ( e.g. , lyons & young 2005 ; krot et al . co self - shielding on the irregular , corrugated outer surface of the disk would also lead to initial spatial heterogeneity , though the process would be continuous in time , rather than a single - shot event like a supernova shock wave . furthermore , the very existence of refractory particles in comet 81p / wild 2 ( brownlee et al . 2006 ; simon et al . 2008 ; nakamura et al . 2008 ) , which are thought to have formed close to the protosun , implies that these small particles experienced large - scale outward transport from the inner solar nebula to the comet - forming regions of the outer solar nebula . mgu disks offer a means to accomplish this early large - scale transport ( e.g. , boss 2008 , 2011 ; boss et al . 2012 ) . marginally gravitationally unstable disks are likely to be involved in the fu orionis outbursts experienced by young solar - type stars ( e.g. , zhu et al . 2010b ; vorobyov & basu 2010 ; martin et al . mgu disk models ( e.g. , boss 2011 ) can easily lead to the high mass accretion rates ( @xmath7 yr@xmath8 ) needed to explain fu orionis events . fu orionis outbursts are believed to last for about a hundred years and to occur periodically for all low mass protostars ( hartmann & kenyon 1996 ; miller et al . mgu models are also capable of offering an alternative mechanism ( disk instability ) for gas giant planet formation ( e.g. , boss 2010 ; meru & bate 2012 ; basu & vorobyov 2012 ) . however , the magnetorotational instability ( mri ) is likely to be involved in fu orionis outbursts as well ( zhu et al . 2009a ) , with mri operating in the ionized innermost disk layers as well as at the disk s surfaces . zhu et al . ( 2009c , 2010a , b ) have constructed one- and two - dimensional ( axisymmetric ) models of a coupled mgu - mri mechanism , with mgu slowly leading to a build - up of mass in the innermost disk , which then triggers a rapid mri instability and an outburst . alternatively , mri may operate in the outermost disk , partially ionized by cosmic rays , leading to a build - up of mass in the dead zone at the intermediate disk midplane , thus triggering a phase of mgu transport . such a coupled mechanism may be crucial for achieving outbursts in t tauri disks , where the disk masses are expected to be smaller than at earlier phases of evolution . we present here several new sets of mgu disk models that examine the time evolution of isotopic heterogeneity introduced in either the inner or outer solar nebula , by either a single - shot event or a continuous injection process , for a variety of disk and central protostar masses , including protostars with m dwarf masses . low mass exoplanets are beginning to be discovered around an increasingly larger fraction of m dwarfs ( bonfils et al . 2013 ; dressing & charbonneau 2013 ; kopparapu 2013 ) , with a number of these being potentially habitable exoplanets , elevating the importance of understanding mixing and transport processes in m dwarf disks . the numerical models were computed with the same three dimensional , gravitational hydrodynamics code that has been employed in previous mgu disk models ( e.g. , boss 2011 ) . complete details about the code and its testing may be found in boss & myhill ( 1992 ) . briefly , the code performs second - order - accurate ( in both space and time ) hydrodynamics on a spherical coordinate grid , including radiative transfer in the diffusion approximation . a spherical harmonic ( @xmath9 ) expansion of the disk s density distribution is used to compute the self - gravity of the disk , with terms up to and including @xmath10 . the radial grid contains 50 grid points for the 10 au disk models and 100 grid points for the 40 au disk models . all models have 256 azimuthal grid points , and effectively 45 theta grid points , given the hemispherical symmetry of the grid . the theta grid is compressed around the disk s midplane to provide enhanced spatial resolution , while the azimuthal grid is uniformly spaced . the jeans length constraint is used to ensure adequate resolution . the inner boundary absorbs infalling disk gas , which is added to the central protostar , while the outer disk boundary absorbs the momentum of outward - moving disk gas , while retaining the gas on the active grid . the central protostar wobbles in such a manner as to preserve the center of mass of the entire system . the time evolution of a color field is calculated ( e.g. , boss 2011 ) in order to follow the mixing and transport of isotopes carried by the disk gas or by small particles , which should move along with the disk gas . the equation for the evolution of the color field density @xmath11 ( e.g. , boss 2011 ) is identical to the continuity equation for the disk gas density @xmath12 @xmath13 where @xmath14 is the disk gas velocity and @xmath15 is the time . the total amount of color is conserved in the same way that the disk mass is conserved , as the hydrodynamic equations are solved in conservation law form ( e.g. , boss & myhill 1992 ) . the initial disk density distributions are based on the approximation derived by boss ( 1993 ) for a self - gravitating disk orbiting a star with mass @xmath16 @xmath17 @xmath18,\ ] ] where @xmath19 and @xmath20 are cylindrical coordinates , @xmath21 is the gravitational constant , @xmath22 is the midplane density , @xmath23 is the surface density , @xmath24 ( cgs units ) and @xmath25 . the initial midplane density is @xmath26 while the initial surface density is @xmath27 the parameters @xmath28 and @xmath29 and the reference radius @xmath30 are defined in table 1 for the various disk models explored in this paper . the total amount of mass in the models does not change during the evolutions ; the initial infalling disk envelope accretes onto the disk , and no further mass is added to the system across the outer disk boundary at @xmath31 . the outer disk surfaces are thus revealed to any potential source of uv irradiation . for the 10 au outer radius disks listed in tables 2 and 3 , the initial disk temperature profiles ( figures 1 and 2 ) are based on the boss ( 1996 ) temperature profiles , with variations in the assumed outer disk temperature @xmath32 , chosen in order to study the effect of varied minimum values of the @xmath2 stability parameter . values of @xmath33 indicate marginally gravitationally unstable disks . the inner disks are all highly @xmath2 stable , with @xmath34 . for the 40 au outer radius disks listed in table 4 , the initial disk temperatures are uniform at the specified outer disk temperature @xmath32 , leading to similar initial @xmath2 values throughout the disks . for all of the models , the temperature of the infalling envelope is 50 k. the initial color field is added to the surface of the initial disk in an azimuthal sector spanning either 45 degrees ( 10 au outer radius disks ) or 90 degrees ( 40 au outer radius disks ) in a narrow ring of width 1 au , centered at the injection radii listed in the tables . these models are intended to represent one - time , single - shot injections of isotopic heterogeneity , such as supernova - induced rayleigh - taylor fingers carrying live @xmath4al ( e.g. , boss & keiser 2012 ) . table 4 lists both single - shot and continuous injection models , where in the latter case the color is added continuously to the same location on the disk surface throughout the evolution , crudely simulating ongoing photodissociation of co ( e.g. , lyons & young 2005 ) possibly leading to stable oxygen isotope fractionation between the gas and solid phases . the color field in the latter case is intended to represent isotopically distinct gas or small particles resulting from the uv photochemistry . note that in both the single - shot and continuous injection models , the total amount of color added is arbitrary ( e.g. , the color field in the injection volume is simply set equal to 1 ) , and is intended to be scaled to whatever value is appropriate for the isotope(s ) under consideration . the color field is a massless , passive tracer that has no effect on the disk s dynamics , so the total amount of color added is irrelevant for the disk s subsequent evolution . the models seek to follow the deviations from uniformity of the color field , not the absolute amounts of color added ; the evolution of the dispersion of the color field about its mean radial value , divided by the mean radial value at each instant of time , is the goal of these models . observations of the dg tau disk by podio et al . ( 2013 ) have shown that dg tau itself irradiates its disk s outer layers from 10 au to 90 au with a strong uv flux , sufficient for significant uv photolysis and the formation of observable water vapor . much higher levels of uv irradiation can occur for protoplanetary disks that form in stellar clusters containing massive stars ( e.g. , walsh et al . 2013 ) , an environment that has been suggested for our own solar system ( e.g. , dauphas & chaussidon 2011 ) in order to explain the evidence for live slris found in primitive meteorites . the fact that molecular hydrogen constitutes the great majority of a disk s mass , yet can not be directly detected , except at the star - disk boundary region , means that estimates of disk masses are uncertain at best ( e.g. , andrews & williams 2007 ) , as they are typically based on an assumed ratio between the amount of mm - sized dust grains and the total disk mass . isella et al . ( 2009 ) estimated that low- and intermediate - mass pre - main - sequence stars form with disk masses ranging from 0.05 to 0.4 @xmath1 . dg tau s disk has a mass estimated to be as high as 0.1 @xmath1 ( podio et al . recently , the mass of the tw hydra disk was revised upward to at least 0.05 @xmath1 ( bergin et al . 2013 ) . these and other observations suggest that the mgu disk masses assumed in these models may be achieved in some fraction of protoplanetary disks , and perhaps in the solar nebula as well . in fact , miller et al . ( 2011 ) detected a fu orionis outburst in the classical t tauri star lkh@xmath35 188-g4 . disk masses are typically thought to be @xmath36 for such stars . the fact that a fu orionis event occurred in lkh@xmath35 188-g4 shows that even the disks around class ii - type objects can experience instabilities leading to rapid mass accretion , e.g. , mgu disk phases . we present results for a variety of protostellar and protoplanetary disk masses , varied initial minimum @xmath2 stability parameters , and varied injection radii , for disks of two different sizes . table 2 shows the initial conditions for the models with a @xmath37 disk in orbit around a @xmath38 protostar . the disks extend from 1 au to 10 au , as in the models by boss ( 2008 , 2011 ) . the main difference from these previous models is that the disk mass ( @xmath37 ) is considerably lower than that of the previous models ( @xmath39 ) . as a result , the initial minimum @xmath2 values are considerably higher than in the previous models , ranging from 2.2 to 3.1 , compared to the previous range of 1.4 to 2.5 . the present models are thus less gravitationally unstable initially than the disks previously considered , with the goal being to learn whether or not the previous results will change for higher values of @xmath40 . figure 1 displays the initial midplane temperature profiles for these models . only the outermost regions of the disks are cool enough to be gravitationally unstable , but the models show that this is sufficient to result in qualitatively similar behavior for all of the table 2 models . figures 3 and 4 show the equatorial plane distribution of the color / gas ratio ( @xmath41 ) for model 1.0 - 2.6 - 9 . this ratio is plotted , as it is equivalent to the @xmath4al/@xmath42al and @xmath5o/@xmath6o ratios measured by cosmochemists , i.e. , the abundance of an injected or photolysis product species , divided by that of a species that was prevalent in the pre - injection disk . figure 3 shows that the initial disk surface injection at 9 au has resulted in the rapid transport of the color field downward to the disk s midplane , as well as inward to close to the inner disk boundary at 2 au . the vigorous three dimensional motions of a mgu disk are responsible for this large - scale transport in just 34 yr . at this time , the color / gas ratio is still highly heterogeneous , but figure 4 shows that only 146 yr later , the color / gas ratio has been strongly homogenized throughout the entire disk midplane . figure 5 shows the evolution of the dispersion of the ratio of the color density to the gas density for models 1.0 - 2.6 - 9 and 1.0 - 2.6 - 2 at two times . these models differ only in the injection radius , either 9 au or 2 au . the dispersion plotted in figure 5 is defined to be the square root of the sum of the squares of the color field divided by the gas density , subtracted by the azimuthal average of this ratio at a given orbital radius , divided by the square of the azimuthal average at that radius , normalized by the number of azimuthal grid points , and plotted as a function of radius in the disk midplane . figure 5 shows that the isotopic dispersion is a strong function of orbital radius and time , with the dispersion initially being relatively large ( i.e. , at 180 yr , in spite of the apparent homogeneity seen in figure 4 at the same time ) as a result of the isotopes traveling downward and radially inward and outward . however , the dispersion decreases dramatically in the outer disks for both models by 777 yr to a value of @xmath0 1% to 2% . in fact , the dispersion in both models 1.0 - 2.6 - 9 and 1.0 - 2.6 - 2 evolves toward essentially the same radial distribution by this time , showing that the exact location of the injection location has little effect on the long term evolution of the distribution : that is controlled solely by the evolution of the underlying mgu disk , which is identical for these two models ( i.e. , the color fields are passive tracers , and have no effect on the disk s evolution ) . note that any small refractory grains present in the initial disk will be carried along with the disk gas , so that some of the grains that start out at 2 au will be transported to the outermost disk , in the same manner that some of the gas is transported outward . most of the gas and dust , however , is accreted by the growing protostar . figure 6 shows the results for three models with varied @xmath40 , i.e. , models 1.0 - 2.6 - 9 , 1.0 - 2.9 - 9 , and 1.0 - 3.1 - 9 , all after 1370 yr . it can be seen that in spite of the variation in the initial degree of instability , the dispersions in the outermost disks all converge to similar values of @xmath0 1% to 2% . this suggests that mgu disk evolutions are not particularly sensitive to the exact choice of the initial @xmath2 profile , a result that was also found by boss ( 2011 ) for somewhat more massive disks . as also found by boss ( 2011 ) , the dispersions in the innermost disks are significantly higher ( @xmath0 10% to 20% ) than in the outermost disks , a direct result of the stronger mixing associated with the cooler outer disks , in spite of the longer orbital periods in the outer disks . table 3 shows the initial conditions for the models with either @xmath43 disks around @xmath44 protostars , or @xmath45 disks around @xmath46 protostars . in either case , the disks extend from 1 au to 10 au . these models are of interest for exploring how conditions might vary between disks around g dwarfs and m dwarfs , with possible ramifications for the habitability of any rocky planets that form ( e.g. , raymond et al . 2007 ) around m dwarfs . figure 2 shows the initial midplane temperature profiles for these models . figure 7 shows the time evolution of the dispersion for model 0.1 - 1.8 - 2 , appropriate for a late m dwarf protostar . as in all the models , it can be seen that the initially highly heterogeneous disk becomes rapidly homogenized , in this case by about 5000 yr . note that this time scale is considerably longer than that for g dwarf disk mixing and transport processes , as a result of the longer keplerian orbital periods for lower mass , m dwarf protostars . as in the g dwarf protostar models ( e.g. , figure 5 ) , the inner disk dispersion is higher than in the outer disk , though in these models ( with a lower initial @xmath47 ) the inner disk dispersion drops to @xmath0 5% to 10% , compared to @xmath0 1% to 2% in the outer disk . figure 8 shows the same behavior for model 0.1 - 1.8 - 9 , which differs from the previous model shown in figure 7 only in having the injection occur at 9 au instead of 2 au . as in the g dwarf disks , the dispersions for both of these models evolve toward essentially identical radial distributions : the underlying mgu disk evolution determines the outcome for the dispersions . similar results hold for the models with @xmath46 protostars , i.e. , early m dwarf disks . we now turn to a consideration of the consequence of single - shot versus continuous injection at the surface of much larger outer radius ( 40 au ) disks than have been considered to date for g dwarf stars ; boss ( 2007 ) considered disks extending from 4 au to 20 au in radius . table 4 shows the initial conditions for the models with a @xmath48 disk around a @xmath38 protostar , with the disks extending from 10 au to 40 au . because of the much larger inner and outer disk radii for this set of models , these models can be calculated for times as long as @xmath49 yr ( table 4 ) . such times are still considerably less than the typical ages ( @xmath50 yr ) of t tauri stars , implying that in order for mgu disks to occur at such late phases , a prior phase of coupled mri - mgu evolution might be required to make the present results relevant . figures 9 and 10 display the evolution of the dispersions for models 1.0 - 1.1 - 40 - 20 and 1.0 - 1.1 - 40 - 20c , differing only in that the former model has single - shot injection while the latter model has continuous injection , intended to simulate a disk with ongoing uv photolysis and fractionation at the outer disk surface . for model 1.0 - 1.1 - 40 - 20 , it can be seen that the evolution is similar to that of the previous single - shot models : a rapid drop in the dispersion , followed by homogenization to @xmath0 1% to 2% away from the inner disk boundary . the higher dispersions seen near the outer disk boundaries ( @xmath0 40 au ) are largely caused by the unphysical pile - up of considerable disk mass at 40 au and should be discounted . however , for the continuous injection model shown in figure 10 , it can be seen that the dispersions throughout the disk even after @xmath3 yr can be as high as @xmath0 20% , consistent with the much larger variation in stable oxygen isotope ratios , compared to slri ratios . in a calculation with finer spatial grid resolution , as well as perhaps sub - grid mixing processes , one might expect even stronger homogenization to occur , so the dispersion levels obtained from the present models should be considered to be upper bounds . the total amount of color added during the continuous injection models is large , compared to single - shot injection models : for model 1.0 - 1.4 - 40 - 20c , for example , after 200 yr , the total amount of color injected has increased by a factor of @xmath0 90 compared to the single - shot total , and by another factor of @xmath0 60 after 27000 yr . figures 11 and 12 compare the results for continuous injection at either 20 au or 30 au , respectively , i.e. , for models 1.0 - 1.1 - 40 - 20c and 1.0 - 1.1 - 40 - 30c . in spite of the different injection radii , figures 11 and 12 show that even at a relatively early phase ( 405 yr ) of evolution , the midplane color / gas ratios look somewhat similar ; as before , the mgu disk evolution is the same for both models , and that is the primary determinant of the long term evolution . finally , similar results as those shown in figures 9 - 12 were obtained for the other models listed in table 4 . these models show that the main factor in determining the radial dispersion profile is whether the injection occurs in a single - shot or continuously ; in the latter case , the mgu disk does its best to homogenize the color field , but the fact that spatial heterogeneity is being continuously injected limits the degree to which this heterogeneity can be reduced . while dust grains in the interstellar medium are overwhelmingly amorphous , crystalline silicate grains have been found in a late m dwarf ( sst - lup3 - 1 ) disk at distances ranging from inside 3 au to beyond 5 au , in both the midplane and surface layers ( mern et al . such crystalline silicate grains are likely to have been produced by thermal annealing in the hottest regions of the disk , well inside of 1 au ( sargent et al . 2009 ) . again , outward transport seems to be required to explain the observations , and the results for the models with a 0.1 @xmath1 protostar suggest that mgu phases in low mass m dwarf disks may be responsible for these observations . in fact , crystalline mass fractions in protoplanetary disks do not appear to correlate with stellar mass , luminosity , accretion rate , disk mass , or the disk to star ratio ( watson et al . these results also appear to be consistent with the results of the present models , which show that mgu disk phases are equally capable of relatively rapid large - scale mixing and transport , regardless of the stellar or disk mass , or the exact value of the @xmath2 stability parameter . these models have shown a rather robust result , namely that a phase of marginal gravitational instability in disks and stars with a variety of masses and disk temperatures can lead to relatively rapid inward and outward transport of disk gas and small grains , as required to drive the protostellar mass accretion associated with fu orionis events , as well as to explain the discovery of refractory grains in comet 81p / wild 2 . a mgu disk phase driving a fu orionis outburst is astronomically quite likely to have occurred for our protosun , and cosmochemically convenient for explaining the relative homogeneity of @xmath4al/@xmath42al ratios derived from a supernova injection event , and the range of @xmath5o/@xmath6o ratios derived from sustained uv self - shielding at the surface of the outer solar nebula . low - mass stars , from g dwarfs to m dwarfs , may well experience a similar phase of mgu disk mixing and transport . in this context , it is worthwhile to note that fu orionis itself , the prototype of the fu orionis outburst phenomenon , has a mass of only @xmath51 ( zhu et al . 2007 , 2009b ; beck & aspin 2012 ) , i.e. , the mass of a m dwarf , suggesting that m dwarf protoplanetary disks may experience evolutions similar to that of the solar nebula , with possible implications for the habitability of any resulting planetary system ( e.g. , raymond et al . 2007 ; bonfils et al . 2013 ; dressing & charbonneau 2013 ; kopparapu 2013 ) . i thank jeff cuzzi for his comments , the referee for a number of suggested improvements , and sandy keiser , michael acierno , and ben pandit for their support of the cluster computing environment at dtm . this work was partially supported by the nasa origins of solar systems program ( nnx09af62 g ) and is contributed in part to the nasa astrobiology institute ( nna09da81a ) . some of the calculations were performed on the carnegie alpha cluster , the purchase of which was partially supported by a nsf major research instrumentation grant ( mri-9976645 ) . 1.0 & 0.019 & @xmath52 & @xmath53 & 1.0 & 1.0 & 10.0 + 0.5 & 0.018 & @xmath54 & @xmath55 & 1.0 & 1.0 & 10.0 + 0.1 & 0.016 & @xmath56 & @xmath57 & 1.0 & 1.0 & 10.0 + 1.0 & 0.13 & @xmath58 & @xmath59 & 4.0 & 10.0 & 40.0 + 1.0 - 2.2 - 2 & 15 & 2.2 & 2 & 2520 + 1.0 - 2.2 - 9 & 15 & 2.2 & 9 & 2520 + 1.0 - 2.6 - 2 & 20 & 2.6 & 2 & 2043 + 1.0 - 2.6 - 9 & 20 & 2.6 & 9 & 2043 + 1.0 - 2.9 - 2 & 25 & 2.9 & 2 & 1200 + 1.0 - 2.9 - 9 & 25 & 2.9 & 9 & 1400 + 1.0 - 3.1 - 2 & 30 & 3.1 & 2 & 1100 + 1.0 - 3.1 - 9 & 30 & 3.1 & 9 & 1300 + 1.0 - 1.1 - 40 - 20 & 30 & 1.1 & 20 & single - shot & 25000 + 1.0 - 1.1 - 40 - 30 & 30 & 1.1 & 30 & single - shot & 24500 + 1.0 - 1.4 - 40 - 20 & 50 & 1.4 & 20 & single - shot & 24000 + 1.0 - 1.4 - 40 - 30 & 50 & 1.4 & 30 & single - shot & 15000 + 1.0 - 1.1 - 40 - 20c & 30 & 1.1 & 20 & continuous & 19500 + 1.0 - 1.1 - 40 - 30c & 30 & 1.1 & 30 & continuous & 19800 + 1.0 - 1.4 - 40 - 20c & 50 & 1.4 & 20 & continuous & 27000 + 1.0 - 1.4 - 40 - 30c & 50 & 1.4 & 30 & continuous & 27000 +

analyses of primitive meteorites and cometary samples have shown that the solar nebula must have experienced a phase of large - scale outward transport of small refractory grains as well as homogenization of initially spatially heterogeneous short - lived isotopes . the stable oxygen isotopes , however , were able to remain spatially heterogenous at the @xmath0 6% level . one promising mechanism for achieving these disparate goals is the mixing and transport associated with a marginally gravitationally unstable ( mgu ) disk , a likely cause of fu orionis events in young low - mass stars . several new sets of mgu models are presented that explore mixing and transport in disks with varied masses ( 0.016 to 0.13 @xmath1 ) around stars with varied masses ( 0.1 to 1 @xmath1 ) and varied initial @xmath2 stability minima ( 1.8 to 3.1 ) . the results show that mgu disks are able to rapidly ( within @xmath3 yr ) achieve large - scale transport and homogenization of initially spatially heterogeneous distributions of disk grains or gas . in addition , the models show that while single - shot injection heterogeneity is reduced to a relatively low level ( @xmath0 1% ) , as required for early solar system chronometry , continuous injection of the sort associated with the generation of stable oxygen isotope fractionations by uv photolysis leads to a sustained , relatively high level ( @xmath0 10% ) of heterogeneity , in agreement with the oxygen isotope data . these models support the suggestion that the protosun may have experienced at least one fu orionis - like outburst , which produced several of the signatures left behind in primitive chondrites and comets .

1306.2220

a recent idea on possible application of topological superconductors to quantum information processing has attracted both theoretical and experimental interest @xcite . according to this idea , a quantum information unit , qubit , can be formed and propagated by means of majorana mode ( see , for e.g. , ref . ) , localized at the end of a one - dimensional ( 1d ) chain hosting a topological superconductor @xcite . recent investigations suggest several detection mechanisms of such a majorana mode @xcite such as an existence of a central peak in the tunneling current through a topological superconductor ( s)- normal metal ( n)junction and fractional period of the josephson current in s - n - s junctions . most recently experimental systems involving 1d semiconductor wire has been shown to host such modes ; the mechanism of the appearance of such modes arise from the combination of strong soi , proximity - induced superconducting gap , chemical potential and applied zeeman field in these wires @xcite . two majorana modes in the josephson junction , formed between two topological insulator edges or one - dimensional superconducting nanowires separated by barrier , hybridize resulting in splitting of the zero energy modes . this splitting energy depends not only on the phase difference of the two superconductors but also on the relative direction of the spin polarization at the two side of the junction . the oscillations of a josephson current between two such superconductors separated by insulator or metal as a function of their phase difference , with @xmath9 periodicity instead of a conventional @xmath10 periodicity due to hybridization of majorana states was predicted by kitaev @xcite for a idealized model of an 1d spinless p - wave superconductor . following this , kwon _ et al_. @xcite proposed that the similar effect can be observed between quasi-1d or 2d unconventional superconducting tunnel barrier junctions where the superconductors are separated by an insulating region , usually modeled by a delta function potential barrier . these systems did not have soi or zeeman field ; majorana - like modes appeared in such systems from the unconventional nature of the pairing potential . further it was realized in ref . that a signature of the fractional josephson effect constitutes in having a halved josephson frequency , @xmath11 , in the presence of a dc voltage @xmath12 applied across the junction . these effects have been interpreted in terms of the josephson current being carried by electrons rather than cooper pairs @xcite . further , it was shown that a fractional josephson effect may be realized at topological insulator edge @xcite . this prediction has later been extended to different systems @xcite . recent activities have established that a topological insulator with proximity - induced coupling to a s - wave superconductor exhibits a superconductivity - magnetism duality @xcite , revealing the fractional periodicity not only with superconducting phase difference but also with the orientation of zeeman magnetic field . in this case , the magnetic field on one side of the junction rotates in the plane normal to the direction of an effective magnetic field of the soi ; consequently , the majorana - mediated josephson current reverses sign after @xmath13 rotation of the magnetic field orientation and reveals an unconventional @xmath9 periodic magneto - josephson oscillation in response to variation of the magnetic field orientation in a topological insulator edge @xcite . furthermore , a dissipationless fractional josephson effect mediated by with @xmath14 periodicity has been also predicted @xcite at the edge of a quantum spin hall insulator . the josephson effect in consisting of topological superconducting ( s ) and normal ( n ) regions , has been reported in @xcite . these works also reveal a signature of majorana bound states located at s - n edges , producing a fractional josephson current with @xmath9 periodicity @xcite . these previous works in the field have pointed out the importance of the fractional josephson and the magneto - josephson effect in 1d superconducting junctions or atop the surface of a topological insulator . however , the role of spin - orbit coupling and the external magnetic field behind these effects has not been systematically investigated in these earlier works . such a systematic study is the main aim of the present work . to this end , we study the josephson effect between two 1d nanowires oriented along @xmath0 with proximity induced @xmath1-wave superconducting pairing and separated by a narrow dielectric with a rashba spin - orbit interaction ( soi ) of strength @xmath2 and zeeman fields ( @xmath3 along @xmath4 and @xmath5 in the @xmath6 plane ) . a schematic representation of the proposed setup is shown in fig . [ josephson ] . the main results of our study are as follows . first , we develop a general method for computing the andreev bound states energy in these junctions . such a method constitutes a generalization of the method of ref . to junctions with zeeman magnetic fields and spin - orbit coupling . second , using this method , we obtain analytical expressions for the energy of the andreev bound states in several asymptotic cases and discuss their implication on the josephson current . for example , we find that in the absence of the magnetic fields the energy gap between these bound states decreases with increasing rashba soi constant leading eventually to level touching while in the absence of rashba soi , they display oscillatory behavior with orientational angle of @xmath15 . third , we present analytic expressions for the dc josephson current charting out their dependence on both @xmath5 and @xmath3 and the soi interaction strength . fourth , we demonstrate the existence of finite spin - josephson current in these junctions in the presence of external magnetic fields and provide analytic expressions for its dependence on @xmath2 , @xmath7 and @xmath3 . finally , we study the ac josephson effect in the presence of the soi ( for @xmath8 ) and an external radiation and show that the width of the resulting shapiro steps in such a system can be tuned by varying @xmath2 . we discuss experiments which can test our theoretical results . the plan of the rest of the paper is as follows . in sec . [ sec2 ] , we describe the model and present explicit form of hamiltonian is presented . the hybridization energy of edge states is calculated in sec . [ sec3 ] , where several asymptotic expressions for the josephson coupling energy are obtained . this is followed by a discussion of the dc josephson effect in sec . [ sec4 ] . the ac josephson effect in these system and the dependence of the shapiro step on soi strength is studied in sec . finally we conclude in sec . some details of our calculations are specified in the appendices . we consider a junction of two 1d nanowires with proximity induced @xmath1-wave pairing symmetry in the presence of rashba spin - orbital interaction and external magnetic fields . the schematic representation of such a junction is shown in fig . [ josephson ] where the proximate superconductors are not shown for clarity . -like dielectric potential under magnetic fields @xmath16 and @xmath5 co - planar and perpendicular to spin - orbit interaction respectively . the bulk s - wave superconductors which induces superconductivity in the wires are not shown for clarity.,height=245 ] in what follows we assume the pairing is induced by two proximate @xmath1-wave superconductors which leads to effective pairing potentials @xmath17 and @xmath18 in the two wires . the hamiltonian for such a system reads @xmath19 where @xmath20 is hamiltonian of the nanowire in the presence of external magnetic fields and @xmath21 represents rashba soi . the former term is given by @xmath22 \sigma_0 + h \sigma_z + b \{[\sigma_x \cos \phi_1 + \sigma_y \sin \phi_1 ] \theta(-x ) + [ \sigma_x \cos \phi_2 + \sigma_y \sin \phi_2 ] \theta(x ) \ } \big ) \psi_{\sigma'}(x ) \nonumber\\ & & + ( \delta_1 \theta(-x ) + \delta_2 \theta(x ) ) \psi_{\uparrow}^{\dag}(x ) \psi_{\downarrow}^{\dag}(x)+ { \rm h.c . } \big\ } , \label{h - sc}\end{aligned}\ ] ] where @xmath23 denotes the electron kinetic energy as measured from the fermi energy @xmath24 , @xmath25 is the electron annihilation operator , @xmath3 and @xmath5 are external zeeman magnetic fields in @xmath26 direction and in the @xmath6 plane respectively , @xmath27 is the heaviside step function , and @xmath28 and @xmath29 denote pauli and identity matrices respectively in spin space . note that the magnetic field @xmath5 forms an angle @xmath30 with wire which can be tuned externally . in what follows , we choose @xmath5 in the left side of the junction to be aligned along the wire ( @xmath31 ) while in the right side it is chosen to make an angle @xmath30 with it ( @xmath32 ) . in eq . ( [ h - sc ] ) , the pairing potential @xmath18 in the right of the junction is chosen to have a phase difference @xmath33 compared to its left counterpart : @xmath34 and @xmath35 . the potential barrier @xmath36 represents the barrier potential between two superconductors located at @xmath37 . the hamiltonian of rashba soi can be written as @xmath38\psi_{\sigma'}(x ) , \label{rashba - o}\end{aligned}\ ] ] where @xmath2 is the strength of rashba soi which is chosen to be the same for both wires . in what follows , we shall look for the localized subgap andreev bound states with @xmath39 for the josephson junction of two nanowires described by eq.([h ] ) . in this section , we first obtain solution for the andreev bound states for junction described by eq . ( [ h ] ) . to do this , it is advantageous to use a four component field operator given by @xmath40 here the third subscript of the annihilation operator ( which we shall designate henceforth as @xmath41 ) labels the right- ( @xmath42 ) and the left - moving @xmath43 ) quasiparticles respectively while the index @xmath44 denotes either right ( @xmath45 ) or left ( @xmath46 ) superconductor . in terms of the field operator given by eq . ( [ op1 ] ) , the hamiltonian ( eq . ( [ h ] ) ) can be written as @xmath47 using the pauli matrices @xmath48 in spin- and @xmath49 in particle - hole spaces . from eqs . ( [ h ] ) and ( [ h - sc ] ) , we find @xmath50 and @xmath51 . in eq.([h2 ] ) , the energy spectrum of the electrons are linearized around the positive and negative fermi momenta leading to @xmath52 , where @xmath53 is the fermi energy . note that the hamiltonians @xmath54 acquires a magnetism - superconductivity duality @xcite in the absence of the kinetic term , implying that it becomes invariant under the transformation @xmath55 . the existence of a magneto - josephson effect in a topological insulator is known to be a result of this duality @xcite . we shall see that for the system we study , the magneto - josephson effect takes place even in the presence of the additional quadratic kinetic energy term of the electrons . the energy spectrum of quasi - particles in a bulk superconductor in the presence of soi and external magnetic fields and its expression for different asymptotic is calculated in appendix [ appa ] . note that in our case , all energies are measured from the fermi energy ; thus the condition for realization of a topological superconducting phase with effective @xmath56-wave pairing is @xmath57 , @xcite . however , the existence of such a topological phase requires strong @xmath58 or @xmath3 and so interaction so that only the electron band of a single spin species remains below the fermi surface . in what follows we shall focus on the other regime where the bands of both spin species are below the fermi surface and the superconductivity is still s - wave . the bogolyubov - de gennes ( bdg ) equations for the superconductors in the right- and left parts of the barrier are written as @xmath59 where @xmath60 denotes the bdg wave function . for a barrier modeled by the delta function potential @xmath61 , they satisfy the boundary condition @xmath62 where @xmath63 and the transmission coefficient @xmath64 is expressed through @xmath65 as @xmath66 . for obtaining the energy of the andreev bound states for our system , we first note that analysis of the energy spectrum of a bulk superconductor ( see , eq . ( [ eo ] ) ) shows that the rashba soi splits the energy spectrum shifting it along the momentum axis leading to four fermi momenta at @xmath67 ( see , eqs.([e100 ] ) and ( [ k100 ] ) ) . the contribution to the andreev bound states comes from momenta around these fermi points . the external magnetic field splits spin - up and spin - down electrons ( see , eqs . ( [ e011 ] ) and ( [ k011 ] ) ) , and the amplitudes of the electron wavefunction are redistributed around four fermi points due to the presence of such a field . finally , the presence of a barrier between the two superconductors leads to superposition of the right and left moving quasiparticles . therefore , the bdg wavefunction @xmath60 can be written as a linear superposition of its right and left moving components around each fermi momentum and with two different spins . since we look for bound state solutions , the general solution of eq . ( [ sch ] ) with ( [ h2 ] ) can be written as @xmath68 % \end{displaymath } \label{wave}\end{aligned}\ ] ] where @xmath69 denotes the localization length of the bound states , and @xmath70 for @xmath71 . henceforth , we shall rename the coefficients as @xmath72 , @xmath73 , and @xmath74 , @xmath75 for clarity . substituting the wave functions ( [ wave ] ) into the boundary conditions ( [ bc ] ) one gets eight linear homogeneous equations for @xmath76 , @xmath77 , @xmath78 , and @xmath79 with @xmath80 which can be represented in terms of a @xmath81 matrix @xmath82 and a column vector @xmath83 as @xmath84 . the details of this procedure is charted out in appendix [ appa2 ] . the energy of the andreev bound states can then be obtained from @xmath85 . we note that since the momentum splitting @xmath86 vanishes in the absence of soi and magnetic field ; in this limit , either @xmath87 and @xmath88 or both @xmath78 and @xmath79 vanish . the elements of four columns of the @xmath81 determinant , depending on @xmath89 become equal to other four column elements as @xmath90 , and the determinant @xmath82 vanishes as @xmath91 and @xmath92 . _ andreev bound states at @xmath93 : _ in this limit , the andreev bound states are determined using @xmath94 determinant written for electron and hole pairs with opposite spins @xcite . the boundary conditions ( [ bc ] ) for the wave function ( [ wave ] ) , written in the absence of the soi induced momentum splitting yield again eight equations for four coefficients @xmath95 and @xmath96 ; these equations are bdg equations for a s - wave superconductor with spin - dependent eigenfunctions @xmath97 and @xmath98 , where the overline of an index ( e.g. , @xmath99 ) means an opposite direction or sign . one chooses four equations corresponding to an electron - hole pair with opposite spins . the determinant corresponding to the matrix ( defined as @xmath100 in appendix[appa2 ] ) in the front of the coefficients @xmath76 and @xmath77 is calculated to give @xmath101 where @xmath102 \left[\frac{\eta^{\ast}_{+ , \uparrow,+}}{\eta_{+,\downarrow,- } } -\frac{\eta^{\ast}_{- , \uparrow,+}}{\eta_{-,\downarrow,-}}\right]-\\ \nonumber ( 1-d ) \left[\frac{\eta^{\ast}_{+ , \uparrow,-}}{\eta_{+,\downarrow,+ } } -\frac{\eta^{\ast}_{- , \uparrow,+}}{\eta_{-,\downarrow,-}}\right ] \left[\frac{\eta^{\ast}_{- , \uparrow,-}}{\eta_{-,\downarrow,+ } } -\frac{\eta^{\ast}_{+ , \uparrow,+}}{\eta_{+,\downarrow,-}}\right ] . \label{energy0}\end{aligned}\ ] ] equating this determinant to zero one gets a condition to find the energy spectrum @xcite . note that the other four equations yields the same expression with only spin being interchanged leading to @xmath103 . it is easy to see that the condition to determine the andreev bound state energy in this limit , where @xmath82 constitutes two @xmath94 blocks , is given by equating @xmath104 to zero . in order to get the explicit expressions for the wave functions @xmath97 and @xmath98 we write eq . ( [ sch ] ) for finite @xmath58 , @xmath3 and @xmath2 as @xmath105 eqs . ( [ sch1]) .. ([sch4 ] ) allow us to calculate all possible ratios @xmath106 , @xmath107 , and @xmath108 , @xmath109 . furthermore , we note that only the ratio @xmath110 is non - zero for @xmath111 . we shall return to this case below . next , we note from eqs . ( [ sch1]) .. ([sch4 ] ) that the dependencies of these equations on @xmath30 and @xmath33 are completely removed by transforming the wave function as @xmath112 in the transformed basis one has @xmath113 the different ratios that appear in the left - side of eqs.[dagup - up] .. [up - up ] can be understood as follows . the ratio @xmath106 corresponds to the amplitude of conventional andreev reflection channel which constitutes reflection of an electron - like quasiparticle to a hole - like quasiparticle with opposite spin on a n - s interface . in contrast , the ratio @xmath114 which is finite only in the presence of soi and/or magnetic field , represents amplitude of andreev reflection channel where the electron - like quasiparticle incident on the interface is reflected to a hole - like quasiparticle state with the same spin orientation . finally , the ratio @xmath115 represents a usual reflection channel of an electron - like quasiparticle on the boundary without creation of a cooper pair in a superconducting part of the junction . since these ratios enter the expressions of @xmath116 , these also represents andreev and normal reflection processes involving electron - like and hole - like quasiparticles in the opposite ( @xmath117 ) and same ( @xmath118 ) spin sector . we note that the ratio of wavefunctions in eq . ( [ dagup - up ] ) depend on both @xmath30 and @xmath33 while those in eqs . ( [ dagup - down ] ) and ( [ up - up ] ) depend on either @xmath33 or @xmath30 . this suggests that the ratios ( [ dagup - up ] ) and ( [ dagup - down ] ) are responsible for the dependence of observable parameters on the order parameter phase difference @xmath33 , whereas the ratios ( [ dagup - up ] ) and ( [ up - up ] ) are responsible for the dependence on the magnetic field orientation angle @xmath30 . the ratios @xmath119 and @xmath120 are determined from eqs . ( [ sch1])-([sch4 ] ) as @xmath121 where the upper ( lower ) sign @xmath122 ( @xmath123 ) corresponds to spin @xmath124 ( @xmath125 ) . using eq . ( [ wave - up - down ] ) , one obtains , after a few lines of algebra , the expressions for @xmath126 and @xmath127 for general @xmath5 , @xmath3 and @xmath2 as @xmath128 ^ 2 \nonumber\\ & & -4d|\delta|^2 ( e^2 + \alpha^2 k^2)^2 \sin^2 \frac{\varphi}{2}\big\ } ( |\delta|^2(e^2+\alpha^2 k^2)^2)^{-1 } , \label{f - up - down } \nonumber\\\end{aligned}\ ] ] at @xmath129 a contribution to the bound state energy comes only from expression of @xmath130 , and all other ratios vanish . by equating @xmath131 ( eq . ( [ f - up - down ] ) ) to zero and using the expressions ( [ e000 ] ) and ( [ k000 ] ) for the energy and momentum in this limit , one gets an expression for the bound state energy in consistent with kwon _ et al_. result @xcite , @xmath132 thus our formalism reproduces the earlier known result in the literature in this limit . in the absence of the magnetic fields a contribution to the bound energy due to soi comes from the _ conventional _ andreev reflection connecting electron - like and hole - like quasiparticles with opposite spins . these can be expressed as @xmath133 where @xmath134 can be obtained using eqs . [ f - up - down ] and [ energy0 ] . in contrast , the main tunneling channel in the presence of the magnetic field constitutes an electron - like quasiparticle with a given spin polarization being andreev reflected to a hole - like quasiparticle state with the same spin . the contribution to the bound state energy from this channel is @xmath135 where @xmath136 is given by @xmath137 we note that @xmath138 ( or @xmath139 ) in eq . ( [ feqss ] ) is determined by eq.([energy0 ] ) after replacing the ratio @xmath140 in @xmath141 by @xmath142 . the expressions for @xmath114 can be obtained from eqs . ( [ sch1])-([sch4 ] ) @xmath143 where the upper(lower ) signs correspond to @xmath144 . these ratios can be used to obtain @xmath145 as @xmath146 finally , the contribution to the bound energy from the channel given by ( [ up - up ] ) can be expressed as @xmath147 where @xmath148 a procedure , similar to the one outlined above yields @xmath149 \left[\frac{\eta_{+ , \uparrow,-}}{\eta_{+,\downarrow,- } } - \frac{\eta_{- , \uparrow,-}}{\eta_{-,\downarrow,-}}\right ] \nonumber\\ & & -(1-d ) \left[\frac{\eta_{+ , \uparrow,+}}{\eta_{+,\downarrow,+ } } -\frac{\eta_{- , \uparrow,-}}{\eta_{-,\downarrow,-}}\right ] \left[\frac{\eta_{- , \uparrow,+}}{\eta_{-,\downarrow,+ } } -\frac{\eta_{+ , \uparrow,-}}{\eta_{+,\downarrow,-}}\right ] \nonumber\\ & = & \frac{16 b^2}{m^2_-(k)}\left[\alpha^2 k^2-d(e^2 + \alpha^2 k^2 ) \sin^2\frac{\phi}{2}\right ] . \label{fmag2}\end{aligned}\ ] ] the expression for @xmath150 differs from @xmath151 by replacing @xmath152 in ( [ fmag2 ] ) . by equating to zero the sum of the expressions ( [ ksy ] ) , ( [ cont1 ] ) , ( [ cont2 ] ) , and ( [ cont3 ] ) yields the andreev bound state energy in the presence of soi and magnetic fields . in what follows , we shall discuss two limiting case where a simple analytical expressions for these bound states can be obtained . _ absence of rashba soi _ : in this case , @xmath153 and @xmath154 , the main contribution , which depends on the magnetic field orientation , yields the expression ( [ cont2 ] ) with ( [ energym ] ) for @xmath155 and @xmath156 . although the contribution from ( [ cont1 ] ) does depend on the magnetic field , it does not depend on the field orientation @xmath30 . a few lines of algebra then leads to the equation for the energy of the andreev bound states , obtained by equating the sum of ( [ ksy ] ) , ( [ cont1 ] ) and ( [ cont2 ] ) to zero , using ( [ e011 ] ) and ( [ k011 ] ) for the energy spectrum and momentum in this limit , given by @xmath157\right\ } = 0 , \label{eq - mag - add}\end{aligned}\ ] ] where the second and third terms come from ( [ cont2 ] ) and ( [ cont1 ] ) corresponding to the reflection mechanisms ( [ dagup - up ] ) and ( [ dagup - down ] ) . if we neglect the third contribution , which can be done for @xmath158 , eq.([eq - mag - add]),can be written @xmath159 we find that eq . ( [ eq - mag ] ) leads to the following features of the andreev bound states . first , @xmath160 decreases with increasing the magnetic field . second , the kwon et al . result @xcite is recovered as @xmath161 . ( [ eq - mag ] ) can be solved approximately . we replace the energy under square root by its zero - approximation value ( [ e0 ] ) , which yields @xmath162 with @xmath163 , where @xmath164 the second term in the bracket of eq . ( [ eq - maga])depends on the magnetic field as @xmath165 for @xmath166 . we note here that @xmath167 oscillates both with the superconducting phase difference @xmath33 and the angle orientation @xmath30 of @xmath5 with a period @xmath10 as shown in fig . [ magnet ] . note that all parameters in the figures presented below are dimensionless ones in the scale of @xmath168 , i.e. @xmath169 , @xmath170 , @xmath171 . at @xmath172 , when kwon et al . @xcite result is recovered for @xmath1-wave superconducting junction and the andreev bound state energy oscillates with @xmath13 periodicity ( see , fig . [ magnet]a ) for barrier transparency @xmath173 . the electron - like and hole - like energy branches corresponding to @xmath174 , touch each other at maximal transmission when @xmath175 , creating a zero - energy state at the center of the brillouin zone . the variation of @xmath58 and @xmath3 changes a character of @xmath33- and @xmath176-dependencies * of @xmath177*. note that since the gap between them vanishes at @xmath178 , it might be possible to have a @xmath9 periodic component of the josephson current in case of landau - zener transitions with a finite transmission probability between two states . this case will be investigated somewhere else . [ h ! ] ) , on the order parameter phase difference at @xmath179 and ( a ) @xmath180 , @xmath181 ; ( b ) @xmath182 , @xmath183 , @xmath184.,title="fig:",height=188 ] ) , on the order parameter phase difference at @xmath179 and ( a ) @xmath180 , @xmath181 ; ( b ) @xmath182 , @xmath183 , @xmath184.,title="fig:",height=188 ] _ absence of in - plane zeeman field _ : next , we consider the andreev bound states for @xmath185 , but @xmath186 . we find that eqs . ( [ sch1])- ( [ sch4 ] ) in this case link only @xmath187 and @xmath188 and are hence greatly simplified . a few lines of algebra shows that the andreev bound states energy in this case can be expressed as @xmath189 ^ 2 + \nonumber \\ f_{\uparrow , \downarrow}^{\ast}(k_+ ) f_{\downarrow , \uparrow}^{\ast}(k_- ) - f_{\uparrow , \downarrow}^{\ast}(k_-)f_{\downarrow , \uparrow}^{\ast}(k_+)=0 . \label{energyb=0}\end{aligned}\ ] ] the expression for @xmath190 in this limit is calculated in appendix [ appb ] and is given by eq.([apf ] ) . the expression for @xmath191 at @xmath111 is obtained from eq.([apf ] ) by replacing @xmath192 and @xmath193 . below we will study two asymptotic solutions of eq . ( [ energyb=0 ] ) at @xmath194 , @xmath195 and @xmath196 , @xmath197 . in the former case , eq . ( [ energyb=0 ] ) with ( [ apf ] ) yields the following equation @xmath198 solution of this equation provides a simple expression for the josephson energy @xmath199 where the sign @xmath200 in the front of the expression signifies an electron and hole energies , whereas the sign @xmath201 characterizes rashba splitting of the electron and hole states . this expression shows that @xmath202 depends nonlinearly on the soi coupling constant @xmath2 , and kwon et al . result @xcite is recovered as @xmath91 . according to ( [ soi ] ) , @xmath203 oscillates still with @xmath13 period for @xmath204 and @xmath205 , which is presented in fig . [ solutions](a ) at @xmath206 and @xmath207 . possible solutions for the energy spectrum according to the expression ( [ soi ] ) as a function of the order parameter phase difference at @xmath208 and @xmath175 is presented in fig.[solutions](b ) . it shows touching of all four branches at @xmath209 . the electron- and hole energy branches approach each other faster for non - zero soi . the dependence of the @xmath210 energy branches on the order parameter phase difference @xmath33 at fixed transmission coefficient @xmath64 and different values of the soi strength @xmath2 , is presented in the left panel of fig . [ e - p - dep](a ) . in fig.[e - p - dep](b ) , we present the dependence of the andreev bound state energies on @xmath64 for fixed @xmath2 . we note that both the branches approach zero as @xmath2 or @xmath64 is varied . _ absence of @xmath58 and @xmath2 _ : next , we consider the case @xmath211 but @xmath212 . in this case , eq . ( [ energyb=0 ] ) reduces to [ h ! ] ) as a function of the order parameter phase difference at ( a ) @xmath206 and @xmath207 , and ( b ) @xmath208 and @xmath175.,title="fig:",height=188 ] ) as a function of the order parameter phase difference at ( a ) @xmath206 and @xmath207 , and ( b ) @xmath208 and @xmath175.,title="fig:",height=188 ] [ h ! ] and different values of the soi strength , and ( b ) @xmath206 and different values of the transmission coefficient @xmath64.,title="fig:",height=226 ] and different values of the soi strength , and ( b ) @xmath206 and different values of the transmission coefficient @xmath64.,title="fig:",height=226 ] [ h ! ] at different @xmath213 . amplitude of the energy oscillation increases with @xmath213 ; ( b ) mutual optimal values of @xmath213 and @xmath64 at which electron- and hole - energy branches are crossed , creating a zero - energy mode ; ( c ) the andreev bound state energies @xmath214 touches at particular values of @xmath215 and @xmath216 ( thin curves ) , and of @xmath215 and @xmath217 ( thick curves ) which may make the oscillation period @xmath9 in the landau - zenner sense.,title="fig:",height=170 ] at different @xmath213 . amplitude of the energy oscillation increases with @xmath213 ; ( b ) mutual optimal values of @xmath213 and @xmath64 at which electron- and hole - energy branches are crossed , creating a zero - energy mode ; ( c ) the andreev bound state energies @xmath214 touches at particular values of @xmath215 and @xmath216 ( thin curves ) , and of @xmath215 and @xmath217 ( thick curves ) which may make the oscillation period @xmath9 in the landau - zenner sense.,title="fig:",height=170 ] at different @xmath213 . amplitude of the energy oscillation increases with @xmath213 ; ( b ) mutual optimal values of @xmath213 and @xmath64 at which electron- and hole - energy branches are crossed , creating a zero - energy mode ; ( c ) the andreev bound state energies @xmath214 touches at particular values of @xmath215 and @xmath216 ( thin curves ) , and of @xmath215 and @xmath217 ( thick curves ) which may make the oscillation period @xmath9 in the landau - zenner sense.,title="fig:",height=170 ] @xmath218 whose solutions read @xmath219 where @xmath201 . we note that the particle - like and the hole - like branches touch at zero energy ; in order to investigate the possible existence of a zero energy mode , which may create a @xmath220 oscillatory component of the josephson current in the landau - zenner sense , we introduce a dimensionless magnetic field @xmath221 . it is easy to see from eq . ( [ h ] ) that the condition for the particle and the hole states to cross at a phase difference @xmath33 is given by @xmath222 which yields @xmath223\sin^2\frac{\varphi}{2}}.\ ] ] for @xmath224 , the value of the critical @xmath225 for spin - up ( @xmath226 ) and spin - down ( @xmath227 ) states are @xmath228 and @xmath229 , correspondingly . we note here that the bands touch each other at @xmath230 but do not cross ; thus the andreev states still have @xmath231 periodic dispersion . the variation of @xmath232 with @xmath213 , the dependence of @xmath213 on @xmath64 , the touching of the @xmath233 and @xmath234 energy branches at @xmath215 and @xmath216 , and that between @xmath235 and @xmath236 energy branches at d=0.5 and @xmath217 are plotted in fig . [ crossing ] . in fig.[crossing](a ) , where the dependence of the spin - up particle energy branch on @xmath33 at different @xmath213 is presented , we find that the amplitude of the energy oscillation increases with @xmath213 , and additionally , the character of dependence around @xmath237 is changed . in fig . [ crossing](b ) we show the mutual optimal values of @xmath213 and @xmath64 at which electron - like and hole - like energy branches touche each other . finally , the touching of the two branches @xmath238 and @xmath239 for @xmath215 and @xmath216 is presented in fig . [ crossing](c ) . as it was mentioned above , these feature might be responsible for a @xmath240 periodicity in case of landau - zener transitions with a finite transmission probability between two states . the contribution of the andreev bound state to the josephson current can be calculated using to the expression @xmath241 where @xmath242 signifies all states which give a contribution to the current , and @xmath243 is the fermi occupation number corresponding to the @xmath242-th states . we note that since only the andreev bound states depend explicitly on the phase difference @xmath33 , their expression can be used to determine the dc josephson current using eq . ( [ current ] ) . in the absence of soi a contribution to the total equilibrium current gives electron and hole states , each of which is split into two levels due to zeeman effect @xmath244 where the expression for @xmath177 is given by eq . ( [ eq - maga ] ) , and @xmath245 \label{des}\end{aligned}\ ] ] with @xmath246 the current - phase relation at magnetic field @xmath182 calculated by using expressions ( [ currentt ] ) , ( [ des ] ) and ( [ cph ] ) is presented in fig . [ 44 ] . we note , that changes in @xmath3 does not make an essential effect at @xmath247 . next , we consider the spin - josephson current which is generated as response to rotation of the magnetic field @xmath248 in @xmath249 plane@xcite . as shown in ref . , the spin current can be defined as a derivative of the tunneling energy with respect to the magnetic field orientation @xmath30 and is given by @xmath250 according to the formulas ( [ currentt])-([cph]),height=188 ] , @xmath182 , @xmath251 , @xmath252 and two values of magnetic filed @xmath183 ( curve 1 ) and @xmath253 ( curve 2 ) . calculations are done according to the formulas ( [ j - mag ] ) , ( [ j - mag2 ] ) and ( [ eq - maga]).,height=188 ] where @xmath254 with @xmath255 for @xmath256 and @xmath257 for @xmath258 . as it is seen from formulas ( [ des ] ) and ( [ j - mag2 ] ) , the product @xmath259 increases with @xmath58 at @xmath166 as @xmath260 . instead in the opposite limit when @xmath261 this product decreases with increasing @xmath3 as @xmath262 . on the other hand , in the high temperature limit , when @xmath263 , one can expand @xmath264 function for small argument @xmath265 as @xmath266 . therefore , the amplitude of the supercurrent @xmath267 , given by eq.([des ] ) , and of the spin current @xmath268 , given by eq.([j - mag2 ] ) , will depend on the magnetic field exactly in the same form as described above for two limiting cases . the change of @xmath269direction can rotate the direction of spin current . spin current as a function of magnetic field orientation at two values of magnetic filed @xmath183 and @xmath253 is shown in fig . [ 47 ] . calculations are done according to the formulas ( [ j - mag ] ) , ( [ j - mag2 ] ) and ( [ eq - maga ] ) . the josephson current in other limiting case when @xmath180 and @xmath195 is calculated by replacing @xmath177 with @xmath270 given by ( [ soi ] ) in the expression ( [ currentt ] ) @xmath271 \sqrt{\left(1-d \sin^2\frac{\varphi}{2}\right)\left[1-s\frac{4v_f\alpha}{(v_f+s\alpha)^2}d\sin^2\frac{\varphi}{2}\right ] } } \tanh\left(\frac{e_s^{soi}}{2k_bt}\right).\ ] ] the corresponding plots demonstrated a strong variation of current - phase relation with parameter of spin - orbital coupling @xmath2 are presented in fig . [ the figure demonstrates a crucial breaking of the sinusoidal current - phase relation with increase in spin - orbital coupling . it shows a singular behavior at small @xmath33 . at @xmath215 ( formula ( 49 ) ) . numbers show the values of parameter spin - orbital coupling.,height=226 ] in this section , we compute the ac josephson effect for the tunnel junctions mentioned above . if there is the voltage in josephson junction @xmath272 , then from josephson relation @xmath273 we get @xmath274 , \label{phaseeq1}\end{aligned}\ ] ] we shall now use this relation to obtain the shapiro step width for @xmath180 and demonstrate that the step - width depends on the strength of the spin - orbit coupling . to do this we first consider the case @xmath251 for which @xmath275 $ ] is given at @xmath276 by @xmath277 substituting eq . ( [ phaseeq1 ] ) into eq . ( [ iexp ] ) , one gets @xmath278 using the identity @xmath279 where @xmath280 means imaginary part , @xmath242 is an integer and @xmath281 denotes bessel function of the first kind , one gets @xmath282/2 } } \label{ieq3}\ ] ] here @xmath283 means the real part . the shapiro steps thus occur when @xmath284 for integer @xmath285 ; at these values of the applied radiation frequency , the ac component of the supercurrent vanishes leading to an extra contribution to the dc current in the circuit . the magnitude of the extra dc current from @xmath286 can be read off from eq . ( [ ieq3 ] ) as @xmath287/2 } } \label{idc1}\ ] ] from eq . ( [ idc1 ] ) , we find that both the shapiro step width and the position of maxima / minima of @xmath288 depends on @xmath64 . let us assume that the maxima and minima occur at @xmath289 . note that @xmath290 can be obtained from the solution of @xmath291 and equals @xmath292 for @xmath293 . in terms of @xmath294 , one obtains the step width as @xmath295/2 } } \label{sstep1}\ ] ] which clearly shows the @xmath64 dependence of the step - width . one can now carry out a similar analysis for the case where @xmath180 and @xmath296 ( eq . ( [ soi ] ) ) . starting from eq.([current ] ) , the ac josephson current at @xmath276 can be obtained as @xmath297^{3/2}\left[1-d ( 1-\cos \varphi(t))/2\right]^{1/2 } } \label{acjos1}\ ] ] where @xmath298 and @xmath299 . similar straightforward algebra , as carried out earlier in this section , leads to steps at @xmath300 with @xmath301^{1/2 } } \sum_{s=\pm } \frac{(1-\eta_s)}{\left[1-\eta_s d ( 1-j_{n_0}(\omega ) \cos \varphi_0)/2\right]^{3/2 } } \label{dcjos1}\ ] ] as before , the minimum and maximum of the dc component of the occurs at @xmath302 which can be obtained as the solution of @xmath303 . the step width can thus be expressed in terms of @xmath304 as @xmath305^{1/2 } } \\ \sum_{s=\pm } \frac{(1-\eta_s)}{\left[1-\eta_s d ( 1-j_{n_0}(\omega ) \cos \varphi_0^{n_0 \alpha})/2\right]^{3/2}}\label{swidthso}\ ] ] thus we find the step width depends on the magnitude of the spin - orbit coupling . indeed , [ plots - width](a ) demonstrates this effect of transparency and spin - orbital coupling on the @xmath33-dependence of the shapiro step width according to formula ( [ swidthso ] ) . we also note that for @xmath293 , the maxima and minima of the dc current occur for @xmath306 and eq . ( [ swidthso ] ) simplifies to yield @xmath307 for small @xmath308 , it is easy to see by expanding @xmath309 in power of @xmath310 , that @xmath311 which demonstrates the dependence of step width on the so coupling @xmath2 . comparison of these three plots according to eqs . ( [ swidthso ] ) , ( [ swidthdll1 ] ) and ( [ swidthdll2 ] ) is presented in fig.[plots - width](b ) . as we can see , the results of approximations ( [ swidthdll1 ] ) and ( [ swidthdll2 ] ) demonstrate more sharper increasing of shapiro step width with @xmath2 in compare with formula ( [ swidthso ] ) . it s clear that the difference disappears in the limit @xmath312 . the obtained dependence of the ss width on the spin - orbit coupling may be used for the experimental estimation of its value . -dependence of the shapiro step width according to the formula ( [ swidthso ] ) ; ( b ) demonstration of @xmath2-dependence of shapiro step width in different approximations according to the formulas ( [ swidthso ] ) , ( [ swidthdll1 ] ) and ( [ swidthdll2]).,title="fig:",height=188 ] -dependence of the shapiro step width according to the formula ( [ swidthso ] ) ; ( b ) demonstration of @xmath2-dependence of shapiro step width in different approximations according to the formulas ( [ swidthso ] ) , ( [ swidthdll1 ] ) and ( [ swidthdll2]).,title="fig:",height=188 ] , @xmath206 without radiation ( curve 1 ) and under external radiation ( curve 2),height=226 ] to investigate the effect of soi on the amplitude dependence of shapiro step width , we have calculated the i - v curves for the junction under external radiation using equation ( [ acjos1 ] ) . this result is presented in fig . [ iv_curve ] , where we show the i - v curve of the junction at @xmath215 , @xmath206 under external electromagnetic radiation with frequency @xmath313 and amplitude @xmath314 . in this figure we include for comparison the i - v characteristics without radiation also . the i - v curve demonstrates the main shapiro step at @xmath315 and its harmonics . [ amp_dep](a ) shows the amplitude dependence of shapiro step width in case @xmath316 ( line 1 ) and @xmath206 ( line 2 ) under external radiation with frequency @xmath313 . calculation is provided for value of transparency @xmath215 . we see that the value of the soi parameter has a noticeable effect on the shapiro step width and its dependence on amplitude of the external radiation . these results of i - v characteristics simulations coincide qualitatively with the conclusion followed from fig . [ plots - width ] . we see that in case with @xmath316 the width of shapiro step is larger than case @xmath206 . the similar effect can be seen in amplitude dependence of critical current @xmath317 , which is shown in fig.[amp_dep](b ) . dependence of @xmath317 for @xmath215 and @xmath64dependence of @xmath317 for @xmath318 at @xmath313 , @xmath314.,height=226 ] the transparency coefficient @xmath64 also effects the critical current value . to distinguish and clarify the effect of soi we have calculated the @xmath2 and @xmath64dependence of @xmath317 , which is demonstrated in figures [ alpha_d - dep ] ( a ) and ( b ) . these results might be used for the comparison with future experimental results . in this paper we study the josephson current between 1d superconducting nanowires separated by an insulating barrier in the presence of rashba soi and the magnetic fields @xmath5 and @xmath3 . the presence of the soi and zeeman magnetic fields leads to four distinct fermi points in each bulk superconductor . therefore , the study of josephson effect in these junctions requires construction of an incident quasiparticle wave function which is in a linear superposition state of plane waves around each fermi points . in our study , we have developed a theoretical method to study josephson effect in such systems ; our work thus constitutes a generalization of analysis of ref . to systems with soi and zeeman fields . we have provided analytical results for the andreev bound states in several asymptotic limits from our analysis , demonstrated the presence of spin - josephson current in these junctions , and studied the dependence of shapiro steps on soi interaction strength @xmath2 in the presence of external radiation . moreover , we have demonstrated the existence of magneto - josephson effect in these systems . we note that although the existence of the magneto - josephson effect in a topological superconductor has been predicted recently @xcite , the question of whether this effect is observable in superconducting junctions with quadratic electronic dispersion and the absence of soi was not addressed before . we show in the paper the magneto - josephson effect takes place even in the absence of soi . experimental verification of our work would require experiments conducted on josephson junctions in 1d nanowires analogous to ones studied in ref . . we predict that the variation of the angle @xmath30 of the in - plane magnetic field @xmath5 would lead to a spin - josephson current as shown in fig . furthermore , ac josephson effect measurement in these junction , analogous to those done in ref . , should reveal a quadratic dependence of the shapiro step - width as a function of @xmath2 for small @xmath319 as shown in fig . [ alpha_d - dep ] . our work allows for several possible future direction . first , a numerical solution of the condition @xmath320 yielding andreev bound state energies in the regime where @xmath321 may lead to a better understanding of the interplay between these parameters to shape the characteristics of the bound state energies . second , the formalism that we develop here may be extended to regime of strong @xmath2 where the presence of majorana bound states shapes the characteristics of the josephson current . third , our formalism may be applied to cases where the superconducting pair - potential is unconventional ( for example p - wave ) ; indeed , interplay of such unconventional pair - potentials and so coupling may lead to additional interesting characteristics in the josepshon current . we intend to explore these issues in future work . in conclusion , we have studied josephson effect in a unction between two 1d nanowires in the presence of soi and zeeman fields . we have analyzed the josephson current in these junctions and provided analytical expressions of the andreev bound states in several limiting cases . we have also demonstrated the presence of magneto - josephson effect in these junctions and studied the shapiro step width in ac josephson effect on the soi strength . our theoretical predictions are shown to be verifiable by straightforward experiments on these systems . the authors thank v. osipov for discussion of this paper and support . the reported study was funded partially by azerbaijan - jinr collaboration , the science development foundation under the president of the republic azerbaijan - grant no eif - ketpl-2 - 2015 - 1(25)-56/01/1 , the rfbr according to the research projects 165245011@xmath322india , 155161011@xmath322egypt , 152901217 and dst - rfbr grant . the expression @xmath323 for the energy spectrum is written @xmath324 where @xmath325 , @xmath326 , @xmath327 , and @xmath328 . calculation of this determinant yields the energy spectrum of a `` bulk '' @xmath329 superconductor @xmath330 this expression contains a linear in energy term , which is a result of an alignment of @xmath16 and the effective magnetic field of the soi @xmath331 . we consider different limiting cases below . * * the case of * @xmath332 . the energy spectrum looks @xmath333 the energy levels of bdg quasi - particles lie in the gap , symmetrical to the fermi level , with momentum @xmath334 * * the case of * @xmath172 , but @xmath197 and @xmath195 . the energy spectrum ( [ eo ] ) in this limiting case is factorized @xmath335\left[(e - h)^2+(v_f+\alpha)^2k^2-|\delta|^2\right]=0 . \label{e100}\ ] ] one gets for the quasi - particles energy @xmath336 where @xmath163 . the momenta is expressed as @xmath337 soi and/or magnetic field @xmath3 split both electron and hole levels due to rashba momentum - shifting and/or zeeman effect . the fermi points around @xmath338 and @xmath339 are split also due to these effects . * * the limit of * @xmath153 , and @xmath340 , @xmath197 . expression ( [ eo ] ) under these conditions reads @xmath341\left[\left(e-\sqrt{b^2+h^2}\right)^2+v_f^2k^2-|\delta|^2\right]=0,\ ] ] yielding the following expression for the energy spectrum @xmath342 the momenta around the fermi points @xmath338 and @xmath339 split also @xmath343 the expressions for the energy and momentum in the limits of @xmath153 , @xmath111 but @xmath197 or of @xmath153 , @xmath194 but @xmath344 are easily obtained from ( [ e011 ] ) and ( [ k011 ] ) . note that a topological superconducting gapped phase is realized when @xmath345 in consistent with ref.@xcite . in this section , we chart out the expression for @xmath82 . substituting the wave functions ( [ wave ] ) into the boundary conditions ( [ bc ] ) one gets eight linear homogeneous equations for @xmath76 , @xmath77 , @xmath78 , and @xmath79 with @xmath80 as explained in the main text . we can represent these equations in terms of a @xmath346 matrix @xmath82 and a column vector @xmath347 as @xmath84 . the energy of the andreev bound states can then be obtained from @xmath348 . the expression for the matrix @xmath82 , obtained from some straightforward algebra , is given by @xmath349 where @xmath350 and @xmath351 takes values @xmath352 . we note that it is difficult to obtain analytical expression of @xmath353 for general values of @xmath58 , @xmath2 and @xmath3 . however , the physical content of the several terms in this determinant can be understood as follows . we define the minors of the selected blocks of @xmath82 as @xmath354 , @xmath355 , @xmath356 , @xmath357 . furthermore we define the @xmath358 matrices @xmath359 the determinants of these matrices are denoted by @xmath360 and @xmath361 . similarly one can also construct expressions for @xmath362 and @xmath363 . note that all these blocks are interpreted to correspond to a definitive physical process as explained in the main text . all of these determinants enter the expressions of the andreev bound states as discussed in sec.[sec3 ] of the main text . in this section we look into the expression of andreev bound states for @xmath185 . equations ( [ sch1])-([sch4 ] ) are strongly simplified in this link providing only a link between @xmath187 and @xmath188 @xmath364 then , one gets for @xmath126 according to eq . ( [ energy0 ] ) @xmath365 \sin^2\frac{\varphi}{2}\right\}. \label{apf}\ ] ] the expression for @xmath366 differs from that for @xmath190 by replacing @xmath367 and @xmath193 in eq . ( [ apf ] ) . the tunneling energy in this case receives its contribution from the expression @xmath368 ^ 2 + f_{\uparrow , \downarrow}^{\ast}(k_+ ) f_{\downarrow , \uparrow}^{\ast}(k_- ) - f_{\uparrow , \downarrow}^{\ast}(k_- ) f_{\downarrow , \uparrow}^{\ast}(k_+)=0 \label{apfeq}\ ] ] with energy spectrum obtained from eq . ( [ sch1b0 ] ) @xmath369 and from eq . ( [ sch2b0 ] ) @xmath370 this expression has been used to analyze eq . ( [ energyb=0 ] ) of the main text . kitaev , annals phys . * 303 * , 2 ( 2003 ) . c. nayak , s. simon , a. stern , m. freedman , and s. das sarma , rev . phys . * 80 * , 1083 ( 2008 ) . j. alicea , rep . prog . phys . * 75 * , 076501 ( 2012 ) . v. mourik , k. zuo , s. m. frolov , s. r. plissard , e. p. a. m. bakkers , and l. p. kouwenhoven , science * 336 * , 1003 ( 2012 ) . k. sengupta , i. zutic , h.j . kwon , v.m . yakovenko , s das sarma physical review b * 63 ( 14 ) * , 144531 ( 2001 ) . kwon , k. sengupta , and v. m. yakovenko , eur . j. b * 37 * , 349 ( 2004 ) . a. das , y. ronen , y. most , y. oreg , m. heiblum , and h. shtrikman , nature phys . * 8 * , 887 ( 2012 ) . l. p. rokhinson , x. liu , and j. k. furdyna , nature phys . * 8 * , 795 ( 2012 ) . m. t. deng , c. l. yu , g. y. huang , m. larsson , p. caroff , and h. q. xu , nano lett . * 12 * , 6414 ( 2012 ) . h. o. h. churchill , v. fatemi , k. grove - rasmussen , m. t. deng , p. caroff , and c. m. markus , phys . b * 87 * , 241401 ( r ) ( 2013 ) . a. d. k. finck , d. j. van harlingen , p. k. mohseni , k. jung , and x. li , phys . . lett . * 110 * , 126406 ( 2013 ) . r. m. lutchyn , j. d. sau , and s. das sarma , phys . lett . * 105 * , 077001 ( 2010 ) . y. oreg , g. refael , and f. von oppen , phys . . lett . * 105 * , 177002 ( 2010 ) . a. kitaev , phys . usp . * 44 * , 131 ( 2001 ) . j. nilsson , a. r. akhmerov , and c. w. j. beenakker , phys . * 101 * , 120403 ( 2008 ) . l. fu and c. l. kane , phys . lett . * 100 * , 096407 ( 2008 ) . l. fu and c. l. kane , phys . b * 79 * , 161408 ( 2009 ) . d. m. badiane , m. houzet , and j. s. meyer , phys . 107 * , 177002 ( 2011 ) . l. jiang , d. pekker , j. alicea , g. refael , y. oreg , and f. von oppen , phys . lett . * 107 * , 236401 ( 2011 ) . p. san - jose , e. prada , and r. aguado , phys . * 108 * , 257001 ( 2012 ) . d. i. pikulin and y. v. nazarov , phys . b * 86 * , 140504 ( 2012 ) . t. ojanen , phys . b * 87 * , 100506(r ) ( 2013 ) . p. san - jose , j. cayao , e. prada , and r. aguado , new journal of physics * 15 * , 075019 ( 2013 ) . lee , k. michaeli , j. alicea , and a. yacoby , phys . * 113 * , 197001 ( 2014 ) . g. campagnano , p. lucignano , d. giuliano , and a. tagliacozzo , j. phys . : matter * 27 * , 205301 ( 2015 ) . l. jiang , d. pekker , j. alicea , g. refael , y. oreg , a. brataas , and f. von oppen , phys . b * 87 * , 075438 ( 2013 ) . p. kotetes , g. schn , and a. shnirman , j. korean phys . * 62 * , 1558 ( 2013 ) . c. w. j. beenakker , d. i. pikulin , t. hyart , s. schomerus , and j. p. dahlhaus , phys . lett . * 110 * , 017003 ( 2013 ) . f. pientika , l. jiang , d. pekker , j. alicea , g. refael , y. oreg , and f. von oppen , new j. physics * 15 * , 115001 ( 2013 ) . f. zhang and c. l. kane , phys . lett . * 113 * , 036401 ( 2014 ) .

we calculate the josephson current between two one - dimensional ( 1d ) nanowires oriented along @xmath0 with proximity induced @xmath1-wave superconducting pairing and separated by a narrow dielectric barrier in the presence of both rashba spin - orbit interaction ( soi ) characterized by strength @xmath2 and zeeman fields ( @xmath3 along @xmath4 and @xmath5 in the @xmath6 plane ) . we formulate a general method for computing the andreev bound states energy which allows us to obtain analytical expressions for the energy of these states in several asymptotic cases . we find that in the absence of the magnetic fields the energy gap between the andreev bound states decreases with increasing rashba soi constant leading eventually to touching of the levels . in the absence of rashba soi , the andreev bound states depend on the magnetic fields and display oscillatory behavior with orientational angle of b leading to magneto - josephson effect . we also present analytic expressions for the dc josephson current charting out their dependence on @xmath5 , @xmath3 , and @xmath2 . we demonstrate the existence of finite spin - josephson current in these junctions in the presence of external magnetic fields and provide analytic expressions for its dependence on @xmath2 , @xmath7 and @xmath3 . finally , we study the ac josephson effect in the presence of the soi ( for @xmath8 ) and an external radiation and show that the width of the resulting shapiro steps in such a system can be tuned by varying @xmath2 . we discuss experiments which can test our theoretical results .

1701.03589

it is presently accepted that most stars form in clusters or groups , although the details of the cluster - formation process , especially the origin of their structural properties , remain a matter of active research ( see , e.g. , the reviews by * ? ? ? * ; * ? ? ? * ; * ? ? ? in particular , in recent years , a number of structural properties of the clusters have been uncovered that still require an adequate theoretical understanding , such as : i ) the existence of a mass segregation in the clusters , with the most massive stars lying closer to the cluster s center @xcite ; ii ) the distribution of protostellar separations , which appears to have no characteristic scale @xcite ; iii ) the likely existence of an age gradient in clusters , with the youngest stars being located in the highest - density regions @xcite ; iv ) the apparent deficit of ob stars in some infrared dark clouds . numerical simulations have begun to offer some insight about these properties . @xcite have concluded , from a suite of simulations of self - gravitating , decaying isothermal turbulence , that the most massive stars form in situ at the cluster centers , rather than `` sinking '' there through dynamical interactions in the cluster itself . however , they gave no physical explanation as to why the most massive stars should form there . more recently , @xcite have suggested , by comparing multi - wavelength observations of stellar clusters with numerical simulations , that clusters form by mergers of `` subcluster '' structures , although again no explanation of why such mergers should occur is provided . observationally , the presence of subunits of somewhat different ages in the clusters has also been pointed out by @xcite . a physical mechanism capable of providing a unifying scenario to these properties is that of global , hierarchical molecular cloud collapse , advanced by . the latter authors noted that , if molecular clouds ( mcs ) are assembled by large - scale colliding streams of warm , atomic gas that rapidly condenses into the cold atomic phase , then they quickly become jeans - unstable and begin to collapse globally . moreover , the turbulence induced by the collision of the streams causes moderately supersonic turbulence ( e.g. , * ? ? ? * ; * ? ? ? * ) in the cold gas , which produces a multi - scale spectrum of density fluctuations , where small - scale , large - amplitude ( ssla ) density fluctuations are superposed on larger - scale , smaller - amplitude ( lssa ) ones . since these density fluctuations are nonlinear , the denser ssla fluctuations have shorter free - fall times than the lssa ones , therefore completing their collapse earlier . this process is therefore similar to @xcite fragmentation , except that the density fluctuations are of turbulent origin and therefore nonlinear . in this sense , the process is also similar to the mechanism of `` gravo - turbulent fragmentation '' , except that the cloud is _ not _ globally supported by turbulence , and the turbulent fluctuations do not collapse directly , but rather just plant the seeds for subsequent , local , scattered collapses as the cloud contracts globally @xcite . in what follows , we will refer to this intermediate scenario between hoyle and gravoturbulent fragmentation as `` global hierarchical collapse '' ( ghc ) . this scenario also predicts that the star formation rate ( sfr ) in mcs evolves ( initially increasing ) over time , as a consequence of the increase of the mean density of the clouds as they go through global gravitational collapse @xcite . assuming that massive stars do not form until the sfr is high enough that the imf is sampled up to high masses , then massive stars form late in the global process , and when they do , they begin to disrupt their parent clouds through their feedcak ( winds , ionising radiation , sn explosions ) , reducing the sfr again . @xcite presented a smoothed - particle hydrodynamics ( sph ) numerical simulation of cloud formation and evolution in the context of ghc that showed the formation of filamentary structures with embedded clumps . in that simulation , the filaments constitute river - like structures , through which material flows from the extended cloud environment to the dense cores where star formation occurs . when the filaments are sufficiently dense , fragmentation occurs within them as the gas flows along them into the most massive cores . this implies that the filaments supply the dense cores with a mixture of stars and gas . this kind of flow was also observed ( although it was not discussed ) in a similar simulation presented by using the adaptive mesh refinement code art @xcite that included a simplified treatment of radiative transfer and a prescription to form stellar particles ) . ] ( sps ) that allows imposing a power - law sp mass function with a slope similar to that of @xcite . this implies that , contrary to the situation in the simulation by @xcite , the clusters formed in the simulation of paper i , as well as their surrounding gas , are subject to realistic dynamics , which allows investigating the evolution of the clusters from their formation to the time when they disperse their surrounding gas . there exist many numerical studies of cluster formation , focusing on issues such as their stellar mass function , the correlation function of the spatial stellar distribution and cluster boundedness , and the formation of binaries ; the effect of feedback on producing massive stars and on destroying their parent clumps ; and the energy balance and rotation of the cluster as a function of the initial turbulence level in the parent cloud @xcite . in this work , instead , we aim to describe the process of assembly and early evolution of the clusters as a consequence of ghc . to this end , we study a cluster formed in the simulation labeled laf1 in paper i , focusing on the resulting spatial structure of the cluster . in sec.[sec : model ] we briefly describe the numerical simulation , and in sec . [ sec : cluster ] we describe the criteria for defining the cluster and stellar groups , both in terms of the origin of their members as well as from their instantaneous positions . next , in sec.[sec : results ] we present our results concerning the assembly from subunits brought in by the ghc , as well as the resulting structure of the clusters . in sec . [ sec : discussion ] we discuss the implications of our results and compare them with existing observations , and in sec.[sec : conclusions ] we give a summary and some conclusions . the numerical simulation used in this work comes from the set performed in paper i with the hydrodynamics+n - body adaptive refinement tree ( art ) code of @xcite . the physical processes included are self - gravity , parameterized heating and cooling ( sec.[sec : hcrates ] ) , star formation with an imposed but realistic imf ( sec.[sec : sf ] ) , and simplified radiative transfer for the feedback from massive - star ionising radiation ( sec . [ sec : feedback ] ) . magnetic fields are neglected . for more details , we refer the reader to paper i. the simulation represents the head - on collision of two cylindrical streams in the warm neutral medium . the streams each have a radius of 64 pc and a length 112 pc , and each is traveling at a speed of @xmath0 . the numerical box ( including the streams ) initially has a uniform density @xmath1 and temperature @xmath2 k , implying an adiabatic sound speed of @xmath3 . thus , the streams move with a mach number of 0.8 with respect to the sound speed in the initial uniform background medium . the simulation uses a base resolution of @xmath4 grid cells , and allows for 5 refinement levels , reaching a maximum resolution equivalent to @xmath5 , with a minimum cell size of 0.0625 pc , or @xmath6 13 000 au . the refinement is based on a `` constant mass '' criterion , so that a cell size is refined when its mass exceeds @xmath7 . this implies that the grid cell size @xmath8 scales with density @xmath9 as @xmath10 . once the maximum refinement level is reached , no further refinement is performed , and the cell s mass can reach much larger values because of the probabilistic sf scheme ( sec . [ sec : sf ] ) . note that this constant - cell - mass refinement criterion does not conform to the so - called _ jeans criterion _ @xcite of resolving the jeans length with at least 4 grid cells . those authors cautioned that failure to do this might result in spurious , numerical fragmentation . however , we do not consider this a cause for concern since , as will be described in sec . [ sec : sf ] , our star formation prescription allows us to choose the stellar - particle mass distribution , and tune it to a @xcite value . we use heating ( @xmath11 ) and cooling ( @xmath12 ) functions of the form @xmath13 these functions are fits to the various heating and cooling processes considered by @xcite , as given by equation ( 4 ) of @xcite , with the typographical corrections noted in @xcite . with these heating and cooling laws , the gas is thermally unstable in the density range @xmath14 . no chemistry tracking is performed , and so the gas is assumed to be all atomic , with a mean particle weight @xmath15 . our simulation employs a probabilistic star formation prescription such that , if a grid cell reaches a number density @xmath16 , where is a density threshold , then an sp of fixed mass @xmath17 may be placed in the cell with a probability @xmath18 every timestep of the root ( coarsest ) grid . if the sp is created , it acquires half of the gas mass of the parent cell . unlike standard sink particles ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , our sps do not accrete . however , since our prescription for sf is probabilistic , once the highest refinement level is reached in a cell , it actually continues to accrete from the surrounding material , and grows in mass and density until an sp is actually placed in it . this implies that the accretion is performed in the gas phase , rather than onto the particles . serendipitously , it was found in paper i that this prescription allows the sps to form with a power - law mass distribution , whose slope can be tuned by varying the value of @xmath18 , at a given maximum resolution . in paper i it was reported that setting @xmath19 at the resolution used in our simulation caused the sp power - law mass distribution to attain an exponent of @xmath20 , thus being similar to the classical salpeter value . we set the threshold density for sp formation at @xmath21 , which corresponds to a cell mass of @xmath22 at the highest refinement level . thus , the minimum possible sp mass is @xmath23 . with this prescription , the most massive sp formed in this simulation has @xmath24 . note , however , that our imposed imf is a strict power law , and thus the most probable stellar mass is also the minimum . this implies that we are missing the low - mass ( @xmath25 ) side of the actual imf , implying that we are missing about half of the total number of stars , and that their corresponding mass ( 2030% of the total mass ) is deposited instead into the stars that do get formed in the simulation . we discuss the implications of this in sec . [ sec : limits ] . our simulation thus contains sps that represent individual stars ( not clusters or groups ) , with a realistic mass distribution above the most probable stellar mass . this allows the dynamics of the sps to be reasonably representative of the forming cluster dynamics , contrary to the case when the particles masses correspond to those of small groups , of up to a few hudreds of solar masses . in that case , low - mass particles can suffer exceedingly strong encounters with the massive particles , and thus acquire unrealistically high velocities . in our simulation , the feedback effect of the ionising stars on the mc is non - local , using a simplified radiative transfer prescription to model the formation and evolution of a hii region around an sp . as a zeroth - order approximation , we assume a uniform `` line - of - sight '' ( los ) characteristic density @xmath26 in the hii region , with a value equal to the geometric mean between the density of the cell containing the sp and that of the target cell . moreover , for any given star we assume an ionising flux @xmath27 taken from the tabulated data provided by @xcite . we then compute the corresponding strmgrem radius , @xmath28 , given by @xmath29 where @xmath30 @xmath31 s@xmath32 is the hydrogen recombination coefficient . at each time step , we search for all cells surrounding every sp particle whose distances to it are smaller than @xmath28 . for each target cell that satisfies this condition , we set its temperature to @xmath33 and turn off the cooling there . the cell will remain so during the whole lifetime of the star , which is determined by its mass @xmath34 as @xmath35 as explained in paper i , for stars more massive than @xmath36 , this time is a fit to the stellar lifetimes by @xcite , while for stars with masses lower than that , it represents the fact that the duration of the stellar - wind phase is @xmath37 myr , roughly independently of mass . this also means that we are representing the effect of the winds and outflows of low mass stars by an ionization prescription . while this is clearly only an approximation , we do not expect it to have much impact on our calculations , since the main source of feedback energy at the level of gmcs is the ionization feedback from massive stars @xcite . although highly simplified , this prescription was tested in paper i against the known analytic formula for the evolution of @xmath38 in a uniform medium @xcite and shown to agree within 30% . this , we believe , is accurate enough given our interest in large - scale molecular cloud evolution and the associated cluster dynamics at early stages of assembly , rather than on the structural details of the ionised regions . one crucial step for the present study is the membership of stars to a given group or cluster . we now discuss our procedure of choice . observationally , the identification and membership definition of embedded stellar clusters is intrinsically a complex task , complicated by ambiguities and uncertainties , since the clusters do not have well defined edges , contamination by field stars is difficult to quantify , and faint members may be difficult to identify . in our case , the origin of the stars in the simulations is known , and this allows a classification of the stars by site of origin . however , this information is unavailable to observations , which instead must proceed by first identifying the cluster as a stellar surface density excess , and then define its members mainly on a statistical basis , by comparison with star counts in nearby control fields off the cluster @xcite . on the other hand , in the simulation there are no `` field '' stars , but only stars from clusters formed at some other locations in the cloud . for the purpose of defining a cluster or group by location at early times , we use the well known friends - of - friends ( fof ) algorithm @xcite . this is convenient at early stages , when the first groups of stars are well separated and the stars are easy to tag as part of a group that has formed at a certain time and location . however , note that , because of the hierarchical structure of the clusters , and [ sec : results ] . ] the number and stellar content of the resulting groups depends on the value of the so - called `` linking parameter '' passed to the algorithm . this parameter is a measure of the distance out to which neighbors are searched . at early times , the separation between groups is easily determined by eye , and the linking parameter can then be chosen to match the visual classification . at later times , however , the groups change their size by dynamical interactions between their members , and moreover new stars continue to form in the same region , or in its outskirts , making it necessary to suitably define the regions size in order to determine stellar membership to a group . we thus define the radius of a group at a given time by performing an iterative process : 1 . considering only the stars that were part of the group in the previous snapshot , we measure the distance from their center of mass to the most distant star . if this distance is less than twice the distance from the center to the second most distant star , we consider the distance to the most distant star as the radius of the group . otherwise , we consider the most distant star as a `` runaway star '' and take the radius of the group as the distance from its center to the second most distant star . next , we assign the new stars ( those formed in the period of time between the actual snapshot and the previous one ) to the group whose center of mass is closest to them . 3 . finally , including the new stars , we recompute the center of mass , size ( @xmath39 ) , total mass ( @xmath40 ) , and total number of stars ( @xmath41 ) of the group . we also compute the age of the group , @xmath42 , as the time elapsed since the moment when the group was first defined . however , neighbouring groups generally approach each other ( as part of the large - scale collapse ) and often they merge . we say that a new group has formed by the merger of two former groups when the distance between their centers of mass is smaller than the larger of the two radii . at that point , we consider that the earlier subgroups disappear . in general , we expect this process to operate at all scales in our simulation , with groups in turn forming from subgroups , and so on , in a hierarchical and self - similar manner . in this simulation , three main clusters form , two of which are sufficiently massive to clear out the dense gas around them on a timescale @xmath43 myr since the formation of the first stars ( from @xmath44 myr to @xmath45 myr in the evolution of the simulation ) . stars begin to form at @xmath46 myr , and the first hii regions appear at @xmath47 myr . specifically , at @xmath48 myr , the dense gas has been cleared from a radius of @xmath49 pc around the clusters . figure [ fig : sim_imgs ] shows one snapshot of the simulation , at @xmath50 myr into the evolution , in which two hii regions , surrounding the two most massive clusters , can be seen , while the filamentary structure of the cloud is still noticeable in general . the clusters themselves have formed along the main filamentary structures in the cloud , and contain @xmath51@xmath52 . note also that the structure of the most massive cluster retains the filamentary shape of its parent cloud , similarly to , for example , the structure observed in the orion a cloud . this similarity in fact is not limited to the filamentary shape , but also includes the fact that the filament has a larger concentration of stars in one end . in a future paper we plan to quantitatively explore the stucture and kinematics of these observed and simulated clusters . the crucial effect of the ghc scenario is that the filaments constitute part of the large - scale gravitational collapse , funneling gas into the cores within them , as observed by @xcite in a numerical simulation of cloud formation and collapse . moreover , these authors observed a hierarchy of collapses within the filaments , so that _ small clumps , which are collapsing locally , and sometimes forming stars already , are themselves falling onto larger - scale ones _ , similarly to the `` conveyor belt '' scenario proposed by @xcite for the gas stream in the central molecular zone . this is illustrated in fig . [ fig : imgs_clus2 ] , which shows a series of snapshots around the second most massive cluster ( hereafter , `` cluster 2 '' , located in the upper left part of the cloud ) among those seen in fig . [ fig : sim_imgs ] , from @xmath53 myr to @xmath54 myr . from @xmath55 18.92 to 19.05 myr it can be seen that new stars have formed in the first group . at @xmath56 myr , a second group is seen to have formed at a distance @xmath57 pc from the first . by @xmath58 myr , this second group is seen to have approached the first , being at a distance @xmath59 pc from it , and to have merged with it by @xmath60 myr . also , at this time , a third group is seen to begin forming in the far left part of the filament , containing only one star at that time , although by @xmath61 myr it already contains several stars ( not shown ) . the effect of the feedback is also worth noting . already by @xmath58 myr , the section of the filament connecting the two clusters is seen to be partially disrupted , and some gas is being expelled from the clump containing the main group . however , accretion along the filament is seen to have replenished this section of the filament at @xmath62 myr , although the other side of the filament is now seen to be in the process of dispersal . finally , by @xmath54 myr , the filament is seen to be in the process of dispersal on both sides of the cluster . figure [ fig : imgs_clus2b ] then shows the late stages of the evolution of this cluster . by @xmath63 myr the filament is seen to have been dispersed out to distances @xmath64 pc , and by @xmath65 myr , a large hii region , of diameter @xmath66 pc has formed , although the group on the left retains some of its gas , and a new group has formed in an `` pillar '' that constitutes the remainder of the filament . also , along this filamentary structure , yet another group is seen to have formed ( at the far right of the image ) . by @xmath67 myr , most of the dense gas has been cleared from the region , and the cluster is seen to consist of four groups , two of them almost devoid of gas ( `` naked '' ) and two of them still embedded in their respective clumps . from the above description , it becomes clear that the clusters in this simulation are assembled by means of a hierarchical process , in which subunits formed at slightly different locations and times merge as they fall into a large - scale potential well , _ feeding the central clump with both stars and gas_. in the following sections we discuss various implications of this mechanism . the resulting evolution of cluster 2 is illustrated in fig.[fig : cluster_evol ] , which shows the projected positions of the stellar particles on the @xmath68 plane at times @xmath69 , 21.44 , 22.40 , and 25.34 myr in the _ top left , top right , bottom left , and bottom right panels _ . in this cluster , stars begin to form at @xmath70 myr and fig . [ fig : cluster_evol ] shows that they are assembled into a cluster in such a way that at @xmath69 myr we identify two subgroups ( hereafter , groups 1 and 2 ; see the _ top left panel _ of fig . [ fig : cluster_evol ] ) of what will later become a larger , merged group ( group 1 - 2 ) . we use square symbols to identify stars that belong to group 1 , and filled circles for stars belonging to group 2 . furthermore , we color them according to their age as shown in the labels and qualitatively represent their masses by the symbol size , with larger symbols indicating larger masses . the green and red circles represent the group radii at each time , computed as explained in sec . [ sec : cluster ] , with the dashed circles representing the size out to the most distant star , and the solid circles representing the distance to the second most distant star . in what follows , we use the distance to the second most distant star as the radius of group 1 - 2 , because the first most distant one is clearly `` running away '' from the group . finally , we use larger symbols to represent more massive stars , and smaller symbols to represent lower - mass stars . thus , even at these early stages , we can see that _ the more massive and younger stars tend to be near the centers of the groups_. from fig . [ fig : cluster_evol ] it is clearly seen that , at these early stages , the groups are undergoing expansion , and that star formation continues within them . by @xmath71 myr , both groups have similar sizes , masses , and number of stars , and they have undergone a large enough expansion that their merger appears inevitable . the formerly `` most distant '' star of group 2 is now far away from the center of either group . by our definition , the merger occurs at @xmath72 myr , forming the new group 1 - 2 . we have kept the circular and square symbols to indicate the original group membership of the stars in the merged group . triangles now represent stars born in the new group 1 - 2 . thus , this new group contains stars that were born at somewhat different times and locations . for later times , we continue to track the evolution of this and other , new groups appearing in a 10-pc box around group 1 - 2 , and by @xmath73 myr , we now find a system of groups with different ages , masses , and sizes ( _ bottom right panel _ of fig . [ fig : cluster_evol ] ) . these groups lie approximately in a straight line , because they formed in the filament feeding the main clump . eventually , as these groups expand and approach each other , they finally become part of what we refer to as the whole cluster 2 , as seen in the _ bottom right panel _ of fig.[fig : imgs_clus2b ] , although the subunits are still partially distinguishable at that time ( @xmath74 myr ) . as mentioned in the introduction , one fundamental implication of the ghc scenario is that the sfr of star - forming clouds must increase in time until the time when massive stars begin to appear and to destroy their parent mc @xcite . figure [ fig : sfr_evol ] shows the evolution of the sfr in groups 1 and 2 , as well as in the merged group 1 - 2 . except for group 2 , in whose parent cloud the sfr starts relatively high and then decreases , for groups 1 and 1 - 2 the trend of increasing sfr is observed . moreover , the sfr for the merged group 1 - 2 is much larger than the maximum value reached in group 2 before the merger , so the increase of the sfr also holds for the system of groups 1 and 2 combined , before and after their merger . assuming that massive stars do not form in a region until the sfr is locally high enough to sample the imf up to large masses , the increase of the sfr implies that the massive stars should tend to appear late in the evolution of star - forming regions . this is illustrated in fig.[fig : age - mass ] , which shows the mass of the stars _ vs. _ their age at @xmath54 and @xmath75 myr . at both times , it is clearly seen that the oldest stars have the lowest masses , and that , as the age decreases , the range of stellar masses extends to larger values . this is also illustrated in fig . [ fig : cum_mass_hist ] , which shows the cumulative mass distribution of groups 1 , 2 , and 1 - 2 at various times . it is seen from this figure that the relative abundance of massive stars increases over time , especially after groups 1 and 2 have merged . the ghc mechanism implies that each star - forming site is locally undergoing collapse while simultaneously it is falling into larger - scale potential wells . moreover , since the small - scale regions are forming stars on their own , this process in turn implies that the larger scales are fed both gas and stars ( cf . [ sec : global_evol ] ) from the surrounding infalling material . also , the small - scale regions fall into the larger - scale ones mostly along filaments @xcite . an important consequence of this `` mixed '' ( gas+stars ) infall is that the ensemble of stars constitutes a non - dissipative `` fluid '' that conserves the kinetic energy of the infalling motions . thus , the stars formed _ before _ reaching the large - scale collapse center should exhibit a velocity dispersion corresponding to their motions as they were infalling to the center of the large - scale potential well . in turn , this implies that older stars should reach somewhat larger distances from the center of a group or cluster that has undergone mergers than the stars formed _ in situ _ in this merged clump , where the gas from which they inherit their velocity dispersion has probably been shocked and thus has lost some of the infall kinetic energy . figure [ fig : age_profile ] shows the age of the stellar particles as a function of their distance to the center of mass of cluster 2 at @xmath76 myr . it is seen that all of the youngest stars ( @xmath77 , and in particular several stars with @xmath78 myr ) , are located within the central 0.5 pc of the cluster . this is qualitatively consistent with the observation that the youngest stars appear tightly clustered around core - like or filamentary star - forming regions . instead , at larger distances ( @xmath79 pc ) , there are no particles with @xmath80 myr . these older stars are the ones that formed in groups 1 and 2 , prior to their merging . this can also be seen in the _ bottom left _ panel of fig . [ fig : cluster_evol ] , which shows that the stars farther from the center of group 1 - 2 are all represented by circles and squares ( i.e. , formed previously to the merger ) and by green and cyan colors , denoting ages between 1 and 4 myr , while the stars clustered near the center have ages @xmath81 myr , as indicated by their black - colored symbols . on the other hand , we also notice in fig . [ fig : age_profile ] that the inner 0.5 pc of the cluster also contains several old stars , and in particular , the oldest ones ( @xmath82 23.5 myr ) . this suggests that some of the stars formed previously to the merger have already had @xmath83-body interactions with the other stars in the region , and thus have transferred their excess kinetic energy to other stars in the cluster . this is facilitated because these old stars have low masses , so that they are strongly affected by interactions with more massive ones . indeed , as discussed in sec . [ sec : sfr ] ( fig . [ fig : age - mass ] ) , the oldest stars have masses near the peak of the imf , which in our case is also the minimum stellar mass ( cf . sec . [ sec : sf ] ) . also , as mentioned in secs . [ sec : intro ] and [ sec : sfr ] , another implication of the ghc is that the sfr increases in each star - forming site until feedback begins to destroy the gas supply onto it , at which point the sfr begins to decline there , and perhaps shut off completely . using a semi - analytic model for the evolution of collapsing clouds and their sfr , @xcite showed that this acceleration of the sf at each star - forming site implies that the age histogram of the site before a few myr of its destruction is characterized by a large peak of very young ( age @xmath84 myr ) and a small tail of older ( up to several myr ) stars , similarly to the age histograms of embedded clusters in various nearby mcs @xcite . figure [ fig : age_hist ] shows the age histogram of the stars in groups 1 and 2 at various times , as the clumps that embed them merge and continue to form stars . at @xmath85 myr , the two clumps and their embedded groups are still separate , and a histogram for each group is shown . at @xmath86 myr , the two clumps and their embedded groups have already merged , and thus a single histogram is shown . it is seen that , at this time , the most numerous stars are those less than 1 myr old . neverthless , there are also a few stars up to nearly 4 myr old . this indicates an acceleration of sf during the first @xmath87 myr of evolution of cluster 2 . at later times ( @xmath88 and @xmath89 myr ) , the number of young stars decreases , while the most numerous stars are those formed around @xmath90 myr . this indicates a decline of the sfr after this time due to the onset of gas dispersal by the feedback . figure [ fig : mass_profile ] shows the radial distribution of the stellar particle masses at @xmath54 myr . it can be seen that all but one of the stars with mass @xmath91 , and all stars with @xmath92 are located within @xmath93 pc of the computed center of the cluster . only one star with @xmath94 is located at @xmath95 pc from the cluster s center of mass . however , close inspection of the time sequence outlined by the various images of fig . [ fig : cluster_evol ] shows that this is a runaway star formed in , and ejected from , group 1 before the merger . this star is represented by the cyan squared symbol lying on the green solid circle in the _ top right _ panel [ @xmath96 myr ] of fig.[fig : cluster_evol ] and on the solid orange circle in the _ bottom left _ [ @xmath97 myr ] panel . thus , this star is just `` flying by '' the cluster . instead , the other massive stars in group 1 - 2 have formed _ in situ _ , as indicated by the triangular shape of their symbols in fig . [ fig : cluster_evol ] . also , the other massive stars are represented by symbols of white and cyan colors , and with triangular shapes , indicating that those stars are younger ( @xmath98 myr ) , and formed _ in situ _ in the merged group 1 - 2 . thus , we conclude that , except for the star that is `` passing by '' , the massive stars have formed after the merger of groups 1 and 2 to form group 1 - 2 , thus being younger and tightly clustered near the group center . another feature of the cluster evolution seen in figs.[fig : age_profile ] and [ fig : mass_profile ] is that the cluster expands as it evolves , increasing its radius from @xmath99 pc at @xmath54 myr to @xmath100 pc at @xmath101 myr . the reason for this is not totally clear . it can be due in part to the fact that the gaseous material is beginning to be expelled from the clump , as can be seen in figs . [ fig : imgs_clus2 ] and [ fig : imgs_clus2b ] , and in part to the fact that the velocity dispersion of the older stars , formed in the scattered ssla regions that have fallen into the lssa one , corresponds to the large - scale potential well , not that of the local star - forming site ( cf . [ sec : age_distr ] ) , causing the cluster to expand to a size representative of this somewhat larger energy . we now discuss the hierarchical structure of the cluster itself , resulting from the process of ghc of its parent cloud . figure [ fig : hier_clus ] shows the stars belonging to the cluster at time @xmath74 myr , first as a whole , and then as identified by a friends - of - friends algorithm using three different values of the `` linking parameter '' @xmath102 , which determines how far neighbours are searched in order to identify a group . we see that , using a small value of the parameter ( @xmath103 , _ top right _ panel ) , 10 tight groups are identified ( denoted by the different colors ) , while using larger values ( @xmath104 and 2 , _ bottom left _ and _ bottom right _ panels ) , respectively 9 and 4 are identified , each group being significantly more extended . we also note that , at the lowest value , @xmath105 , a whole group of moderately scattered stars is left out , which is nevertheless identified as a group at higher values . these results show that the structure of the cluster resulting from the global , hierarchical contraction of the cloud , is inherently `` nested '' , consisting of structures within structures within structures , reflecting the structure of the parent cloud , which in turn is a consequence of its multi - scale , hierarchical collapse . we can summarise the discussion in the previous section as follows . the underlying assumption for interpreting the emerging properties of the clusters formed in the scenario of ghc is that first proposed in @xcite , namely that massive stars only form in sites where the local sfr is high enough that the imf can be sampled to correspondingly large stellar masses . conversely , the most common stars in low - sfr sites should be those at the peak of the imf . this is , of course , a fully probabilistic interpretation of the imf , and it does not address the physical mechanisms responsible for the development of the imf itself . however , it allows a framework for understanding the assembly of the clusters . also , under the assumption that structures of all scales are collapsing , as required by the ghc , at a given density ( and therefore , a given free - fall time ) , more massive clumps are expected to have higher sfrs , if the sfr is given , to first order , by the ratio of the clump mass to its free - fall time . our simulation mimics these hypotheses , due to the probabilistic scheme used for the formation of a stellar particle . the longer it takes for a particle to form , the more massive it will be , since the cell in which it forms continues to accrete and becomes more massive , without further refinement , and then the particle forms with half the mass of the cell . but , as shown in figure 1 of paper i , the probability of being able to wait very long times before forming a stellar particle is very low , and so the probability of forming of a very massive star is also very low . thus , our simulation is expected to generally only form massive stars when the local sfr is very high , although there can be a few exceptions to this pattern . finally , in the ghc scenario , the low - mass , small - scale but large - amplitude ( ssla ) star - forming sites should appear earlier in the evolution of the cloud , because of their shorter free - fall times , while the high - mass , larger - scale , smaller - amplitude ( lssa ) sites should appear later . moreover , the low - mass sites should be falling into the trough of the gravitational potential well of the high - mass ones . however , low - mass sites continue to appear at all times in the filaments that feed the massive clumps until the filaments are destroyed by the feedback . thus , the low mass stars formed in the low - mass , infalling clumps share the clumps infall velocity , and thus should have larger velocity dispersions than the stars formed in the main , massive clump , which can have both low or high masses . the above considerations imply a number of properties of the resulting clusters : 1 . the oldest stars tend to have low masses . however , the converse is not necessarily true : low - mass stars in general can be either young or old . this trend is shown in the _ top _ panel of fig . [ fig : age - mass ] , which shows the mass _ vs. _ the age of the stars in the cluster at times @xmath97 and 23.66 myr . the oldest stars have larger velocities ( fig . [ fig : v - age ] , _ left _ panel ) , and consequently , they may be encountered at larger distances from the local star - forming site ( fig . [ fig : age_profile ] ) . the most massive stars appear later ( a few myr ) in the evolution of the clump ensemble , when various ssla sites have merged to form a lssa site . this is shown in figure [ fig : cum_mass_hist ] , which gives the cumulative stellar mass histogram for groups 1 and 2 , independently and combined , at various times . as time proceeds , progressively larger fractions of the stars are seen to be massive . 4 . the sfr increases during the early stages of the cluster assembly , until the time when massive stars begin to form . after this time , the sfr decreases , as the ionising feedback from the massive stars begins to destroy the clump and its filamentary gas supply . this is seen in the evolution of the age histogram , shown in fig . [ fig : age_hist ] . the oldest stars constitute a minority , since the sfr initially increases with time . however , after the appearance of ob stars that can partially or completely destroy the local clump and its supplying filaments , the sfr decreases again , and thus stars younger than the age of the oldest massive stars should be less abundant as well , as also shown in fig . [ fig : age_hist ] . note that these properties refer to regions of sizes a few parsecs and ages of at least a few myr , and so it specifically does _ not _ refer to the local and instantaneous stellar production in a single clump , for which our simulation can not discern whether massive stars form first or last , or how long does it take to build a massive star . also , note that young star - forming regions may consist of several low - mass local star - forming sites , and contain no large - scale site yet . this situation corresponds to times where the first low - mass sites have not yet merged to form a more massive one . our numerical results are qualitatively consistent with recent observational results on the age and mass distributions of young stellar objects in star - forming regions . in particular , @xcite have recently reported the existence of age gradients in the stellar populations of various massive star - forming regions , such that the youngest stellar objects are located in the obscured regions of molecular clouds , intermediate - age objects are located in revealed clusters , and the oldest objects are found in distributed populations . this is qualitatively consistent with our result that the oldest objects can be found at large distances from the cluster s center due to their larger velocity dispersion , in turn due to their formation far from it , so that their velocities include the component from their infall onto the central massive clump . instead , younger objects formed in the clump lack this component of the velocity , and these are characterised by somewhat lower velocites . also , the larger velocity dispersion of the older objects predicted by the ghc scenario has been observed by @xcite in the orion a cloud . @xcite considered several possible scenarios that could lead to the age segregation they observed . of these , the one closest to the mechanism of hierarchical assembly by ghc is their scenario b , since it invokes the acceleration of sf as proposed by @xcite , which is consistent with our scenario @xcite . however , the similarity between ghc and scenario b of @xcite is not complete , since they did not consider the accretion of both stars and gas onto the trough of the large - scale potential well . in addition , @xcite have concluded that the cloud complex known as the m17 southwest extension ( m17 swex ) appears to be lacking very massive stars in comparison with the number expected from its estimated sfr . this region consists of several scattered star - forming cores , separated by distances of several parsecs from each other ( m.povich , private communication ; see also busquet et al . this is consistent with the hierarchical assembly of the cluster in our simulation , since several isolated , intermediate - mass star - forming clumps may produce a relatively numerous combined population of ysos that may however not yet have formed massive stars , because in the ghc scenario this only occurs after these smaller regions have merged to form a massive one . this is again illustrated in fig.[fig : cum_mass_hist ] , which shows that the relative abundance of massive stars increases over time , especially after groups 1 and 2 have merged . we conclude that the scenario of ghc provides an assembly mechanism for clusters that naturally predicts the observed age segregation and bottom - heavy imf of young clusters . in a future contribution we plan to carry out a detailed quantitative comparison between the clusters in simulations that sample the imf to lower masses ( see sec.[sec : limits ] ) and the observed properties of specific clusters . the present study is based on the underlying hypothesis that our probabilistic stellar particle formation scheme mimics the mechanism leading to the development of the imf . that is , we interpret the imf strictly as a random sampling process , so that the scarcity of massive stars is interpreted to imply that the probability of forming one is very low and given precisely by the imf . interestingly , the fact that our simulation , using a probabilistic criterion for the formation of stellar particles ( cf . sec . [ sec : sf ] ) , qualitatively reproduces several observed properties of young clusters , suggests that our underlying assumption is reasonably realistic . nevertheless , our star formation scheme does have some limitations . most importantly , as described in sec . [ sec : sf ] , the minimmum stellar - particle mass is @xmath106 , which is near the peak of the observed stellar imf , and thus we are missing stars on the low - mass side of the imf . our imposed imf is thus a strict power law , that lacks the turnover at low masses . this implies that we are missing a significant number of low - mass stars , about half of the total number of stars . their mass , @xmath107 of the total mass for a @xcite imf , and @xmath108 for a @xcite imf , is instead deposited into the stars that do form in the simulation , with masses @xmath106 to @xmath109 . this may affect the dynamics of the clusters formed in the simulation , although not very strongly , since the missing stars have low masses , and are not expected to affect the dynamics of the higher - mass ones too much , especially if the tend to form a roughly uniform background . the fact that the structure of our clusters resembles that of observed young clusters reinforces this expectation . we nevertheless plan to improve our star formation scheme in order to produce a more complete stellar - particle mass spectrum , and with this carry out a more detailed and quantitative comparison with observations . meanwhile , we consider that our study conveys an illustrative first approximation to the problem of cluster assembly , capturing the essential aspects of the hierarchical assembly of these objects . other limitations of our simulation are that it does not include magnetic fields nor any form of feedback other than ionisation heating , in particular supernova ( sn ) explosions . the neglect of magnetic fields may not be crucial , since our present understanding is that mcs are generally magnetically supercritical , meaning that the magnetic field is in general unable to support them against their own self - gravity , and therefore they proceed unimpeded to global collapse . the neglect of sn explosions is also not crucial because of two main reasons . first , it is becoming increasingly accepted that sn explosions need to occur within the clouds in order to remove significant amounts of mass from the clouds @xcite , although on the other hand , only sne that explode in low - density environments avoid energy losses that allow efficient coupling with the gas @xcite . also , pre - sn feedback is required to allow blastwaves to propagate efficiently into the ism @xcite . second , and more importantly , our study suggests that the assembly of the clusters is accomplished mostly during the early stages ( first few myr ) of the clouds evoloution , during which few or no sne are expected to occur , especially if massive stars form late in the evolution of the cloud , as suggested by the ghc scenario . so , the ionising feedback included in our simulations is probably the most relevant form of feedback for regulating the gas flow onto the sites where the new stars are forming . finally , our feedback prescription ( sec . [ sec : feedback ] ) is of course crude , solving the radiative transfer only in an approximate way . although it was shown in paper i that the evolution of an hii region with this prescription does track the uniform - medium analytic solution reasonably well , it is possible that our prescription allows evaporation of regions that should be shaded behind dense clumps , since it does not take into account the density of the intervening material between the ionising star and the cell on which the effect of the radiation is being determined . thus , our simulations may somewhat overestimate the evaporation rate of material in filaments containing dense clumps . however , since the filaments themselves are denser than the background medium , and their length far exceeds that of isolated clumps and cores , their column density for situations where the filament is aligned with the ionising object and with the intervening clump ( as in the case of filaments feeding the main clump where the ionising object formed ) may be comparable to or even larger than that of the intervening clump . therefore , the shadowing caused by this clump is likely to be of secondary importance . indeed , figure 10 in paper i shows the gradual evaporation of filaments containing clumps in a way that allows the formation of `` pillars '' , and finally isolated globules within growing hii regions , similarly to objects like the `` pillars of creation '' and dark globules , respectively . on the other hand , our feedback prescription allows the simulations to run orders of magnitude faster than if the radiative transfer is followed in full . we thus conclude that our prescription provides a sufficiently realistic framework for the study of the assembly of stellar clusters . in this paper , we have discussed the mechanism of assembly of a cluster in a numerical simulation of cloud formation and collapse by converging flows . the assembly proceeds in a hierarchical way as a consequence of the hierarchical nature of the collapse of its parent cloud @xcite . the hierarchical collapse regime consists of small - scale , large - amplitude ( ssla ) collapses ( involving small total masses ) within large - scale , small - amplitude ( lssa ) ones ( involving large masses , which are spread out over large scale regions ) . collapse at all scales involves accretion onto the troughs of the potential wells ( the collapse `` centers '' ) . the large - scale collapses involve the generation of filamentary flows that funnel material from the whole large - scale region down to its collapse center , and the small - scale collapsing regions `` ride '' along these large - scale filamentary flows @xcite , in a `` conveyor belt '' fashion @xcite . because the small - scale collapses start forming stars earlier ( due to their larger amplitudes , which imply shorter free - fall times ) , they often reach the large - scale collapse center already containing stars in addition to fresh gas . this mechanism then implies that massive star - forming sites form by the merging of smaller - scale star - forming regions , which supply both stars and gas to the larger sites , and thus these larger - scale sites contain a mixture of locally formed stars and stars that formed at a remote , smaller - scale site and that are brought to the large - scale site by the infalling flow . moreover , the stars that form in the small - scale , scattered sites have an infall velocity corresponding to the larger - scale potential well . however , stars formed in the massive center of the large - scale collapse , form from gas that has probably been shocked upon its arrival there , and so they form with a somewhat lower velocity dispersion . finally , this scenario also implies that the sfr must increase with time during early stages of evolution , and then decrease again as the massive stars begin to destroy their forming sites @xcite . this acceleration of the sf process , combined with a face - value interpretation of the imf as a probability distribution function of stellar masses , suggests that massive stars are only expected to form once the sfr is large enough to sample the imf to high masses . therefore , massive stars should form last in the evolution of the whole ensemble of star - forming sites , once the low - mass ones have merged to form a large - mass , high - sfr one . from these considerations , we have concluded that a cluster that has formed from the hierarchical collapse of its parent cloud must have the following characteristics : * the oldest stars tend to have low masses , because they formed in the low - mass sites early in the evolution of the cloud . however , newer stars can be either low - or high mass , as they form in the high - mass site formed by the merger of the low - mass ones . * the oldest stars tend to have larger velocity dispersions , because they are characterized by the infall velocity of the material onto the high - mass collapse center . however , this effect tends to be washed out by the stellar interaction as the cluster ages . * as a consequence , older stars tend to be at larger distances from the collapse center ( or filament ) . * the most massive stars tend to be younger , because they only form when the sfr of the whole region has increased sufficiently to sample the high - mass regions of the imf . * as a consequence , the low- and intermediate - mass sites that have not concluded the merging process may be defficient in the highest - mass stars . * because of the increasing sfr , most stars are young , although a small fraction may be as old as several myr . these features are in good qualitative agreement with observed properties of young clusters , which seem to have a hierarchical structure , have an age gradient , so that the oldest stars tend to be farther from the forming sites @xcite and to have larger velocities @xcite . also , the scenario implies that ensembles of cores in early phases of evolution will be defficient in massive stars compared to the expected number form their combined stellar production , as observed by @xcite . we conclude that the scenario of ghc allows a clear understanding of the properties of young clusters , in terms of the multi - scale collapse of their parent clouds . e.v .- s . is glad to acknowledge enlightening discussions with matt povich on the observed structure of young clusters . was supported by uc - mexus fellowship .

we discuss the mechanism of cluster formation in a numerical simulation of a molecular cloud ( mc ) undergoing global hierarchical collapse ( ghc ) , to understand how the gas motions in the parent cloud control the assembly of the cluster . the global nature of the collapse implies that the star formation rate ( sfr ) increases over time . the `` hierarchical '' nature of the collapse consists of small - scale collapses within larger - scale ones . the large - scale collapses culminate a few myr later than the small - scale collapses and consist of filamentary flows that accrete onto massive , dense central clumps . in turn , the small - scale collapses form clumps embedded in the filaments , that are falling onto the central clump assembled by the ongoing large - scale collapse . the stars formed in the early , small - scale collapses share the infall motion of their parent clumps towards the larger - scale potential well , so that the filaments feed both gaseous and stellar material to the massive central clump . this leads to the presence of a few older stars in a region where new protostars are forming at a higher rate , and also to a self - similar or fractal - like structure of the clusters , in which each unit is composed of smaller - scale sub - units , which approach each other and may eventually merge , explaining the frequently - observed morphology of cluster - forming regions . moreover , because the older stars formed in the filaments share the infall motion of the gas onto the central clump , they tend to have larger velocities and to be distributed over larger areas than the younger stars formed in the central clump , where the gas from which they form has been shocked and has dissipated some kinetic energy . finally , interpreting the imf as a probability distribution , so that the probability of forming a massive star is much lower than that of forming a low - mass one , implies that massive stars only form once the _ local _ sfr is large enough to sample the imf up to high masses . in combinaton with the increase of the sfr , this implies that massive stars tend to appear late in the evolution of the mc , and only in the central massive clumps . we discuss the correspondence of these features with observed properties of young stellar clusters , finding very good qualitative agreement , thus providing support to the scenario of global , hierarchical collapse of mcs , while explaining the origin of the observed cluster structure . [ firstpage ] galaxies : star clusters , gravitation , hydrodynamics , ism : clouds , stars : formation

1611.00088

as in our previous analysis of run 1a data @xcite , we conduct a general search for new particles with a narrow natural width that decay to dijets . in addition , we search for the following particles summarized in fig . [ fig_particles ] : axigluons @xcite from chiral qcd ( @xmath7 ) , excited states @xcite of composite quarks ( @xmath8 ) , color octet technirhos @xcite ( @xmath9 ) , new gauge bosons ( @xmath10,@xmath11 ) , and scalar @xmath12 diquarks @xcite ( @xmath13 and @xmath14 ) . using four triggers from run 1a and 1b , we combine dijet mass spectra above a mass of 150 gev / c@xmath1 , 241 gev / c@xmath1 , 292 gev / c@xmath1 , and 388 gev / c@xmath1 with integrated luminosities of = 7.5 in = 3.3 in = 3.3 in .089 pb@xmath0 , 1.92 pb@xmath0 , 9.52 pb@xmath0 , and 69.8 pb@xmath0 respectively . jets are defined with a fixed cone clustering algorithm ( r=0.7 ) and then corrected for detector response , energy lost outside the cone , and underlying event . we take the two highest @xmath15 jets and require that they have pseudorapidity @xmath16 and a cms scattering angle @xmath17| < 2/3 $ ] . the @xmath18 cut provides uniform acceptance as a function of mass and reduces the qcd background which peaks at @xmath19 . in fig . [ fig_dijet ] the dijet mass distribution is presented as a differential cross section in bins of the mass resolution ( @xmath20% ) . at high mass the data is systematically higher than a prediction from pythia plus a cdf detector simulation , similar to the inclusive jet @xmath21 spectrum @xcite . to search for new particles we determine the qcd background by fitting the data to a smooth function of three parameters @xcite ; fig . [ fig_dijet ] shows the fractional difference between the data and the fit ( @xmath22 ) . we note upward fluctuations near 200 gev / c@xmath1 ( @xmath23 ) , 550 gev / c@xmath1 ( @xmath24 ) and 850 gev / c@xmath1 ( @xmath25 ) . for narrow resonances it is sufficient to determine the mass resolution for only one type of new particle because the detector resolution dominates the width . in fig . [ fig_resonance ] we show the mass resolution for excited quarks ( q * ) from pythia plus a cdf detector simulation ; the long tail at low mass comes from gluon radiation . for each value of new particle mass in 50 gev / c@xmath1 steps , we perform a binned maximum likelihood fit of the data to the background parameterization and the mass resonance shape . in fig . [ fig_resonance ] we display the best fit and 95% confidence level upper limit for a 550 gev / c@xmath1 resonance . for the mass region @xmath26 gev / c@xmath1 , there are 2947 events in the data , @xmath27 events ( @xmath28 ) in the background for the fit without a resonance , @xmath29 events ( @xmath30 ) in the background for the fit that includes the resonance , and the value of the resonance cross section from the fit is @xmath31 pb ( statistical ) . in fig . [ fig_cos ] we study the angular distribution of the fluctuation in the mass region @xmath26 gev / c@xmath1 . the angular distribution is compatible with both qcd alone , and with = 3.3 in = 3.3 in qcd + 5% excited quark ( best fit ) . this amount of excited quark is coincidentally the same as found in the mass fit . although the fluctuation is interesting , we conclude it is not yet = 3.3 in = 3.3 in statistically significant , and proceed to set limits on new particle production . = 3.3 in = 3.3 in from the likelihood distribution including experimental systematic uncertainties @xcite we obtain the 95% cl upper limit on the cross section for new particles shown in fig . [ fig_dijet_limit ] . we compare this to the cross section for axigluons ( excluding @xmath32 gev / c@xmath1 ) , excited quarks ( excluding @xmath33 gev / c@xmath1 ) , technirhos ( excluding @xmath34 gev / c@xmath1 ) , w@xmath35 ( excluding @xmath36 gev / c@xmath1 ) , z@xmath35 ( excluding @xmath37 gev / c@xmath1 ) , and e6 diquarks ( excluding @xmath38 gev / c@xmath1 ) . the calculations are lowest order @xcite using cteq2l parton distributions @xcite and one - loop @xmath39 and require @xmath16 and @xmath40 . the large mass of the top quark suggests that the third generation may be special . topcolor @xcite assumes that the top mass is large mainly because of a dynamical @xmath5 condensate generated by a new strong dynamics coupling to the third generation . here the @xmath41 of qcd is a low energy symmetry arising from the breaking of an @xmath42 coupling to the third generation and an @xmath43 coupling to the first two generations only . there are then massive color octet bosons , topgluons @xmath44 , which couple largely to @xmath4 and @xmath5 . the topgluon is strongly produced and decays mainly to the third generation ( @xmath45 ) with a relatively large natural width ( @xmath46 ) . here we search for the topgluon in the @xmath4 channel . an additional @xmath47 symmetry is introduced @xcite to keep the @xmath48 quark light while the top quark is heavy ; this leads to a topcolor @xmath6 , which again couples largely to @xmath4 and @xmath5 . the topcolor @xmath6 is electroweakly produced and decays mainly to the third generation ( @xmath49 ) with a narrow natural width ( @xmath50 ) . here we search for the topcolor @xmath6 in both the the @xmath4 and @xmath5 channel ; the @xmath5 channel is the most sensitive because the coupling to @xmath5 is larger . we start with the dijet search in 19 pb@xmath0 of run 1a data @xcite and additionally require at least one of the two leading jets be tagged as a bottom quark . the b - tag requires a displaced vertex in the the secondary vertex detector @xcite . the @xmath4 event efficiency is @xmath51% independent of dijet mass . from fits to the @xmath52 distribution , we estimate that the sample is roughly 50% bottom , 30% charm , and 20% mistags of plain jets . pythia predicts that 1/5 of these bottom quarks are direct @xmath4 , and the rest are from gluon splitting and flavor excitation . consequentially , only about @xmath53% @xmath54 10% of our sample is direct @xmath4 . we expect both the purity and efficiency to increase when we use the run 1b dataset and a new tagging algorithm @xcite . with higher tagging efficiency we should be able to make better use of double b - tagged events like the one in fig . [ fig_btag_event ] . = 2.6 in = 2.6 in in fig . [ fig_btag ] we show the b - tagged dijet mass distribution corrected for the @xmath4 efficiency . also shown is the untagged dijet mass distribution from run 1a , and both are well fit with our standard parameterization @xcite . the b - tagged dijet data has an upward fluctuation near 600 gev / c@xmath1 . we model the shape of a narrow resonance using pythia z@xmath3 production and a cdf detector simulation . in fig . [ fig_btag ] we fit the b - tagged data to a 600 gev / c@xmath1 narrow resonance , and find a cross section of @xmath55 pb ( statistical ) . note that this is comparable to the dijet fluctuation in both mass and rate . however , there are only 8 events in the last two data bins of fig . [ fig_btag ] , and the fluctuation is only a @xmath56 effect , so we proceed to set limits on new particle production . we perform two kinds of fits for the limits . first , narrow resonances are modelled as described above , and the mass resolution in the cdf detector is shown in fig . [ fig_b_bbar_res ] . second , wide resonances characteristic of topgluons @xcite , including interference with normal gluons , was incorporated into pythia and a cdf detector simulation . the mass resolution in fig . [ fig_b_bbar_res ] displays destructive interference to the left of the resonance ; models with destructive interference on the right side of the resonance will be considered in the future . = 3.2 in = 3.2 in limits on new particle production are shown in fig . [ fig_btag_limit ] . the theoretical cross sections are lowest order and use cteq2l parton distributions . for narrow resonances the production cross sections are nt large enough for us to set mass limits at this time . for topgluons the production cross sections @xcite are larger , and we are able to exclude at 95% cl topgluons of width @xmath57 in the mass region @xmath58 gev / c@xmath1 , @xmath59 for @xmath60 gev / c@xmath1 , and @xmath61 for @xmath62 gev / c@xmath1 . = 3.0 in = 3.0 in = 7.7 in to search for new particles decaying to @xmath5 we start with the data sample from the top mass measurement @xcite . there we used top decays to w + four jets with at least one b - tag , and found 19 events on a background of @xmath63 , resulting in a top mass of @xmath64(stat)@xmath65(sys ) gev / c@xmath1 . that analysis fit the entire event for the top hypothesis , discarding events with @xmath66 ( poor fit ) . here we add the additional constraint that the top mass is 176 gev / c@xmath1 , which significantly enhances our resolution of the @xmath5 mass . two of the 19 events fail the @xmath66 cut when the top mass constraint is added to the fit , leaving us with 17 events . the @xmath5 mass distribution expected from a narrow resonance , normalized to the topcolor @xmath6 predicted rate @xcite , is shown in fig . [ fig_ttbar ] . here we used pythia @xmath67 . also in fig . [ fig_ttbar ] is the monte carlo distribution of the background , on the left standard model top production from herwig , and on the right qcd w + jets background from vecbos with parton showers from herwig . all monte carlos include a cdf detector simulation . on the left in fig . [ fig_ttbar ] , the comparison of the topcolor z@xmath3 to sm @xmath5 simulations illustrates that in this data sample we are sensitive to topcolor @xmath6 up to a mass of roughly 600 gev / c@xmath1 . finally , on the right in fig . [ fig_ttbar ] , we present the @xmath5 candidate mass distribution from cdf compared to the total standard model prediction . given the statistics the agreement is quite good overall . the small shoulder of 6 events on a background of @xmath68 in the region @xmath69 gev / c@xmath1 is in an interesting mass region , given the dijet and @xmath4 search results , but is not statistically significant . upper limits on the @xmath5 cross section as a function of @xmath5 mass , and on a topcolor @xmath6 , are currently in progress . = 3.1 in = 3.1 in we have searched for new particles decaying to dijets , @xmath4 , and @xmath5 . in the dijet channel we set the most significant direct mass exclusions to date on the hadronic decays of axigluons ( excluding @xmath70 gev / c@xmath1 ) , excited quarks ( excluding @xmath71 gev / c@xmath1 ) , technirhos ( excluding @xmath34 gev / c@xmath1 ) , w@xmath3 ( excluding @xmath36 gev / c@xmath1 ) , z@xmath35 ( excluding @xmath37 gev / c@xmath1 ) , and for the first time e6 diquarks ( excluding @xmath38 gev / c@xmath1 ) . in the @xmath4 channel we set the first limits on topcolor , excluding a model of topgluons for width @xmath57 in the mass region @xmath58 gev / c@xmath1 , @xmath59 for @xmath60 gev / c@xmath1 , and @xmath61 for @xmath62 gev / c@xmath1 . the search for topcolor in the @xmath5 channel has just begun and limits are in progress . limits are only a consolation prize ; the main emphasis of our search is to explore the possibility of a signal . although we do not have significant evidence for new particle production , the @xmath72 gev / c@xmath1 region shows upward fluctuations in all three channels . we can not ignore the exciting possibility that these apparently separate fluctuations may be the first signs of a new physics beyond the standard model . the remaining integrated luminosity for run 1b , currently being accumulated and analyzed , has the potential to either kill the fluctuations or reveal what may be the most interesting new physics in a generation . + f. abe et al . ( cdf ) , phys . lett . * 74 * , 3538 ( 1995 ) . p. frampton and s. glashow , phys . b190 * , 157 ( 1987 ) . u. baur et al . , int . j. mod . phys a2 , 1285(1987 ) & pr*d42 * , 815(1990 ) . k. lane et al . , pr*d44 * , 2768(1991 ) & phys . lett . * b327 * , 129(1994 ) . j. hewett and t. rizzo , phys . rep . * 183 * , 193 ( 1989 ) . talk by anwar bhatti in these proceedings . parameterization @xmath73 with parameters a , n and p. for new gauge bosons we use a k - factor to account for higher order terms . j. botts et al . ( cteq collaboration ) phys . lett . * b304 * , 159 ( 1993 ) . c. hill and s. parke , phys . rev . * d49 * , 4454 ( 1994 ) . c. hill , phys . b345 * , 483 ( 1994 ) . f. abe et al . ( cdf ) , phys . rev . * d50*,2966 ( 1994 ) . f. abe et al . ( cdf ) , phys . lett . * 74 * , 2626 ( 1995 ) . we model the interference between normal gluons and topgluons using hybrid model c in phys . rev . * d49 * , 4454 ( 1994 ) . g. burdman , c. hill , and s. parke private communication .

we present three searches for new particles at cdf . first , using 70 pb@xmath0 of data we search the dijet mass spectrum for resonances . there is an upward fluctuation near 550 gev / c@xmath1 ( 2.6@xmath2 ) with an angular distribution that is adequately described by either qcd alone or qcd plus 5% signal . there is insufficient evidence to claim a signal , but we set the most stringent mass limits on the hadronic decays of axigluons , excited quarks , technirhos , w@xmath3 , z@xmath3 , and e6 diquarks . second , using 19 pb@xmath0 of data we search the b - tagged dijet mass spectrum for @xmath4 resonances . again , an upward fluctuation near 600 gev / c@xmath1 ( 2 @xmath2 ) is not significant enough to claim a signal , so we set the first mass limits on topcolor bosons . finally , using 67 pb@xmath0 of data we search the top quark sample for @xmath5 resonances like a topcolor @xmath6 . other than an insignificant shoulder of 6 events on a background of 2.4 in the mass region 475 - 550 gev / c@xmath1 , there is no evidence for new particle production . mass limits , currently in progress , should be sensitive to a topcolor z@xmath3 near 600 gev / c@xmath1 . in all three searches there is insufficient evidence to claim new particle production , yet there is an exciting possibility that the upward fluctuations are the first signs of new physics beyond the standard model . fermilab - conf-95/152-e + cdf / pub / exotic / public/3192 + + * search for new particles decaying to dijets , + @xmath4 , and @xmath5 at cdf + * _ fermilab ms 318 + batavia , il 60510 + _

hep-ex9506008

the vast majority ( perhaps all ) of stars are formed in a clustered fashion . however , only a very small percentage of older stars are found in bound clusters . these two observations highlight the importance of clusters in the star - formation process and the significance of cluster disruption . the process of cluster disruption begins soon after , or concurrent with , cluster formation . @xcite found that @xmath3 of stars formed in embedded clusters end up in bound clusters after @xmath4 yr . @xcite and @xcite have shown that at least 20% , but perhaps all , star formation in the merging antennae galaxies is taking place in clusters , the majority of which are likely to become unbound . the case is similar in m51 , with @xmath5 of all young ( @xmath6 myr ) clusters likely to be destroyed within the first 10s of myr of their lives @xcite . on longer timescales , @xcite and @xcite noted a clear lack of older ( @xmath7 few gyr ) open clusters in the solar neighbourhood and @xcite found a strong absence of older clusters in m51 , m33 , smc , and the solar neighbourhood . the lack of old open clusters in the solar neighbourhood is even more striking when compared with the lmc , which contains a significant number of ` blue globular clusters ' with ages well in excess of a gyr ( e.g. @xcite ) . this difference can be understood either as a difference in the formation history of clusters or as a difference in the disruption timescales . this later scenario was suggested by @xcite , who directly compared the age distribution of galactic open clusters and the smc cluster population . he noted that there are @xmath8 times more clusters with an age of 1 gyr in the smc as compared to the solar neighbourhood ( when normalising both populations to an age of @xmath9 yr ) and concluded that disruption mechanisms must be less efficient in the smc . much theoretical work has gone into the later scenario , with both analytic and numerical models of cluster evolution predicting a strong influence of the galactic tidal field on the dissolution of star clusters ( for a recent review see @xcite ) . only recently has there been a large push to understand cluster disruption from an observational standpoint in various external potentials , making explicit comparison with models @xcite . we direct the reader to the review by larsen in these proceedings for a historical look at the observations and theory of cluster disruption . while cluster disruption is a gradual process with several different disruptive agents at work simultaneously , one can distinguish three general phases of cluster mass loss and disruption . as we will see , a large fraction of clusters gets destroyed during the _ primary _ phase . the main phases and corresponding typical timescales of cluster disruption are : _ i ) infant mortality _ ( @xmath0 yr ) , _ ii ) stellar evolution _ ( @xmath1 yr ) and _ iii ) tidal relaxation _ ( @xmath2 yr ) . during all three phases there are additional tidal external perturbations from e.g. giant molecular clouds ( gmcs ) , the galactic disc and spiral arms that heat the cluster and speed up the process of disruption . however , these external perturbations operate on longer timescales for cluster populations and so are most important in phase iii . in fig . [ fig0 ] we schematically illustrate the three phases of disruption and the involved time - scales . note that the number of disruptive agents decreases in time . in this review we will focus on the physics and observations of phase i as well as on recent population studies aimed at understanding phases ii and iii on a statistical basis . for a recent review on the physics of phases ii and iii , we refer the reader to @xcite . before proceeding , it is worthwhile to consider our definition of a cluster . @xcite defines a cluster to be a _ gravitationally bound _ stellar association which will survive for 1020 crossing times . this definition implies that the stars provide enough gravitational potential to bind the cluster and ignores the role of gas in the early evolution of clusters . in this review , we will define a cluster as a collection of gas and stars which was _ initially gravitationally bound_. the reason for this definition will become evident in section [ infantmortality ] recent studies on the populations of young star clusters in m51 @xcite and the antennae galaxies @xcite have shown a large excess of star clusters with ages less than @xmath1010 myr with respect to what would be expected assuming a constant cluster formation rate . the fact that open clusters in the solar neighbourhood display a similar trend @xcite has led to the conclusion that this is a physical effect and not simply that we are observing these galaxies at a special time in their star - formation history . if one adopts this view , then we are forced to conclude that the majority ( between 60 - 90% ) of star clusters become unbound when the remaining gas ( i.e. gas that is left - over from the star formation process ) is expelled . these clusters will survive less than a few crossing times . suppose that a star cluster is formed out of a sphere of gas with an efficiency @xmath11 , where @xmath12 . further suppose that the gas and stars are initially in virial equilibrium . if we define the virial parameter as @xmath13 , with @xmath14 the kinetic energy and @xmath15 the potential energy , virial equilibrium implies @xmath16 . finally , suppose that the remaining gas is removed on a timescale faster than the crossing time of stars in the cluster . in such a scenario the cluster is left in a super - virial state after the gas removal , with @xmath17 , and the star cluster will expand since the binding energy is too low for the stellar velocities . the expanding cluster will reach virial equilibrium after a few crossing times , but only after a ( possibly large ) fraction of the stars have escaped . this process has been shown to remove a significant amount of the stellar mass of a cluster , and if @xmath18 the entire cluster will become unbound on a timescale of 10s of myr @xcite . rapid gas removal of the type discussed above leaves distinct observables . in figure [ fig1 ] we show the surface brightness profiles of three young clusters ( left panels ) as well as two results of @xmath19-body simulations ( right panels ) of clusters including the effects of rapid gas removal . all three young clusters show an excess of light at large radii with respect to the best fitting eff @xcite or @xcite profiles . this is in good agreement with the predictions of the simulations , in which an unbound halo of stars is removed ( although still appearing to be associated with the cluster for 10s of myr ) due to the rapid change of the gravitational potential @xcite . such excess light at large radii has also been found in young clusters in the antennae galaxies @xcite . @xcite show that for values of @xmath11 of 0.1 and 0.6 , clusters will lose 75% and 10% of the stellar mass respectively within the first @xmath20 myr of their lives . thus we see that this is an extremely efficient way to rapidly disperse stars from young clusters into the field . this mechanism provides a natural explanation for the observed diffuse uv light in the field of starburst galaxies @xcite and supports the scenario of these authors that this light is due to rapidly dispersing young clusters . whether or not a cluster survives this phase , and hence more than 1020 crossing times , is largely dependent on the star - formation efficiency of the gmc core in which the cluster formed . thus , two clusters with exactly the same parameters ( radius , mass , metallicity , external potential field , etc ) may experience two radically different evolutionary paths if their star - formation efficiencies are different . @xcite have used the internal dynamical properties of young clusters in order to estimate their @xmath11-values . no clear trend of @xmath11 on cluster ( stellar ) mass or radius was found . -body simulations which include the rapid removal of gas which was left over from a non-100% star - formation efficiency ( right ) . the solid ( red ) and dashed ( blue ) lines are the best fitting eff @xcite and king @xcite profiles respectively . note the excess of light at large radii with respect to the best fitting eff profile in both the observations and models . this excess light is due to an unbound expanding halo of stars caused by the rapid ejection of the remaining gas after the cluster forms . _ hence , excess light at large radii strongly implies that these clusters are not in dynamical equilibrium . _ for details of the modeling and observations see @xcite and @xcite.,height=340 ] even if a cluster survives the gas removal phase , this phase can significantly effect the observed properties of the cluster . hence , deducing the initial properties of a cluster from its current state is not trivial . @xcite have noted the strong effect of residual gas removal on inferring the initial stellar mass of a cluster , while @xcite have refined the mass loss estimates and shown that measurements of the current radii of young clusters may not reflect the initial nor the final value . additionally , @xcite show that this effect can mimic stellar imf variations in young clusters . the clusters that have survived the gas removal phase are subject to disruption phases ii and iii ( [ subsec : phases ] ) as well as tidal effects . disruption due to these effects can be studied on individual clusters , of which the recent observations of the dissolving globular clusters palomar 5 @xcite are probably the most spectacular example . however , much can be learned by approaching this problem from a cluster population point of view . suppose that clusters are formed continuously with a constant cluster formation rate ( a constraint which we can relax later ) . also , we will assume that we know the cluster initial mass function ( here taken to be a power law of the form @xmath21 with @xmath22 ( e.g. @xcite ) and that clusters can be detected down to a known magnitude limit . finally , we will assume that the disruption time of a cluster depends on the cluster mass , such that more massive clusters survive longer ( on average ) than lower mass clusters . for this final assumption we will adopt a function of the form : @xmath23 where @xmath24 is the disruption time of a @xmath25 cluster and @xmath26 @xcite . the beauty of this formulation is that it only has two variables , @xmath24 and @xmath27 , and as we will see , provides extremely good fits to observations . the formulation provided above , when combined with the given assumptions , allows for the parameters @xmath24 and @xmath27 to be found from age and mass distributions of clusters . the first survey using this technique was carried out by @xcite on cluster populations in m51 , m33 , the smc , and the solar neighbourhood . they made a _ sudden disruption _ assumption , meaning that the cluster is in the sample with its initial mass until , when it is disrupted . the somewhat surprising result from this study was that , while @xmath27 had more or less the same value in all environments studied ( @xmath28 ) , @xmath24 varied by over two orders of magnitude , with values of @xmath29 myr in the central regions of m51 to @xmath30 gyr in the smc . the simple _ sudden disruption _ assumption was improved in a more recent model by @xcite , where a gradual loss of cluster mass was implemented . they assumed that the cluster mass decreases exponentially with a time - scale that decreases as the cluster mass decreases . this is done by saying that the mass loss per unit time ( ) relates to as : @xmath31 with @xmath32 from eq . [ eq : ihavebeentypingthisequationtoomuchinmylife ] . this very simple analytical description for cluster mass loss shows remarkably good agreement when compared to the mass loss following from the detailed @xmath19-body simulations of @xcite . in fig . [ dm_lamers ] we show a direct comparison of @xmath33 from the @xmath19-body simulations of clusters with different density profiles and on different orbits ( left ) and the above mentioned analytical model of @xcite ( right ) . in both graphs the time is normalised to and only the mass loss due to stars escaping the cluster is shown , i.e. mass loss due to stellar evolution ( sev ) is not shown . in addition , there is a coupling between the two types of mass loss : if stars loose mass , the cluster will expand and more stars are pushed over the tidal boundary . the simulations of @xcite considered sev , therefore , their results shown in fig . [ dm_lamers ] do include tidal induced by sev . for this reason we can simply add the mass loss due to sev , taken from an ssp model , to eq . [ eq : dm ] . -body simulations of clusters with different number of stars , different concentration and on different orbits ( left ) . the mass loss due to stellar evolution is not shown . in the right panel the analytical model of @xcite is shown.,title="fig:",width=255]-body simulations of clusters with different number of stars , different concentration and on different orbits ( left ) . the mass loss due to stellar evolution is not shown . in the right panel the analytical model of @xcite is shown.,title="fig:",width=255 ] in a series of follow - up works , it has been shown that the similarity of the value of @xmath27 in various environments strongly implies a uniformity in the cluster disruption process , while the varying values of @xmath24 is due to the different tidal field strengths _ ( and gas contents ) _ of the galaxies studied . galaxies with strong tidal fields , as , for example , derived from their rotation curves , having shorter disruption times @xcite . comparison with results of realistic @xmath19-body models performed by @xcite have placed this empirical disruption law on a solid physical footing @xcite . @xcite have also derived a formula for the predicted mass and age distributions of cluster samples that includes both stellar evolution and disruption for any star formation history . @xcite inserted the lamers disruption law into a cluster population synthesis model . this method has two distinct advantages over the earlier formulations . the first is that it removes the requirement of a constant cluster formation rate , and second , it uses the age and mass distributions together to find @xmath27 and @xmath24 . the case of m51 is shown in fig . one first begins by constructing an observed number density grid in age - mass space ( upper - left panel where the shading corresponds to the logarithm of the number of clusters found within that cell ) . then one generates a large number of models with different values of @xmath24 , @xmath27 , ( time dependent ) cluster formation rates , etc . and compares these models with the observed grid . the resultant @xmath34 diagram is shown in the bottom panel of fig . the best fit model is shown in the top right panel of fig . [ fig2 ] . this cluster population synthesis ( cps ) technique , also used in a similar way by @xcite to derive the properties of the cluster population in the galaxy ngc 3627 , holds great promise in disentangling the myriad of effects present in cluster populations . in principle , the dependences of cluster size , galactocentric radius , star - formation efficiency dependent infant mortality rates , or alternative cluster disruption models can be taken into account by this technique . for this technique to be fully exploited one needs large samples of cluster populations with known ages and masses . datasets suitable for these kind of studies are beginning to be collected and released . several face - on spiral galaxies have been imaged in multiple filters with the high resolution / wide field _ hst / acs _ camera ( e.g @xcite for m101 and for m51 ) . and @xmath24 . * bottom : * the @xmath34 diagram in @xmath27-@xmath24 space . the best fitting model is marked with an x , while the accepted fits ( @xmath35 ) are shaded . reproduced from @xcite.,title="fig:",width=226 ] and @xmath24 . * bottom : * the @xmath34 diagram in @xmath27-@xmath24 space . the best fitting model is marked with an x , while the accepted fits ( @xmath35 ) are shaded . reproduced from @xcite.,title="fig:",width=226 ] and @xmath24 . * bottom : * the @xmath34 diagram in @xmath27-@xmath24 space . the best fitting model is marked with an x , while the accepted fits ( @xmath35 ) are shaded . reproduced from @xcite.,title="fig:",width=226 ] it is worth noting possible objections to the lamers disruption law . the first comes from @xcite who find that in the antennae galaxies the number of clusters decreases in time ( @xmath36 ) as @xmath37 , independent of cluster mass . this may be explained if the disruption timescale @xmath24 due to tidal field effects ( e.g. phase ii & iii ) is greater than or similar to the maximum age in the sample . the cluster disruption due to tidal effects would not yet be present in the @xcite sample , instead the decrease in cluster numbers would be the result of infant mortality and the fading of clusters . studies of infant mortality in m51 also suggest that the effect is mass independent @xcite . in fact , if infant mortality was not ( mostly ) independent of cluster mass we would expect the embedded cluster mass function to be significantly different from the optically selected cluster mass function . in fig . [ fig : mf ] we show the dependence of the mass function slope of a multiple age cluster population on the ratio of the @xmath38 and the maximum age of the cluster in the sample ( ) . clusters were created continuously over 1 gyr with an initial mass function of a power - law with index @xmath39 . the lamers disruption law was applied in the same way as in @xcite . the important thing to take away from this figure is that as approaches and exceeds the mass function is less affected by disruption and so it retains its initial form , i.e. the right panel in fig . [ fig : mf ] probably applies to the @xcite sample . ) .,height=151 ] a second observation seemingly contradicting the lamers disruption law is that of @xcite who find an intermediate age ( @xmath40 gyr ) globular cluster in m33 . in m33 , @xcite find a @xmath24 value of @xmath41 myr , implying a disruption time of @xmath42 gyr for a @xmath43 cluster . however , as the authors note , the value of @xmath24 derived by @xcite was presumably of the thin disk of the galaxy , and if the intermediate - age cluster is part of the thick - disk or halo of the galaxy then the expected value of @xmath24 would be significantly larger than that quoted . additionally , it should also be noted that the mass derived by @xcite is the _ present _ mass of the cluster . the cluster presumably started with a much higher mass and disruptive effects have brought this cluster into its current state . if the present mass of the cluster is @xmath44 , then its initial mass ( after infant weight loss ) would have been @xmath45 ( assuming an age of 5 gyr ) , using the value of @xmath24 for m33 given by @xcite . -body models of @xcite ) disruption time of a @xmath46 cluster , @xmath24 , as a function of the mean density @xmath47 in @xmath48pc@xmath49 of the host galaxy . reproduced from @xcite.,width=340 ] as more and more galaxies ( and environments ) have their characteristic disruption timescales measured , it is useful to compare the results to @xmath19-body models in order to check for consistency between the two . this was done in @xcite who compared the @xmath24 values derived for the smc , m33 , m51 and the solar neighbourhood to the @xmath19-body models of @xcite and @xcite which sample a large range in the ambient densities of the host galaxies . their results , shown in fig . [ fig3 ] , are intriguing . while the predicted and observed disruption time of the smc are in excellent agreement , the disruption times of the galaxy , m33 and m51 are observed to be much shorter than predicted by @xmath19-body models . this result is particularly surprising given the fact that the mass loss predictions of a single cluster are in excellent agreement between the lamers empirical description and that given by @xmath19-body models ( @xcite and fig . [ dm_lamers ] ) . = @xmath46 with @xmath50 . * top : * the motion of the gmc is along the x - axis and the line of sight is perpendicular . * bottom : * the motion of the gmc is into the page and the line of sight is along the gmc trajectory . the arrows in the left - hand lower corner of the left - hand panels are parallel to the direction of motion of the gmc . the gmc is shown with grey shades based on the surface density of a gmc with a@xmath51 = 5.8a@xmath52 . the time with respect to the moment of encounter is indicated in each panel of the top row . see @xcite for a description of the methods and parameters used.,width=529 ] thus , we are left to ask , what physical effects are not included in the @xmath19-body models that may be responsible for disrupting clusters ? the @xmath19-body models used in the comparison were carried out in a smooth logarithmic potential which does not realistically represent the thin disk components of disk galaxies . @xcite have attempted to add encounters with giant molecular clouds ( gmcs ) and spiral arm passages to the @xmath19-body models . in fig . [ fig4 ] we show an example of a cluster - gmc encounter ( from @xcite ) . the parameters of this run are for typical open clusters and gmcs in the solar neighbourhood . the top panels show an edge - on view for five different time steps , while the bottom panels show a view along the trajectory of the gmc . encounters with gmcs present the most important external perturbation which cause mass loss of star in clusters . @xcite find that due to encounters with gmcs scales as @xmath53 where @xmath54 for the solar neighbourhood and @xmath55 scales with the surface density of individual gmcs ( @xmath56 ) and the global gmc density ( @xmath57 ) as @xmath58 . the scaling of @xmath55 with @xmath57 implies that it does not matter if the molecular gas is distributed over a large number of low mass clouds or a small number of massive ( giant ) clouds . this makes it easy to estimate from the observed molecular gas density . indeed , for m51 , where the molecular gas density is about an order of magnitude higher than in the solar neighbourhood , a from eq . [ eq : tdis_gmcs ] of 150 myr is predicted . this corresponds well with the value derived from observations of @xmath59 myr @xcite . note that eq . [ eq : tdis_gmcs ] implies a scaling of with the cluster density ( @xmath60 ) . this seems different than the dependence with discussed before . however , there is only a very shallow relation observed between cluster half - mass radius ( ) and , of the form @xmath61 @xcite . with this relation , and eq . [ eq : tdis_gmcs ] , it follows that for external perturbations @xmath62 , i.e. very close to the index of @xmath63 found from observations discussed in [ subsec : appl_pops ] . this suggests that the disruptive effect of the tidal field and additional external perturbations can be added linearly , resulting in a that depends on as @xmath64 . this can explain the large variation found in the @xmath38 value derived from observations and the almost constant @xmath65 @xcite . in [ sec : discussion ] we discuss some of the pitfalls of these results . as seen in the proceeding sections , the observed disruption time of star clusters in the solar neighbourhood is a factor of @xmath66 shorter than predicted by @xmath19-body models . the inclusion of spiral arm passages and gmc encounters into @xmath19-body models is a promising way to bring the predictions into agreement with the observations . this was recently done by @xcite who found excellent agreement after the inclusion gmc encounters and spiral arm passages . they assume that the different mass loss effects ( stellar evolution , tidal field and external perturbations ) can be added linearly . using the mass - radius relation of [ subsec : tdis_external ] and the results from @xcite and @xcite they analytically model the mass loss due to different effects analytically . this is illustrated in the left panel of fig . [ fig5 ] ( from @xcite ) . based on this mass loss description , the age distribution of open clusters in the solar neighbourhood can be predicted ( instead of fitted , as was done hitherto ) . the results are shown in the right panel of fig . [ fig5 ] . cluster due to various disruptive effects . * right : * comparison between the observed age distribution of open clusters ( from ) and the predictions from @xcite for three different maximum masses . , title="fig:",width=255 ] cluster due to various disruptive effects . * right : * comparison between the observed age distribution of open clusters ( from ) and the predictions from @xcite for three different maximum masses . , title="fig:",width=255 ] we showed in [ sec : populations]&[sec : external ] that the simple lamers disruption law can successfully explain the age and mass distribution of young star clusters populations . here we will discuss other observations lending support to the lamers law and some of the standing problems and uncertainties of this scenario which need further attention . @xcite use a variety of studies to look at the cluster population of the lmc . they also find a lack of old clusters ( with respect to what would be expected from a continuous cluster formation rate ) and derive @xmath67 , again in agreement with other galaxies studied by @xcite and @xcite . note that a lower value of @xmath27 is expected to be observed when the typical is of the same order as the oldest clusters in the sample ( fig . [ fig : mf ] ) , as is the case in lmc . outside the local group , the strongly interacting galaxy ngc 6745 has been studied by @xcite who found evidence for mass dependent disruption , with @xmath68 . the rich cluster system of the intermediate - age merger remnant ngc 1316 shows a clear bimodal colour distribution , with the red component presumably being formed during the merger . @xcite showed , using deep _ hst - acs _ images that if one breaks the red component into ` inner ' and ` outer ' regions ( with respect to the galactic centre ) , that the outer region is a continuous power - law while the inner region shows a power - law behavior at the high luminosity end and a flattening at the low luminosity end . the authors interpret this as evidence for mass - dependent cluster disruption , although no attempt was made to find the characteristic disruption timescale or the value of @xmath27 . one standing problem with the lamers disruption law , also present in other studies on disruption , is whether or not an initial power - law cluster initial mass function ( cimf ) can be transformed into a log - normal distribution , which is observed for old globular cluster populations . the lamers law can create such a turnover , however the precise value of the turnover mass should be dependent on the ambient density @xcite , meaning that cluster disruption should be more efficient in the inner regions of a galaxy than in the outer regions . thus , without fine tuning the models ( e.g. having the same disruption time at all radii due to large radially dependent velocity anisotropies ) one would expect a radially dependent turnover peak in the globular cluster mass function , which is not observed . for a more detailed description of this problem , see the review by larsen in these proceedings . additionally , as noted by @xcite the lamers disruption over - predicts the number of low - mass clusters when applied to old globular cluster populations . in [ subsec : tdis_external ] we showed that the scaling of with is a power - law with exponent @xmath63 . this scaling is similar for two - body evaporation in a tidal field with external perturbations , such as shocks by gmcs and spiral arms , and agrees well with the observations . however , there are still some caveats in the theory explaining this , mostly coming from questions regarding the initial conditions of the simulations . 1 . the first caveat stems from the relation between initial mass and radius of the clusters used in the simulations . if we parameterize this relation as @xmath69 @xmath70 , then @xcite use @xmath71 , implying that their clusters fill their tidal radius . however , observations imply that @xmath72 ( with a large scatter ) @xcite , implying that is mostly independent of mass . this shallow relation implies that massive clusters are not filling their tidal radius , which would change the dependence of with @xcite . 2 . in the derivation of @xmath27 for external shocks ( [ subsec : tdis_external ] ) , only clusters in isolation were considered . how would the presence of a tidal field affect this result ? 3 . how does the relation for change if there exists a relation between the concentration parameter and mass of a cluster ( i.e. as seen in the galactic gcs reported by larsen in these proceedings ) ? 4 . could an initial mass - radius relation with @xmath73 be erased during the gas removal phase ? what is the effect of the initial mass / luminosity profile used in the simulations and how does it evolve ? e.g. are clusters born with eff profiles which are converted into king profiles ? does the cluster concentration alter its mass loss evolution ? 6 . how do the external perturbations and the galactic tidal field cooperate ? can the mass loss due to both effects simply be added linearly ? barmby , p. , kuntz , k. d. , huchra , j. p. , & brodie , j. p. 2006 , , 132 , 883 bastian , n. & goodwin , s.p . 2006 , mnras , 369 , l9 bastian , n. , gieles , m. , lamers , h.j.g.l.m . , scheepmaker , r. a. , & de grijs , r. 2005 , a&a 431 , 905 baumgardt , h. , & makino , j. 2003 , mnras , 340 , 227 baumgardt , h. 2006 : globular clusters : guides to galaxies ( astro - ph/0605125 ) boily , c.m . & kroupa , p. 2003a , mnras , 338 , 665 boily , c.m . & kroupa , p. 2003b , mnras , 338 , 673 boutloukos , s.g . & lamers , h.j.g.l.m . 2003 , mnras , 338 , 717 chandar , r. , leitherer , c. , tremonti , c.a . , calzetti , d. , aloisi , a. , meurer , g.r . , & de mello , d. 2005 , apj , 628 , 210 chandar , r. , puzia , t.h . , sarajedini , a. , goudfrooij , p. 2006 , apjl , in press ( astro - ph/0606419 ) de grijs , r. , anders , p. , bastian , n. , lynds , r. , lamers , h.j.g.l.m . , oneil , e.j . 2003 , mnras , 343 , 1285 de grijs , r. , & anders , p. 2006 , mnras , 366 , 295 dolphin , a.e . , & kennicutt , r.c . 2002 , aj , 124 , 158 elson , r.a.w . , fall , m.s . , & freeman , k.c . 1987 , apj 323 , 54 ( eff ) , s. m. & zhang , q. 2001 , , 561 , 751 fall , s.m . , chandar , r. & whitmore , b.c . 2005 , apj , 631 , l133 gascoigne , s. c. b. 1966 , , 134 , 59 gieles , m. , bastian , n. , lamers , h.j.g.l.m . , & mout , j.n . 2005 , a&a , 441 , 949 gieles , m. , athanassoula e. , portegies zwart s. f. , 2006b , mnras , submitted , m. , larsen , s. s. , scheepmaker , r. a. , et al . 2006 , , 446 , l9 gieles , m. , portegies zwart , s.f . , baumgardt , h. athanassoula , e. , lamers , h.j.g.l.m . , sipior , m. , leenaarts , j. 2006c , , 371 , 793 goudfrooij , p. , gilmore , d. , whitmore , b.c . , & schweizer , f. 2004 , apj , 613 , l121 goodwin , s.p . 1997a , mnras , 284 , 785 goodwin , s.p . 1997b , mnras , 286 , 669 goodwn , s.p . & bastian , n. 2006 , mnras , in press ( astro - ph/0609477 ) hodge , p. 1987 , , 99 , 724 , n. v. , piskunov , a. e. , rser , s. , schilbach , e. , & scholz , r .- d . 2005 , , 438 , 1163 king , i. 1962 , aj 67 , 471 kroupa , p. & boily , c.m . 2002 , mnras , 336 , 1188 lada , c.j . & lada , e.a . 2003 , ara&a , 41 , 57 lamers , h.j.g.l.m . , gieles , m. , bastian , n. , baumgardt , h. , & kharchenko , n.v . 2005a , a&a , 117 , 129 lamers , h.j.g.l.m . , gieles , m. , & protegies zwart , s.f . 2005b , a&a , 429 , 173 lamers , h.j.g.l.m . & gieles , m. 2006 , a&a , 455 , l17 , s. s. 2004 , , 416 , 537 odenkirchen , m. , et al . 2001 , , 548 , l165 oort , j. h. 1958 , ricerche astronomiche , 5 , 507 portegies zwart , s.f . , hut , p. , & makino , j. 1998 , a&a , 337 , 363 portegies zwart , s.f . , makino , j. , mcmillan , s.l.w . , & hut , p. 2002 , apj , 565 , 265 schweizer , f. 2006 : globular clusters : guides to galaxies ( astro - ph/0606036 ) , l. 1987 , dynamical evolution of globular clusters ( princeton , nj , princeton university press , 1987 , 191 p. ) tanikawa , a. , & fukushige , t. 2005 , , 57 , 155 tremonti , c.a . , calzetti , d. , leitherer , c. , & heckman , t.m . 2001 , apj , 555 , 322 tutukov , a.v . 1978 , a&a , 70 , 57 , e. 1998 , , 299 , 1019 , e. & zepf , s. e. 2003 , , 587 , l97 waters , c.z . , zepf , s.e . , lauer , t.r , baltz , e.a . , & silk , j. 2006 , apj , in press ( astro - ph/0607238 ) wielen , r. 1971 , a&a , 13 , 309 whitmore , b.c . zhang , q. , leitherer , c. , et al . 1999b , aj , 118 , 1551 whitmore , b. c. 2003 , in a decade of hst science , eds . mario livio , keith noll & massimo stiavelli , ( cambridge : cambridge university press ) , 153 . ( widely referenced as whitmore 2000 , astro - ph/0012546 ) zhang , q. , & fall , s. m. 1999 , , 527 , l81

we review the theory and observations of star cluster disruption . the three main phases and corresponding typical timescales of cluster disruption are : _ i ) infant mortality _ ( @xmath0 yr ) , _ ii ) stellar evolution _ ( @xmath1 yr ) and _ iii ) tidal relaxation _ ( @xmath2 yr ) . during all three phases there are additional tidal external perturbations from the host galaxy . in this review we focus on the physics and observations of phase i and on population studies of phases ii & iii and external perturbations ( concentrating on cluster - gmc interactions ) . particular attention is given to the successes and short - comings of the lamers cluster disruption law , which has recently been shown to stand on a firm physical footing .

astro-ph0609669

even kakutani equivalence is one of the most natural examples in the theory of restricted orbit equivalence of ergodic and finite measure preserving dynamical systems . in this paper we study even kakutani equivalence in the nearly continuous category . a nearly continuous dynamical system is given by a triple @xmath13 , where @xmath2 is a polish space , @xmath14 is a borel probability measure on @xmath2 , and @xmath15 is an ergodic measure preserving homeomorphism . recall that a measurable orbit equivalence between two such systems @xmath13 and @xmath16 is an invertible , bi - measurable , and measure preserving map @xmath17 that sends orbits to orbits . a measurable orbit equivalence @xmath18 is a _ nearly continuous orbit equivalence _ if there exist invariant and @xmath11 subsets @xmath19 and @xmath20 of full measure so that @xmath21 is a homeomorphism . the first result in this category is the celebrated theorem of keane and smorodinsky @xcite that any two bernoulli shifts of equal entropy are finitarily isomorphic , namely , that the isomorphism between them can be made a homeomorphism almost everywhere . in a later paper , denker and keane @xcite established a general framework for studying measure preserving systems that also preserve a topological structure . we refer the reader to a paper by del junco , rudolph , and weiss @xcite for a more complete history of the area . we only mention here that interest in the orbit equivalence theory for this category was more recently revived by the work of hamachi and keane in @xcite where they proved that the binary and ternary odometers are nearly continuously orbit equivalent . their work inspired similar results for other pairs of examples ( see @xcite , @xcite , @xcite , @xcite , @xcite , and @xcite ) . these examples were later subsumed as special cases of a dye s theorem in this category proved by del junco and ahin @xcite . around the same time as a nearly continuous dye s theorem was established , del junco , rudolph , and weiss proved in @xcite that if one does not impose the condition that the invariant sets of full measure on which the orbit equivalence is a homeomorphism are @xmath11 sets , then any restricted orbit equivalence classification is exactly the same as in the measure theoretic case . in particular , they showed that any orbit equivalence can be regularized to be a homeomorphism on a set of full measure , but could not prove that the set of full measure had any topological structure . the importance of the topological structure in the theory is even more striking for the study of even kakutani equivalence . recall that in the measurable category two ergodic and finite measure preserving systems @xmath13 and @xmath16 are even kakutani equivalent if there exists a measurable orbit equivalence @xmath22 , and measurable sets @xmath23 with @xmath24 with the property that @xmath25 is a measurable isomorphism of the induced transformations @xmath7 and @xmath26 . we call the orbit equivalence @xmath6 an _ even kakutani equivalence _ between @xmath0 and @xmath1 . it follows from @xcite that any even kakutani equivalence can be made to be a homeomorphism on a set of full measure . in the same paper they show that if one imposes the additional condition that the sets @xmath12 and @xmath27 be _ nearly clopen _ , meaning within a set of measure zero of an open set and also of a closed set , then there is a new invariant for even kakutani equivalence of nearly continuous dynamical systems called _ near unique ergodicity_. they use this new invariant to show that nearly continuous even kakutani equivalence is stronger than measure theoretic even kakutani equivalence . the example they construct is , in some sense , not natural , and begs the question whether there are any natural examples of nearly continuous systems that are measurably evenly kakutani equivalent but not nearly continuously so . rudolph began looking for examples in the family of zero entropy loosely bernoulli systems . recall that any two zero entropy loosely bernoulli transformations are measurably even kakutani equivalent . furthermore , many natural examples of nearly continuous systems including rotations , all adding machines , and in fact all finite rank transformations , are loosely bernoulli . in @xcite , roychowdhury and rudolph proved that any two adding machines are nearly continuously even kakutani equivalent . shortly after , dykstra and rudolph showed in @xcite that all irrational rotations are nearly continuously kakutani equivalent to the binary odometer . in @xcite , new machinery , called _ templates _ , was introduced to construct the nearly continuous kakutani equivalence . there , templates were defined using the natural topological tower structure present in adding machines . the construction in @xcite showed that the template machinery can be adapted to the case where the underlying system does not have a canonical symbolic structure . more recently , springer @xcite expanded on their ideas and adapted templates further to prove that all minimal isometries of compact metric spaces are nearly continuously kakutani equivalent to the binary odometer . salvi @xcite adapted templates to the setting of @xmath28 actions and used the machinery to prove rudolph s two - step coding theorem in the nearly continuous category . each result mentioned above has required more sophisticated and technically intricate incarnations of templates . on the other hand each proof has also established the usefulness and flexibility of the machinery . in this paper we adapt the template machinery even further to show our main result : the morse minimal system is nearly continuously even kakutani equivalent to the binary odometer . the version of the template machinery in this paper is designed to address the new complication of the additional tower present in the rank two morse system . we believe the generalization we give here is the appropriate starting place to prove more generally that finite rank nearly continuous systems are all nearly continuously kakutani equivalent to the binary odometer . finally we note that this manuscript is a culmination of work that the first author began in 2009 while he was a post - doctoral fellow working with daniel rudolph at colorado state university . the initial architecture of the constructions and the main ideas were all established collaboratively by dykstra and rudolph . the second author joined the project after the untimely death of rudolph in 2010 , and the manuscript was completed in 2014 . in this section , deferring some formal definitions until later , we give an overview of the construction and introduce templates . let @xmath29 denote the morse minimal system and @xmath30 the binary odometer . recall that each system has a canonical refining , generating sequence of clopen partitions that are given by the finite rank structure of each system . the morse system is rank two , so at each stage the partition is defined by a pair of towers . the odometer is rank one , so the sequence of partitions is defined by a sequence of single towers . the construction of the orbit equivalence uses an inductive back and forthprocedure . intuitively , at each stage we need to construct a set map from the levels of the tower of one system to the levels of the tower of the other system , switching the domain and range of the set maps at each stage . the orbit equivalence will be defined on the set of points for which our procedure will converge . in order for this set to be a @xmath11 set , and the map to be a homeomorphism , for each point where we have convergence we need to have the procedure stabilize after a finite number of steps . in other words , once we have defined the set map at a particular stage @xmath31 , we can not modify its domain at any successive stage . this introduces an obvious complication in the construction . at a particular stage @xmath31 , we have to know that our choice of the set map on stage @xmath31 towers will be consistent with the choices that we will make for all stages after . to address this complication , informally speaking , we do not actually choose a particular set map at any stage . instead , at each stage we construct a collection of set maps that are possible extensions of previous stage set maps , and that all agree on a set we call _ the good set_. the convergence then depends on us being able to provide enough choices at each stage @xmath31 so that it is possible to construct sufficiently many choices of maps at stage @xmath32 that extend the @xmath31-stage maps . templates are a combinatorial tool that have been designed to facilitate the intensive book keeping required to describe such a procedure . formally , _ template _ is an ordered multiset . for example , the multiset @xmath33 , together with the ordering @xmath34 , gives a template @xmath35 , which we write as @xmath36 the elements of a template ( with multiplicity ) are called _ levels_. in our work , each level of a template will correspond to a clopen set that is a level of a tower . in particular , the towers themselves can be thought of as templates where each level appears exactly once and the ordering on the levels is exactly the ordering on the sets that is imposed by the underlying dynamics . notice that a set map from one tower to another can be thought of as a re - ordering of the levels of the domain tower according to the levels that they are being mapped to in the image tower . we replace the notion of a set map with maps between templates , where the re - ordering is not described by the map , but rather the ordering given by the image template . more formally , given templates @xmath35 and @xmath37 , written @xmath38 we _ represent _ @xmath35 and @xmath37 with the intervals @xmath39 \subset { \mathbb z}\hspace{1 cm } \mbox{and } \hspace{1 cm } j = [ 0 , 1 , \ldots , m-1 ] \subset { \mathbb z}\ ] ] via the correspondences @xmath40 and @xmath41 . a _ partial interval bijection _ is an ordered quintuple @xmath42 $ ] , where @xmath43 , @xmath44 , and @xmath45 is some bijection . this perspective allows for an explicit combinatorial understanding of how many maps need to be defined at each stage in order to construct maps at later stages , on larger domains . it also allows for an explicit combinatorial description of how maps from one stage are constructed from maps of a previous stage , so it is easy to prove that the construction has indeed stabilized for points on a @xmath11 set of full measure . in our proof we will construct an increasing sequence @xmath46 , templates for each system and partial interval bijections between template sets that constitute the back and forth " diagram given in figure [ thefigure ] below . the objects in the diagram are template sets . the template sets on the left ( @xmath47 , @xmath48 , @xmath49 , and @xmath50 ) belong to the morse minimal system , while those on the right ( @xmath51 , @xmath52 , @xmath53 , and @xmath54 ) belong to the binary odometer . the maps @xmath55 are partial interval bijections . a key ingredient of the diagram is its _ almost commutative _ nature , as introduced by roychowdhury and rudolph in @xcite . interpreting the levels of templates as levels of towers that form a refining sequence of partitions , we see that every level at stage @xmath31 of a tower is a subset of a level from a previous stage template , and the maps @xmath56 and @xmath57 are the natural inclusion maps . the consistency of set maps from one level to another is achieved by requiring that on the good set , all partial interval bijections agree when composed with @xmath56 and @xmath57 . there are two key differences our work here and that of @xcite or @xcite in how we use templates from a particular stage to construct later stage templates . in particular , we can not use the notion of concatenation as was defined in the earlier papers , instead we define _ overlapping concatenations_. in addition , to accommodate the combinatoric structure of the towers of the morse minimal system we introduce a new family of partial interval bijections called _ reordering maps_. once the diagram is built up , however , the argument that it produces a well - defined nearly continuous kakutani equivalence is nearly identical to the arguments in both @xcite and @xcite . we include it here for completeness ( see sections [ convergenceofpibs ] and [ kakequivalence ] ) . it is our hope that one day we might discover a more general machine that could characterize broad classes of systems , perhaps even all zero entropy loosely bernoulli systems . but at the moment it is not clear how such a machine , if one exists , could be sufficiently general to account for the differences between systems . the paper is organized as follows : section [ overview ] : : we define the tools for constructing templates and partial interval bijections that we will use throughout the construction . sections [ morsepreliminaries ] - [ binarypreliminaries ] : : we give preliminary definitions of the morse minimal system and the binary odometer . in particular , we define the template sets @xmath58 , @xmath59 , @xmath60 , and @xmath61 , for @xmath62 . these template sets are given by the towers in the respective system . section [ omegaksection ] : : we define the template sets @xmath63 and @xmath64 , for @xmath62 . these are templates in the odometer system that are rearrangements of the tower templates , reflecting set maps that map morse towers to odometer towers . section [ stage2 ] : : we construct stage @xmath65 of the diagram by explicitly defining @xmath66 , along with the partial interval bijections @xmath67 . section [ tksection ] : : we define the template sets @xmath68 and @xmath69 , for @xmath62 . these are new templates in the morse system , describing how odometer towers will be mapped to morse towers . section [ sequences ] : : we define the sequence @xmath46 recursively . sections [ freqnotation ] - [ stage8 ] : : we assume the diagram has been built down to stage @xmath31 , where @xmath31 is even , and show how to build it down to stage @xmath70 . because the construction depends on whether @xmath71 or @xmath72 , we proceed as follows : + * in sections [ freqnotation ] - [ blockpartitionsoftauhat ] , we introduce notation and machinery that is used for every even @xmath31 . * in sections [ thegoodset ] - [ stage4missingextra ] , we illustrate the @xmath73 case by building the diagram down to stage @xmath74 . * in sections [ stage6bigpicture ] - [ stage6missingextra ] , we illustrate the @xmath72 case by building the diagram down to stage @xmath75 . * in section [ stage8 ] , we indicate how the induction looks in stages @xmath76 . sections [ convergenceofpibs ] - [ kakequivalence ] : : we use the properties of the diagram to prove that our procedure produces a nearly continuous kakutani equivalence between the two systems . in this section we introduce the terminology and tools necessary to build and extend partial interval bijections . [ domainandrange ] given templates @xmath35 and @xmath37 , and a partial interval bijection @xmath42 $ ] from @xmath35 to @xmath37 , the _ domain _ of @xmath77 consists of those levels in @xmath35 that are represented by @xmath12 . the _ range _ of @xmath77 consists of those levels in @xmath37 that are represented by @xmath27 . for the next definition , suppose @xmath78 is another template which , like @xmath37 , is represented by @xmath79 . suppose there is a partial interval bijection @xmath80 $ ] from @xmath35 to @xmath81 . note that @xmath82 and @xmath83 agree in their first three components ( @xmath84 , @xmath85 , and @xmath12 ) . [ match ] suppose @xmath86 and @xmath87 are two partial interval bijections , given by @xmath42 $ ] and @xmath88 $ ] . then @xmath77 and @xmath89 _ match _ if , for each integer @xmath90 , the level in @xmath37 that is represented by @xmath91 is identical to the level in @xmath81 that is represented by @xmath92 . two partial interval bijections are _ equivalent _ if one is a translate of the other . more precisely , @xmath93 \sim [ i ' , j ' , a ' , b ' , f']$ ] if there exist @xmath94 and @xmath95 , @xmath96 , @xmath97 , @xmath98 , and @xmath99 . given a partial interval bijection @xmath42 $ ] , the _ inverse _ of @xmath77 is the partial interval bijection @xmath100 $ ] . suppose @xmath101 $ ] for @xmath102 are two partial interval bijections , and assume each @xmath103 and @xmath104 begin at @xmath105 . let @xmath106 and @xmath107 . define the _ simple concatenation _ of @xmath108 and @xmath109 by @xmath110,\ ] ] where @xmath111 note that this is an associative semigroup action on the space of all partial interval bijections . also note that , if @xmath108 and @xmath109 are partial interval bijections , then so is @xmath112 . for our construction , some partial interval bijections will need to be decomposed into a form @xmath113 where @xmath114 and @xmath115 make up a very small portion of the overall map . we will refer to @xmath114 and @xmath115 as the _ bottom _ and _ top sticky notes _ of @xmath77 . the _ body _ is @xmath116 . we will often want to glue " partial interval bijections together . top and bottom sticky notes will be our means of doing this via the following definition . [ overlapconcatenation1 ] suppose @xmath101 $ ] for @xmath102 are two partial interval bijections that are decomposed via ( [ stage2pibstructure ] ) as @xmath117 if @xmath118 , then define the _ overlapping concatenation _ of @xmath108 and @xmath109 , denoted @xmath119 , by @xmath120 the sticky notes described in section [ stickynotes ] will be used only in the early stages of the construction . from then on , top and bottom sticky notes will _ overlap _ with the body , so that a typical partial interval bijection will need to be decomposed into the form @xmath121 where again @xmath114 and @xmath115 make up a very small portion of the overall map . here again , we will refer to @xmath114 and @xmath115 as the bottom and top sticky notes , and @xmath116 as the body , of @xmath77 . once we are far enough along in the construction that sticky note decompositions take the form ( [ stage4pibstructure ] ) , we will no longer be able to use definition [ overlapconcatenation1 ] to glue partial interval bijections together . we will instead use the following . [ overlapconcatenation2 ] suppose @xmath101 $ ] for @xmath102 are two partial interval bijections that are decomposed via ( [ stage4pibstructure ] ) as @xmath122 if @xmath118 , then define the _ overlapping concatenation _ of @xmath108 and @xmath109 , denoted @xmath119 , by @xmath123 a _ reordering map _ is a partial interval bijection of the form @xmath124 $ ] . unlike arbitrary partial interval bijections , in a reordering map each of the first four components is the same interval . therefore it is possible to compose two reordering maps , as follows . if @xmath125 and @xmath126 are two bijections , then , as a usual composition of functions , @xmath127 is a bijection . therefore we can define @xmath128 _ composed with _ @xmath129 to be the reordering map @xmath130 $ ] . reordering maps will be used to move certain levels in the bottom part " of a template up to the top part , " while shifting all levels in the middle part " down . for example , consider the template @xmath131 and think of @xmath132 as the middle part " . suppose we wish to shift this middle part down by two positions . we could accomplish this , for example , by moving @xmath133 and @xmath134 from the bottom part to the top part , as follows . let @xmath135 $ ] and define @xmath125 by @xmath136 and @xmath137 by @xmath138 notice that @xmath139 is the ( new ) template @xmath140 and @xmath141 is the ( new ) template @xmath142 we can also compose a reordering map @xmath124 $ ] with an arbitrary partial interval bijection @xmath42 $ ] by defining @xmath143 $ ] , where @xmath144 is the restriction of @xmath145 to @xmath27 . given a binary word @xmath146 of length @xmath147 , the _ flip _ of @xmath27 is the word @xmath148 , where @xmath149 if @xmath150 and @xmath151 if @xmath152 . let @xmath153 be the substitution rule on the symbols @xmath105 and @xmath154 given by @xmath155 and @xmath156 . iterating @xmath153 on the symbol @xmath105 determines , for each @xmath157 , a word @xmath158 of length @xmath159 : @xmath160 observe that @xmath161 and that @xmath162 . morse sequence _ is the sequence in @xmath163 whose first @xmath159 symbols are the word @xmath164 . let @xmath2 denote the set of all doubly infinite sequences @xmath165 in @xmath166 such that every finite subword of @xmath167 occurs as a subword of the morse sequence . given @xmath167 and @xmath168 in @xmath2 , let @xmath169 if @xmath170 ; otherwise , define @xmath171 , where @xmath31 is maximal such that @xmath172 for all @xmath173 . then @xmath174 is a metric on @xmath2 that determines a borel sigma algebra @xmath175 . the _ morse minimal system _ is then the system @xmath176 , where @xmath177 is the left shift , and @xmath14 is the unique complete ergodic borel probability measure . the following results are proved in @xcite . [ partitionprop ] for each @xmath178 and @xmath157 , there exists a unique partition of @xmath179 into intervals of length @xmath180 so that the subword of @xmath167 on each interval of this partition is either @xmath181 or @xmath182 . [ partitioncor ] for each @xmath178 and @xmath157 , there exists a unique partition of @xmath179 into intervals of length @xmath159 so that the subword of @xmath167 on each interval of this partition is either @xmath164 or @xmath183 . [ 0coordinate]for @xmath178 and @xmath157 , partition @xmath179 into intervals as in corollary [ partitioncor ] and let @xmath184 be the position occupied by @xmath105 in its interval , @xmath185 . the functions @xmath186 are continuous . given @xmath178 and @xmath157 , let @xmath184 be as defined in proposition [ 0coordinate ] . for @xmath187 , let @xmath188 denote the @xmath189th symbol in @xmath164 , and let @xmath190 denote the @xmath189th symbol in @xmath191 . define sets @xmath192 and @xmath193 then @xmath194 and @xmath195 are obtained by taking the clopen set where @xmath196 and splitting it according to whether the symbol at the origin is @xmath105 or @xmath154 . therefore @xmath194 and @xmath195 are clopen . these @xmath180 sets , all of equal measure , are called the _ @xmath147-canonical cylinders _ in @xmath2 . we will often refer to @xmath147-canonical cylinders as _ levels_. define the _ k - canonical templates _ in @xmath2 by : @xmath197 and @xmath198 note that each @xmath147-canonical template has height @xmath159 . the order on the @xmath147-canonical templates is simply the order given by the action of @xmath0 . the set of levels in the @xmath147-canonical templates gives a partition of @xmath2 . let @xmath199 . given @xmath200 , each level @xmath201 is a subset of a unique level @xmath202 . sending @xmath203 then gives a map @xmath204 . observe that @xmath57 is measure preserving in the sense that the measure of the pull back of a set in @xmath205 is the same as its measure . define three modified versions of each @xmath147-canonical template : one in which the bottom level is removed , a second in which there is an extra copy of @xmath206 tacked on to the top , and a third in which there is an extra copy of @xmath207 tacked on to the top . the superscripts @xmath208 , @xmath209 , and @xmath210 will denote missing bottom level " , extra copy of @xmath206 " , and extra copy of @xmath207 " , respectively : @xmath211 define @xmath212 , and @xmath213 by analogy . let @xmath214 be the set of all modified canonical templates at stage @xmath147 . let @xmath215 . then @xmath3 is compact and metrizable ; the metric @xmath216 , where @xmath217 , induces the topology and determines a borel sigma algebra @xmath218 . define @xmath219 by @xmath220 , where @xmath221 and the addition is coordinate - wise mod @xmath222 with right carry . the _ binary odometer _ is the system @xmath223 , where @xmath224 is the unique complete ergodic borel probability measure . @xmath147-canonical cylinder _ is a set @xmath225 , where @xmath226 is a binary word of length @xmath147 . these @xmath147-canonical cylinders are the levels of templates for the binary odometer . the _ k - canonical template _ in @xmath3 , denoted @xmath227 , is the set of @xmath147-canonical cylinders together with the order @xmath228 inherited by the action of @xmath1 , where @xmath229 is an element of the first cylinder . for example , the @xmath230-canonical template is @xmath231 in general , let @xmath232 denote the @xmath189-th level in @xmath227 , so that @xmath233 note that @xmath227 has height @xmath159 , and the set of levels in @xmath227 gives a partition of @xmath3 . let @xmath234 . given @xmath200 , each level @xmath235 is a subset of a unique level @xmath236 . sending @xmath237 then gives a map @xmath238 . observe that @xmath57 is measure preserving in the sense that the measure of the pull back of a set in @xmath239 is the same as its measure . define two modified versions of the @xmath147-canonical template @xmath227 : one in which the level @xmath240 is removed , and another in which there is an extra copy of @xmath240 tacked on to the top . the superscripts @xmath208 and @xmath241 will denote missing @xmath240 " and extra copy of @xmath240 " , respectively : @xmath242 let @xmath243 be the set of all modified canonical templates at stage @xmath147 . define @xmath244 , so that both @xmath245 and @xmath246 are the trivial canonical templates for the binary odometer ( each consisting of just one level ) as indicated in figure [ thefigure ] . the template sets @xmath63 and @xmath247 , defined in section [ omegakdefinition ] consist of templates of the following types : basic , diminished , augmented , missing and extra . we begin by describing these types of templates . a _ basic template _ at stage @xmath147 is any template that satisfies all of the following conditions : it has height @xmath159 ; its elements are @xmath147-canonical cylinders ; and its ordering is allowed " by the action of @xmath1 . for the ordering of a template @xmath248 to be allowed " by the action of @xmath1 , @xmath248 must have one of the following forms . either it is a special basic template @xmath249 that we will refer to as the _ zero - template _ , or for some ( unique ) @xmath250 , @xmath248 is : @xmath251 let @xmath252 be the set of all basic templates for the binary odometer at stage @xmath147 . in each basic template , the level @xmath240 occurs once and only once . call this level the _ global cut_. if @xmath253 is a basic template , define the _ predecessor _ template for @xmath35 , denoted @xmath254 , to be the basic template whose global cut is one position higher ( mod @xmath159 ) , and define the _ successor _ template for @xmath248 , denoted @xmath255 , to be the basic template whose global cut is one position lower ( mod @xmath159 ) . for example , if @xmath253 is : @xmath256 then @xmath254 is @xmath257 and @xmath255 is @xmath258 given a basic template @xmath253 , define two additional , _ diminished _ templates @xmath259 and @xmath260 as follows : * @xmath259 is @xmath248 with the global cut removed , and with an extra level tacked on at the bottom ( the level that would naturally precede the bottom level in @xmath248 , namely the pre - image of the bottom level of @xmath248 under the map @xmath1 ) . * @xmath260 is @xmath248 with the global cut removed , and with an extra level tacked on at the top ( the level that would naturally follow the top level in @xmath248 , namely the image of the top level in @xmath248 under the map @xmath1 ) . for example , if @xmath253 is given by @xmath256 then @xmath259 is given by @xmath261 the letters d " and u " refer to down " and up " , respectively , for reasons that will be made clear later . let @xmath262 be the set of all diminished templates for the binary odometer at stage @xmath147 . note that all diminished templates in @xmath262 have height @xmath159 . given a basic template @xmath253 , define two additional , _ augmented _ templates as follows : * @xmath263 is @xmath248 with an extra copy of the global cut , @xmath240 , inserted right next to the actual global cut , and with the bottom level deleted . * @xmath264 is @xmath248 with an extra copy of the global cut , @xmath240 , inserted right next to the actual global cut , and with the top level deleted . for example , if @xmath253 is given by @xmath256 then @xmath263 is given by @xmath265 let @xmath266 be the set of all augmented templates for the binary odometer at stage @xmath147 . note that all augmented templates in @xmath266 have height @xmath159 . given a basic template @xmath253 , define one additional , _ missing _ template , denoted @xmath267 , to be @xmath248 with its bottom level removed . for example , if @xmath248 is the zero - template , then @xmath268 let @xmath269 be the set of all missing templates for the binary odometer at stage @xmath147 . note that all missing templates have height @xmath270 . given a basic template @xmath253 , define one additional , _ extra _ template , denoted @xmath271 , to be @xmath248 with an one extra level tacked on at the top ( the level that would naturally follow the top level ) . for example , if @xmath253 is given by @xmath256 then @xmath271 is given by @xmath272 let @xmath273 be the set of all extra templates for the binary odometer at stage @xmath147 . note that all extra templates in @xmath273 have height @xmath274 . define @xmath276 , so that the canonical templates in @xmath277 have height @xmath74 . for each canonical template @xmath278 for the morse system , and for each template @xmath279 in the binary odometer , we will define a partial interval bijection @xmath280 from a subset of the levels in @xmath281 to a subset of the levels in @xmath248 . only after we have done this will we define partial interval bijections from the modified canonical templates in @xmath282 to the modified templates in @xmath283 . there are exactly four basic templates in @xmath284 : * @xmath285 , * @xmath286 , * @xmath287 , and * @xmath288 . for @xmath289 and @xmath290 , define @xmath291 to be the partial interval bijection @xmath292 $ ] , where : * @xmath293 \subset { \mathbb z}$ ] , * @xmath294 , * @xmath295 , * @xmath296 , and * each @xmath297 is the obvious bijection @xmath298 . note that each @xmath299 is equivalent to @xmath300 as a formal map between intervals in @xmath179 . but of course @xmath301 and @xmath302 are different as set maps because the levels represented by @xmath12 in @xmath303 and @xmath304 are different . now recall definition [ domainandrange ] , where the domain and range of a partial interval bijection are defined . the following propositions are obvious . [ nobottomortop ] for each canonical template @xmath278 , neither the bottom level nor the top level is in the domain of any @xmath305 . [ nobottomtoporcut ] given @xmath306 , the global cut in @xmath248 is not in the range of @xmath305 . we now define the good set " at stage @xmath222 , and establish its most important property ( proposition [ stage2goodsetprop ] ) , which is obvious at this stage because @xmath245 is the trivial canonical template . let @xmath307 . recall the vertical map @xmath308 which was defined in section [ pkpi ] . we similarly define @xmath309 . the following is immediate since @xmath310 . [ stage2goodsetprop ] if @xmath311 and @xmath312 , then @xmath313 is in the domain of both @xmath305 and @xmath314 , and @xmath315 . let @xmath248 be a basic template in @xmath284 . define @xmath316 to match with @xmath305 , and define @xmath317 to match with @xmath318 , as in definition [ match ] . that these bijections are well defined follows from proposition [ nobottomtoporcut ] ; for example , suppose @xmath248 is given by @xmath319 where the box indicates the level that is in the range of @xmath305 . then , using this same box " notation to indicate the levels in the ranges of the corresponding partial interval bijections , we have @xmath320 @xmath321 and @xmath322 now observe that @xmath323 , and its corresponding partial interval bijection , are : @xmath324 that @xmath325 matches with @xmath317 is not a coincidence : [ stage2diminishedmatching ] given any basic template @xmath306 , @xmath317 matches with @xmath326 . both @xmath317 and @xmath326 are defined to match with @xmath318 . similarly , given @xmath306 , define @xmath327 to match with @xmath305 , and define @xmath328 to match with @xmath318 . that these definitions are possible again follows from proposition [ nobottomtoporcut ] . [ stage2augmentedmatching ] given any basic template @xmath306 , @xmath328 matches with @xmath329 . both @xmath328 and @xmath329 are defined to match with @xmath318 . recall the modified template sets @xmath282 and @xmath330 defined in and . given @xmath331 and @xmath332 , we know from propositions [ nobottomortop ] and [ nobottomtoporcut ] that the bottom level of @xmath281 is not in the domain of @xmath333 , nor is the bottom level of @xmath248 in the range therefore , if @xmath334 $ ] , where @xmath335 $ ] , then we may define @xmath336 by @xmath337 $ ] , where @xmath338 $ ] . in other words , @xmath339 is identical to @xmath340 except that the bottom level of @xmath248 is technically missing . similarly , given @xmath341 and @xmath342 , if again @xmath334 $ ] , then we may define @xmath343 by @xmath344 $ ] , where @xmath345 $ ] . in other words , @xmath343 is identical to @xmath280 except that there is technically an extra level at the top . similar to @xmath63 and @xmath247 , the template sets @xmath68 and @xmath69 , defined in section [ tkdefinition ] below , consist of templates of the following types : basic , diminished , augmented , missing , and extra . we define these types here . a _ basic template _ at stage @xmath147 is any template that satisfies the all of the following conditions : it has height @xmath159 , its elements are @xmath147-canonical cylinders , and its ordering is allowed " by the action of @xmath0 . for the ordering of a template @xmath35 to be allowed " by the action of @xmath0 , @xmath35 must have one of the following forms : 1 . this is a special basic template we call the _ zero - template_. 2 . this is a special basic template we call the _ one - template_. 3 . [ fourforms ] for some ( unique ) @xmath250 , @xmath35 is one of the following four templates : @xmath348 @xmath349 @xmath350 @xmath351 let @xmath352 be the set of all basic templates for the morse system at stage @xmath147 . in each basic template , exactly one of the levels is either @xmath206 or @xmath207 . call that level the _ global cut_. if @xmath353 is a basic template whose global cut is neither the bottom level nor the top level , then @xmath35 must be one of the four templates listed in item [ fourforms ] of section [ morsebasictemplates ] above . in this case , define the _ predecessor _ template for @xmath35 , denoted @xmath354 , to be the basic template of that same form whose global cut is one position higher , and define the _ successor _ template for @xmath35 , denoted @xmath355 , to be the basic template of that same form whose global cut is one position lower . for example , if @xmath353 is the template @xmath356 then @xmath357 is @xmath358 and @xmath359 is @xmath360 if @xmath353 is a basic template whose global cut is the bottom level , then @xmath35 is either the zero - template or the one - template . in this case , define two predecessor templates for @xmath35 , denoted @xmath361 and @xmath362 , as follows : * @xmath361 is @xmath35 with the top level removed and @xmath363 tacked on at the bottom . * @xmath362 is @xmath35 with the top level removed and @xmath364 tacked on at the bottom . also in this case , define two successor templates for @xmath35 , denoted @xmath365 and @xmath366 , as follows : * @xmath365 is @xmath35 with the bottom level removed and @xmath206 tacked on at the top . * @xmath366 is @xmath35 with the bottom level removed and @xmath207 tacked on at the top . if @xmath353 is a basic template whose global cut is the top level ( there are exactly two such basic templates ) , then define two predecessor templates for @xmath35 , denoted @xmath361 and @xmath362 , as follows : * @xmath361 is @xmath35 with the top level ( the global cut ) removed and @xmath206 tacked on at the bottom . * @xmath362 is @xmath35 with the top level ( the global cut ) removed and @xmath367 tacked on at the bottom . also in this case , define a single successor template for @xmath35 , denoted @xmath355 , to be the basic template that is @xmath35 with its bottom level removed , and with the level that would naturally follow the top level tacked on at the top . for example , if @xmath353 is @xmath368 then @xmath355 is @xmath369 because certain templates have multiple predecessor / successor templates , while others have only one , the following definition will be useful . if @xmath353 is a basic template , then a _ predecessor template for @xmath35 _ is any basic template of the form @xmath354 , @xmath361 , or @xmath362 , as defined above . successor template for @xmath35 _ is any basic template of the form @xmath355 , @xmath365 , or @xmath366 , as defined above . if @xmath353 is a basic template , then all predecessor templates for @xmath35 agree in every level except possibly the bottom . also , all successor templates for @xmath35 agree in every level except possibly the top . given a basic template @xmath353 that is neither the zero - template nor the one - template , define two additional , _ diminished _ templates @xmath370 and @xmath371 as follows : * @xmath370 is @xmath35 with the global cut removed , and with an extra level tacked on at the bottom ( the level that would naturally precede the bottom level ) . * @xmath371 is @xmath35 with the global cut removed , and with an extra level tacked on at the top ( the level that would naturally follow the top level ) . for example , suppose @xmath353 is given by @xmath372 then the global cut is @xmath207 , and the bottom level is @xmath194 . the level that would naturally precede this bottom level is @xmath373 . therefore the diminished template @xmath370 is given by @xmath374 now suppose @xmath353 is either the zero - template or the one - template . then there are two levels that could naturally precede the bottom level ( either @xmath363 or @xmath364 ) , as well as two levels that could naturally follow the top level ( either @xmath206 or @xmath207 ) . for this reason , we define four diminished templates , as follows : * @xmath375 is @xmath35 with the global cut ( which , in this case , is also the bottom level ) removed , and with the level @xmath363 tacked on in its place . * @xmath376 is @xmath35 with the global cut ( which , in this case , is also the bottom level ) removed , and with the level @xmath364 tacked on in its place . * @xmath377 is @xmath35 with the global cut ( which , in this case , is also the bottom level ) removed , and with the level @xmath206 tacked on at the top . * @xmath378 is @xmath35 with the global cut ( which , in this case , is also the bottom level ) removed , and with the level @xmath207 tacked on at the top . for example , if @xmath35 is the zero template , then @xmath376 is @xmath379 let @xmath380 be the set of all diminished templates for the morse system at stage @xmath147 . note that all diminished templates in @xmath380 have height @xmath159 . given a basic template @xmath353 , define four additional , _ augmented _ templates as follows : * @xmath381 is @xmath35 with an extra copy of @xmath206 inserted directly before the global cut , and with the bottom level deleted . * @xmath382 is @xmath35 with an extra copy of @xmath207 inserted directly before the global cut , and with the bottom level deleted . * @xmath383 is @xmath35 with an extra copy of @xmath206 inserted directly before the global cut , and with the top level deleted . * @xmath384 is @xmath35 with an extra copy of @xmath207 inserted directly before the global cut , and with the top level deleted . for example , if @xmath353 is given by @xmath356 then @xmath381 is given by @xmath385 let @xmath386 be the set of all augmented templates for the morse system at stage @xmath147 . note that all augmented templates in @xmath386 have height @xmath159 . given a basic template @xmath353 , define one additional , _ missing _ template , denoted @xmath387 , to be @xmath35 with its bottom level removed . for example , if @xmath35 is the zero - template , then @xmath388 let @xmath389 be the set of all missing templates for the morse system at stage @xmath147 . note that all missing templates have height @xmath270 . given a basic template @xmath353 that is neither the zero - template nor the one - template , define one additional , _ extra _ template , denoted @xmath390 , to be @xmath35 with one extra level tacked on at the top ( the level that would naturally follow the top level ) . for example , if @xmath353 is given by @xmath391 then @xmath390 is given by @xmath392 if @xmath35 is the zero - template or the one - template , then define two extra templates , as follows : * @xmath393 is @xmath35 with the level @xmath206 tacked on at the top . * @xmath394 is @xmath35 with the level @xmath207 tacked on at the top . let @xmath395 be the set of all extra templates for the morse system at stage @xmath147 . note that all extra templates in @xmath395 have height @xmath274 . let @xmath397 be a summable sequence . recall that we defined @xmath244 and @xmath276 . now define @xmath398 , @xmath399 , by the following two - step recursion . given @xmath398 , for @xmath31 even , define @xmath400 and @xmath401 as follows . first choose @xmath402 large enough so that @xmath403 and pick @xmath404 large enough so that @xmath405 then choose @xmath406 large enough so that @xmath407 and pick @xmath408 large enough so that @xmath409 note that inequalities ( [ m2choice ] ) and ( [ knplustwochoice ] ) are the same as ( [ m1choice ] ) and ( [ knplusonechoice ] ) , only with @xmath398 , @xmath400 , @xmath410 , and @xmath411 replaced with @xmath400 , @xmath401 , @xmath412 , and @xmath413 . assume the diagram in figure [ thefigure ] has been built down to stage @xmath31 , where @xmath31 is even . using our choices of @xmath400 and @xmath401 from section [ sequences ] , we then build the diagram down to stage @xmath70 . the construction is largely the same whether @xmath71 or @xmath72 ; sections [ freqnotation ] - [ blockpartitionsoftauhat ] apply in either case . define @xmath414 by @xmath415 \text { and } j ' = [ 0 , 1 , \ldots , 2^{k_{n+1}}-1].\ ] ] define the _ bottom and top global safe zones in @xmath85 _ to be the subintervals @xmath416 \text { and } [ 2^{k_{n+2}}-(2^{k_{n+1}}+2^{k_{n+1 } } \cdot 2^{k_{n+1 } } ) , \ldots , 2^{k_{n+2}}-1]\ ] ] of @xmath85 , respectively . by ( [ knplustwochoice ] ) the global safe zones are well - defined , and by ( [ m2choice ] ) the fraction of @xmath85 in the global safe zones is less than @xmath413 . similarly , define the _ bottom and top intermediate safe zones in @xmath417 _ to be the subintervals @xmath418 \text { and } [ 2^{k_{n+1}}-(2^{k_{n}}+2^{k_{n } } \cdot 2^{k_{n } } ) , \ldots , 2^{k_{n+1}}-1]\ ] ] of @xmath417 , respectively . by the intermediate safe zones are well - defined , and by the fraction of @xmath417 in the intermediate safe zones is less than @xmath419 . let @xmath420 be the set of basic templates at stage @xmath70 . let @xmath421 and @xmath422 . fix a basic template @xmath423 , represent @xmath35 with the interval @xmath85 , and let @xmath424 be the position in @xmath85 where the global cut occurs . let @xmath425 and @xmath426 be such that @xmath427 and @xmath428 . let @xmath429 . in this section we define two partitions of @xmath35 , which we call the intermediate and local block partitions of @xmath35 . we also define the intermediate and local cuts in @xmath35 . depending on @xmath430 , the _ intermediate block partition of @xmath85 _ is either ( [ simpleintblockpartition ] ) or ( [ complicatedintblockpartition ] ) below : if @xmath431 , then @xmath432 where @xmath433 $ ] for @xmath434 . if @xmath435 , then @xmath436 where : * @xmath437 $ ] * @xmath438 $ ] for @xmath439 * @xmath440 $ ] . whether the intermediate block partition of @xmath85 takes the form ( [ simpleintblockpartition ] ) or ( [ complicatedintblockpartition ] ) , the sub - intervals @xmath441 are called the _ intermediate blocks in @xmath85_. the _ intermediate block partition of @xmath35 _ is either @xmath442 or @xmath443 depending on whether the intermediate block partition of @xmath85 takes the form ( [ simpleintblockpartition ] ) or ( [ complicatedintblockpartition ] ) , respectively . either way , the sub - templates @xmath444 consist of those levels in @xmath35 that occur in positions from @xmath445 , and are called the _ intermediate blocks in @xmath35_. the first level in an intermediate block @xmath446 is called an _ intermediate cut in @xmath35_. [ jlocalblocks ] the _ local block partition of @xmath417 _ is @xmath447 where @xmath448 $ ] for @xmath449 . the sub - intervals @xmath450 , each of which has length @xmath451 , are called the _ local blocks in @xmath417_. [ localblocks1 ] if @xmath444 is an intermediate block of height @xmath452 , then the _ local block partition of @xmath446 _ is @xmath453 where the sub - templates @xmath454 consist of those levels in @xmath446 that occur in positions from @xmath455 , and are called the _ local blocks in @xmath446_. given an interval @xmath456 \subset { \mathbb z}$ ] of length @xmath457 and @xmath458 , denote the subinterval of @xmath84 consisting of the last @xmath459 integers in @xmath84 by @xmath460^d$ ] and the subinterval of @xmath84 consisting of the first @xmath461 integers by @xmath460_d$ ] . namely , @xmath460_d = i\setminus[i]^d$ ] . [ jalocalblocks ] the _ local block partitions of @xmath462_a$ ] and @xmath462^a$ ] _ are @xmath463_a = j'(0 ) \cup j'(1 ) \cup \cdots \cup j'(l'-c-2 ) \cup \left [ j'(l'-c-1 ) \right]_b\ ] ] and @xmath464^a = \begin{cases } \left[j'(l'-c-1 ) \right]^b \cup j'(l'-c ) \cup \cdots \cup j'(l'-1 ) & \mbox { if } a \neq 0 \\ \emptyset & \mbox { if } a = 0.\end{cases}\ ] ] the sub - intervals in the partitions ( [ j_apartition ] ) and ( [ j^apartition ] ) are called the _ local blocks in @xmath462_a$ ] and @xmath462^a$]_. if @xmath431 , then @xmath465 , so the local block partition ( [ j_apartition ] ) of @xmath462_0 $ ] is identical to the local block partition ( [ jpartition ] ) of @xmath417 . also @xmath466_0 $ ] . therefore definition [ jlocalblocks ] is just a special case of definition [ jalocalblocks ] . [ localblocks2 ] intermediate blocks of height @xmath467 can only exist if @xmath435 ; in this case , @xmath468 and @xmath469 are the only two such . represent @xmath468 with @xmath462_a$ ] and @xmath469 with @xmath462^a$ ] . then the _ local block partitions of @xmath468 and @xmath469 _ are @xmath470_b\ ] ] and @xmath471^b \cup \tau(0 , l'-c ) \cup \cdots \cup \tau(0 , l'-1),\ ] ] where the sub - templates in the partitions ( [ taulpartition ] ) and ( [ tau0partition ] ) consist of those levels in @xmath468 and @xmath469 that occur in positions from the corresponding local blocks in @xmath462_a$ ] and @xmath462^a$ ] , and are called the _ local blocks in @xmath468 and @xmath469_. a _ local block in @xmath35 _ is any local block from definitions [ localblocks1 ] or [ localblocks2 ] . the _ local block partition of @xmath35 _ is the partition of @xmath35 into its local blocks . the first level in a local block is called a _ local cut in @xmath35_. recall that each basic template @xmath423 has four variation types : diminished , augmented , missing , and extra . and within a given variation type , there may be multiple templates . but any such template is constructed by applying one or both of the following operations to @xmath35 : 1 . remove one level from the bottom of a local block in @xmath35 , 2 . insert one new level at the top ( or bottom ) of a local block in @xmath35 . therefore , if @xmath37 is a diminished , augmented , missing , or extra version of @xmath35 , then the intermediate and local block partitions of @xmath35 determine intermediate and local block partitions of @xmath37 , as follows : 1 . suppose a level , @xmath459 , is removed from the bottom of a local block @xmath27 . let @xmath472 be the intermediate block that contains @xmath27 . then , in the intermediate and local block partitions of @xmath37 , replace @xmath27 and @xmath472 with @xmath473 and @xmath474 . leave all other intermediate and local blocks alone . 2 . suppose a level , @xmath459 , is inserted at the top ( resp . , bottom ) of a local block @xmath27 . let @xmath472 be the intermediate block that contains @xmath27 . then , in the intermediate and local block partitions of @xmath37 , replace @xmath27 and @xmath472 with @xmath475 and @xmath476 , where , in the new order , @xmath459 is the top ( resp . , bottom ) level in @xmath475 . leave all other intermediate and local blocks alone . recall the definition of a reordering map in section [ reorderingmaps ] . we will employ two types of reordering maps : _ global _ and _ intermediate_. roughly speaking , the _ global _ reordering map takes a small number of levels in @xmath35 that occur in positions from the bottom global safe zone in @xmath85 and moves them , one by one , up to the top global safe zone . this has the effect of sliding all levels in @xmath35 that do not occur in the global safe zones ( those in the middle part " ) down . here is the formal definition : the _ global reordering map for @xmath35 _ is a map @xmath477 $ ] where 1 . if @xmath478 , then @xmath479 and therefore @xmath129 is the identity . 2 . if @xmath480 then @xmath479 is defined so that @xmath129 takes the bottom levels in @xmath481 and inserts them , in order , directly after the top levels in @xmath482 . observe that this means @xmath129 shifts all levels in @xmath35 that do not occur in the global safe zones down by exactly @xmath430 positions . more formally when @xmath480 , for each intermediate block @xmath445 in @xmath85 , let @xmath483 and @xmath484 denote the smallest and largest integers in @xmath445 , respectively . then @xmath485 , where , for @xmath486 , @xmath487 [ intcutspostp1 ] the intermediate cuts in @xmath139 that do not occur in the global safe zones occur in positions in @xmath85 that are congruent to @xmath105 mod @xmath452 . this follows immediately from the definitions and implies that the intermediate cuts in @xmath139 that do not occur in the global safe zones line up " with intermediate cuts in the zero template @xmath488 . this in turn implies that the intermediate blocks in @xmath139 that do not occur in the global safe zones also line up with intermediate blocks in the zero template . if @xmath431 , then the intermediate block partition of @xmath35 is given by ( [ simpleinttau ] ) and @xmath489 . in this case we define the intermediate block partition of @xmath139 to be identical to the intermediate block partition of @xmath35 . formally @xmath490(0 ) \cup [ \hat p_1(\tau)](1 ) \cup \cdots \cup [ \hat p_1(\tau)](l-1),\ ] ] where each @xmath491(m ) = \tau(m)$ ] . in this @xmath478 case we define the the local block partition of @xmath139 to be identical to the local block partition of @xmath35 ( see ( [ talltaupartition ] ) ) . formally , for @xmath434 , @xmath492(m ) = [ \hat p_1(\tau)](m , 0 ) \cup [ \hat p_1(\tau)](m , 1 ) \cup \cdots \cup [ \hat p_1(\tau)](m , l'-1)\ ] ] where each @xmath491(m , i ) = \tau(m , i)$ ] . now suppose @xmath435 , so that the intermediate block partition of @xmath35 is given by ( [ complicatedinttau ] ) . then @xmath129 takes the bottom levels in @xmath481 and inserts them , in order , directly after the top levels in @xmath482 . then we define the intermediate block partition of @xmath139 by @xmath493(0 ) \cup [ \hat p_1(\tau)](1 ) \cup \cdots \cup [ \hat p_1(\tau)](l)\ ] ] where : * for @xmath494 , @xmath491(m ) = \tau(m)$ ] , and * for @xmath495 , @xmath491(m)$ ] is @xmath446 with its bottom level removed , and @xmath491(l - a+m-1)$ ] is @xmath496 with the bottom level of @xmath446 inserted at the top . if @xmath435 and @xmath494 , then define the local block partition of @xmath491(m)$ ] to be identical to the local block partition of @xmath35 . formally , if @xmath497 , then the local block partition of @xmath491(m)$ ] is given by ( [ tallinttaupartition ] ) . the local block partitions of @xmath491(l)$ ] and @xmath491(0)$ ] are @xmath498(l ) = [ \hat p_1(\tau)](l , 0 ) \cup \cdots \cup [ \hat p_1(\tau)](l , l'-c-2 ) \cup \left [ [ \hat p_1(\tau)](l , l'-c-1)\right]_b\ ] ] and @xmath499(0 ) = \left [ [ \hat p_1(\tau)]\right(0 , l'-c-1)]^b \cup [ \hat p_1(\tau)](0 , l'-c ) \cup \cdots \cup [ \hat p_1(\tau)](0 , l'-1)\ ] ] where @xmath491(l , i ) = \tau(l , i)$ ] for each @xmath189 . finally , if @xmath435 and @xmath495 , then define the local block partitions of @xmath491(m)$ ] and @xmath491(l - a+m-1)$ ] by @xmath500(m ) = [ \hat p_1(\tau)](m , 0 ) \cup [ \hat p_1(\tau)](m , 1 ) \cup \cdots \cup [ \hat p_1(\tau)](m , l'-1)\ ] ] and @xmath500(l - a+m-1 ) = [ \hat p_1(\tau)](l - a+m-1 , 0 ) \cup \cdots \cup [ \hat p_1(\tau)](l - a+m-1 , l'-1)\ ] ] where : * for @xmath501 , @xmath491(m , i ) = \tau(m , i)$ ] , * for @xmath502 , @xmath491(l - a+m-1 , i ) = \tau(l - a+m-1 , i)$ ] , and * @xmath491(m , 0)$ ] is @xmath503 with the bottom level removed , and @xmath491(l - a+m-1 , l'-1)$ ] is @xmath504 with the bottom level of @xmath503 inserted at the top . [ twocases ] let @xmath491(m)$ ] be an intermediate block in @xmath139 that does not occur in a global safe zone . then the levels in @xmath491(m)$ ] occur in the same positions as those in the intermediate block @xmath505 in the zero template where @xmath506 therefore if @xmath35 is an odometer template then @xmath507(m ) ) = \zeta(\tau^\star(\ell))$ ] . if @xmath35 is a morse template then either @xmath507(m ) ) = \zeta(\tau^\star(\ell))$ ] or @xmath507(m ) ) = \overline{\zeta(\tau^\star(\ell))}$ ] . follows from proposition [ intcutspostp1 ] . in this section we define a reordering map @xmath508 for each intermediate block @xmath491(m)$ ] in @xmath139 . we then define the _ intermediate reordering map _ to be the concatenation , denoted @xmath128 , of the maps @xmath508 ( either @xmath509 or @xmath510 ) . here is the formal definition . [ intermediatereorderingmap ] depending on whether @xmath431 or @xmath435 , define the _ intermediate reordering map for @xmath35 _ to be the concatenation @xmath509 or @xmath511 , respectively , where each @xmath512 $ ] is defined as follows . if @xmath491(m)$ ] is an intermediate block in @xmath139 that occurs in a global safe zone , then @xmath513 identity . if @xmath491(m)$ ] is an intermediate block in @xmath139 that does not occur in a global safe zone then , letting @xmath514 proposition [ twocases ] determines two cases : 1 . if @xmath507(m ) ) = \zeta(\tau^\star(x))$ ] , then @xmath515 is the identity . if @xmath507(m ) ) = \overline{\zeta(\tau^\star(x))}$ ] , then @xmath516 is defined so that @xmath508 takes the bottom levels in @xmath491(m , 1 ) , \ldots , [ \hat p_1(\tau)](m , 2^{k_n})$ ] and inserts them , in order , directly after the top levels in @xmath491(m , l ' - 2^{k_n } ) , \ldots , [ \hat p_1(\tau)](m , l'-1)$ ] . this shifts all other local blocks down by exactly @xmath451 positions . more formally , for @xmath517 , let @xmath518 and @xmath519 denote the smallest and largest integers in @xmath520 , respectively . then let @xmath521 where , for @xmath522 , @xmath523 let @xmath525 and , depending on whether @xmath431 or @xmath435 , define the intermediate block partition of @xmath526 to be either @xmath527 respectively , where , for each @xmath208 , @xmath528(m ) \right)$ ] . given an intermediate block @xmath529 in @xmath526 , if @xmath513 identity , then define the local block partition of @xmath529 to be identical to the local block partition of @xmath491(m)$ ] . denote the local blocks in @xmath529 by @xmath530 , @xmath531_b$ ] , or @xmath531^b$ ] depending on the form that @xmath532(m)$ ] takes ( the various possible forms are described in section [ p1hatpartitions ] ) . if @xmath533 identity , then @xmath534 , as in definition [ intermediatereorderingmap ] . in this case , define the local block partition of @xmath529 to be @xmath535 where : * @xmath536 consists of the @xmath451 consecutive levels in @xmath529 that occur in positions from @xmath537 $ ] in @xmath417 ; * for @xmath538 , @xmath530 consists of the @xmath539 consecutive levels in @xmath529 that occur in positions from @xmath540 $ ] in @xmath417 ; * for @xmath541 , @xmath530 consists of the @xmath451 consecutive levels in @xmath529 that occur in positions from @xmath542 $ ] in @xmath417 ; * for @xmath543 , @xmath530 consists of the @xmath544 consecutive levels in @xmath529 that occur in positions from @xmath545\ ] ] in @xmath417 ; and * @xmath546 consists of the @xmath451 consecutive levels in @xmath529 that occur in positions from @xmath547 $ ] in @xmath417 . all the machinery we have defined up to this point will be used to construct the partial interval bijections from figure [ thefigure ] . the global and intermediate reordering maps in particular are defined to guarantee the existence of good sets " . recall from the introduction that these are subsets of @xmath2 and @xmath3 defined for each stage of the construction on which all partial interval bijections match . define the _ good set in @xmath417 _ to be the subset @xmath548 given by @xmath549 then @xmath550 consists of every other local block in @xmath417 that does not occur in the intermediate safe zones . for @xmath551 , define @xmath552 and @xmath553 note that @xmath554 ; we call @xmath555 the _ good set within @xmath85 at stage @xmath70_. for @xmath31 congruent to @xmath556 ( @xmath557 ) we set @xmath558 ( @xmath559 ) to consist of those levels in @xmath560 ( @xmath561 ) that occur in positions from @xmath562 . we call @xmath563 or @xmath564 the _ good set at stage @xmath70_. note that , because the global and intermediate safe zones make up a very small proportion of @xmath85 , @xmath555 consists of roughly half of @xmath85 . therefore @xmath565 and @xmath566 are both approximately @xmath567 . [ everyotherproperty ] let @xmath568 and @xmath569 be basic templates in @xmath570 @xmath571 , and let @xmath572 @xmath573 . let @xmath574 and @xmath575 be the levels that occur in position @xmath576 within @xmath577 and @xmath578 . then @xmath579 . the global reordering map was defined so that , after global reordering , the intermediate block structures of all templates line up outside of the global safe zone ( see proposition [ twocases ] ) . for the odometer system , then , after global reordering , every local block that is not in the global safe zones matches the corresponding local block in the zero template , in the sense that it has the same image under @xmath580 . for the morse system , the intermediate reordering map was defined so that , after both global and intermediate reordering , every other local block that is neither in the global nor intermediate safe zones moved down by exactly one complete local block . as a consequence of the combinatoric structure of the morse system , now every other such local block matches the corresponding local block in the zero template . to see why this is the case , consider the morse sequence and its flip : @xmath581 notice that , if we shift the flip to the left by one coordinate , then it matches the morse sequence in every other coordinate . the same would be true for sequences of substitution blocks ( just replace @xmath105 s and @xmath154 s with blocks @xmath582 and @xmath583 ) . [ independenceofgoodsets ] let @xmath584 be natural numbers congruent to @xmath585 @xmath586 . then @xmath587 and @xmath588 @xmath589 and @xmath590 are independent events in @xmath3 @xmath591 . we only give the proof in the case where @xmath31 and @xmath208 are congruent to @xmath556 . the other case follows similarly . since @xmath584 , we have @xmath592 , where @xmath593 . then @xmath594 is partitioned into @xmath145 copies " of @xmath51 : @xmath595 where each @xmath596 is a union of exactly @xmath451 consecutive levels in @xmath594 , and @xmath597 for each @xmath189 . because @xmath598 is a union of local blocks from stage @xmath599 and @xmath600 , we can write @xmath598 as a disjoint union @xmath601 for some @xmath602 . because ( [ qmpartition ] ) is obtained by cutting and stacking @xmath51 @xmath145 times , we have @xmath603 , and therefore @xmath604 , for each @xmath189 . since ( [ gmdisjoint ] ) is a disjoint union , @xmath605 therefore @xmath598 and @xmath606 are independent . suppose @xmath607 and @xmath608 . then the local block partitions of @xmath609 and @xmath610 are @xmath611^b \cup \hat \tau(0 , l'-c ) \cup \cdots \hat \tau(0 , l'-1)\ ] ] and @xmath612_b,\ ] ] as described section [ blockpartitionsoftauhat ] , and in ( [ localpartitionof0th ] ) and ( [ localpartitionoflth ] ) . notice that the very first local block in @xmath526 has height @xmath613 , i.e. , it is a _ partial _ block . correspondingly , the bottom @xmath614 levels in @xmath615 can be thought of as the top @xmath614 levels in a collection of templates @xmath279 . define @xmath616 similarly , define @xmath617 if @xmath618 , then define @xmath619 . given @xmath607 , in this section we define a collection of partial interval bijections of the form @xmath620 there will be one such partial interval bijection for each pair @xmath621 . we call @xmath622 and @xmath623 the _ bottom _ and _ top sticky notes _ of @xmath624 . if @xmath618 , then each @xmath624 is defined on an interval of length @xmath625 . however , if @xmath608 , then each @xmath624 is defined on an interval of length approximately @xmath626 . if @xmath618 , then define just one top sticky note and just one bottom sticky note , namely , the trivial partial interval bijection @xmath93 $ ] where @xmath629 . if @xmath608 , then , for each @xmath630 , define @xmath631 . to conserve notation , since the particular choice of @xmath248 will not matter for our construction , the expression @xmath627 does not indicate dependence on @xmath248 . similarly , for each @xmath632 , define @xmath633 . the local block partition of @xmath526 determines a partition of @xmath635 $ ] into subintervals , which in turn determines a partition of @xmath615 , which we call the _ local block partition of @xmath615_. the local block partition of @xmath615 has the form @xmath636 where the @xmath637 are consecutively occurring sets of levels in @xmath615 . if @xmath618 , then , for each @xmath189 , @xmath638 is a template from @xmath639 . in this case , define @xmath640 if @xmath608 , then for @xmath641 , @xmath638 is a template from @xmath639 . in this case , define @xmath642 [ stage4matching ] let @xmath643 be a level in @xmath615 from the good set at stage @xmath74 . then , given @xmath568 and @xmath569 in @xmath644 , either both @xmath645 and @xmath646 are undefined on @xmath313 , or else @xmath647 . by construction , together with proposition [ everyotherproperty ] . it follows from lemma [ stage4matching ] that it is possible to extend the domain of definition of @xmath648 to a partial interval bijection , call it @xmath649 , between the set of all levels in @xmath650 and those levels in @xmath526 that occur in positions from @xmath651 . moreover , the same bijection can be used for all reordered templates @xmath526 so that , given @xmath568 and @xmath569 in @xmath644 and @xmath652 , @xmath653 . finally , define @xmath654 where @xmath655 denotes the restriction of @xmath656 to the subinterval of @xmath657 corresponding to @xmath658 . ( note that @xmath655 is well - defined because @xmath656 is the identity outside of @xmath659 . ) [ stage4nobottomortop ] neither the bottom level nor the top level of @xmath615 is in the domain of any @xmath624 . given @xmath624 of the form @xmath660 determined by @xmath621 , the bottom level in @xmath615 occurs as the cut in @xmath661 . but by proposition [ nobottomtoporcut ] , this level is not in the range of the stage 2 map @xmath662 . therefore it is not in the domain of @xmath622 . [ stage4nocut ] given @xmath607 , the global cut in @xmath35 is not in the range of any @xmath624 . follows from proposition [ nobottomortop ] . [ stage4matchingprop ] let @xmath643 . then , given @xmath568 and @xmath569 in @xmath644 and partial interval bijections @xmath663 and @xmath664 of the form ( [ basicpibform ] ) , both @xmath663 and @xmath664 are defined on @xmath313 , and @xmath665 . follows from lemma [ stage4matching ] together with the definition of the partial interval bijections @xmath624 . we now define maps to diminished , augmented , missing , and extra templates , and establish analogues of lemmas [ stage2diminishedmatching ] and [ stage2augmentedmatching ] , which are needed to establish analogues of propositions [ possiblebodyoverlaps ] and [ bodyoverlapsvalid ] in stage 8 . given @xmath607 that is neither the zero - template nor the one - template , define @xmath666 to match with @xmath624 , and define @xmath667 to match with @xmath668 . here we mean that the top and bottom sticky notes as well as the body maps all match . similar to stage 2 , the fact that these definitions are possible follows from proposition [ stage4nocut ] . if @xmath607 is either the zero - template or the one - template , then for @xmath331 , define @xmath669 to match with @xmath624 , and define @xmath670 to match with @xmath671 . [ stage4dimlemma ] if @xmath607 is neither the zero - template nor the one - template , then @xmath667 matches with @xmath672 . if @xmath607 is either the zero - template or the one - template , then for @xmath331 , @xmath670 matches with @xmath673 . by construction . given @xmath607 and @xmath331 , define @xmath674 to match with @xmath624 . if @xmath35 is neither the zero - template nor the one - template , then define @xmath675 to match with @xmath668 . if @xmath35 is the zero - template or the one - template , then define @xmath675 to match with @xmath671 . [ stage4auglemma ] given @xmath331 , and @xmath607 , if @xmath35 is neither the zero - template nor the one - template , then @xmath675 matches with @xmath676 . if @xmath35 is the zero - template or the one - template , then @xmath675 matches with @xmath677 . by construction . given @xmath678 , define @xmath679 . note that , if @xmath680 , then @xmath681 . define @xmath682 note that @xmath683 , but @xmath684 since , in particular , if @xmath630 , then @xmath685 , but @xmath686 . define @xmath687 as follows . for each @xmath688 , define @xmath689 , and , for each @xmath690 , define @xmath691 . define @xmath692 . similarly , given @xmath607 that is neither the zero - template nor the one - template , define @xmath693 . note that , if @xmath694 , then @xmath695 . define @xmath696 note that , if @xmath632 , then @xmath697 . stage 6 is analogous to stage 4 , but with one new layer of complexity : whereas the maps in stage 4 were defined as _ concatenations _ of maps from stage 2 , the maps in stage 6 will be _ overlapping concatenations _ of the maps from stage 4 . the top and bottom sticky notes defined in stage 4 are used to glue these overlapping concatenations together . we need to verify that there are enough sticky notes defined in stage 4 to choose from so that these overlapping concatenations are well - defined . the global reordering map in stage 6 , denoted @xmath129 , is the same as it was in stage 4 except , of course , now @xmath698 $ ] . in stage 4 , an intermediate reordering map was also used to ensure that every other local block that does not occur in a safe zone has the same image under @xmath580 as the corresponding local block in the zero - template ( see proposition [ everyotherproperty ] ) . but an intermediate reordering map is not needed in stage 6 because the odometer has just one canonical tower at each stage not two . so in stage 6 , let the intermediate reordering map @xmath128 be simply the identity . given @xmath699 , let @xmath700 . define the intermediate and local block partitions of @xmath701 , as well as the intermediate block partition of @xmath702 , in exactly the same way that they were defined in stage 4 ( see sections [ p1hatpartitions ] and [ blockpartitionsoftauhat ] ) except , of course , that @xmath35 is replaced with @xmath248 . given an intermediate block @xmath703 in @xmath702 , because the intermediate reordering map @xmath128 is simply the identity , just like in stage 4 , define the local block partition of @xmath703 to be identical to the local block partition of @xmath704(m)$ ] . the good set within @xmath85 at stage @xmath75 , @xmath705 , is defined in section [ thegoodset ] . let @xmath706 , called the _ good set at stage @xmath75 _ , consist of those levels in @xmath707 that occur in positions from @xmath705 . note that proposition [ everyotherproperty ] holds in stage 6 . given @xmath699 , the sets @xmath708 and @xmath709 are defined analogously to the definitions in section [ headsandtails ] ( just replace @xmath35 , @xmath248 , @xmath66 , and @xmath710 with @xmath248 , @xmath35 , @xmath710 , and @xmath711 , respectively ) . however , there is an added layer of complexity in stage 6 : if @xmath712 ( or @xmath713 ) , then @xmath35 has head and tail sets of its own , @xmath714 and @xmath715 . much like in stage 6 , given @xmath699 , the partial interval bijections @xmath305 take the form @xmath716 where @xmath717 represents _ overlapping concatenation _ as defined in definition [ overlapconcatenation1 ] . there is one such partial interval bijection for each pair @xmath718 . the maps @xmath719 , @xmath720 , and @xmath721 are defined analogously to the definitions in sections [ stage4stickynotes ] and [ stage4bodymap ] except that within the body map @xmath720 ( section [ stage4bodymap ] ) , concatenations ( @xmath722 ) are replaced with overlapping concatenations ( @xmath723 ) . we now show that the overlapping concatenations within @xmath720 can be glued together . suppose @xmath608 ( the case that @xmath618 is nearly identical ) . then , following what was done in section [ stage4bodymap ] , we define @xmath724 to take the form @xmath725 [ possiblebodyoverlaps ] with the templates @xmath726 from ( [ phiomegabodydecomp ] ) listed in order of overlapping concatenation ( @xmath727 ) , the following six types of successive pairs can occur : 1 . @xmath728 where @xmath729 , 2 . @xmath730 where @xmath731 , 3 . @xmath732 , where @xmath331 and @xmath733 , 4 . @xmath734 , where @xmath731 , 5 . @xmath735 , where @xmath341 and @xmath733 , and 6 . @xmath736 , where @xmath731 . by construction . [ bodyoverlapsvalid ] in any of the six cases of proposition [ possiblebodyoverlaps ] , top and bottom sticky notes can be chosen so that the overlapping concatenation of the corresponding partial interval bijections @xmath737 are well - defined . case 1 is trivial because partial interval bijections to the zero- and one - templates do not have sticky notes ( there is no overlap to worry about ) . for case 2 , observe that if @xmath306 is a basic template from stage 2 such that @xmath630 , then @xmath632 as well , so we can glue @xmath738 together with itself by picking such @xmath739 on the overlap . for case 3 , observe that there exists @xmath306 such that @xmath740 and @xmath741 . lemma [ stage2diminishedmatching ] then guarantees that there is a bottom sticky note on @xmath624 that matches with a top sticky note on @xmath742 . cases 4 - 6 are similar . the following shows that is well defined . top and bottom sticky notes can be chosen so that the overlapping concatenations @xmath743 and @xmath744 are well - defined . similar to the proof of proposition [ bodyoverlapsvalid ] . the following propositions are analogous to propositions from stage 4 . neither the bottom level nor the top level of @xmath745 or @xmath746 is in the domain of any @xmath305 . follows from proposition [ stage4nocut ] . given @xmath699 , the global cut in @xmath248 is not in the range of any @xmath305 . follows from proposition [ stage4nobottomortop ] . let @xmath747 and let @xmath748 be a level that occurs in position @xmath576 , where @xmath331 . then , given @xmath661 and @xmath749 in @xmath750 and partial interval bijections @xmath662 and @xmath751 of the form ( [ phiomegabodydecomp ] ) , both @xmath662 and @xmath751 are defined on @xmath313 , and @xmath752 . proposition [ everyotherproperty ] can be used to show that lemma [ stage4matching ] holds in stage 6 ( with appropriate notational modifications ) . the proposition follows . the definitions of these modified maps are analogous to those in stage 4 . maps to missing and extra templates are needed to define the maps at stage 8 in places where individual levels have been deleted or inserted . analogues of lemmas [ stage4dimlemma ] and [ stage4auglemma ] hold in stage 6 , and are used to glue sticky notes together in stage 10 . the essential components of the induction have now been established . stages @xmath753 are analogous to stage @xmath74 , while stages @xmath754 are analogous to stage @xmath75 . in each stage @xmath76 , concatenations of the ( inverses of ) the partial interval bijections at stage @xmath755 are glued together using definition [ overlapconcatenation2 ] . given a level @xmath643 , proposition [ stage4matchingprop ] guarantees that @xmath756 is a level in @xmath277 that does not depend on which @xmath757 is used . moreover , proposition [ stage4matchingprop ] holds in every stage @xmath72 , which permits us to define @xmath758 to be the restriction of @xmath759 to @xmath606 . similarly , for @xmath73 , we can define @xmath760 to be the restriction of @xmath761 to @xmath606 . given @xmath72 and @xmath762 , let @xmath763 denote the unique @xmath398-canonical cylinder in @xmath3 that contains @xmath764 . let @xmath765 be the set of @xmath762 such that @xmath766 for infinitely many @xmath31 . similarly , given @xmath73 and @xmath178 , let @xmath767 denote the unique @xmath398-canonical cylinder in @xmath2 that contains @xmath167 . let @xmath768 be the set of @xmath178 such that @xmath769 for infinitely many @xmath31 . the sets @xmath770 and @xmath771 are @xmath11 sets of full measure . by proposition [ independenceofgoodsets ] , the sets @xmath606 for @xmath72 are independent with respect to @xmath224 . therefor @xmath772 by the borel - cantelli lemma . moreover , the sets @xmath606 are open , so @xmath770 is a @xmath11 subset of @xmath3 . the argument for @xmath771 is similar . [ containmentlemma ] given @xmath773 , if @xmath72 and @xmath774 and @xmath775 are levels such that @xmath776 , then @xmath777 . also , if @xmath73 , @xmath778 , @xmath779 , and @xmath780 , then @xmath781 . given @xmath782 , the map @xmath624 at stage @xmath783 is an extension of a concatenation of the maps @xmath624 at stage @xmath31 it follows that @xmath784 where , depending on the context , @xmath57 refers either to the map @xmath785 or to the map @xmath786 . the argument when @xmath73 is nearly identical . given @xmath787 , let @xmath788 be the increasing sequence of indices , each congruent to @xmath585 , such that each @xmath789 . then , by lemma [ containmentlemma ] , the levels @xmath790 form a nested sequence . it follows that there is a unique point in the intersection @xmath791 . similarly , if @xmath792 is the analogous sequence of indices for @xmath793 , then there is a unique point in the intersection @xmath794 . this permits the following definition . given @xmath793 , let @xmath795 be the unique point in the intersection @xmath794 . given @xmath787 , let @xmath796 be the unique point in the intersection @xmath791 . the maps @xmath6 and @xmath797 are continuous in the relative topologies on @xmath770 and @xmath771 . let @xmath793 and @xmath798 . choose @xmath73 large enough so that @xmath799 and such that @xmath800 . choose @xmath801 small enough so that @xmath802 implies @xmath803 . in particular , @xmath802 implies @xmath804 . it then follows from lemma [ containmentlemma ] that , for all @xmath773 such that @xmath805 , @xmath806 and , for all @xmath807 such that @xmath808 , @xmath809 therefore @xmath810 the continuity of @xmath797 is proved similarly . the maps @xmath6 and @xmath797 are measure preserving . fix a natural number @xmath811 and a level @xmath459 in @xmath594 . we wish to show that @xmath812 . given @xmath813 , let @xmath814 where @xmath56 is the map @xmath815 . let @xmath816 and @xmath817 . observe that @xmath818 as discussed in section [ thegoodset ] , for large @xmath31 , @xmath819 consists of roughly half of the levels in @xmath820 , so we can ( conservatively ) assume @xmath819 consists of at least @xmath821 of them . then @xmath822 for @xmath823 , let @xmath824 denote the set of levels in @xmath820 that are contained in levels from @xmath825 , and recursively define @xmath826 and @xmath827 . observe that @xmath828 and @xmath829 ( again , @xmath821 is a conservative lower bound it is actually closer to @xmath830 ) . it follows from ( [ dnequation ] ) and ( [ dnminusaequation ] ) that @xmath831 it now follows from ( [ enequation ] ) and ( [ enminusaequation ] ) that @xmath832 this implies that @xmath833 letting @xmath834 gives @xmath835 . this being true for each level @xmath836 trivially implies that @xmath837 . if @xmath793 and @xmath838 , then @xmath839 . similarly , if @xmath787 and @xmath840 , then @xmath841 . let @xmath798 and let @xmath73 be such that @xmath842 and @xmath843 . let @xmath844 be such that @xmath845 . then @xmath846 where , depending on the context , @xmath57 refers either to the map @xmath847 or to the map @xmath848 . since @xmath849 , we have @xmath850 , which implies @xmath851 . therefore @xmath852 . the second statement is proved similarly . let @xmath853 then @xmath854 and @xmath855 are invariant @xmath11 subsets of full measure . let @xmath856 and @xmath857 . then @xmath9 and @xmath10 are full measure subsets because @xmath6 is measure preserving . and @xmath9 and @xmath10 are @xmath11 subsets because @xmath6 is continuous in the relative topology on @xmath854 . in this section we show that @xmath9 and @xmath10 are invariant and that @xmath4 is an orbit equivalence that is a conjugacy when restricted to @xmath858 , the good set at stage @xmath222 . [ orbitsintoorbits ] for @xmath859 , @xmath6 maps the @xmath0-orbit of @xmath167 into the @xmath1-orbit of @xmath795 . and for @xmath860 , @xmath797 maps the @xmath1-orbit of @xmath764 into the @xmath0-orbit of @xmath796 . let @xmath861 and @xmath862 for some @xmath863 . recall that the bottom global safe zone at stage @xmath31 has height @xmath864 . pick @xmath865 such that @xmath866 and @xmath867 . then @xmath868 and @xmath869 are in the same tower in @xmath870 and @xmath871 . moreover , since each @xmath872 consists of complete local blocks from stage @xmath755 , for each @xmath873 ( @xmath73 ) such that @xmath874 , we have @xmath875 and @xmath876 . fix such @xmath31 . for @xmath877 , let @xmath878 be such that @xmath879 . let @xmath880 be such that @xmath881 and @xmath882 . let @xmath883 ( @xmath884 ) such that @xmath885 . for @xmath877 , let @xmath886 be levels such that @xmath887 . then because the partial interval bijections @xmath305 at stage @xmath208 are extensions of concatenations of those at stage @xmath31 , we have @xmath888 . since @xmath208 was arbitrary , it follows that @xmath889 . therefore @xmath6 maps the backward @xmath0-orbit of @xmath167 into the @xmath1-orbit of @xmath795 . by similar argument , @xmath6 maps the forward @xmath0-orbit of @xmath167 into the @xmath1-orbit of @xmath795 . the argument for @xmath797 is also similar . we have already seen that @xmath9 and @xmath10 are @xmath11 subsets of full measure . let @xmath892 . then @xmath859 and @xmath893 by definition let @xmath894 for some @xmath895 . then @xmath896 because @xmath854 is @xmath0-invariant . and @xmath897 by lemma [ orbitsintoorbits ] ( and because @xmath893 ) . therefore @xmath898 , so @xmath9 is @xmath0-invariant . similarly , @xmath10 is @xmath1-invariant . now suppose @xmath764 is a point in the @xmath1-orbit of @xmath795 . then @xmath899 , so @xmath796 is in the orbit of @xmath167 . hence @xmath840 . therefore @xmath6 carries the orbit of @xmath167 onto the orbit of @xmath795 . the map @xmath6 is a conjugacy between the two induced maps @xmath900 and @xmath901 . both @xmath858 and @xmath902 are nearly clopen . therefore @xmath6 is a nearly continuous kakutani equivalence of @xmath0 and @xmath1 . let @xmath868 and @xmath903 be two points in @xmath904 . then , by theorem [ x0theorem ] , @xmath905 for some @xmath906 . we wish to show that @xmath907 and @xmath147 have the same sign . this will imply that @xmath6 restricted to @xmath904 is order preserving on orbits , and hence a conjugacy between the induced maps . as we saw in the proof of lemma [ orbitsintoorbits ] , @xmath908 for all sufficiently large @xmath73 . let @xmath909 be minimal among such @xmath31 . if @xmath910 , then @xmath907 and @xmath147 automatically have the same sign because each partial interval bijection @xmath305 at stage @xmath222 maps the levels in @xmath858 in an order - preserving way , and this is then carried through the diagram via concatenations . if @xmath911 , then because @xmath909 is minimal , @xmath912 and @xmath913 must lie in different towers in @xmath914 . and the partial interval bijections at stage @xmath909 are extensions of concatenations of those at stage @xmath915 . so if @xmath863 , then the stage-(@xmath915 ) partial interval bijection that acts on @xmath868 in stage @xmath909 comes before the stage-(@xmath915 ) partial interval bijection that acts on @xmath869 in stage @xmath909 . this order - preservation at stage @xmath909 is then carried through the diagram via concatenations , so @xmath916 . similarly , @xmath917 implies @xmath918 . m. k. roychowdhury , _ @xmath919-odometer and the binary odometer are finitarily orbit equivalent _ , ergodic theory and related fields , contemp . 430 , amer . soc . , providence , ri , 2007 , 123 - 134 .

ergodic homeomorphisms @xmath0 and @xmath1 of polish probability spaces @xmath2 and @xmath3 are _ evenly kakutani equivalent _ if there is an orbit equivalence @xmath4 between full measure subsets of @xmath2 and @xmath3 such that , for some @xmath5 of positive measure , @xmath6 restricts to a measurable isomorphism of the induced systems @xmath7 and @xmath8 . the study of even kakutani equivalence dates back to the seventies , and it is well known that any two zero - entropy loosely bernoulli systems are evenly kakutani equivalent . but even kakutani equivalence is a purely measurable relation , while systems such as the morse minimal system are both measurable and topological . recently del junco , rudolph and weiss studied a new relation called _ nearly continuous kakutani equivalence_. a nearly continuous kakutani equivalence is an even kakutani equivalence where also @xmath9 and @xmath10 are invariant @xmath11 sets , @xmath12 is within measure zero of both open and closed , and @xmath6 is a homeomorphism from @xmath9 to @xmath10 . it is known that nearly continuous kakutani equivalence is strictly stronger than even kakutani equivalence , and nearly continuous kakutani equivalence is the natural strengthening of even kakutani equivalence to the _ nearly continuous _ category the category where maps are continuous after sets of measure zero are removed . in this paper we show that the morse minimal substitution system is nearly continuously kakutani equivalent to the binary odometer .

1404.0246

it is common accepted that braking of pulsars is caused by the magneto - dipole radiation of the rotating magnetic star . in this case the rate of losses of the neutron star rotation energy can be equated to the power of its magneto - dipole radiation : @xmath1 + where _ i _ is the moment of inertia of the neutron star , @xmath2 - the angular speed of its rotation , @xmath3 - its magnetic moment , @xmath0 - the angle between the rotation axis and the magnetic moment , _ c _ - speed of light . for standard parameters of neutron stars : masses of order of the solar mass ( @xmath4 ) and radii _ r _ of order of @xmath5 cm we can put _ i _ = @xmath6 . for the magnetic moment we have @xmath7 + here @xmath8 is the magnetic induction at the magnetic pole , @xmath9 ? the induction at the magnetic equator . instead of @xmath2 the rotation period @xmath10 is usually measured and we can obtain from ( 1 ) and ( 2 ) : @xmath11 + this equality is used usually to calculate magnetic inductions of pulsars assuming that @xmath12 for all objects . the known catalogs ( see , for example manchester et al . , 2005 ) contain as a rule @xmath9 instead of @xmath8 . here we propose to decline the assumption on the constancy of @xmath13 and use some estimations of this parameter to calculate more accurate values of pulsar magnetic inductions . in a number of our works ( malov & nikitina , 2011a , b , 2013 ) some methods for calculations of the angle @xmath0 have been put forward and applied to some catalogs of pulsars ( keith et al . , 2010 ; van ommen et al . , 1997 ; weltevrede & johnston , 2008 ) at approximately 10 , 20 and 30 cm . basic equations for this aim are ( manchester & taylor , 1977 ) : @xmath14 @xmath15 + here @xmath16 is the angle between the line of sight and the rotation axis , @xmath17 - the angular radius of the emission cone , @xmath18 - a half of the angular width of the observed pulse , @xmath19 - the position angle of the linear polarization , @xmath20 - longitude . the simplest case for the calculations of the angle @xmath0 is realized when the line of sight passes through the center of the emission cone , i.e. @xmath21 + in this case we can use the dependence of the observed pulse width @xmath22 at the @xmath23 level on the rotation period and determine the lower boundary in the corresponding diagram to obtain @xmath24 + as the result we have from ( 4 ) , ( 5 ) and ( 7 ) ( malov & nikitina , 2011a ) : @xmath25 + the values of angles calculated by this method are denoted as @xmath26 and given in the table 1 . usually polarization measurements are made inside the pulse longitudes only . in this case we can use the maximal derivative of the position angle . from ( 5 ) we have @xmath27 we can obtain from the dependence of @xmath22 on _ p _ by the least squares method @xmath28 + the third equation for the calculations of the angle @xmath0 is ( 4 ) . from these three equations we obtain @xmath29y^2 + 2c(d - b^2)y+c^2d^2-b^2(1+c^2)=0.\\\ ] ] + here @xmath30 + we can transform the equation ( 9 ) to the following form @xmath31 + then finding the value of y from the equation ( 11 ) we can calculate @xmath0 from ( 13 ) . we have calculated values of @xmath0 by this method and list them in the table 1 as @xmath32 . here we correct the misprint in the equation ( 11 ) made in our papers ( malov & nikitina , 2011a , b , 2013 ) . there is an additional way to calculate angles @xmath0 . this way uses observable values of position angles and shapes of average profiles for individual pulsars . in this case , original equations form the closed system for calculations of the angles @xmath17 , @xmath16 and @xmath0 : @xmath33 as the observed pulsar profiles have various forms , the coefficient _ n _ has a different value depending on a profile structure . we put arbitrary the following values of _ n _ ( fig.1 ) . if the ratio of the intensity @xmath34 in the center of the pulse to the maximal intensity @xmath35 is zero then @xmath36 . for @xmath37 @xmath38 , @xmath39 @xmath40 , @xmath41 @xmath42 , and for @xmath43 @xmath44 . it is worth noting that the solution of the system ( 14 ) can be obtained numerically for any value of _ n_. for example , if @xmath45 , the solution for @xmath46 can be obtained from the equation : @xmath47 at n = 2 : @xmath48 y^4 + 2c \left [ c^2 ( 1 + d - 2d^2 ) - 2 - d \right ] y^3 + \left [ 2dc^4 ( 1 - d ) - \right . . - c^2 ( 2d^2 - 6d + 7 ) + 5 \right ] y^2 + 2c \left [ c^2 d^2 + d(1 + c^2 ) - 2 ( c^2 - 1 ) \right ] y + c^2 d^2 ( 1 + c^2 ) - ( c^2 - 1)^2 = 0;\\ \end{array}\ ] ] at n = 3/2 : @xmath49 \sqrt{\frac{1 + \frac{c + y}{\sqrt{c^2 + 2cy + 1}}}{2 } } - c y^2 ( 1 - d ) - y - cd = 0;\ ] ] at n = 5/4 : @xmath50 this method gives angles @xmath51 ( see the table 1 ) . for some pulsars calculations were made by one method only . when it was possible we used two or all three methods . in these cases , the mean value of the angle @xmath0 has been calculated . the resulting values @xmath52 are listed in the table 1 . some other authors ( for example , kuzmin & dagkesamanskaya , 1983 ; kuzmin et al . , 1984 ; lyne & manchester , 1988 ) carried out calculations of the angle @xmath0 earlier for the shorter samples of pulsars using some additional assumptions . we will use further our estimations to calculate magnetic inductions at the surface of the neutron stars . the distribution of the angles @xmath0 from the table 1 ( fig.2 ) shows that the majority of pulsars have rather small inclinations of the magnetic moments . these pulsars are old enough , and we can conclude that they evolve to the aligned geometry . the average characteristic age for our sample of pulsars is @xmath53 years . we must note however that the angles calculated by the method * _ 1 ) _ * are the lower limits of this parameter . this explains partly the predominance of the small values of @xmath0 . from the table 1.,width=453 ] .values of the angle @xmath0 ( deg ) . [ cols="^,^,^,^,^,^,^,^,^,^,^ " , ] 1 . some methods for calculations of the angle @xmath0 between rotation and magnetic axes were applied to obtain the values of @xmath0 for 376 radio pulsars . the distribution of these values shows the predominance of small inclinations of the magnetic axes . 2 . magnetic inductions at the surface of 375 pulsars considered were calculated . there is no the measured derivative @xmath54 for the pulsar j1713 - 3949 and it is excluded from the consideration . the distribution of the calculated magnetic inductions can be described by the gaussian with the maximal value of @xmath55 and the width in the logarithmic scale nearly 1 . the calculated inductions are higher than the catalog equatorial inductions with the mean value of the ratio of these quantities of 5 . for the pulsar j1410 - 7404 @xmath56 . the maximal value of the ratio @xmath57 for the pulsar j2007 + 0809 . this work has been carried out with the financial support of basic research program of the presidium of the russian academy of sciences * _ transitional and explosive processes in astrophysics _ * ( p-41 ) . we thank a.v.biryukov for very useful comments and discussions . 99 keith m.j . , johnston s. , weltevrede p. and kramer m. , 2010 , mnras , 402 , 745 kuzmin a.d . , dagkesamanskaya i.m . , 1983 , soviet astron . letters , 9 , 80 kuzmin a.d . , dagkesamanskaya i.m . , pugachev v.d . , 1984 , soviet astron . letters , 10 , 357 lyne a.g . , manchester r.n . , 1988 , mnras , 243 , 477 manchester r.n . , taylor j.h . , 1977 , pulsars . w.h.freeman and company , san francisco manchester r.n . et al . , 2005 , j. , 129 , 1993 . malov i.f . , nikitina e.b . , 2011a , astron.rep . , 55 , 19 malov i.f . , nikitina e.b . , 2011b , astron.rep . , 55 , 878 malov i.f . , nikitina e.b , 2013 , astron.rep . , 57 , 833 van ommen t.d . et al . , 1997 , mnras , 287 , 1210 weltevrede p. , johnston s. , 2008 , mnras , 391 , 1210

we used the magneto - dipole radiation mechanism for the braking of radio pulsars to calculate the new values of magnetic inductions at the surfaces of neutron stars . for this aim we estimated the angles @xmath0 between the rotation axis and the magnetic moment of the neutron star for 376 radio pulsars using three different methods . it was shown that there was the predominance of small inclinations of the magnetic axes . using the obtained values of the angle @xmath0 we calculated the equatorial magnetic inductions for pulsars considered . these inductions are several times higher as a rule than corresponding values in the known catalogs . * keywords * magnetic fields ; methods : data analysis ; methods : statistical ; _ ( stars : ) _ pulsars : general

1608.08525

the exotic hydrogen like atoms are formed in highly excited states , when negative particles ( @xmath2 ) are stopped in hydrogen . the deexcitation of exotic atoms proceeds via many intermediate states until the ground state is reached or a nuclear reaction takes place . despite a long history of theoretical and experimental studies the kinetics of this atomic cascade is not yet fully understood . the present experiments with the exotic hydrogen - like atoms addresses a number of fundamental problems using precision spectroscopy methods , the success of which relies crucially on a better knowledge of the atomic cascade . the experimental data are mainly appropriate to the processes at the last stage of the atomic cascade ( x - ray yields and the products of the weak or strong interaction of the exotic particle in the low angular momentum states with hydrogen isotopes ) . so the reliable theoretical backgrounds on the processes both in low - lying and in highly excited states are required for the detailed and proper analysis of the experimental data . in this paper we present the _ ab initio _ quantum - mechanical treatment of non - reactive scattering processes of the excited exotic hydrogen atom in collisions with the hydrogenic atom in the ground state : @xmath3 elastic scattering(@xmath4 ) , stark transitions ( @xmath5 ) , and coulomb deexcitation ( cd ) ( @xmath6 ) . here @xmath7 are hydrogen isotopes and @xmath8 ; @xmath9 are the principal and orbital quantum numbers of exotic atom . while the deexcitation processes are obviously essential for the atomic cascade , the role of the collisional processes preserving the principal quantum number @xmath10 is also very important . the stark transitions affect the population of the @xmath11 sublevels and together with the elastic scattering they decelerate the exotic atoms thus influencing their energy distribution during the cascade . starting from the classical paper by leon and bethe @xcite , stark transitions has been treated in the semiclassical straight - line - trajectory approximation ( see @xcite and references therein ) . the fully quantum - mechanical treatment of the elastic scattering and stark transitions based on the adiabatic description was given for the first time in @xcite . recently @xcite , the processes have also been studied in a close - coupling approach with using dipole approximation for interaction potential and taking the electron screening effect into account by the model . the cross sections calculated in approaches @xcite are in good agreement . concerning the cd process , the situation is much less defined , especially for low @xmath10 . the first work on the cd process was performed by bracci and fiorentini @xcite in frame of the semiclassical approach with some additional approximations . in the following numerous papers ( see @xcite and references therein ) the cd process is considered within the asymptotic approaches using the adiabatic hidden crossing theory . the cd calculations were also performed in the classical - trajectory monte - carlo ( ctmc ) approach@xcite . while the coulomb deexcitation cross sections obtained in ctmc approach are in fair agreement with the semiclassical ones of bracci and fiorentini @xcite , the more elaborated advanced adiabatic approach ( aaa ) @xcite gives too small cd cross sections to explain the experimental data @xcite . the reasons of such a strong discrepancy are not clear . one can only assume that the semiclassical model of bracci and fiorentini as well as the ctmc approach are not valid for low - lying states and at low energies . the processes ( 1 ) have been treated recently by authors in a unified manner in the framework of the close - coupling ( cc ) approach ( see for detail @xcite ) . the differential and integral cross sections for the processes ( 1 ) have been calculated for muonic , pionic and antiprotonic hydrogen atoms in excited states with @xmath12 and in a kinetic energy range relevant for cascade calculations . the energy shifts of the @xmath0 states due to vacuum polarization and strong interaction ( for hadronic atoms ) are included in the close - coupling method . this approach allows to obtain the self - consistent description of all the processes ( 1 ) and is free from the additional approximations used in previous studies . the calculated differential and integral cross sections presented in this paper mainly refer to the cd process to illustrate some of our new results obtained quite recently . all open channels corresponding to the exotic atom states with @xmath13 ( @xmath14 is principal quantum number in the entrance channel ) have been included in the cc calculations . the effect of the closed channels with @xmath15 was also studied and will be discussed below . ( throughout the whole paper the cross sections are given in atomic units . ) the hamiltonian of the system @xmath16 ( after separation of the c.m . motion ) is given by @xmath17 where @xmath18 is the reduced mass of the system , @xmath19 is the radius vector between the c.m . of the colliding atoms , @xmath20 and @xmath21 are their inner coordinates . the interaction potential , @xmath22 , is a sum of four coulomb pair interactions between the projectile atom and the target atom particles . @xmath23 and @xmath24 are the hydrogen - like hamiltonians of the free exotic and hydrogen atom , whose eigenfunctions together with the angular wave function @xmath25 of the relative motion form the basis states , @xmath26 , with the conserving total angular momentum ( @xmath27 ) and parity @xmath28 . in the present consideration we use the `` frozen '' electron approximation . the cc approach can be extended in a straightforward manner to include the target electron excitations . the total wave function of the system are expanded in terms of the basis states as follows @xmath29 the expansion ( 3 ) leads to the close - coupling second order differential equations for the radial functions of the relative motion , @xmath30 , @xmath31 the channel wave number is defined as @xmath32 , where @xmath33 and ( @xmath34 ) are the energy of the relative motion and the exotic atom quantum numbers in the entrance channel , respectively . the bound energy of the exotic atom , @xmath35 includes the eigenvalue of @xmath23 , @xmath36 , and the energy shift , @xmath37 , due to the vacuum polarization and strong interaction ( in case of adronic atom ) . hereafter , the energy @xmath33 will be referred to @xmath38 in the entrance channel ( we assume here that @xmath39 ) the matrix elements of the interaction potential , @xmath22 , @xmath40 are obtained by averaging it over the electron wave function of the @xmath41-state and then applying the multipole expansion . the integration over ( @xmath42 reduces the matrix elements ( 5 ) to the multiple finite sum . at fixed @xmath33 the coupled differential equations ( 4 ) for the given @xmath43 and @xmath44 values are solved numerically by the numerov method with the standing - wave boundary conditions involving the real symmetrical @xmath45-matrix related to @xmath46-matrix by the equation @xmath47 . all open channels corresponding to the exotic atom states with @xmath13 have been included in the close - coupling calculations . the effect of closed channels with @xmath15 was also investigated and will be discussed below . in the next sections we present and discuss the following total cross sections of the scattering processes : the partial cross section @xmath48 @xmath49 the total cross section of the @xmath50 transition @xmath51 , and the @xmath52-averaged cross section @xmath53 , and also the analogous differential cross sections . the total cross sections calculated in adiabatic @xcite and present cc approaches are as a whole in good agreement at energies @xmath54 ev . the angular distributions obtained in the present cc and adiabatic @xcite calculations coincide in the region of the diffraction maximums but demonstrate significant differences in backward hemisphere ( see the left side of fig.1 ) . -averaged differential ( elastic and stark ) cross sections ( left ) : adiabatic model @xcite(dashed lines ) and present cc ( solid lines ) . the @xmath52-averaged differential stark cross sections for @xmath55 collisions vs. cms scattering angle @xmath56 at @xmath57 ev ( right).,title="fig:",scaledwidth=45.0% ] -averaged differential ( elastic and stark ) cross sections ( left ) : adiabatic model @xcite(dashed lines ) and present cc ( solid lines ) . the @xmath52-averaged differential stark cross sections for @xmath55 collisions vs. cms scattering angle @xmath56 at @xmath57 ev ( right).,title="fig:",scaledwidth=45.0% ] the typical angular distributions of the stark @xmath58 transitions for @xmath59 are shown in fig . 1 ( right ) . it is well known @xcite that cross sections of these processes are similar to the diffraction scattering ( at energies more @xmath60 ev ) with a strong forward peak enhancing with increasing energy and a set of maxima and minima . while the elastic cross sections always have a strong peak at @xmath61 , the first maximum position in the stark cross section depends on the @xmath62 value . in particular , for @xmath63 this maximum is at finite scattering angles as it is also remarked in @xcite . + some of our cc results for @xmath64 collision are shown in figs . 2,3 : the e - dependence of @xmath52-averaged stark cross sections for @xmath12 ( fig.2 ) and cross sections of the @xmath58 transitions for @xmath65 at @xmath66 ev ( fig.3 ) . it is seen from fig.3 that @xmath0-state energy shift due to strong interaction leads to essential suppression of both stark @xmath67 and elastic @xmath68 transitions ( we used the value @xmath69 ev for the @xmath41-state shift and @xmath70 for @xmath0-states ) at energies compared with the shift value . the same effect is observed for all hadronic atoms . the nature of the cd process is quite different from the one of the elastic or stark processes . in contrast to elastic ( stark ) scattering cd process is accompanied by the large energy release ( tens and hundreds ev ) and occur at smaller distances , so the details of the short - range interaction are more important for the treatment of cd process than for elastic processes . this difference between elastic and deeply inelastic processes is illustrated by fig . 4 where the @xmath43 dependence of the partial - wave l - averaged cross sections @xmath71 for @xmath72 at three fixed energies 0.1 , 2 , and 50 ev is shown . it is seen that a substantial part of the cd cross section ( @xmath73 ) comes from the partial waves with rather a low @xmath43 in contrast to the elastic ( stark ) process . in spite of the value of the total cd cross section constitutes about few per cent of the total elastic cross section it is incorrect to treat cd in the framework of the perturbation theory . in each significant for cd partial wave the value of the cd cross section is comparable with the elastic cross section . the cd process is determined by the short - range behaviour of the wave function which changes when new channels are included in calculation . therefore , to calculate the transition @xmath74 in a proper way it is impossible to be restricted with the two - level approximation ( @xmath10 and @xmath75 ) and the states with other nearest @xmath10 should be involved . we studied the dependence of the results on the number of included channels and found that the inclusion of the channels with @xmath76 leads to a strong suppression of the main @xmath74 transitions in comparison with the two - level cc approximation and due to this the total cd cross section is also suppressed @xcite . so in all our cd calculations we included all the open channel with @xmath13 . it should be noted that all the previous calculations of cd realized within semiclassical or adiabatic approaches used a two - level approximation . it is obvious that two - level approximation is not absolutely suitable for the treatment of transitions with @xmath77 . in contrast to the elastic scattering and stark transitions where the `` dipole '' approximation ( @xmath78 ) and even more rough dipole potential ( used in @xcite ) gives reasonable results ( at not too low energies ) , in case of the deeply inelastic process such as cd the full interaction must be used as it is clear from the present study . it is commonly believed @xcite that the cd cross sections at low energies behave like @xmath79 . in order to reveal more explicitly the distinction from the @xmath79 behaviour the present @xmath52-averaged cd cross sections multiplied by energy are shown in fig.5 in comparison with the results of the sc model @xcite for @xmath80 and ctmc calculations @xcite for @xmath81 . as it is seen , the energy dependence of the cc cross sections in the region @xmath82 ev , as a whole , is in a qualitative agreement with the results @xcite and @xcite . at lower energies the present cc results reveal @xmath83 dependence in accordance with the wigner threshold law ( the similar behaviour is seen in the ctmc results @xcite for @xmath81 ) and in disagreement with @xmath79 dependence obtained in the sc model @xcite and aa approach @xcite . the distribution over the final states @xmath84 is strongly different from the sc results @xcite as illustrated in fig . the cc calculations predict that the transitions with @xmath85 are strongly enhanced as compared with the results of the two - level approaches @xcite . the @xmath85 transitions make up a substantial fraction ( 16% - 37% ) of the total cd cross section for @xmath86 . the angular distributions of cd was calculated for the first time in @xcite . earlier in the cascade calculations the angular distributions of the cd process are presumed to be isotropic . the calculated cross sections for individual @xmath87 transitions with @xmath88 and 2 at energy @xmath89 ev are shown in fig . 7 . in fig . 8 the @xmath52-averaged cross sections for the @xmath90 transition at different values of the relative energy from 0.01 up to 15 ev are presented . ( left ) and with @xmath91 ( right ) for @xmath59 at @xmath92 ev.,title="fig:",scaledwidth=45.0% ] + ( left ) and with @xmath91 ( right ) for @xmath59 at @xmath92 ev.,title="fig:",scaledwidth=45.0% ] we found that the angular distributions both of the individual and @xmath52-averaged cross sections ( excluding very low energies ) are far from isotropic : as a whole the scattering at @xmath93 and @xmath94 is noticeably enhanced . the cross sections for @xmath95 transitions ( see fig . 7 ) have ( as for elastic scattering ) a more pronounced diffraction structure with sharp maxima and minima and a strong peak at zero angle as compared with the smoother angular dependence for other cd transitions . the increase of kinetic energy enhances asymmetry in the angular dependence of the @xmath52-averaged cross sections and decreases the role of the backward scattering ( see fig . in order to illustrate the influence of the @xmath0 state energy shifts on the cd cross sections , we calculated the cd cross sections for @xmath96 collisions both with and without taking energy shifts into account . the effect is the most pronounced for the low - lying states and is illustrated in fig . one can see that the maximal suppression due to the energy shift of @xmath97 state is about two times at very low energy both for @xmath98 and @xmath99 transitions , while at @xmath100 ev does not exceed 15% ( for details see @xcite ) . the energy dependence of the cd cross sections for the collisions of the @xmath101 atom ( @xmath65 ) with the hydrogen atom obtained in the cc approach is shown in fig . 10 for @xmath65 and the different values of @xmath102 = 1 , 2 and 3 . the special features of these cross sections are the following : the similar energy dependence but sharper than that of the elastic scattering and stark transitions ( see also in fig . 10 ) ; the contribution of the transitions with @xmath103 is comparable with the one for @xmath102 = 1 and is equal about 50% . the effect of the @xmath0 state shifts in the @xmath52-averaged cd cross sections is negligible due to small statistical weight of the @xmath0-state . in fig . 10 we also compare our results with those obtained in the semiclassical model for the @xmath104 transition . the satisfactory agreement is observed , but this agreement is quite occasional and takes no place for other @xmath10 values . the distribution over the final states @xmath84 is completely different from the sc results @xcite as it was illustrated in fig . the presented above results were obtained by the solution of the close - coupling equations including all the open channels . although the channels with @xmath15 for @xmath105 and at energies less or @xmath106 ev are strongly closed , they can essentially change the open - channel wave functions at short ranges determining cd process . in the present studies we included in the calculations all the open channels with @xmath107 and step by step added closed channels with @xmath108 to achieve the convergency of results . in fig . 11 some of our preliminary results are shown for the @xmath52-averaged cd cross sections at @xmath109 ev . according to our investigation , the closed channel effect on cd cross sections depends crucially on the transition considered . as it is seen from fig . 11 in case of @xmath110 atom the inclusion in the basis set of the closed channels results in a more pronounced effect for low - lying states . the investigations of convergency with the increase of number of closed channels are very time - consuming and are continuing now . the unified treatment of the elastic scattering , stark transitions and coulomb deexcitation is presented within the quantum - mechanical close - coupling approach . the differential and integral cross sections for the above processes are calculated for the excited muonic , pionic and antiprotonic hydrogen atoms with @xmath12 and relative energies relevant to the cascade calculations . the new results for cd process are obtained : anisotropy of the angular distribution , substantial fraction of @xmath103 transitions up to @xmath111 ( for @xmath112 at all energies under consideration ) , and a proper threshold behaviour of the cd cross - section . the calculated cross - sections are very important for the kinetics of the atomic cascade and give a more reliable theoretical input for the improved version of the cascade model @xcite . v.p.popov and v.n.pomerantsev , hyp . interact . * 101/102 * , 133 ( 1996 ) ; * 119 * , 133 ( 1999 ) ; * 119 * , 137 ( 1999 ) . v.v.gusev , v.p.popov and v.n.pomerantsev , hyp . interact . * 119 * , 141 ( 1999 ) . t.s.jensen and v.e.markushin , eur.journ.phys . d , * 19 * , 165 ( 2001 ) . l.bracci and g.fiorentini , nuovo cim . * 43a * , 9 ( 1978 ) . l.i.ponomarev and e.a.solovov , yad.fiz . * 65 * , 1615 ( 2002 ) . a.v . kravtsov and a.i . mikhailov , yad.fiz . * 69 * , 395 ( 2006 ) t.s.jensen and v.e.markushin , eur.journ.phys . d , * 21 * , 261 ( 2001 ) . a.badertscher _ et al_. , europhys . lett . * 54 * ( 3 ) , 313 ( 2001 ) . korenman , v.n . pomerantsev , and v.p . popov , jetp lett . * 81 * , 543 ( 2005 ) ; nucl - th/0501036 . v.p.popov and v.n.pomerantsev , nucl - th/051207 . v.n.pomerantsev and v.p.popov , jetp lett . * 83 * , 331 ( 2006 ) . v.n.pomerantsev and v.p.popov , phys . a * 73 * , 040501(r ) ( 2006 ) . t.s.jensen , v.p.popov and v.n.pomerantsev , arxiv:0712.3010 ( 2007 ) .

the scattering processes of exotic atoms in excited states from hydrogen such as elastic scattering , stark transitions and coulomb de - excitation are studied within a close coupling approach . the vacuum polarization and the strong interaction shifts of @xmath0-states ( in case of hadronic atoms ) are taken into account . the differential and integral cross sections of the above processes are calculated to use them as the input in cascade calculations . the effect of closed channels on the scattering processes is investigated . * scattering processes of excited exotic atoms : close - coupling approach * + _ institute of nuclear physics , moscow state university , 119992 moscow , russia@xmath1 institution1 _

0712.3111

studies of massive galaxy clusters and groups at @xmath5 typically find environments with little - to - no star formation activity , in sharp contrast with the field . over - dense regions are dominated by red , passively - evolving s0 and elliptical galaxies , whereas more sparsely - populated regions tend to have galaxies with spiral morphologies , younger stellar populations , and systematically higher star formation rates @xcite . an observed trend of increasing blue galaxy fraction with redshift ( the butcher - oemler effect ; * ? ? ? * ) has been interpreted as evidence for higher star formation activity and stellar mass build - up in higher redshift clusters or alternatively , that star formation is quenched more recently by one or more processes in over - dense regions . several physical mechanisms can account for the quenching of star formation in over - dense regions ( for a review , see * ? ? ? galaxies in environments with sufficiently low velocity dispersions can be strongly perturbed by mergers . galaxies can also be transformed more gradually by an ensemble of small perturbations with neighbours , a process called harassment @xcite . tidal forces can strip away a galaxy s halo gas ( starvation ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) , cutting off a fuel source for future star formation and leading to a gradual decline in sf activity . in the high - density cores of massive clusters , the hot ( @xmath6 ) intra - cluster medium ( icm ) can quench star formation by removing gas from galaxies via ram - pressure stripping @xcite . the relative strengths of these physical mechanisms are strongly dependent on the cluster or group properties ( dynamical state , mass , and intra - cluster or intra - group medium ) and environment . targeted studies of galaxy clusters or groups at @xmath7 have revealed overwhelming evidence that galaxy transformation occurs not just in dense cluster cores , but at lower densities characteristic of cluster outskirts or galaxy groups @xcite . studies with star formation tracers in the ir @xcite , uv @xcite , and optical emission - line measures @xcite have shown evidence of _ pre - processing _ , whereby infalling galaxies undergo changes prior to their arrival in the galaxy cluster , or galaxies are transformed entirely in the group environment @xcite . the pre - processing hypothesis has also been supported by studies of the environmental dependence on galaxy morphology @xcite and colour @xcite . numerical simulations have also been used to study the causes and implications of galaxy pre - processing . @xcite showed that the dominant physical processes galaxies are likely subjected to in group environments , specifically the frequent weak tidal interactions of harassment , are capable of transforming late - type , disk - dominated galaxies into bulge - dominated , early - types . furthermore , @xcite used simulations of dark matter halo merger trees , with semi - analytic models ( sams ) to populate the haloes with galaxies , and traced the histories of the simulated galaxies that ended up accreting onto cluster - mass haloes in different epochs . in doing so , @xcite determined what fraction of those cluster galaxies had resided in haloes characteristic of group - masses for a long enough time to have been pre - processed prior to entering the cluster . the results of their simulation showed that at low redshift a large fraction of cluster galaxies could have been affected by their environment prior to entering the cluster , while at earlier epochs the fraction of pre - processed galaxies in clusters should steadily decline . the fraction of cluster galaxies affected by pre - processing in the @xcite simulation depends on the assumed timescale for the physical process(es ) in group environments to affect galaxies , and also has a stellar mass dependence . although many assumptions go into this simulation , the result highlights a key point that the role of pre - processing has likely varied significantly over cosmic time , and that at @xmath8 pre - processing should be extremely prevalent . recent studies have suggested that the quenching of sf activity in cosmic history is primarily driven by two distinct , and possibly separable , components : secular evolution ( or ` mass quenching ' ) and environmentally - driven processes ( or environment quenching ; * ? ? ? * ; * ? ? ? . however , see also @xcite for a discussion about how _ history bias _ affects one s ability to disentangle mass- and environment - quenching . nevertheless , any attempt to examine the environmental dependence on galaxy evolution must include a careful account for the possibility that one s galaxy selection function has mass biases , particularly since the galaxy stellar mass function is known to vary with environment @xcite . concerns about biases introduced by the galaxy selection function are compounded when examining galaxies over a wide range in redshift , as one s sensitivity , in galaxy mass and in other properties , like sfr , will undoubtedly also vary with @xmath9 . as a result , in many of these studies that extend to higher-@xmath9 one must restrict one s sample to only massive galaxies with high sfrs , and thereby have a less complete picture of the effects of environment on galaxy evolution . furthermore , studies extending to higher-@xmath9 tend to sample a smaller dynamic range of environments , which similarly reduces one s ability to draw general conclusions about environmentally - driven processes . a comprehensive view of galaxy evolution in different environments must be sensitive to a large dynamic range of local densities in order to capture not just the dense regions , like clusters and groups , but the more diffuse filament and void regimes . a key challenge faced when interpreting the many results examining galaxy evolution , in addition to the aforementioned sources of potential bias , is the wide range of methods employed to characterise environment . recently , @xcite used an array of different environmental mapping techniques , which could be roughly grouped into two categories : nearest - neighbour methods , which measure galaxy density with an aperture that changes depending on the local galaxy density , and fixed - aperture techniques , whose apertures do not vary , to examine a mock galaxy catalogue . @xcite found that these techniques can analyse the same data set and get different results , but that the nearest - neighbour methods appear to be optimal for mapping the density fields within massive haloes , while the fixed - aperture methods are better suited for probing superhalo distance scales . therefore , the technique that is optimal to identify large scale structures ( lss ) , like clusters , groups , and filaments , is not necessarily the best choice for measuring the density fields within those structures . in this work , we seek to quantify the role of pre - processing in the local universe by analysing the rest - frame colour and star - formation activity of galaxies as a function of environment over about three orders of magnitude in projected density in the coma supercluster . by focusing on a low-@xmath9 field , we ensure that our sample of galaxies , taken from the sloan digital sky survey ( sdss ; * ? ? ? * ) , is spectroscopically complete down to dwarf masses ( @xmath10 ) . furthermore , we do not have to rely on photometric redshift ( photo-@xmath9 ) measurements , which would introduce additional contamination due to interlopers in our sample and significant smearing along the line - of - sight . to map the environments of the supercluster , we employ two complementary techniques : voronoi tessellation ( vt ) and the minimal spanning tree ( mst ) . the former is a nearest - neighbour - based approach , which can measure the local density field effectively over the large dynamic range of densities that we find in the coma supercluster . the latter technique is most effective at characterising continuous structures , like clusters , groups , and filaments , and therefore we use the mst to differentiate the types of environment extending over super - halo scales . our combined vt and mst approach allows us to select discrete components of the cosmic web by exploiting the fundamental density contrasts of the cluster , group , filament , and void environments . another benefit of the proximity of our target field is sensitivity to low sfrs , as our combined approach of using the _ galaxy evolution explorer _ ( galex ; * ? ? ? * ) and wide - field infrared survey explorer ( wise ; * ? ? ? * ) to recover unobscured and dust - obscured star - formation activity , respectively , across the entire coma supercluster down to 0.02@xmath11 . section [ sample ] describes the coma supercluster and our sample selection process , with our data from sdss , _ galex _ , and wise . in section [ lss ] we outline our techniques for mapping the lss in the coma supercluster , and in section [ results ] we present our resulting sfrs and comparisons of sf activity and colour versus environment . in section [ discussion ] we discuss the implications of our results , and compare our work to previous studies . throughout this paper we use cosmological parameters @xmath12 , @xmath13 , and @xmath14 km s@xmath15 mpc@xmath15 , where pertinent cosmological quantities have been calculated using the online cosmology calculator of e. l. wright @xcite . throughout we assume a kroupa imf @xcite , and hereafter we will refer to galaxies with stellar masses m@xmath1610@xmath17 as dwarf galaxies , and those with m@xmath1810@xmath17 as massive galaxies . the coma supercluster is an ideal field to observe signatures of galaxy transformation in different environments . it contains two rich galaxy clusters , abell 1656 and abell 1367 , and several galaxy groups distributed in a filamentary pattern between the two clusters @xcite . furthermore , the two clusters are in very different dynamical states , with a1656 being relaxed and a1367 still undergoing significant merging @xcite . the close proximity of the supercluster ( @xmath19 ) allows us to probe its galaxy population down to dwarf masses ( @xmath20 ) with a spectroscopically complete sample , and the geometric alignment of the supercluster , with the galaxy distribution extending largely perpendicular to our line - of - sight @xcite , makes it an ideal case study to examine galaxies in a wide range of environments with minimal projection effects . past studies of the coma supercluster have been primarily focused on the most massive cluster , a1656 . its low redshift , high galactic latitude ( @xmath21 ) , and richness ensured that it received a great deal of attention from observers in early extragalactic studies ( see * ? ? ? * and references therein ) . a significant substructure @xmath01 mpc sw of the centre of a1656 , which has since been positively identified as an infalling group @xcite , was noticed first by the high local concentration of galaxies centred on the galaxy ngc 4839 , and was later confirmed by a diffuse x - ray profile and radial velocities of member galaxies . @xcite found a large number of ` post - starburst ' ( or k+a ) galaxies coincident with the ngc 4839 group , leading to the conclusion that the ngc 4839 group had experienced a burst of star formation @xmath22 gyr ago @xcite , possibly triggered by tidal effects of the group - cluster merging @xcite . a study by @xcite , examining emission - line and k+a galaxies in the core region of a1656 , found a spatial correlation between these galaxies and known x - ray sub - structures from @xcite , which indicates that stripping from the shocked icm might be an important factor in triggering starbursts , and subsequent quenching , for infalling galaxies . studies of the entire supercluster population had to wait for new all - sky surveys with sufficient sensitivity to detect galaxies down to dwarf masses . furthermore , a positive identification of supercluster members necessitates spectroscopic redshifts , which , prior to the sdss catalog , only existed for the most massive galaxies and those immediately around the two clusters . @xcite were the first to conduct an extensive survey of the environmental dependence of star - formation activity in galaxies using the sdss , by comparing the sfrs , among other spectroscopic and photometric measures of galaxy properties , to the local density around @xmath0122 000 galaxies in the sdss data release one ( dr1 ) . they found that for galaxies at fixed stellar masses , the sfrs sharply decline at higher densities , and that the presence of an active galactic nucleus ( agn ) is also much more common in galaxies with greater local density . @xcite used @xmath23 low-@xmath9 galaxies in sdss dr4 , and showed a strong bimodal distribution between active and passive sf activity , using the equivalent width ew(h@xmath24 ) , and determined that the passive galaxies preferentially lie in regions with higher local galaxy density . one of the first major systematic studies of the environmental dependence of galaxy properties across coma , using spectroscopically - confirmed members , was by @xcite . they used sdss dr7 to select @xmath04000 supercluster members , and characterised the local environment around each galaxy by measuring the volume density of galaxies within a cylinder of radius @xmath25mpc and a half - length of 1000 km s@xmath15 ( but with a slightly modified treatment of galaxies associated with the clusters , as described in appendix [ appendix_b ] ) . they examined the optical colours , morphologies , and frequency of post - starburst galaxies in different environments , and found a weak dependence of galaxy colour and morphology with environment for the most massive galaxies and a strong dependence of colour and morphology on environment for dwarf galaxies , and also that almost all post - starburst galaxies reside in higher - density regions . @xcite , using a similar sample of sdss - selected members of coma , examined the local fraction of star - forming and agn - hosting galaxies ( characterised using sdss spectral line measurements ) vs local projected density . they found a similar broad trend as @xcite , whereby star formation in dwarf galaxies is strongly quenched at higher densities , and more massive galaxies show a weaker dependence on local density . @xcite also used observations from _ spitzer _ mips at 24@xmath26@xmath27 , which are available only for the core regions of a1656 and a1367 , to obtain a complete accounting of star formation activity in the highest - density regions of coma . for a1656 , the more massive of the two clusters , they found significant ir detections only in the infalling regions , and recovered the expected correlation between dust - obscured and unobscured sfr with cluster - centric radius . however , for the less massive a1367 they found the reverse radial dependence for the fraction of star - forming galaxies when sf activity is derived by the optical measure vs ir measure . this result seems to indicate that an accounting of all tracers of sf activity , un - obscured and dust - obscured , may be required to get a clear picture of the quenching of sfr in galaxies . with the release of the wise all - sky survey , we now have access to the ir component of sf activity throughout the entire coma supercluster , and so this work presents the first look sensitive to un - obscured and dust obscured sf activity for virtually all galaxies in all environments of the coma supercluster , down to quiescent sfrs and with a sample of galaxies spanning over two orders of magnitude in stellar mass . with our sfr sensitivity , and our techniques for mapping the components of the cosmic web , we are in a position to put significant quantitative constraints on the degree to which pre - processing affects galaxies at @xmath8 . our sample of supercluster galaxies is selected from dr9 of the sloan digital sky survey ( sdss ; * ? ? ? * ) , which has mapped a large fraction of the sky in @xmath28 bands and performed an extensive optical spectroscopic campaign complete ( over the sdss coverage areas ) for galaxies with @xmath29 mag . we select coma supercluster members from the sdss dr9 galaxy sample following the selection criteria used by @xcite , choosing galaxies with positions ( @xmath30 , @xmath31 ) consistent with the coma supercluster and line - of - sight velocities , @xmath32 , within @xmath332000 km s@xmath15 of either a1656 ( @xmath34 km s@xmath15 ) or a1367 ( @xmath35 km s@xmath15 ) , where the central velocity of each cluster comes from @xcite . to ensure no duplicate objects in our sample , we use only galaxies with the specprimary designation set . to be sure that our sample is spectroscopically complete over the whole supercluster , we select only galaxies with @xmath3617.77 mag . we have also excluded any galaxies with a zwarning flag to indicate a poor redshift determination ( which affects less than one per cent of galaxies in the sample ) . we also require detections in the wise 3.4 and 4.5@xmath26 m bands for our entire sample ( less than 1.5 per cent of our sample lacks detections in these two bands ) , and we apply a stellar mass cut - off ( using stellar masses calculated with wise photometry , see section [ wise ] ) such that all galaxies in our sample have m@xmath37 . the m@xmath38 cut - off excludes just 130 galaxies , but it s necessary to prevent our sample , which is @xmath39-band selected , from being biased at the lowest galaxy masses towards only those dwarf galaxies which are the most actively star - forming . these selection criteria result in a sample of 3505 galaxies over the supercluster region covering @xmath0500 deg@xmath1 on the sky . figure [ fig : coma_galaxies ] plots the galaxy positions over the supercluster . the virial radii for the two clusters plotted in figure [ fig : coma_galaxies ] come from the @xmath40 ( the radius within which the density is equal to 200 times that of the critical density ) values determined by @xcite . having sdss spectra for all of our galaxies , we can also mitigate the contributions of galaxies dominated by an agn , which can otherwise contaminate our sfr estimates . @xcite used the emission lines of the sdss galaxy spectra to classify galaxies according to a baldwin , phillips , & terlevich ( bpt ) diagram @xcite , which cleanly delineates those galaxies which host liner and agn emission compared to those dominated by emission from hii regions . we use the @xcite classifications to identify galaxies dominated by agn or liner emission , and we exclude the wise 22@xmath26 m observations from these galaxies when measuring their sf activity throughout . another benefit of having sdss spectral information on our galaxies is access to the ` 4000 break ' index , @xmath41 , which is a measurement of the ratio of the average flux density in two narrow continuum bands , 3850 - 3950 and 4000 - 4100 @xcite . this index correlates strongly with the age of the stellar population in a galaxy , and has been shown to be a robust proxy for separating quiescent ` red sequence ' galaxies from those that are more actively star - forming in the ` blue cloud ' , with the approximate dividing line between these galaxy populations at @xmath42 @xcite . hereafter , we use measurements of @xmath41 for sdss galaxies from @xcite . we also make use of the h@xmath43 line , specifically the index h@xmath44 @xcite which is described in greater detail in section [ kpa_vs_environ ] . we correct the optical - to - nir band photometry for reddening using the @xcite extinction maps , assuming the extinction curve of @xcite with r@xmath45=3.1 . for the _ galex _ bands we used the extinction corrections of @xcite . we applied _ k_-corrections to our photometry using kcorrect v4_2 @xcite to get all uv - through - nir photometry into the rest frame . . the two massive clusters ( ne : a1656 , sw : a1367 ) are identified by the red circles , which have radii equal to their virial radii . the virial radii come from @xmath40 measurements in @xcite , width=326 ] . [ fig : coma_galaxies ] we measure the un - obscured component of sf activity in coma supercluster galaxies from the _ galaxy evolution explorer _ ( galex ; * ? ? ? * ) gr6/gr7 data release , which includes mappings of the supercluster in near - uv ( nuv ; 1750 - 2750 ) and far - uv ( fuv ; 1350 - 1750 ) bands . matching the sdss galaxy catalogue to galex was done using the mikulski archive for space telescopes ( mast ) database , with a 4@xmath46 search radius centred on the sdss galaxy positions . in cases of multiple galex matches within the search radius , the galex match with a position closest to that of the sdss galaxy coordinate was used . the galex bands are most sensitive to the photospheric emission of stars with masses @xmath47 , and thus the uv continuum measurements provide an excellent tracer of recent star formation . under the assumption of a star formation timescale that s long relative to the ages of these massive stars ( @xmath48 yr ) , and a chosen imf , one can derive a sfr@xmath49 corresponding to a given l@xmath49 . the conversion between an observed luminosity and a sfr will be accurate as long as the emission picked up in one s uv band is dominated by the light of stars younger than 10@xmath50 years . although the fuv and nuv bands are both dominated by emission from young stars , if there is recent or on - going sf activity , the nuv band contains a greater fraction of contaminating flux from stars as old as @xmath010@xmath51 years @xcite . our survey covers roughly 500 sq . degrees , and so it is unsurprising that the _ galex _ observation depths vary greatly across the supercluster . almost the entire supercluster has been mapped with _ galex _ at various depths , but about 5 per cent of our galaxy sample of supercluster members do not lie in a _ galex _ coverage area . the galaxies that are outside of _ galex _ coverage regions are flagged so that they are excluded from further analysis involving sf activity in the supercluster , as the sfrs we measure from wise alone will necessarily be lower limits . the shallowest observation with _ galex _ in coma has an exposure time of just 60 seconds , while the deepest is about 3@xmath5210@xmath53 seconds . therefore , to ensure that the sensitivity of our catalogue to sfr@xmath49 is uniform across the supercluster , we must carefully account for the variation in completeness due to differences in survey depth . we measure completeness in a representative sample of _ galex _ nuv and fuv maps in coma , including the shallowest maps , by extracting a supercluster galaxy detected in the map and re - inserting that galaxy , with a range of normalisations , into the maps . for each normalisation , we insert 100 of these ` fake ' galaxies into a fuv and nuv map with random positions , and then repeat 100 times for a total of 10@xmath54 randomly placed galaxies per flux bin . we then run source extractor @xcite on each of the 100 maps per flux bin to determine the fraction of the ` fake ' galaxies that we recover as a function of flux density . figure [ fig : galex_completeness ] shows the completeness that we measure for the shallowest fuv and nuv map ( with 60 second exposure time ) of the coma supercluster . we have taken the fluxes corresponding to 75 per cent completeness in the shallowest fuv and nuv maps ( indicated by vertical dashed lines in figure [ fig : galex_completeness ] ) , and we exclude any uv data for galaxies detected with fluxes below these completeness thresholds from our results . the completeness limit in nuv indicates that our coma supercluster catalogue is 75% complete to sfr@xmath55 in all environments . at the mean redshift of our coma supercluster sample.,width=307 ] as figure [ fig : galex_completeness ] shows , our sfr sensitivity in the coma supercluster is far greater in the nuv than in the fuv , and therefore we choose to use the nuv band to derive sfrs . to avoid having our sfr estimates significantly skewed by the presence of an older stellar population , which can contaminate the nuv band to a greater degree than in the fuv , we will ignore nuv - based sfr estimates for any galaxy whose sdss spectrum shows a strong ` 4000 break ' , based on @xmath56 . this is explored in detail in section [ galexsfrs ] . a complete measure of sfr , especially for galaxies with reasonably high dust content , must include dust - obscured ( indirect ) tracers of sf activity . a significant step forward in the measurement of star - formation activity in the coma supercluster has recently been made possible with the all - sky data release from the wide - field infrared survey explorer ( wise ; * ? ? ? * ) , which has mapped the mid - infrared sky in four bands ( w1-w4 ) centred at 3.4 , 4.6 , 12.0 , and 22.0@xmath26 m . of particular interest to our analysis is the fourth band , which probes the blue - ward side of the dust emission curve in star - forming galaxies . we match our sdss catalogue to the wise point source catalogue by searching in a 5@xmath46 radius around each sdss galaxy position , and selecting the wise match whose position is closest to that of the sdss galaxy . for galaxies matched to the wise point source catalogue , we ignore w4 fluxes whose signal - to - noise in w4 is less than three . we measured completeness in w4 for a representative sample of the coma supercluster , following the procedure outlined in section [ galex ] . for wise w4 we are 75% complete to f@xmath57}$ ] @xmath58 4.7 mjy , which means our measurements of sfr@xmath59 are complete to [email protected] @xmath11 for members of the coma supercluster ( at l@xmath61 ) . for comparison , the only survey prior to wise capable of detecting dust - obscured star formation activity across the entire supercluster was iras @xcite , whose completeness limit for l@xmath59 at coma with its 25@xmath26 m band is a full two orders of magnitude higher ( l@xmath62 ) . the two shortest - wavelength wise bands , which sample the red side of the 1.6 stellar photospheric feature , can be used to robustly estimate total galactic stellar masses ( m@xmath38 ) . recent work by @xcite gives a calibration between flux densities measured in _ irac ch1 and ch2 and a galaxy s stellar mass ( assuming a salpeter imf ) . we have applied the @xcite calibration , converted to a kroupa imf to be consistent with the rest of our study , using the wise bands w1 and w2 . in figure [ fig : st_masses ] we plot a comparison between the stellar masses of coma supercluster galaxies using the @xcite calibration with w1 + w2 photometry and the stellar mass estimates from the sdss @xcite , where we find excellent agreement between these two independent measures of stellar mass . a key component of our analysis is to characterise the local environment in a physically meaningful way . typically , this approach has involved a calculation of the local surface ( 2d ) or volume ( 3d ) density of galaxies to describe the environment near a given galaxy based on the local density . a proper characterization of environment requires not just a measurement of the local galaxy density , but a technique to resolve the structures ( groups , filaments , etc . ) traced by galaxies . the latter task can be very difficult , as lss mapping techniques are often susceptible to biases introduced by a characteristic shape or size scale one is examining . we have developed a technique to map lss and characterise galaxy environments in a manner which is independent of the shape and size scale of the structure , and allows one to estimate local projected galaxy density over an arbitrarily large dynamic range of densities , by using a combination of voronoi tessellation ( to calculate local surface density ) and the minimal spanning tree ( to resolve continuous structures ) . vt is a method of decomposing a set of points into polygonal cells ( voronoi cells ) , where each cell corresponds to one point and the boundaries of the cell enclose all of the surrounding space closest to that point . vt has proven to be a very powerful tool for characterizing the galaxy density field ( e.g. , * ? ? ? * ; * ? ? ? when using a 2d surface density measurement to define the local galaxy density , one must be careful to not allow projection effects to significantly contaminate the density estimates . to help mitigate this we are using only spectroscopically - confirmed members . we show in appendix [ appendix_b ] that our surface densities in the coma supercluster correlate strongly with volume density estimates calculated following the procedures of @xcite . for an additional strong demonstration of the effectiveness of surface density measurements as tracers of the volume density , see figure 1 of @xcite , which shows a tight correlation between surface density and volume density over about two orders of magnitude in density being probed in the abell 901/902 supercluster . a common challenge when trying to measure the galaxy density field is determining the area ( or volume ) over which to measure the density at each galaxy s position . often the density measurements are made on a size scale set by the n@xmath63 nearest neighbour or by using a characteristic kernel with an adaptive size scale . although these methods have flexible size scales , the shape of the region used to calculate the density field is generally fixed . calculating densities reliably over a very large dynamic range of environments requires a technique that can adjust to arbitrary size scales _ and _ local geometry . vt addresses the difficulty of needing both adaptive size and shape by using cells which automatically adjust to the nearby density , and which assume no _ a priori _ shape . in regions of lower source density the cells are larger on average , and the cells get progressively smaller in higher density regions . with vt , one can calculate the local density around a given galaxy s position by taking the inverse of the area of the cell that encloses that galaxy . we compute the vt of the coma supercluster using the qhull function @xcite in * idl * , which calculates convex hulls for the 2d distribution of points . a common issue with vt , and with any method of measuring the local density field , is spurious density estimates arising near the edges of the map , where many cells can be artificially large or even unbounded . we avoid this issue entirely by adding galaxies from the sdss dr9 in a 10 degree - wide ` buffer ' surrounding our coma supercluster map , which all have redshifts consistent with coma and the same selection criteria defined in section [ sdss ] , when calculating our vt . we then exclude these buffer galaxies from further analysis , so the only purpose of these galaxies is to ensure that we do not suffer any edge effects in our supercluster dataset . in figure [ fig : tessduo ] ( left ) we plot the voronoi cells over the supercluster . we compare the distribution of cell densities in the supercluster to a set of 1500 maps generated with source positions randomly distributed , but each with an equal number of sources and an area equal to the coma supercluster map . in figure [ fig : tessduo ] ( right ) we show the cumulative distribution of cell densities observed in coma compared to the mean cumulative distribution of projected voronoi cell densities for the set of random realizations . these random maps are necessary to establish a baseline density with which to compare our observed voronoi cell densities in coma . figure [ fig : tessduo ] ( right ) highlights a couple of key differences between the cell density distribution of the observed supercluster population and that of the random realizations . as one would expect , since the supercluster contains regions of extremely high galaxy density , there is a much larger fraction of cells with densities upwards of 100 - 1000 gal h@xmath64 mpc@xmath65 than in the random distributions . but we also find that there is a larger fraction of cells in the coma supercluster at very low densities , around 1 - 10 gal h@xmath64 mpc@xmath65 , as a more clustered population implies that one finds more prominent voids as well . the cumulative distribution of projected cell densities in our full coma catalogue gradually increases over a density range of log(@xmath66)=0.53.5 [ gal h@xmath64 mpc@xmath65 ] , meaning that our supercluster map samples about three orders of magnitude in projected galaxy density . the random distributions tend to sample about one order of magnitude in galaxy density with any appreciable number of cells . vt measures the density field across the huge dynamic density range of the supercluster , but it is less effective at resolving continuous structures . therefore , we now turn to the complementary approach of the mst . [ cols="^,^ " , ] @xcite , hereafter m2010 , presented a table of 110 dwarf k+a galaxies in the coma supercluster , which come from a parent supercluster galaxy sample very similar to that of the present work , and with an identical set of criteria to select the post - starburst galaxies ( see section [ kpa_vs_environ ] ) . however , we find only 62 such k+a galaxies . we matched the published m2010 k+a catalogue to our own , and find that 91 of the 110 galaxies proposed by m2010 are indeed in our parent supercluster sample . however , only 41 of the 110 proposed k+a galaxies from m2010 show spectral characteristics , based on our sdss line measurements , indicative of being k+a . figure [ fig : h_alpha_delta_compare ] shows the comparison between the m2010 ew(h@xmath24 ) and ew(h@xmath43 ) measurements , and those from the present study , for the 91 proposed k+a galaxies matched to our sample . it s abundantly clear that there are significant differences in the spectral line measurements for both lines relevant for selecting k+a galaxies , with a prominent systematic offset for h@xmath24 and a large scatter in the measurements of h@xmath43 . the reasons for these inconsistencies stem from the different techniques used to obtain the line measurements . the data used in the m2010 catalogue come from the * specline * products , but for the present work we use * galspecline * and * galspecindx * , which are based on the mpa - jhu analysis ( see * ? ? ? for the * specline * data products , the spectrum continuum is fit using a sliding mean / median filter ( c. tremonti , private comm . ) , while for * galspecline * and * galspecindx * the continuum is fit with stellar population synthesis models @xcite . the differences in continuum fitting techniques are responsible for most of the systematic offset between h@xmath24 measurements in figure [ fig : h_alpha_delta_compare ] , as the underlying h@xmath24 from the stellar population has not been subtracted from the measurements used in m2010 . there is also a factor of @xmath68 difference because the * specline * measurements were not converted to the rest - frame , but for the redshift of coma this is a relatively small effect . the differences seen in h@xmath43 come from a combination of the lack of robust continuum fitting and the fact that the * specline * products measure h@xmath43 using a simple gaussian . the h@xmath43 measurement used for our present study comes from the lick index h@xmath44 , first proposed by @xcite , and also described in @xcite , which is designed to optimally capture the h@xmath43 absorption feature in the atmospheres of a - type stars .

we examine the effects of pre - processing across the coma supercluster , including 3505 galaxies over @xmath0500 deg@xmath1 , by quantifying the degree to which star - forming ( sf ) activity is quenched as a function of environment . we characterise environment using the complementary techniques of voronoi tessellation , to measure the density field , and the minimal spanning tree , to define continuous structures , and so we measure sf activity as a function of local density _ and _ the type of environment ( cluster , group , filament , and void ) , and quantify the degree to which environment contributes to quenching of sf activity . our sample covers over two orders of magnitude in stellar mass ( 10@xmath2 to 10@xmath3 ) , and consequently we trace the effects of environment on sf activity for dwarf and massive galaxies , distinguishing so - called ` mass quenching ' from ` environment quenching ' . environmentally - driven quenching of sf activity , measured relative to the void galaxies , occurs to progressively greater degrees in filaments , groups , and clusters , and this trend holds for dwarf and massive galaxies alike . a similar trend is found using @xmath4 colours , but with a more significant disparity between galaxy mass bins driven by increased internal dust extinction in massive galaxies . the sfr distributions of massive sf galaxies have no significant environmental dependence , but the distributions for dwarf sf galaxies are found to be statistically distinct in most environments . pre - processing plays a significant role at low redshift , as environmentally - driven galaxy evolution affects nearly half of the galaxies in the group environment , and a significant fraction of the galaxies in the more diffuse filaments . our study underscores the need for sensitivity to dwarf galaxies to separate mass - driven from environmentally - driven effects , and the use of unbiased tracers of sf activity . [ firstpage ] galaxies : clusters : general galaxies : evolution infrared : galaxies ultraviolet : galaxies .

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precise doppler surveys of main sequence stars have revealed eight companions that have masses under 5 m@xmath6/ @xmath7 , with the orbital inclination , @xmath8 , remaining unknown ( mayor et al . 1999 , marcy & butler 1998 , noyes et al . 1997 , cochran et al . 1997 ) . these `` planetary '' companions exhibit both circular and eccentric orbits , consistent with formation in dissipative circumstellar disks , followed by gravitational perturbations ( cf . lin et al . 1995 , artymowicz 1997 , levison et al . 1998 ) . the semimajor axes are all less than 2.5 au , with most being less than 0.3 au . this `` piling up '' of planets near their host stars appears to be a real effect , although enhanced by the selection effect that favors detection of small orbits . jupiters orbiting between 0.5 and 1.5 au would be easily detected with our current doppler precision of 5 , but none has been found . this distribution of orbits supports models in which orbital migration in a gaseous protoplanetary disk drags jupiter mass planets inward ( lin et al . 1995 , trilling et al . 1998 ) . the distribution of the masses of substellar companions reveals two populations . our survey of 107 gk dwarfs revealed none that had @xmath9= 10 80 m@xmath6@xmath10(marcy & butler 1998 ) . thus , `` brown dwarf '' companions occur with a frequency less than @xmath111% , within 5 au . similarly , mayor et al . ( 1997 , 1999 ) surveyed @xmath11500 gk dwarfs , and found at most 4 companions between 10 80 m@xmath6 . ( hipparcos astrometry has shown that seven previously suspected brown dwarfs from that sample are actually h burning stars . ) in contrast , at least 5% of gk stars harbor companions with masses from 0.5 5 m@xmath6 . for example , in our doppler survey of 107 main sequence stars at lick observatory , we found 6 companions that have @xmath12 = 0.5 5 m@xmath6@xmath10(marcy and butler 1998 , this paper ) . thus , relative to this well populated planetary decade of masses , there exists a brown dwarf `` desert '' at masses 10 80 m@xmath6 , within 5 au . the efforts described above have focussed on g and k type main sequence stars having masses between 0.8 and 1.2 m@xmath5 . the question arises regarding the prevalence of planets around the m dwarfs which constitute 70% of the stars in the galaxy . here we describe the detection of the first apparent planetary companion to an m dwarf , gliese 876 , located 4.7 pc from the sun . gliese 876 ( = hip 113020 ) has v magnitude of 10.1 , a spectral type of m4v and a parallax from hipparcos of 0.213 ( perryman et al . 1997 ) . adopting this parallax and the bolometric correction of delfosse et al . ( 1998 ) gives @xmath13 = 9.52 , which implies a luminosity of , @xmath14 = 0.0124 l@xmath5 . the mass of the star gliese 876 can be derived from its k - band apparent magnitude ( k=5.04 ) and parallax , along with an empirical mass luminosity relation ( henry & mccarthy 1993 ) . this gives @xmath15 = 0.32 @xmath3 0.03 m@xmath5 . gliese 876 is chromospherically inactive ( delfosse et al . 1998 ) , which suggests that it is older than @xmath111 gyr . however its space motion is slow which suggests that its age is less than 10 gyr . its metalicity is not known well , though a preliminary synthesis of the spectrum indicates that it is metal poor by a factor of 23 relative to the sun ( valenti , 1998 ) . doppler shifts for gliese 876 have been obtained at both lick and keck observatories , using the hamilton and hires echelle spectrometers , respectively ( vogt 1987 , vogt et al . the first observations were made in 1994.9 ( at lick ) and in 1997.4 ( at keck ) , and both data sets extend to the present . the calibration of wavelength and the measurement of the spectrometer psf was determined for each exposure and for each 2 chunk of spectrum by using iodine absorption lines superimposed on the stellar spectrum ( butler et al . 1996 ) . figures 1 and 2 show all of the individual velocity measurements as a function of time , along with the separate keplerian fits . the velocities from lick observatory have typical uncertainties of 30 and those from keck are 6 m@xmath10s@xmath16 . poisson statistics of the photons dominate the velocity errors for this relatively faint ( v=10.1 ) star . error bars on all points are the uncertainty in the mean of the velocities ( @xmath17 ) from the many 2 wide chunks into which the spectrum was divided . doppler measurements of gliese 876 at haute provence by delfosse et al . ( mayor et al . 1999 ) also show an amplitude and periodicity in agreement with those reported here , thus constituting an immediate confirmation . it remains to be seen if their orbital parameters agree with those quoted here . the lick and keck data each carry independent and arbitrary velocity zero - points . the relative zero point has been determined by combining the two data sets and adjusting the velocity offset until the keplerian fit ( see 3 ) yields a minimum in the @xmath18 statistic . thus , the lick and keck velocities were forced to have the same zero - point . independent keplerian fits were determined from the lick and keck data sets , and the resulting curves and orbital parameters are shown in figures 1 and 2 . the final orbital parameters are given in table [ orbit ] , based on an orbital fit to the combined data set . the uncertainties reflect the differences in the two independent orbital fits . the two solutions agree within their uncertainties . the joint orbital period is @xmath19 = 60.85 @xmath3 0.15 d , and the eccentricity is @xmath2 = 0.27 @xmath3 0.03 . the orbital solution implies a planetary orbital semi - major axis of 0.21 @xmath3 0.01 au , and a minimum mass of @xmath9= 2.1 @xmath3 0.2 m@xmath6 . this inferred @xmath9is proportional to the assumed mass of the host star ( 0.32 @xmath30.03 m@xmath5 ) which contributes most of the uncertainty in the companion mass . the periodic repetition of an asymmetric radial velocity variation is apparent from the raw data and from the fits in figures 1 and 2 . the orbit is clearly not circular . there is no pattern in the residuals , thus excluding the presence of any second planet with a mass greater than 1 jupiter mass and a period of 4 years or less in the gliese 876 system . the lick and keck velocities can be merged to yield a final fit , as shown in figure 3 . this shows that the two sets share a common orbital phase in addition to similar best fit orbital parameters . we note that two points from lick sit off the keplerian curve by 2@xmath20 , and we suspect that the quoted errors of @xmath1130 in those cases may be underestimated due to the low signal to noise ratios of those spectra . the large velocity amplitude of 220 for gliese 876 leaves orbital motion as the probable cause of the velocity variations . spots on a rotating star can , in principle , cause artifical velocity variations . but for gliese 876 , the equatorial rotation velocity is less than 2 km@xmath10s@xmath16 , and the star is photometrically stable to within @xmath110.02 mag ( marcy and chen 1992 , weiss 1996 , delfosse et al . therefore , spots can not alter the apparent velocity by more than @xmath110.02 @xmath21 2000 = 40 . we have not checked for stellar pulsations , but the photometric stability suggests that any pulsations are not significant here . moreover , acoustic oscillations and g modes for a 0.3 m@xmath5 dwarf would have time scales of minutes and hours , respectively , unlike the observed 60 day velocity period . the companion to gliese 876 , with @xmath9= 2.1 @xmath30.2 m@xmath6 , has a likely mass of 2 to 4 m@xmath6 , assuming unbiased orbital inclinations . for an assumed companion mass of 2.1 m@xmath6 , the astrometric semimajor axis would be 0.28 mas . hipparcos astrometry exhibits no wobble at a 2@xmath20 upper limit of 4 mas ( perryman et al . thus , the upper limit to the companion mass is 29 m@xmath6 . at 4.7 pc , this is the closet known extrasolar planet . the semimajor axis implies an angular separation 0.045 arcsec , with a greatest separation of 0.062 arcsec . it is thus a prime candidate for direct imaging with ir adaptive optics and with interferometry ( i.e. , keck , lbt , sim , vlti ) . astrometric detection is also favored due to : 1 . its close proximity to the sun , 2 . the large mass of the planet , 3 . the low mass of the star , 4 . the small orbital period which permits many cycles to be monitored within a season . gliese 876 is apparently the first m dwarf with a known planetary companion . we have surveyed only 24 m dwarfs from lick observatory during the past 4 yr ( with poor precision of 25 ) , which implies that the occurrence of jupiter mass planets within 2 au of m dwarfs could be a few percent , based on this one detection . the duration and paucity of keck observations render them not yet adequate ( @xmath111 yr ) to add information on the occurrence of planets around m dwarfs . the small orbital semi - major axis of @xmath4=0.21 au and the eccentricity of @xmath2=0.27 pose two profound puzzles regarding the origin of such planetary orbits . there is too little mass within a planetary feeding zone in a nominal protoplanetary disk at distances of 0.2 au to provide 2 jupiter masses of material to a growing planet ( cf . lissauer 1995 ) . one suggestion is that giant planets form several au from the star and then migrate inwards . orbital migration can be induced by interactions between the planet and the gas in the protoplanetary disk , bringing the planet inwards ( lin et al . 1995 , trilling _ et al . _ 1998 ) . however , it is not clear what would cause the planet around gliese 876 to cease its migration at 0.2 au . neither tidal interactions with the star nor a magnetospherically cleared hole at the disk center would extend to 0.2 au , and thus they can not halt the migration . a similar , as yet - unidentified parking mechanism appears needed for the planets around 55 cancri and @xmath22 cor bor ( noyes et al . 1997 , butler et al . 1997 ) . the non circular orbits for both @xmath22 cor bor ( @xmath2=0.16 @xmath3 0.06 ) and for this planet around gliese 876 ( @[email protected] ) imply that significant orbital eccentricities are common for jupiter mass companions orbiting between 0.1 and 0.3 au from their star . some physical mechanism must be identified which generally produces sizable eccentricities , in contrast to the inexplicably low eccentricities of the giant planets in our solar system . infrared speckle reveals no companions to gliese 876 from 1 au outward ( henry & mccarthy 1990 ) , and the lack of large variations in the velocities rule out stellar companions within 1 au . thus , the eccentricity of the planetary companion around gliese 876 could not have been pumped by a stellar companion . apparently , migration , if necessary , did not enforce circularity in the final orbits of gliese 876 or @xmath22 cor bor . one possible explanation is that gravitational scattering of planetary cores ( of earth mass and larger ) can dominate the orbital evolution ( rasio and ford 1996 , weidenschilling and marzari 1997 , lin and ida 1996 ) . orbit crossings and global instabilities among planetesimals in the disk can lead to dramatic orbit changes and large eccentricities ( levison et al . 1998 ) . long lived gas in a protoplanetary disk may lead to circular orbits in such planetary systems . other systems that lose their gas may suffer dynamical instabilities , leading to eccentric orbits at a variety of semimajor axes . however , the latter scenario , if common , does not explain the apparent paucity of jupiters from 0.5 to 1.5 au , and it remains to be seen if jupiters are common farther out . the equilibrium temperature at optical depth unity in the atmosphere of the planet around gliese 876 is estimated to be -70 c , too cold for water in liquid form ( saumon 1998 ) . temperatures would be higher at deeper layers in the atmosphere . any bodies orbiting interior to 0.2 au would have surface temperatures above -70 c. it would be interesting to determine if planets could reside in stable orbits within 0.2 au , perhaps in mean motion resonances with the giant planet discovered here . we thank kevin apps for analysis of hipparcos astrometry . we thank xavier delfosse , michel mayor , and didier queloz for communicating their velocities for gliese 876 . we thank m. duncan , d. lin , and g.basri for useful conversations . we acknowedge support by nasa grant nagw-3182 and nsf grant ast95 - 20443 ( to gwm ) , and by nsf grant ast-9619418 and nasa grant nag5 - 4445 ( to ssv ) and by sun microsystems . we thank the nasa and uc telescope assignment committees for allocations of telescope time . lcc orbital period @xmath19 ( days ) & 60.85 & 0.15 + velocity semi - amplitude @xmath23 ( ms@xmath16 ) & 239 & 5 + eccentricity @xmath2 & 0.27 & 0.03 + longitude of periastron @xmath24 ( deg ) & 24 & 6 + periastron date @xmath25 ( julian date ) & 2450301.0 & 1.0 + @xmath26 ( m@xmath6 ) & 2.11 & 0.20 + semimajor axis a ( au ) & 0.21 & 0.01 +

doppler measurements of the m4 dwarf star , gliese 876 , taken at both lick and keck observatory reveal periodic , keplerian velocity variations with a period of 61 days . the orbital fit implies that the companion has a mass of , @xmath0 = 2.1 m@xmath1 , an orbital eccentricity of , @xmath2 = [email protected] , and a semimajor axis of , @xmath4 = 0.21 au . the planet is the first found around an m dwarf , and was drawn from a survey of 24 such stars at lick observatory . it is the closest extrasolar planet yet found , providing opportunities for follow up detection . the presence of a giant planet on a non - circular orbit , 0.2 au from a 1/3 m@xmath5 star , presents a challenge to planet formation theory . this planet detection around an m dwarf suggests that giant planets are numerous in the galaxy .

astro-ph9807307

let us begin with the following facts : 1 . for any differential graded algebra @xmath1 , the hochschild cohomology @xmath2 has a gerstenhaber algebra structure . 2 . let @xmath3 be a closed , oriented @xmath4-dimensional @xmath0-manifold , @xmath5 be the free loop space . then , @xmath6 has a batalin - vilkovisky ( in particular , gerstenhaber ) algebra structure . 3 . let @xmath7 denote the differential graded algebra of differential forms on @xmath3 . there exists a linear map @xmath8 defined by iterated integrals of differential forms , which preserves the gerstenhaber structures . \(a ) is originally due to gerstenhaber @xcite . ( b ) is due to chas - sullivan @xcite , which is the very first paper on string topology . ( c ) relates the geometric construction ( b ) to the algebraic construction ( a ) . it seems that ( c ) is also well - known to specialists ( see remark [ 150205_2 ] ) . ( a)(c ) concern algebraic structures on homology level , and it is an important and interesting problem to define chain level refinements of these structures . for ( a ) , so called deligne s conjecture claims that a certain chain model of the little disks operad acts on the hochschild cochain complex . various affirmative solutions to this conjecture and its variations are known ; see @xcite part i section 1.19 , @xcite section 13.3.15 , and the references therein . the aim of this paper is to propose a chain level algebraic structure which lifts ( b ) ( the batalin - vilkovisky ( bv ) algebra structure in string topology ) , and compare it with a solution to deligne s conjecture via a chain map which is a chain level lift of ( c ) . let us briefly describe our main result ( see theorem [ 150628_1 ] for the rigorous statement ) . first of all , for any closed , oriented @xmath0-manifold @xmath3 , we define a chain complex @xmath9 over @xmath10 , which is a chain model of @xmath11 . we also define a differential graded operad @xmath12 and its suboperad @xmath13 . these operads are chain models of the framed little disks operad and the little disks operad , and defined independently from @xmath3 . we establish the following statements : 1 . for any differential graded algebra @xmath1 , the hochschild complex @xmath14 admits an action of @xmath13 , which lifts the gerstenhaber structure on @xmath15 in ( a ) . 2 . for any closed , oriented @xmath0-manifold @xmath3 , the chain complex @xmath9 admits an action of @xmath12 , thus @xmath16 has the bv algebra structure . there exists an isomorphism @xmath17 preserving the bv structures . 3 . there exists a @xmath18-equivariant chain map @xmath19 , such that @xmath20 coincides the map in ( c ) . there may be several different ways to work out chain level structures in string topology , based on choices of chain models of the free loop space . the singular chain complex has the transversality trouble , namely string topology operations are defined only for chains transversal to each other . the hochschild complex of differential forms ( used e.g. in @xcite ) avoids this trouble , however it is not always a correct chain model of the free loop space ( see remark [ 150811_1 ] ) , and loses some geometric informations ( e.g. lengths of loops , see section 1.5.3 ) . our chain model @xmath9 is an intermediate one of these two . this section is organized as follows . in sections 1.11.4 , we recall several basic definitions and facts , fixing various notations an signs . in section 1.5 , we state theorem [ 150628_1 ] , which is our main result , and a few supplementary results . section 1.6 discusses previous works and 1.7 discusses potential applications to symplectic topology . section 1.8 explains the plan of the rest of this paper . first we briefly recall the notion of ( nonsymmetric ) operads . the main aim is to fix conventions , and we refer @xcite part ii section 1.2 for details . let @xmath21 be any symmetric monoidal category with a multiplication @xmath22 and a unit @xmath23 . a _ nonsymmetric operad _ @xmath24 in @xmath21 consists of the following data : * an object @xmath25 for every integer @xmath26 . * a morphism @xmath27 for every @xmath28 and @xmath29 . these morphisms are called _ ( partial ) compositions_. * a morphism @xmath30 called a _ unit _ of @xmath24 . we require that compositions satisfy associativity , and @xmath31 is a two - sided unit for compositions . when @xmath25 admits a right action of the symmetric group @xmath32 ( @xmath33 is the trivial group ) for each @xmath26 , such that these actions are compatible with compositions , @xmath24 is called an _ operad _ in @xmath21 . for any ( nonsymmetric ) operads @xmath24 and @xmath34 , a morphism of ( nonsymmetric ) operads @xmath35 is a sequence of morphisms @xmath36 which preserves the above structures . when @xmath37 are monics for all @xmath26 , we say that @xmath24 is a suboperad of @xmath34 . throughout this paper , all vector spaces are defined over @xmath10 . a graded vector space @xmath38 is a sequence @xmath39 of vector spaces . a differential graded ( or dg ) vector space ( or chain complex ) is a pair @xmath40 of a graded vector space @xmath38 and @xmath41 satisfying @xmath42 . we may consider any graded vector space as a dg vector space with @xmath43 . one can define the symmetric monoidal structure on the category of dg vector spaces as follows : @xmath44 the unit @xmath45 is defined as @xmath46 and @xmath43 . in this paper , we mainly work on the category of graded and dg vector spaces . operads in these categories are called _ graded operads _ and _ dg operads _ , respectively . for any dg vector spaces @xmath38 and @xmath47 , @xmath48 has the structure of a dg vector space : @xmath49 for any dg vector space @xmath38 , @xmath50 has the structure of a dg operad defined as follows ( @xmath51 , @xmath52 , and @xmath53 ) : @xmath54 this dg operad is called the _ endomorphism operad _ of @xmath38 . for any dg operad @xmath24 , a dg @xmath24-algebra is a chain complex @xmath38 with a morphism @xmath55 of dg operads . for each @xmath26 we have a chain map @xmath56 for any dg @xmath24-algebras @xmath38 and @xmath47 , a chain map @xmath57 is called a morphism of dg @xmath24-algebras if @xmath58 for any @xmath59 and @xmath60 . for any graded operad @xmath24 , the notions of graded @xmath24-algebras and their morphisms are defined in a similar way . for any dg operad @xmath61 , a _ dg ideal _ of @xmath24 is a sequence @xmath62 such that the following conditions hold : * for every @xmath26 , @xmath63 is a chain subcomplex of @xmath25 , which is preserved by the @xmath32-action on @xmath25 . * for any @xmath59 , @xmath64 and @xmath28 , @xmath65 for any dg ideal @xmath66 , the quotient @xmath67 has a natural structure of a dg operad , and there exists a natural morphism of dg operads @xmath68 . for any graded operad @xmath24 , the notions of its graded ideals and associated quotient graded operads are defined in the obvious way ( see @xcite section 5.2.14 ) . gerstenhaber and bv operads are graded operads , which play central roles in this paper . we recall definitions of these operads using generators and relations , partially following @xcite sections 13.3.12 and 13.7.4 . the _ gerstenhaber operad _ @xmath69 is generated by @xmath70 , @xmath71 , @xmath72 with the following relations : 1 . @xmath73 . 2 . @xmath74 . 3 . @xmath75 . 4 . @xmath76 , @xmath77 . more precisely , we consider the free operad @xmath78 ( see @xcite section 5.5 ) generated by @xmath79 , @xmath80 , @xmath81 , and a graded ideal @xmath82 generated by relations ( a ) , ( b ) , ( ab ) , ( u ) . then , we define @xmath83 . for any graded @xmath69-algebra @xmath38 , we define operations @xmath84 and @xmath85 on @xmath38 as @xmath86 then , @xmath87 is a graded commutative , associative algebra , and @xmath88 is a graded lie algebra ( with grading shifted by @xmath89 ) . the triple @xmath90 is called a _ gerstenhaber algebra_. in the above definition , any gerstenhaber algebra @xmath38 has a unit @xmath91 of the multiplication @xmath84 . it seems that the existence of a unit is usually not assumed . the _ bv operad _ @xmath92 is generated by @xmath93 , @xmath94 , @xmath95 , @xmath96 with the relations ( a ) , ( b ) , ( ab ) , ( u ) and @xmath97 obviously , there exists a natural morphism of graded operads @xmath98 . for any graded @xmath92-algebra @xmath38 , we define an operation @xmath99 on @xmath38 by @xmath100 . the triple @xmath101 is called a _ bv algebra_. the bracket @xmath85 is recovered by the formula @xmath102 . for any integer @xmath103 , let @xmath104 be the set of tuples @xmath105 such that * for each @xmath106 , @xmath107 is a closed disk of positive radius contained in @xmath108 . we denote its center by @xmath109 . @xmath110 is a point on @xmath111 . * @xmath112 are disjoint . the set @xmath104 has a natural topology . let @xmath113 denote the subspace of @xmath104 , which consists of @xmath105 such that @xmath114 for every @xmath106 . we define @xmath115 to be the space consists of a point . then , @xmath116 has a natural structure of a topological operad , and @xmath117 is its suboperad . @xmath118 ( resp . @xmath119 ) is called the _ little disks _ ( resp . _ framed little disks _ ) operad . there are isomorphisms of graded operads @xmath120 ( @xcite ) and @xmath121 ( @xcite ) , which are compatible with the inclusion maps . a _ differential graded associative algebra _ is a dg vector space @xmath1 with a degree @xmath122 product @xmath123 , which is associative , and satisfies the leibniz rule . we also assume that it has a unit @xmath124 . we abbreviate the term `` differential graded associative algebra '' as `` dga algebra '' . here the letter `` a '' stands for `` associative '' , not for algebra ( see @xcite pp . let @xmath1 be any dga algebra . a dg @xmath125-bimodule is a dg vector space @xmath3 with degree @xmath122 left and right @xmath125-actions @xmath126 and @xmath127 , which satisfy the leibniz rule and associativity . for every @xmath128 and @xmath129 , we define a chain map @xmath130 by @xmath131 we set @xmath132 by @xmath133 , and define the hochschild cochain complex @xmath134 by @xmath135 . notice that @xmath136 _ decreases _ the degree by @xmath89 . the cohomology of this complex is denoted by @xmath137 , and called the _ hochschild cohomology_. notice that @xmath1 itself has a natural structure of a dg @xmath125-bimodule . @xmath138 has natural dga and dg lie algebra structures , with operations @xmath139 and @xmath85 defined below . the product @xmath139 is defined as @xmath140 the bracket @xmath141 is defined as @xmath142 where @xmath143 is defined as @xmath144 the operations @xmath139 and @xmath141 induce the gerstenhaber structure on @xmath2 . this result is originally due to gerstenhaber @xcite . throughout this paper , we set @xmath145 . let @xmath3 be a closed , oriented @xmath0-manifold of dimension @xmath4 . @xmath5 is equipped with the @xmath0-topology . we often abbreviate @xmath146 as @xmath147 . also , we often use the notation @xmath148 . in @xcite , chas - sullivan introduced the _ loop product _ on @xmath149 . let us briefly recall its definition . let us consider the evalutation map @xmath150 , and the fiber product @xmath151 let @xmath152 be a tubular neighborhood of @xmath153 , and @xmath154 be the thom isomorphism . the gysin map @xmath155 is defined as the composition of the following maps : @xmath156 let @xmath157 denote the concatenation map . precisely , it is defined as follows ( see the remark on pp . 780 @xcite ) . let us take an increasing @xmath158-function @xmath159 \to [ 0,1]$ ] such that @xmath160 and @xmath161 ( @xmath162 denotes the @xmath163-th derivative ) for any @xmath164 . then , @xmath157 is defined by @xmath165 it is easy to see that @xmath166 does not depend on choices of @xmath167 . the loop product @xmath168 is defined as the composition of the following three maps . the first map is the cross product and the second map is the gysin map . @xmath169 ^ -{\times } & h_{*+2d}(\l^{\times 2 } ) \ar[r ] & \h_*(\l { \,_{e}\!\!\times_{e } } \l ) \ar[r]^-{\h_*(c ) } & \h_*(\l ) . } \ ] ] let us consider the map @xmath170 . let @xmath171 denote the intersection product . then , @xmath172 on the other hand , @xmath147 admits a natural @xmath173-action @xmath174 , which is defined by @xmath175 . we define @xmath176 by @xmath177 \times x)$ ] , where @xmath178 \in h_1(s^1)$ ] is represented by the singular chain @xmath179 $ ] ( see the next subsection for the definition of @xmath180 ) . [ thm : ch - su ] for any closed , oriented @xmath4-dimensional @xmath158-manifold @xmath3 , @xmath181 is a bv algebra . this result is the starting point of string topology . the bracket @xmath85 of this bv structure is called the _ loop bracket_. there is a relation between gerstenhaber structures on loop space homology ( theorem [ thm : ch - su ] ) and hochschild cohomology ( section 1.2 ) . we explain this relation via iterated integrals of differential forms , which originates in @xcite . to discuss iterated integrals of differential forms , it is convenient to work with @xmath0-singular chains on @xmath146 . let us define the @xmath182-dimensional simplex @xmath183 by @xmath184 a map @xmath185 is said to be of class @xmath0 , if there exists an open neighborhood @xmath152 of @xmath186 and a map @xmath187 , such that @xmath188 , and @xmath189 is of class @xmath0 . let @xmath190 denote the @xmath10-vector space generated by all @xmath0-maps @xmath191 . it is easy to see that , any @xmath0-map @xmath192 is continuous with respect to the @xmath0-topology on @xmath146 . therefore , @xmath193 is a subcomplex of the singular chain complex of @xmath146 . in section 4 , we show that this inclusion map is a quasi - isomorphism ( theorem [ 150219_1 ] ) . therefore , @xmath194 . for any @xmath195 , let us define @xmath196 then , @xmath197 is a dga algebra , where @xmath4 denotes the exterior derivative , and @xmath198 denotes the exterior product . we denote it by @xmath7 . we define a dg @xmath199-bimodule structure on @xmath200_*:=\hom({{\mathcal{a}}}^{*+d}(m ) , \r)$ ] as follows : @xmath201 a morphism of @xmath7-bimodules @xmath202 ; \ , \alpha \mapsto ( \beta \mapsto \int_m \alpha \wedge \beta)$ ] is a quasi - isomorphism ( this is an obvious consequence of the poincar duality ) . for any @xmath0-map @xmath203 and @xmath129 , we define @xmath204 by @xmath205 and define @xmath206)$ ] by @xmath207 it is easy to see that @xmath208 ) ; \quad \sigma \mapsto ( i_k(\sigma))_k\ ] ] is a chain map ( signs are checked in section 6.3 ) . taking homology , we obtain a map @xmath209 ) \cong h^*({{\mathcal{a}}}_m , { { \mathcal{a}}}_m).\ ] ] this map preserves the gerstenhaber structures on @xmath210 and @xmath211 . [ 150205_2 ] the fact that ( [ 141218_01 ] ) preserves the gerstenhaber structures seems to be known ; see @xcite for the product , and @xcite section 7 for the bracket . we can recover this fact as a consequence of theorem [ 150628_1 ] , see remark [ 150703_1 ] . we state our main result theorem [ 150628_1 ] and some supplementary results . let us recall some notations : @xmath69 and @xmath92 denote the gerstenhaber and bv operads . for any @xmath0-manifold @xmath3 of dimension @xmath4 , @xmath5 , and @xmath212 . @xmath7 denotes the dga algebra of differential forms on @xmath3 . [ 150628_1 ] there exists a dg operad @xmath12 and its suboperad @xmath13 , satisfying the following properties . 1 . there exist isomorphisms of graded operads @xmath213 and @xmath214 , such that the following diagram commutes : @xmath215 \ar[d]_-{\cong } & h_*(f\tilde{\lambda } ) \ar[d]^-{\cong } \\ \ger \ar[r ] & \batvil . } \ ] ] 2 . for any dga algebra @xmath1 , the hochschild complex @xmath138 has a dg @xmath13-algebra structure , which lifts the gerstenhaber structure on @xmath2 . 3 . for any closed , oriented @xmath0-manifold @xmath3 , there exist : * a dg @xmath12-algebra @xmath9 . * an isomorphism @xmath216 of bv algebras . * a morphism of dg @xmath13-algebras @xmath19 such that @xmath20 is equal to the map ( [ 141218_01 ] ) . [ 150212_1 ] it seems that ( although there are some technical issues the author has not confirmed ) the following stronger version of ( i ) is also true : there exists a zig - zag of quiasi - isomorphisms ( of dg operads ) connecting @xmath13 ( resp . @xmath12 ) and @xmath217 ( resp . @xmath218 ) , which is the singular chain operad of @xmath118 ( resp . @xmath119 ) . [ 150703_1 ] since @xmath219 is a map of dg @xmath13-algebras , and @xmath214 , ( iii)-(c ) shows that the map ( [ 141218_01 ] ) preserves the gerstenhaber structures . [ 150811_1 ] as ( iii)-(c ) suggests , the chain complex @xmath9 is a `` geometric analogue '' of the hochschild complex @xmath220 . an important difference is that the isomorphism @xmath221 holds for any closed oriented @xmath3 , while we need some assumptions ( e.g. @xmath222 ) to show that the map ( [ 141218_01 ] ) is an isomorphism . as the next proposition shows , one can define chain level loop product and bracket on @xmath9 . [ 150801_1 ] the loop product @xmath84 and the loop bracket @xmath85 on @xmath223 lift to operators on @xmath9 , denoted by the same symbols . @xmath84 defines a dga algebra structure , @xmath85 defines a dg lie algebra structure on @xmath9 , and the chain map @xmath224 preserves these structures . moreover , there exists an injective chain map @xmath225 such that @xmath226 and the following diagram commutes : @xmath227 ^ -{\h_*(i_m)}\ar[d]_{\cong } & \h_*({{\mathcal{l}}}m ) \ar[d]^{\cong } \\ h^{-*}_\dr(m ) \ar[r]_{h_*(\iota_m ) } & h_*(c^{{{\mathcal{l}}}m } ) . } \ ] ] as a consequence , one can define @xmath228-refinements of the loop product . namely , by the homotopy transfer theorem ( see section 10.3 @xcite ) , one can define an @xmath228-structure @xmath229 on @xmath223 , such that @xmath230 , @xmath231 and @xmath232 is homotopy equivalent to the dga algebra @xmath233 . moreover , we may take @xmath229 so that @xmath234 for every @xmath128 , and @xmath235 is homotopy equivalent to the dga algebra @xmath236 . in particular , @xmath229 recovers the classical massey product on @xmath237 . by same arguments , one can define an @xmath238-structure @xmath239 on @xmath223 , such that @xmath240 , @xmath241 , and @xmath242 is homotopy equivalent to the dg lie algebra @xmath243 . moreover , we may take @xmath239 so that @xmath244 for any @xmath128 . when @xmath3 has a riemannian metric , @xmath223 is equipped with the length filtration , and it is shown in @xcite that the loop product preserves this filtration . one of features of our chain model @xmath9 is that , one can define the length filtration on this chain model so that the filtration is preserved by string topology operations . for any @xmath245 , let us denote @xmath246 . for any @xmath247 $ ] , we define @xmath248 . in general , a filtration ( indexed by @xmath249 $ ] ) on a chain complex @xmath250 is a family @xmath251}$ ] of subcomplexes of @xmath250 , such that @xmath252 . for any @xmath253 , we set @xmath254 . [ 141005_2 ] let @xmath3 be a closed , oriented riemannian manifold . then , the chain complex @xmath9 has a filtration @xmath255}$ ] such that the following conditions hold : 1 . there exists an isomorphism @xmath256 for every @xmath247 $ ] , such that the following diagram is commutative for any @xmath257 , where vertical maps are induced by inclusions : @xmath258 ^ -{\cong } \ar[d ] & h_*(f^a c^{{{\mathcal{l}}}m } ) \ar[d ] \\ \h_*(\l^b m ) \ar[r]^-{\cong } & h_*(f^b c^{{{\mathcal{l}}}m } ) . } \ ] ] 2 . the @xmath259-algebra structure on @xmath9 preserves the filtration . namely , @xmath260 rich algebraic structures in chain level string topology were outlined in @xcite by d. sullivan , and there have been several papers working out details . x. chen @xcite introduced a chain model of the free loop space using whitney differential forms , and defined several string topology operations ( the loop product , loop bracket and rotation ) on that chain model , recovering the bv - algebra structure on homology level ( @xcite also studies the @xmath173-equivariant case ) . it is not clear whether these operations extend to actions of a dg operad on this chain model . on the other hand , a recent paper @xcite by g. drummond - cole , k. poirier and n. rounds proposed a more geometric approach using short geodesic segments and diffuse intersection classes . @xcite defines operations on the singular chain complex of the free loop space , recovering the homology level structure defined by cohen - godin @xcite . in particular , @xcite covers operations with multiple outputs and those corresponding to surfaces of higher genus . however , the operations in @xcite are associative only up to homotopy , and it seems that the resulting algebraic structure is yet to be fully worked out . let us discuss some potential applications of results in this paper to symplectic topology . for any @xmath0-manifold @xmath3 , the cotangent bundle @xmath261 has the natural sympelctic structure . when @xmath3 is closed , oriented and spin , the floer homology @xmath262 has a bv algebra structure , and there exists an isomorphism of bv algebras @xmath263 ( see @xcite and the references therein ) . there should be chain level refinements of this correspondence , and we expect that our chain level structures in string topology fit into this picture . more specifically , we expect that one can define an @xmath228 ( resp . @xmath238 ) structure on @xmath262 via counting solutions of appropriate floer equations , which is homotopy equivalent to the @xmath228 ( resp . @xmath238 ) refinement of the loop product ( resp . bracket ) on @xmath223 defined in section 1.5.2 . on the other hand , fukaya @xcite used chain level loop bracket for compactifications of the moduli space of pseudo - holomorphic disks with lagrangian boundary conditions , and obtained restrictions on topological types of lagrangian submanifolds . we expect that our definition of chain level loop bracket could be used to work out details of this approach . finally , there is a very interesting program by cieliebak - latschev @xcite , which compares symplectic field theory of sphere cotangent bundles and string topology of @xmath173-equivariant chains on free loops modulo constant loops . we hope that the construction in this paper will be a first step towards working out details of the string topology side of this program . the rest of this paper is split into two parts . the goal of the first part ( sections 26 ) is to define the chain complex @xmath9 . this is a chain model of the free loop space @xmath11 , carefully designed to define string topology operations on it . our construction of the chain complex @xmath9 is based on two ideas . the first idea is to introduce a notion of _ de rham chains _ , which is a certain hybrid of singular chains and differential forms . we also need a notion of _ differentiable spaces _ , on which de rham chains are defined . these notions are introduced in section 2 . we also define de rham chain complex ( the chain complex which consists of de rham chains ) for any differentiable space . an important step in our argument is to show that homology of the de rham chain complex is naturally isomorphic to the usual singular homology . we prove it for finite - dimensional manifolds in section 3 , and for the free loop space in section 4 . the second idea is to use ( moore ) loops with marked points , which is inspired by the theory of iterated integrals . in section 5 , we introduce a space of moore loops with marked points , and show that the collection of de rham chain complexes of these spaces has a natural structure of a dg operad . moreover , this dg operad is cyclic and has a multiplication and a unit . based on results in section 5 , we define the chain complex @xmath9 in section 6 . we also prove results presented in section 1.5 assuming theorem [ 150624_8 ] , which is a purely algebraic result on operads . the second part ( sections 711 ) is devoted to the proof of theorem [ 150624_8 ] . * acknowledgements . * the author appreciates professors k. fukaya and k. ono for their encouragements and comments on this project . he also appreciates b. ward for useful communications , and an anonymous referee for many suggestions on the preliminary version of this paper . this research is supported by jsps kakenhi grant number 25800041 . in this section , we introduce notions of _ differentiable spaces _ and _ de rham chain complexes _ , which are basic for the arguments in this paper . the notion of differentiable spaces is introduced in section 2.2 . , and de rham chain complexes for these spaces are defined in section 2.3 . the rest of this section ( 2.42.8 ) is devoted to establishing several basic results about de rham chain complexes of differentiable spaces . let @xmath264 be a @xmath4-dimensional @xmath0-manifold . throughout this paper , all manifolds are without boundary , unless otherwise specified . let @xmath265 be an integer , and @xmath266 be an @xmath267-bundle . we define @xmath268 . when @xmath269 , we define @xmath270 to be the trivial @xmath10-bundle on @xmath264 . any exact sequence @xmath271 of real vector bundles induces an isomorphism @xmath272 . an orientation of @xmath78 is a section of the double cover @xmath273 . an orientation of @xmath274 is called an orientation of @xmath264 . recall that we denote @xmath275 when @xmath276 . let @xmath277 denote its subspace which consists of compactly supported @xmath278-forms . when @xmath279 , we set @xmath280 . when @xmath264 is oriented , we define @xmath281 let @xmath282 , @xmath283 be oriented @xmath0-manifolds , and @xmath284 be a submersion ( i.e. @xmath285 is of class @xmath0 and @xmath286 is surjective for any @xmath287 ) . let @xmath288 . the _ integration along fibers _ is a chain map @xmath289 , which is defined in the following way . first we consider the case @xmath290 , @xmath291 and @xmath292 . we assume that @xmath293 and @xmath294 are positive with respect to orientations on @xmath295 and @xmath296 . for @xmath297 where @xmath298 and @xmath299 , we define @xmath300 by @xmath301 in the general case , @xmath302 is defned by taking local charts and partitions of unity on @xmath282 . below is a list of some basic properties of the integration along fibers . * if @xmath283 is a positively oriented point , then @xmath303 . * @xmath302 is a chain map , i.e. there holds @xmath304 . * for any @xmath305 , @xmath306 . * for any submersion @xmath307 and @xmath308 , there holds @xmath309 . for any integers @xmath310 , let @xmath311 denote the set of oriented @xmath163-dimensional submanifolds in @xmath296 . we set @xmath312 . let @xmath313 be a set . differentiable structure _ on @xmath313 is a family of maps called _ plots _ , which satisfies the following conditions : * every plot is a map from @xmath314 to @xmath313 . * if @xmath315 is a plot , @xmath316 and @xmath317 is a submersion , then @xmath318 is a plot . a _ differentiable space _ is a pair of a set and a differentiable structure on it . for any differentiable space @xmath313 , let @xmath319 . a map @xmath320 between differentiable spaces @xmath313 and @xmath321 is called _ smooth _ , if there holds @xmath322 the term `` plot '' is originally used in the theory of chen s differentiable spaces ( @xcite ) and the theory of diffeological spaces ( @xcite , @xcite ) . our notion of differentiable spaces is weaker than these notions . in particular , in axioms of both of these spaces , it is required that all constant maps are plots , while we do not require this condition ( see example [ 150623_1 ] ( i)-(b ) below ) . [ 150623_1 ] let us explain some examples of differentiable structures . 1 . let @xmath3 be a @xmath0-manifold . one can consider the following differentiable structures on @xmath3 : * @xmath323 is a plot if @xmath324 is of class @xmath0 . we denote the resulting differentiable space as @xmath3 . * @xmath323 is a plot if @xmath324 is a submersion ( we always assume that any submersion is of class @xmath0 ) . we denote the resulting differentiable space as @xmath325 . + the identity map @xmath326 is smooth , but @xmath327 is not . 2 . @xmath328 has the following differentiable structure : a map @xmath329 is a plot if @xmath330 is of class @xmath0 . 3 . let @xmath313 be a differentiable space , @xmath321 a subset of @xmath313 , and @xmath331 be the inclusion map . one can define the following differentiable structure on @xmath321 : a map @xmath332 is a plot if @xmath333 is a plot of @xmath313 . 4 . let @xmath334 be a family of differentiable spaces parametrized by the nonempty set @xmath335 . the product @xmath336 has the following differentiable structure : @xmath337 is a plot if @xmath338 is a plot of @xmath339 for any @xmath340 ( @xmath341 denotes the projection to @xmath339 ) . two smooth maps @xmath342 are called _ smoothly homotopic _ , if there exists a smooth map @xmath343 such that @xmath344 @xmath345 is called a _ smooth homotopy _ between @xmath346 and @xmath347 . the differentiable structure on @xmath10 is defined as in example [ 150623_1 ] ( i)-(a ) , i.e. @xmath348 . the differentiable structure on @xmath349 is defined as in example [ 150623_1 ] ( iv ) . it seems that transitivity of @xmath350 does not hold in general . unless otherwise specified , any @xmath0-manifold @xmath3 will be equipped with the differentiable structure in example [ 150623_1 ] ( i)-(a ) . when @xmath0-manifolds @xmath3 , @xmath351 are equipped with these differentiable structures , a map @xmath352 is smooth if and only if @xmath346 is of class @xmath0 . let @xmath313 be a differentiable space . for any @xmath353 , we set @xmath354 notice that @xmath355 for any @xmath356 . for any @xmath357 and @xmath358 , let @xmath359 denote the image of @xmath360 by the natural injection @xmath361 . let @xmath362 denote the subspace of @xmath363 , which is generated by @xmath364 we define @xmath365 . for every @xmath353 , @xmath366 \mapsto [ ( u , \ph , d\omega)]\ ] ] is well - defined , since @xmath304 . moreover , @xmath42 since @xmath367 . we call @xmath368 the _ de rham chain complex _ of @xmath313 , and denote its homology as @xmath369 . elements of @xmath370 are called _ de rham chains _ of @xmath313 . our notion of de rham chains is inspired by the notion of _ approximate de rham chains _ by k. fukaya ( @xcite definition 6.4 ) . however , an explicit definition of a chain complex is not given in @xcite . the _ augmentation map _ @xmath371 is defined as @xmath372):= \int_u \omega.\ ] ] @xmath373 vanishes on @xmath374 by stokes theorem . next we define the _ fiber product _ on de rham chain complexes . let @xmath3 be an oriented @xmath0-manifold of dimension @xmath4 . let us consider the differentiable space @xmath325 in example [ 150623_1 ] ( i)-(b ) . let @xmath313 , @xmath321 be differentiable spaces , and @xmath375 , @xmath376 be smooth maps . we define a differentiable structure on @xmath377 as a subset of @xmath378 ( see example [ 150623_1 ] ( iii ) and ( iv ) ) . we are going to define a chain map @xmath379 which we call the fiber product on de rham chain complexes . let @xmath357 , @xmath380 . then , @xmath381 and @xmath382 are submersions . thus , @xmath383 is a @xmath0-manifold , moreover it is a submanifold of the euclidean space ( since @xmath384 ) . the map @xmath385 is also a submersion . let us define an orientation on @xmath386 . let @xmath387 , @xmath388 . there exist exact sequences @xmath389 then we obtain isomorphisms @xmath390 since @xmath3 , @xmath152 , @xmath38 are oriented , one can define an orientation on @xmath391 ( resp . @xmath392 ) so that the first ( resp . second ) isomorphism is orientation preserving . then , one can define an orientation of @xmath386 so that the third isomorphism is orientation preserving . now , @xmath386 is an oriented submanifold of the euclidean space . therefore , @xmath386 is an element of @xmath393 . let @xmath394 , @xmath395 , @xmath396 , @xmath397 be projection maps . then * since @xmath398 is a submersion , @xmath399 . * since @xmath400 is a submersion , @xmath401 . therefore @xmath402 . now , let us define ( [ 150127_1 ] ) by @xmath403 \times_m [ ( v , \psi , \eta)]:= ( -1)^{l(\dim u - d ) } [ ( u \times_m v , \ph \times \psi , \omega \times \eta)].\ ] ] by direct computations , one can check that this is a well - defined chain map . the product is associative , and a smooth map @xmath404 satisfies @xmath405 . when @xmath3 is a positively oriented point ( thus @xmath406 ) , the chain map ( [ 150127_1 ] ) is @xmath407 which we call the _ cross product _ on de rham chain complexes . let @xmath408 be a set which has a unique element . for any @xmath314 , there exists a unique map @xmath409 . we define a differentiable structure on @xmath408 by @xmath410 . let us consider @xmath411 with the positive orientation . for any @xmath314 , let us denote the unique map @xmath412 by @xmath398 . for any @xmath413 , we obtain @xmath414 = [ ( \{0\ } , \ph_{\{0\ } } , ( \pi_u ) _ ! \omega ) ] = [ ( \{0\ } , \ph_{\{0\ } } , \int_u \omega)].\ ] ] therefore , @xmath415 if @xmath416 , and the augmentation map @xmath417 is an isomorphism . in particular , @xmath418 for any differentiable spaces @xmath313 , @xmath321 and a smooth map @xmath419 , @xmath420 \mapsto [ ( u , f \circ \ph , \omega)]\ ] ] is a well - defined chain map . [ prop : homotopy ] let @xmath313 , @xmath321 be differentiable spaces , and @xmath342 be smooth maps . if @xmath346 , @xmath347 are smoothly homotopic , then @xmath421 are chain homotopic . let @xmath422 be a smooth homotopy between @xmath346 and @xmath347 , i.e. , @xmath345 is a smooth map such that @xmath423 if @xmath424 and @xmath425 if @xmath426 . take @xmath427 so that @xmath428 on @xmath429 $ ] for some @xmath430 . let @xmath431 \in c^\dr_1(\r)$ ] , and define a linear map @xmath432 by @xmath433 . since @xmath434 and the cross product are chain maps , for any @xmath435 , we have @xmath436 thus , it is enough to show that @xmath437 . since both sides are linear on @xmath438 , we may assume that @xmath439 $ ] for some @xmath357 and @xmath413 . let @xmath440 , @xmath441 be the inclusion maps , and @xmath442 , @xmath443 . define @xmath444 by @xmath445 $ ] , @xmath446 $ ] . then , since @xmath447 is supported on @xmath448 , we have @xmath449 . let @xmath450 \in c^\dr_*(x)$ ] . then , @xmath451.\ ] ] there holds @xmath452 , where @xmath453 is the projection map . moreover , @xmath454 , since @xmath455 . thus , @xmath456 . a similar argument shows @xmath457 . hence , we get @xmath458 let @xmath313 be a differentiable space . for any function @xmath459 and @xmath460 , let us define @xmath461 this is a subspace of @xmath313 truncated by the inequality @xmath462 . we prove some technical results about de rham chain complexes of these truncated spaces . a function @xmath463 is called smooth , if @xmath464 for any @xmath357 . @xmath346 is called _ approximately smooth _ , if there exists a decreasing sequence @xmath465 of smooth functions on @xmath313 such that @xmath466 for any @xmath467 . if @xmath463 is approximately smooth , @xmath468 is upper semi - continuous for any @xmath357 . this is because @xmath469 for any @xmath460 . an important example of an approximately smooth ( but not smooth ) function is the length functional ( see section 1.5.3 ) on the free loop space . let @xmath3 be a riemannian manifold , and consider the differentiable structure on @xmath11 as in example [ 150623_1 ] ( ii ) . then , @xmath470 is approximately smooth . it is easy to check that , for any @xmath471 , a functional @xmath472 is smooth . let us take a decreasing sequence @xmath473 on @xmath474 , such that @xmath475 for any @xmath476 . then , @xmath477 is a decreasing sequence of smooth functions on @xmath146 , such that @xmath478 . [ 150210_1 ] let @xmath465 be a decreasing sequence of approximately smooth functions on @xmath313 , such that @xmath479 for any @xmath467 . then , the chain map @xmath480 , which is induced by the inclusion maps , is surjective . let @xmath357 and @xmath413 . setting @xmath481 for each @xmath482 , @xmath483 is a decreasing sequence of upper semi - continuous functions on @xmath152 . since @xmath484 is compact , there exists @xmath278 such that @xmath485 for any @xmath486 . setting @xmath487 , the chain map @xmath488 maps @xmath489 $ ] to @xmath490 $ ] . [ 150210_2 ] let @xmath463 be an approximately smooth function . then , the chain map @xmath491 , which is induced by the inclusion map , is injective . first let us consider the case when @xmath346 is a smooth function on @xmath313 . for any @xmath492 , @xmath493 denote the chain maps induced by the inclusion maps . suppose that @xmath494 satisfies @xmath495 . by the previous lemma , there exists @xmath496 such that @xmath497 . take @xmath498 so that @xmath499 . if there exists a linear map @xmath500 such that @xmath501 , we can prove @xmath502 by @xmath503 to define such @xmath219 , we fix @xmath504)$ ] so that @xmath505 on @xmath506 and @xmath507 . for any @xmath357 , we set @xmath508 . then , a linear map @xmath509 \mapsto [ ( u_{f,0 } , \ph|_{u_{f,0 } } , ( ( \kappa \circ f \circ \ph ) \cdot \omega)|_{u_{f,0}})]\ ] ] is well - defined ( it is _ not _ a chain map ) . @xmath501 is obvious since @xmath505 on @xmath506 . this completes the proof when @xmath346 is a smooth function on @xmath313 . finally , we consider the case when @xmath346 is any approximately smooth function on @xmath313 . by definition , there exists a decreasing sequence of smooth functions @xmath465 such that @xmath510 . the previous lemma shows that the chain map @xmath511 is surjective . on the other hand , since each @xmath512 is smooth , @xmath488 is injective . thus @xmath513 is injective . [ 150213_1 ] let @xmath465 be any decreasing sequence of approximately smooth functions on @xmath313 , such that @xmath514 for any @xmath467 . then , the chain map @xmath480 , which is induced by the inclusion maps , is an isomorphism . we already prove the surjectivity in lemma [ 150210_1 ] . by lemma [ 150210_2 ] , @xmath488 is injective for every @xmath278 , thus @xmath480 is also injective . we define _ smooth singular chains _ on differentiable spaces , and compare them with de rham chains . let @xmath313 be a differentiable space , and @xmath265 be an integer . a map @xmath515 is called _ strongly smooth _ , if there exists an open neighborhood @xmath152 of @xmath186 , and a smooth map @xmath516 such that @xmath517 . @xmath183 and @xmath152 are equipped with the differentiable structures as subsets of @xmath267 . for any @xmath265 , let @xmath518 denote the @xmath10-vector space generated by all strongly smooth maps @xmath519 . for any @xmath356 , we set @xmath520 . a differential on @xmath521 is defined in the same way as in the singular chain complex . to fix notations , let us spell out details . for any @xmath128 and @xmath522 , we define a map @xmath523 by @xmath524 in particular , @xmath525 is defined as @xmath526 for @xmath527 . for any @xmath128 , a differential @xmath528 is defined as @xmath529 we call the chain complex @xmath530 _ smooth singular chain complex _ , and its homology @xmath531 _ smooth singular homology_. for any smooth map @xmath320 between differentiable spaces , one can define the chain map @xmath532 in the obvious way . if smooth maps @xmath342 are smoothly homotopic , @xmath533 are chain homotopic . for any @xmath128 , we define @xmath534 by @xmath535 . then , @xmath536 for any @xmath128 . [ 150624_10 ] 1 . there exists a sequence @xmath537 such that @xmath538 for any @xmath265 , @xmath539 is characterized by @xmath540 , and @xmath541 2 . suppose that @xmath537 and @xmath542 satisfy the conditions in ( i ) . then there exists a sequence @xmath543 such that @xmath544 for any @xmath128 , @xmath545 , and @xmath546 for any @xmath265 , @xmath183 is smoothly homotopic to @xmath408 , thus @xmath547 . using this fact , the assertions are easy to prove by induction on @xmath182 . for any @xmath548 which satisfies lemma [ 150624_10 ] ( i ) , one can define a natural transformation @xmath549 as follows ( we set @xmath550 when @xmath356 ) : @xmath551 the equation @xmath552 shows that @xmath553 is a chain map . lemma [ 150624_10 ] ( ii ) shows that the homotopy equivalence class of @xmath553 does not depend on choices of @xmath81 . in particular , the linear map @xmath554 does not depend on @xmath81 . finally , we define the cross product on @xmath555 . let us take @xmath556 for all @xmath557 , such that @xmath558 is characterized by @xmath559 , and the following equation holds for any @xmath557 : @xmath560 we define the cross product @xmath561 by @xmath562 the homotopy equivalence class of this map does not depend on choices of @xmath563 . on the other hand , we defined the cross product for de rham chains ( see section 2.3 ) . the next lemma is proved by the standard method of acyclic models . [ 150627_1 ] for any differentiable spaces @xmath313 , @xmath321 and @xmath548 satisfying lemma [ 150624_10 ] ( i ) , the following diagram commutes up to homotopy , where horizontal maps are cross products : @xmath564 \ar[d]_{\iota^u(x ) \otimes \iota^u(y ) } & c^\sm_*(x \times y ) \ar[d]^{\iota^u(x \times y ) } \\ c^\dr_*(x ) \otimes c^\dr_*(y ) \ar[r ] & c^\dr_*(x \times y ) . } \ ] ] let @xmath3 be a @xmath0-manifold , and @xmath26 . we define @xmath565 by @xmath566 \rangle : = \int_u \ph^ * \alpha \wedge \omega.\ ] ] it induces a linear map @xmath567 , which we also denote by @xmath568 . for any subset @xmath569 , let @xmath570 where @xmath152 runs over all open neighborhoods of @xmath335 , and let @xmath571 . then , one can define @xmath572 . in the next lemma , we consider the case @xmath573 . [ 150629_2 ] suppose that @xmath537 satisfies lemma [ 150624_10 ] ( i ) . for any nonnegative integers @xmath574 and @xmath575 , @xmath576 we fix @xmath163 and prove the lemma by induction on @xmath182 . when @xmath269 , i.e. @xmath577 , the lemma can be directly checked . if the lemma is established for @xmath578 , the case @xmath579 is proved as follows . let us take @xmath580 such that @xmath581 ( this is always possible since @xmath582 ) . then , @xmath583 the second equality follows from the induction hypothesis , and the last equality follows from stokes theorem . let @xmath3 be an oriented @xmath0-manifold , and consider the differentiable structure as in example [ 150623_1 ] ( i)-(a ) , i.e. @xmath584 . with this differentiable structure , a map @xmath585 is strongly smooth if it extends to a @xmath0-map @xmath586 for some open neighborhood @xmath152 of @xmath186 . in particular , @xmath587 is a subcomplex of @xmath588 ( the usual singular chain complex of @xmath3 ) . it is known that @xmath589 is a quasi - isomorphism ( see @xcite theorem 18.7 ) . therefore , we obtain the natural isomorphism @xmath590 . in section 2.7 , we defined the map @xmath591 for any differentiable space @xmath313 . the goal of this section is to prove the following theorem [ 150623_2 ] . as an immediate consequence , we obtain the natural isomorphism @xmath592 . [ 150623_2 ] for any oriented @xmath0-manifold @xmath3 , the map @xmath593 is an isomorphism . let us denote the map @xmath593 by @xmath594 . the proof of theorem [ 150623_2 ] is separated into two steps . let @xmath595 . * in section 3.1 , we define an isomorphism @xmath596 . * in section 3.2 , we define an isomorphism @xmath597 via the poincar duality , and show that @xmath598 . let us consider the differentiable space @xmath325 ( see example [ 150623_1 ] ( i)-(b ) ) . it is easy to check that @xmath599 \mapsto \ph_!\omega\ ] ] is a well - defined chain map . on the other hand , for any @xmath600 , let us take @xmath314 and an orientation - preserving open embedding @xmath601 such that @xmath602 . then , @xmath603 \in c^\dr_*(m_\reg)$ ] does not depend on choices of @xmath152 and @xmath324 . thus , one can define a chain map @xmath604,\ ] ] and this is the inverse of ( [ 150210_3 ] ) . therefore , ( [ 150210_3 ] ) is an isomorphism of chain complexes . in particular , @xmath605 . @xmath326 is a map of differentiable spaces , as noted in example [ 150623_1 ] ( i ) . the goal of this subsection is to prove the next proposition . [ 150210_4 ] @xmath326 induces an isomorphism @xmath606 . as an immediate consequence , we obtain an isomorphism @xmath596 . to prove proposition [ 150210_4 ] , let us take a proper @xmath0-function @xmath607 , and set @xmath608 for any integer @xmath128 . since @xmath346 is a smooth function on @xmath3 and @xmath325 , corollary [ 150213_1 ] implies isomorphisms @xmath609 here we need the following lemma . [ 150210_5 ] let @xmath351 be an oriented @xmath0-manifold , and let @xmath47 be an open set in @xmath351 with a compact closure . then , there exists a chain map @xmath610 such that the following diagram of chain maps commutes up to homotopy . @xmath611^{i_\reg } \ar[d]_-{(\id_w ) _ * } & c^\dr_*(n_\reg)\ar[d]^-{(\id_n ) _ * } \\ c^\dr_*(w ) \ar[r]_{i } \ar[ru]_{j } & c^\dr_*(n)}\ ] ] @xmath612 and @xmath613 are induced by the inclusion map @xmath614 . let us apply lemma [ 150210_5 ] for @xmath615 , @xmath616 , and take a chain map @xmath617 as in lemma [ 150210_5 ] . then , @xmath618 is the inverse of @xmath619 , thus proposition [ 150210_4 ] is proved . to prove lemma [ 150210_5 ] , we need the following lemma . [ 150210_6 ] let @xmath351 be a @xmath0-manifold , and @xmath620 be a compact set in @xmath351 . there exists an integer @xmath621 and a @xmath0-map @xmath622 such that the following conditions hold . * for any @xmath623 , @xmath624 is a diffeomorphism . * @xmath625 . * for any @xmath626 , @xmath627 is a submersion . let @xmath628 denote the space of compactly supported @xmath158-vector fields on @xmath351 . for any @xmath629 , let @xmath630 denote the flow generated by @xmath631 . let us take a sequence @xmath632 on @xmath628 , such that @xmath633 spans @xmath634 for any @xmath626 . for @xmath635 , we set @xmath636 . let us define a @xmath0-map @xmath637 by @xmath638 . then , @xmath639 is onto for any @xmath626 and @xmath640 for some @xmath430 . finally , take any diffeomorphism @xmath641 preserving the origin . then , @xmath642 satisfies the requirements of the lemma . [ 150626_4 ] when @xmath351 is a riemannian manifold , for any @xmath643 we may further require the following condition : for any @xmath644 and @xmath623 , @xmath645 . let us apply lemma [ 150210_6 ] for @xmath646 , and take an integer @xmath621 and @xmath622 . for any @xmath647 , @xmath648 is a submersion . we take @xmath649 so that @xmath650 , and define @xmath651 \mapsto [ ( u \times \r^d , f \circ ( \ph \times \id_{\r^d } ) , \omega \times \nu_d)].\ ] ] it is easy to see that @xmath219 is a well - defined chain map . to show that @xmath652 and @xmath612 are chain homotopic , let us take @xmath653)$ ] so that * @xmath654 for any @xmath655 , and @xmath656 for any @xmath657 . * @xmath658 is compact , and there exists @xmath430 such that @xmath659 for any @xmath660 $ ] . for any @xmath661 , we define @xmath662 by @xmath663 since @xmath664 is a submersion , and @xmath665 is a diffeomorphism on @xmath351 for any @xmath666 , @xmath667 is also a submersion . therefore , @xmath668 . now , it is easy to see that @xmath669 \mapsto ( -1)^{|\omega|+d } [ ( u \times \r^d \times \r , \phi , \omega \times \nu_d \times b(s))]\ ] ] is a well - defined linear map , and @xmath670 . similar arguments show that @xmath671 and @xmath613 are chain homotopic . the homotopy operator is given by exactly the same formula as @xmath620 . this case is easier , since we do not have to care about the submersion condition . let us define an isomorphism @xmath597 . when @xmath672 is finite - dimensional , it is defined as @xmath673 the first isomorphism is defined by integrations of differential forms on smooth chains , and the second one follows from the poincar duality . to define @xmath674 in the general case , let us define a set @xmath675 by @xmath676 then , we define @xmath674 by @xmath677 to prove that @xmath594 is an isomorphism , it is enough to check that @xmath598 . we may assume that @xmath672 is finite - dimensional , since the general case follows from this case by taking limits . let us consider the following diagram : @xmath678 ^ -{\cong}\ar[rd]_{i_0 } & ( h^*_\dr(m))^ * & \ar[l]_-{\cong}\ar[ld]^{i_1 } h^{d-*}_{c,\dr}(m ) \\ & h^\dr_*(m ) . \ar[u ] & } \ ] ] the vertical map @xmath679 is defined by the pairing @xmath680 ( see section 2.8 ) . to show that @xmath681 , it is enough to check that the above diagram commutes up to sign . the commutativity of the right triangle is easy to check from the definitions . the commutativity of the left triangle follows from lemma [ 150629_2 ] ( the case @xmath682 ) . let @xmath3 be a closed , oriented riemannian manifold . we abbreviate @xmath683 as @xmath684 . we consider the differentiable structure on @xmath684 as in example [ 150623_1 ] ( ii ) . for any @xmath247 $ ] , we set @xmath685 , and consider the differentiable structure as a subset of @xmath684 ( see example [ 150623_1 ] ( iii ) ) . any strongly smooth map @xmath686 is continuous with respect to the @xmath158-topology on @xmath687 . therefore , we obtain a map @xmath688 , where the rhs denotes the singular homology . on the other hand , for any differentiable space @xmath313 , we defined the map @xmath591 . the aim of this section is to prove the following result . [ 150219_1 ] for any closed , oriented riemannian manifold @xmath3 and @xmath689 $ ] , the maps @xmath690 and @xmath688 are isomorphisms . as an immediate consequence , we obtain an isomorphism @xmath691 . the proof of theorem [ 150219_1 ] uses finite - dimensional approximations of the free loop space @xmath692 , which we explain in section 4.1 . recall that the rotation operator @xmath693 is defined as @xmath694 \times x)$ ] , where @xmath695 denotes the rotation . via isomorphisms @xmath696 , one can define the rotation operators on @xmath697 and @xmath698 in the same way . it is easy to see that the isomorphism @xmath699 preserves the rotation operators , since @xmath521 is a subcomplex of @xmath700 for @xmath701 . the isomorphism @xmath702 also preserves the rotation operators , since @xmath703 is functorial and commutes with the cross product ( lemma [ 150627_1 ] ) . thus we have proved the following corollary . [ 150627_2 ] the isomorphism @xmath704 preserves the rotation operators . let us define @xmath705 by @xmath706 . @xmath707 is smooth as a function on the differentiable space @xmath147 . for any @xmath708 $ ] , we define @xmath709 by @xmath710 . for any positive integer @xmath351 , let us define @xmath711 @xmath712 is smooth as a map between differentiable spaces . for any @xmath713 , let us define ( @xmath4 denotes the distance on @xmath3 ) : @xmath714 for any @xmath715 $ ] , we define @xmath716 it is easy to see that @xmath717 for any @xmath718 $ ] . let @xmath719 be the injectivity radius of @xmath3 ( since @xmath3 is closed , @xmath720 ) . for any @xmath721 such that @xmath722 , there exists a unique shortest geodesic @xmath723 \to m$ ] such that @xmath724 , @xmath725 . we denote it by @xmath726 . suppose that @xmath351 is sufficiently large , such that @xmath727 . for any @xmath728 and @xmath729 , there holds @xmath730 . for any integer @xmath731 , we define @xmath732 by @xmath733 for any @xmath734 and @xmath735 , we define @xmath736 by @xmath737 . this is a @xmath0-map between @xmath0-manifolds . [ 150625_1 ] for any positive real numbers @xmath738 and @xmath739 , there exists @xmath740 such that the following holds : for any integer @xmath741 and any integer @xmath731 , there exists a map @xmath742 such that the following diagram commutes up to homotopy ( @xmath743 denotes the inclusion map ) : @xmath744 ^ -{i}\ar[d]_-{f_n } & \l^{a',e ' } \ar[d]^-{f_{mn } } \\ { { \mathcal{f}}}^{a , e}_n \ar[r]_-{i_m}\ar[ru]^-{g}&{{\mathcal{f}}}^{a',e'}_{mn}. } \ ] ] @xmath347 and the homotopies are both continuous ( @xmath745 , @xmath746 are equipped with the @xmath0-topology ) and smooth ( as maps between differentiable spaces ) . to prove lemma [ 150625_1 ] , we need the following preliminary lemma [ 150625_2 ] . we define @xmath747 \times \{(p , q ) \in m^{\times 2 } \mid d(p , q)<r_m \ } \to m$ ] by @xmath748 . for any @xmath749 $ ] , we define a map @xmath750 by @xmath751 . for any @xmath752 such that @xmath753 , we define @xmath754 by @xmath755 . [ 150625_2 ] for any @xmath643 , there exists @xmath756 such that the following holds : if @xmath757 satisfy @xmath758 , for any @xmath759 there holds @xmath760 the following assertion is easy to prove by contradiction : there exists @xmath756 such that , if @xmath721 satisfy @xmath761 , then @xmath762).\ ] ] take @xmath763 as above . then , if @xmath752 satisfy @xmath764 , there holds @xmath765 for any @xmath749 $ ] and @xmath766 . the lemma follows from this estimate . let us take @xmath643 so that @xmath767 and @xmath768 . let us also take a @xmath0-function @xmath769 \to [ 0,1]$ ] with the following properties : * @xmath770 for any @xmath771 $ ] . * @xmath772 for any integer @xmath773 . * @xmath774 is constant on some neighborhoods of @xmath122 and @xmath89 . let us take an integer @xmath351 so that @xmath775 . for any @xmath728 , we define @xmath776 by @xmath777 then @xmath778 satisfies @xmath779 and @xmath780 , thus one can define @xmath781 by @xmath782 . it is clear that @xmath783 . @xmath347 is both smooth ( as a map between differentiable spaces ) and continuous ( @xmath784 is equipped with the @xmath0-topology ) . let us define a homotopy between @xmath743 and @xmath785 . for any @xmath786 , we set @xmath787 , and @xmath788 . then @xmath789 satisfies @xmath790 and @xmath791 . when @xmath351 is sufficiently large , @xmath792 thus , for any @xmath759 , @xmath793 satisfies @xmath794 and @xmath795 . finally , take @xmath796)$ ] so that @xmath797 for any @xmath655 , and @xmath798 for any @xmath657 . then , @xmath799 is a homotopy between @xmath743 and @xmath785 . @xmath345 is both smooth and continuous . [ rem : a = infty ] as is clear from the above proof , lemma [ 150625_1 ] holds even when @xmath800 . in this case , we denote @xmath801 by @xmath802 . let us take strictly increasing sequences of positive real numbers @xmath803 and @xmath804 , such that @xmath805 , @xmath806 . then , @xmath807 is an increasing sequence of open sets ( with respect to the @xmath0-topology ) of @xmath687 , and @xmath808 . thus , we have isomorphisms @xmath809 since the length functional on @xmath147 is approximately smooth , corollary [ 150213_1 ] implies @xmath810 . thus , it is enough to show that the following maps are isomorphisms : @xmath811 now we apply lemma [ 150625_1 ] for each @xmath278 . let us take a sequence @xmath812 of positive integers so that @xmath813 and @xmath814 for every @xmath278 . then , there exists a map @xmath815 such that the following diagram commutes up to homotopy : @xmath816\ar[d]_-{f_{n_j } } & \l^{a_{j+1 } , e_{j+1 } } \ar[d]^-{f_{n_{j+1}}}\\ { { \mathcal{f}}}^{a_j , e_j}_{n_j } \ar[r ] \ar[ru]^-{g_j}&{{\mathcal{f}}}^{a_{j+1 } , e_{j+1}}_{n_{j+1}}. } \ ] ] then @xmath817 is an isomorphism , since @xmath818 is its inverse . the same argument also works for @xmath819 and @xmath820 , and we obtain isomorphisms @xmath821 these isomorphisms fit into the following commutative diagram : @xmath822 ^ -{\cong } & \vlim_j h^\sm_*(\l^{a_j , e_j } ) \ar[d]^-{\cong } \ar[r ] \ar[l ] & \vlim_j h^\dr_*(\l^{a_j , e_j } ) \ar[d]^-{\cong } \\ \vlim_j h_*({{\mathcal{f}}}^{a_j , e_j}_{n_j } ) & \vlim_j h^\sm_*({{\mathcal{f}}}^{a_j , e_j}_{n_j } ) \ar[r ] \ar[l ] & \vlim_j h^\dr_*({{\mathcal{f}}}^{a_j , e_j}_{n_j } ) . } \ ] ] since @xmath823 is an oriented _ finite - dimensional _ manifold for every @xmath278 , @xmath824 and @xmath825 are isomorphisms . thus , the horizontal maps on the second row are isomorphisms . therefore , the horizontal maps on the first row are also isomorphisms . this completes the proof of theorem [ 150219_1 ] . constructions of string topology operations ( e.g. the loop product ) require ( at least ) two steps : * fiber products of ( de rham ) chains of the loop space with respect to evaluation maps . * concatenations of loops . the differentiable space @xmath826 , which we studied in the previous section , is not adequate for both steps . to avoid this trouble , in this section we introduce _ moore loops with marked points_. we explain the plan of this section . let @xmath3 denote a closed , oriented riemannian manifold of dimension @xmath4 . in section 5.1 , we introduce the space @xmath827 , which consists of moore paths on @xmath3 . in section 5.2 , we introduce the space @xmath828 , which consists of moore loops with @xmath829 marked points . we define two differentiable structures on @xmath828 , and denote the resulting differentiable spaces by @xmath828 and @xmath830 . the latter is adequate to define fiber products on de rham chain complexes , and we show that a sequence of de rham chain complexes @xmath831 has a natural structure of a cyclic dg operad with a multiplication and a unit ( see definitions [ 150723_1 ] , [ 150624_4 ] ) . this is a key geometric input in the definition of the chain complex @xmath9 , which we explain in section 6 . sections 5.35.5 are devoted to proofs of technical lemmas [ 150624_2 ] , [ 150624_3 ] , [ 150626_5 ] , which we state in the end of section 5.2 . let @xmath3 be a closed , oriented @xmath0-manifold . we define the set of moore paths on @xmath3 as follows : @xmath832 , m ) , \ , \gamma^{(m)}(0)=\gamma^{(m)}(t)=0 \,(\forall m \ge 1 ) \}.\ ] ] @xmath833 denotes the @xmath163-th derivative of @xmath778 . the last condition is required to take concatenations of @xmath0-paths . we define @xmath834 by @xmath835 , @xmath836 . for any @xmath837 , let us define a map @xmath838 and @xmath839 by @xmath840 the concatenation map @xmath841 is defined by @xmath842 functionals @xmath843 and @xmath844 on @xmath827 are defined by @xmath845 to define a differentiable structure on @xmath827 , we need the following definition . [ 150625_3 ] let @xmath313 and @xmath321 be @xmath0-manifolds , and @xmath335 be any subset of @xmath313 . then , a map @xmath846 is of class @xmath0 , if there exists an open neighborhood @xmath152 of @xmath847 and a @xmath0-map @xmath848 such that @xmath849 . we define two differentiable structures on @xmath827 , and denote the resulting differentiable spaces as @xmath827 and @xmath850 . the set of plots @xmath851 and @xmath852 are defined as follows : * let @xmath314 and @xmath853 . denote @xmath854 for any @xmath855 . then , @xmath856 if @xmath857 and @xmath858 is of class @xmath0 in the sense of definition [ 150625_3 ] ( @xmath859 , @xmath860 ) . * @xmath852 consists of @xmath856 such that @xmath861 is a submersion for @xmath527 . the identity map @xmath862 is smooth as a map of differentiable spaces . the functional @xmath844 is smooth , and @xmath843 is approximately smooth with both differentiable structures ( @xmath827 and @xmath850 ) . the goal of this subsection is to prove the next lemma . [ 150626_1 ] the concatenation maps @xmath863 are smooth as maps of differentiable spaces . set theoretically , the two maps in lemma [ 150626_1 ] are same . however , differentiable structures on the domain and the target of the map are different . first we prove the following lemma , which is a subtle point of the proof . [ 150626_2 ] let @xmath152 be a @xmath0-manifold , and @xmath864 . let @xmath865 , and suppose that @xmath866 satisfies the following conditions : * @xmath346 is of class @xmath0 in the sense of definition [ 150625_3 ] . * for any @xmath855 and an integer @xmath29 , @xmath867 . then , @xmath868 defined by @xmath869 is of class @xmath0 . let @xmath870 . a nontrivial part of the lemma is to check differentiability of @xmath871 at @xmath872 such that @xmath873 and @xmath874 . since @xmath346 is of class @xmath0 , there exists a @xmath0-function @xmath875 , which is defined on an open neighborhood of @xmath876 and @xmath877 . let us prove @xmath878 for any @xmath879 and @xmath874 . it is obvious when @xmath880 , since @xmath881 for any @xmath29 and @xmath882 which is near @xmath81 . even when @xmath873 , there exists a sequence @xmath883 such that @xmath884 and @xmath885 , since @xmath874 . thus , @xmath886 . to prove the lemma , it is enough to prove the claim @xmath887 for every integer @xmath29 : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ @xmath887 : for any @xmath879 such that @xmath888 , @xmath889 is totally differentiable , and @xmath890 unless @xmath891 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ let us prove @xmath892 . it is enough to check @xmath893 for any @xmath81 such that @xmath873 and @xmath874 . if this does not hold , there exists a sequence @xmath894 such that @xmath895 and @xmath896 . since @xmath897 , we may assume @xmath898 . then , @xmath899 , and thus , @xmath900 . on the other hand @xmath901 , and thus , @xmath902 . this contradicts @xmath903 , and @xmath892 is proved . let us prove @xmath904 . it is enough to check @xmath905 for any @xmath81 such that @xmath873 and @xmath874 . if this does not hold , there exists a sequence @xmath894 such that @xmath895 and @xmath906 . by @xmath907 , @xmath908 . thus , we may assume that @xmath909 . by @xmath907 , this implies @xmath899 , and thus , @xmath910 . on the other hand @xmath878 , and thus , @xmath911 . this contradicts @xmath912 , and @xmath887 is proved . we first prove that @xmath913 is smooth . let @xmath914 , and denote @xmath915 . we need to show that @xmath916 is a plot on @xmath827 . it is enough to show that @xmath917 is of class @xmath0 in the sense of definition [ 150625_3 ] . we may assume that @xmath3 is embedded in @xmath918 for some integer @xmath351 . for @xmath919 , we define @xmath920 by @xmath921 since @xmath922 for any @xmath855 and @xmath731 , lemma [ 150626_2 ] shows that @xmath923 . let us define @xmath924 by @xmath925 then , it is easy to see that @xmath926 for any @xmath855 and @xmath927 . this shows that @xmath928 is of class @xmath0 , hence @xmath913 is smooth . now , it is easy to see that @xmath929 is smooth , since any @xmath930 satisfies @xmath931 and @xmath932 . for any integer @xmath265 , we define @xmath933 in particular , @xmath934 . for any @xmath935 , we define @xmath936 we also define @xmath937 by @xmath938 . [ 150701_1 ] we can identify @xmath828 with @xmath939 , m ) , \ , 0 \le t_1 \le \cdots \le t_k \le t , \\ & \qquad \gamma(0)=\gamma(t ) , \quad \gamma^{(m)}(0)=\gamma^{(m)}(t)= \gamma^{(m)}(t_j)=0 \ , ( \forall m \ge 1 , \ , 1 \le \forall j \le k ) \}. \end{aligned}\ ] ] indeed , for any @xmath940 , one can assign @xmath941 . recall that we defined two differentiable structures on @xmath827 , and denote the resulting differentiable spaces @xmath827 and @xmath850 . since @xmath828 is a subset of @xmath942 , we can define two differentiable structures on @xmath828 ( see example [ 150623_1 ] ( iii ) and ( iv ) ) . we denote the resulting differentiable spaces by @xmath828 and @xmath830 . let us define @xmath943 by @xmath944 . the function @xmath843 is approximately smooth with both differentiable structures ( @xmath828 and @xmath830 ) . for any @xmath689 $ ] , we define @xmath945 . we define differentiable structures on @xmath946 as a subspace of @xmath828 and @xmath830 , and denote the resulting differentiable spaces as @xmath946 and @xmath947 . the chain maps @xmath948 and @xmath949 are injective . for any @xmath950 and @xmath951 , we define a map @xmath952 by @xmath953 then , @xmath954 is a smooth map by lemma [ 150626_1 ] . also , @xmath955 is a smooth map . now we need some definitions on algebraic operads . our notion of a multiplication of an operad ( definition [ 150624_4 ] ) is similar to the one in @xcite . [ 150723_1 ] let @xmath956 be a nonsymmetric dg operad . a _ cyclic structure _ on @xmath957 is a sequence @xmath958 with the following properties . * for any @xmath265 , @xmath959 is a chain map on @xmath960 of degree @xmath122 , satisfying @xmath961 . * @xmath962 is cyclically invariant , i.e. @xmath963 . * for any @xmath950 , @xmath951 , @xmath964 and @xmath965 , there holds @xmath966 a pair @xmath967 is called a _ nonsymmetric cyclic dg operad_. [ 150624_4 ] let @xmath968 be a nonsymmetric dg operad . @xmath969 is called a _ multiplication _ of @xmath957 , if @xmath970 and @xmath971 . @xmath972 is called a _ unit _ of @xmath774 , if @xmath973 and @xmath974 . let @xmath975 be a nonsymmetric dg operad with a multiplication and a unit . then , @xmath976 has the structure of a cosimplicial chain complex with chain maps @xmath977 defined as @xmath978 also , suppose that @xmath957 has a cyclic structure @xmath958 , and @xmath969 is cyclically invariant , i.e. @xmath979 . then , @xmath957 is a cocyclic chain complex . now let us return to the moore loop spaces . for any @xmath265 , let @xmath980 . then , @xmath981 has the structure of a nonsymmetric cyclic dg operad with a multiplication and a unit , defined as follows . * for any @xmath950 and @xmath951 , we define @xmath982 by @xmath983 . * for any @xmath265 , we define @xmath984 by @xmath985 . we define @xmath986 to be the identity map on @xmath987 . * let us take @xmath988 and an orientation - preserving diffemorphism @xmath989 . then , we define @xmath990 \in { { \mathcal{cl}}}(0)_0 , \quad 1_{{{\mathcal{cl}}}}:= [ ( m ' , i_1 \circ \ph , 1 ) ] \in { { \mathcal{cl}}}(1)_0 , \\ \mu&:= [ ( m ' , i_2 \circ \ph , 1 ) ] \in { { \mathcal{cl}}}(2)_0 . \end{aligned}\ ] ] it is easy to check that the elements @xmath373 , @xmath991 and @xmath774 are well - defined , i.e. not depend on choices of @xmath992 and @xmath324 . also , @xmath991 and @xmath774 are cyclically invariant . in particular , @xmath993 has the structure of a cocyclic chain complex . notice that this structure preserves the length filtration . namely , for any @xmath689 $ ] , @xmath994 has the structure of a cocyclic chain complex . the rest of this section is devoted to proofs of the following lemmas . [ 150624_2 ] for any @xmath265 , the identity map @xmath995 induces an isomorphism @xmath996 . let us recall notations @xmath997 and @xmath998 . [ 150624_3 ] for any @xmath265 , let us define @xmath999 and consider the differentiable structure on @xmath1000 as a subset of @xmath1001 . then , the inclusion map @xmath1002 induces an isomorphism @xmath1003 . [ 150626_5 ] for any @xmath265 , let us define the map @xmath1004 by @xmath1005 , @xmath1006 ( we set @xmath1007 , @xmath1008 ) . then , the map @xmath1009 is smooth , and induces an isomorphism @xmath1010 . summarizing these lemmas , we have the following zig - zag of the quasi - isomorphisms : @xmath1011 & c^\dr_*(\bl^a_k ) & c^\dr_*({{\mathcal{l}}}^a_k ) \ar[r]\ar[l ] & c^\dr_*({{\mathcal{l}}}^a \times \delta^k ) . \\ } \ ] ] the sequences @xmath1012 , @xmath1013 , @xmath1014 have natural structures of cocyclic chain complexes , and ( [ 150628_2 ] ) induces quasi - isomorphisms of these cocyclic chain complexes . let us take an increasing sequence of positive real numbers @xmath803 , such that @xmath1015 . since the length functional is approximately smooth on @xmath828 and @xmath830 , corollary [ 150213_1 ] implies @xmath1016 now , the key technical step is the next lemma . [ 150626_3 ] for any integer @xmath482 , there exists a chain map @xmath1017 such that the following diagram commutes up to chain homotopy : @xmath1018 ^ -{(\id_j ) _ * } \ar[d]_-{(i_\reg)_*}&c^\dr_*(\bl^{a_j}_k ) \ar[d]^-{i_*}\ar[ld]^-{j } \\ c^\dr_*(\bl^{a_{j+1}}_{k,\reg } ) \ar[r]_-{(\id_{j+1 } ) _ * } & c^\dr_*(\bl^{a_{j+1}}_k ) . } \ ] ] in the above diagram , @xmath1019 , @xmath1020 are identity maps , and @xmath613 , @xmath612 are inclusion maps . lemma [ 150626_3 ] implies @xmath1021 , then we obtain @xmath996 , that is lemma [ 150624_2 ] . let @xmath1022 . by lemma [ 150210_6 ] and remark [ 150626_4 ] , there exists an integer @xmath1023 and a @xmath0-map @xmath1024 such that * for any @xmath623 , @xmath1025 is a diffeomorphism . moreover , there holds @xmath1026 for any @xmath1027 . * @xmath1028 . * for any @xmath1029 , @xmath1030 is a submersion . let us define @xmath1031 by @xmath1032 then , @xmath1033 for any @xmath1034 . let us take @xmath1035 such that @xmath1036 . it is easy to see that @xmath1037 \mapsto [ ( u \times \r^d , { { \mathcal{f } } } \circ(\ph \times \id_{\r^d } ) , \omega \times \nu)]\ ] ] is a well - defined chain map . we show that @xmath219 satisfies the requirement in lemma [ 150626_3 ] . to show that @xmath1038 and @xmath1039 are homotopic , we take @xmath653)$ ] so that * @xmath1040 for any @xmath655 , and @xmath1041 for any @xmath657 . * @xmath1042 is compact , and there exists @xmath430 such that @xmath659 for any @xmath660 $ ] . for any @xmath1043 , we define @xmath1044 by @xmath1045 let us show that it is a plot of @xmath830 , namely @xmath1046 is a submersion for every @xmath935 . it is easy to check that @xmath1047 . since @xmath1048 , @xmath1049 is a submersion . on the other hand , @xmath665 is a diffeomorphism on @xmath3 . thus , @xmath1050 is a submersion . hence @xmath1051 . it is easy to see that the linear map @xmath1052 \mapsto ( -1)^{|\omega|+d } [ ( u \times \r^d \times \r , \phi , \omega \times \nu \times b(s))]\ ] ] is well - defined , and satisfies @xmath1053 . similar arguments show that @xmath1054 is homotopic to @xmath1055 . the homotopy operator is given by exactly the same formula as @xmath620 . this case is easier , since we do not have to care about the submersion condition . let us take strictly increasing sequences @xmath803 , @xmath804 of positive real numbers , such that @xmath1056 , @xmath1057 . we set @xmath1058 [ 150630_1 ] for every @xmath482 , there exists a smooth map @xmath1059 such that the following diagram commutes up to smooth homotopy : @xmath1060 \ar[d ] & \l^{a_j , e_j } \times \delta^k \ar[ld]^-j \ar[d ] \\ \l^{a_{j+1 } , e_{j+1}}_k \ar[r ] & \l^{a_{j+1 } , e_{j+1 } } \times \delta^k . } \ ] ] all maps other than @xmath219 are inclusion maps . assuming lemma [ 150630_1 ] , we can prove lemma [ 150624_3 ] as @xmath1061 to prove lemma [ 150630_1 ] , we need the following sublemma . [ 150630_2 ] for any @xmath643 , there exists a @xmath0-map @xmath1062 \to [ 0,1]$ ] such that the following properties hold for any @xmath1063 . * @xmath1064 , @xmath1065 . * @xmath1066 $ ] for any @xmath1067 $ ] . * @xmath1068 for any @xmath1067 $ ] . * @xmath1069 for any integer @xmath731 and @xmath1070 . we may assume @xmath1071 . let us take @xmath1072 . we also take @xmath504)$ ] such that @xmath1073 for any integer @xmath29 , and @xmath1074 if @xmath1075 . for any @xmath771 $ ] , we set @xmath1076 , and we define @xmath1077)$ ] by @xmath1078 it is clear that @xmath774 satisfies ( i ) and ( iv ) . it is also easy to check @xmath1079 , then ( ii ) and ( iii ) are verified as @xmath1080 let us fix @xmath1081 , and take @xmath643 so that @xmath1082 for the definition of @xmath1083 in the second inequality , see lemma [ 150625_2 ] . we take @xmath1062 \to [ 0,1]$ ] as in lemma [ 150630_2 ] . for any @xmath1084 and @xmath1085 , we define @xmath1086 by @xmath1087 ( this is well - defined by ( i ) ) . then @xmath1088 , and @xmath1089 by ( ii ) . by ( iv ) , for any @xmath731 and @xmath1090 , there holds @xmath1091 . therefore , @xmath1092 . since @xmath1093 , one can define @xmath219 by @xmath1094 by lemma [ 150630_2 ] ( iii ) and @xmath1095 , there holds @xmath1096 for any @xmath749 $ ] , let @xmath1097 . the map @xmath750 is defined right before lemma [ 150625_2 ] . applying lemma [ 150625_2 ] for @xmath1098 , we obtain @xmath1099 for any @xmath749 $ ] . if @xmath1100 , there holds @xmath1101 for any @xmath731 , @xmath759 , and @xmath1102 . thus , @xmath1103 for any @xmath759 . let us take @xmath1104)$ ] so that @xmath1105 for @xmath655 and @xmath1106 for @xmath657 . we define @xmath1107 by @xmath1108 obviously , this is a smooth homotopy between @xmath219 and the inclusion map @xmath1109 . finally , the restriction of @xmath1110 to @xmath1111 is a smooth homotopy between @xmath1112 and the inclusion map @xmath1113 . it is easy to see that the map @xmath1009 is smooth , thus it is enough to show that the map induces an isomorphism on @xmath820 . the proof consists of three steps . * step 1 . * we identify @xmath828 with the set consists of tuples @xmath1114 ( see remark [ 150701_1 ] ) . setting @xmath1115 \to \r/\z$ ] by @xmath1116 $ ] , the map @xmath1009 is given by @xmath1117 . the image of this map is contained in @xmath1118 now , @xmath1119 induces an isomorphism on @xmath820 , since @xmath1120 is its smooth homotopy inverse , where @xmath1121 . therefore , it is enough to show that the inclusion map @xmath1122 gives an isomorphism on @xmath820 . * step 2 . * let @xmath1123 . for any @xmath1124 , let us set @xmath1125 since @xmath707 is smooth as a function on @xmath946 , we obtain isomorphisms @xmath1126 , @xmath1127 . thus , it is enough to show that the inclusion map @xmath1128 induces an isomorphism on @xmath820 for every @xmath1124 . * step 3 . * for any @xmath1129 , let us set @xmath1130 consider the following commutative diagram , where all maps are inclusion maps : @xmath1131^{j_3 } \ar[d]_{j_1 } & \bl^{a , e}_{k , \delta } \ar[d]^{j_2 } \\ \bl^{a , e}_{k , t>0 } \ar[r]_{j_4 } & \bl^{a , e}_k . } \ ] ] then , it is easy to check the following claims : * there exists @xmath1132 , depending only on @xmath78 , such that if @xmath1133 then @xmath1134 , @xmath1135 induce isomorphisms on @xmath820 . this is because one can define homotopy inverses of @xmath1134 and @xmath1135 by contracting a loop @xmath1136 to the constant loop at @xmath1137 whenever @xmath1138 ( notice that @xmath1139 ) . * for any @xmath643 , @xmath1140 induces an isomorphism on @xmath820 , since its homotopy inverse is given by @xmath1141 , such that * * @xmath1142 for any @xmath1143 , @xmath1144 for any @xmath1145 , and @xmath1146 . * * @xmath1147 is defined as @xmath1148 therefore , we can conclude that @xmath1149 induces an isomorphism on @xmath820 . this completes the proof of lemma [ 150626_5 ] . the first goal of this section is to define the chain complex @xmath9 , which is our chain model of the free loop space @xmath11 . the second goal is to prove the results presented in section 1.5 assuming theorem [ 150624_8 ] , which is a purely algebraic result on dg operads . the proof of theorem [ 150624_8 ] occupies the rest of this paper ( sections 711 ) . after some algebraic preliminaries in section 6.1 , we define the chain complex @xmath9 in section 6.2 . in section 6.3 , we establish the relation between @xmath9 and the hochschild complex of @xmath7 . in section 6.4 , we state theorem [ 150624_8 ] , and prove most results presented in section 1.5 , except that the isomorphism @xmath221 preserves the bv structures . we prove this result in sections 6.56.8 . a double complex @xmath1150 consists of a sequence @xmath1151 of chain complexes and anti - chain maps @xmath1152 for each @xmath128 , such that @xmath536 for any @xmath128 . we denote @xmath1153 as @xmath1154 if necessary . for any double complexes @xmath1150 and @xmath1023 , a morphism @xmath1155 is a sequence @xmath1156 such that , @xmath1157 is a chain map and @xmath1158 for every @xmath128 . for any double complex @xmath1150 , we define a chain complex @xmath1159 by @xmath1160 which we call the _ total complex _ of @xmath1150 . a morphism @xmath1155 of double complexes induces a chain map @xmath1161 . [ 150629_1 ] let @xmath1162 be a double complex . if the sequence @xmath1163 & h_q(c(0 ) ) \ar[r]_-{h_q(\delta_1 ) } & h_q(c(1 ) ) \ar[r]_-{h_q(\delta_2 ) } & h_q(c(2 ) ) \ar[r]_-{h_q(\delta_3 ) } & \cdots } \ ] ] is exact for every @xmath1164 , then the total complex @xmath1165 is acyclic . for any @xmath951 , let @xmath1166 . then , @xmath1167 is a decreasing filtration on @xmath1165 which is complete , i.e. @xmath1168 . let us consider the spectral sequence of this filtered complex . then , the assumption implies that all @xmath1169-terms vanish . now , the convergence theorem 5.5.10 ( 2 ) in @xcite pp.139 shows @xmath1170 . [ 150627_5 ] let @xmath1155 be a morphism of double complexes . suppose that @xmath1157 is a quasi - isomorphism for every @xmath265 . then , the chain map @xmath1171 is a quasi - isomorphism . let us consider the filtrations @xmath1167 and @xmath1172 as in the proof of the previous lemma . then , @xmath1171 induces a morphism of the spectral sequences , and the assumption implies that it induces isomorphisms on @xmath1173-terms . then , the comparison theorem 5.5.11 in @xcite pp . 141 shows that @xmath1174 is an isomorphism . let @xmath1175 be any cosimplcial chain complex . for any @xmath128 , let us define @xmath1152 by @xmath1176 . then , @xmath1150 is a double complex . for any dga algebra @xmath1 , the endomorphism operad @xmath1177 has a multiplication @xmath1178 and a unit @xmath1179 , defined by @xmath1180 and @xmath1181 . in particular , @xmath1182 has the structure of a cosimplicial chain complex . the total complex @xmath1183 is isomorphic to the hochschild cochain complex @xmath14 . as a consequence of lemma [ 150629_1 ] , we obtain the next lemma . [ 150627_4 ] let @xmath1175 be a cosimplicial chain complex , and suppose that the chain map @xmath1184 , which is induced by the cosimplicial structure , is a quasi - isomorphism for any @xmath265 . then , the projection map @xmath1185 is a quasi - isomorphism . since @xmath1186 is surjective , it is enough to show that @xmath1187 is acyclic . the assumption shows that , for any @xmath1164 the sequence @xmath1163 & h_q(c(1 ) ) \ar[r]_-{h_q(\delta_2 ) } & h_q(c(2 ) ) \ar[r]_-{h_q(\delta_3 ) } & h_q(c(3 ) ) \ar[r]_-{h_q(\delta_4 ) } & \cdots } \ ] ] is exact , thus lemma [ 150629_1 ] shows that @xmath1188 is acyclic . let @xmath3 be a closed , oriented riemannian manifold of dimension @xmath4 . we define the chain complex @xmath9 to be the total complex of the cosimplicial chain complex @xmath993 . explicitly , @xmath1189 for any @xmath247 $ ] , we define a subcomplex @xmath1190 by @xmath1191 . then , @xmath1192}$ ] is the length filtration on @xmath9 . we are going to show that , there exists an isomorphism @xmath1193 for each @xmath689 $ ] , such that proposition [ 141005_2 ] ( i ) holds . in particular , when @xmath1194 we obtain an isomorphism @xmath221 . let us abbreviate @xmath1195 by @xmath692 , as usual . let us recall the zig - zag ( [ 150628_2 ] ) of quasi - isomorphisms from section 5.2 : @xmath1196 & c^\dr_{*+d}(\bl^a_k ) & \ar[l ] c^\dr_{*+d}({{\mathcal{l}}}^a_k ) \ar[r ] & c^\dr_{*+d}({{\mathcal{l}}}^a \times \delta^k ) . } \ ] ] it extends to a zig - zag of quasi - isomorphisms of cocylic chain complexes . let @xmath1197 denote the total complex of the cosimplicial chain complex @xmath1198 . then , lemma [ 150627_5 ] implies the isomorphism @xmath1199 . for every @xmath265 the projection @xmath1200 induces a quasi - isomorphism on @xmath1201 . then , lemma [ 150627_4 ] shows that @xmath1202 is a quasi - isomorphism , thus @xmath1203 . the next lemma is confirmed by short computations . [ 150629_3 ] let @xmath548 be as in lemma [ 150624_10 ] ( i ) . then , @xmath1204 is a chain map such that @xmath1205 . on the other hand , we have the isomorphism @xmath1206 by theorem [ 150219_1 ] . in conclusion , for every @xmath247 $ ] , we obtain a sequence of isomorphisms @xmath1207 in particular , we obtain @xmath1208 . to define a chain map @xmath225 , we take @xmath988 and an orientation - preserving diffeomorphism @xmath989 . let us recall the map @xmath1209 , and define @xmath1210 by @xmath1211 & ( k=0 ) , \\ 0 & ( k \ge 1 ) . \end{cases}\ ] ] it is easy to check that this is well - defined ( i.e. not depend on @xmath992 and @xmath324 ) , and the diagram ( [ 150801_3 ] ) commutes . for any @xmath265 , @xmath1212 and @xmath1213 , let @xmath1214 . by definition , @xmath1215 are submersions for all @xmath278 . we define a chain map @xmath1216 by @xmath1217)(\eta_1 \otimes \cdots \otimes \eta_k):= ( -1)^{(\dim u - d)(|\eta_1| + \cdots + |\eta_k| ) } ( \ph_0)_!(\omega \wedge \ph_1^*\eta_1 \wedge \cdots \wedge \ph_k^*\eta_k).\ ] ] recall that we defined nonsymmetric dg operads with multiplications and units @xmath1218 @xmath1219 is a morphism of nonsymmetric dg operads preserving multiplications and units . in general , if @xmath1220 is a morphism of nonsymmetric dg operads preserving multiplications and units , then @xmath324 is a morphism of cosimplicial chain complexes , thus induces a chain map @xmath1221 . in particular , we obtain a chain map @xmath19 . we are going to show the following lemma ( see theorem [ 150628_1 ] ( iii)-(c ) ) . [ 150725_1 ] @xmath20 is equal to the map ( [ 141218_01 ] ) . for any @xmath1213 , we define @xmath1222 by @xmath1223 for any @xmath1224 , we set @xmath1214 . let us define a chain map @xmath1225)$ ] by @xmath1226(\eta_1 \otimes \cdots \otimes \eta_k)(\eta_0 ) : = ( -1)^{(\dim u - d)(|\eta_0|+|\eta_1| + \cdots + |\eta_k| ) } \int_u \omega \wedge \ph_1^*\eta_1 \wedge \cdots \wedge \ph_k^ * \eta_k \wedge \ph_0^*\eta_0.\ ] ] then , one can define a chain map @xmath1227)$ ] . as is obvious from the construction , the following diagram commutes : @xmath1228 ^ -{h_*(j ) } \ar[d]_-{\cong } & h^*({{\mathcal{a}}}_m , { { \mathcal{a}}}_m ) \ar[d]^-{\cong } \\ h_*(c^{{{\mathcal{l}}}\delta } ) \ar[r]_-{h_*(j ' ) } & h^*({{\mathcal{a}}}_m , { { \mathcal{a}}}^\vee_m[d ] ) . } let us consider the chain map @xmath1229)$ ] in ( [ 141222_4 ] ) . we need to show that @xmath1230 corresponds to @xmath1231 via the isomorphism @xmath1232 . let us take @xmath548 as in lemma [ 150624_10 ] ( i ) , and consider chain maps @xmath1233 the first map is defined right after lemma [ 150624_10 ] , and the second map is defined in lemma [ 150629_3 ] . then , it is enough to show that the following diagram commutes : @xmath1234 ^ -{h_*(i_u)}_-{\cong } \ar[rd]_-{h_*(i ) } & h^\dr_{*+d}({{\mathcal{l } } } ) \ar[r]^-{h_*(e_u)}_-{\cong } & h_*(c^{{{\mathcal{l}}}\delta } ) \ar[ld]^-{h_*(j ' ) } \\ & h^*({{\mathcal{a}}}_m , { { \mathcal{a}}}_m^\vee[d ] ) . & } \ ] ] the commutativity of this diagram follows from lemma [ 150629_2 ] and short computations . first we state the following result , which we prove in sections 711 . [ 150624_8 ] there exists a dg operad @xmath12 and its suboperad @xmath13 with the following conditions . 1 . there exist isomorphisms of graded operads @xmath1235 and @xmath214 , such that the following diagram commutes : @xmath215 \ar[d]_-{\cong } & h_*(f\tilde{\lambda } ) \ar[d]^-{\cong } \\ \ger \ar[r ] & \batvil . } \ ] ] 2 . let @xmath957 be any nonsymmetric dg operad , with a multiplication @xmath969 and a unit @xmath972 . then , @xmath1236 has a dg @xmath18-algebra structure , in particular @xmath1237 has the gerstenhaber algebra structure . the operators @xmath84 and @xmath85 are defined in chain level by the following formulas , where @xmath1238 : @xmath1239 these operators induce dga and dg lie algebra structures on @xmath1236 . finally , for any morphism @xmath1240 of nonsymmetric dg operads which preserves multiplications and units , the chain map @xmath1241 is a morphism of dg @xmath18-algebras . 3 . let @xmath957 be as ( ii ) , and assume that @xmath957 has a cyclic structure @xmath958 such that @xmath979 . then , @xmath1236 has a dg @xmath1242-algebra structure , which extends the dg @xmath18-algebra structure in ( ii ) . in particular , @xmath1237 has the bv algebra structure . the operator @xmath99 is defined at chain level by the following formula : @xmath1243 @xmath99 is an anti - chain map on @xmath1236 . [ 150811_2 ] there are several results similar to theorem [ 150624_8 ] in the literature , e.g. @xcite theorem 5.17 , @xcite theorems a and b. actually , @xcite considers operads with maurer - cartan elements , which are generalizations of operads with multiplications . however , for the following reasons we can not directly apply results in @xcite in our argument : * @xcite assumes @xmath1244 , which we can not assume in our argument . * in theorem b in @xcite , actions of operads are defined only for _ normalized _ chains ( see section 3.1 @xcite ) , while we would like to consider arbitrary chains . therefore , we give a self - contained proof of theorem [ 150624_8 ] in sections 711 . as a consequence of theorem [ 150624_8 ] , we obtain the following : * for any dga algebra @xmath1 , the hochschild complex @xmath138 has a dg @xmath18-algebra structure ; apply theorem [ 150624_8 ] ( ii ) for @xmath1245 . * for any closed , oriented @xmath0-manifold @xmath3 , the chain complex @xmath9 has a dg @xmath1242-algebra structure ; apply theorem [ 150624_8 ] ( iii ) for @xmath1246 . also , this @xmath1242-algebra structure on @xmath9 preserves the length filtration ; see remark [ 150801_4 ] . * the chain map @xmath19 ( see the previous subsectoin ) is a map of dg @xmath18-algebras ; since @xmath1219 is a morphism of nonsymmetric dg operads preserving multiplications and units , we can apply theorem [ 150624_8 ] ( ii ) . * the chain map @xmath225 satisfies ( [ 150801_2 ] ) ; this follows form explicit formulas of @xmath84 and @xmath1247 in theorem [ 150624_8 ] ( ii ) . now , we have verified most results presented in section 1.5 . the only statement we have not verified is that the isomorphism @xmath17 preserves the bv structures , i.e. the operators @xmath84 and @xmath99 are preserved by this isomorphism . in the rest of this section , we prove that @xmath667 preserves the operator @xmath99 ( section 6.5 ) and @xmath84 ( sections 6.66.8 ) . the arguments in sections 6.6 - 6.8 are less detailed than the other parts of this paper . the aim of this subsection is to show that the isomorphism @xmath17 preserves the rotation operator @xmath99 . for any cocyclic chain complex @xmath1175 and its total complex @xmath1165 , let us define an anti - chain map @xmath1248 by @xmath1249 this is a generalization of the definition of @xmath99 in theorem [ 150624_8 ] ( iii ) . in particular , we can define @xmath99 on @xmath1250 so that the isomorphism @xmath1251 preserves @xmath99 . since we already proved that the isomorphism @xmath1252 preserves @xmath99 ( corollary [ 150627_2 ] ) , it is enough to show that the isomorphism @xmath1253 preserves @xmath99 . the isomorphism @xmath1253 is induced by @xmath1254 , and its inverse is induced by @xmath1255 ( lemma [ 150629_3 ] ) , where @xmath548 is as in lemma [ 150624_10 ] ( i ) . therefore , it is enough to prove @xmath1256 on @xmath1257 . it is easy to check that @xmath1258 for any @xmath1259 . the chain map @xmath1260 is induced by @xmath1261 ^ -{\id_{{{\mathcal{l } } } } \times p } & { { \mathcal{l } } } \times s^1 \ar[r ] & s^1 \times { { \mathcal{l } } } \ar[r]^-{r } & { { \mathcal{l } } } , } \ ] ] where @xmath1262 is defined by @xmath1116 $ ] , the second map is the inversion , and the last map @xmath1263 is the rotation . therefore @xmath1264 for any @xmath1259 . since @xmath1265 is a cycle which represents @xmath178 $ ] , for any cycle @xmath438 we obtain @xmath1266 ) = - h_*(r)([s^1 ] \times [ x])$ ] . this completes the proof . we are going to show that the isomorphism @xmath17 preserves the operator @xmath84 . in this subsection , we reduce the proof to lemma [ 150629_4 ] , which is proved in the next two subsections . let us consider the concatenation map @xmath1267 and define a chain map @xmath1268 by @xmath1269 . then , @xmath1270 intertwines the operators @xmath84 and @xmath1271 : @xmath1272 therefore , it is enough to prove the next proposition . [ 150627_3 ] the isomorphism @xmath1273 intertwines the operator @xmath1271 on @xmath1274 and the loop product @xmath84 on @xmath1275 . let us define @xmath1276 by @xmath1277 . a key step in the definition of the loop product on @xmath1275 is to define the fiber product @xmath1278 via thom isomorphism ( see section 1.3 ) . on the other hand , let us consider the following differentiable structure on @xmath684 , and denote the resulting differentiable space by @xmath1279 : @xmath1280 then , as in section 2.3 , one can define the fiber product @xmath1281 we have isomorphisms @xmath1282 ; the first isomorphism is by theorem [ 150219_1 ] , and the second isomorphism is obtained by similar arguments as the proof of lemma [ 150624_2 ] . we also have isomorphisms @xmath1283 by similar arguments . let us state the key technical result : [ 150629_4 ] the following diagram commutes : @xmath1284 \ar[d]_{\cong } & h^\dr_{*-d}(\l_\reg { \,_{e}\!\!\times_{e } } \l_\reg ) \ar[d]^{\cong } \\ h_*(\l)^{\otimes 2 } \ar[r ] & h_{*-d}(\l { \,_{e}\!\!\times_{e } } \l ) , } \ ] ] where horizontal maps are fiber products . it is easy to deduce proposition [ 150627_3 ] from lemma [ 150629_4 ] , and details are left to the reader . the rest of this section is devoted to the proof of lemma [ 150629_4 ] . in the next subsection , we prove lemma [ 150726_1 ] , which is a preliminary result in the finite - dimensional setting . let @xmath313 be a @xmath0-manifold , and @xmath1285 be a submersion . then , one can define the fiber product @xmath1286 via the thom isomorphism for the tubular neighborhood of @xmath1287 . on the other hand , let us consider the following differentiable structure on @xmath313 , and denote the resulting differentiable space by @xmath1288 : @xmath1289 then , one can define the fiber product @xmath1290 . it is obvious that the identity maps on @xmath313 and @xmath1291 induces smooth maps @xmath1292 and @xmath1293 . these maps induce isomorphisms @xmath1294 and @xmath1295 . this fact is proved by similar arguments as proposition [ 150210_4 ] , and details are omitted . then , we obtain isomorphisms @xmath1296 [ 150726_1 ] the following diagram commutes : @xmath1297 \ar[d]^{\cong } & h^\dr_{*-d}(x_{\reg / m } { \,_{e}\!\!\times_{e } } x_{\reg / m } ) \ar[d]^{\cong } \\ h_*(x)^{\otimes 2 } \ar[r ] & h_{*-d}(x { \,_{e}\!\!\times_{e } } x ) . } \ ] ] via isomorphisms @xmath1298 and @xmath1299 both horizontal maps in ( [ eq : fbp_finite ] ) are identified with the map @xmath1300 \otimes [ \eta ] \mapsto [ \omega \times \eta|_{x { \,_{e}\!\!\times_{e } } x}],\ ] ] hence ( [ eq : fbp_finite ] ) is commutative . we use notations in section 4.1 . we abbreviate @xmath1301 by @xmath1302 , and @xmath1303 by @xmath1304 . let @xmath804 be a strictly increasing sequence of positive real numbers , such that @xmath1305 . let us take a sequence @xmath812 of positive integers , so that @xmath814 and @xmath1306 for every @xmath482 ( see remark [ rem : a = infty ] ) . by lemma [ 150625_1 ] , there exists a continuous map @xmath1307 such that @xmath1308\ar[d]_-{f_{n_j}}&\l^{e_{j+1 } } \ar[d]^-{f_{n_{j+1}}}\\ { { \mathcal{f}}}^{e_j}_{n_j } \ar[r ] \ar[ru]_-{g_j}&{{\mathcal{f}}}^{e_{j+1}}_{n_{j+1 } } } \ ] ] commutes up to homotopy . then , @xmath1309 . in the following arguments , we abbreviate @xmath1310 by @xmath1311 , and @xmath1312 by @xmath512 . let @xmath1313 . as is clear from the proof of lemma [ 150625_1 ] , one may take @xmath1314 so that @xmath1315 ( thus @xmath1316 is well - defined ) , and the following diagram commutes up to homotopy : @xmath1317\ar[d]_-{f_j \times f_j}&\l^{e_{j+1 } } { \,_{e}\!\!\times_{e } } \l^{e_{j+1 } } \ar[d]^-{f_{j+1 } \times f_{j+1 } } \\ { { \mathcal{f}}}^j { \,_{e_j}\!\!\times_{e_j } } { { \mathcal{f}}}^j \ar[r ] \ar[ru]_-{g_j \times g_j } & { { \mathcal{f}}}^{j+1 } { \,_{e_{j+1}}\!\!\times_{e_{j+1 } } } { { \mathcal{f}}}^{j+1}. } \ ] ] then , @xmath1318 . since @xmath1319 is a submersion , one can define the differentiable space @xmath1320 as in the previous subsection . then , @xmath512 maps plots of @xmath1321 to plots of @xmath1320 . also , one may take @xmath1314 so that it maps plots of @xmath1320 to plots of @xmath1322 , and the diagrams ( [ eq : ejgj ] ) and ( [ eq : ejgj_2 ] ) commute up to smooth homotopy with these differentiable structures . hence we obtain @xmath1323 for every @xmath482 , let us consider the following diagram : @xmath1324\ar[ddd]\ar[rd]^{\star } & & & h^\dr_{*-d } ( \l^{e_j}_\reg { \,_{e}\!\!\times_{e } } \l^{e_j}_\reg)\ar[ld]_{\star}\ar[ddd ] \\ & h^\dr_*\big({{\mathcal{f}}}^j_{\reg / m } \big)^{\otimes 2}\ar[r]\ar[d]&h^\dr_{*-d } \big ( { { \mathcal{f}}}^j_{\reg / m } { \,_{e_j}\!\!\times_{e_j } } { { \mathcal{f}}}^j_{\reg / m } \big)\ar[d ] & \\ & h_*\big({{\mathcal{f}}}^j \big)^{\otimes 2}\ar[r]&h_{*-d } \big({{\mathcal{f}}}^j { \,_{e_j}\!\!\times_{e_j } } { { \mathcal{f}}}^j \big ) & \\ h_*(\l^{e_j})^{\otimes 2}\ar[ru]^{\star}\ar[rrr]&&&h_{*-d } ( \l^{e_j } { \,_{e}\!\!\times_{e } } \l^{e_j}).\ar[lu]_{\star } } \ ] ] the center square in ( [ eq : bigsquare ] ) is commutative by lemma [ 150726_1 ] . the commutativity of the other four squares in ( [ eq : bigsquare ] ) is easy to check from definitions . taking direct limits as @xmath1325 , all maps in ( [ eq : bigsquare ] ) pass to the limit . moreover , the limits of the maps with @xmath1326 are isomorphisms . therefore , the limit of the big square in ( [ eq : bigsquare ] ) is commutative , and this completes the proof of lemma [ 150629_4 ] . the rest of this paper is devoted to the proof of theorem [ 150624_8 ] . in this section , we fix definitions and terminologies about ( ribbon ) graphs , partially following @xcite and @xcite . as a definition of a graph , we adopt the one in @xcite . a _ graph _ is a quadruple @xmath1327 where @xmath1328 and @xmath1329 are finite sets , @xmath1330 is a map @xmath1331 , @xmath1332 is a map @xmath1333 such that @xmath1334 . when there is no risk of confusion , the subscript @xmath1335 will be dropped . * the elements of @xmath38 are called the _ vertices _ of @xmath1335 . * the elements of @xmath875 are called the _ flags _ of @xmath1335 . * for any @xmath1336 , the elements of @xmath1337 are called the flags at @xmath882 . @xmath1338 is called the _ valence _ of @xmath882 , and denoted by @xmath1339 . * an _ edge _ of @xmath1335 is a set @xmath1340 such that @xmath1341 and @xmath1342 . the set of all edges of @xmath1335 is denoted by @xmath1343 . we say that the edge @xmath1340 connects vertices @xmath1344 and @xmath1345 . * a _ tail _ of @xmath1335 is @xmath1341 such that @xmath1346 . the set of all tails of @xmath1335 is denoted by @xmath1347 . an isomorphism of graphs consists of bijections between the sets of vertices and flags , compatible with maps @xmath1348 and @xmath1349 . to any graph @xmath1335 , one can associate a @xmath89-dimensional cw complex @xmath1350 in the obvious way ( see @xcite , pp.1502 ) . @xmath1335 is called _ connected _ if @xmath1350 is connected as a topological space . for any @xmath1351 , let @xmath1352 . let us denote a graph @xmath1353 by @xmath1354 . @xmath78 is called _ acyclic _ if @xmath1355 . in this paper , a _ tree _ is a connected graph @xmath1335 which has no tails , and @xmath1356 . it is easy to see that @xmath1350 is contractible . in particular , for any vertices @xmath1357 , there exists at most one edge which connects @xmath882 and @xmath1358 . when @xmath1359 , there is no such edge . a _ ribbon graph _ is a graph with a cyclic order on the set of flags at @xmath882 for each vertex @xmath882 . formally , a ribbon graph is a quintuple @xmath1360 , where @xmath1361 is a graph and @xmath1362 is a bijection , such that @xmath1363 , and @xmath1364 acts transitively on @xmath1365 for every @xmath1366 . an isomorphism of ribbon graphs consists of bijections between the sets of vertices and flags , compatible with maps @xmath1348 , @xmath1349 and @xmath351 . in all figures in this paper , the map @xmath1364 is a _ clockwise _ rotation of flags . orbits of the bijection @xmath1367 are called _ cycles _ of @xmath1335 . for any cycle @xmath1368 , we define a cyclic order on @xmath1368 by @xmath1369 . the set of all cycles of @xmath1335 is denoted by @xmath1370 . when @xmath1335 is connected , the _ genus _ of @xmath1335 , denoted by @xmath1371 , is defined by the formula @xmath1372 . let @xmath1335 be a ribbon graph . for any @xmath1373 , we define a ribbon graph @xmath1374 by removing elements of @xmath875 . here we present a formal definition . first , we set @xmath1375 . next , we consider the case that @xmath875 consists of a single flag @xmath346 . we define a ribbon graph @xmath1376 by @xmath1377 , @xmath1378 , @xmath1379 , and @xmath1380 finally , in the general case , we define so that there holds @xmath1381 for any @xmath1373 and @xmath1382 . let @xmath1335 be a ribbon graph . for any @xmath1351 which is acyclic , we define a ribbon graph @xmath1383 by contracting elements of @xmath78 . here we present a formal definition . first , we set @xmath1384 . next , we consider the case that @xmath78 consists of a single edge @xmath1385 . since @xmath78 is acyclic , @xmath1385 connects different vertices . namely , let @xmath1386 , @xmath1387 , @xmath1388 , then @xmath1389 . we consider a new vertex @xmath1390 , and define @xmath1391 . we define @xmath1392 , @xmath1393 , and @xmath1394 contracting an edge @xmath1385 . finally , in the general case we define so that there holds @xmath1395 for any @xmath1351 and @xmath1396 such that @xmath1397 is acyclic . the next lemma is easy to check and its proof is omitted . [ lem : removing_contracting ] let @xmath1335 be a ribbon graph , @xmath1351 , @xmath1373 . suppose that @xmath78 is acyclic and @xmath1398 . then , @xmath1399 . in this section we introduce a notion of a _ decorated cactus _ , which is a ribbon graph with additional data . we also define the framed version ( _ framed decorated cactus _ ) , and define compositions of framed decorated cacti . in the next section , we show that the collection of framed decorated cacti has a natural structure of a @xmath1400-colored operad . [ 141121_1 ] let @xmath1335 be a ribbon graph , @xmath1401 be cycles of @xmath1335 , and @xmath1402 . a tuple @xmath1403 is called a _ decorated cactus _ , if the following conditions hold : 1 . @xmath1335 is connected , and @xmath1404 . 2 . @xmath1328 is a disjoint union of @xmath1405 . 3 . for any @xmath1406 , there holds @xmath1407 . in particular , @xmath1408 . an isomorphism of decorated cacti @xmath1409 is an isomorphism @xmath1410 of ribbon graphs , which maps @xmath1411 to @xmath1412 , and @xmath1413 to @xmath1414 for every @xmath1415 . for any @xmath1416 , @xmath1417 denotes the set of isomorphism classes of decorated cacti @xmath1403 such that @xmath1418 , @xmath1419 . given a decorated cactus @xmath1420 , we define a graph @xmath1421 by @xmath1422 , @xmath1423 , and each @xmath1424 connects @xmath1425 and @xmath1426 . ( here @xmath1425 and @xmath1426 are considered as vertices of @xmath1421 . ) [ lem : dual_bw_tree ] @xmath1421 is a tree . it is easy to see that @xmath1421 is connected . therefore , it is enough to show that @xmath1427 . it is obvious that @xmath1428 . on the other hand , definition [ 141121_1 ] ( ii ) , ( iii ) imply @xmath1429 . therefore , we obtain @xmath1430 . we call @xmath1421 the _ dual tree _ of the decorated cactus @xmath1420 . [ cor : ci_is_cycle ] for any @xmath1431 and @xmath1432 , if @xmath1433 , then @xmath1434 . let @xmath1435 . then , both @xmath346 and @xmath1436 connect @xmath882 and @xmath1413 in @xmath1421 . ( here @xmath882 and @xmath1413 are considered as vertices of @xmath1421 . ) since @xmath1421 is a tree , we obtain @xmath1434 . the figure below shows an example of a decorated cactus . this is an element in @xmath1437 , and let us denote it by @xmath1438 . for each @xmath1439 , @xmath1413 is a boundary of the region @xmath743 . notice that there are @xmath1440 vertices on @xmath1413 for each @xmath1439 . the elements of @xmath1347 are depicted as the eight segments . in particular , @xmath1402 is depicted as the arrow . the configuration of cycles @xmath1441 is tree - like ( this holds in general by lemma [ lem : dual_bw_tree ] ) , which is often called a `` cactus '' ( see @xcite , @xcite ) . additionally , there are @xmath1440 vertices on each cycle @xmath1413 , and @xmath829 tails ( hence the name `` decorated '' cactus ) . the tail @xmath1411 corresponds to the `` global zero '' of a cactus ( see @xcite section 2.2.6 ) . in most papers , a cactus is a tree - like configuration of _ parametrized _ circles . on the other hand , cycles @xmath1442 of our decorated cactus are _ not _ parametrized . in the next subsection , we introduce a notion of a _ framing _ on a decorated cactus , which corresponds to a parametrization of a circle . the next lemma would be clear from geometric intuitions , however we give a detailed proof . [ lem:140916_1 ] let @xmath1431 and @xmath1443 . for each @xmath1444 , let @xmath1445 . then , @xmath1446 with respect to the cyclic order on @xmath1447 . it is enough to prove the following claim : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ suppose that @xmath1448 and @xmath128 satisfy @xmath1449 and @xmath1450 for any @xmath1451 . then , @xmath1452 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ let @xmath1453 . since @xmath1432 , by corollary [ cor : ci_is_cycle ] , it is enough to show that @xmath1454 . for any @xmath1455 , let @xmath1456 . then , @xmath1457 , and @xmath1458 therefore , we have to show @xmath1459 . since this is obvious for @xmath1460 , we may assume that @xmath1461 . let @xmath1421 be the dual tree of @xmath1335 . for each @xmath1451 , let us take @xmath278 so that @xmath1462 . then , @xmath1463 , and @xmath1464 . therefore , if @xmath1465 , then @xmath1466 is a path on @xmath1421 . if @xmath1467 , then @xmath1468 is a tail , since @xmath1469 . therefore , @xmath1470 . in conclusion , there exists a path from @xmath1471 to @xmath1472 on @xmath1421 , which is disjoint from @xmath1413 . on the other hand , since @xmath1473 and @xmath1474 , @xmath1475 is a path on @xmath1421 . since @xmath1421 is a tree , we conclude that @xmath1459 . a _ framing _ on a decorated cactus @xmath1420 is a map @xmath1476 , such that @xmath1477 for every @xmath1431 . a tuple @xmath1478 is called a _ framed decorated cactus_. an isomorphism of framed decorated cacti is an isomorphism of decorated cacti which preserves framing . for any @xmath1416 , @xmath1479 denotes the set of isomorphism classes of framed decorated cacti @xmath1480 such that @xmath1481 \in \lambda(k : l_1 , \ldots , l_r)$ ] . let @xmath1420 be a decorated cactus , and @xmath1421 be its dual tree . there exists a unique framing @xmath1476 such that , for every @xmath1431 , @xmath1482 is on the shortest path of @xmath1421 connecting @xmath1413 and @xmath1483 ( here @xmath1482 is considered as an edge of @xmath1421 , and @xmath1413 , @xmath1483 are considered as vertices of @xmath1421 ) . we call it the _ canonical framing_. by considering canonical framings , we define an injective map @xmath1484 . in the rest of this paper , we consider @xmath1417 as a subset of @xmath1479 . when we draw a figure of a framed decorated cactus @xmath1478 , we draw an arrow pointing at a vertex @xmath1485 for each @xmath1431 . for any @xmath1486 and @xmath1487 , we set @xmath1488 , where @xmath1489 for every @xmath1431 . then , a map @xmath1490 is a bijection , and there holds @xmath1491 for any @xmath1492 . as a restriction of this map , we obtain a bijection @xmath1493 . here we present examples of framed decorated cacti . for any decorated cactus @xmath1494 , we denote @xmath1329 , @xmath1328 etc . as @xmath1495 , @xmath1496 etc . moreover , @xmath1497 are denoted as @xmath1498 , and @xmath1411 is denoted as @xmath1499 . for any @xmath265 , @xmath1500 consists of a unique element , which we denote by @xmath1501 . more explicitly , @xmath1501 is defined as follows : * @xmath1502 , @xmath1503 . * for any @xmath129 , @xmath1504 * @xmath1505 , @xmath1506 . @xmath1507 ( left ) and @xmath1508 ( right ) for any @xmath265 , there exists a unique element @xmath1509 , such that @xmath1510 is a bijection . more explicitly , @xmath1511 is defined as follows : * @xmath1512 , @xmath1513 . * for any @xmath1514 , we define @xmath1515 * @xmath1516 , @xmath1517 . * @xmath1518 . @xmath1519 ( left ) and @xmath1520 ( right ) for any @xmath128 , let us consider a noncanonical framing @xmath1521 on @xmath1511 so that @xmath1522 . we obtain an element in @xmath1523 , which we denote by @xmath959 . @xmath1524 ( left ) and @xmath1525 ( right ) finally , for any @xmath1526 and @xmath951 , we define @xmath1527 . in general , for any @xmath1528 where @xmath1529 , @xmath1530 consists of a unique vertex ( consider its dual tree ) , which we denote by @xmath1531 . let us consider the following condition : _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ( @xmath143 ) : @xmath1532 , and @xmath1533 is a bijection . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ for any @xmath557 , @xmath1534 has exactly @xmath1535 elements which satisfy ( @xmath143 ) , depending on choices of @xmath1536 . in particular , for every @xmath1537 , there exists a unique element @xmath1538 which satisfies ( @xmath143 ) , @xmath1539 and @xmath1540 . we denote this element as @xmath1541 . @xmath1542 ( left ) and @xmath1543 ( right ) for any @xmath1544 and @xmath1431 , we define a composition map @xmath1545 namely , given @xmath1546 we define @xmath1547 . we first define a ribbon graph @xmath1548 as follows : * @xmath1549 . * for every @xmath1550 , let @xmath1551 , and insert a new flag @xmath1314 at @xmath1552 as in the figure below . + precisely , we define @xmath1553 * for every @xmath1550 , we define @xmath1554 ( the set of tails of @xmath1555 ) so that @xmath1556 and @xmath1557 with respect to the cyclic order on @xmath1558 . we `` glue '' @xmath1559 and @xmath1314 for every @xmath1550 . precisely , we define @xmath1560 this completes the definition of @xmath1548 . for every @xmath1550 , we define @xmath1561 by @xmath1562 . we define a new ribbon graph @xmath1563 by removing flags in @xmath1564 , and contracting edges @xmath1565 : @xmath1566 there is a natural bijection @xmath1567 . let us identify the lhs and the rhs via this bijection . [ lem:150119_1 ] 1 . for any @xmath1568 , @xmath1425 is a cycle of @xmath1563 . 2 . for any @xmath1569 , @xmath1570 is a cycle of @xmath1563 . @xmath1571 is a cycle of @xmath1563 . \(i ) holds since @xmath1572 for any @xmath1573 . ( ii ) holds since @xmath1574 for any @xmath1575 . ( iii ) is a consequence of lemma [ lem:140916_1 ] . by lemma [ lem:150119_1 ] , we obtain @xmath1576 . in particular , @xmath1577 . on the other hand , there holds @xmath1578 then , we obtain @xmath1579 . moreover , @xmath1580 for any @xmath1581 . also , @xmath1582 . in conclusion , @xmath1583 is a decorated cactus . finally , we define a framing @xmath1584 by @xmath1585 now we define @xmath1586 to be the resulting framed decorated cactus . this completes the definition of compositions of framed decorated cacti . [ lem:150119_2 ] 1 . for any @xmath1587 , there holds @xmath1588 . 2 . for any @xmath1589 and @xmath1587 , @xmath1590 , @xmath1591 , there holds @xmath1592 . 3 . for any @xmath1593 , @xmath1594 and @xmath1587 , @xmath1590 , @xmath1595 , there holds @xmath1596 . if @xmath1597 and @xmath1598 , then @xmath1599 ( ii ) , ( iii ) are consequences of lemma [ lem : removing_contracting ] . ( i ) , ( iv ) are easy to verify , and proofs are omitted . we end this section with a few more examples of decorated cacti . for any @xmath128 and @xmath129 , we define @xmath1600 by @xmath1601 @xmath1602 ( left ) , @xmath1603 ( middle ) and @xmath1604 ( right ) for any @xmath265 and @xmath129 , we define @xmath1605 by @xmath1606 @xmath1607 ( left ) and @xmath1608 ( right ) for any @xmath557 , we define @xmath1609 by @xmath1610 @xmath1611 ( left ) and @xmath1612 ( right ) section 9.1 is devoted to some preliminaries on colored operads . in section 9.2 , we show that the collection of framed decorated cacti has a natural structure of a @xmath1400-colored operad , and we give a presentation of this colored operad by generators and relations ( lemma [ 20141204_5 ] ) . let @xmath620 be any set , and @xmath21 be any symmmetric monoidal category . a @xmath620-colored operad in @xmath21 , which we denote by @xmath24 , consists of the following data : * @xmath1613 for any @xmath1614 and @xmath1615 . * a morphism @xmath1616 for any @xmath1617 , which we call a unit . * for any @xmath106 , @xmath1618 and @xmath1619 , a composition morphism @xmath1620 * for any @xmath1614 , @xmath1615 and @xmath1486 , a morphism @xmath1621 we require that @xmath1622 , and @xmath1623 . we require that each @xmath1624 is a two - sided unit for composition morphisms , and there holds the associativity and equivariance properties , which are defined in the obvious ways . notions of morphisms and suboperads for @xmath620-colored operads are also defined in the obvious ways . colored operads in the category of dg ( resp . graded ) vector spaces are called dg ( resp . graded ) @xmath620-colored operads . let @xmath1625 be a family of dg vector spaces . setting @xmath1626 @xmath1627 has the natural structure of a dg @xmath620-colored operad . for any dg @xmath620-colored operad @xmath24 , a dg @xmath24-algebra consists of @xmath1625 and a morphism @xmath55 of dg @xmath620-colored operads . we obtain the following chain map for any @xmath1615 : @xmath1628 for any dg @xmath24-algebras @xmath38 and @xmath47 , a morphism @xmath1629 of dg @xmath24-algebras is a family of chain maps @xmath1630 such that @xmath1631 for any @xmath1632 and @xmath1633 . [ 20141219_3 ] let @xmath1634 be a family of sets . there exists a unique ( up to isomorphism ) @xmath620-colored operad @xmath1635 of sets , which satisfies the following conditions : * for any @xmath1615 , there exists a map @xmath1636 * let @xmath24 be any @xmath620-colored operad of sets , such that for any @xmath1615 , a map @xmath1637 is given . then , there exists a unique morphism of @xmath620-colored operads @xmath1638 , such that @xmath1639 . we call @xmath1635 the free @xmath620-colored operad generated by @xmath335 . uniqueness is obvious . existence is proved by an explicit construction using planar trees with all edges colored by @xmath620 . the construction is standard ( see e.g. @xcite ) and omitted . let @xmath24 be a @xmath620-colored operad of sets . a binary relation on @xmath24 is a family @xmath1640 , where each @xmath1641 is a binary relation on @xmath1642 . if @xmath1641 is an equivalence relation for any @xmath1643 , and there holds @xmath1644 for any @xmath1645 , @xmath1646 , @xmath1647 and @xmath1648 , @xmath1649 is called an equivalence relation on @xmath24 . for any equivalence relation @xmath1649 on @xmath24 , @xmath1650 has a natural @xmath620-colored operad structure . let @xmath1634 be a family of sets , and @xmath1651 be a binary relation on the free @xmath1652-colored operad @xmath1635 . let @xmath1653 denote the equivalence relation on @xmath1635 generated by @xmath1649 ( i.e. the minimal equivalence relation which contains @xmath1649 ) . then , we define @xmath1654 . let us denote @xmath1655 we can rephrase lemma [ lem:150119_2 ] as follows . ( compatibility between symmetric group actions and composition maps is not verified there , however this is obvious . ) [ 140917_2 ] @xmath1656 is a @xmath1400-colored operad with the composition maps and symmetric group actions defined in section 8 , and unit elements are @xmath1657 . @xmath1658 is its suboperad . the aim of this subsection is to give explicit presentations of @xmath1658 and @xmath1656 by generators and relations . for any @xmath1416 , let us define @xmath1659 it is easy to check the following relations : @xmath1660 @xmath1661 @xmath1662 @xmath1663 @xmath1664 @xmath1665 @xmath1666 let us define @xmath1400-colored operads @xmath1667 , @xmath1668 , @xmath1669 by generators and relations : @xmath1670 there exist natural morphisms of @xmath1400-colored operads @xmath1671 and @xmath1672 [ 20141204_5 ] @xmath1673 , @xmath1674 are isomorphisms of @xmath1400-colored operads . the proof consists of six steps . * step 1 . * for any decorated cactus @xmath1675 , let @xmath1676 denote the number of @xmath1366 which satisfies neither of the following conditions : * @xmath1677 and @xmath1678 . * @xmath1679 and @xmath1680 . by induction on @xmath1263 , we can show that @xmath1681 if and only if @xmath438 is in the image of @xmath1682 . * step 2*. for any @xmath1683 , there exist integers @xmath1684 and @xmath1685 , such that @xmath1686 and @xmath1687 since @xmath1688 is in the image of @xmath1689 , @xmath438 is in the image of @xmath1673 . therefore , @xmath1673 is surjective . the above presentation of @xmath438 is unique in the following sense : for any integers @xmath1690 and @xmath1691 such that @xmath1692 and @xmath1693 there exists @xmath1694 such that @xmath1695 is the identity , @xmath1696 , and @xmath1697 for every @xmath1698 . * step 3*. for any @xmath1699 , using relations ( [ eq:20141204_1 ] ) and ( [ eq:20141204_2 ] ) , it is possible to obtain a presentation @xmath1700 where @xmath1701 , and @xmath1486 . this presentation is uniquely determined by @xmath1702 . therefore , @xmath1682 is injective . since @xmath1689 factors through @xmath1703 , this map is also injective . thus , we can consider @xmath1667 as a suboperad of @xmath1668 . * step 4*. for any @xmath1704 , there exist integers @xmath1705 and @xmath1706 such that @xmath1707 let @xmath1708 denote the minimal possible value of @xmath4 . it is easy to see that @xmath1709 . we show that in fact @xmath1710 . suppose that @xmath1711 . then , either ( a ) or ( b ) holds : 1 . there exists @xmath1712 such that @xmath1713 . 2 . there exist @xmath1714 such that @xmath1715 , where @xmath1716 , @xmath1717 are cycles on @xmath1718 . in case ( a ) , one can use ( [ eq:20141204_3 ] ) to decrease @xmath4 by @xmath89 . in case ( b ) , by replacing @xmath1719 with @xmath1720 for some @xmath1721 if necessary , we may assume that @xmath1722 , @xmath1723 and @xmath1724 , where @xmath1725 and @xmath1726 . since @xmath1727 and @xmath1689 is injective ( step 3 ) , we get @xmath1728 . then , @xmath1729 where the second equality follows from ( [ eq:20141204_4 ] ) . hence we can decrease @xmath4 by @xmath89 . therefore , we have shown that @xmath1730 . * step 5*. suppose @xmath1731 satisfy @xmath1732 . let us set @xmath1733 . by step 4 , there exist integers @xmath1734 , @xmath1735 and @xmath1736 , such that @xmath1737 by step 2 , there exists @xmath1721 , such that @xmath1695 is the identity , @xmath1738 , and @xmath1739 for every @xmath1740 . since @xmath1689 is injective ( step 3 ) , @xmath1741 . hence @xmath1742 , thus @xmath1673 is injective . since we already proved that @xmath1673 is surjective ( step 2 ) , we have proved that @xmath1673 is an isomorphism . * step 6*. finally , we show that @xmath1743 is an isomorphism . for any @xmath1744 , there exist @xmath1745 and integers @xmath1746 , such that @xmath1747 and @xmath1748 . moreover , @xmath1749 , @xmath1746 are uniquely determined by @xmath438 . then , surjectivity of @xmath1674 immediately follows from surjectivity of @xmath1673 . on the other hand , using relations ( [ eq:20141206_1 ] ) , ( [ eq:20141206_2 ] ) , ( [ eq:20141206_3 ] ) , we can show the following : for any @xmath1750 , there exist @xmath1751 and integers @xmath1746 , such that @xmath1752 , and @xmath1753 . since @xmath1754 and @xmath1746 are uniquely determined by @xmath1755 , injectivity of @xmath1674 follows from injectivity of @xmath1673 . for any set @xmath335 , let @xmath1756 $ ] denote the @xmath10-vector space freely generated by @xmath335 , i.e. @xmath1756:= \bigoplus_{s \in s } \r s$ ] . for any @xmath620-colored operad of sets @xmath1757 , we define a @xmath620-colored graded operad @xmath1756 $ ] by @xmath1758(k : l_1 , \ldots , l_r)_*:= \begin{cases } \r[s(k : l_1,\ldots , l_r ) ] & ( * = 0 ) \\ 0 & ( * \ne 0 ) \end{cases}\ ] ] with the natural composition maps and symmetric group actions . we call dg @xmath1756$]-algebras simply as dg @xmath335-algebras . the next corollary is an immediate consequence of lemma [ 20141204_5 ] . [ 150702_1 ] let @xmath1759 be a sequence of dg vector spaces . * the following structures on @xmath957 are equivalent : * * a structure of a nonsymmetric dg operad with a multiplication @xmath969 and a unit @xmath972 . * * a structure of a dg @xmath1658-algebra . + the correspondence is given by the following formulas : @xmath1760 * the following structures on @xmath957 are equivalent : * * a structure of a cyclic nonsymmetric dg operad with a multiplication @xmath969 and a unit @xmath972 , such that @xmath774 is cyclically invariant . * * a structure of a dg @xmath1656-algebra . + the correspondence is given by the formulas in ( i ) and @xmath1761 . [ 150801_4 ] by corollary [ 150702_1 ] ( ii ) , @xmath1762 has a dg @xmath1656-algebra structure . for any @xmath1763 , let @xmath1764 . then , it is easy to see that @xmath1765 using this fact and the definition of the dg @xmath1242-algebra structure on @xmath1236 ( see the next section ) , we conclude that the dg @xmath1242-algebra structure on @xmath9 preserves the length filtration , as we claimed in proposition [ 141005_2 ] ( ii ) . in this section , we define dg operads @xmath12 and @xmath13 , and reduce theorem [ 150624_8 ] to theorem [ 150702_2 ] and lemma [ 150703_2 ] , which we prove in the next section . for any integer @xmath1614 , we define graded vector spaces @xmath1766 and @xmath1767 by @xmath1768 , \quad \tilde{\lambda}(r)_*:= \prod_{\substack { k , l_1 , \ldots , l_r \in \z_{\ge 0 } \\ l_1 + \cdots + l_r -k = * } } \r[\lambda(k : l_1 , \ldots , l_r)].\ ] ] obviously , @xmath1769 . for any @xmath1770 , we denote its @xmath1771-component by @xmath1772 . as is clear from the definition , the graded vector space @xmath1773 is unbounded . in fact , @xmath1774 for every @xmath1775 . when @xmath103 , @xmath1776 for every @xmath1777 . we show that @xmath1778 has a natural dg operad structure , and @xmath1779 is its suboperad . * composition map : * for any @xmath106 and @xmath1618 , we define a degree @xmath122 linear map @xmath1780 by @xmath1781 where @xmath1782 . * associativity : * for any @xmath1770 , @xmath1783 and @xmath1784 , there holds @xmath1785 this follows from associativity of composition maps on @xmath1656 and sign computations . * unit : * we define @xmath1786 $ ] by @xmath1787 . then , for any @xmath1770 , there holds @xmath1788 and @xmath1789 for any @xmath1431 . * differential : * for any integer @xmath128 , let us define @xmath1790 $ ] by @xmath1791 . then , @xmath1792 for any @xmath128 . we define @xmath1793 $ ] by @xmath1794 . then , @xmath1795 . we define a differential @xmath1796 by @xmath1797 by associativity of composition maps , it is easy to check @xmath1798 and the leibniz rule @xmath1799 * symmetric group actions : * for any @xmath1614 , we define a linear map @xmath1800 \to f\tilde{\lambda}(r ) _ * ; \ , x \otimes \sigma \mapsto x^\sigma$ ] by @xmath1801 by direct computations on signs , we can check that this is a chain map , and compatible with composition maps . * suboperad @xmath13 : * since @xmath1658 is a suboperad of @xmath1656 , it follows that @xmath13 is a suboperad of @xmath12 . we show that for any dg @xmath1658 ( resp . @xmath1656 ) algebra @xmath957 , the total complex @xmath1236 has the dg @xmath13 ( resp . @xmath12 ) algebra structure . [ 20141221_1 ] 1 . let @xmath1802 be a dg @xmath1658 ( resp . @xmath1656 ) algebra . then , @xmath1803 has the natural dg @xmath13 ( resp . @xmath12 ) algebra structure . 2 . let @xmath1804 be a morphism of dg @xmath1658 ( resp . @xmath1656 ) algebras . then , @xmath1805 is a morphism of dg @xmath13 ( resp . @xmath12)-algebras . let @xmath1806 be a dg @xmath1807-algebra . then , for any @xmath1808 , we have a chain map @xmath1809(k : l_1 , \ldots , l_r ) \otimes { { \mathcal{o}}}(l_1 ) \otimes \cdots \otimes { { \mathcal{o}}}(l_r ) \to { { \mathcal{o}}}(k ) ; x \otimes y_1 \otimes \cdots \otimes y_r \mapsto x \cdot ( y_1 \otimes \cdots \otimes y_r).\ ] ] for every @xmath1614 , we define @xmath1810 by @xmath1811 notice that the rhs of the first formula is a finite sum . direct computations show that this is a dg @xmath18-algebra structure on @xmath1236 . thus we have verified lemma [ 20141221_1 ] ( i ) for @xmath1658 . ( ii ) is straightforward from the construction . the case for @xmath1656 is completely parallel , and omitted . let us define @xmath1812 , @xmath1813 , @xmath1814 by the following formulas : @xmath1815 let us define operators @xmath84 , @xmath1247 and @xmath99 on @xmath1236 by @xmath1816 now , theorem [ 150624_8 ] follows from the next two results , which are proved in the next section . [ 150703_2 ] 1 . @xmath879 satisfies @xmath1817 and @xmath1818 . in particular , @xmath84 defines a dga algebra structure on @xmath1236 . @xmath1819 satisfies @xmath1820 and @xmath1821 , @xmath1822 . in particular , @xmath141 defines a dg lie algebra structure on @xmath1236 . @xmath1823 satisfies @xmath1824 . in particular , @xmath99 is an anti - chain map on @xmath1236 . [ 150702_2 ] there exist isomorphisms @xmath214 and @xmath213 , compatible with the inclusion maps . moreover , @xmath1825 \in h_0(\tilde{\lambda}(2))$ ] , @xmath1826 \in h_1(\tilde{\lambda}(2))$ ] , @xmath1827 \in h_1(f\tilde{\lambda}(1))$ ] correspond to @xmath70 , @xmath71 , @xmath95 via these isomorphisms . the goal of this section is to prove theorem [ 150702_2 ] . in the course of the proof , we also verify lemma [ 150703_2 ] ( see remark [ 150703_3 ] ) . here we explain an outline of our proof assuming lemmas [ lem : q ] , [ 141022_1 ] , [ 141103_1 ] . for any @xmath1614 and @xmath265 , we define @xmath1828.\ ] ] obviously @xmath1829 , and @xmath1830 is a subcomplex of @xmath1766 for any @xmath265 . we denote the quotient @xmath1831 by @xmath1832 . @xmath1833 and @xmath1834 are defined in the same way . let us consider quotient maps @xmath1835 our first step is to prove the following lemma in section 11.1 . [ 141022_1 ] @xmath1840 has a dg operad structure , such that @xmath1841 is its suboperad . there exists a morphism of dg operads @xmath1842 , such that @xmath1843 and @xmath1844 . in particular , @xmath264 and @xmath1845 are quasi - isomorphisms of dg operads . since @xmath1837 is surjective , it is enough to show that @xmath1850 is acyclic . for each integer @xmath265 , we denote the differential on @xmath1834 by @xmath1851 . in other words , let us define an anti - chain map @xmath1853 by @xmath1854 . then , the chain complex @xmath1855 is identified with @xmath1856 , where the differential is @xmath1857 . now , let us assume the following lemma . * @xmath1859 is a chain ( resp . anti - chain ) map , if @xmath182 is even ( resp . * for every @xmath182 , @xmath1859 induces an isomorphism on homology . * for every @xmath182 , there holds @xmath1860 , where @xmath1861 . by lemma [ lem : jk ] , the sequence @xmath1163 & h_*(\tilde{\lambda}_1(r ) ) \ar[r]_-{h_*(d_1 ) } & h_{*-1}(\tilde{\lambda}_2(r ) ) \ar[r]_-{h_*(d_2)}&h_{*-2}(\tilde{\lambda}_3(r ) ) \ar[r]_-{h_*(d_3 ) } & \cdots } \ ] ] is exact . hence , lemma [ 150629_1 ] shows that @xmath1862 is acyclic . therefore , it is enough to prove lemma [ lem : jk ] . the proof consists of four steps . we define @xmath1858 by @xmath1869 . ( briefly speaking , @xmath1859 removes all tails other than @xmath1411 . ) then , there holds @xmath1870 hence , @xmath1859 is a chain ( resp . anti - chain ) map if @xmath182 is even ( resp . odd ) . on the other hand , since @xmath1871 , there holds @xmath1860 . * step 2 . * to show that @xmath1859 induces an isomorphism on homology , we define a filtration on the chain complex @xmath1872 . for any @xmath1873 , we define positive integers @xmath1874 so that @xmath1875 . for every @xmath1876 , we define @xmath1877.\end{aligned}\ ] ] @xmath1878 is a subcomplex of @xmath1834 for every @xmath29 , and we obtain a filtration @xmath1879 . since @xmath1880 is a bijection , @xmath1881 is an isomorphism . therefore , to show that @xmath1859 is a quasi - isomorphism , it is enough to show that @xmath1882 is acyclic for every @xmath1883 . * step 4 . * we define a degree @xmath89 linear map @xmath620 on @xmath1889 . let @xmath1890 . then @xmath1891 for @xmath1892 , and @xmath1893 . let @xmath1894 . then , @xmath1895 since @xmath1896 . let us take @xmath1897 so that @xmath1898 , and define @xmath1899 as follows : @xmath1900 * let us define an order on @xmath1922 by @xmath1924 , and denote the corresponding order on @xmath78 as @xmath1921 . * let us define an order on @xmath1925 by * * when @xmath1448 , @xmath1926 and @xmath1927 , then @xmath1928 . * * on each @xmath1413 , @xmath1929 . + let us denote the corresponding order on @xmath78 as @xmath1920 . * @xmath264 is a chain map . namely , for any @xmath1937 , there holds @xmath1938 * for any @xmath1939 , @xmath106 and @xmath1940 , there holds @xmath1941 . * for any @xmath1939 and @xmath1486 , there holds @xmath1942 . for any @xmath106 and @xmath1618 , let us define a composition map on @xmath1840 by @xmath1943 by the above assertions , @xmath1840 is a dg operad with these composition maps ( unit is @xmath1519 ) , and @xmath1944 is a morphism of dg operads . as is obvious from definitions , @xmath1945 is the identity on @xmath1840 , and @xmath1843 . this completes the proof of lemma [ 141022_1 ] . let @xmath1946 . @xmath1366 is called a _ free vertex _ , if @xmath1947 , and @xmath1948 for any @xmath1431 . we call @xmath438 _ degenerate _ , if there exists a free vertex in @xmath1329 . let @xmath1949 denote the set of degenerate elements in @xmath1950 . we also set ( @xmath1951 stands for nondegenerate ) : @xmath1952 since @xmath1953 is nondegenerate , there holds @xmath1954 for every @xmath1614 , let us define @xmath1955 , \qquad \tilde{\lambda}^\deg_0(r)_*:= \prod_{l_1 + \cdots + l_r= * } \r[\lambda^\deg(0 : l_1 , \ldots , l_r)].\ ] ] notice that @xmath1956 . it is easy to see that , @xmath1957 and @xmath1958 are dg ideals of @xmath1840 and @xmath1841 , respectively . for any @xmath103 and @xmath1959 , let @xmath1960 denote the framed decorated cactus obtained by removing all free vertices of @xmath438 let @xmath1961 denote the number of edges of @xmath1962 . it is easy to see that @xmath1963 for any @xmath1959 , and @xmath1964 for any @xmath1965 . the lemma is obvious for @xmath1909 , thus we may assume @xmath103 . we prove the lemma only for @xmath1967 , since the proof for @xmath1966 is similar . for @xmath1968 , let @xmath1969 be the subspace of @xmath1967 , which is generated by @xmath1970 such that @xmath1971 . it is easy to see that this is a subcomplex of @xmath1967 , and we obtain @xmath1972 to prove that @xmath1967 is acyclic , it is sufficient to show that @xmath1973 is acyclic for every @xmath1968 . for any @xmath1974 such that @xmath1975 , let @xmath1976 denote the subcomplex of @xmath1973 , which is generated by @xmath1977 . it is easy to see that @xmath1976 is acyclic for every @xmath1978 . therefore , @xmath1979 is also acyclic . since @xmath1980 and @xmath1981 are dg ideals of @xmath1840 and @xmath1841 , the quotients @xmath1982 and @xmath1983 are dg operads . as graded vector spaces , @xmath1984 , \qquad \tilde{\lambda}^\nondeg_0(r)_*= \prod_{l_1 + \cdots + l_r= * } \r[\lambda^\nondeg(0 : l_1 , \ldots , l_r)].\ ] ] by lemma [ 141103_3 ] , the morphisms of dg operads @xmath1985 and @xmath1986 are quasi - isomorphisms . on the other hand , one can define morphisms of graded operads @xmath1987 and @xmath1988 by @xmath1989 , \quad a \mapsto [ \alpha_{0,0 } ] , \quad b \mapsto -[\beta_{1,1,0 } + \beta_{1,1,0}^{(12 ) } ] , \quad \delta \mapsto [ \sigma_{0,0 } \circ_1 \tau_1].\ ] ] the relevant relations are checked by direct computations . now , lemma [ 141103_1 ] is reduced to the following lemma . it seems that the dg operad @xmath1992 ( resp . @xmath1993 ) is isomorphic ( up to sign ) to the dg operad @xmath1994 ( resp . @xmath1995 ) , which is introduced in @xcite ( resp . these dg operads are defined by cw - decompositions of the cacti and spineless cacti operads . in papers @xcite , @xcite , @xcite , it is proved that these operads are chain models for the ( framed ) little disks operad , therefore their homology are isomorphic to the gerstenhaber and bv operads . lemma [ 141103_2 ] may follow from these results , however we give a direct proof in the next subsection . in this subsection , we abbreviate @xmath1996 and @xmath1997 by @xmath1998 and @xmath1999 for simplicity . for every integer @xmath103 , we define a chain complex @xmath2000 by @xmath2001 and differential is zero . we define a chain map @xmath2002 by @xmath2003 where @xmath2004 is defined as @xmath2005 it is easy to see that @xmath2016 for every @xmath29 . thus , @xmath2006 induces a map on quotients @xmath2017 to show that @xmath2006 is a quasi - isomorphism , it is enough to show that ( [ 150120_1 ] ) is a quasi - isomorphism for every @xmath29 . @xmath2018 is generated by @xmath2019 . on the other hand , @xmath2020 , where @xmath1976 is generated by @xmath2021 . ( [ 150120_1 ] ) maps @xmath2022 \otimes \gamma^r_*$ ] to @xmath1976 , and it is easy to see that this is a quasi - isomorphism for every @xmath1978 . therefore , ( [ 150120_1 ] ) is a quasi - isomorphism . by induction on @xmath1263 , we show that @xmath2023 is an isomorphism . this is directly checked for @xmath2024 . suppose that @xmath2025 is an isomorphism for some @xmath103 . by lemma [ 141225_1 ] , @xmath2026 is surjective , and @xmath2027 . on the other hand , @xmath2028 for any @xmath657 ( see @xcite and @xcite ) . therefore @xmath2029 . thus , @xmath2026 is an isomorphism . this completes the proof that @xmath1990 is an isomorphism . finally , we show that @xmath1991 is an isomorphism . for every @xmath103 , @xmath2030 is an isomorphism , since a map between topological spaces @xmath2031 is a homotopy equivalence . on the other hand , the next lemma holds . for any @xmath2033 , let @xmath1901 denote the decorated cactus obtained by forgetting the framing of @xmath438 . for every integer @xmath29 , let @xmath2034 denote the subspace of @xmath1999 , which is generated by @xmath2035 . this is a subcomplex of @xmath1999 . we set @xmath2036 . we need to show that @xmath2040 is an isomorphism . this is directly checked for @xmath2024 . since @xmath2041 is an isomorphism , the isomorphism ( [ 141226_1 ] ) and lemma [ 141226_2 ] imply that @xmath2042 is an isomorphism for every @xmath103 . c. berger , i. moerdijk , _ resolution of coloured operads and rectification of homotopy algebras _ , categories in algebra , geometry and mathematical physics , 3158 , contemp . , 431 , amer . soc . , providence , ri , 2007 . k. cieliebak , j. latschev , _ the role of string topology in symplectic field theory _ , new perspectives and challenges in symplectic field theory , 113146 , crm proc . lecture notes , 49 , amer . soc . , providence , ri , 2009 . r. l. cohen , v. godin , _ a polarized view of string topology _ , topology , geometry and quantum field theory , 127154 , london math . lecture note ser . 308 , cambridge univ . press , cambridge , 2004 . k. fukaya , _ application of floer homology of lagrangian submanifolds to symplectic topology _ , morse theoretic methods in nonlinear analysis and in symplectic topology , 231276 , nato sci . ser . , 217 , springer , dordrecht , 2006 .

the aim of this paper is to define a chain level refinement of the batalin - vilkovisky ( bv ) algebra structure on homology of the free loop space of a closed @xmath0-manifold . namely , we propose a new chain model of the free loop space , and define an action of a certain chain model of the framed little disks operad on it , recovering the original bv structure on homology level . we also compare this structure to a solution of deligne s conjecture for hochschild cochain complexes of differential graded algebras . to define the chain model of the loop space , we introduce a notion of de rham chains , which is a hybrid of singular chains and differential forms .

1404.0153

there is evidence for the existence of supermassive black holes ( smbhs ) over a range of epochs ; smbhs with masses @xmath3 m@xmath1 are ubiquitous in local galactic bulges ( e.g. magorrian et al . 1998 : ferrarese 2002 ) , while the smbhs powering quasars [ which have been detected at redshifts as high as @xmath4 ( fan et al . 2001 ; fan et al . 2003 ) ] are estimated to range between @xmath5 m@xmath1 ( see e.g. shields et al . evidence for the early assembly of smbhs , when interpreted within a hierarchical cosmology , suggests that smbh coalescence may be a frequent event . if this is the case , the gravitational waves ( gws ) generated during smbh coalescence are a prime candidate for detection by the _ laser interferometer space antennae _ ( lisa , folkner et al . 1998 ; flanagan & hughes 1998 ; hughes et al . 2001 ) . estimates of the smbh coalescence rate depend crucially on the occupation fraction of smbhs in halos , and therefore on the adopted model for the formation and growth of smbhs . within a hierarchical cdm cosmology , both seeded smbh formation models [ in which a primordial population of smbhs grow via accretion and/or merging ( eg . volonteri et al . 2003 ) ] and ongoing formation models [ where smbhs form jointly with bulges in halos ( eg . kauffmann & haehnelt 2000 ) ] are consistent with the present - day ubiquity of smbhs in galactic bulges ( menou et al . 2001 ) . the formation of a smbh may be limited to potential wells above a minimum depth . if so , then there exists a critical halo virial temperature ( @xmath6 ) below which a halo can not host smbh formation . if @xmath6 is low ( eg . @xmath7 k ) , then smbhs are abundant in small halos , and predicted event rates are in the order of 100 s per year ( wyithe & loeb 2003 ; haehnelt 2004 ) . this event rate may be significantly lower , less than one per year , if @xmath8 k and smbhs form only at the centres of massive galaxies ( haehnelt 2004 ) . hence the detection rate has the potential to constrain the global smbh population [ with the important caveat of the unknown efficiency of binary black hole ( bbh ) coalescence , see [ snr ] ] . the rate of coalescence detectable by lisa depends on the form of the gw signal and the instrument s sensitivity curve . previous estimates of the event rate have used characteristic ( rather than time - dependent ) gw signals to determine approximate detection limits ( wyithe & loeb 2003 ) or have estimated the number of detectable events by comparing the sensitivity curve with the total gravitational wave background due to coalescing bbhs ( sesana et al . recently sesana et al . ( 2004b ) estimated the expected event rate for a detailed physical model for smbh growth . we perform the first empirically motivated calculations for the lisa detection rate of smbh mergers in a hierarchical cosmology that use accurate signal to noise ratios ( snrs ) to determine detection criteria . in [ snr ] we describe the calculation of snrs for the lisa detection of bbh coalescence . in [ merger ] we discuss our calculation of the smbh merger rate , including our halo merger rate predictions , estimate for the occupation fraction of smbhs in halos and the @xmath9@xmath10 relationship . finally , we present our event rate predictions as a function of @xmath6 ( [ merger ] ) before concluding in [ concl ] . throughout this work we assume @xmath11 , @xmath12 , @xmath13 , @xmath14 km s@xmath15 mpc@xmath16 km s@xmath15 mpc@xmath15 , @xmath17 and a primordial power spectrum with slope @xmath18 as determined by the _ wilkinson microwave anisotropy probe _ ( wmap , spergel et al . the efficiency with which bbhs coalesce is highly uncertain . initially , the smbhs sink independently toward the centre of a merged system due to dynamical friction from the dark matter background until they form a bound binary ( begelman , blandford & rees 1980 ) . the efficiency of this process depends on the orbital parameters of the merging dark matter halos ( van den bosch et al . 1999 ; colpi , mayer & governato 1999 ) . as the orbital separation ( @xmath19 ) decreases , 3-body interactions with stars that pass within @xmath20 of the bbh centre of mass ( the `` loss cone '' ) increasingly dominate the energy loss . depending on the orbital parameters of the binary and the background distribution of stars , this process may result in a hardened [ orbital velocity ( @xmath21 ) @xmath22 stellar velocity dispersion ( @xmath23 ) ] binary system . if hardening continues until a binary separation where energy losses are dominated by gws , the binary will coalesce . yu ( 2002 ) argues that the efficiency of bbh coalescence hinges on the time - scale for bbh hardening during the hard binary phase . during this stage , three - body interactions between the bbh and individual stars eject stars into highly elliptical orbits , lowering the inner stellar density and slowing further hardening ( eg . volonteri et al . the deceleration of the hardening rate is compounded by the preferential depletion of stars within the loss cone ( region of parameter space where angular momentum is low enough that stars pass near the bbh ) , resulting in a deficiency of stars that can extract energy from the binary system ( see yu 2002 ) . uncertainties in the efficiency of processes which may replenish loss cone stars make this effect difficult to analyse . two - body stellar relaxation is expected to result in some diffusion of stars back into the loss cone ( binney & tremaine 1987 ) but it is unclear that this process alone can support sufficient hardening . numerical n - body simulations have been used to include the effect of the bbh wander within the star field ( due to 3-body interactions with stars ) , which increases the effective size of the loss cone and may prevent the coalescence from stalling ( quinlan & hernquist 1997 ; milosavljevi ' c & merritt 2001 ; chatterjee , hernquist & loeb 2003 ) . other scenarios which may aid evolution of the bbh into the gw dominated regime have been proposed , including the effects of gaseous disks ( gould & rix 2000 ) , flattened or triaxial stellar distributions ( yu 2002 ) and disruption by a third smbh ( hut & rees 1992 ) . the efficiency of bbh coalescence is of prime concern for lisa detection rates . whilst undoubtably still an open question , we note that there are plausible mechanisms to extract the required energy from the binary system . for the purposes of our calculation ( [ merger ] ) we therefore assume that hardening of the binary to the gw dominated regime occurs with an efficiency ( @xmath24 ) of unity . all event rates quoted in this paper are proportional to @xmath24 . the gravitational wave dominated portion of bbh coalescence may be divided into three main phases ( hughes 2002 ) . the binary begins in the _ inspiral phase _ , with the smbhs slowly spiralling into tighter orbits due to the adiabatic loss of gw energy . eventually the dynamics become relativistically unstable and the smbhs violently plunge to form a single object ( _ merger phase _ ) . the final gw signal can be described by modelling the merged system as a perturbed kerr smbh ( _ ringdown phase _ ) . the dynamics of the inspiral and ringdown phases are well understood and theoretical waveforms for the gws have been derived . however , certain parameters of the ringdown solution depend on unknown details of the merger phase and must be guided by the results of numerical simulation ( see [ ringdown ] ) . detection of gws from a single binary source will be complicated by the presence of galactic and extra - galactic gw foregrounds , and by lisa s sensitivity to gws from all sky directions . matched filter template searches will therefore be necessary to detect the gw signal from an individual event ( hughes et al . 2001 ) . the snr for a matched filter detection ( @xmath25 ) is defined by the ratio of the coherently folded signal and noise powers : @xmath26 where @xmath27 is the fourier transform of the dimensionless strain and @xmath28 is the spectral power of the noise . we calculate snrs for the final year ( before the merger phase ) of the inspiral signal . the effective duration of the ringdown phase is much shorter than one year and we therefore calculate ringdown snrs over the entire phase . estimates of lisa s noise spectrum are evolving with plans for the instrument design . a recently suggested sensitivity goal ( bender 2003 ) has a ( dimensionless ) threshold sensitivity [ defined as @xmath29 , for @xmath30 and 1 yr of observation ] with a low frequency power - law slope of @xmath31 for frequencies ranging between @xmath32 and @xmath33 hz , and a power - law slope of @xmath34 for frequencies of @xmath35 to @xmath32 hz . bender ( 2003 ) suggests a hard low frequency cut off at @xmath36 hz , roughly corresponding to the lowest resolvable frequency for a year long mission . lisa will be sensitive to frequencies up to @xmath37 hz . as a general rule @xmath38 is required for the confident detection of a signal ( hughes et al . 2001 ) . as the orbital separation of the binary shrinks , both the amplitude and frequency of the inspiralling strain increase . to calculate accurate inspiral phase snrs , it is important to include the frequency dependence of the strain in the integration of signal to noise . we achieve this using the technique outlined by flanagan & hughes ( 1998 , henceforth fh98 ) . fh98 show that sky averaged squared snr ( @xmath39 can be re - expressed in terms of the emitted gw energy spectrum @xmath40 @xmath41 where @xmath42 is the frequency of the gw . this is related to the observed gw frequency ( @xmath43 ) via the cosmological redshift of the source [ @xmath44 . the inspiral energy spectrum may be easily calculated to first order from the quadrupole approximation for the gravitational wave luminosity ( fh98 ) @xmath45 where @xmath9 and @xmath46 are the individual bbh masses corresponding to a reduced mass @xmath47 and total mass @xmath48 . numerical simulations of bbh coalescence indicate that this approximation is reasonable until @xmath43 reaches @xmath49 hz ( fh98 ) which roughly corresponds to the observed frequency when equal mass bbhs have an orbital separation of 3 schwarzchild radii ( @xmath50 ) . this frequency forms the upper limit of integration in equation ( [ en_spec_snr ] ) . since the gw frequency of bbh coalescence typically increases with time , this frequency may be used to delineate the inspiral and merger phases . the observed gw frequency of the inspiral signal at an observed time @xmath51 prior to the merger phase is given by ( fh98 ) @xmath52^{5/3 } t^{\prime}.\end{aligned}\ ] ] depending on the binary characteristics , @xmath53 may lie below the lisa waveband . we therefore use the greater of @xmath54 and @xmath55 as the lower limit of integration in equation ( [ en_spec_snr ] ) . we use the technique described by hughes ( 2002 ) to calculate snrs for the ringdown phase . bbh coalescence is expected to result in a rotating smbh with a bar - like excitation of the event horizon ( hughes 2002 ) . this distortion of the kerr solution may be modelled by the bar - like [ @xmath56 quasi - normal mode ( kokkotas & schmidt 1999 ) . the corresponding time dependent strain is described by an exponentially damped sinusoid with total ( observed ) amplitude ( @xmath57 ) whose value depends on the fraction of the bbh mass - energy ( @xmath58 ) radiated during the ringdown phase ( @xmath59 ) , the binary mass , and the spin parameter of the merger product ( @xmath60 ) . the amplitude is @xmath61 ^2,\ ] ] where @xmath62 is the comoving distance to a source at redshift @xmath63 . the quality factor ( @xmath64 ) is related to the damp - time of the gw signal ( @xmath65 ) by @xmath66 . the final factor in equation ( [ amp ] ) accounts for the reduced signal amplitude for unequal mass bbh coalescence ( fh98 ) . numerical simulations suggest that @xmath67 ( baker 2001 ; fh98 and references therein ) . we adopt @xmath68 . assuming that the distribution of the strain between polarisations ( @xmath69 ) mimics that of the inspiral phase , the dependence of @xmath70 and @xmath71 on the orientation of the system s angular momentum ( @xmath72 ) and sky position [ @xmath73 is given by ( fryer , holtz & hughes 2002 ) @xmath74 where @xmath75;\\ \nonumber \mathcal{a}_{\times}&=&-2 \mathcal{a}_{\rm rd}(\hat{l}.\hat{n}),\end{aligned}\ ] ] @xmath76 is the observed ringdown gw frequency and @xmath77 is the initial phase . leaver ( 1985 ) , echeverria ( 1989 ) and fryer holtz & hughes ( 2002 ) have produced fitting formulae for @xmath78 and @xmath64 in terms of @xmath79 and @xmath60 : @xmath80,\ ] ] @xmath81 which are accurate to @xmath82 per cent . hughes & blandford ( 2003 ) note that the value of @xmath60 will depend on the role that smbh merging plays in smbh evolution . we follow hughes ( 2002 ) and adopt @xmath83 . the observed strain [ @xmath84 is the sum of the strain in each polarisation weighted by the corresponding detector response functions [ @xmath85 ; thorne 1987 ] for a source with polarisation axes rotated by an angle @xmath86 to @xmath87 @xmath88 for @xmath89 , the quality factor is large and most of the energy radiated during the ringdown phase is observed at the central ringdown frequency ( @xmath90 ) . in this limit the snr may be approximated by ( hughes 2002 ) @xmath91 for comparison with the sky averaged inspiral snr we average the snr in equation ( [ ring_snr ] ) over randomly generated sets of @xmath92 . more massive bbh systems radiate gws at lower frequencies and for a given binary system the ringdown signal is at a higher frequency [ see equations ( [ f_oneyr ] ) & ( [ rd_fits1 ] ) ] . therefore if the sensitivity curve has a low frequency cut off and/or degrades rapidly with frequency , searches for the ringdown signal will probe more massive bbhs . we find that ringdown searches can detect coalescence between bhs with masses up to @xmath93 m@xmath1 ( @xmath94 m@xmath1 ) at @xmath95 ( @xmath96 ) . in the left hand panel of figure [ snr_results ] we plot the snr for the inspiral ( _ dot - dashed line _ ) and ringdown ( _ solid line _ ) phases of equal mass , @xmath95 bbh coalescence as a function of single bh mass . the right hand panel shows the ratio of ringdown to inspiral snrs , with the hatched region denoting the mass range over which the ringdown signal is detectable . the snr is greater in the ringdown signal for @xmath97 m@xmath1 . we note that there exists a range of smbh masses ( @xmath98 m@xmath1 at @xmath99 ) for which bbh coalescence is detectable only in the ringdown phase . since characteristic frequencies decay with @xmath100 , this mass range decreases with redshift . by @xmath101 , the mass range is comparable to the typical smbh mass observed in the local universe ( @xmath102 m@xmath1 ) . these results are consistent with the calculations of fh98 and reinforce the point that searches for gravitational radiation from the ringdown phase may generate constraints on the high mass smbh population . the higher frequency of the ringdown signal from smbh binaries offers limited advantage for their detection if the low frequency sensitivity has a power - law slope of @xmath103 and no lower cut - off ( corresponding to the optimal scenario of constant accelerometer dominated noise , see white 2002 ) . however , it is expected that frequency dependent sources of spurious acceleration will result in a steeper low - frequency sensitivity curve ( bender 2003 ) . in addition to the instrumental noise , unresolved gws from various classes of binary systems are expected to limit lisa s sensitivity ( see sesana et al . however , sesana et al . ( 2004b ) calculate that bbh coalescence only dominates the sky noise below a frequency of @xmath104 hz , and this noise is still an order of magnitude lower than the suggested lisa sensitivity . therefore the unresolved bbh background should not significantly affect lisa s sensitivity to supermassive bbh coalescence . estimates of the merger rate of dark matter halos are provided by the extended press - schechter ( 1974 ) formalism ( lacey & cole 1993 ) . we denote the merger rate per halo of mass @xmath105 per unit time with halos of mass between @xmath106 and @xmath107 by @xmath108 . the local ( all - sky ) detection rate of smbh mergers ( @xmath109 ) occurring in the redshift interval ( @xmath110 ) may be estimated ( e.g. wyithe & loeb 2003 ) by integrating the halo merger rate ( @xmath111 ) over the press - schechter dark matter halo mass function ( @xmath112 ) as below : @xmath113 where @xmath114 is the comoving volume per unit solid angle between @xmath63 and @xmath115 and the factor @xmath116 accounts for time - dilation . the product of the halo smbh occupation fractions [ @xmath117 relates the expected smbh merger rate to the halo merger rate . equation ( [ event_rate ] ) has the following additional features . we assume an observationally motivated dependence of smbh mass on host halo mass [ @xmath118 , see [ mbh_reln ] ] . the merger rate @xmath109 is weighted by the probability for detection of the central smbh coalescence , denoted @xmath119 , where the detection @xmath120 is determined through calculation of the snr @xmath121 we exclude halo mergers where the accreted satellite halo takes longer than the hubble time to sink to the centre of the merged system , since the bbh will not have time to form . this is achieved by limiting the merger rate contribution to halo pairs with mass ratios smaller than 3 ( colpi , mayer & governato 1999 ) . we assume efficient coalescence of smbh binaries in halo mergers , making our prediction of the coalescence rate a maximum . the merger rate scales with the fraction of bbhs that evolve into the gw dominated regime ( @xmath24 ) . we further assume coalescence to be rapid . this last assumption may affect the redshift distribution of our expected counts ( although this effect may be small , see sesana et al . 2004a ) , but should have little effect on estimates of the total event rate . shields et al . ( 2003 ) investigated the evolution of the relationship between @xmath9 and the velocity dispersion of the host bulge ( @xmath124 ) out to @xmath125 using a sample of radio - quiet agn . they found no evidence for variation from the locally determined relationship ( ferrarese & merritt 2000 ; gebhardt 2000 ) @xmath126 ferrarese ( 2002 ) found that the velocity dispersion of the local galactic bulges correlates with the circular velocity in the flat region of the halos rotation curve @xmath127 , indicating that @xmath128 . assuming that there is no evolution in the @xmath124@xmath129 relation , and relating @xmath129 to the virial velocity of a virialised halo of mass @xmath105 @xmath130^{1/6 } \\ & \times & \left ( \frac{1+z}{10 } \right)^{1/2}{\rm kms}^{-1},\end{aligned}\ ] ] [ where @xmath131 , @xmath132 and @xmath133 is the overdensity of a virialised halo at redshift @xmath63 ] we can determine the redshift dependent @xmath9@xmath10 relationship ( normalised to the local relationship from ferrarese 2002 ) @xmath135^{5/2}.\ ] ] the @xmath9@xmath10 relationship has been established locally for halo masses in the range @xmath136 ( ferrarese 2002 ) . fits to gravitational lens separation distributions [ porciani & madau 2000 ; kochanek & white 2001 hereafter kw2001 ] and simple models for baryon cooling ( cole et al . 2000 ; kw2001 ) indicate that halos with @xmath137 m@xmath1 tend to form groups and clusters of galaxies . this will suppress the number of very massive smbhs relative to the number of massive dark matter halos . in addition , limitations on accretion from the igm , and feedback from massive galactic winds are expected to suppress starformation in small mass halos ( see barkana & loeb 2001 and references therein ) , and may also inhibit smbh formation . in the following section , we construct an observationally motivated smbh occupation fraction ( @xmath138 ) that empirically accounts for these effects . a model which assumes that all dark - matter halos contain smbhs ( e.g. wyithe & loeb 2003 ) may substantially overestimate the smbh coalescence rate . here we assume that all galaxies residing in halos with virial temperatures larger than a critical value host a central smbh [ with mass determined by the @xmath9@xmath10 relationship , equation ( [ mbh - mhalo ] ) ] . we construct the galaxy occupation fraction of dark - matter halos at @xmath139 by comparing the observed velocity distribution of local galaxies with a velocity function that is generated from the press - schechter mass - function , and which assumes a relationship between cooled baryonic and virial velocities ( see [ gal_vel ] ) . motivated by the absence of evolution in the observed correlation between @xmath9 and the velocity dispersion of the host bulge , we assume that the fraction of halos of mass ( @xmath105 ) and redshift ( @xmath63 ) that host a central smbh is determined by @xmath140 independently of redshift [ ie . @xmath141 . measurements of the circular velocities of galaxies reflect the dynamics of the stars in the inner region of the halo where the density profile differs significantly from the profile predicted by navarro , frenk & white ( nfw , 1997 ) for cdm . simple galaxy formation models consider the effect of adiabatic cooling of baryons on the radial density and velocity profiles of galaxies but neglect heating due to starformation . adiabatic cooling and condensation of the halo s baryonic component into a rotationally supported disk steepens the inner density profile , resulting in a higher maximal circular velocity ( @xmath142 ) than implied by the initial profile ( dalcanton , spergel & summers 1997 ; mo , mao & white 1998 ; gonzalez et al . 2000 ; kw2001 ) . to model the cooled halo density profile , mo et al . ( 1998 ) assume that the baryonic and dark matter components are initially uniformly mixed in an nfw density profile with concentration parameter @xmath143 , spin parameter @xmath77 and specific angular momentum @xmath144 . the peak circular velocity of the modified density profile may be determined by self - consistently solving ( with angular momentum and energy conservation ) for the scale factor of the exponential disk . for a disk with specific angular momentum equal to that of the original halo , the cooled velocity is related to the virial velocity by ( mo et al . 1998 ) @xmath145 numerical n - body simulations of hierarchical structure formation suggest the average halo spin to be roughly independent of halo mass and equal to about @xmath146 ( bullock et al . furthermore , the concentration parameter of the nfw profile may be related to the halo virial mass @xmath147 ( at @xmath139 ) by @xmath148 ( bullock et al . we assume the fraction of halo mass in the cooled disk to be @xmath149 , independent of halo mass . the cooled baryon fraction is thought to be smaller in more massive halos ( see kw2001 ) , however we find that allowing @xmath150 to vary over halo mass as suggested by the cooling model of kw2001 has little effect on the profile of the adiabatically cooled velocity function . the halo cooled velocity function may then be computed from the psmf through the change of variable @xmath151 for a cooled baryon fraction ( @xmath150 ) that is independent of mass , this prescription results in a @xmath152 dependence of cooled velocity on halo mass for @xmath153 m@xmath154 m@xmath1 . consequently , the net effect of this adiabatic cooling model is to translate ( and re - scale ) the press - schechter distribution of halo virial velocities . to construct our occupation fraction we assume that there is one circular velocity for which galaxies have an occupation fraction of unity . we then find the ( linear ) relationship between cooled and virial velocity that enforces this condition . we have adopted the observational galaxy velocity function used by kw2001 for this purpose . kw2001 provide data down to @xmath155 km s@xmath15 . below this we extrapolate with a single power law @xmath156 with @xmath157 ( appropriate for a luminosity function with faint - end slope @xmath158 ) . this process returns a halo cooled velocity ( @xmath142 ) which , at the peak of the occupation fraction , is consistent with the velocity derived using the prescription of mo et al . ( 1998 ) to within @xmath159 per cent . moreover , our relationship is consistent with the ratio of @xmath160 measured from weak lensing studies ( seljak 2002 ) . the resulting occupation fraction ( see figure [ fig2 ] ) peaks at @xmath161 km s@xmath15 ( @xmath162 km s@xmath15 ) . this cooled velocity is also consistent with the circular velocity at which press - schechter and observed galaxy velocity functions agree in the works by gonzalez et al . ( 2000 ) and kw2001 . van den bosch et al . ( 2002 ) provide another estimate of the dependence of galaxy formation efficiency on host halo mass via the average halo mass - to - light ratio . this quantity may be estimated by constructing halo mass conditional luminosity functions using the relationship between ( luminosity dependent ) observed galaxy - galaxy correlation lengths and mass - scale for a hierarchical cosmology ( see van den bosch et al . their calculation yields a minimum mass - to - light ratio ( indicative of maximum galaxy formation efficiency ) for dark matter halos with @xmath163 m@xmath1 , which is comparable to the peak in @xmath164 at @xmath165 m@xmath1 . the occupation fraction decays rapidly for cooled velocities larger than @xmath166 km s@xmath15 . we note that errors in @xmath138 at high @xmath140 will not effect our predicted event rates due to the exponential tail of the psmf . indeed wyithe & loeb ( 2003 ) found that imposing a sharp cut - off to @xmath138 at a virial velocity of @xmath167 km s@xmath15 had little impact on their predictions . prior to the reionisation of the igm , halos required a virial temperature larger than @xmath168 k for the accreted gas to cool via atomic hydrogen transitions within a hubble time . after reionisation of the igm , the higher igm temperature inhibited the accretion of gas into halos with virial temperatures lower than @xmath169 k. these virial temperatures specify the minimum mass of a halo inside of which a smbh may form ( e.g. barkana & loeb 2001 ) @xmath170 ^{-1.5}\\ \times \left [ \frac{\omega_m}{\omega_m^z}\frac{\delta_c}{18\pi^2}\right ] ^{-1/2}\left ( \frac{1+z}{10 } \right)^{-3/2},\end{aligned}\ ] ] equivalently , a minimum virial temperature specifies a minimum depth for the potential well ( or velocity , or mass ) of a halo in which a smbh can exist . given the @xmath9@xmath10 relationship , equation ( [ mvir ] ) implies a minimum smbh mass ( @xmath171 ) which is independent of redshift . for @xmath172 k ( @xmath173 k ) this minimum smbh mass corresponds to @xmath174 m@xmath1 ( @xmath175 m@xmath1 ) . in contrast , the locally detected population of smbhs reside in more massive halos with virial temperatures greater than @xmath176 k , although this lower limit may be partially attributable to the difficulties in obtaining dynamical evidence for bhs with @xmath177 ( e.g. kormendy & richstone 1995 ) . the predicted event rates are plotted in figure [ fig3 ] as a function of the minimum virial temperature ( lower axis ) and minimum smbh mass ( upper axis ) . the event rates per year decline with increasing @xmath6 . for @xmath178 k ( @xmath179 m@xmath180 we predict @xmath181 detections per year in each of the ringdown and inspiral phases . these predictions are significantly below those of wyithe & loeb ( 2003 ) , with the discrepancy due to our inclusion of the smbh occupation fraction in the merger rate . our smbh occupation fraction and accurate detection criteria result in predictions that decay more rapidly with increasing @xmath6 than previous calculations suggested . smbhs powering luminous quasars at @xmath182 weigh @xmath183 m@xmath1 . if @xmath184 m@xmath185 the predicted event - rates are very low ; falling below @xmath186 yr@xmath15 for each phase . hence lisa will not detect coalescence from the major merges of bright quasar hosts . however , there is evidence for smbhs as small as @xmath0 m@xmath1 in local galactic bulges ( ferrarese 2002 ) and low - luminosity , low - redshift agn ( greene & ho 2004 ) . a reasonable estimate for the minimum smbh mass is therefore @xmath187 m@xmath1 , which corresponds to predictions of @xmath188 events per year in each phase . figure [ fig4 ] shows the redshift distribution of events as a function of the minimum virial temperature . most of the events originate at @xmath189 due to mergers between halos more massive than @xmath190 m@xmath1 . sesana et al . ( 2004a&b ) make predictions for the detection rate of inspiral phase bbh coalescence for a seeded smbh formation model ; estimating @xmath191 `` bursts '' in 3 years of observation due to events at @xmath192 . this calculation employs a specific model for the formation and evolution of bbhs embedded in an isothermal stellar distribution . sesana et al . ( 2004a ) use the seeded smbh growth model of volonteri et al . ( 2003 ) in which ( large ) stellar mass bhs form in @xmath193 peaks at @xmath194 and grow via accretion and coalescence during major halo mergers . while we also rely on the merger rate of dark matter halos for the basis of our coalescence rate , our calculation differs by employing empirically motivated estimates for the population of smbhs in halos . we do not assume the physics leading to the absence of galaxies ( and by extension smbhs ) in large and small halos , however we have shown that the inclusion of the occupation fraction significantly affects the coalescence rate . we have assumed an efficient coalescence of smbh binaries , while sesana et al . ( 2004a ) specify a detailed model for binary evolution that results in some binaries being ejected from the galaxy core ( where they do not coalesce ) . we estimate an upper limit for the rate of supermassive bbh coalescence ( where both binary members weigh more than @xmath195 m@xmath1 ) of @xmath196 yr@xmath15 , which is larger than , but comparable to , the @xmath197 yr@xmath15 estimated by sesana et al . 2004a&b for binary coalescence with one member heavier than @xmath195 m@xmath1 . if intermediate mass black holes exist , then our estimate for the detectable coalescence rate increases only modestly [ due to the declining occupation fraction for lower mass halos ( see figure [ fig2 ] ) ] to @xmath22 tens per year ( see figure [ fig3 ] ) . we have computed the snr of both the inspiral and ringdown phases of smbh coalescence , and calculated mass and redshift ranges over which bbh coalescence will be detectable by lisa . the higher frequency of the ringdown signal allows detection of the coalescence of more massive binary systems . in particular , searches for gravitational waves for the ringdown signal offer a superior probe of bbh coalescence between smbhs which are more massive than @xmath198 m@xmath1 , and allow confident detection of the coalescence of smbhs in the mass range @xmath98 m@xmath1 at @xmath95 should they exist ( or @xmath199 m@xmath1 at @xmath96 ) . we have predicted the rate of binary black hole coalescence detectable by lisa assuming a hierarchical cosmology in which black holes form in halos with virial temperature above a critical value and obey an observationally motivated scaling with halo mass and redshift of the form @xmath200 . prompted by the observation that the numbers of both large and small galaxies fall short of the expected numbers of dark - matter halos , we compute the empirical galaxy occupation fraction . this is obtained by comparing the locally observed galaxy velocity function with a cooled press - schechter ( 1974 ) velocity function . our predicted event rate declines rapidly with increased estimates for the smallest smbh that can form . if we are guided by the smallest nuclear bh mass inferred in local galactic bulges and nearby low - luminosity agn ( @xmath201 m@xmath1 ) , then we expect @xmath188 detections per year . most of these events will be detectable via both their final - year inspiral and ringdown signals . our observationally motivated estimates for the event rate of bbh coalescence therefore indicate that observations of coalescing bbhs are a realistic target for lisa . the authors wish to thank rachel webster , andrew melatos and avi loeb for helpful discussions during the course of this work . sesana , a. , haardt , f. , madau , p. , & volonteri , m. 2004b , apj , accepted ( astro - ph/0409255 ) shields , g. a. , gebhardt , k. , salviander , s. , wills , b. j. , xie , b. , brotherton , m. s. , yuan , j. , & dietrich , m. 2003 , apj , 583 , 124

the gravitational waves generated during supermassive black hole ( smbh ) coalescence are prime candidates for detection by the satellite lisa . we use the extended press - schechter formalism combined with empirically motivated estimates for the smbh dark matter halo mass relation and smbh occupation fraction to estimate the maximum coalescence rate for major smbh mergers . assuming efficient binary coalescence , and guided by the lowest nuclear black hole mass inferred in local galactic bulges and nearby low - luminosity active galactic nuclei ( @xmath0 m@xmath1 ) we predict approximately 15 detections per year at a signal to noise greater than five , in each of the inspiral and ringdown phases . rare coalescences between smbhs having masses in excess of @xmath2 m@xmath1 will be more readily detected via gravitational waves from the ringdown phase . [ firstpage ] black hole physics , cosmology : theory , gravitational waves

astro-ph0503210

connectivity and network design problems play an important role in combinatorial optimization and algorithms both for their theoretical appeal and their many real - world applications . an interesting and large class of problems are of the following type : given a graph @xmath5 with edge or node costs , find a minimum - cost subgraph @xmath6 of @xmath2 that satisfies certain connectivity properties . for example , given an integer @xmath7 , one can ask for the minimum - cost spanning subgraph that is @xmath8-edge or @xmath8-vertex connected . if @xmath9 then this is the classical minimum spanning tree ( mst ) problem . for @xmath10 the problem is np - hard and also apx - hard to approximate . more general versions of connectivity problems are obtained if one seeks a subgraph in which a subset of the nodes @xmath11 referred to as _ terminals _ are @xmath8-connected . the well - known steiner tree problem is to find a minimum - cost subgraph that ( @xmath12-)connects a given set @xmath13 . many of these problems are special cases of the survivable network design problem ( sndp ) . in sndp , each pair of nodes @xmath14 specifies a connectivity requirement @xmath15 and the goal is to find a minimum - cost subgraph that has @xmath15 disjoint paths for each pair @xmath16 . given the intractability of these connectivity problems , there has been a large amount of work on approximation algorithms . a number of elegant and powerful techniques and results have been developed over the years ( see @xcite ) . in particular , the primal - dual method @xcite and iterated rounding @xcite have led to some remarkable results including a @xmath1-approximation for edge - connectivity sndp @xcite . an interesting class of problems , related to some of the connectivity problems described above , is obtained by requiring that only @xmath0 of the given terminals be connected . these problems are partly motivated by applications in which one seeks to maximize profit given a upper bound ( budget ) on the cost . for example , a useful problem in vehicle routing applications is to find a path that maximizes the number of vertices in it subject to a budget @xmath17 on the length of the path . in the exact optimization setting , the profit maximization problem is equivalent to the problem of minimizing the cost / length of a path subject to the constraint that at least @xmath0 vertices are included . of course the two versions need not be approximation equivalent , nevertheless , understanding one is often fruitful or necessary to understand the other . the most well - studied of these problems is the @xmath0-mst problem ; the goal here is to find a minimum - cost subgraph of the given graph @xmath2 that contains at least @xmath0 vertices ( or terminals ) . this problem has attracted considerable attention in the approximation algorithms literature and its study has led to several new algorithmic ideas and applications @xcite . we note that the steiner tree problem can be relatively easily reduced in an approximation preserving fashion to the @xmath0-mst problem . more recently , lau _ et al . _ @xcite considered the natural generalization of @xmath0-mst to higher connectivity . in particular they defined the @xmath18-subgraph problem to be the following : find a minimum - cost subgraph of the given graph @xmath2 that contains at least @xmath0 vertices and is @xmath8-edge connected . we use the notation @xmath0-@xmath8ec to refer to this problem . in @xcite an @xmath19 approximation was claimed for the @xmath0-@xmath1ec problem . however , the algorithm and proof in @xcite are incorrect . more recently , and in independent work from ours , the authors of @xcite obtained a different algorithm for @xmath0-@xmath1ec that yields an @xmath20 approximation . we give later a more detailed comparison between their approach and ours . it is also shown in @xcite that a good approximation for @xmath0-@xmath8ec when @xmath8 is large would yield an improved algorithm for the @xmath0-densest subgraph problem @xcite ; in this problem one seeks a @xmath0-vertex subgraph of a given graph @xmath2 that has the maximum number of edges . the @xmath0-densest subgraph problem admits an @xmath21 approximation for some fixed constant @xmath22 @xcite , but has resisted attempts at an improved approximation for a number of years now . in this paper we consider the vertex - connectivity generalization of the @xmath0-mst problem . we define the @xmath0-@xmath8vc problem as follows : given an integer @xmath0 and a graph @xmath2 with edge costs , find the minimum - cost @xmath8-vertex - connected subgraph of @xmath2 that contains at least @xmath0 vertices . we also consider the _ terminal _ version of the problem where the subgraph has to contain @xmath0 terminals from a given terminal set @xmath3 . it can be easily shown that the @xmath0-@xmath8ec problem reduces to the @xmath0-@xmath8vc problem for any @xmath23 . we also observe that the @xmath0-@xmath8ec problem with terminals can be easily reduced , as follows , to the uniform problem where every vertex is a terminal : for each terminal @xmath24 , create @xmath25 dummy vertices @xmath26 and attach @xmath27 to @xmath28 with @xmath8 parallel edges of zero cost . now set @xmath29 in the new graph . one can avoid using parallel edges by creating a clique on @xmath26 using zero - cost edges and connecting @xmath8 of these vertices to @xmath28 . note , however , that this reduction only works for edge - connectivity . we are not aware of a reduction that reduces the @xmath0-@xmath8vc problem with a given set of terminals to the @xmath0-@xmath8vc problem , even when @xmath30 . in this paper we consider the @xmath0-@xmath1vc problem ; our main result is the following . [ thm : kv ] there is an @xmath31 approximation for the @xmath0-@xmath1vc problem where @xmath32 is the number of terminals . [ cor : ke ] there is an @xmath31 approximation for the @xmath0-@xmath1ec problem where @xmath32 is the number of terminals . one of the technical ingredients that we develop is the theorem below which may be of independent interest . given a graph @xmath2 with edge costs and weights on terminals @xmath3 , we define @xmath33 for a subgraph @xmath6 to be the ratio of the cost of edges in @xmath6 to the total weight of terminals in @xmath6 . [ thm : cycle ] let @xmath2 be an @xmath1-vertex - connected graph with edge costs and let @xmath34 be a set of terminals . then , there is a simple cycle @xmath35 containing at least @xmath1 terminals ( a non - trivial cycle ) such that the density of @xmath35 is at most the density of @xmath2 . moreover , such a cycle can be found in polynomial time . using the above theorem and an lp approach we obtain the following . [ cor : cycle ] given a graph @xmath5 with edge costs and @xmath32 terminals @xmath36 , there is an @xmath37 approximation for the problem of finding a minimum - density non - trivial cycle . note that theorem [ thm : cycle ] and corollary [ cor : cycle ] are of interest because we seek a cycle with at least _ two _ terminals . a minimum - density cycle containing only one terminal can be found by using the well - known min - mean cycle algorithm in directed graphs @xcite . we remark , however , that although we suspect that the problem of finding a minimum - density non - trivial cycle is np - hard , we currently do not have a proof . theorem [ thm : cycle ] shows that the problem is equivalent to the dens-@xmath1vc problem , defined in the next section . * remark : * the reader may wonder whether @xmath0-@xmath1ec or @xmath0-@xmath1vc admit a constant factor approximation , since the @xmath0-mst problem admits one . we note that the main technical tool which underlies @xmath38 approximations for @xmath0-mst problem @xcite is a special property that holds for a lp relaxation of the prize - collection steiner tree problem @xcite which is a lagrangian relaxation of the steiner tree problem . such a property is not known to hold for generalizations of @xmath0-mst including @xmath0-@xmath1ec and @xmath0-@xmath1vc and the @xmath0-steiner forest problem @xcite . thus , one is forced to rely on alternative and problem - specific techniques . we consider the rooted version of @xmath0-@xmath1vc : the goal is to find a min - cost subgraph that @xmath1-connects at least @xmath0 terminals to a specified root vertex @xmath39 . it is relatively straightforward to reduce @xmath0-@xmath1vc to its rooted version ( see section [ sec : k2vc ] for details . ) we draw inspiration from algorithmic ideas that led to poly - logarithmic approximations for the @xmath0-mst problem . to describe our approach to the rooted @xmath0-@xmath1vc problem , we define a closely related problem . for a subgraph @xmath6 that contains @xmath39 , let @xmath40 be the number of terminals that are @xmath1-connected to @xmath39 in @xmath6 . then the _ density _ of @xmath6 is simply the ratio of the cost of @xmath6 to @xmath40 . the dens-@xmath1vc problem is to find a 2-connected subgraph of minimum density . an @xmath37 approximation for the dens-@xmath1vc problem ( where @xmath32 is the number of terminals ) can be derived in a some what standard way by using a bucketing and scaling trick on a linear programming relaxation for the problem . we exploit the known bound of @xmath1 on the integrality gap of a natural lp for the sndp problem with vertex connectivity requirements in @xmath41 @xcite . the bucketing and scaling trick has seen several uses in the past and has recently been highlighted in several applications @xcite . our algorithm for @xmath0-@xmath1vc uses a greedy approach at the high level . we start with an empty subgraph @xmath42 and use the approximation algorithm for dens-@xmath1vc in an iterative fashion to greedily add terminals to @xmath42 until at least @xmath43 terminals are in @xmath42 . this approach would yield an @xmath44 approximation if @xmath45 . however , the last iteration of the dens-@xmath1vc algorithm may add many more terminals than desired with the result that @xmath46 . in this case we can not bound the quality of the solution obtained by the algorithm . to overcome this problem , one can try to _ prune _ the subgraph @xmath6 added in the last iteration to only have the desired number of terminals . for the @xmath0-mst problem , @xmath6 is a tree and pruning is quite easy . we remark that this yields a rather straightforward @xmath20 approximation for @xmath0-mst and could have been discovered much before a more clever analysis given in @xcite . one of our technical contributions is to give a pruning step for the @xmath0-@xmath1vc problem . to accomplish this , we use two algorithmic ideas . the first is encapsulated in the cycle finding algorithm of theorem [ thm : cycle ] . second , we use this cycle finding algorithm to repeatedly merge subgraphs until we get the desired number of terminals in one subgraph . this latter step requires care . the cycle merging scheme is inspired by a similar approach from the work of lau _ et al . _ @xcite on the @xmath0-@xmath1ec problem and in @xcite on the directed orienteering problem . these ideas yield an @xmath47 approximation . we give a slightly modified cycle - merging algorithm with a more sophisticated and non - trivial analysis to obtain an improved @xmath31 approximation . some remarks are in order to compare our work to that of @xcite on the @xmath0-@xmath1ec problem . the combinatorial algorithm in @xcite is based on finding a low - density cycle or a related structure called a bi - cycle . the algorithm in @xcite to find such a structure is incorrect . further , the cycles are contracted along the way which limits the approach to the @xmath0-@xmath1ec problem ( contracting a cycle in @xmath1-node - connected graph may make the resulting graph not @xmath1-node - connected ) . in our algorithm we do not contract cycles and instead introduce dummy terminals with weights to capture the number of terminals in an already formed component . this requires us to now address the minimum - density non - trivial simple cycle problem which we do via theorem [ thm : cycle ] and corollary [ cor : cycle ] . in independent work , lau _ et al . _ @xcite obtain a new and correct @xmath48-approximation for @xmath0-@xmath1ec . they also follow the same approach that we do in using the lp for finding dense subgraphs followed by the pruning step . however , in the pruning step they use a completely different approach ; they use the sophisticated idea of no - where zero @xmath49-flows @xcite . although the use of this idea is elegant , the approach works only for the @xmath0-@xmath1ec problem , while our approach is less complex and leads to an algorithm for the more general @xmath0-@xmath1vc problem . we work with graphs in which some vertices are designated as _ terminals_. given a graph @xmath2 with edge costs and terminal weights , we define the _ density _ of a subgraph @xmath6 to be sum of the costs of edges in @xmath6 divided by the sum of the weights of terminals in @xmath6 . henceforth , we use @xmath1-connected graph to mean a @xmath1-vertex - connected graph . the goal of the @xmath0-@xmath1vc problem is to find a minimum - cost 2-connected subgraph on at least @xmath0 terminals .. for simplicity of exposition , however , we stick to the more restricted version . ] recall that in the rooted @xmath0-@xmath1vc problem , the goal is to find a min - cost subgraph on at least @xmath0 terminals in which every terminal is 2-connected to the specified root @xmath39 . the ( unrooted ) @xmath0-@xmath1vc problem can be reduced to the rooted version by _ guessing _ 2 vertices @xmath16 that are in an optimal solution , creating a new root vertex @xmath39 , and connecting it with 0-cost edges to @xmath50 and @xmath28 . it is not hard to show that any solution to the rooted problem in the modified graph can be converted to a solution to the unrooted problem by adding 2 minimum - cost vertex - disjoint paths between @xmath50 and @xmath28 . ( since @xmath50 and @xmath28 are in the optimal solution , the cost of these added paths can not be more than @xmath51 . ) we omit further details from this extended abstract . in the dens-@xmath1vc problem , the goal is to find a subgraph @xmath6 of minimum density in which all terminals of @xmath6 are 2-connected to the root . the following lemma is proved in section [ subsec : lp ] below . it relies on a @xmath1-approximation , via a natural lp , for the min - cost @xmath1-connectivity problem due to fleischer , jain and williamson @xcite , and some standard techniques . [ lem : densv ] there is an @xmath37-approximation algorithm for the dens-@xmath1vc problem , where @xmath32 is the number of terminals in the given instance . let @xmath51 be the cost of an optimal solution to the @xmath0-@xmath1vc problem . we assume knowledge of @xmath51 ; this can be dispensed with using standard methods . we pre - process the graph by deleting any terminal that does not have 2 vertex - disjoint paths to the root @xmath39 of total cost at most @xmath51 . the high - level description of the algorithm for the rooted @xmath0-@xmath1vc problem is given below . @xmath52 , @xmath53 is the empty graph . + while ( @xmath54 ) : + use the approximation algorithm for dens-@xmath1vc to find a subgraph @xmath6 in @xmath2 . + if ( @xmath55 ) : + @xmath56 , @xmath57 + mark all terminals in @xmath6 as non - terminals . + else : + _ prune _ @xmath6 to obtain @xmath58 that contains @xmath59 terminals . + @xmath60 , @xmath61 + output @xmath42 at the beginning of any iteration of the while loop , the graph contains a solution to the dens-@xmath1vc problem of density at most @xmath62 . therefore , the graph @xmath6 returned always has density at most @xmath63 . if @xmath55 , we add @xmath6 to @xmath42 and decrement @xmath59 ; we refer to this as the _ augmentation _ step . otherwise , we have a graph @xmath6 of good density , but with too many terminals . in this case , we prune @xmath6 to find a graph with the required number of terminals ; this is the _ pruning step_. a simple set - cover type argument shows the following lemma : [ lem : greedy ] if , at every augmentation step , we add a graph of density at most @xmath64 ( where @xmath59 is the number of additional terminals that must be selected ) , the total cost of all the augmentation steps is at most @xmath65 . therefore , we now only have to bound the cost of the graph @xmath58 added in the pruning step ; we prove the following theorem in section [ sec : pruning ] . [ thm : avekv ] let @xmath66 be an instance of the rooted @xmath0-@xmath1vc problem with root @xmath39 , such that every vertex of @xmath2 has @xmath1 vertex - disjoint paths to @xmath39 of total cost at most @xmath67 , and such that @xmath68 . there is a polynomial - time algorithm to find a solution to this instance of cost at most @xmath69 . we can now prove our main result for the @xmath0-@xmath1vc problem , theorem [ thm : kv ] . theorem [ thm : kv ] let @xmath51 be the cost of an optimal solution to the ( rooted ) @xmath0-@xmath1vc problem . by lemma [ lem : greedy ] , the total cost of the augmentation steps of our greedy algorithm is @xmath70 . to bound the cost of the pruning step , let @xmath59 be the number of additional terminals that must be covered just prior to this step . the algorithm for the dens-@xmath1vc problem returns a graph @xmath6 with @xmath71 terminals , and density at most @xmath72 . as a result of our pre - processing step , every vertex has 2 vertex - disjoint paths to @xmath39 of total cost at most @xmath51 . now , we use theorem [ thm : avekv ] to prune @xmath6 and find a graph @xmath58 with @xmath59 terminals and cost at most @xmath73 . therefore , the total cost of our solution is @xmath74 . it remains only to prove lemma [ lem : densv ] , that there is an @xmath75-approximation for the dens-@xmath1vc problem , and theorem [ thm : avekv ] , bounding the cost of the pruning step . we prove the former in section [ subsec : lp ] below . before the latter is proved in section [ sec : pruning ] , we develop some tools in section [ sec : cycles ] ; chief among these tools is theorem [ thm : cycle ] . recall that the dens-@xmath1vc problem was defined as follows : given a graph @xmath5 with edge - costs , a set @xmath76 of terminals , and a root @xmath77 , find a subgraph @xmath6 of minimum density , in which every terminal of @xmath6 is 2-connected to @xmath39 . ( here , the density of @xmath6 is defined as the cost of @xmath6 divided by the number of terminals it contains , not including @xmath39 . ) we describe an algorithm for dens-@xmath1vc that gives an @xmath37-approximation , and sketch its proof . we use an lp based approach and a bucketing and scaling trick ( see @xcite for applications of this idea ) , and a constant - factor bound on the integrality gap of an lp for sndp with vertex - connectivity requirements in @xmath41 @xcite . we define * lp - dens * as the following lp relaxation of dens-@xmath1vc . for each terminal @xmath78 , the variable @xmath79 indicates whether or not @xmath28 is chosen in the solution . ( by normalizing @xmath80 to 1 , and minimizing the sum of edge costs , we minimize the density . ) @xmath81 is the set of all simple cycles containing @xmath78 and the root @xmath39 ; for any @xmath82 , @xmath83 indicates how much ` flow ' is sent from @xmath28 to @xmath39 through @xmath35 . ( note that a pair of vertex - disjoint paths is a cycle ; the flow along a cycle is 1 if we can 2-connect @xmath78 to @xmath39 using the edges of the cycle . ) the variable @xmath84 indicates whether the edge @xmath85 is used by the solution . @xmath86 @xmath87 it is not hard to see that an optimal solution to * lp - dens * has cost at most the density of an optimal solution to dens-@xmath1vc . we now show how to obtain an integral solution of density at most @xmath88 , where @xmath89 is the cost of an optimal solution to * lp - dens * . the linear program * lp - dens * has an exponential number of variables but a polynomial number of non - trivial constraints ; it can , however , be solved in polynomial time . fix an optimal solution to * lp - dens * of cost @xmath89 , and for each @xmath90 ( for ease of notation , assume @xmath91 is an integer ) , let @xmath92 be the set of terminals @xmath78 such that @xmath93 . since @xmath94 , there is some index @xmath95 such that @xmath96 . since every terminal @xmath97 has @xmath98 , the number of terminals in @xmath92 is at least @xmath99 . we claim that there is a subgraph @xmath6 of @xmath2 with cost at most @xmath100 , in which every terminal of @xmath92 is 2-connected to the root . if this is true , the density of @xmath6 is at most @xmath101 , and hence we have an @xmath37-approximation for the dens-@xmath1vc problem . to prove our claim about the cost of the subgraph @xmath6 in which every terminal of @xmath92 is 2-connected to @xmath39 , consider scaling up the given optimum solution of * lp - dens * by a factor of @xmath102 . for each terminal @xmath103 , the flow from @xmath78 to @xmath39 in this scaled solution ] is at least 1 , and the cost of the scaled solution is @xmath104 . in @xcite , the authors describe a linear program @xmath105 to find a minimum - cost subgraph in which a given set of terminals is 2-connected to the root , and show that this linear program has an integrality gap of 2 . the variables @xmath84 in the ` scaled solution ' to * lp - dens * correspond to a feasible solution of @xmath105 with @xmath92 as the set of terminals ; the integrality gap of 2 implies that there is a subgraph @xmath6 in which every terminal of @xmath92 is 2-connected to the root , with cost at most @xmath106 . therefore , the algorithm for dens-@xmath1vc is : 1 . find an optimal fractional solution to * lp - dens * . 2 . find a set of terminals @xmath92 such that @xmath107 . 3 . find a min - cost subgraph @xmath6 in which every terminal in @xmath92 is 2-connected to @xmath39 using the algorithm of @xcite . @xmath6 has density at most @xmath37 times the optimal solution to dens-@xmath1vc . a cycle @xmath108 is _ non - trivial _ if it contains at least 2 terminals . we define the min - density non - trivial cycle problem : given a graph @xmath5 , with @xmath3 marked as terminals , edge costs and terminal weights , find a minimum - density cycle that contains at least 2 terminals . note that if we remove the requirement that the cycle be non - trivial ( that is , it contains at least 2 terminals ) , the problem reduces to the min - mean cycle problem in directed graphs , and can be solved exactly in polynomial time ( see @xcite ) . algorithms for the min - density non - trivial cycle problem are a useful tool for solving the @xmath0-@xmath1vc and @xmath0-@xmath1ec problems . in this section , we give an @xmath75-approximation algorithm for the minimum - density non - trivial cycle problem . first , we prove theorem [ thm : cycle ] , that a 2-connected graph with edge costs and terminal weights contains a simple non - trivial cycle , with density no more than the average density of the graph . we give two algorithms to find such a cycle ; the first , described in section [ subsec : nonpoly ] , is simpler , but the running time is not polynomial . a more technical proof that leads to a strongly polynomial - time algorithm is described in section [ subsec : strong ] ; we recommend this proof be skipped on a first reading . to find a non - trivial cycle of density at most that of the 2-connected input graph @xmath2 , we will start with an arbitrary non - trivial cycle , and successively find cycles of better density until we obtain a cycle with density at most @xmath109 . the following lemma shows that if a cycle @xmath35 has an ear with density less than @xmath110 , we can use this ear to find a cycle of lower density . [ lem : goodear ] let @xmath35 be a non - trivial cycle , and @xmath6 an ear incident to @xmath35 at @xmath50 and @xmath28 , such that @xmath111 . let @xmath112 and @xmath113 be the two internally disjoint paths between @xmath50 and @xmath28 in @xmath35 . then @xmath114 and @xmath115 are both simple cycles and one of these is non - trivial and has density less than @xmath110 . @xmath35 has at least 2 terminals , so it has finite density ; @xmath6 must then have at least 1 terminal . let @xmath116 , @xmath117 and @xmath118 be , respectively , the sum of the costs of the edges in @xmath112 , @xmath113 and @xmath6 , and let @xmath119 , @xmath120 and @xmath121 be the sum of the weights of the terminals in @xmath112 , @xmath113 and @xmath122 . assume w.l.o.g . that @xmath112 has density at most that of @xmath113 . ( that is , @xmath123 . ) and @xmath113 has cost 0 and weight 0 . in this case , let @xmath112 be the component with non - zero weight . ] @xmath112 must contain at least one terminal , and so @xmath114 is a simple non - trivial cycle . the statement @xmath124 is equivalent to @xmath125 . @xmath126 therefore , @xmath114 is a simple cycle containing at least 2 terminals of density less than @xmath110 . [ lem:2conncomp ] given a cycle @xmath35 in a @xmath1-connected graph @xmath2 , let @xmath42 be the graph formed from @xmath2 by contracting @xmath35 to a single vertex @xmath28 . if @xmath6 is a connected component of @xmath127 , @xmath128 is @xmath1-connected in @xmath42 . let @xmath6 be an arbitrary connected component of @xmath127 , and let @xmath129 . to prove that @xmath58 is 2-connected , we first observe that @xmath28 is 2-connected to any vertex @xmath130 . ( any set that separates @xmath131 from @xmath28 in @xmath58 separates @xmath131 from the cycle @xmath35 in @xmath2 . ) it now follows that for all vertices @xmath132 , @xmath131 and @xmath133 are 2-connected in @xmath58 . suppose deleting some vertex @xmath50 separates @xmath131 from @xmath133 . the vertex @xmath50 can not be @xmath28 , since @xmath6 is a connected component of @xmath127 . but if @xmath134 , @xmath28 and @xmath131 are in the same component of @xmath135 , since @xmath28 is 2-connected to @xmath131 in @xmath58 . similarly , @xmath28 and @xmath133 are in the same component of @xmath135 , and so deleting @xmath50 does not separate @xmath131 from @xmath133 . we now show that given any 2-connected graph @xmath2 , we can find a non - trivial cycle of density no more than that of @xmath2 . [ thm : cycleexists ] let @xmath2 be a @xmath1-connected graph with at least @xmath1 terminals . @xmath2 contains a simple non - trivial cycle @xmath136 such that @xmath137 . let @xmath35 be an arbitrary non - trivial simple cycle ; such a cycle always exists since @xmath2 is @xmath1-connected and has at least 2 terminals . if @xmath138 , we give an algorithm that finds a new non - trivial cycle @xmath139 such that @xmath140 . repeating this process , we obtain cycles of successively better densities until eventually finding a non - trivial cycle @xmath136 of density at most @xmath109 . let @xmath42 be the graph formed by contracting the given cycle @xmath35 to a single vertex @xmath28 . in @xmath42 , @xmath28 is not a terminal , and so has weight 0 . consider the 2-connected components of @xmath42 ( from lemma [ lem:2conncomp ] , each such component is formed by adding @xmath28 to a connected component of @xmath127 ) , and pick the one of minimum density . if @xmath6 is this component , @xmath141 by an averaging argument . @xmath6 contains at least 1 terminal . if it contains 2 or more terminals , recursively find a non - trivial cycle @xmath139 in @xmath6 such that @xmath142 . if @xmath139 exists in the given graph @xmath2 , it has the desired properties , and we are done . otherwise , @xmath139 contains @xmath28 , and the edges of @xmath139 form a ear of @xmath35 in the original graph @xmath2 . the density of this ear is less than the density of @xmath35 , so we can apply lemma [ lem : goodear ] to obtain a non - trivial cycle in @xmath2 that has density less than @xmath110 . finally , if @xmath6 has exactly 1 terminal @xmath50 , find any 2 vertex - disjoint paths using edges of @xmath6 from @xmath50 to distinct vertices in the cycle @xmath35 . ( since @xmath2 is 2-connected , there always exist such paths . ) the cost of these paths is at most @xmath143 , and concatenating these 2 paths corresponds to a ear of @xmath35 in @xmath2 . the density of this ear is less than @xmath110 ; again , we use lemma [ lem : goodear ] to obtain a cycle in @xmath2 with the desired properties . we remark again that the algorithm of theorem [ thm : cycleexists ] does not lead to a polynomial - time algorithm , even if all edge costs and terminal weights are polynomially bounded . in section [ subsec : strong ] , we describe a strongly polynomial - time algorithm that , given a graph @xmath2 , finds a non - trivial cycle of density at most that of @xmath2 . note that neither of these algorithms may directly give a good approximation to the min - density non - trivial cycle problem , because the optimal non - trivial cycle may have density much less than that of @xmath2 . however , we can use theorem [ thm : cycleexists ] to prove the following theorem : [ thm : equivalence ] there is an @xmath144-approximation to the ( unrooted ) dens-@xmath1vc problem if and only if there is an @xmath144-approximation to the problem of finding a minimum - density non - trivial cycle . assume we have a @xmath145-approximation for the dens-@xmath1vc problem ; we use it to find a low - density non - trivial cycle . solve the dens-@xmath1vc problem on the given graph ; since the optimal cycle is a 2-connected graph , our solution @xmath6 to the dens-@xmath1vc problem has density at most @xmath145 times the density of this cycle . find a non - trivial cycle in @xmath6 of density at most that of @xmath6 ; it has density at most @xmath145 times that of an optimal non - trivial cycle . note that any instance of the ( unrooted ) dens-@xmath1vc problem has an optimal solution that is a non - trivial cycle . ( consider any optimal solution @xmath6 of density @xmath146 ; by theorem [ thm : cycle ] , @xmath6 contains a non - trivial cycle of density at most @xmath146 . this cycle is a valid solution to the dens-@xmath1vc problem . ) therefore , a @xmath147-approximation for the min - density non - trivial cycle problem gives a @xmath147-approximation for the dens-@xmath1vc problem . theorem [ thm : equivalence ] and lemma [ lem : densv ] imply an @xmath37-approximation for the minimum - density non - trivial cycle problem ; this proves corollary [ cor : cycle ] . we say that a graph @xmath5 is minimally 2-connected on its terminals if for every edge @xmath148 , some pair of terminals is not 2-connected in the graph @xmath149 . section [ subsec : strong ] shows that in any graph which is minimally 2-connected on its terminals , every cycle is non - trivial . therefore , the problem of finding a minimum - density non - trivial cycle in such graphs is just that of finding a minimum - density cycle , which can be solved exactly in polynomial time . however , as we explain at the end of the section , this does not directly lead to an efficient algorithm for arbitrary graphs . in this section , we describe a strongly polynomial - time algorithm which , given a 2-connected graph @xmath5 with edge costs and terminal weights , finds a non - trivial cycle of density at most that of @xmath2 . we begin with several definitions : let @xmath35 be a cycle in a graph @xmath2 , and @xmath42 be the graph formed by deleting @xmath35 from @xmath2 . let @xmath150 be the connected components of @xmath42 ; we refer to these as _ earrings _ of @xmath35 . were simply a path , it would be an ear of @xmath35 , but @xmath151 may be more complex . ] for each @xmath151 , let the vertices of @xmath35 incident to it be called its _ clasps_. from the definition of an earring , for any pair of clasps of @xmath151 , there is a path between them whose internal vertices are all in @xmath151 . we say that a vertex of @xmath35 is an _ anchor _ if it is the clasp of some earring . ( an anchor may be a clasp of multiple earrings . ) a _ segment _ @xmath13 of @xmath35 is a path contained in @xmath35 , such that the endpoints of @xmath13 are both anchors , and no internal vertex of @xmath13 is an anchor . ( note that the endpoints of @xmath13 might be clasps of the same earring , or of distinct earrings . ) it is easy to see that the segments partition the edge set of @xmath35 . by deleting a segment , we refer to deleting its edges and internal vertices . observe that if @xmath13 is deleted from @xmath2 , the only vertices of @xmath152 that lose an edge are the endpoints of @xmath13 . a segment is _ safe _ if the graph @xmath153 is 2-connected . arbitrarily pick a vertex @xmath154 of @xmath35 as the _ origin _ , and consecutively number the vertices of @xmath35 clockwise around the cycle as @xmath155 . the first clasp of an earring @xmath6 is its lowest numbered clasp , and the last clasp is its highest numbered clasp . ( if the origin is a clasp of @xmath6 , it is considered the first clasp , not the last . ) the _ arc _ of an earring is the subgraph of @xmath35 found by traversing clockwise from its first clasp @xmath156 to its last clasp @xmath157 ; the length of this arc is @xmath158 . ( that is , the length of an arc is the number of edges it contains . ) note that if an arc contains the origin , it must be the first vertex of the arc . figure [ fig : earring ] illustrates several of these definitions . ( 0,0 ) circle ( 2 cm ) ; at ( 0,0 ) @xmath35 ; ( 30:2 cm ) ( 30:3 cm ) ; ( 2,0 ) ( 3,0 ) ; ( -45:2 cm ) ( -45:3 cm ) ; at ( 38:2.35 cm ) @xmath159 ; at ( 2.35,0.25 ) @xmath160 ; at ( -53:2.35 cm ) @xmath161 ; at ( 0,2.35 ) @xmath162 ; ( 30:2.8 cm ) arc(30:-45:2.8 cm ) ; ( 30:4.2 cm ) arc ( 30:-45:4.2 cm ) ; ( 30:2.8 cm ) arc ( -150:-330:0.7 cm ) ; ( -45:2.8 cm ) arc ( 135:315:0.7 cm ) ; at ( 15:3.5 cm ) @xmath6 ; ( 30:2 cm ) circle ( 1 mm ) ; ( -45:2 cm ) circle ( 1 mm ) ; ( 2,0 ) circle ( 1 mm ) ; ( 0,2 ) circle ( 1 mm ) ; ( 30:1.75 cm ) arc ( 30:-45:1.75 cm ) ; ( 0,0 ) circle ( 2 cm ) ; at ( 0,0 ) @xmath35 ; ( 0,2 ) ( 0,3.6 ) arc(90:-90:3.6 cm ) ( 0,-2 ) ; at ( 0,1.6 ) @xmath163 ; at ( 0,-1.6 ) @xmath164 ; at ( 0,3.9 ) [ font= ] @xmath165 ; ( 30:2 cm ) ( 30:3 cm ) arc(30:-30:3 cm ) ( -30:2 cm ) ; at ( 30:1.6 cm ) @xmath156 ; at ( -30:1.6 cm ) @xmath157 ; ( 30:2.2 cm ) arc(30:330:2.2 cm ) ; ( 25:2.2 cm ) ( 25:2.8 cm ) arc(25:-25:2.8 cm ) ( -25:2.2 cm ) ; ( 30:2 cm ) circle ( 1 mm ) ; ( -30:2 cm ) circle ( 1 mm ) ; ( 0,2 ) circle ( 1 mm ) ; ( 0,-2 ) circle ( 1 mm ) ; ( 150:2 cm ) circle ( 1 mm ) ; at ( 150:1.6 cm ) @xmath162 ; ( 0,0 ) circle ( 2 cm ) ; at ( 0,0 ) @xmath35 ; ( 0,2 ) ( 0,3.6 ) arc(90:-90:3.6 cm ) ( 0,-2 ) ; at ( 0,1.6 ) @xmath163 ; at ( 0,-1.6 ) @xmath164 ; at ( 0,3.9 ) [ font= ] @xmath166 ; at ( 30:1.6 cm ) @xmath156 ; at ( -30:1.6 cm ) @xmath157 ; ( 30:2 cm ) ( 30:3.6 cm ) ; ( 88:2 cm ) ( 85:3 cm ) arc(85:-85:3 cm ) ( -88:2 cm ) ; at ( 85:3.3 cm ) [ font= ] @xmath167 ; ( -30:2 cm ) ( -30:3 cm ) ; ( 25:2.2 cm ) ( 25:3.4 cm ) arc ( 25:-88:3.4 cm ) ( -88:2.2 cm ) arc ( -88:-35:2.2 cm ) ; ( -25:2.2 cm ) ( -25:2.8 cm ) arc ( -25:80:2.8 cm ) ( 80:2.2 cm ) arc ( 80:35:2.2 cm ) ; ( 30:2 cm ) circle ( 1 mm ) ; ( -30:2 cm ) circle ( 1 mm ) ; ( 0,2 ) circle ( 1 mm ) ; ( 0,-2 ) circle ( 1 mm ) ; ( 150:2 cm ) circle ( 1 mm ) ; at ( 150:1.6 cm ) @xmath162 ; ( 0,0 ) circle ( 2 cm ) ; at ( 0,0 ) @xmath35 ; ( 0,2 ) ( 0,3.6 ) arc(90:-90:3.6 cm ) ( 0,-2 ) ; at ( 0,1.6 ) @xmath163 ; at ( 0,-1.6 ) @xmath164 ; at ( 0,3.9 ) [ font= ] @xmath165 ; at ( 30:1.6 cm ) @xmath156 ; at ( -30:1.6 cm ) @xmath157 ; ( 30:2 cm ) ( 30:3.1 cm ) arc ( 30:-105:3.1 cm ) ( -105:2 cm ) ; at ( 38:2.55 cm ) [ font= ] @xmath166 ; ( -30:2 cm ) ( -30:2.5 cm ) arc(-30:-135:2.5 cm ) ( -135:2 cm ) ; at ( -143:2.5 cm ) [ font= ] @xmath167 ; ( 35:2.15 cm ) arc ( 35:85:2.15 cm ) ( 85:3.45 cm ) arc ( 85:-85:3.45 cm ) ( -85:2.15 cm ) arc ( -85:-35:2.15 cm ) ; ( 25:2.2 cm ) ( 25:2.95 cm ) arc(25:-100:2.95 cm ) ( -100:2.15 cm ) arc ( -100:-130:2.15 cm ) ( -130:2.65 cm ) arc ( -130:-25:2.65 cm ) ( -25:2.15 cm ) ; ( 30:2 cm ) circle ( 1 mm ) ; ( -30:2 cm ) circle ( 1 mm ) ; ( 0,2 ) circle ( 1 mm ) ; ( 0,-2 ) circle ( 1 mm ) ; ( 150:2 cm ) circle ( 1 mm ) ; at ( 150:1.6 cm ) @xmath162 ; ( 0,0 ) circle ( 2 cm ) ; at ( 0,0 ) @xmath35 ; ( 0,2 ) ( 0,3.6 ) arc(90:-90:3.6 cm ) ( 0,-2 ) ; at ( 0,1.6 ) @xmath163 ; at ( 0,-1.6 ) @xmath164 ; at ( 0,4.1 ) [ font= ] @xmath168 ; at ( 30:1.6 cm ) @xmath156 ; at ( -30:1.6 cm ) @xmath157 ; ( 30:2 cm ) ( 30:3.6 cm ) ; ( -30:2 cm ) ( -30:3 cm ) arc(-30:-120:3 cm ) ( -120:2 cm ) ; at ( -125:3 cm ) [ font= ] @xmath167 ; ( -35:2.2 cm ) arc ( -35:-85:2.2 cm ) ( -85:3.4 cm ) arc ( -85:25:3.4 cm ) ( 25:2.2 cm ) ; ( -35:2.2 cm ) ( -35:2.8 cm ) arc(-35:-115:2.8 cm ) ( -115:2.2 cm ) arc ( -115:-330:2.2 cm ) ; ( 30:2 cm ) circle ( 1 mm ) ; ( -30:2 cm ) circle ( 1 mm ) ; ( 0,2 ) circle ( 1 mm ) ; ( 0,-2 ) circle ( 1 mm ) ; ( 150:2 cm ) circle ( 1 mm ) ; at ( 150:1.6 cm ) @xmath162 ; ( 0,0 ) circle ( 2 cm ) ; at ( 0,0 ) @xmath35 ; ( 0,2 ) ( 0,3.6 ) arc(90:-90:3.6 cm ) ( 0,-2 ) ; at ( 0,1.6 ) @xmath163 ; at ( 0,-1.6 ) @xmath164 ; at ( 0,4.1 ) [ font= ] @xmath168 ; at ( 30:1.6 cm ) @xmath156 ; at ( -30:1.6 cm ) @xmath157 ; ( -30:2 cm ) ( -30:3.6 cm ) ; ( 30:2 cm ) ( 30:3 cm ) arc(30:-120:3 cm ) ( -120:2 cm ) ; at ( -125:3 cm ) [ font= ] @xmath167 ; ( -25:2.2 cm ) ( -25:3.4 cm ) arc ( -25:85:3.4 cm ) ( 85:2.2 cm ) arc ( 85:35:2.2 cm ) ; ( -35:2.2 cm ) arc ( -35:-115:2.2 cm ) ( -115:2.8 cm ) arc(-115:35:2.8 cm ) ( 35:2.2 cm ) ; ( 30:2 cm ) circle ( 1 mm ) ; ( -30:2 cm ) circle ( 1 mm ) ; ( 0,2 ) circle ( 1 mm ) ; ( 0,-2 ) circle ( 1 mm ) ; ( 150:2 cm ) circle ( 1 mm ) ; at ( 150:1.6 cm ) @xmath162 ; [ thm : earringproof ] let @xmath6 be an earring of minimum arc length . every segment contained in the arc of @xmath6 is safe . let @xmath169 be the set of earrings with arc identical to that of @xmath6 . since they have the same arc , we refer to this as the arc of @xmath169 , or the _ critical arc_. let the first clasp of every earring in @xmath169 be @xmath163 , and the last clasp of each earring in @xmath169 be @xmath164 . because the earrings in @xmath169 have arcs of minimum length , any earring @xmath170 has a clasp @xmath171 that is not in the critical arc . ( that is , @xmath172 or @xmath173 . ) we must show that every segment contained in the critical arc is safe ; recall that a segment @xmath13 is safe if the graph @xmath153 is 2-connected . given an arbitrary segment @xmath13 in the critical arc , let @xmath156 and @xmath157 ( @xmath174 ) be the anchors that are its endpoints . we prove that there are always 2 internally vertex - disjoint paths between @xmath156 and @xmath157 in @xmath152 ; this suffices to show 2-connectivity . we consider several cases , depending on the earrings that contain @xmath156 and @xmath157 . figure [ fig : earringproof ] illustrates these cases . if @xmath156 and @xmath157 are contained in the same earring @xmath58 , it is easy to find two vertex - disjoint paths between them in @xmath153 . the first path is clockwise from @xmath175 to @xmath176 in the cycle @xmath35 . the second path is entirely contained in the earring @xmath58 ( an earring is connected in @xmath177 , so we can always find such a path . ) otherwise , @xmath156 and @xmath157 are clasps of distinct earrings . we consider three cases : both @xmath156 and @xmath157 are clasps of earrings in @xmath169 , one is ( but not both ) , or neither is . 1 . we first consider that both @xmath156 and @xmath157 are clasps of earrings in @xmath169 . let @xmath156 be a clasp of @xmath166 , and @xmath157 a clasp of @xmath167 . the first path is from @xmath157 to @xmath163 through @xmath167 , and then clockwise along the critical arc from @xmath163 to @xmath156 . the second path is from @xmath157 to @xmath164 clockwise along the critical path , and then @xmath164 to @xmath156 through @xmath166 . it is easy to see that these paths are internally vertex - disjoint . 2 . now , suppose neither @xmath156 nor @xmath157 is a clasp of an earring in @xmath169 . let @xmath156 be a clasp of @xmath166 , and @xmath157 be a clasp of @xmath167 . the first path we find follows the critical arc clockwise from @xmath157 to @xmath164 ( the last clasp of the critical arc ) , from @xmath164 to @xmath163 through @xmath165 , and again clockwise through the critical arc from @xmath163 to @xmath156 . internal vertices of this path are all in @xmath6 or on the critical arc . let @xmath178 be a clasp of @xmath166 not on the critical arc , and @xmath179 be a last clasp of @xmath167 not on the critical arc . the second path goes from @xmath156 to @xmath178 through @xmath166 , from @xmath180 to @xmath181 through the cycle @xmath35 outside the critical arc , and from @xmath179 to @xmath157 through @xmath167 . internal vertices of this path are in @xmath182 , or in @xmath35 , but not part of the critical arc ( since each of @xmath178 and @xmath179 are outside the critical arc ) . therefore , we have 2 vertex - disjoint paths from @xmath156 to @xmath157 . finally , we consider the case that exactly one of @xmath183 is a clasp of an earring in @xmath169 . suppose @xmath156 is a clasp of @xmath168 , and @xmath157 is a clasp of @xmath184 ; the other case ( where @xmath185 and @xmath186 is symmetric , and omitted , though figure [ fig : earringproof ] illustrates the paths . ) let @xmath181 be the index of a clasp of @xmath167 outside the critical arc . the first path is from @xmath157 to @xmath164 through the critical arc , and then from @xmath164 to @xmath156 through @xmath166 . the second path is from @xmath157 to @xmath179 through @xmath167 , and from @xmath179 to @xmath156 clockwise through @xmath35 . note that the last part of this path enters the critical arc at @xmath163 , and continues through the arc until @xmath156 . internal vertices of the first path that are in @xmath35 are on the critical arc , but have index greater than @xmath175 . internal vertices of the second path that belong to @xmath35 are either not in the critical arc , or have index between @xmath163 and @xmath156 . therefore , the two paths are internally vertex - disjoint . we now describe our algorithm to find a non - trivial cycle of good density , proving theorem [ thm : cycle ] : _ let @xmath2 be a @xmath1-connected graph with edge - costs and terminal weights , and at least @xmath1 terminals . there is a polynomial - time algorithm to find a non - trivial cycle @xmath136 in @xmath2 such that @xmath137 . _ theorem [ thm : cycle ] let @xmath2 be a graph with @xmath32 terminals and density @xmath146 ; we describe a polynomial - time algorithm that either finds a cycle in @xmath2 of density less than @xmath146 , or a proper subgraph @xmath42 of @xmath2 that contains all @xmath32 terminals . in the latter case , we can recurse on @xmath42 until we eventually find a cycle of density at most @xmath146 . we first find , in @xmath187 time , a minimum - density cycle @xmath35 in @xmath2 . by theorem [ thm : cycleexists ] , @xmath35 has density at most @xmath146 , because the minimum - density _ non - trivial _ cycle has at most this density . if @xmath35 contains at least 2 terminals , we are done . otherwise , @xmath35 contains exactly one terminal @xmath28 . since @xmath2 contains at least 2 terminals , there must exist at least one earring of @xmath35 . let @xmath28 be the origin of this cycle @xmath35 , and @xmath6 an earring of minimum arc length . by theorem [ thm : earringproof ] , every segment in the arc of @xmath6 is safe . let @xmath13 be such a segment ; since @xmath28 was selected as the origin , @xmath28 is not an internal vertex of @xmath13 . as @xmath28 is the only terminal of @xmath35 , @xmath13 contains no terminals , and therefore , the graph @xmath188 is 2-connected , and contains all @xmath32 terminals of @xmath2 . the proof above also shows that if @xmath2 is minimally 2-connected on its terminals ( that is , @xmath2 has no 2-connected proper subgraph containing all its terminals ) , every cycle of @xmath2 is non - trivial . ( if a cycle contains 0 or 1 terminals , it has a safe segment containing no terminals , which can be deleted ; this gives a contradiction . ) therefore , given a graph that _ is _ minimally 2-connected on its terminals , finding a minimum - density non - trivial cycle is equivalent to finding a minimum - density cycle , and so can be solved exactly in polynomial time . this suggests a natural algorithm for the problem : given a graph that is not minimally 2-connected on its terminals , delete edges and vertices until the graph is minimally 2-connected on the terminals , and then find a minimum - density cycle . as shown above , this gives a cycle of density no more than that of the input graph , but this may not be the minimum - density cycle of the original graph . for instance , there exist instances where the minimum - density cycle uses edges of a safe segment @xmath13 that might be deleted by this algorithm . in this section , we prove theorem [ thm : avekv ] . we are given a graph @xmath2 and @xmath3 , a set of at least @xmath0 terminals . further , every terminal in @xmath2 has 2 vertex - disjoint paths to the root @xmath39 of total cost at most @xmath67 . let @xmath32 be the number of terminals in @xmath2 , and @xmath189 its total cost ; @xmath190 is the density of @xmath2 . we describe an algorithm that finds a subgraph @xmath6 of @xmath2 that contains at least @xmath0 terminals , each of which is 2-connected to the root , and of total edge cost @xmath191 . we can assume @xmath192 , or the trivial solution of taking the entire graph @xmath2 suffices . the main phase of our algorithm proceeds by maintaining a set of 2-connected subgraphs that we call _ clusters _ , and repeatedly finding low - density cycles that merge clusters of similar weight to form larger clusters . ( the weight of a cluster @xmath136 , denoted by @xmath193 , is ( roughly ) the number of terminals it contains . ) clusters are grouped into _ tiers _ by weight ; tier @xmath95 contains clusters with weight at least @xmath194 and less than @xmath102 . initially , each terminal is a separate cluster in tier 0 . we say a cluster is _ large _ if it has weight at least @xmath0 , and _ small _ otherwise . the algorithm stops when most terminals are in large clusters . we now describe the algorithm mergeclusters ( see next page ) . to simplify notation , let @xmath144 be the quantity @xmath195 . we say that a cycle is _ good _ if it has density at most @xmath144 ; that is , good cycles have density at most @xmath196 times the density of the input graph . : + for ( each @xmath95 in @xmath197 ) do : + if ( @xmath198 ) : + every terminal has weight 1 + else : + mark all vertices as non - terminals + for ( each small 2-connected cluster @xmath136 in tier @xmath95 ) do : + add a ( dummy ) terminal @xmath199 to @xmath2 of weight @xmath193 + add ( dummy ) edges of cost 0 from @xmath199 to two ( arbitrary ) distinct vertices of @xmath136 + while ( @xmath2 has a non - trivial cycle @xmath35 of density at most @xmath144 in @xmath2 ) : + let @xmath200 be the small clusters that contain a terminal * or an edge * of @xmath35 . + ( note that the terminals in @xmath35 belong to a subset of @xmath201 . ) + form a new cluster @xmath202 ( of a higher tier ) by merging the clusters @xmath203 + @xmath204 + if ( @xmath198 ) : + mark all terminals in @xmath202 as non - terminals + else : + delete all ( dummy ) terminals in @xmath202 and the associated ( dummy ) edges . we briefly remark on some salient features of this algorithm and our analysis before presenting the details of the proofs . 1 . in iteration @xmath95 , the terminals correspond to tier @xmath95 clusters . clusters are 2-connected subgraphs of @xmath2 , and by using cycles to merge clusters , we preserve 2-connectivity as the clusters become larger . 2 . when a cycle @xmath35 is used to merge clusters , all small clusters that contain an edge of @xmath35 ( regardless of their tier ) are merged to form the new cluster . therefore , at any stage of the algorithm , all currently small clusters are edge - disjoint . large clusters , on the other hand , are _ frozen _ ; even if they intersect a good cycle @xmath35 , they are not merged with other clusters on @xmath35 . thus , at any time , an edge may be in multiple large clusters and up to one small cluster . 3 . in iteration @xmath95 of mergeclusters , the density of a cycle @xmath35 is only determined by its cost and the weight of terminals in @xmath35 corresponding to tier @xmath95 clusters . though small clusters of other ( lower or higher ) tiers might be merged using @xmath35 , we do _ not _ use their weight to pay for the edges of @xmath35 . 4 . the @xmath95th iteration terminates when no good cycles can be found using the remaining tier @xmath95 clusters . at this point , there may be some terminals remaining that correspond to clusters which are not merged to form clusters of higher tiers . however , our choice of @xmath144 ( which defines the density of good cycles ) is such that we can bound the number of terminals that are `` left behind '' in this fashion . therefore , when the algorithm terminates , most terminals are in large clusters . by bounding the density of large clusters , we can find a solution to the rooted @xmath0-@xmath1vc problem of bounded density . because we always use cycles of low density to merge clusters , an analysis similar to that of @xcite and @xcite shows that every large cluster has density at most @xmath205 . we first present this analysis , though it does not suffice to prove theorem [ thm : avekv ] . a more careful analysis shows that there is at least one large cluster of density at most @xmath206 ; this allows us to prove the desired theorem . we now formally prove that mergeclusters has the desired behavior . first , we present a series of claims which , together , show that when the algorithm terminates , most terminals are in large clusters , and all clusters are 2-connected . [ rem : cluster ] throughout the algorithm , the graph @xmath2 is always 2-connected . the weight of a cluster is at most the number of terminals it contains . the only structural changes to @xmath2 are when new vertices are added as terminals ; they are added with edges to two distinct vertices of @xmath2 . this preserves 2-connectivity , as does deleting these terminals with the associated edges . to see that the second claim is true , observe that if a terminal contributes weight to a cluster , it is contained in that cluster . a terminal can be in multiple clusters , but it contributes to the weight of exactly one cluster . we use the following simple proposition in proofs of 2-connectivity ; the proof is straightforward , and hence omitted . [ prop : shareedge ] let @xmath207 and @xmath208 be @xmath1-connected subgraphs of a graph @xmath5 such that @xmath209 . then the graph @xmath210 is @xmath1-connected . [ lem : clusters2conn ] the clusters formed by mergeclusters are all @xmath1-connected . let @xmath202 be a cluster formed by using a cycle @xmath35 to merge clusters @xmath200 . the edges of the cycle @xmath35 form a 2-connected subgraph of @xmath2 , and we assume that each @xmath211 is 2-connected by induction . further , @xmath35 contains at least 2 vertices of each @xmath211 may be a singleton vertex ( for instance , if we are in tier 0 ) , but such a vertex does not affect 2-connectivity . ] , so we can use induction and proposition [ prop : shareedge ] above : we assume @xmath212 is 2-connected by induction , and @xmath35 contains 2 vertices of @xmath213 , so @xmath214 is 2-connected . note that we have shown @xmath215 is 2-connected , but @xmath35 ( and hence @xmath202 ) might contain dummy terminals and the corresponding dummy edges . however , each such terminal with the 2 associated edges is a ear of @xmath202 ; deleting them leaves @xmath202 2-connected . [ lem : fewleftbehind ] the total weight of small clusters in tier @xmath95 that are not merged to form clusters of higher tiers is at most @xmath216 . assume this were not true ; this means that mergeclusters could find no more cycles of density at most @xmath144 using the remaining small tier @xmath95 clusters . but the total cost of all the edges is at most @xmath189 , and the sum of terminal weights is at least @xmath216 ; this implies that the density of the graph ( using the remaining terminals ) is at most @xmath217 . but by theorem [ thm : cycleexists ] , the graph must then contain a good non - trivial cycle , and so the while loop would not have terminated . [ cor : weightlargeclusters ] when the algorithm mergeclusters terminates , the total weight of large clusters is at least @xmath218 . each terminal not in a large cluster contributes to the weight of a cluster that was not merged with others to form a cluster of a higher tier . the previous lemma shows that the total weight of such clusters in any tier is at most @xmath219 ; since there are @xmath220 tiers , the total number of terminals not in large clusters is less than @xmath221 . so far , we have shown that most terminals reach large clusters , all of which are 2-connected , but we have not argued about the density of these clusters . the next lemma says that if we can find a large cluster of good density , we can find a solution to the @xmath0-@xmath1vc problem of good density . [ lem : segment ] let @xmath202 be a large cluster formed by mergeclusters . if @xmath202 has density at most @xmath222 , we can find a graph @xmath223 with at least @xmath0 terminals , each of which is @xmath1-connected to @xmath39 , of total cost at most @xmath224 . let @xmath200 be the clusters merged to form @xmath202 in order around the cycle @xmath35 that merged them ; each @xmath211 was a small cluster , of weight at most @xmath0 . a simple averaging argument shows that there is a consecutive segment of @xmath211s with total weight between @xmath0 and @xmath225 , such that the cost of the edges of @xmath35 connecting these clusters , together with the costs of the clusters themselves , is at most @xmath226 . let @xmath227 be the `` first '' cluster of this segment , and @xmath228 the `` last '' . let @xmath28 and @xmath229 be arbitrary terminals of @xmath227 and @xmath228 respectively . connect each of @xmath28 and @xmath229 to the root @xmath39 using 2 vertex - disjoint paths ; the cost of this step is at most @xmath230 . ( we assumed that every terminal could be 2-connected to @xmath39 using disjoint paths of cost at most @xmath67 . ) the graph @xmath223 thus constructed has at least @xmath0 terminals , and total cost at most @xmath231 . we show that every vertex @xmath232 of @xmath223 is 2-connected to @xmath39 ; this completes our proof . let @xmath232 be an arbitrary vertex of @xmath223 ; suppose there is a cut - vertex @xmath131 which , when deleted , separates @xmath232 from @xmath39 . both @xmath28 and @xmath229 are 2-connected to @xmath39 , and therefore neither is in the same component as @xmath232 in @xmath233 . however , we describe 2 vertex - disjoint paths @xmath234 and @xmath235 in @xmath223 from @xmath232 to @xmath28 and @xmath229 respectively ; deleting @xmath131 can not separate @xmath232 from both @xmath28 and @xmath229 , which gives a contradiction . the paths @xmath234 and @xmath235 are easy to find ; let @xmath211 be the cluster containing @xmath232 . the cycle @xmath35 contains a path from vertex @xmath236 to @xmath237 , and another ( vertex - disjoint ) path from @xmath238 to @xmath239 . concatenating these paths with paths from @xmath240 to @xmath28 in @xmath227 and @xmath241 to @xmath229 in @xmath228 gives us vertex - disjoint paths @xmath242 from @xmath243 to @xmath28 and @xmath244 from @xmath245 to @xmath229 . since @xmath211 is 2-connected , we can find vertex - disjoint paths from @xmath232 to @xmath243 and @xmath245 , which gives us the desired paths @xmath234 and @xmath235 . may not be in any cluster @xmath211 . in this case , @xmath234 is formed by using edges of @xmath35 from @xmath232 to @xmath237 , and then a path from @xmath240 to @xmath28 ; @xmath235 is formed similarly . ] we now present the two analyses of density referred to earlier . the key difference between the weaker and tighter analysis is in the way we bound edge costs . in the former , each large cluster pays for its edges separately , using the fact that all cycles used have density at most @xmath246 . in the latter , we crucially use the fact that small clusters which share edges are merged . roughly speaking , because small clusters are edge - disjoint , the average density of small clusters must be comparable to the density of the input graph @xmath2 . once an edge is in a large cluster , we can no longer use the edge - disjointness argument . we must pay for these edges separately , but we can bound this cost . first , the following lemma allows us to show that every large cluster has density at most @xmath205 . [ lem : tiercost ] for any cluster @xmath202 formed by mergeclusters during iteration @xmath95 , the total cost of edges in @xmath202 is at most @xmath247 . we prove this lemma by induction on the number of vertices in a cluster . let @xmath248 be the set of clusters merged using a cycle @xmath35 to form @xmath202 . let @xmath249 be the set of clusters in @xmath248 of tier @xmath95 , and @xmath250 be @xmath251 . ( @xmath250 contains clusters of tiers less or greater than @xmath95 that contained an edge of @xmath35 . ) the cost of edges in @xmath202 is at most the sum of : the cost of @xmath35 , the cost of @xmath249 , and the cost of @xmath250 . since all clusters in @xmath250 have been formed during iteration @xmath95 or earlier , and are smaller than @xmath202 , we can use induction to show that the cost of edges in @xmath250 is at most @xmath252 . all clusters in @xmath249 are of tier @xmath95 , and so must have been formed before iteration @xmath95 ( any cluster formed during iteration @xmath95 is of a strictly greater tier ) , so we use induction to bound the cost of edges in @xmath249 by @xmath253 . finally , because @xmath35 was a good - density cycle , and only clusters of tier @xmath95 contribute to calculating the density of @xmath35 , the cost of @xmath35 is at most @xmath254 . therefore , the total cost of edges in @xmath202 is at most @xmath255 . let @xmath202 be an arbitrary large cluster ; since we have only @xmath220 tiers , the previous lemma implies that the cost of @xmath202 is at most @xmath256 . that is , the density of @xmath202 is at most @xmath205 , and we can use this fact together with lemma [ lem : segment ] to find a solution to the rooted @xmath0-@xmath1vc problem of cost at most @xmath257 . this completes the ` weaker ' analysis , but this does not suffice to prove theorem [ thm : avekv ] ; to prove the theorem , we would need to use a large cluster @xmath202 of density @xmath206 , instead of @xmath205 . for the purpose of the more careful analysis , implicitly construct a forest @xmath258 on the clusters formed by mergeclusters . initially , the vertex set of @xmath258 is just @xmath13 , the set of terminals , and @xmath258 has no edges . every time a cluster @xmath202 is formed by merging @xmath200 , we add a corresponding vertex @xmath202 to the forest @xmath258 , and add edges from @xmath202 to each of @xmath203 ; @xmath202 is the parent of @xmath259 . we also associate a cost with each vertex in @xmath258 ; the cost of the vertex @xmath202 is the cost of the cycle used to form @xmath202 from @xmath203 . we thus build up trees as the algorithm proceeds ; the root of any tree corresponds to a cluster that has not yet become part of a bigger cluster . the leaves of the trees correspond to vertices of @xmath2 ; they all have cost 0 . also , any large cluster @xmath202 formed by the algorithm is at the root of its tree ; we refer to this tree as @xmath260 . for each large cluster @xmath202 after mergeclusters terminates , say that @xmath202 is of type @xmath95 if @xmath202 was formed during iteration @xmath95 of mergeclusters . we now define the _ final - stage _ clusters of @xmath202 : they are the clusters formed during iteration @xmath95 that became part of @xmath202 . ( we include @xmath202 itself in the list of final - stage clusters ; even though @xmath202 was formed in iteration @xmath95 of mergeclusters , it may contain other final - stage clusters . for instance , during iteration @xmath95 , we may merge several tier @xmath95 clusters to form a cluster @xmath136 of tier @xmath261 . then , if we find a good - density cycle @xmath35 that contains an edge of @xmath136 , @xmath136 will merge with the other clusters of @xmath35 . ) the _ penultimate _ clusters of @xmath202 are those clusters that exist just before the beginning of iteration @xmath95 and become a part of @xmath202 . equivalently , the penultimate clusters are those formed before iteration @xmath95 that are the immediate children in @xmath260 of final - stage clusters . figure 1 illustrates the definitions of final - stage and penultimate clusters . such a tree could be formed if , in iteration @xmath262 , 4 clusters of this tier merged to form @xmath263 , a cluster of tier @xmath264 . subsequently , in iteration @xmath95 , clusters @xmath6 and @xmath265 merge to form @xmath266 . we next find a good cycle containing @xmath267 and @xmath2 ; @xmath266 contains an edge of this cycle , so these three clusters are merged to form @xmath17 . note that the cost of this cycle is paid for the by the weights of @xmath267 and @xmath2 only ; @xmath266 is a tier @xmath264 cluster , and so its weight is not included in the density calculation . finally , we find a good cycle paid for by @xmath268 and @xmath35 ; since @xmath17 and @xmath263 share edges with this cycle , they all merge to form the large cluster @xmath202 . = [ circle , draw , inner sep=0pt , minimum size=6 mm ] ; = [ circle , draw , inner sep=0pt , minimum size=7 mm ] ; = [ font= ] ; \(y ) at ( 6,5.5 ) [ high ] @xmath202 ; \(a ) at ( 1.5,4 ) [ vertex ] @xmath95 ; ( b ) at ( 4.5,4 ) [ high ] @xmath269 ; ( c ) at ( 7.5,4 ) [ vertex ] @xmath95 ; ( d ) at ( 10.5,4 ) [ vertex ] @xmath264 ; ( a ) ( y ) ( b ) ; ( c ) ( y ) ( d ) ; at ( 0.7,4 ) @xmath268 ; at ( 3.6,4 ) @xmath17 ; at ( 6.7,4 ) @xmath35 ; at ( 9.7,4 ) @xmath263 ; \(e ) at ( 3,2.5 ) [ vertex ] @xmath95 ; ( f ) at ( 4.5,2.5 ) [ high ] @xmath264 ; ( g ) at ( 6,2.5 ) [ vertex ] @xmath95 ; ( e ) ( b ) ( f ) ; ( b ) ( g ) ; at ( 2.3,2.7 ) @xmath267 ; at ( 3.8,2.7 ) @xmath266 ; at ( 5.35,2.7 ) @xmath2 ; \(h ) at ( 3.5,1 ) [ vertex ] @xmath95 ; ( j ) at ( 5.5,1 ) [ vertex ] @xmath95 ; ( h ) ( f ) ( j ) ; at ( 2.9,1.3 ) @xmath6 ; at ( 4.9,1.3 ) @xmath265 ; at ( y ) [ vertex ] ; at ( b ) [ vertex ] ; at ( f ) [ vertex ] ; an edge of a large cluster @xmath202 is said to be a _ final edge _ if it is used in a cycle @xmath35 that produces a final - stage cluster of @xmath202 . all other edges of @xmath202 are called _ penultimate edges _ ; note that any penultimate edge is in some penultimate cluster of @xmath202 . we define the _ final cost _ of @xmath202 to be the sum of the costs of its final edges , and its _ penultimate cost _ to be the sum of the costs of its penultimate edges ; clearly , the cost of @xmath202 is the sum of its final and penultimate costs . we bound the final costs and penultimate costs separately . recall that an edge is a final edge of a large cluster @xmath202 if it is used by mergeclusters to form a cycle @xmath35 in the final iteration during which @xmath202 is formed . the reason we can bound the cost of final edges is that the cost of any such cycle is at most @xmath144 times the weight of clusters contained in the cycle , and a cluster does not contribute to the weight of more than one cycle in an iteration . ( this is also the essence of lemma [ lem : tiercost ] . ) we formalize this intuition in the next lemma . [ lem : final ] the final cost of any large cluster @xmath202 is at most @xmath270 , where @xmath271 is the weight of @xmath202 . let @xmath202 be an arbitrary large cluster . in the construction of the tree @xmath260 , we associated with each vertex of @xmath260 the cost of the cycle used to form the corresponding cluster . to bound the total final cost of @xmath202 , we must bound the sum of the costs of vertices of @xmath260 associated with final - stage clusters . the weight of @xmath202 , @xmath271 is at least the sum of the weights of the penultimate tier @xmath95 clusters that become a part of @xmath202 . therefore , it suffices to show that the sum of the costs of vertices of @xmath260 associated with final - stage clusters is at most @xmath144 times the sum of the weights of @xmath202 s penultimate tier @xmath95 clusters . ( note that a tier @xmath95 cluster must have been formed prior to iteration @xmath95 , and hence it can not itself be a final - stage cluster . ) a cycle was used to construct a final - stage cluster @xmath136 only if its cost was at most @xmath144 times the sum of weights of the penultimate tier @xmath95 clusters that become a part of @xmath136 . ( larger clusters may become a part of @xmath136 , but they do not contribute weight to the density calculation . ) therefore , if @xmath136 is a vertex of @xmath260 corresponding to a final - stage cluster , the cost of @xmath136 is at most @xmath144 times the sum of the weights of its tier @xmath95 immediate children in @xmath260 . but @xmath260 is a tree , and so no vertex corresponding to an penultimate tier @xmath95 cluster has more than one parent . that is , the weight of a penultimate cluster pays for only one final - stage cluster . therefore , the sum of the costs of vertices associated with final - stage clusters is at most @xmath144 times the sum of the weights of @xmath202 s penultimate tier @xmath95 clusters , and so the final cost of @xmath202 is at most @xmath270 . [ lem : penultimate ] if @xmath272 and @xmath273 are distinct large clusters of the same type , no edge is a penultimate edge of both @xmath272 and @xmath273 . suppose , by way of contradiction , that some edge @xmath85 is a penultimate edge of both @xmath272 and @xmath273 , which are large clusters of type @xmath95 . let @xmath274 ( respectively @xmath275 ) be a penultimate cluster of @xmath272 ( resp . @xmath273 ) containing @xmath85 . as penultimate clusters , both @xmath274 and @xmath275 are formed before iteration @xmath95 . but until iteration @xmath95 , neither is part of a large cluster , and two small clusters can not share an edge without being merged . therefore , @xmath274 and @xmath275 must have been merged , so they can not belong to distinct large clusters , giving the desired contradiction . [ thm : goodlargecluster ] after mergeclusters terminates , at least one large cluster has density at most @xmath206 . we define the _ penultimate density _ of a large cluster to be the ratio of its penultimate cost to its weight . consider the total penultimate costs of all large clusters : for any @xmath95 , each edge @xmath276 can be a penultimate edge of at most 1 large cluster of type @xmath95 . this implies that each edge can be a penultimate edge of at most @xmath220 clusters . therefore , the sum of penultimate costs of all large clusters is at most @xmath277 . further , the total weight of all large clusters is at least @xmath278 . therefore , the ( weighted ) average penultimate density of large clusters is at most @xmath279 , and hence there exists a large cluster @xmath202 of penultimate density at most @xmath280 . the penultimate cost of @xmath202 is , therefore , at most @xmath281 , and from lemma [ lem : final ] , the final cost of @xmath202 is at most @xmath270 . therefore , the density of @xmath202 is at most @xmath282 . theorem [ thm : goodlargecluster ] and lemma [ lem : segment ] together imply that we can find a solution to the rooted @xmath0-@xmath1vc problem of cost at most @xmath191 . this completes our proof of theorem [ thm : avekv ] . we list the following open problems : * can the approximation ratio for the @xmath0-@xmath1vc problem be improved from the current @xmath44 to @xmath283 or better ? removing the dependence on @xmath32 to obtain even @xmath284 could be interesting . if not , can one improve the approximation ratio for the easier @xmath0-@xmath1ec problem ? * can we obtain approximation algorithms for the @xmath0-@xmath8vc or @xmath0-@xmath8ec problems for @xmath285 ? in general , few results are known for problems where vertex - connectivity is required to be greater than 2 , but there has been more progress with higher edge - connectivity requirements . * given a 2-connected graph of density @xmath146 with some vertices marked as terminals , we show that it contains a non - trivial cycle with density at most @xmath146 , and give an algorithm to find such a cycle . we have also found an @xmath37-approximation for the problem of finding a minimum - density non - trivial cycle . is there a constant - factor approximation for this problem ? can it be solved _ exactly _ in polynomial time ? b. awerbuch , y. azar , a. blum and s. vempala . new approximation guarantees for minimum weight @xmath0-trees and prize - collecting salesmen . , 28(1):254262 , 1999 . preliminary version in _ proc . of acm stoc _ , 1995 . m. x. goemans and d. p. williamson . the primal - dual method for approximation algorithms and its application to network design problems . in d. s. hochbaum , editor , _ approximation algorithms for np - hard problems_. pws publishing company , 1996 .

in the @xmath0-@xmath1vc problem , we are given an undirected graph @xmath2 with edge costs and an integer @xmath0 ; the goal is to find a minimum - cost 2-vertex - connected subgraph of @xmath2 containing at least @xmath0 vertices . a slightly more general version is obtained if the input also specifies a subset @xmath3 of _ terminals _ and the goal is to find a subgraph containing at least @xmath0 terminals . closely related to the @xmath0-@xmath1vc problem , and in fact a special case of it , is the @xmath0-@xmath1ec problem , in which the goal is to find a minimum - cost 2-edge - connected subgraph containing @xmath0 vertices . the @xmath0-@xmath1ec problem was introduced by lau _ et al . _ @xcite , who also gave a poly - logarithmic approximation for it . no previous approximation algorithm was known for the more general @xmath0-@xmath1vc problem . we describe an @xmath4 approximation for the @xmath0-@xmath1vc problem .

0802.2528

one of the most promising solutions to the hierarchy problem is the randall - sundrum ( rs ) model @xcite . in this model there is a single extra dimension compactified on @xmath1 with the non - factorizable metric of @xmath2 @xmath3 here @xmath4 is the extra - dimensional coordinate , and @xmath5 is the inverse of the @xmath6 curvature , @xmath7 . two branes define the boundaries of the extra dimension . one , at @xmath8 , is called the uv , or planck , brane . the other , at @xmath9 , is the ir , or tev , brane . picking @xmath10 , which is natural in realistic stabilization mechanisms , solves the hierarchy problem @xcite . in the original rs model the sm was confined to the ir brane , and only gravity propagated in the bulk @xcite . it has since been realized that both gauge and fermion fields can live in the bulk in a realistic model @xcite . these models are realistic , but the parameter space can be strongly reduced by precision electroweak constraints . much of this problem can be traced to the fact that the massive gauge fields receive a contribution to their mass from the bulk geometry which does not respect the custodial @xmath11 . this can be fixed by expanding the gauge group to @xmath12 , which dramatically improves the electroweak fit @xcite . the breaking of this extended electroweak symmetry proceeds in two stages : on the uv brane @xmath13 ; on the ir brane @xmath14 , where @xmath15 is the diagonal of the @xmath16 groups . this paper investigates the properties of the higgs sector that accomplishes this breaking . we now ask what drives the breaking on each brane . on the planck brane all degrees of freedom will have planck scale masses , so we can ignore them . we can then implement the breaking with boundary conditions to good approximation . this leads to the boundary conditions at @xmath8 @xmath17 here @xmath18 and @xmath19 are ratios of 5d gauge couplings : @xmath20 , and @xmath21 . on the tev brane , the masses will be tev scale , so we should look at the higgs sector in detail . the simplest structure that will create the breaking pattern is a real higgs that is a bidoublet under @xmath22 . this leads to the boundary conditions at @xmath9 @xmath23 note that in the @xmath24 limit we obtain the usual higgsless boundary conditions , and this model reduces to the higgsless model in @xcite . we will use this parameter , @xmath25 to interpolate between the sm limit ( @xmath26 ) , and the higgsless limit . [ fig : wroot ] to write down the effective 4d theory , we expand the 5d fields into kaluza klein ( kk ) fields , @xmath27 we can now obtain the gauge boson wavefunctions by solving the equation of motion subject to the boundary conditions ( [ eq : gaugebcr ] ) and ( [ eq : gaugebcrp ] ) . this produces a spectrum of eigenvalues corresponding to the excitations of the gauge fields . the lowest masses in each of the charged and neutral sectors will correspond to the @xmath28 and @xmath29 bosons . the neutral sector also contains a zero mode , corresponding to the photon . fig [ fig : wroot ] shows the eigenvalue for the @xmath28 as a function of the parameter @xmath25 . [ fig : wcoup ] one interesting feature of this model is that the @xmath28 and @xmath29 wavefunctions are suppressed near the ir brane , as can be seen by inspecting the boundary conditions . this suppression increases for increasing @xmath25 . this means that the coupling of massive gauge bosons to the higgs will generically be suppressed . [ fig : wcoup ] shows the coupling of the @xmath28 to the higgs . for values of @xmath25 near unity the lep bounds on the higgs mass can be dramatically reduced . ( for larger values of @xmath25 the model is effectively higgsless . ) the fermion sector of this model is more complicated . again , the higgs vev induces mixed boundary conditions that link left and right handed fields to give the fermions masses . however , there are two new degrees of freedom . first , since 5d fermions are achiral , ther can always be a mass term in the bulk @xmath30 . the main effect of this term is to shift the location of the fermion zero mode in the bulk . by changing this parameter we can cause the zero mode to be localized either near the uv or ir brane , and also can change the degree of this localization . this allows us to control the overlap of the zero mode with the ir brane , and consequently the strength with which the fermion interacts with the higgs . in this way the hierarchy of fermion masses can be generated by order one changes in the 5d masses . the second complication arises from the @xmath31 symmetry which enforces that , for example , @xmath32 if unbroken . this mass relation can be modified by mixing with new fermions localized to the planck brane , where the @xmath31 is broken . for full details , see @xcite . note that there are tree - level corrections to precision electroweak observables , coming largely from the kk excitations of the gauge bosons . unfortunately , the magnitude of these corrections is highly sensitive to the configuration of the fermion sector . for the specific configuration studied in @xcite we find the constraint @xmath33 . there are , however , special points in the fermion parameter space where the constraint becomes trivial , so a wide range of @xmath25 should be considered . the final interesting shift in higgs properties is in the couplings to massless gauge bosons , _ i.e. _ gluons and photons . the coupling of the higgs to gluon pairs is induced through top loops . however , in this model the kk excitations also couple to the higgs . furthermore , note that we have arranged small 4d yukawa couplings for the other fermions by small wavefunction overlaps with the would - be zero modes . the 5d yukawa couplings are all order 1 , and the excited states have _ no _ wavefunction suppression . hence there are large contributions to the higgs - glue - glue coupling from the kk excitations of _ all _ colored fermions . this leads to an enhancement in that coupling , as seen in fig . [ fig : hgg ] . there are similar corrections to the higgs - gamma - gamma coupling . the situation there is more complicated , however , since there are also contributions from @xmath28 boson loops , which are dominant in the sm , and the higgs coupling to @xmath28s is suppressed . we can now look at the behavior of the higgs branching ratios , as shown in fig . [ fig : widthbr ] . note in particular the dominance of @xmath34 over a wide range , and the late onset of @xmath35 . this is driven by the @xmath31 symmetry , which gives a large enhancement of the @xmath36-quark yukawa , and the suppression of the gauge boson couplings . note also the reduction in the @xmath37 mode . this will make discovery at the lhc difficult . the suppression of the coupling to gauge bosons also means that production at the ilc will be reduced , making this higgs a particularly difficult one to find . however , it will be essential to measure the higgs couplings with precision to identify a warped extra dimension as the correct theory of new physics , if indeed this is what is realized in nature . 9 b. lillie , arxiv : hep - ph/0505074 . l. randall and r. sundrum , phys . lett . * 83 * , 3370 ( 1999 ) [ arxiv : hep - ph/9905221 ] . w. d. goldberger and m. b. wise , phys . * 83 * , 4922 ( 1999 ) [ arxiv : hep - ph/9907447 ] . h. davoudiasl , j. l. hewett and t. g. rizzo , phys . lett . * 84 * , 2080 ( 2000 ) [ arxiv : hep - ph/9909255 ] . h. davoudiasl , j. l. hewett and t. g. rizzo , phys . b * 473 * , 43 ( 2000 ) [ arxiv : hep - ph/9911262 ] and phys . d * 63 * , 075004 ( 2001 ) [ arxiv : hep - ph/0006041 ] ; a. pomarol , phys . b * 486 * , 153 ( 2000 ) [ arxiv : hep - ph/9911294 ] ; y. grossman and m. neubert , phys . b * 474 * , 361 ( 2000 ) [ arxiv : hep - ph/9912408 ] ; t. gherghetta and a. pomarol , nucl . b * 586 * , 141 ( 2000 ) [ arxiv : hep - ph/0003129 ] . k. agashe , a. delgado , m. j. may and r. sundrum , jhep * 0308 * , 050 ( 2003 ) [ arxiv : hep - ph/0308036 ] . c. csaki , c. grojean , h. murayama , l. pilo and j. terning , phys . d * 69 * , 055006 ( 2004 ) [ arxiv : hep - ph/0305237 ] . c. csaki , c. grojean , l. pilo and j. terning , phys . * 92 * , 101802 ( 2004 ) [ arxiv : hep - ph/0308038 ] . g. cacciapaglia , c. csaki , c. grojean and j. terning , phys . d * 70 * , 075014 ( 2004 ) [ arxiv : hep - ph/0401160 ] and phys . d * 71 * , 035015 ( 2005 ) [ arxiv : hep - ph/0409126 ] ; g. cacciapaglia , c. csaki , c. grojean , m. reece and j. terning , arxiv : hep - ph/0505001 . y. nomura , jhep * 0311 * , 050 ( 2003 ) [ arxiv : hep - ph/0309189 ] . r. barbieri , a. pomarol and r. rattazzi , phys . b * 591 * , 141 ( 2004 ) [ arxiv : hep - ph/0310285 ] . h. davoudiasl , j. l. hewett , b. lillie and t. g. rizzo , phys . d * 70 * , 015006 ( 2004 ) [ arxiv : hep - ph/0312193 ] and jhep * 0405 * , 015 ( 2004 ) [ arxiv : hep - ph/0403300 ] ; j. l. hewett , b. lillie and t. g. rizzo , jhep * 0410 * , 014 ( 2004 ) [ arxiv : hep - ph/0407059 ] ; r. s. chivukula , e. h. simmons , h. j. he , m. kurachi and m. tanabashi , phys . d * 70 * , 075008 ( 2004 ) [ arxiv : hep - ph/0406077 ] and phys . b * 603 * , 210 ( 2004 ) [ arxiv : hep - ph/0408262 ] . a. birkedal , k. matchev and m. perelstein , phys . * 94 * , 191803 ( 2005 ) [ arxiv : hep - ph/0412278 ] . c. csaki , c. grojean , j. hubisz , y. shirman and j. terning , phys . d * 70 * , 015012 ( 2004 ) [ arxiv : hep - ph/0310355 ] .

we study the corrections to higgs physics in a model of a single warped extra dimension with all fields except the higgs in the bulk , and a gauge symmetry extended to @xmath0 . we find that generically the higgs coupling to electroweak gauge boson pairs is suppressed , the coupling to gluons is enhanced , and the coupling to photons is often suppressed , but can be enhanced .

hep-ph0508266

essentially all astronomical measurements are performed via electromagnetic waves . the availability of accurate gravitational wave measurements within the next decade or so will thus be a significant development for astronomy . in particular , since the propagation of photons and gravitons could differ at a fundamental level , gravitational waves emitted by cosmologically - distant `` space - time sirens , '' such as coalescing pairs of massive black holes , could be used as valuable new probes of physics on cosmological scales . black holes with masses @xmath0 are present at the center of numerous nearby galaxies ( e.g. * ? ? ? * ; * ? ? ? as such galaxies collide over cosmic times , their central black holes coalesce , releasing @xmath1 ergs of binding energy in the form of gravitational waves ( hereafter gws ) . to measure the gws emitted by these cosmologically - distant space - time sirens , esa and nasa will build the laser interferometer space antenna , lisa . gws emitted by black hole binaries have the unfamiliar property of providing a direct measure of the luminosity distance , @xmath2 , to the black holes , without extrinsic calibration . owing to the highly coherent nature of gw emission @xcite , the amplitude ( or strain ) , @xmath3 , frequency , @xmath4 , and frequency derivative , @xmath5 , of the leading order ( quadrupolar ) gw inspiral signal scale as @xmath6^{5/3 } f^{2/3}}{d_l } , \\ \dot f ( t ) & \propto & \left [ ( 1+z ) m_c \right]^{5/3 } f^{11/3},\end{aligned}\ ] ] where @xmath7 represents the two transverse gw polarizations , @xmath8 is the black hole pair `` chirp '' mass and @xmath9 its redshift . provided the gw source can be reasonably well localized on the sky , an extended observation of the chirping signal leads to precise measurements of @xmath3 , @xmath4 , @xmath5 and thus @xmath2 , independently . as illustrated in fig . [ fig : one ] , lisa s orbital configuration allows for a `` triangulation '' of gw sources on the sky , to within a solid angle @xmath10 deg@xmath11 typically @xcite . this permits very accurate measurements , e.g. distances with errors @xmath12 at @xmath13 typically @xcite . masses are independently determined to very high accuracy ( typically @xmath14 ; e.g. , @xcite ) = in principle , the same sky localization that helps determine the distance to a source accurately can be used to find the host galaxy of a pair of merging black holes seen by lisa . the secure identification of the host galaxy would enable a wide variety of new galactic black hole studies ( see [ sec : newbh ] ) . initially , the prospects for finding the host galaxy of a pair of merging black holes were considered to be poor , simply because of the large number of galactic candidates located in the @xmath10 deg@xmath11 lisa sky error - box ( e.g. , * ? ? ? * ; * ? ? ? * ) recently , however , this possibility has been reconsidered , with more optimistic conclusions @xcite . given a cosmology , it is possible to translate the accurate luminosity distance measurement to the gw source into a narrow redshift slice in which the host galaxy must be located @xcite . various contributions to the redshift errors that arise in performing this conversion are shown in fig . [ fig : two ] , for a representative equal - mass binary , as a function of the gw source redshift @xcite . at redshifts @xmath15 , where most black hole binary sources are expected to be found , weak lensing errors due to line - of - sight inhomogeneities ( on top of the smooth average cosmology ) are the main limitation to an accurate determination of the redshift slice in which the host galaxy ought to be located . @xcite have studied in detail the possibility that the three - dimensional information available ( sky localization + redshift slice ) could be used to single out a quasar , or any other unusually rare object ( such as a star - bust galaxy ) , in the lisa error box , after coalescence . finding such a statistically rare object post - merger would make it a good host galaxy candidate for the newly - coalesced pair of black holes . = however , it maybe much more advantageous to use a pre - merger strategy to identify the host galaxy of a pair of coalescing black holes seen by lisa . indeed , one can use near real - time gw information on the sky localization , in combination with the accurate timing of the inspiral event , to predetermine well in advance where on the sky the merger is located . a unique host galaxy identification could then proceed through coordinated observations with traditional telescopes , by monitoring in real time the sky area for unusual electromagnetic emission , as the coalescence proceeds . a variety of mechanisms exist through which disturbed gas in the vicinity of black hole pairs will power electromagnetic emission during and after coalescence @xcite . for example , at the time of coalescence , @xmath16 ergs of kinetic energy are delivered to the recoiling black hole remnant and its environment , for typical recoil velocities @xmath17 km / s ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) . this may lead to detectable signatures @xcite and permit the coincident identification of a unique host galaxy . the detailed nature of such electromagnetic counterparts remains largely unknown , however . to a large extent , lisa s ability to localize a long - lived source on the sky is related to the gw signal being modulated as a result of the detector s revolution and change of orientation when the constellation orbits around the sun ( fig . [ fig : one ] ) . even though most of the gw snr accumulates during the final stages of inspiral / coalescence for typical gw sources , reasonably good information on sky localizations must be available well before final coalescence since this information accumulates slowly , over the long signal modulation ( orbital ) timescale . because of significant cross - correlations between sky localization and distance errors , it turns out that this argument is also largely valid for luminosity distance errors @xcite . figure [ fig : three ] shows the pre - merger time evolution of luminosity distance and angular sky localization errors for a representative black hole pair at @xmath18 . errors improve quickly at early times but their evolution slows down considerably at late times . according to both panels , even accounting for random orientations of various source and detector angles ( shown as best , typical and worst cases ) , significant information is available days to weeks prior to the final coalescence @xcite . including black hole spins in the analysis has been shown to result in significant improvements on the errors during the last few days to hours prior to coalescence @xcite . with the expected availability , by the time lisa is operational , of sensitive large field - of - view ( fov ) astronomical instruments for weak lensing and supernova studies , it becomes interesting to estimate the amount of time prior to merger during which the lisa sky localization falls within the fov of such an instrument . when this happens , continuous monitoring of the designated sky area , until final coalescence , becomes possible . @xcite have performed a detailed analysis of this possibility , using lsst and its @xmath19 deg@xmath11 fov as a reference . figure [ fig : four ] shows results for representative equal - mass binaries , as a function of their total mass and redshift . the various contours show that prospects for electromagnetic monitoring days to weeks before the coalescence are good for sources at redshifts @xmath13@xmath20 . monitoring for the best gw sources out to @xmath21-@xmath22 may even be possible @xcite . a large variety of new galactic black hole astrophysics would be enabled by successful identifications of the host galaxies of coalescing black hole pairs . we mention only a few possibilities here and refer the interested reader to @xcite and @xcite for additional discussions . from the black hole masses , spins and binary orientation , all accurately constrained by the gw signal , one would be able to study the physics of the post - merger accretion flow onto the remnant black hole @xcite with unprecedented accuracy . this would include precise constraints on the eddington ratio of the accreting source , its emission and absorption geometries and possibly its jet phenomenology . similarly , studies of the galactic host might tell us about the nature ( dry / wet ) and the timing of the galactic merger that resulted in the black hole binary coalescence . finally , measuring velocity dispersions , @xmath23 , for several host galaxies , together with the black hole masses known from the gw signal , would allow us to accurately map the evolution of the @xmath24@xmath23 relation with cosmic time , at least for such transitional objects as the hosts of coalescing black hole pairs . another consequence of successfully identifying the host galaxies of coalescing black hole pairs is the possibility to draw a gravitational hubble diagram , i.e. one that relates the gravitational luminosity distances , @xmath2 , of these gw sources to the electromagnetic redshifts , @xmath9 , of their host galaxies . one of the main interests of a gravitational hubble diagram arises from its immunity to common systematics affecting electromagnetic measurements . indeed , a gravitational hubble diagram , which is based on gravitational distance measurements with self - calibrated sources , is not susceptible to any significant bias from absorption , scattering or reddening of gws . in practice , however , the value of such a diagram is limited by line - of - sight matter inhomogeneities , which generate weak lensing uncertainties on the gravitational @xmath2 measurement ( see fig . [ fig : two ] ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? while the lensing effect can in principle be averaged out over many random lines - of - sights , it may not be possible to do so for coalescing pairs of massive black holes if lisa merger event rates are modest ( e.g. , a few tens per year at @xmath25 ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? weak lensing errors on individual measurements amount to distance uncertainties ranging from @xmath26 at @xmath27 to @xmath28 at @xmath29 ( e.g. * ? ? ? * ) , which makes a gravitational hubble diagram imprecise even at moderate redshifts . the extent to which lisa events can be used to draw a meaningful hubble diagram will thus depend strongly on the actual distribution of massive black hole merger events with redshifts and the corresponding efficiency of host galaxy identifications . as we describe in [ sec : wd ] , however , white dwarf spiraling into massive black holes may offer a practical avenue for precision cosmology with lisa . the possibility that the accelerated expansion of the universe results from a failure of general relativity has fueled much theoretical work on large scale modifications of gravity over the past few years . since building a satisfactory theory of modified relativistic gravity is a formidable task , any insight that can be gained from direct observational constraints on the linearized gw regime can not be overlooked . lisa , with its ability to measure the gw signal from cosmologically - distant sources , may thus be one of our best probes of modified gravity on cosmological scales @xcite . one may expect gravity modifications to contain a new length scale , @xmath30 , beyond which gravity deviates from general relativity . in order to explain the observed accelerated expansion of the universe , this scale is expected to be of the order of the current hubble radius , @xmath31 . an existence proof of modifications of this type is given by dgp gravity @xcite , a braneworld model with an infinite extra dimension . @xcite discuss the possibility that extra - dimensional leakage of gravity in dgp - like scenarios may lead to cosmologically - distant gw sources appearing dimmer than they truly are , from the loss of gw energy flux to the extra - dimensional bulk . indeed , in the presence of large distance leakage , flux conservation over a source - centered hypersphere requires that the gw amplitude scales with distance @xmath32 from the source as @xmath33 where @xmath34 is the total number of space - time dimensions accessible to gravity modes . thus , for @xmath35 , it deviates from the usual @xmath36 scaling . in principle , black hole merger events and associated host galaxies could thus reveal the leakage of gravity over scales of order a few hubble distances , by comparison to purely electromagnetic hubble diagrams , which are immune to such leakage effects . this is only one of several possible modified gravity signatures in the gws from cosmologically - distant sources @xcite . another class of signatures is related to the gw polarization signal , with possibly additional polarizations beyond the two transverse quadrupolar ( @xmath7 ) modes of general relativity ( e.g. , * ? ? ? . signatures also exist in relation to the gw propagation velocity which , in modified gravity scenarios , can differ from the speed of light . in this respect , the possibility to time a cosmological gw , relative to an electromagnetic signal causally associated with the black hole merger , may offer unique diagnostics of large - scale modified gravity . this could reveal , for instance , that the phase of the gw signal deviates from general relativistic expectations , once propagated over cosmological distances . @xcite have explored further the possibilities of measuring photon and graviton arrival times from a same cosmological source . a general difficulty with this approach is that there will be a systematic and a priori unknown delay in the emission of photons , relative to the emission of gravitons , since the former must causally lag behind the perturbing gravitational event . this difficulty could be overcome if it were possible to calibrate the relative timing of the photon and graviton signals at the source . prior to coalescence , gas present in the near environment of the black hole binary would be gravitationally perturbed in such a way that it could radiate a variable electromagnetic signal with a period closely matching that of the leading - order quadrupolar perturbation induced by the coalescing binary ( see fig . [ fig : five ] ) . this would help identify the electromagnetic counterparts of specific gw events . in addition , it may be possible to match the variability frequencies of the electromagnetic and gw signals . the offset in phase between the fourier components of the two signals with similar frequencies could be used to effectively calibrate the intrinsic delay in electromagnetic emission at the source . late inspiral and coalescence can be tracked via the gw signal , so that the relative timing of the gravitational and electromagnetic signals may be known to within a fraction of the binary s orbital time . any drift in arrival - time with frequency between the gravitational and electromagnetic chirping signals , as the source spans about a decade in gw frequency during the last 2 weeks before merger , could then be attributed to a fundamental difference in the way photons and gravitons propagate over cosmological distances . for instance , such a drift could occur if the graviton is massive , resulting in a frequency - dependent propagation velocity ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? this tracking possibility is illustrated graphically in fig . [ fig : five ] . interestingly , while lorentz invariance has been extensively tested for standard model fields , lorentz symmetry could be violated in the gravity sector , especially on cosmological scales ( e.g. , * ? ? ? * ; * ? ? ? * ) . with a good enough understanding of the source , electromagnetic counterparts to black hole binary mergers may offer unique tests of lorentz violations in the gravity sector , via the opportunity to match and track the gravitational and electromagnetic signals in frequency and phase . it may be possible , as the black hole binary decays toward final coalescence , spanning a range of frequencies , to measure the delays in graviton vs. photon arrival times as a function of increasing frequency of the chirping signal . the consistency expected if lorentz symmetry is satisfied in the gravity sector could be tested explicitly for gravitons propagated over cosmological scales . to have any chance to perform such new tests of gravitational physics , one will need to identify the electromagnetic counterparts of coalescing pairs of massive black hole binaries as early as possible . this may be one of the strongest motivations behind ambitious efforts to localize these rare , transient events well before final coalescence . as seen before , a large number of sources must be accumulated to turn a gravitational hubble diagram into a high precision tool for cosmology . white dwarf frequently spiraling into ( moderately ) massive black holes may offer unique opportunities in this respect . an additional goal of the lisa mission is the detection of gws emitted by compact objects being captured by massive black holes ( the so - called extreme mass ratio inspirals , or emris ) . @xcite present a detailed parameter estimation analysis for this class of lisa events and show that they could be detected with good snr out to a distance @xmath37 gpc . in addition , these authors found that the sky localization errors for these events ( @xmath38 steradians at @xmath37 gpc ) are comparable to the case of black hole binary merger events . no electromagnetic counterpart is expected from emris of dense neutron stars or stellar - mass black holes . on the other hand , partial ( or total ) disruptions of inspiraling low - density white dwarfs ( wds ) could produce such counterparts and thus help identify host galaxies for such events . to the best of our knowledge , the possibility to find the electromagnetic counterparts of this subclass of lisa emris and use them for precision cosmology has not been previously discussed in the literature . as a result of its relatively low density , a typical wd suffers complete disruption around a non - spinning black hole of mass @xmath39 . this mass limit is increased to perhaps @xmath40 if the black hole is spinning rapidly @xcite . even if the wd were not disrupted , some level of partial shedding of its lower density outer layers ( e.g. * ? ? ? * ) may be expected as the inspiral proceeds . as the stream of debris from a partially stripped wd shocks against itself or the inspiraling wd ( e.g. , * ? ? ? * ) , some level of electromagnetic emission during such emris is expected . a tidally - triggered detonation of the wd is yet another possibility @xcite . all these arguments point to the possibility that electromagnetic counterparts may exist for a subset of all the emris detected by lisa ( the one corresponding to wd inspirals ) . the fraction of all emris which involve wds is , indeed , expected to be substantial @xcite . if electromagnetic counterparts for a subset of all lisa emris can be detected , the rewards will be significant . just like black hole binary merger events , emris with uniquely identified counterparts and host galaxies could be used to draw a gravitational hubble diagram , with the significant advantages that , at redshifts as low as @xmath41@xmath42 , weak lensing due to line - of - sight inhomogeneities would be small or negligible ( fig . [ fig : two ] ) and that the @xmath2@xmath9 relation at these low redshifts is strongly sensitive to the dominant dark energy content . dark energy becomes dynamically significant , affecting the expansion rate and geometry of the universe , and modifying @xmath2 , at @xmath43 ( see , e.g. * ? ? ? * for a review ) . indeed , as emphasized by @xcite , once cosmic microwave background anisotropies are measured by planck , @xmath2 will be known accurately at @xmath44 , and a low redshift ( @xmath45 ) measurement will provide the best complement to constrain dark energy parameters . wd emris detections with lisa will occur frequently , and they could perhaps be singled out on the basis of the comparatively low mass of the inspiraling compact object . it remains to be determined how the lisa sky localization error evolves with pre - merger ( or pre - disruption ) time for such events and what is the nature of their electromagnetic counterparts . but altogether , the various advantages that we have outlined point to the need for a detailed assessment of the potential use of wd emris for precision cosmology with lisa . for centuries , astronomers have measured distances exclusively with light . direct gravitational measurements , gravitational hubble diagrams and comparisons between the propagation of electromagnetic and gravitational signals offer fundamentally new ways to probe physics on cosmological scales . the novelty involved in joint , time - constrained electromagnetic and gravitational measurements will require that special efforts be made to reach out across the gw and astronomy communities .

the notion that microparsec - scale black holes can be used to probe gigaparsec - scale physics may seem counterintuitive , at first . yet , the gravitational observatory _ lisa _ will detect cosmologically - distant coalescing pairs of massive black holes , accurately measure their luminosity distance and help identify an electromagnetic counterpart or a host galaxy . a wide variety of new black hole studies and a gravitational version of hubble s diagram become possible if host galaxies are successfully identified . furthermore , if dark energy is a manifestation of large - scale modified gravity , deviations from general relativistic expectations could become apparent in a gravitational signal propagated over cosmological scales , especially when compared to the electromagnetic signal from a same source . finally , since inspirals of white dwarfs into massive black holes at cosmological distances may permit pre - merger localizations , we suggest that careful monitoring of these events and any associated electromagnetic counterpart could lead to high - precision cosmological measurements with _ lisa_.

0803.3627

narrow - line seyfert 1 galaxies ( nls1s ) are a special and interesting group of active galactic nuclei ( agns ) . they show narrow optical balmer emission lines [ fwhm(h@xmath0 ) @xmath1 2000 km s@xmath2 , weak [ ] @xmath35007 emission ( [ ] /h@xmath4 ) , strong emission , and soft x - ray excess @xcite . nls1s show remarkable radio - loud / radio - quiet bimodality @xcite . only @xmath5 of nls1s are radio - loud objects @xcite . the fraction is much smaller than that found in qsos . very radio - loud nls1s ( rl - nls1s , @xmath6 ) are even much fewer ( @xmath7 ) @xcite , where the radio loudness @xmath8 is commonly defined as the flux ratio of radio to optical at @xmath9 @xcite . so far , it is still a puzzle why rl - nls1s are so scarce . at present , the origin of rl - nls1s is also still poorly understood . a few efforts have been made in the past few years to understand the nature of rl - nls1s . yuan et al . ( 2008 ) found that the broadband spectra of some rl - nls1s are similar to those of high - energy - peaked bl lac objects , and suggested that some of them may be bl lac objects actually . basing upon the recent observation taken by _ satellite , some rl - nls1s display a hard x - ray component suggesting the presence of relativistic jets on the line of sight @xcite . the presence of the relativistic jets motivates us to search for intranight optical variability in some rl - nls1s , because of the well - known beaming effect ( e.g. , wagner & witzel 1995 ) . @xcite argued that sdssj094857.3 + 002225 is a right candidate for searching for rl - nls1s with beaming effect . the object is a very radio - loud nls1 at @xmath10 . the reported radio loudness derived from the radio flux at 5 ghz ranges from 194 to 1982 @xcite . it is in the crates catalog as a flat - spectral radio source @xcite . the simultaneous observations taken by both _ swift _ and _ fermi _ also suggest that the broadband spectral energy distribution is similar to those of flat - spectral radio quasars @xcite . recent photometry from the guide star catalogs 2.21 is @xmath11 mag @xcite . previous studies revealed multi - wavelength variabilities in the object at timescales from day to year . previous radio observations indicate its fluctuation in the radio band on the timescale from weeks to years @xcite . @xcite also said that the object shows long - term variability in both the radio and optical bands . the amplitude of the variation in the radio can be @xmath12 within a year . the long - term variability amplitude may be about 1 mag in the optical band . the latest multi - wavelength campaign carried out by @xcite discovered an optical variability on day timescales . dramatic flux variabilities in both x - rays and radio 37 ghz were also found in the study . in this letter , we report an optical monitor for the rl - nls1 sdssj094857.3 + 002225 . the monitor was designed to search for intranight optical variability ( inov ) in the object . the inov should be detected if the object indeed hosts a relativistic jet beaming toward the observers . our observations were carried out at the xinglong observatory of national astronomical observatories , chinese academy of sciences ( naoc ) , using the 80 cm tnt telescope . the telescope is a cassegrain system with a @xmath13 beam . a liquid nitrogen cooled pi va1300b 1300@xmath141340 ln ccd was used as the detector that covers @xmath1511 arcmin@xmath16 of the sky . each pixel of the ccd corresponds to @xmath170.5 arcsec@xmath16 . gain and readout noise of the ccd is 2.3 electrons adu@xmath18 and 5 electrons , respectively . the standard johnson _ b_- and _ r_-band filters were used in the observations . we monitored the object on seven moonless nights in 2009 . they are february 27 , march 1 , 5 , and april 2426 , and 28 . the typical exposure time is 600 s for each frame . continuous monitoring for this object was run as long as possible in each night . the sky flat - field frames in both _ b _ and _ r _ passbands were obtained before and after each observation run during the twilight time . dark frames are not needed because the temperature of the detector is so low ( @xmath19c ) that the dark electrons can be entirely ignored . the observed data are preliminarily reduced through the standard routine by iraf package , including bias and flat - field corrections . several bright comparison stars are selected from the same ccd frame to calculate differential light curve . because the comparison stars are brighter than the object , several check stars with brightness comparable to the object are selected to assess the errors in photometry . the instrumental magnitudes of the object and of those selected stars are calculated by the apphot task . the aperture photometry is adopted because the object is a point - like source without extended emission . in each frame , the fwhm of the object is comparable with those of the field stars . the circular aperture radius twice of the mean fwhm of the field stars was therefore adopted in our calculations . all the results reported below are based on these radii . our observations can be divided into two parts . both of them contain about 1 week . the source was well monitored on the nights of 2009 february 27 , march 1 , march 5 , april 25 , and april 28 . ( the corresponding dates on the time - axis of figure 1 are 3345 , 3347 , 3351 , 3402 , and 3405 , respectively . ) there were no or only scarce data on the other nights because of the bad weather . the intrinsic brightness of the comparison stars was obtained by the formulae given by lupton ( 2005 ) and the sloan digital sky survey ( sdss ) database was used . then the apparent magnitudes of the object can be calculated from the differential instrumental magnitudes . the light curves of the observations are plotted in figure 1 . the upper two light curves show the variation of the object in the _ b _ ( by blue solid squares ) and _ r _ ( by red solid circles ) bands . the corresponding variations of the comparison stars are plotted by the bottom to light curves . the fluctuations of the comparison stars are not larger than 0.05 mag . the error bars overplotted on the light curve are estimated from the selected check stars with brightness comparable to that of the object . in addition to a long - term variation with amplitude about 1 mag , the obtained light curves indicate that there were several nights during which the inov can be clearly identified in the object in both the _ b _ and _ r _ bands . the variations in both bands are similar to each other . the amplitudes of the rapid variations are so large that the short - term variability is quite obvious on 2009 march 1 , 5 , and april 25 . in particular , the weather was relatively good on 2009 march 1 and april 25 , which results in relatively smaller error bars . for example , the typical error bars on april 25 are 0.05 mag and 0.02 mag in the _ b _ and _ r _ bands , respectively . the brightness of the object changes about 0.50.6 mag in both bands within several hours on the same night . the inset in figure 1 shows the details of the variation within 4 hr on the night of april 25 . although the errors are relatively large on the nights of march 5 and april 28 because of the relatively poor weather , the presence of inov can still be identified from the observations . when we do aperture photometry , there is a problem that whether the contamination from the host galaxy of the target agn contributes to the light variability . some authors argued that the fluctuations in the seeing may result in spurious variable contributions from the host galaxy within the photometric aperture , especially when the apertures are small @xcite . we argue that the contamination from the host galaxy is not important in the current study . first of all , no clear features of the host galaxy could be identified from the images taken by sdss , likely because the object is far away from us ( at @xmath20 ) . thus , the host galaxy is much fainter than the agn . second , as described above , the photometry apertures we adopted in this study are twice of the fwhm of field stars , which is large enough to include most of the emission from the underlying host galaxy . the particular rl - nls1 galaxy sdssj094857.3 + 002225 was monitored in optical bands by naoc 80 cm tnt telescope to search for its inov phenomenon . our optical monitoring indeed provides clear evidence for the presence of inov in both the _ b _ and _ r _ bands in the object . the object exhibits optical variability not only on the timescale of a week , but also on several hours . the detection of the inov indicates that the object contains a relativistic jet on the line of sight of an observer , which confirms the conclusion drawn from the high - energy observations ( e.g. , abdo et al . 2009a , 2009b ) and from the inverted radio spectrum and high brightness temperature ( zhou et al . 2003 ) . sdssj094857.3 + 002225 is particular for its observational properties . on the one hand , its optical spectrum with strong emission is typical of nls1s . the narrow h@xmath0 emission yields a relatively low black hole ( bh ) mass @xmath21 and a high eddington ratio ( zhou et al . 2003 ) . on the other hand , some observational behaviors are characteristic of blazars with the relativistic jets close to the line of sight , such as the inov detected here , flat radio spectrum , high brightness temperature , and variable @xmath22-ray emission ( see citations in section 1 ) . so far , outstanding rl - nls1s with blazer - like radio emission have been revealed by multi - wavelength observations in several cases including the object sdssj094857.3 + 002225 . we refer the reads to yuan et al . ( 2008 ) for a brief summarization . with the successful launch of _ fermi _ satellite , @xmath22-ray emission was detected in four rl - nls1s , including the object studied here , which suggests the presence of fully developed jets in these objects @xcite . the authors argued that the four rl - nls1s may form a new class of @xmath22-ray agns because of their small bh masses , large eddington ratios , and possibly disk - like morphology of the host galaxies . the intrinsic mechanism of rl - nls1s is an attractive field . there are two possible models for interpreting rl - nls1s . the first one is the inclination model @xcite . the model suspects that the ( at least a fraction of ) rl - nls1s are preferentially viewed pole - on . the observed narrow width of the balmer emission lines could be resulted from small inclination if the broad - line region ( blr ) is constrained to a plane @xcite . in fact , there is some evidence supporting a flat blr in some rl - agns @xcite . in this scenario , the bh mass is largely underestimated in these objects since the current available estimation of the bh mass of agn from single - epoch spectroscopic observation comes from an assumption of an isotropic distribution of the broad - line clouds with random orbital inclinations @xcite . although the inclination model sounds reasonable because it is able to shift the location of rl - nls1s on the @xmath8@xmath23 plane to the massive bh end @xcite , the massive bhs are not supported by the lack of massive bulges in several cases in which the host galaxies can be resolved . the second is the accretion mode model @xcite . the rl - nls1s with small bh masses are accreting close to or even above the eddington limit . low - mass bhs may lead to narrow emission lines when keplerian velocities are considered mostly @xcite . the accretion is thought related to the radio emission . different accretion modes may result in different phenomena and maybe can explain the differences in the radio loudness . accretion processes are known related to the spin of the accreting bhs . so the rapid spin of bhs may also affect the radio loudness of nls1s @xcite . it is possible that accretion mode combining with bh spin can explain the nature of the rl - nls1s . although our monitor indicates that the extremely high radio emission in rl - nls1 sdssj0948 + 57.3 + 002225 is mainly contributed from the beamed non - thermal jet with a small viewing angle ( can also be found in the aforementioned other studies ) , more information is needed in the future to investigate the origin of the relativistic jet . we are very grateful to dr . s. komossa for her helpful discussion and useful suggestions . this work was supported by the chinese natural science foundation through grants nsfc 10803008 and nsfc 10873017 . it was also supported by the 973 program ( 2009cb824800 ) . boller , t. 1997 , astron . nachr . , 318 , 209 cellone , s. a. , romero , g. e. , & combi , j. a. 2000 , , 119 , 1534 foschini , l. , maraschi , l. , tavecchio , f. , ghisellini , g. , gliozzi , m. , & sambruna , r. m. 2009 , adv . space res . , 43 , 889 healey , s. e. , romani , r. w. , taylor , g. b. , sadler , e. m. , ricci , r. , murphy , t. , ulvestad , j. s. , & winn , j. n. 2007 , , 171 , 61 komossa , s. , voges , w. , xu , d. , mathur , s. , adorf , h. m. , lemson , g. , duschl , w. , & grupe , d. 2006 , , 132 , 531 komossa , s. 2008 , revista mexicana de astronomia y astrofisica conference series , 32 , 86 lacy , m. , laurent - muehleisen , s. a. , ridgway , s. e. , becker , r. h. , & white , r. l. 2001 , , 551 , l17

sdssj094857.3 + 002225 is a very radio - loud narrow - line seyfert 1 ( nls1 ) galaxy . here , we report our discovery of the intranight optical variability ( inov ) of this galaxy through the optical monitoring in the _ b _ and _ r _ bands that covered seven nights in 2009 . violent rapid variability in the optical bands was identified in this rl - nls1 for the first time , and the amplitudes of the inov reaches 0.5 mag in both the _ b _ and _ r _ bands on the timescale of several hours . the detection of the inov provides a piece of strong evidence supporting the fact that the object carries a relativistic jet with a small viewing angle , which confirms the conclusion drawn from the previous multi - wavelength studies .

1005.0916

in the study of physics , the stability criteria of a system or configuration is one of the main interesting aspects . unstable system or configurations are generally not realizable in nature and they are generally an intermediate stage in the dynamical evolution of a system . a black hole system in general relativity can also be put in the above mentioned category : the question one asks there is whether a black hole which is stable under some perturbation , i.e. if we perturb the black hole from outside , whether it comes back to its original state after some time or whether the perturbation grows unbound making the black hole unstable . the study of black hole perturbations remains an extremely intriguing topic which has enormous effect on various important properties of a black hole @xcite . in general , the dynamical evolution of perturbations of a black hole background can be classified into three stages , the first of which consists of an initial outburst of wave , depending completely on the initial perturbing field , the second stage consists of damped oscillations , known in the literature as the quasinormal modes ( qnm ) whose frequency turns out to be a complex number , the real part representing the oscillation frequency and the imaginary part representing damping . qn frequencies completely depend on the background and not on the field which is causing the perturbation and thereby giving immense importance to these modes which are used to determine the black hole parameters ( mass , charge and angular momentum ) . the third is a power law tail behaviour at very late times . the equations governing the black hole perturbations in most of the cases can be cast into a schrdinger like wave equation . the qnms are solutions to the wave equation with complex frequencies with a boundary condition which are completely ingoing at the horizon and purely outgoing at asymptotic infinity . with the first ever detection of the gravitational wave @xcite the interest in studying black hole perturbations has gained another peak . apart from the fact that the qn frequencies contain important information about the black hole parameters , they were also found to be important from the point of view of ads / cft correspondence . it has been found that qnms in ads space - time appear naturally in the description of the dual conformal field theory on the boundary @xcite , thereby directing the study of qnms towards ads black holes @xcite . although the qnms are classical in origin , they have been shown to provide glimpses to quantum nature of black holes @xcite . however , in the present work we will be focussing on the second of the above three stages of evolution of black hole perturbation in a regular black hole background in asymptotically de sitter space - time . the importance of studying black holes in de sitter space lies in the fact that our universe looks like asymptotically de sitter at very early and late times . recent observational data also indicates that our universe is going through a phase of accelerated expansion @xcite , thereby providing the existence of a positive cosmological constant . in general de sitter space turns out to be a maximally symmetric solution to the vacuum einstein equations with a positive cosmological constant . just like the ads / cft correspondence , a holographic ds / cft duality exists between gravity in de sitter space and conformal field theory on the boundary in one less dimension @xcite . coming back to the perturbations and stability of black holes in de sitter space , there have been a lot of work @xcite-@xcite on quasinormal modes of scalar , electromagnetic , gravitational , dirac perturbations , decay of charged fields , asymptotic quasinormal modes , and signature of quantum gravity etc . on another front , it is well known that general relativity is plagued with the appearance of singularities . the problem of avoiding the singularities in general relativity , therefore , is one of the most fundamental ones and it is a very old problem . in this regard , `` regular black hole '' solutions play a central role . when a black hole does not have a space - time singularity at the origin , it is termed as a `` regular black hole '' in the literature . the first solution of such regular black holes with non - singular geometry satisfying the weak energy condition were obtained by bardeen @xcite , which is now known as the bardeen black hole . however , the solution bardeen proposed lacked physical motivation because the solution was not a vacuum solution , rather gravity was modified by introducing some form of matter and thereby introducing an energy momentum tensor in the einstein s equation . the introduction of the energy - momentum tensor was done in an _ ad hoc _ manner . much later , ayn - beato and garca @xcite showed the energy momentum tensor to be the gravitational field of some magnetic monopole arising out of a specific form of non - linear electrodynamics . subsequently , many other solutions @xcite-@xcite , motivating the avoidance of singularity was proposed in the literature . there were many works published regarding such regular black holes : stability properties @xcite , qnms @xcite , thermodynamics @xcite and geodesic structure @xcite of regular black holes to mention a few . very recently fernando @xcite has proposed a de sitter branch for the regular bardeen black hole and calculated the grey body factor for such a black hole . in this paper , we will be discussing the qnms of the bardeen de sitter ( henceforth bds ) black hole due to scalar ( both massless and massive ) and dirac perturbations . although study of scalar field perturbations in a black hole background and its corresponding qnms is not new , the dirac field perturbations , on the other hand , are relatively less studied . therefore , apart from the scalar perturbations , it will also be interesting to study the dirac perturbations in the regular black hole backgrounds in de sitter space . the plan of the paper is as follows : in the next section we briefly discuss the bds black hole . in section 3 we present a brief discussion of wkb method along with a study of the scalar qnms of the bds black holes . section 4 deals with the dirac quasinormal modes of the bds black hole . finally , in section 5 we conclude the paper with a brief discussion on future directions . in this section , we will briefly discuss the bardeen de sitter ( bds ) black hole following the works of fernando @xcite . the author of this paper modified the works of @xcite to incorporate a positive cosmological constant in the action : @xmath0 where @xmath1 is the ricci scalar and @xmath2 is function of the field strength tensor of the non - linear electrodynamics @xmath3 and its form is given by @xmath4 in the above , the parameter @xmath5 is related to the magnetic charge and the mass of the black hole in the following manner : @xmath6 . if one derives the equations of motion from the above action([action ] ) , then following equations will be arrived at @xmath7 it was shown in @xcite that a static spherically symmetric solution for the above set of equations exist : @xmath8 with @xmath9 being given @xmath10 the solution of @xmath11 gives the horizon and in the particular case of bds black hole there may be three real roots implying three horizons : the black hole inner and outer horizons along with the cosmological horizon . there lies the possibility of getting either one real root corresponding to cosmological horizon only for a set of parameters of this theory or a possibility of getting degenerate roots corresponding to a merger of the inner and outer black hole horizons for a range of parameters @xmath12 and @xmath13 . structurally the bds black hole is similar to the reissner - nordstrm - de sitter ( rnds ) or born - infeld de sitter ( bids ) black holes which also admits a possibility of three distinct horizons as well as a single or degenerate horizons . however , it was shown in @xcite that the event horizon is larger in the case of rnds black hole compared to a bds one . the interesting nature of bds geometry is its non - singular structure everywhere . it can be checked by direct calculation that all the scalar curvatures @xmath1 , @xmath14 , @xmath15 are finite everywhere except for the electromagnetic field invariant @xmath16 which is singular at @xmath17 @xcite . in this section we will consider the massless and massive scalar field perturbations of the bds black hole geometry to study the behaviour of the qnms in bds background with the given black hole parameters . as discussed in section 2 , bds background metric is given by equations ( [ metric ] ) and ( [ fr ] ) . the klein - gordon equation for a massless scalar field @xmath18 is @xmath19 which explicitly takes the form @xmath20 as usual , we introduce the ansatz for @xmath18 as , @xmath21 with the above ansatz , we have the standard schrdinger - like wave equation for the perturbation of the bds metric by a scalar field is given by @xmath22 where , @xmath23 . the coordinate @xmath24 is the standard tortoise coordinate related to radial coordinate @xmath25 as @xmath26 . the advantage of using the tortoise coordinate lies in the fact that the range of the coordinate now extends between @xmath27 to @xmath28 , whereas in the old radial coordinate @xmath25 , the physically accessible region lies between the black hole and cosmological horizon . note also that the potential @xmath29 as @xmath30 . it can be easily seen by plotting the scalar field potential against the radial coordinate ( ) for various values of the multipole number @xmath31 that the @xmath32 mode has a distinct local minimum between the black hole outer horizon and the cosmological horizon ( see fig . 1 ) , which was also pointed out in @xcite . for this reason , the method used in this paper to evaluate the qnms for the bds black hole , namely , the wkb approach is not a valid one to evaluate qnms for @xmath32 modes . therefore , from now on , we will only talk about @xmath33 modes for the massless scalar qnms of bds black hole . as already stated , we will solve the wave equation for complex qn frequencies semi - analytically , using the sixth order wkb method developed in @xcite . it has been shown extensively in literature that wkb method works extremely well for determining qn frequencies . the sixth order wkb method is more accurate than the third order method and the former in fact gives results practically coinciding with those obtained from full numerical integration of the wave equation @xcite for low overtones , i.e. for modes with small imaginary parts , and for all multipole numbers @xmath34 . the sixth order formula for a general black hole potential @xmath35 is mentioned below @xmath36 where @xmath37 is peak value of @xmath35 , @xmath38 , @xmath39 is the value of the radial coordinate corresponding to the maximum of the potential @xmath35 and @xmath40 is the overtone number . qn frequencies @xmath41 would be of the form @xmath42 . in equation([qnmeqn ] ) , @xmath43 and @xmath44 are given by @xcite @xmath45\\ \lambda_3=&\frac{(n+\frac{1}{2})}{2v^{''}(r_0)}\big[\frac{5}{6912 } \left ( \frac{v_0^{(3)}}{v^{''}(r_0 ) } \right)^4(77 + 188b^2)\nonumber\\ & -\frac{1}{384 } \left ( \frac{(v_0^{(3)})^2 v_0^{(4)}}{(v^{''}(r_0))^3}\right ) ( 51 + 100b^2)\nonumber\\ & + \frac{1}{2304}\left ( \frac{v_0^{(4)}}{v^{''}(r_0)}\right)^2(67 + 68b^2)\nonumber\\ & + \frac{1}{288}\left(\frac{v_0^{(3)}v_0^{(5)}}{(v^{''}(r_0))^2}\right)(19 + 28b^2)\nonumber\\ & -\frac{1}{288 } \left(\frac{v_0^{(6)}}{v^{''}(r_0)}\right)(5 + 4b^2)\big].\end{aligned}\ ] ] in the above expression @xmath46 , @xmath47 at @xmath48 and @xmath49 , @xmath50 and @xmath51 can be found in the appendix of @xcite . the above method also works extremely well in the eikonal limit of large @xmath31 corresponding to large quality factor , which will also be discussed in the paper . using eqn ( [ qnmeqn ] ) , we computed the qnms and in we plotted re @xmath41 and magnitude of i m @xmath41 vs black hole mass . both re @xmath41 and i m @xmath41 decreases when mass @xmath52 is increased . in table i , we list the values of the qn frequencies obtained by using third order and sixth order wkb approach for the parameter range @xmath53 and @xmath54 . the data from the table suggests that the value of the real part of the frequency shows a steady increase over its third order outcome but on the other hand , the negative imaginary part obtained using sixth order wkb method shows a steady decline when compared to the third order result . p3cmp3cmp3.5cmp3.5 cm multipole number & overtone & 3rd order wkb & 6th order wkb + + @xmath31=1&n=0 & 0.300446 -0.089967i & 0.302242 -0.090150i + & n=1 & 0.278912 -0.278097i & 0.282993 -0.277074i + + + & n=0 & 0.499385 -0.088861i & 0.499841 -0.088903i + @xmath31=2&n=1 & 0.485040 -0.269800i & 0.486281 -0.269658i + & n=2 & 0.461291 -0.456456i & 0.462177 -0.458553i + + + & n=0 & 0.698242 -0.088552i & 0.698417 -0.088563i + @xmath55=3&n=1 & 0.687778 -0.267316i & 0.688277 -0.267273i + & n=2 & 0.669085 -0.449812i & 0.669173 -0.450547i + & n=3 & 0.644523 -0.635942i & 0.643394 -0.640730i + + + & n=0 & 0.897184 -0.088421i & 0.897268 -0.088426i + & n=1 & 0.888985 -0.266272i & 0.889230 -0.266255i + @xmath31=4&n=2 & 0.873746 -0.446645i & 0.873717 -0.446952i + & n=3 & 0.853024 -0.629992i & 0.851862 -0.632179i + & n=4 & 0.828001 -0.815925i & 0.825285 -0.823178i + + + & n=0 & 1.09619 -0.088354i & 1.09624 -0.088356i + & n=1 & 1.08946 -0.265738i & 1.08960 -0.265730i + & n=2 & 1.07666 -0.444914i & 1.07663 -0.445061i + @xmath31=5&n=3 & 1.05883 -0.626438i & 1.05796 -0.627545i + & n=4 & 1.03691 -0.810304i & 1.03452 -0.814191i + & n=5 & 1.01159 -0.996158i & 1.00748 -1.005740i + + in and we plot the behaviour of low lying qn frequencies vs @xmath13 and @xmath56 for different @xmath31 . both the plots reveals that re @xmath41 and i m @xmath41 decreases with increasing @xmath13 . real part of frequencies still increasing steadily with g increased and imaginary part decreases in magnitude . we have also computed the qn frequencies for larger multipole number @xmath31 with overtone @xmath57 only . we plot for @xmath31 ranging between 1 to 40 in while we have fixed the values of @xmath58 , @xmath59 and @xmath57 . re(@xmath41 ) increases linearly with @xmath31 @xcite while magnitude of im(@xmath41 ) first decreases and remains constant for larger @xmath31 . to examine the field oscillations , we will define the quality factor(q.f ) as @xmath60 we plotted the q.f versus the parameters @xmath13 and @xmath56 in . quality factor increases with increasing @xmath56 and decreases with an increase in @xmath13 . thus , q.f implies that oscillations will be more with larger magnetic charge @xmath56 and decay faster for small @xmath13 . it is worth mentioning here that by computing the lyapunov exponent ( the inverse of the instability timescale associated with the geodesic motion ) , one can show that , in the eikonal limit , qnms of black holes in any dimensions are determined by the parameters of the circular null geodesics @xcite . this is a very strong result and is independent of the field equations . the only assumption goes into the calculation is the fact that the black hole space - time is static , spherically symmetric and asymptotically flat . however a non - trivial example of non - asymptotically flat near - extremal schwarzschild de sitter black hole space time was also discussed in this context . the same argument can be applied in case of bds black holes too in the limit of near - extremal nariai or cold black holes where either the black hole horizon and the cosmological horizon merges or the inner and outer horizon coincides . in these limits , it may be possible to get the eikonal limit using the wkb method following @xcite . for massive scalar perturbation the klein gordon equation is given by @xmath61 where @xmath62 is scalar field mass . similarly , we chose the ansatz as in equation ( [ ansatz ] ) and finally , we have the schrdinger - like equation and modified effective potential as @xmath63 where the tortoise coordinate @xmath24 is related to @xmath25 by @xmath26 . in , we plot the effective potential @xmath64(@xmath25 ) vs @xmath25 for different scalar mass ( @xmath62 ) . we have chosen the parameters @xmath65 , @xmath66 , @xmath67 and @xmath68 vs @xmath25 for various masses(@xmath62),width=302 ] notice that the peak of the potential depends on the scalar field mass @xmath62 with other parameters fixed . since qnms are known to be the waves trapped within the peak of this potential @xcite . as discussed in @xcite , we expect similar behaviour for bds black hole that the imaginary part of the quasinormal modes frequencies will decrease for large @xmath62 . however , the real part of qnms will increase as @xmath62 increases . in and , we have plotted the variations of imaginary and real part of @xmath41 versus scalar field mass @xmath69 for different fixed values of parameters @xmath13 , @xmath52 , @xmath56 and @xmath70 . we have plotted all the data obtained by 3rd , 4th , 5th and 6th order wkb calculations simultaneously to compare the accuracy between different orders . we observed from the plots of both im(@xmath41 ) and re(@xmath41 ) that for low overtone number @xmath40 , the accuracy between lower and higher order wkb is not much significant but for large @xmath40 deviation is more . indeed , the magnitude of im(@xmath41 ) decreases with increasing scalar mass as expected , on the contrary , the magnitude of re(@xmath41 ) increases with increasing field mass . in table ii , we present the numerical values of qn frequencies with corresponding parameters . since it is well known that wkb method is more accurate for @xmath71 , we have tabulated the qnms frequencies for @xmath71 only . [ cols="<,<,<,<",options="header " , ] [ ii ] as in massless case , we plot in and the behaviour of qn frequencies with @xmath13 and @xmath56 for @xmath57 and @xmath72 respectively . re(@xmath41 ) increases with increasing value of magnetic charge @xmath56 while magnitude of im(@xmath41 ) decreases . this behaviour of qnms can be well understood from the form of the potential . as the height of the potential peak increases with @xmath56 , therefore real part of qnms increases . on the other hand , shows that re(@xmath41 ) decreases with an increase in cosmological constant(@xmath13 ) but im(@xmath41 ) increases with @xmath13 in magnitude . similarly if we plot the variation of potential with @xmath13 the height decreases , thus i m ( @xmath41 ) increases . hence , we can say for scalar field perturbations with the scalar mass included , the oscillations decay faster with large cosmological constant @xmath13 and oscillates better for large magnetic charge @xmath56 . in this section , we will extend our discussion to massless dirac perturbations for bds black holes . as in @xcite , by starting from dirac equation in spherically symmetric curved background , the schrdinger - like equation we finally arrived at is given by @xmath73 where @xmath24 is the tortoise coordinate given by @xmath74 , @xmath75 is the energy . the effective potentials is given by @xmath76 it is worth mentioning here that the potentials @xmath77 and @xmath78 corresponding to dirac particles and anti - particles are supersymmetric to each other and derived from the same superpotential @xmath79 . we will evaluate the quasinormal modes by solving equation [ [ sc1 ] ] taking only @xmath77 as it is well known that both dirac particles and anti - particles have the same quasi - normal spectra @xcite . in , we showed the behaviour of the effective potential ( @xmath77 only ) for bds black hole with spherical harmonic @xmath70 for parameters @xmath80 ) vs @xmath25 for different values of @xmath31 for the massless dirac perturbations , width=226 ] we have computed the massless fermion qnms semi - analytically using sixth order wkb method.the plots are shown below . in , we showed the variation of real and imaginary part of @xmath41 with cosmological constant @xmath13 and in , the variation with magnetic charge @xmath56 for different values of @xmath31 with fixed overtone number are shown . we can clearly see from the plots that re(@xmath41 ) slowly increases with an increase in the magnetic charge @xmath56 of the bds black hole whereas it slowly decreases with increasing value of @xmath13 . whereas the behaviour of the imaginary part of the frequency reverses its role , i.e. as we increase the cosmological constant , the imaginary part increases , however , it decreases if we increase the magnetic charge , keeping all other parameters fixed . in this paper , we have discussed the massless and massive scalar field perturbations and the massless fermionic perturbations for a regular bds black hole . we have used sixth order wkb approximation method to calculate the qnms frequencies . we studied how the frequencies vary as a function of the scalar field mass ( @xmath62 ) , multipole number ( @xmath31 ) as well as with the parameters like the cosmological constant ( @xmath13 ) , black hole mass ( @xmath52 ) and magnetic charge ( @xmath56 ) . we found that the qn frequencies decrease with an increase in black hole mass @xcite . the plots of frequencies versus the scalar mass show that re @xmath41 increases with mass @xmath62 while i m @xmath41 decreases . the figures also suggested that if we plot the frequencies from low to higher overtones taking into account different wkb orders , we see that comparative accuracy is better for @xmath81 . we also found that re @xmath41 decreases with an increase in cosmological constant @xmath13 for scalar ( both massless and massive ) perturbations as well as with dirac perturbations but i m @xmath41 decreases in massless and fermionic case however , increases for the massive case when @xmath13 is increased . we have also studied the behaviour of how the q - factor for the massless scalar field varies with @xmath13 and @xmath56 . for massive scalar perturbations , we see that mass @xmath62 enhances the field oscillations and decreases the damping for small @xmath13 , unlike in the massless case where it is just the opposite . in all the three scenarios real frequency of oscillations re @xmath41 increases steadily with magnetic charge ( @xmath56 ) but the damping denoted by i m @xmath41 decreases . for future directions , it would be interesting to study the time evolution of perturbations for this particular black hole . apart from that , in @xcite , the authors have used the conformal properties of the spinor field to obtain the dirac qnms for a higher dimensional schwarzschild - tangherlini black hole . they have described these modes in the light of the so - called split fermion models , where quarks and leptons exist on different branes in order to keep proton stability . such split fermion theories also have massive fermions in the bulk and it will be interesting to study such massive dirac perturbations in the context of higher - dimensional generalization of the bds black holes . spphys kokkotas k d and schmidt b g 1999 _ living rev . * 2 * 2 nollert h - p 1999 _ class . quantum grav . _ * 16 * r159 berti e , cardoso v and starinets a o 2009 _ class.quant.grav . _ * 26 * 163001 konoplya r a and zhidenko a 2011 _ rev.mod.phys . _ * 83 * 793 - 836 abott b p et al . 2016 _ phys . * 116 * 241102 birmingham d , sachs i and solodukhin s n 2002 _ phys . lett . _ * 88 * 151301 birmingham d , sachs i and solodukhin s n 2003 _ phys . rev . _ * d67 * 104026 horowitz g t and hubeny v e 2000 _ phys . _ * d62 * 024027 ; konoplya r a 2002 _ phys . rev . _ * d66 * 044009 cardoso v and lemos j p s 2001 _ phys _ * d63 * 124015 hod s 1998 _ phys . _ * 81 * 4293 motl l and neitzke a 2003 _ adv . theor . math . phys . _ * 7 * 307 maggiore m 2008 _ phys.rev.lett . _ * 100 * 141301 perlmutter s _ etal _ 1997 _ astrophys . j. _ * 483 * 565 riess a g et al . 1998 _ astron . j. _ * 116 * 1009 tegmark m et al . 2004 _ phys . _ * d69 * 103501 strominger a 2001 _ jhep _ * 0110 * 034 strominger a 2001 _ jhep _ * 0111 * 049 abdalla e , castello - branco k h c and lima - santos a 2002 _ phys.rev . _ * d66 * 104018 konoplya r a 2003 _ phys . rev . _ * d68 * 124017 konoplya r a and zhidenko a 2004 _ jhep _ * 06 * 037 ; zhidenko a 2004 _ class . quant . grav . _ * 21 * 273 cardoso v and lemos j p s 2003 _ phys . rev . _ * d67 * 084020 molina c 2003 _ phys . rev . _ * d68 * 064007 lopez - ortega a 2006 _ gen . * 38 * 1565 lopez - ortega a 2006 _ gen . grav . _ * 38 * 743 lopez - ortega a 2007 _ gen . _ * 39 * 1011 lopez - ortega a 2008 _ gen . _ * 40 * 1379 smirnov a a 2005 _ class . * 22 * 4021 bardeen j m 1968 _ proceedings of gr5 ( tbilisi ) _ 174 ayn - beato e and garcia a 2000 _ phys . _ * b493 * 149 ayn - beato and garca a 1998 _ phys . rev . lett . _ * 80 * 5056 bronnikov k a 2001 _ phys . * d63 * 044005 hayward s a 2006 _ phys . lett . _ * 96 * 031103 dymnikova i 2004 _ class . * 21 * 4417 bronnikov k a and fabris j a 2006 _ phys . _ * 96 * 251101 moreno c and sarbach o 2003 _ phys . rev . _ * d67 * 024028 flachi a and lemos j 2013 _ phys . * d87 * 024034 fernando s and correa j 2012 _ phys . * d86 * 64039 man j and cheng h 2014 _ gen . grav . _ * 46 * 1559 zhou s , chen j and wang y 2012 _ int . _ * d21 * 1250077 fernando s , arxiv : 1611.05337 [ gr - qc ] fernando s , correa juan 2012 _ phys . rev _ * d86 * 064039 cardoso v , berti e , witek h , zanchin v t 2009 _ phys.rev . _ * d79 * 064016 akira ohashi and masa - aki sakagami _ classical and quantum gravity _ * 21 * 3973 d. r. brill and j. a. wheeler , rev * 29 * , 465 ( 1957 ) . cho h t , cornell a s , doukas j and naylor w 2007 _ phys . rev _ * d75 * 104005 schutz b and will c m 1988 _ astrophys . j. _ * 291 * l33 iyer s and will c m 1987 _ phys . rev _ * d35 * 3621 konoplya r a 2003 _ phys . rev._*d68 * 024018 jing j l 2004 _ phys . rev _ * d69 * 084009

we compute the quasi - normal ( qn ) frequencies for the regular bardeen de sitter ( bds ) black hole due to massless and massive scalar field perturbations as well as the massless dirac perturbations . we analyze the behaviour of both real and imaginary parts of quasinormal frequencies by varying the black hole parameters .

1703.04286

magneto - optical ( mo ) effects have found important applications in data storage , telecommunications , imaging , spectroscopy,@xcite as well as in magnonics . @xcite an increasingly broad application of nanotechnologies in these areas sets new requirements for novel mo devices by demanding , among other things , a stronger mo response and smaller dimensions . for example , small dimensions are important for @xmath2d imaging systems based on holographic principles.@xcite such systems require a spatial light modulator ( slm ) a device used to modulate amplitude , phase , or polarisation of light waves in space and time . basically , an slm device can produce high quality @xmath2d images if it possesses pixels sizes @xmath3 @xmath1 m and also is fast enough to address a large number of pixels within a single image frame . slm devices exploiting mo effects have a very fast response time . however , their typical pixel sizes are @xmath4 @xmath1m.@xcite at the nanoscale , mo effects can be enhanced by using subwavelength diffraction gratings @xcite , magneto - photonic crystals , @xcite and nanoantennas , @xcite which are made of pure magnetic materials or consist of alternating magnetic - nonmagnetic layers . however , as a typical grating period is @xmath5 nm and there should be tens of periods to achieve optimal optical properties , the footprint of the grating is large . magneto - photonic crystals often have comparable dimensions and suffer from similar disadvantages . typical dimensions of ferromagnetic / normal metal nanoantennas @xcite can be much smaller than @xmath6 @xmath1 m because of magneto - plasmonic resonances which give an additional degree of freedom for light manipulation at the nanoscale.@xcite however , due to huge absorption losses in ferromagnetic metals , plasmonic resonance properties of such nanoantennas are not very strong as compared with nanoantennas fabricated of gold or silver ( nonmagnetic metals with relatively low absorption losses ) . consequently , measures have to be taken to mitigate absorption losses , which can be done by reducing the amount of ferromagnetic metals and tailoring fano resonance effects.@xcite . alternatively , by analogy with subwavelength gratings,@xcite losses in nanoantennas can be reduced by using magneto - dielectrics instead of ferromagnetic metals . normally , magnetic gratings , photonic crystals and nanoantennas are used to enhance the faraday rotation and kerr effect in different configurations . @xcite for instance , if the incident light is @xmath7-polarised and the static external magnetic field is applied perpendicularly to the plane of incidence , the so - called transverse mo kerr effect ( tmoke ) @xcite is observed in the reflection mode . less well - known are transverse mo effects observed in the transmission mode . these effects are similar to the tmoke but occur in transparent magnetic films or plates @xcite . an example of a transverse mo effect in the transmission mode is the transverse faraday effect ( tfe ) . @xcite this effect can be used in imaging , data storage systems and magnonics . the tfe has been observed in transparent magnetic plates and fabry - perot resonators . @xcite because the tfe response is amplified as a result of the wave propagating back and forth within a magnetised medium , it is enhanced in a fabry - perot resonator . however , a typical fabry - perot resonator consists of a @xmath8 @xmath1m - thick plate of a transparent magnetic material sandwiched between a pair of mirrors , which in turn consist of @xmath9 alternating layers of high- and low - refractive - index quarter - wave - thick dielectrics.@xcite consequently , the dimensions of the resonator are also very large , which makes it unsuitable for miniaturisation . in this paper , we predict and demonstrate theoretically a large multifrequency tfe in single microspheres made of bi@xmath10-substituted yttrium - iron magnetic garnet ( bi : yig ) . bi : yig is a magneto - dielectric exhibiting large mo activity and high transparency in the visible and infrared spectral ranges.@xcite by exploiting high - quality factor resonances of a bi : yig microsphere we show a strong tfe response at multiple wavelengths at the same time . the strength of the predicted tfe is comparable with the strength of the tmoke in subwavelength magnetic gratings . @xcite moreover , the observed tfe response is higher than that in micron - thick transparent magnetic plates . @xcite however , the dimensions of a single sphere are significantly smaller as compared with the area occupied by a subwavelength grating or the thickness of a plate used in modern mo devices . the problem of light scattering by a homogeneous sphere of arbitrary diameter and dielectric permittivity is exactly soluble by using the mie theory.@xcite whereas the mie theory can be extended to calculate the scattering by homogeneous magnetic spheres described by their magnetic permeability @xcite , the problem of light scattering by a sphere magnetised along a certain coordinate direction is more difficult . this is because the dielectric permittivity becomes a tensor describing the interaction between the light and the static external magnetic field ( or the internal magnetisation of the medium)@xcite @xmath11 by considering bi : yig as an isotropic material , the three diagonal elements of @xmath12 become identical , and in the presence of a static external magnetic field along the _ y_-axis , there is a non - zero off - diagonal element @xmath13 , which couples the _ x_- and _ z_-components of the optical electric field @xmath14 due to low absorption losses in bi : yig the refractive index @xmath15 and @xmath13 of bi : yig can be assumed to be real and also frequency - independent over a narrow range of wavelengths : @xmath16 and @xmath17 . @xcite because bi : yig is a ferrimagnetic material , the spin - orbit coupling is the dominant source of the mo interaction and it makes @xmath13 proportional to the magnetisation @xmath18 of the medium . @xcite by considering the off - diagonal elements of @xmath12 in the mie theory , one can take into account the physical mechanisms responsible for the mo kerr and faraday effects contribution to the optical response of a sphere . @xcite it has been shown that the mo kerr effect in a homogeneous cobalt sphere is small but detectable.@xcite the small mo response of cobalt spheres is due to large optical@xcite and magneto - optical@xcite absorption effects in this ferromagnetic metal . considerable optical absorption losses are typical of all metals , including gold and silver . these losses have long been known to be the drawback of metallic nanostructures used in photonic devices such as , e.g. , nanoantennas . @xcite consequently , a large and growing body of research investigates all - dielectric nanostructures , e.g. , low - loss single - sphere nanoantennas . @xcite in this context , the use of bi : yig as the model material of the microsphere opens up opportunities to overcome the drawback of metallic ferromagnetic spheres . it is also worth noting brillouin light scattering ( bls ) from spin - wave modes in uniformly magnetised ferromagnetic and ferrimagnetic spheres . @xcite the fluctuation of magnetisation due to spin - wave modes causes a time - dependent change of the dielectric permittivity tensor @xmath19 of the material of the sphere . the contribution of the transverse magnetisation to @xmath19 allows generating a description of bls by processes in which a single spin - wave quantum ( magnon ) is created or destroyed . in this case , bi : yig still can be used as the model material of the microsphere because bls has been used to study nonlinear spin wave phenomena in bi : yig structures.@xcite furthermore , microspheres can also be made of pure yttrium iron garnet ( yig ) , which is a very well - known material used in magnonics.@xcite finally , both semiconductor and ferromagnetic properties have been established in some of the rare earth mononitrides , which thus attract interest for the potential to exploit the spin of charge carriers in spintronics . @xcite the refractive index of these materials is @xmath20 and they are also transparent in the visible and infrared spectral ranges.@xcite most significantly , these materials also exhibit significant mo activity.@xcite thus , they might be employed instead of bi : yig . fig . @xmath6(a ) schematically shows the scattering by an arbitrary homogeneous nonmagnetic sphere . by applying the mie theory for nonmagnetic spheres @xcite we show that the intensity of the forward scattered light [ @xmath21 in fig . @xmath6(a ) ] has multiple peaks in the spectral range @xmath22 , where @xmath23 is the diameter of the sphere and @xmath24 is the wavelength in the free space [ fig . @xmath6(b ) ] . the intensity of the backscattered light ( @xmath25 ) in this spectral range is low . this implies that at the resonance the single microsphere has a very directive far - field emission pattern [ fig . @xmath6(c ) ] . for example , in order to achieve the maximum forward - scattering intensity in the visible spectral range a bi : yig sphere with the diameter @xmath26 @xmath1 m is needed . it is noteworthy that a nonmagnetised sphere with @xmath27 @xmath1 m will be in the so - called multi - stripe domain state.@xcite basically , light can be scattered by the stripe domain structure.@xcite however , the strength of this process is very low and it can be neglected , i.e. we will assume that a nonmagnetised bi : yig sphere has the optical properties of a homogeneous nonmagnetic sphere with the same refractive index . the application of the static magnetic field orientated along the _ y_-axis [ fig . @xmath6a ] will magnetically saturate the sphere by aligning the direction of magnetisation inside the sphere along the _ y_-axis . therefore , in the saturated state the mo properties of the sphere can be modelled by using the tensor @xmath12 [ eq . ( @xmath6 ) ] . in this case , the conventional mie theory can not be applied ; it must be extended employing , e.g. , a perturbation approach.@xcite alternatively , approximate approaches such as , e.g. , a modified discrete dipole approximation ( dda ) @xcite may be employed . however , these methods have disadvantages . the perturbation approach requires the application of the green s function in spherical coordinates.@xcite the application of the dda to large mo spheres may require unaffordable computational efforts.@xcite consequently , we will use a finite - difference time - domain ( fdtd ) method . @xcite although the fdtd method is also computationally demanding because it uses the staircase approximation of the surface of the sphere and thus requires a very fine finite - difference grid , its application to large mie scattering problems is known to produce accurate results.@xcite the computing power of a modern desktop computer is enough for these simulations . furthermore , the fdtd can be applied to scatterers with a more complex shape than the sphere . of course , in this case for scatterers of a complex shape one also needs to solve a micromagnetics problem to find the distribution of the magnetisation inside the scatterer . the solid line in fig . @xmath28 shows the forward - scattering intensity differential @xmath29 quantifying the strength of the tfe in the microsphere as a function of the normalised wavelength @xmath30 . in this case , @xmath31 is the intensity of the forward - scattered light ( shown for reference in fig . @xmath28 by the dashed line ) and @xmath32 is the saturation magnetisation for bi : yig . the change in the sign of @xmath32 implies the change in the direction of the static external magnetic field by @xmath33 and it leads to the change in the sign of @xmath13.@xcite recall that a similar differential @xmath34 , being @xmath35 the reflectivity , is used to quantify the strength of the tmoke in gratings and other mo devices operating in the reflection mode.@xcite however , in contrast to all - magneto - dielectric gratings operating at a single wavelength,@xcite in fig . @xmath28 the multiple maxima of @xmath36 are observed at a multiple wavelength corresponding to the sharp resonances supported by the microsphere . we note that the peak values of @xmath36 are of the same order of magnitude as the value of @xmath37 in the gratings.@xcite but the footprint of a single microsphere is much smaller than the area occupied by a grating . quantifying the strength of the tfe in the plate as a function of the angle of incidence @xmath38 for the wavelength in the free space @xmath39 nm . dashed line the transmittance @xmath40 of the nonmagnetised plate . thanks to the one - dimensional character of the problem , these results were obtained using the fresnel formula for _ p_-polarised light . @xcite , width=321 ] to further demonstrate the advantages of the microsphere over the existing mo devices used to enhance transverse mo effects , we calculate the tfe in a @xmath8 @xmath1m - thick plate made of bi : yig [ fig . @xmath2(a ) ] . such plates have practical applications in fabry - perot resonator - based mo devices.@xcite the considered plate is a one - dimensional scatterer with the spatial variation along the _ z_-direction so that @xmath41 . therefore , for a _ x_- or _ y_-polarised incident wave , there is no _ z_-component of the optical electric and magnetic fields , and for the incident _ p_-polarised light the medium of the plate exhibits an effective refractive index @xmath42^{1/2}$ ] ( the medium does not exhibit mo activity for the _ s_-polarised light).@xcite due to the dependence of @xmath43 on @xmath13 the magnitude of the transmitted _ p_-light is sensitive to the magnetisation . importantly , the tfe is not bipolar , so changing the direction of the static external magnetic field from @xmath44 to @xmath45 ( i.e. changing @xmath46 ) does not alter the magnitude of the transmission @xmath47 . hence , the tfe is quantified by the differential @xmath48.@xcite importantly , the tfe can take place only in scatterers where a nonhomogeneous or leaky refracted wave is induced by a wave obliquely incident at the interface , i.e. the tfe vanishes at both the normal and grazing incidence.@xcite consequently , in fig . @xmath2(b ) we plot @xmath49 as a function of the angle of incidence @xmath38 for the wavelength in the free space @xmath39 nm . we note that at this wavelength a sphere with @xmath50 @xmath1 m would produce the maximum of the tfe response ( fig . @xmath28 ) . figure @xmath2(b ) shows that in the absence of the static external magnetic field the transmission curve @xmath40 for the _ p_-polarised light has multiple maxima due to interference effects . the curve of the differential @xmath49 follows the behaviour of @xmath40 . around the brewster angle @xmath51 one observes @xmath52 because vanishing surface reflectivity results in minimal interference effects . it is noteworthy that for the chosen thickness of the slab , which is @xmath8 times larger than the diameter of the reference sphere @xmath50 @xmath1 m , the strength of the tfe is ten times smaller than in the sphere . furthermore , in a @xmath6 @xmath1m - thick plate ( not shown ) the strength of the tfe plummets down because the interference effect is significantly weaker . of course , the fabrication techniques for magnetic films and plates are very much well - established and less effort demanding than those for single magnetic microspheres . however , the recent advances in making ultra - fine magnetic garnet particles make it possible to fabricate single scatterers with a close - to - spherical shape.@xcite yig powders with controllable individual particle sizes of @xmath53 @xmath1 m can be obtained by a microwave heating method.@xcite furthermore , a magnetic garnet microsphere can be fabricated using direct laser writing @xcite . finally , laser - based printing techniques can potentially be applied.@xcite we proposed an efficient scheme for enhancing the transverse faraday effect in single magneto - dielectric microspheres . we investigated the scenario of a sphere made of a typical magnetic garnet exhibiting a realistically low absorption and high magneto - optical activity in the visible and infrared spectral ranges . we demonstrated the transverse faraday effect of the order of @xmath54 , which is larger or comparable with typical values attainable with the modern mo devices which , however , have a significantly larger footprint as compared with microspheres . our findings may find applications in nanophotonics , imaging , and magnonics . this work was supported by the australian research council . ism gratefully acknowledges a postdoctoral research fellowship from the university of western australia . the authors thanks ass./prof . p. metaxas for valuable discussions .

we propose using a single magneto - dielectric microsphere as a device for enhancing the transverse faraday effect at multiple wavelengths at the same time . although the diameter of the sphere can be @xmath0 @xmath1 m , the numerically predicted strength of its magneto - optical ( mo ) response can be an order of magnitude stronger than in mo devices based on thick magnetic plates . the mo response of a microsphere is also comparable with that of subwavelength magneto - dielectric gratings which , however , operate at a single wavelength and occupy a large area . in contrast to gratings and thick plates , the compact size of the microsphere and its capability to support spin - wave excitations make it suitable for applications in nanophotonics , imaging systems , and magnonics .

1405.1149

the proper - motion observations of pulsars show that the pulsars had the kick velocity in the formation stage . the young pulsars have proper velocity of @xmath4 @xcite . the physical mechanism of such kick velocity may be due to the harrison tademaru mechanism @xcite , anisotropic emission of neutrinos , anisotropic explosion and so on ( see lorimer @xcite for the review ) . therefore , it is also reasonable to assume the existence of the proper motion of the pulsars in the formation process of pop iii nss , although there is no direct evidence since no pop iii star or pulsar is observed . while , repetto et al . @xcite suggest that bhs also have a natal kick velocity comparable to pulsars from the galactic latitude distribution of the low mass x - ray binaries in our galaxy . but , first , this is not the direct observation of proper motion of bhs , and second , since the mass of pop iii bhs is larger than pop i and pop ii bhs , their kick velocity might be so small that it can be neglected . therefore , we take into account the natal kick for pop iii nss but not for pop iii bhs in this paper . the kick speed @xmath5 obeys a maxwellian distribution as @xmath6 \,,\ ] ] where @xmath7 is the dispersion . the details of the method how to calculate the natal kick are shown in ref . @xcite . in this paper , we perform population synthesis monte carlo simulations of pop iii binary stars . we calculate the pop iii ns - bh and pop i and ii ns - bh for comparison . pop i and pop ii stars mean solar metal stars and metal poor stars whose metallicity is less than 10% of solar metallicity , respectively . in this paper , we consider five metallicity cases of @xmath8 ( pop iii ) , @xmath9 and @xmath10 ( pop i ) . there are important differences between pop iii and pop i and ii . pop iii stars are ( 1 ) more massive , @xmath11 , ( 2 ) smaller stellar radius compared with that of pop i and ii , and ( 3 ) no stellar wind mass loss . these properties play key roles in binary interactions . in order to estimate the event rate of ns - bh mergers and the properties of ns - bh , we use the binary population synthesis method @xcite which is the monte calro simulation of binary evolution . first , we choose the binary initial conditions such as the primary mass @xmath12 , the mass ratio @xmath13 , the separation @xmath14 , and the eccentricity @xmath15 when the binary is born . these binary initial conditions are chosen by the monte calro method and the initial distribution functions such as the initial mass function ( imf ) , the initial mass ratio function ( imrf ) , the initial separation function ( isf ) , and the initial eccentricity distribution function ( ief ) . we adopt these distribution functions for pop iii stars and pop i and ii stars as table [ idf ] . [ cols="^,^,^",options="header " , ] this work was supported by mext grant - in - aid for scientific research on innovative areas , `` new developments in astrophysics through multi - messenger observations of gravitational wave sources '' , no . 24103006 ( tn , hn ) , by the grant - in - aid from the ministry of education , culture , sports , science and technology ( mext ) of japan no . 15h02087 ( tn ) , and jsps grant - in - aid for scientific research ( c ) , no . 16k05347 ( hn ) . a. g. lyne and d. r. lorimer , nature * 369 * , 127 ( 1994 ) . b. m. s. hansen and e. s. phinney , mon . not . soc . * 291 * , 569 ( 1997 ) [ astro - ph/9708071 ] . e. r. harrison and e. p. tademaru , astrophys . j. * 201 * , 447 ( 1975 ) . d. r. lorimer , living rev . * 11 * , 8 ( 2008 ) [ arxiv:0811.0762 [ astro - ph ] ] . s. repetto , m. b. davies and s. sigurdsson , mon . not . 425 * , 2799 ( 2012 ) [ arxiv:1203.3077 [ astro-ph.ga ] ] . j. r. hurley , c. a. tout and o. r. pols , mon . not . roy . astron . soc . * 329 * , 897 ( 2002 ) [ astro - ph/0201220 ] . t. kinugawa , k. inayoshi , k. hotokezaka , d. nakauchi and t. nakamura , mon . not . soc . * 442 * , 2963 ( 2014 ) [ arxiv:1402.6672 [ astro-ph.he ] ] . t. kinugawa , a. miyamoto , n. kanda and t. nakamura , mon . not . soc . * 456 * , 1093 ( 2016 ) [ arxiv:1505.06962 [ astro-ph.sr ] ] . c. e. rhoades , jr . and r. ruffini , phys . rev . lett . * 32 * , 324 ( 1974 ) . j. b. hartle , phys . rep , * 46 * , 201 ( 1978 ) r. m. kulsrud , r. cen , j. p. ostriker and d. ryu , astrophys . j. * 480 * , 481 ( 1997 ) [ astro - ph/9607141 ] . l. m. widrow , rev . phys . * 74 * , 775 ( 2002 ) [ astro - ph/0207240 ] . m. langer , j. l. puget and n. aghanim , phys . d * 67 * , 043505 ( 2003 ) [ astro - ph/0212108 ] . k. doi and h. susa , astrophys . j. * 741 * , 93 ( 2011 ) [ arxiv:1108.4504 [ astro-ph.co ] ] . h. nieuwenhuijzen and c. de jager , astron . astrophys . * 231 * , 134 ( 1990 ) . e. vassiliadis and p. r. wood , astrophys . j. * 413 * , 641 ( 1993 ) . r. m. humphreys and k. davidson , publ . . soc . pac . * 106 * , 1025 ( 1989 ) . n. smith , ann . astrophys . * 52 * , 487 ( 2014 ) [ arxiv:1402.1237 [ astro-ph.sr ] ] . k. belczynski , t. bulik , c. l. fryer , a. ruiter , j. s. vink and j. r. hurley , astrophys . j. * 714 * , 1217 ( 2010 ) [ arxiv:0904.2784 [ astro-ph.sr ] ] . j. s. vink and a. de koter , astron . astrophys . * 442 * , 587 ( 2005 ) [ astro - ph/0507352 ] . r. s. de souza , n. yoshida and k. ioka , astron . astrophys . * 533 * , a32 ( 2011 ) [ arxiv:1105.2395 [ astro-ph.co ] ] . e. visbal , z. haiman and g. l. bryan , mon . not . * 453 * , 4456 ( 2015 ) [ arxiv:1505.06359 [ astro-ph.co ] ] . t. hartwig , m. volonteri , v. bromm , r. s. klessen , e. barausse , m. magg and a. stacy , mon . not . * 460 * , l74 ( 2016 ) [ arxiv:1603.05655 [ astro-ph.ga ] ] . k. inayoshi , k. kashiyama , e. visbal and z. haiman , mon . not . soc . * 461 * , 2722 ( 2016 ) [ arxiv:1603.06921 [ astro-ph.ga ] ] . p. madau and m. dickinson , ann . rev . astron . astrophys . * 52 * , 415 ( 2014 ) [ arxiv:1403.0007 [ astro-ph.co ] ] . x. ma , p. f. hopkins , c. a. faucher - giguere , n. zolman , a. l. muratov , d. keres and e. quataert , mon . not . soc . * 456 * , 2140 ( 2016 ) [ arxiv:1504.02097 [ astro-ph.ga ] ] . a. fontana _ et al . _ , astron . astrophys . * 459 * , 745 ( 2006 ) [ astro - ph/0609068 ] . k. kawaguchi , k. kyutoku , h. nakano , h. okawa , m. shibata and k. taniguchi , phys . d * 92 * , 024014 ( 2015 ) [ arxiv:1506.05473 [ astro-ph.he ] ] . d. keppel and p. ajith , phys . d * 82 * , 122001 ( 2010 ) [ arxiv:1004.0284 [ gr - qc ] ] . j. aasi _ et al . _ [ ligo scientific and virgo collaborations ] , living rev . rel . * 19 * , 1 ( 2016 ) [ arxiv:1304.0670 [ gr - qc ] ] . t. nakamura _ et al . _ , arxiv:1607.00897 [ astro-ph.he ] . n. dalal , d. e. holz , s. a. hughes and b. jain , phys . rev . d * 74 * , 063006 ( 2006 ) [ astro - ph/0601275 ] . p. ajith _ et al . . rev . lett . * 106 * , 241101 ( 2011 ) [ arxiv:0909.2867 [ gr - qc ] ] .

in the population synthesis simulations of pop iii stars , many bh ( black hole)-bh binaries with merger time less than the age of the universe @xmath0 are formed , while ns ( neutron star)-bh binaries are not . the reason is that pop iii stars have no metal so that no mass loss is expected . then , in the final supernova explosion to ns , much mass is lost so that the semi major axis becomes too large for pop iii ns - bh binaries to merge within @xmath1 . however it is almost established that the kick velocity of the order of @xmath2 exists for ns from the observation of the proper motion of the pulsar . therefore , the semi major axis of the half of ns - bh binaries can be smaller than that of the previous argument for pop iii ns - bh binaries to decrease the merging time . we perform population synthesis monte carlo simulations of pop iii ns - bh binaries including the kick of ns and find that the event rate of pop iii ns - bh merger rate is @xmath3 . this suggests that there is a good chance of the detection of pop iii ns - bh mergers in o2 of advanced ligo and advanced virgo from this autumn .

1610.00305

it has been known for several decades that bright low - mass x - ray binaries ( lmxbs ) are @xmath1 times overabundant in globular clusters ( gcs ) relative to the galactic field ( katz 1975 ; clark 1975 ) . more specifically , gcs contain 13 of the @xmath1 bright galactic lmxbs , but only @xmath20.01% of the total stellar mass content of the galaxy . the reason for this is thought to be the existence of _ dynamical _ lmxb formation channels , which are only available in dense gc cores . potential channels include the direct collision of a neutron star ( ns ) with a red giants ( verbunt 1987 , davies et al . 1992 , ivanova et al . 2005 ) , the tidal capture of a main sequence star by a ns ( fabian , pringle & rees 1975 ; bailyn & grindlay 1987 ) and exchange interactions between nss and primordial binaries ( hilles 1976 ; rasio et al . 2000 ) . if the dominant lmxb formation channels are different in gcs and the galactic field , the properties of their respective lmxb populations may also be different . in particular , most of the bright lmxbs in gcs might be ultracompact x - ray binaries ( ucxbs ; bildsten & deloye 2004 , ivanova et al . ucxbs , which are interacting binaries with extremely small binary separations ( @xmath3 cm ) and short orbital periods ( @xmath4 1 hr ) , appear to be rare amongst the galactic field lmxb population : the list of confirmed ucxbs ( with measured @xmath5 ) in int zand et al . ( 2007 ) contains only 5 objects that belong to this population . by contrast , 3 of the 13 gc lmxbs are confirmed ucxbs ; these are 4u1820 - 30 in ngc6624 ( @xmath6 min , stella et al . 1987 ) , 4u1850 - 087 in ngc6712 ( @xmath7 min , homer et al . 1996 ) and cxoj212958.1 + 121002 in m15 ( = m15-x2 ; dieball et al . 2005 ) , with several more suggested to be ultracompact x - ray binaries on the basis of more indirect evidence ( see e.g. verbunt & lewin 2006 for a review ) . since the period distribution of gc lmxbs may be a direct tracer of the dynamical close encounters taking place in gc cores , it is important to establish orbital periods for as many of these sources as possible . doing so could also lead to a significant increase in the size of the total ucxb sample . this is desirable , because ucxbs are astrophysically important systems in their own right . this is because they are laboratories for accretion and binary evolution in extreme settings , and because they are strong gravitational wave sources that may be detectable by lisa ( nelemans & jonker 2006 ; nelemans 2009 ) . here , we present time - resolved , far - uv photometry of the lmxb 4u 0513 - 40 in ngc 1851 , which was suspected to be a ucxb based on several strands of circumstantial evidence ( deutsch etal 2000 ; verbunt 2005 ; nelemans & jonker 2006 ; int zand etal . 2007 ) . our far - uv data of this system contain a @xmath8 17 min periodic signal that is present in all four observing epochs , is consistent with being coherent and is probably caused by a reflection effect associated with the irradiated surface of the donor star in this system . based on all this , we argue that the observed periodic variability is an orbital signature , and thus that 4u 0513 - 40 should be regarded as a confirmed ucxb with @xmath9 min . ngc 1851 was observed three times with the f140lp filter in the solar blind channel ( sbc ) of the advanced camera for surveys ( acs ) on board the _ hst_. this instrument / detector / filter combination has a plate scale of 0.032 pixel@xmath10 , a pivot wavelength of @xmath11 , and an rms bandwidth of @xmath12 . all of the observations took place in august of 2006 . each observing epoch consisted of 4 _ hst _ orbits , broken up into a series of 90 second exposures . in total , we obtained 273 of these exposures . in addition , we also examined archival data taken in march of 1999 with the space telescope imaging spectrograph ( stis ) , using the fuv - mama / f25qtz detector / filter combination , with a plate scale of 0.025 pixel@xmath10 , @xmath13 and @xmath14 . a full description of the data , as well as their reduction and analysis will be provided in a separate publication ( zurek et al . 2009 , in preparation ) . briefly , all of the fuv count rates and magnitudes presented in this paper were calculated via standard aperture photometry techniques , as implemented in the _ daophot _ package within _ iraf_. for the photometry on our acs / sbc ( stis / fuv - mama ) images , we used an aperture radius of 4 ( 7 ) pixels and a sky annulus extending from 10 to 20 ( 15 to 35 ) pixels . aperture photometry is sufficient for our purposes because the fuv image is not particularly crowded ( see figure 1 ) . the wavelength - dependent throughput curves of the acs / sbc / f140lp and stis / fuv - mama / f25qtz instrument / detector / filter combinations are very similar , though not identical . therefore we checked for far - uv variability by comparing the acs and stis count rates , after correcting for throughput differences and the different photometric aperture sizes and background regions that were used . we have calculated this correction factor from a set of ( mostly blue horizontal branch ) stars that are common to both sets of images . we find that for these stars , our acs / sbc count rates are 3.3 times larger than our stis / f25qtz ones . homer et al . ( 2001 ) have already used the hst / stis / f25qtz observations to identify the optical / far - uv counterpart of 4u 0513 - 40 . they confirm the suggestion of deutsch et al . ( 2000 ) that `` star a '' ( in the nomenclature of deutsch et al . ) is the correct counterpart to the lmxb , while two other blue sources previously suggested as possible counterparts by aurire , bonnet - bidaud & koch - miramond ( 1994 ) , designated as x-1 and x-2b , are inconsistent with the precise chandra position of 4u 0513 - 40 . figure 1 shows the location of these 3 sources in our acs / sbc images . since homer et al . ( 2001 ) had already reported a negative search for far - uv variability associated with star a in the stis observations , we started by focusing on the newer , higher signal - to - noise acs data . the acs - based far - uv light curves of all 3 objects are shown in figure 2 . it is immediately obvious that star a , the counterpart proposed by deutsch et al . ( 2000 ) and homer et al . ( 2001 ) , does , in fact , exhibit strong far - uv variability , especially between the 3 observing epochs . for example , the mean count rate drops by a factor of about 2 between epochs 2 and 3 . no similar change in count rate is seen in either x-1 or x-2b , which bracket star a in far - uv brightness . we then searched for _ periodic _ signals in the data by carrying out a power spectral analysis . figure 3 shows the lomb - scargle power spectra calculated for all 3 sources from the combined acs data sets . it is immediately obvious that only star a shows clear evidence of a signal at a frequency other than hst s orbital frequency ( @xmath15 ) . more specifically , there is an obvious peak at a frequency of about @xmath16 c d@xmath10 ( which corresponds to a period of @xmath17 min ) . in an effort to better understand the star a power spectrum ( specifically the power excess at @xmath15 and the sidebands at @xmath18 ) , we have created a simple model light curve . this model has exactly the same time sampling as the observations , contains a single 17-min periodic signal with amplitude 0.75 c s@xmath10 and accounts for the dominant long - term trend in the data by setting the simulated mean count rate in each epoch equal to the observed one . the power spectrum generated from this simple noise free model is shown in the second panel of figure 3 ( labelled `` star a simulation '' ) , and clearly captures all of the main features of the data . this shows that both the sidebands around @xmath19 and the strong apparent signal at @xmath15 are sampling artifacts . the latter , in particular , is due to leakage from the low - frequency power excess associated with the long - term variability . in order to test if the 17 min signal is persistent , we carried out power spectral analysis on each epoch independently , where we now also included the stis data that was obtained 7 years before the acs observations . the result is shown in figure 4 . the periodicity was indeed present in all 4 epochs ( 1 stis + 3 acs ) , but was noticeably weaker in the stis data and in the third acs epoch . considered in isolation , the power excess at @xmath19 in each of those epochs would be marginally significant at best . both figures 3 and 4 show that there are several possible aliases associated with the 17 min signal . in order to establish the relative likelihoods for each of these and get estimates of the errors associated with each of them , we carried out a bootstrap analysis . only the acs data was used for this purpose , since the signal is weakest in the stis data and since no unique cycle count can be assigned to the large time interval between the stis and acs observations . we began by removing long - term trends from the data stream by estimating the mean count rate in each hst orbit and subtracting these averages from the data . next , we created 1000 mock time series by sampling with replacement from the orbit - mean - subtracted acs data . finally , we created power spectra for all of these fake data sets and recorded the frequencies corresponding to peak flux . the histogram created from this set of peak frequencies is shown in figure 5 ( superposed on the orbit - mean subtracted power spectrum for all of the data , which is shown as the shaded region ) . as expected from the power spectrum , there are clearly four dominant plausible aliases . the relative number of bootstrap trials associated with each alias is a measure of the likelihood that this alias is the correct one ( see southworth et al 2006 , 2007 , 2008 and dillon et al . 2008 ) . similarly , the location and width of the histogram peak associated with each alias provides a simple estimate of the period and period error for this alias . our final results for the four viable aliases are : @xmath20 we also tried to establish more directly whether the periodic signal is consistent with being coherent across all of our epochs and whether the evidence for a changing amplitude is compelling . to this end , we carried out a least - squares fit to the orbit - mean - subtracted stis+acs data set for star a. in this fit , the period was kept fixed at @xmath21 that are present in the power spectrum of the _ combined _ stis+acs data . ] , so the free parameters were the phase and amplitude ( both assumed to be constant across the data ) . the result of this fit is shown in figure 6 . it illustrates rather clearly that the amplitude is not constant across epochs . as already noted above , the signal is strongest in acs epochs 1 and 2 and weakest in the stis data and acs epoch 3 . the fit also shows that the assumption of constant phase is reasonable , i.e. the observed signal is consistent with being coherent across the entire data set . in order both to test further the coherence properties of the signal , and to determine its average waveform , we have folded the entire stis+acs data stream for star a onto the same ( sub-)alias that was adopted for the fit in figure 6 . the result is shown in figure 7 , along with the phase - binned average waveform . again , the data seem to be consistent with the idea that the signal is fully coherent . moreover , the average waveform is fairly simple and roughly sinusoidal . as a final step , we carried out separate fits with fixed period to each individual epoch in order to quantify the changing amplitude of the 17-min signal . table 1 lists the mean count rates and absolute as well as fractional amplitudes of the signal as estimated from these fits . the fractional amplitude varies by over a factor of 3 between epochs , from about 3% to 10% , with the largest change happening across the 6-day gap between acs epochs 2 and 3 . it is also worth noting that there appears to be a correlation between the fuv brightness of the system ( i.e. the mean count rate ) and the amplitude of the 17-min signal , in the sense that the amplitude is highest when the system is brightest . this correlation holds regardless of whether the amplitude is expressed in absolute or fractional terms . it is worth noting that there is no inconsistency between our claim that the 17-min signal is present in the stis observations , and the non - detection of a periodicity in the same data set by homer et al . ( 2001 ) . as mentioned above , the signal is only marginally detectable if this data set is analysed in isolation . what makes the weak power excess in the stis observations convincing is that it is located at the same frequency as the obvious signal in the acs data set . indeed , homer et al . ( 2001 ) place a formal upper limit of 5% on the amplitude of any signal between 5 min and 6 hrs . this is entirely consistent with our own estimate of a 4% amplitude for the 17 min signal in the stis data ( see table 1 ) . we have shown that the fuv counterpart to 4u 0513 - 40 in ngc 1851 exhibits a clear periodic signal with @xmath0 min . this signal is roughly sinusoidal , has an amplitude of 3%-10% , is present in all four fuv observing epochs and is consistent with being fully coherent . these properties are in line with those of the orbital signals seen in uv / optical observations of other ucxb ( e.g. homer et al . 1996 ; anderson et al . 1997 ; dieball et al . 2005 ) and with a simple model in which these signals are due to a `` reflection effect '' associated with the irradiated donor star in the system ( arons & king 1993 ) . in the arons & king ( 1993 ) model , there are two components that contribute to the fuv / optical light : the irradiated front face of the donor star ( which is likely to be a low - mass , helium - core white dwarf ) , and the accretion disk , whose energy budget is also dominated by irradiation . the observed orbital signal is due to the changing projected area of the irradiated face of the donor star , with the fractional amplitude being set by the relative contributions of the disk and donor to the phase - averaged fuv / optical light . we have checked that this simple irradiation model matches roughly the fractional fuv signal using reasonable choices of parameters ( including a moderate - to - high inclination ) . the dependence on total fuv brightness is explained straightforwardly by different fuv responses of the donor and disk to variations in the irradiating x - ray luminosity . in this context , a factor of @xmath8 2 change in total fuv and fractional amplitude would requires roughly an order of magnitude variation in @xmath22 . this is consistent with the x - ray variability of the source in the rxte data base , where factor of 10 changes in count rate are seen on time - scales as short as weeks . for comparison , the largest previously reported variations in x - ray luminosity from this source were a factor of 5 ( grindlay & hertz 1983 ) . it seems likely that additional effects such as changes in radius , changes in reprocessing efficiency , and disk shielding , play a significant role in the variations of the amplitude of the uv oscillations . based on all this , we are confident that the fuv signal we have discovered is orbital in nature and probably due to a simple reflection effect . we therefore confirm 4u 0513 - 40 as a ucxb with @xmath17 min , making it the fourth confirmed ucxb in a galactic gc . the identification of this system as a ucxb is entirely consistent with existing circumstantial evidence regarding its nature . more specifically , the optical brightness , x - ray spectrum and burst properties of 4u 0513 - 40 have been known for some time to point towards a ucxb classification ( e.g. verbunt 2005 ) . given the extremely short orbital period we have measured , it is interesting to ask if the gravitational wave signal produced by this source would be detectable by lisa . the gravitational radiation strain for a circular orbit is given in a convenient form by nelemans , yungelson & portegies zwart ( 2001 ) as : @xmath23 where @xmath24 , @xmath25 is the mass of the neutron star primary , and @xmath26 is the mass of the white dwarf donor , and with the power coming out entirely in the second harmonic , for a circular orbit . if we adopt @xmath27 , @xmath28 kpc ( harris 1996 ) and @xmath29 ( suggested for this @xmath5 by the mass - radius relations in deloye & bildsten [ 2003 ] ) , we find a strain of @xmath30 . this is at approximately the 1-@xmath31 level for a one year integration with lisa , suggesting that the system may be detectable if the mission lifetime is several years . in practice , this will , however , also depend on the strength and frequency - dependence of the gravitational - wave background due to double wds , which may become dominant in the relevant frequency regime ( @xmath32 mhz ; e.g. nelemans et al . an additional challenge for detecting gravitational radiation from this object will be dealing with the possible effects of acceleration of the binary in the gravitational potential of the globular cluster , which could lead to a measurable frequency drift over several years , for which a correction would have to be made in order to keep all the gravitational wave power in a single frequency bin . the fact that the position and period of the source are already known ( and the latter can still be substantially improved before lisa starts ) will be helpful in this context and also ameliorate the hidden trials problem . we finally point out that , as more ucxb periods are being determined , it seems increasingly likely that there are significant differences between the period distributions of field and gc ucxbs . all four of the confirmed gc ucxbs have @xmath33 min , whereas all five of the confirmed field ucxbs have @xmath34 min ( nelemans & jonker 2006 ; int zand , jonker & markwardt 2007 ) . clearly , this comparison can not yet be taken at face value : the present numbers are still too small and selection effects have not been taken into account . moreover , there is probably no absolute dividing line between the two orbital period distributions . for example , the field lmxb 4u 1543 - 624 probably has an orbital period of around 18 minutes ( wang & chakrabarty 2004 ) , even though it is not yet included in the list of confirmed ucxbs given by nelemans & jonker ( 2006 ) and int zand et al . ( 2007 ) . nevertheless , the period distributions are in line with the expectation that different ucxb formation channels should dominate in the two different environments . if so , it may be possible to use the period distribution of field ucxbs as a tracer of binary evolution , and that of gc ucxbs as a tracer of stellar dynamics in dense environments . this work was supported by nasa through grant go-10184 from the space telescope science institute , which is operated by aura , inc . , under nasa contract nas5 - 26555 . we thank frank verbunt for pointing out that all known x - ray binaries with orbital periods less than half an hour are in globular clusters . arons , j. & king , i. r. 1993 , apj , 413 , l121 aurire , m. , le fvre , o. & terzan , a. 1984 , a&a , 138 , 415 bailyn , c. d. & grindlay , j. e. 1987 , apj , 316 , l25 bildsten , l. & deloye , c. j. 2004 , apj , 607 , l119 clark , g. w. 1975 , apj , 199 , l143 davies , m. b. , benz , w. & hills , j. g. 1992 , apj , 401 , 246 deutsch , e. w. , margon , b. , & anderson , s. f. 2000 , , 530 , l21 dieball a. , knigge c. , zurek d.r . , shara m.m . , long k.s . , charles p.a . , hannikainen d.c . , van zyl l. , 2005 , apj , 634 , l105 dillon , m. , gansicke , b.t . , augwerojwit , a. , rodriguez - gil , p. , marsh , t.r . , barros , s.c.c . , szkody , p. , brady , s. , krajci , t. & oksanen , a. 2008 , mnras , 386 , 1568 fabian a. c. , pringle j. e. , rees m. j. , 1975 , mnras , 172 , 15p harris , w.e . 1996 , aj , 112 , 1487 hills j.g . , 1976 , mnras , 175p , 1 homer , l. , charles , p. a. , naylor , t. et al . 1996 , mnras , 282 , l37 homer l. , anderson s.f . , margon b. , deutsch e.w . , downes r.a . , 2001 , apj , 550 , l155 homer , l. 2003 , kitp workshop : `` the physics of ultracompact stellar binaries '' ( feb 1 - 2 , 2003 ) , coordinators : l. bildsten , d. chakrabarty , g. nelemans , http://online.itp.ucsb.edu/online/ultra_c03/homer int zand j. j. m. , jonker p. g. , markwardt c. b. , 2007 , a&a , 465 , 953 ivanova , n. , rasio , f. a. , lombardi , j. c. , dooley , k. l. & proulx , z. f. 2005 , apj , 621 , l109 katz , j. i. 1975 , nature , 253 , 698 nelemans , g. 2009 , arxiv:0901.1778 nelemans g. , jonker p. g. , 2006 , astro , arxiv : astro - ph/0605722 rasio , f. a. , pfahl , e. d. & rappaport s. 2000 , apj , 532 , l47 stella , l. , priedhorsky , w. , & white , n. e. 1987 , apj , 312 , l17 southworth , j. , gansicke , b.t . , marsh , t.r . , de martino , d. , hakala , p. , littlefair , s. , rodriguez - gil , p. & szkody , p. 2006 , mnras , 373 , 687 southworth , j. , marsh , t.r . , gansicke , b.t . , augwerojwit , a. , hakala , p. , de martino , d. & lehto , h. 2007 , mnras , 382 , 1145 southworth , j. , gansicke , b.t . , marsh , t.r . , torres , m.a .. p . , steeghs , d. , hakala , p. , copperwheat , c.m . , aungwerojwit , a. , & mukadam , a. 2008 , mnras , 391 , 591 verbunt , f. 1987 , apj , 312 , l23 verbunt f. , 2005 , in interacting binaries : accretion , evolution , and outcomes , aip conference proceedings , aip conference series , 797 , 30 verbunt f. , lewin w.h.g . , 2006 , in _ compact stellar x - ray sources _ , cambridge university press : cambridge , uk ( w.lewin , m. van der klis eds . )

we present far - ultraviolet photometry obtained with the _ hubble space telescope _ of the low - mass x - ray binary 4u 0513 - 40 in the globular cluster ngc 1851 . our observations reveal a clear , roughly sinusoidal periodic signal with @xmath0 min and amplitude 3%-10% . the signal appears fully coherent and can be modelled as a simple reprocessing effect associated with the changing projected area presented by the irradiated face of a white dwarf donor star in the system . all of these properties suggest that the signal we have detected is orbital in nature , thus confirming 4u 0513 - 40 as an ultracompact x - ray binary ( ucxb ) . all four confirmed ucxbs in globular clusters have orbital periods below 30 minutes , whereas almost all ucxbs in the galactic field have orbital periods longer than this . this suggests that the dynamical formation processes dominate ucxb production in clusters , producing a different orbital period distribution than observed among field ucxbs . based on the likely system parameters , we show that 4u 0513 - 40 should be a strong gravitational wave source and may be detectable by lisa over the course of a multi - year mission .

0905.0145

suppose that at @xmath9 a gas of diffusing particles of constant density @xmath10 is brought in contact with a spherical absorber of radius @xmath0 in @xmath3 dimensions . the particles are absorbed upon hitting the absorber . remarkably , this simple setting captures the essence of many diffusion - controlled chemical kinetic processes @xcite . the evolution of the _ average _ coarse - grained particle density of the gas is described by the diffusion equation @xmath11 , \label{difeq}\ ] ] where @xmath12 is the gas diffusivity . here we will be interested in large fluctuations rather than in the average behavior . one important fluctuating quantity is the number of particles @xmath13 that is absorbed during a long time @xmath2 . we will focus on two questions : ( i ) what is the probability that @xmath14 , that is no particle hit the absorber until time @xmath2 ? ( ii ) what is the most likely history of the particle density of the gas conditional on the non - hitting until time @xmath2 ? these questions also appear in the context of a search for an immobile target by a swarm of diffusing searchers , see e.g. ref . @xcite and references therein . this process has been studied extensively in the simplest case when the searchers are non - interacting random walkers ( rws ) . in this case @xmath15 , and the probability that the target survives until time @xmath2 , @xmath16 was found to exhibit the following long - time behavior @xcite : - , & @xmath17 , [ survivaldecay1 ] + , & @xmath18 , [ survivaldecay2 ] + ( d-2 ) _ dr^d-2 d_0 t , & @xmath7 , [ survivaldecay3 ] where @xmath19 is the surface area of the @xmath3-dimensional unit sphere , and @xmath20 is the gamma function . equations ( [ survivaldecay2 ] ) and ( [ survivaldecay3 ] ) give the leading terms of the corresponding asymptotics at long times , when @xmath21 , i.e. , the characteristic diffusion length @xmath22 is very large compared to the target radius @xmath0 . equation ( [ survivaldecay1 ] ) is independent of @xmath0 , and the parameter @xmath6 is irrelevant . as a result , eq . ( [ survivaldecay1 ] ) becomes valid as soon as @xmath2 is much larger than the inverse microscopic hopping rate . the target survival problem is a particular case of a more general problem of finding the complete statistics of particle absorption by the absorber . for the rws , this problem has been recently studied in ref . @xcite . here we extend the target problem in several directions . first , we consider a lattice gas of _ interacting _ searchers . throughout most of the paper , we assume that the searchers interact via exclusion . this can be a good simplistic model for studying diffusion - controlled chemical reactions in crowded environments such as a living cell @xcite . specifically , we will consider a lattice gas described by the symmetric simple exclusion process ( ssep ) . in this process each particle can hop to a neighboring lattice site if that site is unoccupied by another particle . if it is occupied , the move is disallowed . the average behavior of this gas is still described by the diffusion equation ( [ difeq ] ) with @xmath23 @xcite , so the ssep and the rws are indistinguishable at the level of averages . however , as we show here , the long - time asymptotic of the target survival probability @xmath1 for the ssep behaves differently from that for the rws : @xmath24 this expression has the same structure as eq . ( [ survivaldecay3 ] ) , but it increases much faster with the gas density @xmath10 @xcite ; see fig . [ arcsinsquared ] . we note that previous results for the ssep only included bounds on @xmath25 @xcite . second , we show that , for @xmath17 , the survival probability @xmath1 depends strongly on the initial condition . this effect does not require inter - particle interaction , and it also occurs for the rws , as we show below . in particular , the asymptotic ( [ survivaldecay1 ] ) is only valid after averaging over random initial distributions of particles , that is , for the annealed setting @xcite . we find a different result for a deterministic initial condition , also called a quenched setting @xcite . for the rws , the two results for @xmath8 differ by a numerical factor . for the ssep , even their @xmath10-dependence is different for @xmath17 . third , we demonstrate that the two basic one - dimensional solutions , annealed and quenched , play a central role in higher dimensions when one is interested in _ intermediate _ asymptotics of @xmath1 for @xmath26 , that is when the diffusion length @xmath22 is much longer than the lattice constant , but much shorter than the absorber radius @xmath0 . fourth , in addition to evaluating @xmath1 in different regimes , we also find the most likely history of the gas density conditional on the target survival until time @xmath2 . we achieve this result , and most of the others , by employing the macroscopic fluctuation theory ( mft ) @xcite . this coarse - grained large - deviation formalism was unavailable when most of the studies of the target survival probability were performed . the mft is well suited for the analysis of large deviations in lattice gases , including the ( unlikely ) target survival at long times that we are dealing with here . one of our central findings for the ssep is that , in the long - time regime , @xmath27 , the most likely gas density profile for @xmath7 is , for most of the time , almost stationary : @xmath28 , \ ] ] where the radial coordinate is rescaled by the diffusion length @xmath22 . it is this density profile that determines the leading term ( [ actresultd ] ) of the survival probability . last but not least , we extend our approach to a whole class of additional interacting diffusive gases . in the next section we present the mft formulation of the target survival problem . section [ steady ] deals with @xmath29 for @xmath27 . here @xmath8 is mostly contributed to by a _ stationary _ solution of the mft equations , independently of whether the setting is annealed or quenched . we derive these solutions , evaluate @xmath30 and verify the results for @xmath31 by solving the mft problem numerically . in sec . [ nonstat ] we study analytically and numerically the survival probability in non - stationary settings , deterministic and random , in all dimensions and at different densities . in sec . [ extension ] we extend our results for @xmath29 to a broad class of interacting lattice gases . our main results are summarized in sec . [ discussion ] . in appendix we present , for non - interacting rws , exact microscopic derivations of @xmath1 for the annealed and quenched settings and for @xmath17 , @xmath32 and @xmath33 . both the microscopic derivation and the mft calculations show that , for @xmath34 , the leading contribution to @xmath8 is sensitive to the initial condition only in one dimension . the macroscopic fluctuation theory ( mft ) was developed for the analysis of non - equilibrium steady states of diffusive lattice gases @xcite . subsequently it was extended to a host of non - stationary settings @xcite . the mft , and its extensions to reacting particle systems @xcite , have proven to be highly efficient and versatile . here we outline the mft formulation , referring the reader to the above references for further details . the starting point for the derivation of the mft can be a langevin equation that provides a faithful large - scale description to a broad family of diffusive gases : @xmath35 + \nabla \cdot \left[\sqrt{\sigma(n ) } \,\text{\boldmath$\eta$ } ( \mathbf{x},t)\right],\ ] ] where @xmath36 is a zero - average gaussian noise , delta - correlated both in space and in time @xcite . as one can see , a fluctuating diffusive gas is fully characterized by @xmath12 and another coefficient , @xmath37 , that comes from the shot noise and is equal to twice the mobility of the gas @xcite . essentially , the mft formalism is a wkb theory ( after wentzel , kramers and brillouin ) of the functional fokker - planck equation following from the langevin equation ( [ lang ] ) . the wkb theory employs , in a smart way , the typical number of particles in the relevant region of space as a large parameter @xcite . in the mft formalism , the particle number density field @xmath38 and the canonically conjugate momentum " density field @xmath39 obey the hamilton equations @xmath40 , \label{d1 } \\ \partial_t p & = & - d(q ) \nabla^2 p-\frac{1}{2 } \,\sigma^{\prime}(q ) ( \nabla p)^2 , \label{d2}\end{aligned}\ ] ] where the prime denotes the derivative with respect to the argument . equations and can be written in terms of variational derivatives : @xmath41 here @xmath42= \int d\mathbf{x}\,\mathcal{h}\ ] ] is the hamiltonian , and @xmath43 is the hamiltonian density . the spatial integration in eq . ( [ hamiltonian ] ) , and everywhere in the following , is performed over the whole space outside the target . because of the rotational symmetry of the problem , we assume that the solution only depends on the radial coordinate and time . we will consider the target survival problem in an arbitrary dimension @xmath3 . the boundary conditions on the target are @xmath44 @xcite , where the condition on @xmath45 just fixes an arbitrary constant . far away from the target the gas is unperturbed , so @xmath46 . the boundary conditions in time are the following . at @xmath9 we prescribe @xmath47 where , for the ssep , @xmath48 . this is a deterministic , or quenched , initial condition , see refs . a random initial condition ( that is , an annealed setting ) is considered in sec . [ app : annealed ] . before focusing on the target survival problem , let us consider for a moment a slightly different setting where @xmath13 , the specified number of absorbed particles by time @xmath49 , is arbitrary . this condition , @xmath50 = n,\ ] ] imposes an integral constraint on the solution . this constraint is identical to the one arising in the problem of statistics of integrated current during a specified time @xcite . a similar derivation yields the following boundary condition for @xmath51 at @xmath49 : @xmath52 where @xmath53 is the heaviside step function , and @xmath54 is an a priori unknown lagrange multiplier that is ultimately set by eq . ( [ number ] ) accordingly , we demand @xmath55 . the particular case of @xmath14 in which we are interested here corresponds to @xmath56 @xcite . in this case the total particle flux to the target vanishes at all times @xmath57 . the solution of the mft equations for @xmath58 yields the optimal trajectory : the most likely density history of the system conditional on the number of absorbed particles @xmath13 . once @xmath58 and @xmath59 are found , we can calculate the mechanical action @xmath60 which yields @xmath61 up to a pre - exponential factor : @xmath62 for the ssep @xmath63 and @xmath64 @xcite , and eq . ( [ actionmain ] ) becomes @xmath65 upon rescaling @xmath66 by @xmath2 and @xmath67 by @xmath22 @xcite , we can effectively put @xmath68 in eqs . ( [ number ] ) and ( [ pt ] ) and replace @xmath0 by @xmath69 and @xmath13 by @xmath70 everywhere . equation ( [ actionmainssep ] ) for the ssep becomes @xmath71 where @xmath72 we are interested in the limit of @xmath73 as @xmath74 . in one spatial dimension , @xmath17 , the parameter @xmath0 ( and hence @xmath75 ) is irrelevant because of the translational symmetry of the ensuing mft problem . we will consider this case in sec . [ 1d ] . for @xmath29 there are two natural limiting cases : of small and large @xmath6 . a small @xmath6 in the _ deterministic _ theory , described by eq . ( [ difeq ] ) , means that @xmath2 is much longer than the characteristic diffusion time @xmath76 needed for the gas density to approach a steady state around the target . as a result , the average particle flux to the target can be determined by using the _ stationary _ solution of the diffusion equation . for @xmath15 this reduces to solving the laplace equation @xmath77 with the boundary conditions @xmath78 and @xmath79 , leading to @xmath80 we argue that same logic holds for _ fluctuations _ , including those responsible for the survival probability . hence , when @xmath27 , the leading order contribution to the action @xmath60 from eq . ( [ actionmain ] ) comes from the _ stationary _ solution of the mft equations that obeys the boundary conditions in space , but not the boundary conditions in time . for such solutions eqs . ( [ d1 ] ) and ( [ d2 ] ) become @xmath81= j= \text{const } , \label{zeroflux}\\ & & \frac{d(q)}{r^{d-1 } } \,\frac{d}{dr}\left(r^{d-1 } v\right)+\frac{1}{2}\,\sigma^{\prime}(q ) v^2= 0 , \label{steadyv}\end{aligned}\ ] ] where @xmath82 . the target survival implies that the particle flux at @xmath83 vanishes at all times @xmath57 . therefore @xmath84 , and from eq . ( [ zeroflux ] ) @xmath85 . plugging this into eq . ( [ steadyv ] ) we obtain @xmath86 where @xmath87 is the spherically symmetric laplace operator in @xmath3 dimensions . for the ssep eq . ( [ steadyeqgen ] ) reads @xmath88 remarkably , the substitution @xmath89 reduces the nonlinear ordinary differential equation ( [ steadyssep ] ) to the spherically symmetric laplace equation in @xmath3 dimensions : @xmath90 the boundary conditions @xmath91 and @xmath92 become @xmath93 and @xmath94 . solving this problem and returning to @xmath95 , we obtain eq . ( [ qd ] ) . this is the most likely density profile conditional on survival of the target until time @xmath2 . now we can calculate @xmath96 : @xmath97}.\end{aligned}\ ] ] in particular , for @xmath31 @xmath98 , \label{q3}\\ v(r ) & = & \frac{2\ell \arcsin\sqrt{n_0}}{r^2 \sin\left[2 \left(1-\frac{\ell}{r}\right)\,\arcsin \sqrt{n_0}\right]}. \label{v3}\end{aligned}\ ] ] the asymptotic of @xmath99 near the target , @xmath100 is quadratic in @xmath101 . also notable is a diverging asymptotic of @xmath102 near the target : @xmath103 which is independent of @xmath10 . the asymptotic behaviors near the target assure that the particle flux to the target vanishes . furthermore , each of the two terms in the flux , see eq . ( [ zeroflux ] ) , vanish separately . as it turns out , these features , including the one over the distance asymptotic ( [ vdiverge ] ) , are quite universal : they are observed , for @xmath104 , in the quenched and annealed settings and in all dimensions ( including @xmath17 where the mft solution is _ non_-stationary ) for all lattice gases that behave as non - interacting rws at low densities . an example of the stationary gas density profile for @xmath31 is shown in fig . [ 3dthnum ] . in spite of the singularity of @xmath96 at @xmath83 , the action ( [ actionscaled2 ] ) is bounded , and we obtain @xmath105 and arrive at eq . ( [ actresultd ] ) . in particular , for @xmath31 @xmath106 and @xmath107 notice that as @xmath10 approaches @xmath108 , the asymptotic survival probability goes down rapidly but remains non - zero . as the solution ( [ qd ] ) and ( [ vd ] ) is stationary , the survival probability is independent , in the leading order , of whether the particles are distributed randomly or deterministically at @xmath9 . here for very long times , @xmath109 , the optimal fluctuation becomes unconstrained by the process duration , and details of the initial condition become irrelevant . as we will see in sec . [ nonstat ] , the situation changes for @xmath17 , and for any @xmath3 when @xmath34 . for @xmath110 , eq . ( [ s3d ] ) reduces to eq . ( [ survivaldecay3 ] ) for the rws . further , eqs . ( [ qd ] ) and ( [ vd ] ) become @xmath111^{-1}. \label{vdrw}\end{aligned}\ ] ] these low - density asymptotics for the ssep represent exact solutions for the rws , where @xmath63 and @xmath112 @xcite . the stationary solution ( [ qd ] ) and ( [ vd ] ) , or ( [ qdrw ] ) and ( [ vdrw ] ) , does not satisfy the boundary conditions in time . to accommodate these boundary conditions , the full time - dependent solutions of the mft problem must develop narrow boundary layers in time at @xmath9 and @xmath113 , cf . the boundary layers only give a subleading contribution to @xmath114 . we verified this scenario numerically for the ssep . figure [ 3dthnum ] shows the density history obtained by solving the full non - stationary mft problem numerically for a sufficiently small @xmath6 and @xmath31 . the numerical algorithm is described in sec . [ numeric1d ] . one can see that @xmath115 stays almost constant for most of the time and , at these times , agrees very well with the theoretical prediction ( [ q3 ] ) . the rescaled action , found numerically from eq . ( [ actionscaled2 ] ) , is also close to the theoretical prediction , eq . ( [ s3d ] ) . in the marginal case of @xmath18 logarithmic corrections appear . for @xmath18 all non - constant solutions of the circularly - symmetric laplace s equation ( [ laplace ] ) diverge logarithmically with @xmath67 . as a result , a reasonable stationary solution of eq . ( [ steadyssep ] ) can only be obtained if we introduce a cutoff distance @xmath116 ( to remind the reader , all lengths are rescaled by @xmath22 ) : @xmath117 correspondingly , @xmath118 although there is a derivative jump in @xmath99 at @xmath119 , and divergence of @xmath96 at @xmath83 and @xmath119 , the flux is continuous ( and equal to zero ) everywhere . the cutoff @xmath116 in these formulas should be chosen @xmath120 : in the original variables it is of the order of the characteristic diffusion length @xmath22 where the stationary solution crosses over to a time - dependent one . the uncertainty the cutoff introduces only affects the argument of the logarithm . as a result , eqs . ( [ q2 ] ) and ( [ v2 ] ) are correct with logarithmic accuracy . the same happens @xcite if one circumvents the tedious exact time - dependent solution of the two - dimensional ( @xmath121 ) diffusion equation ( [ difeq ] ) and solves instead the laplace equation @xmath77 for the purpose of computing the average particle flux onto the absorber . the rescaled action is @xmath122^{-1}\arcsin^2\ ! \sqrt{n_0}$ ] with the same logarithmic accuracy . thus @xmath123 for @xmath110 , eq . ( [ action2 ] ) reduces to eq . ( [ survivaldecay2 ] ) as expected . in the short - time limit , @xmath26 , the deterministic theory of diffusion - controlled absorption is non - stationary . for @xmath17 the non - stationarity holds , in the deterministic theory , for any @xmath6 . again , we argue that the same features hold in the context of survival probability . for the ssep in one dimension eqs . ( [ d1 ] ) and ( [ d2 ] ) can be written as @xmath124 , \label{qt } \\ \partial_t p & = & - \partial_{x}^2 p+(2 q-1)(\partial_x p)^2 , \label{pt}\end{aligned}\ ] ] whereas the hamiltonian density ( [ ham ] ) becomes @xmath125 here , and in most of the following exposition on the ssep and rw , we put @xmath126 . we will consider a one - sided problem and put the absorbing wall at @xmath127 , so that @xmath128 . we assume a deterministic initial condition , @xmath129 and demand @xmath130 . upon rescaling @xmath66 by @xmath2 and @xmath131 by @xmath132 eq . ( [ pt ] ) becomes @xmath133 and we also have @xmath134 . we remind the reader that @xmath54 is ultimately set by the number of absorbed particles : when this number goes to zero , @xmath135 @xcite . once @xmath115 and @xmath136 are found , we obtain @xmath137 where the subscript in @xmath138 refers to @xmath17 . we have been unable to solve this problem exactly for arbitrary @xmath10 . in the following we solve it in the limit of @xmath110 , when the ssep reduces to rws . based on these results , we then compute the next - order correction in @xmath10 perturbatively . at the end of this subsection we solve the problem numerically for a range of values of @xmath10 . in the limit of @xmath110 we can drop @xmath139 in the hamiltonian density ( [ w ] ) , and the corresponding terms in the mft equations , arriving at the rw model . as in other examples @xcite , the mft problem for the rw is solvable by the hopf - cole transformation @xmath140 and @xmath141 . this is because , in the new variables , the hamilton equations are decoupled : @xmath142 we can solve the anti - diffusion equation ( [ pt ] ) backward in time , with the initial condition @xmath143 and the boundary conditions @xmath144 and @xmath145 . the solution is @xmath146 where @xmath147 , and @xmath148 is the error function . at @xmath9 we obtain @xmath149 this expression is the initial condition for the diffusion equation ( [ qt ] ) forward in time . the boundary conditions are @xmath150 and @xmath151 . the solution is @xmath152 transforming back to @xmath95 and @xmath51 , and taking the limit of @xmath135 @xcite , we obtain @xmath153 figure [ rwquenchedq ] depicts the density history of the system as described by eq . ( [ qrw ] ) . the lower panel shows a density void " that forms immediately . also noteworthy is a density peak that accompanies the void formation . to the right of the density peak @xmath154 is very small , and the dynamics is essentially governed by the deterministic equation ( [ difeq ] ) and corresponds to a diffusive outflow of the gas . at @xmath113 we obtain @xmath155 as one can see , @xmath156 behaves linearly in @xmath131 at small @xmath131 . at @xmath157 , however , @xmath156 is quadratic at small @xmath131 , as in the stationary solution derived above . now , @xmath158 , as described by eq . ( [ vrw ] ) , again exhibits the universal one over the distance asymptotic . indeed , at @xmath159 , @xmath160 independent of time . this asymptotic already holds at @xmath9 . the character of singularity at @xmath127 only changes at @xmath113 , as @xmath161 is equal to @xmath162 with an infinite prefactor @xmath135 . to compute the rescaled action @xmath163 we write @xmath164 which follows from eq . ( [ actionscaled1d ] ) at @xmath165 . it is more convenient , however , to use the formula ( derived in @xcite ) which only includes spatial integration . for @xmath14 this formula simplifies to @xmath166.\ ] ] after cancelations we obtain @xmath167 where @xmath168 . as a result , @xmath169 has the same scaling with @xmath10 and @xmath2 as in eq . ( [ survivaldecay1 ] ) . the coefficient @xmath170 , however , is different from the coefficient @xmath171 obtained for the random initial condition ( see ref . @xcite , sec . [ app : annealed ] and appendix ) . equation ( [ 1dquenched ] ) can also be deduced from an exact microscopic derivation when the particles are initially distributed periodically in space , see ref . @xcite and appendix . the final density distribution ( [ qt1 ] ) can also be obtained from microscopic arguments . now let us go back to the ssep and consider a small but finite @xmath10 . we can calculate a small correction @xmath172 to the action ( [ actionrw2 ] ) by treating the term @xmath173 of the ssep hamiltonian ( [ w ] ) perturbatively . in the first order of perturbation theory we have @xmath174 \nonumber \\ & = & \int_0 ^ 1 dt \int_0^{\infty } dx\,q_0 ^ 2(x , t ) v_0 ^ 2(x , t ) , \label{s1a}\end{aligned}\ ] ] where @xmath175 and @xmath176 are the _ unperturbed _ solutions , given by the rw formulas ( [ qrw ] ) and ( [ vrw ] ) . plugging eqs . ( [ qrw ] ) and ( [ vrw ] ) into eq . ( [ s1c ] ) we obtain @xmath177 ^ 2 , \label{s1c}\end{aligned}\ ] ] where @xmath147 . to evaluate the above integral , we first replace the square of the integral over @xmath178 by a product of two identical integrals over @xmath179 and @xmath180 . the integration over @xmath181 reduces to calculating gaussian integrals : @xmath182 \nonumber \\ & \times & \!\!\!\left[e^{-\frac{\left(x-\mu_2\right){}^2}{t}}-e^{-\frac{\left(x+\mu_2\right){}^2}{t}}\right ] \nonumber \\ & = & \!\!\!\sqrt{\frac{\pi\ , t(1-t ) } { 2}}\,e^{-\frac{\left(\mu_1+\mu_2\right){}^2 ( t+1)}{2 t}}\left(e^{\frac{2 \mu_1 \mu_2}{t}}-e^{2 \mu_1 \mu_2}\right ) . \label{intx}\end{aligned}\ ] ] now we perform integration over @xmath66 in eq . ( [ s1c ] ) : @xmath183 \equiv i(\mu_1 , \mu_2 ) , \label{intt}\end{aligned}\ ] ] where @xmath184 . the remaining double integral over @xmath179 and @xmath180 is evaluated numerically to yield @xmath185 where @xmath186 . therefore , @xmath187 we solved the mft equations using a modification of the iteration algorithm , originally developed by chernykh and stepanov @xcite for evaluating the probability density of large negative velocity gradients in the burgers turbulence . variants of this algorithm have been used in the context of mft of lattice gases , with and without on - site reactions @xcite . the algorithm iterates the diffusion - type equation ( [ d1 ] ) forward in time from @xmath9 to @xmath113 , and the anti - diffusion - type equation ( [ d2 ] ) backward in time from @xmath113 to @xmath9 . as in ref . @xcite , our implementation of this algorithm involved an implicit finite difference scheme , which is beneficial for iteration convergence . at fixed @xmath10 and @xmath54 we continued iterations until local convergence of the solutions was achieved with a high accuracy . then we increased @xmath54 and repeated the solution until the action ( [ actionscaled2 ] ) converged to 1 per cent . we also verified that , for large @xmath54 that we achieved , the mass loss to the absorber was negligible . figure [ ssepnum ] shows an example of our numerical solution for the deterministic initial condition and @xmath17 . at small and moderately large @xmath10 , the density history of the system is similar to that for rws , with a rapidly forming density void accompanied by a density peak . the density peak is lower than for the rws , and it becomes progressively lower and broader as @xmath10 approaches @xmath108 . the numerically found @xmath188 exhibits , at small @xmath131 , the universal asymptotic ( [ sing1d ] ) . figure [ snumeric ] shows the numerically found @xmath189 for the deterministic initial condition and @xmath17 . for small @xmath10 , there is an excellent agreement with the rw asymptotic ( [ actionrw2 ] ) . for moderate @xmath10 , the results agree with the weakly - nonlinear asymptotic ( [ actionweak ] ) . as @xmath10 continues to grow , @xmath138 grows more rapidly . it must diverge at @xmath190 , because in this case @xmath191 scales with time as @xmath2 rather than @xmath132 , as follows from simple microscopic arguments . our numerical solution becomes prohibitive at @xmath10 very close to 1 . the available data indicate the @xmath192 divergence of @xmath138 as @xmath193 . a spherically symmetric three - dimensional version of the iteration algorithm was used for the verification of the stationary solution for @xmath31 , presented in sec . [ larged ] . when @xmath26 , eq . ( [ actionscaled2 ] ) simplifies to @xmath195 the remaining double integral is equal to the rescaled action @xmath189 in the ( rescaled ) _ one - dimensional _ problem , @xmath196 , with an absorber at @xmath197 . because of the translational invariance , @xmath198 is independent of @xmath6 . as a result , @xmath199 here @xmath200 is proportional to @xmath132 , rather than @xmath2 . in particular , for @xmath31 @xmath201 the case of @xmath18 is not special here , and eq . ( [ sfast ] ) holds : @xmath202 in the annealed setting , that we consider here , one allows equilibrium fluctuations in the initial condition and averages over them . in a stochastic realization of the process , the initial density profile is chosen from the equilibrium probability distribution corresponding to density @xmath10 . as a consequence , the most likely initial density profile , conditional on the target survival until time @xmath2 , is different from the flat profile @xmath203 . the cost " of optimal fluctuation now includes the cost of creating the optimal initial density profile . still , the total cost is less than the cost for the quenched ( deterministic ) initial condition , so the survival probability for the annealed setting is higher than for the quenched setting . in the mft formalism , the annealed setting is described , in one dimension , by the initial condition that involves a combination of @xmath204 and @xmath205 @xcite : @xmath206 for the ssep , @xmath207 and @xmath208 , this becomes @xmath209 } = \lambda \,\theta(x).\ ] ] for the rws , @xmath207 and @xmath210 , we have @xmath211 equation ( [ inannrw ] ) replaces eq . ( [ t01 ] ) in sec . when @xmath115 and @xmath136 are found , one can evaluate @xmath212 here @xmath138 is the action given by eq . ( [ actionscaled1d ] ) [ but with a different @xmath115 , see below ] , whereas @xmath213 is the cost of creating the optimal initial condition @xmath214 . this cost is given by the boltzmann - gibbs formula @xcite . for the ssep @xmath215 } \right\},\ ] ] whereas for the rws @xmath216.\ ] ] for the rws , the annealed problem can be solved via the hopf - cole transformation . in the new variables @xmath217 and @xmath218 , the initial condition ( [ inannrw ] ) yields : @xmath219 solving the diffusion equation ( [ qt ] ) with this initial condition and the boundary conditions @xmath220 and @xmath221 , we obtain @xmath222 where @xmath147 as before . now , @xmath223 is still described by eq . ( [ stepp ] ) . therefore , we can calculate @xmath224 . sending @xmath54 to infinity , we arrive at @xmath225 a symmetric function of @xmath226 . there is no density peak in the annealed setting : the density is monotonically increasing with @xmath131 at all times . interestingly , at @xmath113 and @xmath9 the optimal density @xmath227 is the same as predicted by the deterministic theory , eq . ( [ difeq ] ) , at @xmath113 . at times @xmath157 , the optimal density profile @xmath115 is a quadratic function of @xmath131 at small @xmath131 as before . the action @xmath138 is given by eq . ( [ actionrw1 ] ) with the same @xmath158 as in the quenched case , eq . ( [ vrw ] ) , and with @xmath115 given by eq . ( [ qann ] ) . as in the quenched setting , it is more convenient to calculate @xmath138 using eq . ( [ s ] ) that is equally valid in the annealed case . the cost of the initial condition @xmath213 can be evaluated from eq . ( [ freeenergy ] ) . adding up @xmath213 and @xmath138 , we obtain after cancelations @xmath228 so @xmath229 with @xmath230 , in agreement with previous results @xcite , see also appendix . to our knowledge , the optimal density history ( [ qann ] ) that contributes most to this survival probability , has been previously unknown . now we return to the ssep . assuming @xmath110 , we can calculate a small correction @xmath231 to the expression @xmath232 from eq . ( [ s0s1 ] ) . the correction to @xmath138 is again calculated from eq . ( [ s1a ] ) , where @xmath176 is still given by eq . ( [ vrw ] ) , but @xmath175 is now given by the annealed history , eq . ( [ qann ] ) . we obtain @xmath233 the integral over @xmath131 can be evaluated using the formula @xmath234 the remaining integral over @xmath66 is elementary , @xmath235 and we obtain @xmath236 there is also a small correction to @xmath213 that comes from the difference of free energies of the ssep and the rws . we calculate this correction by expanding the integrand of eq . ( [ freeenssep ] ) in small @xmath10 and @xmath214 up to , and including , the quadratic terms . the resulting correction is @xmath237 ^ 2= \frac{(2-\sqrt{2 } ) n_0 ^ 2}{\sqrt{\pi}},\ ] ] where we used the zero - order result ( [ q0ann ] ) for @xmath214 . adding up @xmath238 and @xmath239 , we finally obtain , for the annealed setting , @xmath240 . \label{actionweakan}\end{aligned}\ ] ] the @xmath241 correction agrees with the results of santos and schtz @xcite . they solved a different problem for the ssep , which involved particle injection from the boundary into a semi - infinite line . remarkably , that problem can be mapped , already at the exact microscopic level , into the target survival problem we are dealing with here . as a result , the @xmath241 correction in the annealed setting , described by eq . ( [ actionweakan ] ) , corresponds to the second cumulant of the statistics of the total number of injected particles at time @xmath49 , when the system is empty at @xmath9 @xcite . overall , eqs . ( [ actionweak ] ) and ( [ actionweakan ] ) show that , in one dimension , the survival probability exhibits different @xmath10-dependences in the quenched and annealed settings . when @xmath26 , the @xmath242 results for the annealed setting represent an essential building block " in all dimensions @xmath194 . here one obtains @xmath243 these equations resemble eqs . ( [ sfast ] ) and ( [ sfast ] ) , except that the rescaled one - dimensional action for the annealed setting @xmath244 is different from the corresponding quantity @xmath189 for the quenched setting . for very small densities @xmath244 is described by eq . ( [ annrw1 ] ) . for small but finite densities it is given by eq . ( [ actionweakan ] ) . for arbitrary @xmath10 , it can be found numerically . importantly , the steady - state equation ( [ steadyeqgen ] ) can be solved analytically for general @xmath245 and @xmath246 , thus extending our long - time results for @xmath29 to a whole family of diffusive gases of interacting particles . indeed , by denoting @xmath247}{\sqrt{\sigma[q(r)]}}\ ] ] we can recast eq . ( [ steadyeqgen ] ) into a linear first order ordinary differential equation ( ode ) , @xmath248 whose general solution is @xmath249,\ ] ] where @xmath250 . using eq . ( [ uf ] ) , we obtain one more first - order ode that can be easily integrated . using the boundary conditions @xmath91 and @xmath92 to determine the two integration constants , we obtain the solution for @xmath99 in implicit form : @xmath251 this solution exists for all lattice gases for which the integrals in eq . ( [ qgeneral ] ) are bounded . this puts a limitation on the behaviors of @xmath245 and @xmath246 at @xmath252 . for example , let @xmath253 and @xmath254 . then the integrals converge at @xmath255 if and only if @xmath256 for the ssep and rws one has @xmath257 and @xmath258 . therefore , the condition ( [ crit ] ) is satisfied , and the solution ( [ qgeneral ] ) exists . the condition ( [ crit ] ) is also satisfied for a family of repulsion processes @xcite . when the solution ( [ qgeneral ] ) exists , the action is bounded leading to a nonzero target survival probability . the rescaled action is the following : @xmath259^{2 } , \label{scaledgeneral}\end{aligned}\ ] ] where eq . ( [ u(r ) ] ) and the steady - state relation @xmath85 have been used in the last step . as a result , @xmath260^{2}.\ ] ] this closed - form result solves the target survival problem for a broad class of diffusive lattice gases . it has the same structure as eq . ( [ survivaldecay3 ] ) except the @xmath10-dependence which is model - specific . when specialized to the rw and ssep , eq . ( [ pgeneral ] ) yields eqs . ( [ survivaldecay3 ] ) and ( [ actresultd ] ) , respectively . for @xmath18 we obtain , with logarithmic accuracy , the long - time asymptotic @xmath261^{2}.\ ] ] as an additional illustration of the general results ( [ qgeneral ] ) and ( [ pgeneral ] ) , we consider a family of zero range processes ( zrp ) . a zrp describes interacting ( but not excluding ) random walkers on a lattice : a particle at site @xmath262 can hop to a neighboring site with a rate @xmath263 that only depends on the number of particles @xmath264 on the departure site @xmath262 . naturally , @xmath265 . if @xmath266 , the zrp is described at the macroscopic level by @xmath267 and @xmath268 , see e.g. ref . therefore , @xmath269 . evaluating the integrals in eq . ( [ qgeneral ] ) , we obtain for the stationary density profile : @xmath270 ^ 2.\ ] ] in its turn , eq . ( [ pgeneral ] ) yields the long - time asymptotic of the target survival probability for the zrp : @xmath271 in this work we evaluated the survival probability @xmath1 of a spherical target of radius @xmath0 in a gas of unbiased diffusive particles ( searchers " ) , with density @xmath10 , that interact with each other via exclusion as described by the ssep . we also determined the most likely particle density history conditional on the target survival until time @xmath2 . the results depend on the dimension of space @xmath3 and on the basic rescaled parameter @xmath4 . when @xmath6 is small and @xmath7 , @xmath1 is mostly contributed to by an exact _ stationary _ solution of the macroscopic fluctuation theory ( mft ) that we obtained . for large @xmath6 , and for any @xmath6 in one dimension , the relevant mft solutions are non - stationary . in this case @xmath8 scales differently with @xmath2 , @xmath0 , @xmath3 and @xmath10 , and it also depends on whether the initial condition is deterministic or random . these effects ( for large @xmath6 , and for any @xmath6 in one dimension ) are also observed in the absence of exclusion : for non - interacting random walkers . in the special case of @xmath34 and @xmath18 logarithmic corrections to @xmath1 appear . table [ table1 ] can serve as a quick guide to our main results for the survival probability for the ssep in different limits . the long - time asymptotics of the survival probability for a whole class of interacting lattice gases for @xmath7 and @xmath18 are given by eqs . ( [ pgeneral ] ) and ( [ pgeneral2d ] ) , respectively . .@xmath1 for the ssep in different limits [ cols="<,<,<",options="header " , ] the mft provides a simple interpretation to the fact that , at @xmath7 , @xmath272 scales as @xmath2 at very long times , whereas at @xmath17 it scales as @xmath132 . the difference in behavior is related to the existence or non - existence of a proper stationary solution of the mft equations . in summary , the macroscopic fluctuation theory opens new directions in the classical problem of target survival probability . we are grateful to davide gabrielli for a helpful comment and to gunter schtz for a useful discussion of ref . @xcite and its connection to the target survival probability of the ssep in one dimension . two of us ( bm and plk ) thank the galileo galilei institute for theoretical physics for the hospitality and the infn for partial support during the completion of this work . this research was supported by grant no . 2012145 from the united states israel binational science foundation ( bsf ) . here we present microscopic derivations of the target survival probability @xmath1 for the non - interacting rws in one , two and three dimensions . we verify that , for @xmath17 , @xmath1 depends on the initial conditions ( random or deterministic ) . we also show that , for @xmath18 and @xmath33 , the leading term of the asymptotic of @xmath1 is independent of the initial conditions . for @xmath31 we reproduce , both in microscopic calculations and in the mft framework , an exact result @xcite for @xmath1 in the random setting . finally , we derive , for @xmath31 , a more accurate asymptotic of @xmath1 for the deterministic setting . in all these calculations we set @xmath126 . we start with random initial conditions and first consider rws on a large but finite interval @xmath273 . a rw starts in the interval @xmath274 , where @xmath275 , with probability @xmath276 . the probability that this rw does not hit the target ( which is at the origin ) until time @xmath2 is @xmath277 . averaging over random particle locations at @xmath9 we obtain the average single - particle non - hitting probability @xmath278 for sufficiently large @xmath279 we can assume that the number of rws on the interval is equal to @xmath280 . since all the @xmath281 particles are independent , the probability that none of them hits the target is @xmath282^{n_0l},\ ] ] which in the @xmath283 limit becomes @xmath284 \nonumber \\ & = & \exp\!\left(-\frac{2 n_0 \sqrt{t}}{\sqrt{\pi}}\right ) , \label{anmicro}\end{aligned}\ ] ] in agreement with eq . and ref . @xcite . now let the initial positions of our rws be deterministic . one example of deterministic setting is a periodic one , with exactly one particle on each site @xmath285 of a lattice with lattice constant @xmath286 . the probability that the particles that is initially located at site @xmath287s does not hit the target until time @xmath2 is @xmath288 $ ] . the probability that neither of the particles hit the target is @xmath289 taking the logarithm we obtain @xmath290 we are interested in the regime of @xmath291 , when the characteristic diffusion length is much larger than the lattice constant . the leading - order result can be obtained by replacing the summation in eq . ( [ summation ] ) by integration . here we present a more accurate result that also includes a pre - exponential factor . we use the asymptotic @xcite : @xmath292 here @xmath293 , see eq . ( [ actionrw2 ] ) . as a result , @xmath294 the exponential factor is independent of details of the deterministic initial condition . it coincides with our mft result ( [ 1dquenched ] ) and differs from the annealed result , eqs . ( [ anmicro ] ) and and ref . @xcite . the pre - exponential factor is non - universal : it depends on details of the initial condition . this dependence is quite sensitive , as can be seen if we change the periodic arrangement of rws at @xmath9 by putting exactly 2 particles on each _ even _ lattice cite @xmath295 , @xmath296 of the same lattice as before , leaving all odd sites empty . repeating the calculations , we arrive at @xmath297 with the same exponent as in eq . ( [ rw : periodic ] ) but a much larger pre - exponent . in two dimensions , the probability @xmath298 that a rw starting at the radial coordinate @xmath299 does not hit the target by the time @xmath2 has a cumbersome exact expression . in the long time limit , @xmath27 , it suffices to use the following asymptotic that is valid with logarithmic accuracy ( see e.g. @xcite ) : @xmath300 here @xmath301 is the incomplete gamma function . we now employ the same line of reasoning as in one dimension . for the random initial condition we average the probability ( [ prob:2d ] ) over the random locations of the particles in the annulus @xmath302 and obtain the average single - particle non - hitting probability @xmath303 when @xmath304 , the number of rws in the annulus is approximately equal to @xmath305 . therefore , the probability that no rw hit the target is @xmath306^{\pi n_0(l^2-r^2)}\\ & & \to\exp\!\left[- \frac{4\pi n_0 t}{\ln\frac{4}{\ell^2}}\,\int_0^\infty dz\,\gamma(0,z)\right],\end{aligned}\ ] ] where we have simplified the limits of integration by recalling that @xmath34 and taking the limit of @xmath283 . computing the integral @xmath307 , we recover eq . ( [ survivaldecay2 ] ) . in the deterministic setting the probability is @xmath308\ ] ] the product is taken over initial positions @xmath309 which are deterministic . we assume that , at @xmath9 , there is exactly one particle on each site of a square grid with lattice spacing @xmath310 outside of the circular target of radius @xmath0 . we take the logarithm of and , ignoring pre - exponential factors in the final result , expand the logarithm to the leading order and replace the summation by integration by virtue of @xmath34 . we obtain @xmath311 again arriving at eq . ( [ survivaldecay2 ] ) . that is , in contrast to @xmath17 , here the leading - order results for @xmath312 for random and deterministic initial conditions coincide in the limit of @xmath27 . in three dimensions , the probability @xmath298 that a rw starting at the radial coordinate @xmath299 does not hit the target until time @xmath2 can be found by solving the backward diffusion equation ( which is mathematically identical to the forward diffusion equation ) @xmath313 subject to @xmath314 in contrast to two dimensions , the solution now has a simple form : @xmath315 when the initial locations are random , we start with a spherical annulus @xmath302 and average over initial locations to yield the average single - particle non - hitting probability @xmath316 the probability @xmath312 that no rw hit the target is given by @xmath317\nonumber\\ & & \!\!\!\!\!\!\to- 4\pi n_0 r \sqrt{4t}\int_r^\infty dr\ , r\,\text{erfc}\!\left(\frac{r - r}{\sqrt{4t}}\right)\nonumber\\ & & \!\!\!\!\!\!=- 4\pi n_0 r\ , \sqrt{4t}\int_0^\infty dx\ , ( x \sqrt{4t}+r)\,\text{erfc}\ , x \nonumber \\ & & \!\!\!\!\!\!= - 4\pi n_0 r\,t-8\sqrt{\pi}\ , n_0 r^2 \sqrt{t},\end{aligned}\ ] ] in agreement with previous results @xcite . equation ( [ twoterms ] ) is valid for any @xmath6 . when @xmath318 , the first term is the leading one and yields eq . with @xmath31 . in the opposite case of @xmath319 it is the second term that is the leading one , and it yields eq . ( [ sfastan ] ) with @xmath31 . even in the long - time limit , @xmath318 , eq . ( [ twoterms ] ) is more accurate then the leading - order asymptotic that stems from the steady - state mft solution . importantly , the final result ( [ twoterms ] ) can also be obtained from the mft formalism if one solves the full time - dependent problem . the problem formulation is almost identical to that for @xmath17 , see sec . [ app : annealed ] , except that eq . ( [ inann ] ) is replaced by @xmath320 and all integrations over @xmath131 from @xmath321 to @xmath322 are replaced by integrations over the whole space outside the target . the calculations proceed along the lines of sec . [ app : annealed ] . the most likely gas density history , in the original ( not rescaled ) variables , is @xmath323 \nonumber \\ & \times & \left\{1-\frac{r}{r}\,\text{erfc } \left[\frac{r - r}{\sqrt{4 ( t - t)}}\right]\right\}. \label{qann3d}\end{aligned}\ ] ] the calculation of the target survival probability ultimately reduces to evaluating the same integral as in eq . ( [ twoterms ] ) , giving the same result . in the long - time limit , @xmath27 , this integral is mostly contributed by the region @xmath324 . in this region ( [ qann3d ] ) can be approximated , up to small corrections , by @xmath325 , which coincides with the steady solution ( [ qdrw ] ) for @xmath31 . the deviations from the steady - state solution are responsible for the second term on the right - hand side of eq . ( [ twoterms ] ) , which is a subleading term in this limit . in the deterministic case the microscopic calculation , similar to that in @xmath121 , boils down to evaluating the integral @xmath326.\ ] ] exactly the same expression follows from the mft . for @xmath27 , we expand the logarithm to the second order and arrive at @xmath327 the leading term coincides with that for the annealed setting , eq . ( [ twoterms ] ) . the subleading term is different . for @xmath319 , we can replace @xmath67 by @xmath0 everywhere in the integrand of eq . ( [ deter3d ] ) except under the @xmath328 , thus arriving at eq . ( [ sfast ] ) ( but for the rws ) with @xmath329 . here the initial conditions affect the leading - order result . s. f. burlatsky , m. moreau , g. oshanin , and a. blumen , phys . lett . * 75 * , 585 ( 1995 ) ; d. p. bhatia , m. a. prasad , and d. arora , phys . rev . lett . * 75 * , 586 ( 1995 ) ; k. seki and m. tachiya , phys . e * 80 * , 041120 ( 2009 ) ; k. seki , m. wojcik , and m. tachiya , j. chem . 134 * , 094506 ( 2011 ) . l. bertini , a. de sole , d. gabrielli , g. jona - lasinio , and c. landim , phys . * * 8**7 , 040601 ( 2001 ) ; j. stat . * 107 * , 635 ( 2002 ) ; phys . lett . * 94 * , 030601 ( 2005 ) ; j. stat . * 1**23 , 237 ( 2006 ) .

suppose that a lattice gas of constant density , described by the symmetric simple exclusion process , is brought in contact with a target " : a spherical absorber of radius @xmath0 . employing the macroscopic fluctuation theory ( mft ) , we evaluate the probability @xmath1 that no gas particle hits the target until a long but finite time @xmath2 . we also find the most likely gas density history conditional on the non - hitting . the results depend on the dimension of space @xmath3 and on the rescaled parameter @xmath4 , where @xmath5 is the gas diffusivity . for small @xmath6 and @xmath7 , @xmath1 is determined by an exact stationary solution of the mft equations that we find . for large @xmath6 , and for any @xmath6 in one dimension , the relevant mft solutions are non - stationary . in this case @xmath8 scales differently with relevant parameters , and it also depends on whether the initial condition is random or deterministic . the latter effects also occur if the lattice gas is composed of non - interacting random walkers . finally , we extend the formalism to a whole class of diffusive gases of interacting particles .

1406.3502

during the last few years applications of gauge / gravity duality @xcite to hadronic physics attracted a lot of attention , and various holographic dual models of qcd were proposed in the literature ( see , e.g. , @xcite ) . these models were able to incorporate such essential properties of qcd as confinement and chiral symmetry breaking , and also to reproduce many of the static hadronic observables ( decay constants , masses ) , with values rather close to the experimental ones . amongst the dual models , a special class is the so - called `` bottom - up '' approaches ( see , e.g. , @xcite ) , the goal of which is to reproduce known properties of qcd by choosing an appropriate theory in the 5-dimensional ( 5d ) ads bulk . within the framework of the ads / qcd models , by modifying the theory in the bulk one may try to explain / fit experimental results in different sectors of qcd . in the present paper , we will be interested in the hard - wall ads / qcd model @xcite , where the confinement is modeled by sharp cutting off the ads space along the extra fifth dimension at a wall located at some finite distance @xmath0 . in the framework of this hard - wall model , it is possible to find form factors and wave functions of vector mesons ( see , e.g. , @xcite ) . to reproduce the general features of the spectrum for the higher states ( `` linear confinement '' ) , a soft - wall model was proposed in @xcite . the @xmath1-meson form factors for this model were calculated in ref . @xcite . in general , the vector sector is less sensitive to the infrared ( ir ) effects , since this symmetry is not broken in qcd . however , the axial - vector sector appears to be very sensitive to the particular way the chiral symmetry is broken or , in other words , to the bulk content and the shape of the ir wall @xcite . in this respect , one of the interesting objects to study in the holographic dual models of qcd is the pion . the properties of the pion were studied in various holographic approaches , ( see e.g. refs . in particular , the approach of ref . @xcite ( see also recent papers @xcite ) managed to reproduce the ( gell - mann oakes renner ) relation @xmath2 between the quark mass @xmath3 and mass of the pion @xmath4 and also the @xmath5 coupling ( the coupling between @xmath1 meson and two pions ) . in ref . @xcite , the solution of the pion wave - function equation was explicitly written for the @xmath6 limit . in this paper , working in the framework of the model proposed in @xcite ( hard - wall model ) , we describe a formalism to calculate the form factor and wave functions ( and also the density function ) of the pion . since the fits of ref . @xcite give a very small @xmath7mev value for the explicit chiral symmetry breaking parameter @xmath8 , we consider only the chiral limit @xmath6 of the hard - wall holographic dual model of two - flavor qcd . resorting to the chiral limit allows us to utilize one of the main advantages of ads / qcd - the possibility to work with explicit analytic solutions of the basic equations of motion . expressing the pion form factor in terms of these solutions , we are able , in particular , to extract and analyze the behavior of the pion electric radius in various regions of the holographic parameters space . on the numerical side , we come to the conclusion that the radius of the pion is smaller than what is known from experiment . however , we suggest that , as in case of the radius of the @xmath1 meson , smoothing the ir wall may increase the pion radius . in our analysis , we introduce and systematically use two types of holographic wave functions @xmath9 and @xmath10 , which are conjugate to each other and basically similar to the analogous objects introduced in our papers @xcite , where we studied vector mesons . the paper is organized in the following way . we start with recalling , in section ii , the basics of the hard - wall model and some results obtained in ref . @xcite , in particular , the form of the relevant action , the eigenvalue equations for bound states and their solutions . in section iii , we describe a formalism for calculating the pion form factor and express it in terms of the two wave functions mentioned above . in section iv , we discuss the relation of our ads / qcd results to experimental data . we express the values of the pion decay constant and the pion charge radius in terms of the fundamental parameters of the theory and study their behavior in different regions of the parametric space . at the end , we study the behavior of the pion form factor at large momentum transfer . finally , we summarize the paper . in the holographic model of hadrons , qcd resonances correspond to kaluza - klein ( kk ) excitations in the sliced ads@xmath11 background . in particular , vector mesons correspond to the kk modes of transverse vector gauge field in this background . since the gauge symmetry in the vector sector of the h - model is not broken , the longitudinal component of the vector gauge field is unphysical , and only transverse components correspond to physical mesons . similarly , the axial - vector mesons are the modes of the transverse part of the axial - vector gauge field . however , because the axial - vector gauge symmetry is broken in the 5d background , the longitudinal components have physical meaning and are related to the pion field . this should be taken into account if we want to treat the pion in a consistent way . the standard prescription of the holographic model is that there is a correspondence between the 4d vector and axial - vector currents and the corresponding 5d gauge fields : @xmath12 where @xmath13 , ( @xmath14 and @xmath15 are usual pauli matrices ) . in general , one can write @xmath16 , where @xmath17 and @xmath18 are transverse and longitudinal components of the axial - vector field . the spontaneous symmetry breaking causes @xmath19 to be physical and associated with the goldstone boson , pion in this case . the longitudinal component may be written in the form : @xmath20 . then @xmath21 corresponds to the pion field . physics of the axial - vector and pseudoscalar sectors is described by the action @xmath22 \ , \end{aligned}\ ] ] where @xmath23 , ( @xmath24 ) and @xmath25 is taken as a product of the chiral field @xmath26 and the function @xmath27 containing the chiral symmetry breaking parameters @xmath8 and @xmath28 , with @xmath8 playing the role of the quark mass and @xmath28 that of the quark condensate . expanding @xmath29 in powers of @xmath30 gives the relevant piece of the action @xmath31.\end{aligned}\ ] ] this higgs - like mechanism breaks the axial - vector gauge symmetry by bringing a @xmath32-dependent mass term in the @xmath33-part of the lagrangian . varying the action with respect to the transverse part of the axial - vector gauge field @xmath34 and representing the fourier image of @xmath34 as @xmath35 we will get the following equation of motion @xmath36_{\perp } = 0 \ , \end{aligned}\ ] ] that determines physics of the axial - vector mesons , like @xmath37 . the axial - vector bulk - to - boundary propagator @xmath38 is introduced by the relation @xmath39 . it satisfies eq . ( [ axvec ] ) with boundary conditions ( b.c . ) @xmath40 and @xmath41 . similarly , variation with respect to the longitudinal component @xmath42 gives @xmath43 finally , varying with respect to @xmath44 produces @xmath45 the pion wave function is determined from eqs . ( [ phieq ] ) and ( [ pieq ] ) with b.c . @xmath46 , @xmath47 and @xmath48 . within the framework of the model of ref . @xcite , it is possible to derive the gell - mann oakes renner relation @xmath49 producing massless pion in the @xmath6 limit . taking @xmath50 in eq . ( [ pieq ] ) gives @xmath51 a perturbative solution in the form of @xmath52 expansion was proposed in ref @xcite , with @xmath53 in the lowest order . then it was shown that , in the @xmath54 limit , @xmath55 tends to @xmath56 or , roughly speaking , @xmath57 in this limit . since our goal is to calculate the pion form factor in the chiral limit , this approximation will be sufficient for us . the spectrum in the axial - current channel consists of the pseudoscalar pion @xmath58 and axial - vector mesons @xmath59 , where @xmath60 correspond to the @xmath61 axial - vector meson decay constant ( and we ignored the flavor indexes ) . thus , the two - point function for the axial - vector currents has the form : @xmath62 where the meson polarization tensor is given by @xmath63 the representation for the two - point function can be also written as @xmath64 in which the second term on the rhs is explicitly transverse to @xmath65 . as noted in ref . @xcite , using holographic correspondence one can relate the two - point function to @xmath66_{z=0}$ ] and derive that @xmath67 for large spacelike @xmath68 , eq . ( [ axvec ] ) gives the same solution as in case of vector mesons , and the same asymptotic logarithmic behavior , just as expected from qcd . the longitudinal component of the axial - vector gauge field was defined as @xmath69 . in the chiral limit , when @xmath70 , we have @xmath71 , and the basic equation for @xmath72 , eq . ( [ phieq ] ) can be rewritten as the equation @xmath73 for the function @xmath74 . in the chiral limit , when @xmath75 , the value of @xmath76 tends to 1 as @xmath77 . this value and the b.c . @xmath78 are the same as those for @xmath79 and , furthermore , eq . ( [ pioneq ] ) coincides with the @xmath80 version of equation ( [ axvec ] ) for @xmath79 . hence , the solution for @xmath81 coincides with @xmath82 : @xmath83 and we may write @xmath84 in our analysis of @xmath1-meson wave functions in refs . @xcite , we emphasized that it makes sense to consider also the conjugate functions @xmath85 of the corresponding sturm - liouville equation . as we observed , they are closer in their structure to the usual quantum mechanical bound state wave functions than the solutions of the original equation . in the pion case , it is convenient to define the @xmath86 function as @xmath87 it vanishes at the ir boundary @xmath88 and , according to eq . ( [ piondecay1 ] ) , is normalized as @xmath89 at the origin . note also that using eq . ( [ pioneq ] ) we can express @xmath90 as derivative of @xmath86 : @xmath91 to obtain the pion form factor , we need to consider three - point correlation functions . the correlator should include the external em current @xmath92 and currents having nonzero projection onto the pion states , e.g. the axial currents @xmath93 @xmath94 where @xmath95 are the corresponding momenta , with the momentum transfer carried by the em source being @xmath96 ( as usual , we denote @xmath97 , @xmath98 ) . the spectral representation for the three - point function is a two - dimensional generalization of eq . ( [ 2axial ] ) @xmath99 where the first term , longitudinal both with respect to @xmath100 and @xmath101 contains the pion electromagnetic form factor @xmath102 @xmath103 ( normalized by @xmath104 ) , while other pole terms contain the contributions involving axial - vector mesons and are transverse either with respect to @xmath100 or @xmath101 , or both . hence , the pion form factor can be extracted from the three - point function using @xmath105 to obtain form factor from the holographic model , we need the action of the third order in the fields . there are two types of terms contributing to the pion electromagnetic form factor : @xmath107 term and @xmath106 terms . let us consider first the contribution from @xmath106 terms . they contain @xmath108 , @xmath109 and @xmath110 interactions and may be written as @xmath111 + v_{mn}[a^{m},a^{n } ] + a_{mn}[v^{m},a^{n } ] \right ) \ , \end{aligned}\ ] ] where @xmath112 and @xmath113 . taking @xmath114 gauge , we pick out the part of the action which is contributing to the 3-point function @xmath115 : @xmath116 + a_{\mu\nu}[v^{\mu},a^{\nu } ] \right).\end{aligned}\ ] ] introducing fourier transforms of fields , we define , as usual , @xmath117 for the vector field , where @xmath118 is the fourier transform of the 4-dimensional field @xmath119 and @xmath120 is the bulk - to - boundary propagator satisfying the equation @xmath121 with b.c . @xmath122 and @xmath123 . it can be written as the sum @xmath124 involving all the bound states in the @xmath125-channel , with @xmath126 being the mass of the @xmath127th bound state and @xmath128 its wave function given by a solution of the basic equation of motion in the vector sector . the projection ( [ proj0 ] ) picks out only the longitudinal part @xmath129 of the axial - vector field . taking into account that @xmath130 , we may write @xmath131 where @xmath132 and @xmath133 are the fourier transforms of @xmath134 and @xmath135 , respectively . furthermore , there is only one particle in the expansion over bound state in this case , namely , the massless pion . thus , we have @xmath136 and , therefore , @xmath137 this allows us to rewrite @xmath138 in the form @xmath139 involving the longitudinal projector @xmath140 and the pion wave function @xmath141 , which is the solution of the basic equation ( [ phieq ] ) . using this representation and making fourier transform of @xmath142 gives @xmath143 \int^{z_0}_{\epsilon } dz~\frac{1}{z}\ , { \cal v } ( w , z)\ , \psi^2(z ) \end{aligned}\ ] ] ( notice that the second term in eq.([xnia ] ) vanishes for longitudinal axial - vector fields ) . varying this functional with respect to sources produces the following 3-point function : @xmath144 where , anticipating the limit @xmath145 , we took @xmath146 in the numerator factors . now , representing @xmath147 and applying the projection suggested by eq . ( [ proj0 ] ) , we will have @xmath148 where @xmath149 is the dynamic factor given by the convolution @xmath150 the vector bulk - to - boundary propagator @xmath151 for spacelike momenta , entering into the dynamic factor @xmath149 , satisfies the equation @xmath152 with b.c . @xmath153 and @xmath154 . its explicit form is given by @xmath155 \ .\ ] ] one can easily see that @xmath156 . combining all the factors , we get @xmath157 integrating by parts and using equations of motion both for @xmath158 and @xmath159 gives @xmath160 \ .\end{aligned}\ ] ] we need also to add the @xmath161 contribution from the @xmath107 term of the ads action ( [ ads ] ) . it is generated by @xmath162,\end{aligned}\ ] ] and its inclusion changes @xmath163 into @xmath164 in eq . ( [ almost ] ) . the total result ( see also ref . @xcite ) may be now conveniently expressed in terms of the @xmath165 wave function @xmath166 \ .\end{aligned}\ ] ] using equation of motion for @xmath167 , one can see that the expression in square brackets coincides with @xmath168 and write the form factor as @xmath169 this representation allows one to easily check the normalization @xmath170 where we took into account that @xmath171 and @xmath172 . we can also represent our result for the pion form factor as @xmath173 \equiv \int_{0}^{z_0 } dz \ z \ { \cal j } ( q , z)\ , \rho ( z ) \ , \end{aligned}\ ] ] and interpret the function @xmath174 as the radial distribution density , as it was done in refs . @xcite . note that keeping only the first term in square brackets gives an expression similar to our result @xcite for the @xmath1-meson form factor @xmath175 in terms of the function @xmath176 conjugate to the solution of the basic equation of motion . the value of @xmath177 at the origin is proportional to the @xmath1-meson decay constant @xmath178 ( experimentally , @xmath179 @xcite ) , namely , @xmath180 . thus , the pion wave function @xmath181 is a direct analog of the @xmath1-meson wave function @xmath182 . main difference is that , in the pion case , there is also the second term in the form factor expression . the latter , in fact , is necessary to secure correct normalization of the form factor at @xmath183 . in eq . ( [ ff ] ) , this term is written in terms of the @xmath184 wave function , but using eq . ( [ phiz ] ) we can rewrite it also in terms of @xmath9 or @xmath185 : @xmath186 explicit form of the @xmath90 wave function follows from the solution of eq . ( [ pioneq ] ) : @xmath187 \left(\frac{\alpha}{2}\right)^{1/3 } } \left [ i_{-1/3}\left(\alpha z^3\right ) - i_{1/3}\left(\alpha z^3\right ) \frac{i_{2/3}\left(\alpha z^3_0\right ) } { i_{-2/3}\left(\alpha z^3_0\right)}\right ] \ , \end{aligned}\ ] ] where @xmath188 ( recall that @xmath189 , see e.g. ref.@xcite ) . as a result , @xmath190 is given by @xmath191 \left(\frac{\alpha^4}{2}\right)^{1/3 } \left[- i_{2/3}\left(\alpha z^3\right ) + i_{-2/3}\left(\alpha z^3\right ) \frac{i_{2/3}\left(\alpha z^3_0\right ) } { i_{-2/3}\left(\alpha z^3_0\right)}\right ] \ .\end{aligned}\ ] ] this formula , combined with eq . ( [ psi0 ] ) , establishes the relation @xmath192}{\gamma[1/3]}\ , \frac{i_{2/3}\left(\alpha z^3_0\right ) } { i_{-2/3}\left(\alpha z^3_0\right ) } \frac{\alpha^{2/3}}{g_5 ^ 2 } \end{aligned}\ ] ] for @xmath193 in terms of the condensate parameter @xmath194 and the confinement radius @xmath195 . since @xmath28 appears in the solutions only through @xmath194 , we will use @xmath194 in what follows . note also that @xmath196 . realizing that the equations of motion for the vector sector in this holographic model are not affected by the chiral symmetry - breaking effects expressed through the function @xmath197 , it is natural to set the value of @xmath195 from the vector sector spectrum , i.e. , by the @xmath1-meson mass . the numerical value of @xmath195 ( call it @xmath198 ) is then @xmath199 . as given by eq . ( [ fpi2 ] ) , @xmath193 looks like a rather complicated function of two scales , @xmath195 and @xmath194 . note , however , that the ratio @xmath200 is very close to 1 for @xmath201 and practically indistinguishable from 1 for @xmath202 . hence , for sufficiently large values of the confinement radius , @xmath203 , the value of @xmath193 is determined by the value of @xmath194 alone . this limiting value of @xmath193 is given by @xmath204}{\gamma[1/3 ] } } = \frac{3^{1/2}}{2^{1/3 } \pi } \ , \sqrt { \frac { \gamma[2/3]}{\gamma[1/3 ] } } \ , \alpha^{1/3 } \ , \approx \frac{\alpha^{1/3}}{3.21 } \ .\end{aligned}\ ] ] requiring that @xmath205 coincides with the experimental value , @xmath206 , one should take @xmath207 . for such @xmath194 , the value of @xmath208 is close to @xmath198 , i.e. , we are in the region @xmath209 and we may expect that , even if we use exact formula ( [ fpi2 ] ) with @xmath210 , the value of @xmath193 would not change much . indeed , to get @xmath206 from eq . ( [ fpi2 ] ) for @xmath211 , we should take @xmath212 . thus , in this range of parameters , the value of @xmath193 is practically in one - to - one correspondence with the value of @xmath194 . it is convenient to introduce a dimensionless variable @xmath213 then the values @xmath214 and @xmath215 correspond to @xmath216 . as one can see from fig.([fpi ] ) , the dependence of @xmath193 is practically flat for @xmath201 . the confinement radius @xmath195 presents a natural scale to measure length , so it makes sense to rewrite the form factor formula ( [ ff ] ) as an integral over the dimensionless variable @xmath217 : @xmath218 \equiv \int_{0}^{1 } d\zeta \ , \zeta \ { \cal j } ( q,\zeta , z_0 ) \ , \rho ( \zeta , a ) \ , \end{aligned}\ ] ] where the mass scale @xmath194 is reflected by the dimensionless parameter @xmath219 . the factor @xmath220 takes care of the correct normalization of the form factor . it is given by @xmath221}{\gamma[1/3 ] } \ , \frac{i_{2/3 } ( a)}{i_{-2/3}(a ) } \ .\end{aligned}\ ] ] for small @xmath219 , it may be approximated by @xmath222 . for large @xmath219 , using the fact that @xmath223 is very close to 1 for @xmath201 , we may approximate @xmath224 in this region . in terms of @xmath220 , the pion decay constant can be written as @xmath225 for large @xmath219 , this gives @xmath226 for small @xmath219 , we have @xmath227 the functions @xmath228 are just the @xmath86 and @xmath90 wave functions written in @xmath229 and @xmath219 variables . for @xmath230 , the limiting forms are @xmath231 and @xmath232 . as @xmath219 increases , both functions become more and more narrow ( see fig.[psiphi ] ) . for density , we have @xmath233 in the @xmath234 limit , a function that vanishes at the origin ( see fig.([rhodense ] ) ) . for nonzero @xmath219 , the value of @xmath235 monotonically increases with @xmath219 , and the function itself narrows . the increase of @xmath235 with @xmath219 is generated by the monotonically increasing function @xmath220 . it is interesting to compare the pion density @xmath236 ( taken at the `` experimental '' value @xmath237 ) with the @xmath1-meson density @xmath238 of ref . these densities are rather close for @xmath239 , but strongly differ for small @xmath240 . in particular , the @xmath1-meson density is more than two times larger for @xmath241 , which corresponds to the hard - wall model result that @xmath242 is essentially larger than @xmath193 . it is interesting to investigate how well these values @xmath243 and @xmath244 describe another important low - energy characteristics of the pion its charge radius . using the @xmath245-expansion of the vector source @xcite @xmath246 + \ldots\end{aligned}\ ] ] and explicit form of the density @xmath247 \ , \end{aligned}\ ] ] where @xmath248 , we obtain for the pion charge radius : @xmath249 \ , \rho ( \zeta , a ) = \frac43 \ , z_0 ^ 2 \ , \left \ { 1 - \frac{a^2}{4 } + { \cal o } ( a^4 ) \right \ } \ .\end{aligned}\ ] ] hence , for fixed @xmath195 and small @xmath219 , when @xmath250 , the pion radius is basically determined by the confinement scale @xmath195 . in particular , @xmath251 for @xmath252 . numerically , taking we obtain @xmath253 . this result is very close to the value @xmath254 that we obtained in the hard - wall model for the @xmath1-meson electric radius determined in @xcite from the slope of the @xmath255 form factor . however , since @xmath255 involves kinematic - type terms @xmath256 , it seems more appropriate to compare @xmath257 with the @xmath258 form factor ( [ calf ] ) given directly by a wave function overlap integral . the slope of @xmath258 is smaller than that of @xmath255 , and the corresponding radius is also smaller : @xmath259 . thus , for @xmath260 , the pion r.m.s . radius is about 1.4 times larger than the @xmath1-meson size determined by @xmath261 . with the increase of @xmath194 , the pion becomes smaller ( see fig.[radius ] ) . the experimental value of @xmath262 @xcite is reached for @xmath263 . however , the corresponding value @xmath264 is too small . if we take @xmath265 , then @xmath266 . thus , if we insist on using @xmath210 dictated by the hard - wall model calculation of the @xmath1-meson mass , and the value of @xmath194 producing the experimental @xmath193 ( note that then @xmath267 ) , the pion radius is smaller than the experimental value . in linear units , the difference , in fact , does not look very drastic : just 0.58fm instead of 0.66 fm . given that the hard - wall model for confinement is rather crude , the agreement may be considered as encouraging . furthermore , one may expect that , in a more realistic softer model of confinement , the size of the pion will be larger . such an expectation is supported by our soft - wall model calculation of the @xmath1-meson electric radius , for which we obtained @xmath2680.66@xmath269 ( 0.40@xmath269 for @xmath270 ) , i.e. , the result by 0.13@xmath269 larger than in the hard - wall model . if @xmath271 would increase by a similar amount , the result will be very close to the quoted experimental value . to find @xmath272 for large @xmath219 ( i.e. , when @xmath273 for fixed @xmath195 , or when @xmath274 for fixed @xmath194 ) , we use first the observation that , in the region @xmath201 , we may approximate @xmath275 . then the factor in square brackets in eq . ( [ rhozeta ] ) becomes a function of the combination @xmath276 ( call it @xmath277 ) , and we can write @xmath278 \ .\end{aligned}\ ] ] for @xmath201 , the upper limit of integration in this expression may be safely substituted by infinity producing @xmath279 which gives @xmath280 \ .\end{aligned}\ ] ] using eq . ( [ fpiinfty ] ) , we can express the coefficient in front of the square bracket in terms of @xmath193 : @xmath281 thus , @xmath282 in the @xmath283 region consists of two componens : a fixed term @xmath284 and a term logarithmically increasing with @xmath195 . as @xmath285 , the pion charge radius becomes infinite , reflecting the fact that the pion in this model is massless . a similar structure in the expression for the pion charge radius was obtained @xcite in the nambu - jona - lasinio ( njl ) model @xmath286 it also has the logarithmic term @xmath287 @xcite resulting in the infinite radius for massless pion and the infrared - finite piece @xmath288 @xcite . the latter , however , is twice larger than that in our result ( [ rpifpi ] ) and contributes 0.34@xmath269 to @xmath282 , with the chiral logarithm term producing the extra 0.11@xmath269 required for agreement with experiment . in our case , the logarithmic term taken for @xmath289 is approximately equal to @xmath284 , thus almost doubling the outcome value for @xmath282 . more precisely , we can write @xmath290 \ .\end{aligned}\ ] ] for @xmath237 , the modified logarithmic term gives a very small contribution , and our net result is very close to the value given by the njl fixed term . numerically , though , this prediction of the hard - wall ads / qcd model , as we have seen , is essentially smaller than the experimental value . in the large-@xmath245 limit , the source @xmath291 is given by its free - field version @xmath292 that behaves asymptotically like @xmath293 . as a result , only small values @xmath294 are important in the form factor integral , and the large-@xmath245 asymptotic behavior of the form factor is determined by the value of @xmath295 at the origin @xcite , namely , @xmath296 note that the combination @xmath297 frequently appears in the pion studies . in particular , it is the basic scale of the pion wave function in the local quark - hadron duality model @xcite , where it corresponds to the `` pion duality interval '' . the leading contribution comes entirely from the @xmath298 term of the form factor integral ( [ ff2 ] ) while the @xmath299 term contribution behaves asymptotically like @xmath300 since it is accompanied by extra @xmath301 factor . note , however , that it is quite visible in the experimentally interesting region @xmath302 : it is responsible for more than @xmath303 of the form factor value in this region ( moreover , at @xmath183 , the @xmath299 term contributes about 40% into the normalization of the form factor ) . from a phenomenological point of view , different ads / qcd - like models for the pion form factor differ in the shape of the density @xmath304 that they produce . if we require that the density @xmath305 equals @xmath306 at the origin , the asymptotic behavior is @xmath307 in any such model . for @xmath183 , the form factor is normalized to one , so basically the models would differ in how they interpolate between these two limits . in particular , the simplest interpolation is provided by the monopole formula @xmath308 while our hard - wall calculation gives a curve that goes above @xmath309 : the ratio @xmath310 is larger than 1 for all @xmath311 , slowly approaching unity as @xmath312 ( see fig.[ffact ] ) . in fact , a purely monopole form factor was obtained in our paper @xcite , where we studied the @xmath1-meson form factors in the soft - wall holographic model , in which confinement is generated by @xmath313 oscillator - type potential . it was shown in @xcite that the form factor integral @xmath314 is the lowest bound state wave function , and @xmath315\end{aligned}\ ] ] is the bulk - to - boundary propagator of this oscillator - type model , is exactly equal to the magnitude of the oscillator scale @xmath316 was fixed in our paper @xcite by the value of the @xmath1-meson mass : @xmath317 . as a result , the form factor @xmath318 had the @xmath1-dominance behavior @xmath319 . if we take @xmath320 both for @xmath321 and @xmath322 , the integral ( [ calf00 ] ) gives @xmath323 . the relevant wave function @xmath324 has the expected correct normalization however , the slope @xmath325 of @xmath323 at @xmath183 ( corresponding to for the radius squared ) is smaller than that of the experimental pion form factor . furthermore , @xmath326 tends to @xmath327gev@xmath328 for large @xmath245 , achieving values about @xmath329gev@xmath328 for @xmath330gev@xmath328 , and thus exceeding by more than 25% the experimental jlab values @xcite measured for @xmath331 and 2.45gev@xmath328 . the authors of ref . @xcite proposed to use eqs . ( [ calf00]),([caljo ] ) as an ads / qcd model for the pion form factor , with @xmath332mev chosen so as to fit these high-@xmath245 data . however , such a choice underestimates the value of @xmath333 by almost 30% . our opinion is that the ads / qcd models should describe first the low - energy properties of hadrons , and the basic low - energy characteristics , such as @xmath334 and @xmath193 , should be used to fix the model parameters . on the other hand , if the form factor calculations based on these parameters disagree with the large-@xmath245 data , it is quite possible that this is just an indication that one is using the model beyond its applicability limits . furthermore , as we have seen in the hard - wall model , to correctly describe the pion one needs to include the chiral symmetry breaking effects absent in the vector channel . as a result , equations for pion wave functions are rather different from those in the @xmath1-meson case . similarly , there are no reasons to expect that , in a soft - wall model , the pion density should have the same shape as the @xmath1-meson one . unfortunately , the procedure of bringing in the chiral symmetry breaking effects that was used in the hard - wall model of ref . @xcite faces serious difficulties when applied to the ads / qcd model @xcite with the @xmath301 soft wall . as discussed in ref . @xcite , the solution of the equation for the @xmath335 field in this model requires that chiral condensate @xmath28 and the mass parameter @xmath8 are proportional to each other , so that @xmath336 can not be varied independently of @xmath8 . moreover , if one takes the chiral limit @xmath6 , the chiral condensate should also vanish . this difficulty may be avoided by switching to more sophisticated recent models ( cf . @xcite ) in which the chiral condensate is generated dynamically . however , such a consideration goes well beyond the scope of the present paper . thus , we just resort to an idea that whatever the mechanism is involved , the net practical outcome is a particular shape of the density @xmath337 that eventually determines the pion form factor and other pion characteristics . below , we give an example of a density @xmath338 that is normalized at the origin by the experimental value of @xmath193 , i.e. , @xmath339 , but which is also capable to reproduce the experimental value of the pion charge radius . evidently , to increase the radius , we should take a density which is larger for large @xmath32 than @xmath340 . since the overall integral normalization of the density is kept fixed , this can be achieved only by decreasing the density for small @xmath32 values . consider a simple ansatz ( see fig.[rhomod ] ) @xmath341 \ , \end{aligned}\ ] ] with @xmath342 . it has both the desired value for @xmath343 and satisfies the normalization condition @xmath344 integrating it with @xmath345 taken at @xmath346 produces the model form factor given by the following sum of contributions of the three lowest vector states : @xmath347 the slope of @xmath348 at @xmath183 is given by @xmath349 \ .\end{aligned}\ ] ] taking @xmath350 , one obtains the experimental value 0.45 @xmath269 for @xmath351 . it is interesting to note that the model density providing this value , has an enhancement for larger values of @xmath32 ( see fig.[rhomod ] ) , just like the pion densities in the hard - wall model ( see fig.([rhodense ] ) ) . due to a larger slope , @xmath348 decreases faster than the simple monopole interpolation @xmath352 and , as a result , is in better agreement with the data . in fact , it goes very close to @xmath353gev@xmath328 data , but exceeds the values of the jlab @xmath3541.6 and 2.45gev@xmath328 points by roughly 10% and 20% , respectively . this discrepancy has a general reason . the asymptotic ads / qcd prediction is which is @xmath355gev@xmath328 for experimental value of @xmath193 . on the other hand , jlab experimental points correspond to @xmath356gev@xmath328 , which is much smaller than the theoretical value quoted above . the pre - asymptotic effects , as we have seen , reduce the discrepancy , but there still remains a sizable gap . as we already stated , such a disagreement may be just a signal that we are reaching a region where ads / qcd models should not be expected to work . in particular , ads / qcd models of refs . @xcite describe the pion in terms of an effective field or current , without specifying whether the current is built from spin-1/2 fields , or from scalar fields , etc . for @xmath245 above 1gev@xmath328 , the quark substructure of the pion may be resolved by the electromagnetic probe ( which is a wide - spread belief ) , and the description of the pion `` as a whole '' may be insufficient . in this paper , we studied the pion in the chiral limit of two flavor qcd . to this end , we described a formalism that allows to extract pion form factor within the framework of the holographic dual model of qcd with hard - wall cutoff . following ref . @xcite , we identified the pion with the longitudinal component of the axial - vector gauge field . we defined two ( sturm - liouville ) conjugate wave functions @xmath9 and @xmath10 that describe the structure of the pion along the 5@xmath357 dimension coordinate @xmath358 . these wave functions provide a very convenient framework to study the holographic physics of the pion . we demonstrated that , just like in the @xmath1-meson case @xcite , the pion form factor is given by an integral involving the function @xmath337 that has the meaning of the charge density inside the pion . however , in distinction to the @xmath1-case , when the density was simply given by @xmath359 , the pion density has an additional term proportional to @xmath360 and entering with the @xmath32-dependent coefficient reflecting the mechanism of the spontaneous symmetry breaking . both terms are required for normalization of the form factor at @xmath361 . we found an analytic expression for the pion decay constant in terms of two parameters of the model : @xmath362 and @xmath363 , similar to those used in ref . @xcite . analyzing the results , we found it convenient to work with two combinations @xmath364 and @xmath365 of the basic parameters . in particular , we found @xmath265 for the value of @xmath219 corresponding to the experimental @xmath1-meson mass @xmath334 and pion decay constant @xmath193 . the importance of the parameter @xmath219 is that its magnitude determines the regions , where the pion properties are either governed by the confinement effects or by the effects from the spontaneous chiral symmetry breaking . for example , in the practically important domain @xmath366 , the pion decay constant is determined primarily by @xmath362 , with negligibly tiny corrections due to @xmath195 value . however , when @xmath367 the pion decay constant is proportional to the ratio @xmath368 . besides , for small @xmath369 , the radius of the pion is given by @xmath370 , i.e. , as one may expect , the pion size is completely determined by the confinement radius . on the other hand , for @xmath366 the radius is basically determined by @xmath371 , slowly increasing with @xmath195 due to the @xmath372 correction . we also found that the pion rms charge radius @xmath373fm in the hard - wall model is smaller than that measured experimentally . in a sense , the hard wall at the distance @xmath374fm ( fixed from the @xmath1-meson mass ) , `` does not allow '' the pion to get larger . so , we argued that if the ir wall is `` softened '' , the size of the pion may be increased by an amount sufficient to accomodate the data . a straightforward idea is to use the soft - wall model of ref . @xcite and treat the pion in a way similar to what was done in @xcite for the @xmath1-meson case . unfortunately , there are prohibiting complications with directly introducing the chiral symmetry effects within the ads / qcd model with the @xmath301 soft wall . as explained in ref . @xcite , the chiral condensate @xmath28 in such a model is proportional to the mass parameter @xmath8 , so that in the chiral limit the condensate vanishes together with the quark mass . to illustrate a possible change in the form factor predictions due to the softening of the ir wall , we proposed an ansatz for the pion density function and used the vector current source from the soft - wall model considered in ref . @xcite . we demonstrated that this ansatz is capable to fit the experimental value of the pion charge radius . it also closely follows the data in the @xmath375gev@xmath328 region , while still overshoots available data in the @xmath376gev@xmath328 region . the basic source of this discrepancy is very general : the asymptotic ads / qcd prediction for the pion form factor is @xmath377 , and if one takes the experimental value for @xmath193 , one obtains @xmath378gev@xmath328 , which is much larger than the 0.4gev@xmath328 value given by @xmath376gev@xmath328 jlab data . for this reason , we argued that the disagreement mentioned above may be a signal that the region @xmath379gev@xmath328 is beyond the applicability region of ads / qcd models . finishing the write - up of this paper , we have learned that the paper @xcite addressing the same problem was posted into the arxive . we did not observe , however , essential overlaps with our ideas and results . h.g . would like to thank a. w. thomas for valuable comments and support at jefferson laboratory , j. p. draayer for support at louisiana state university . notice : authored by jefferson science associates , llc under u.s . doe contract no . de - ac05 - 06or23177 . government retains a non - exclusive , paid - up , irrevocable , world - wide license to publish or reproduce this manuscript for u.s . government purposes . j. m. maldacena , adv . theor . math . * 2 * , 231 ( 1998 ) [ int . j. theor . phys . * 38 * , 1113 ( 1999 ) ] ; s. s. gubser , i. r. klebanov and a. m. polyakov , phys . lett . b * 428 * , 105 ( 1998 ) ; e. witten , adv . theor . math . phys . * 2 * , 253 ( 1998 ) j. polchinski and m. j. strassler , phys . lett . * 88 * , 031601 ( 2002 ) ; jhep * 0305 * , 012 ( 2003 ) h. boschi - filho and n. r. f. braga , jhep * 0305 * , 009 ( 2003 ) ; eur . j. c * 32 * , 529 ( 2004 ) s. j. brodsky and g. f. de tramond , phys . b * 582 * , 211 ( 2004 ) ; g. f. de teramond and s. j. brodsky , phys . * 94 * , 201601 ( 2005 ) t. sakai and s. sugimoto , prog . phys . * 113 * , 843 ( 2005 ) ; * 114 * , 1083 ( 2006 ) j. erlich , e. katz , d. t. son and m. a. stephanov , phys . rev . * 95 * , 261602 ( 2005 ) j. erlich , g. d. kribs and i. low , phys . d * 73 * , 096001 ( 2006 ) l. da rold and a. pomarol , nucl . b * 721 * , 79 ( 2005 ) ; jhep * 0601 * , 157 ( 2006 ) a. karch , e. katz , d. t. son and m. a. stephanov , phys . d * 74 * , 015005 ( 2006 ) c. csaki and m. reece , jhep * 0705 * , 062 ( 2007 ) t. hambye , b. hassanain , j. march - russell and m. schvellinger , phys . d * 74 * , 026003 ( 2006 ) ; arxiv : hep - ph/0612010 j. hirn and v. sanz , jhep * 0512 * , 030 ( 2005 ) ; j. hirn , n. rius and v. sanz , phys . d * 73 * , 085005 ( 2006 ) k. ghoroku , n. maru , m. tachibana and m. yahiro , phys . b * 633 * , 602 ( 2006 ) s. j. brodsky and g. f. de teramond , phys . * 96 * , 201601 ( 2006 ) n. evans , a. tedder and t. waterson , jhep * 0701 * , 058 ( 2007 ) r. casero , e. kiritsis and a. paredes , arxiv : hep - th/0702155 u. gursoy and e. kiritsis , arxiv:0707.1324 [ hep - th ] u. gursoy , e. kiritsis and f. nitti , arxiv:0707.1349 [ hep - th ] o. bergman , s. seki and j. sonnenschein , arxiv:0708.2839 [ hep - th ] j. erdmenger , k. ghoroku and i. kirsch , arxiv:0706.3978 [ hep - th ] a. dhar and p. nag , arxiv:0708.3233 [ hep - th ] a. v. radyushkin , phys . b * 642 * , 459 ( 2006 ) h. r. grigoryan and a. v. radyushkin , phys . b * 650 * , 421 ( 2007 ) h. r. grigoryan and a. v. radyushkin , arxiv:0706.1543 [ hep - ph ] s. eidelman _ et al . _ [ particle data group ] , phys . lett . b * 592 * , 1 ( 2004 ) s. j. brodsky and g. f. de teramond , arxiv:0707.3859 [ hep - ph ] h. j. hippe , phys . c * 52 * 2172 ( 1995 ) m. a. b. beg and a. zepeda , phys . d * 6 * , 2912 ( 1972 ) m. k. volkov and v. n. pervushin , phys . b * 51 * , 356 ( 1974 )

we describe a formalism to calculate form factor and charge density distribution of the pion in the chiral limit using the holographic dual model of qcd with hard - wall cutoff . we introduce two conjugate pion wave functions and present analytic expressions for these functions and for the pion form factor . they allow to relate such observables as the pion decay constant and the pion charge electric radius to the values of chiral condensate and hard - wall cutoff scale . the evolution of the pion form factor to large values of the momentum transfer is discussed , and results are compared to existing experimental data .

0709.0500

in cold dark matter cosmology , the initially smooth distribution of matter in the universe is expected to collapse into a complex network of filaments and voids , structures which have been termed the `` cosmic web '' . the filamentary distribution of galaxies in the nearby universe has been revealed in detail by recent large galaxy redshift surveys such as the 2dfgrs ( colless et al . 2001 , baugh et al . 2004 ) , the sloan digital sky survey ( sdss , stoughton et al . 2002 , doroshkevich et al . 2004 ) and the 2@xmath19 all sky survey ( 2mass , maller et al . numerical simulations successfully reproduce this network ( jenkins et al . 1998 ; colberg et al . 2004 ) and indicate that galaxies are only the tip of the iceberg in this cosmic web ( katz et al . 1996 ; miralda - escud et al . hydrodynamic simulations suggest that at the present epoch , in addition to dark matter and galaxies , the filaments are also composed of a mixture of cool , photoionised gas ( the low@xmath0 remnants of the forest ) and a shock heated , low - density gaseous phase at temperatures between @xmath20 k and @xmath21 k that contains most of the baryonic mass , the `` warm - hot '' intergalactic medium ( whim , cen & ostriker 1999 ; dav et al . 1999 ) . observational constraints on the physical conditions , distribution , a nd metal enrichment of gas in the low - redshift cosmic web are currently quite limited . the existence of the whim appears to be a robust prediction of cosmological simulations ( dav et al . thus , observational efforts are increasingly being invested in the search for whim gas and , more generally , the gaseous filamentary structures predicted by the models . large - scale gaseous filaments have been detected in x - ray emission ( wang et al . 1997 ; scharf et al . 2000 ; tittley & henriksen 2001 ; rines et al 2001 ) . however , x - ray emission studies with current facilities predominantly reveal gas which is hotter and denser than the whim ; this x - ray emitting gas is not expected to contain a substantial portion of the present - epoch baryons ( dav et al . the most promising method for observing the whim in the near term is to search for uv ( o@xmath4roman6 , ne@xmath4roman8 ) and x - ray ( o@xmath4roman7 , o@xmath4roman8 , ne@xmath4roman9 ) absorption lines due to whim gas in the spectra of background qsos / agns ( tripp et al . 2000 , 2001 ; savage et al . 2002,2005 ; nicastro et al . 2002 ; bergeron et al . 2002 ; richter et al . 2004 ; sembach et al . 2004 ; prochaska et al . 2004 ; danforth & shull 2005 ) . while absorption lines provide a sensitive and powerful probe of the whim , the pencil - beam nature of the measurement along a sight line provides little information on the context of the absorption , e.g. , whether the lines arise in an individual galaxy disk / halo , a galaxy group , or lower - density regions of a large - scale filament or void . thus , to understand the nature of highly ionised absorbers at low redshifts , several groups are pursuing deep galaxy redshift surveys and observations of qsos behind well - defined galaxy groups or clusters . for example , to study gas located in large - scale filaments , bregman et al . ( 2004 ) have searched for absorption lines indicative of the whim in regions between galaxy clusters / superclusters and have identified some candidates . in this paper , we carry out a similar search as part of a broader program that combines a large _ hst _ survey of low@xmath0 o@xmath4roman6 absorption systems observed on sight lines to low@xmath0 quasars ( tripp et al . 2004 ) and a ground based survey to measure the redshifts and properties of the galaxies foreground to the background qsos . the ground based survey is done in two steps : first , multi - band ( u , b , v , r and i ) imagery is obtained to identify the galaxies and to estimate their photometric redshifts . then , spectroscopic redshifts are obtained for the galaxies that are potentially ( according to the photometric redhshifts ) at lower redshift that the background object . as part of the large _ hst _ survey , we have observed the quasar hs0624 + 6907 ( @xmath22 = 0.3700 ) with the e140 m echelle mode of the space telescope imaging spectrograph ( stis ) on board the _ hubble space telescope_. we have also obtained multiband images and spectroscopic redshifts of galaxies in the 0624 field . the sight line to 0624 passes by several foreground abell clusters ( [ sec : abell_clusters ] ) and provides an opportunity to search for gas in large - scale filaments . we shall show that gas ( absorption systems ) and galaxies are detected at the redshifts of the structures delineated by the abell clusters in this direction . while the absorbing gas is intergalactic , and it is likely that we are probing gas in cosmic web filaments , the properties of these absorbers are surprising . instead of low - metallicity whim gas , we predominantly find cool , photoionised , and high - metallicity gas in these large - scale structures . this paper is organized as follows . the observations and data reduction procedures are described in 2 , including _ hst_/stis and _ far ultraviolet spectroscopic explorer _ ( ) observations as well as ground - based imaging and galaxy redshift measurements . in 3 , we present information on the foreground environments probed by the 0624 sight line , derived from the literature on abell clusters and from our new galaxy redshift survey . the absorption - line measurement methods are described in 4 , and we investigate the physical state and metallicity of the absorbers in 5 . section 6 reviews the properties of the full sample of ly@xmath16 lines derived from the stis spectrum with emphasis on the search for broad ly@xmath16 lines . section 7 discusses the implications of this study , and we summarize our conclusions in 8 . throughout this paper , we use the following cosmological parameters : @xmath23 , @xmath24 and @xmath25 . 0624 was observed with stis on 2 jan . 2002 and 23 - 24 feb . 2002 as part of a cycle 10 _ hst _ observing program ( id=9184 ) . the echelle spectrograph was used with the e140 m grating which provides a resolution of 7 fwhm and covers the 1150@xmath261730 range with only a few small gaps between orders at wavelengths greater than 1630 . the @xmath27 entrance aperture was used to minimize the effect of the wings of the line spread function . the total exposure time was 61.95 ksec . the data were reduced as described in tripp et al . ( 2001 ) using the stis team version of calstis at the goddard space flight center . the final signal - to - noise ( s / n ) per resolution element is 3 at 1150 , increases linearly to 14 at 1340 and then decreases to 7 at 1730 . for further information on the design and performance of stis , see woodgate et al . ( 1998 ) and kimble et al . ( 1998 ) . hs0624 + 6907 was also observed by the pi team on several occasions between 1999 november and 2002 february ( program ids p1071001 , p1071002 , s6011201 , and s6011202 ) . records spectra with four independent spectrographs ( `` channels '' ) , two with sic coatings for coverage of the 905@xmath261105 wavelength range , and two with lif coatings optimized to cover 1100@xmath261187 ( see moos et al . 2000,2002 for details about design and performance ) . the spectrograph resolutions range from 20@xmath2630 km s@xmath28 ( fwhm ) . for hs0624 + 6907 , the total integration time in the lif1 channel was 110 ksec ; the other channels had somewhat lower integration times due to channel coalignment problems during some of the observations . we have retrieved the spectra from the archive and have reduced the data using calfuse version 2.4.0 as described in tripp et al . ( 2005 ) . because the spectra in the individual channels have modest s / n ratios , we have aligned and combined all available lif channels to form the final spectra that we used for our measurements ( we find that combining all available lif data does not degrade the spectral resolution ) . for the spectral range uniquely covered by the sic channels , we used only the sic2a data . finally , we compared absorption lines of comparable strength ( e.g. , fe@xmath4roman2 @xmath291144.94 vs. fe@xmath4roman2 @xmath291608.45 ) observed by _ fuse _ and stis in order to align the _ fuse _ spectrum with the stis spectrum and thereby correct the wavelength zero point of the data . one of the primary goals of our low@xmath0 qso absorption line program is to study the connections between galaxies and absorption systems . these studies require good imaging ( for galaxy target selection and information on individual galaxies of interest ) followed by optical spectroscopy for accurate redshift measurements . to initiate the galaxy - absorber study toward 0624 , we first obtained a @xmath30 mosaic of images centered on the qso with spicam on the apache point observatory ( apo ) 3.5 m telescope on 2002 october 5 . subsequently , we obtained images of a larger field in better seeing with the noao 8k@xmath318k ccd mosaic camera ( mosa , muller et al . 1998 ) , on the kitt peak national observatory ( kpno ) 4 m telescope . the spicam images were used to select targets for the first spectroscopic observing run , but thereafter we only used the better - quality mosa images . [ cols="<,^,^,^,^,^ " , ] 0624 was observed with mosa on the 4 m on 2003 january 29 - 30 . the field of view is @xmath32 with a scale of @xmath33/pixel . as summarized in table [ tab : obslog ] , images were recorded in @xmath34 and @xmath35 with a standard dithering pattern for filling in gaps between the ccds and for rejection of cosmic rays . photometric standard stars from landolt ( 1992 ) were also observed at regular intervals . during these observations , the seeing ranged from @xmath360 to 1@xmath373 . the data were reduced with the iraf software package mscred following standard procedures . the final r - band mosa image of 0624 is shown in figures [ fig : field ] and [ fig : fieldzoom ] . galaxy targets for follow - up spectroscopy were selected from the images using the sextractor software package ( bertin & arnouts 1996 ) . redshifts of 29 galaxies were obtained using the double imaging spectrograph ( dis ) on the apo 3.5 m telescope on the following dates : 2002 november 12 , 2003 january 29@xmath2631 , 2003 april 03 , 2003 april 21 , and 2003 december 25 . spectra were recorded using a single 1.5 arcsec wide slit with total exposure times ranging from 360 to 1800 s per object . the data were processed in the conventional manner , and were wavelength calibrated using helium - neon - argon arc - lamp exposures . small zero - point offsets in wavelength were applied as needed , after comparing observed skyline wavelengths with their rest values . the spectra were typically recorded at resolutions of @xmath38 fwhm . llccccccccc & & & & & & & & & + & & & & & & & & & & & qso & 06:30:02.50 & 69:05:03.99 & 0.3700 & 0.0 & 0.000 & 13.8 & 14.2 & 0.1 & -27.7 + 01 & se12 & 06:30:41.70 & 68:58:32.71 & 0.0327 & 7.4 & 0.290 & 16.6 & 17.8 & 1.0 & -19.2 + 02 & se3 & 06:30:55.32 & 69:02:41.99 & 0.0424 & 5.3 & 0.265 & 19.6 & 20.5 & 0.9 & -16.7 + 03 & ne1 & 06:30:56.14 & 69:08:00.90 & 0.0547 & 5.6 & 0.358 & 15.9 & 17.0 & 0.9 & -21.0 + 04 & nw2 & 06:29:46.66 & 69:08:03.59 & 0.0560 & 3.3 & 0.216 & 18.6 & 19.5 & 0.9 & -18.4 + 05 & se4 & 06:30:33.00 & 68:53:02.00 & 0.0622 & 12.3 & 0.887 & 16.8 & 17.9 & 0.9 & -20.5 + 06 & se5 & 06:32:55.20 & 68:56:59.99 & 0.0637 & 17.4 & 1.282 & 16.8 & 18.3 & 1.3 & -20.5 + 07 & se8 & 06:31:01.79 & 68:57:35.89 & 0.0638 & 9.2 & 0.675 & 16.0 & 17.1 & 0.9 & -21.2 + 08 & se1 & 06:30:11.22 & 69:02:09.61 & 0.0640 & 3.0 & 0.222 & 18.8 & 19.4 & 0.7 & -18.5 + 09 & sw3 & 06:29:07.80 & 69:03:32.01 & 0.0650 & 5.1 & 0.384 & 17.4 & 18.3 & 0.9 & -20.0 + 10 & se13 & 06:30:58.30 & 69:04:34.11 & 0.0650 & 5.0 & 0.375 & 16.9 & 18.5 & 1.3 & -20.4 + 11 & se6 & 06:32:50.70 & 68:56:03.00 & 0.0652 & 17.6 & 1.318 & 15.9 & 17.3 & 1.2 & -21.5 + 12 & ne3 & 06:30:21.40 & 69:05:39.70 & 0.0655 & 1.8 & 0.135 & 16.6 & 18.1 & 1.3 & -20.8 + 13 & nw11 & 06:29:23.48 & 69:22:43.29 & 0.0660 & 18.0 & 1.367 & 16.3 & 17.9 & 1.3 & -21.1 + 14 & se7 & 06:32:49.20 & 68:56:00.39 & 0.0664 & 17.5 & 1.334 & 17.0 & 18.0 & 0.7 & -20.4 + 15 & ne2 & 06:32:25.55 & 69:20:05.81 & 0.0733 & 19.7 & 1.646 & 16.5 & 18.0 & 1.3 & -21.1 + 16 & nw1 & 06:29:43.65 & 69:09:35.33 & 0.0760 & 4.8 & 0.417 & 15.9 & 17.3 & 1.1 & -21.8 + 17 & sw2 & 06:28:33.03 & 68:59:26.30 & 0.0763 & 9.8 & 0.849 & 16.6 & 18.3 & 1.3 & -21.1 + 18 & se9 & 06:30:14.81 & 68:49:44.79 & 0.0764 & 15.4 & 1.334 & 16.4 & 18.0 & 1.3 & -21.3 + 19 & nw12 & 06:29:11.59 & 69:07:07.89 & 0.0764 & 5.0 & 0.433 & 16.4 & 18.0 & 1.4 & -21.3 + 20 & se10 & 06:32:52.40 & 68:57:59.01 & 0.0764 & 16.8 & 1.457 & 16.5 & 18.1 & 1.3 & -21.2 + 21 & nw3 & 06:29:53.77 & 69:08:20.51 & 0.0766 & 3.4 & 0.293 & 18.3 & 19.6 & 1.1 & -19.4 + 22 & sw1 & 06:29:33.24 & 69:05:01.00 & 0.0903 & 2.6 & 0.264 & 17.1 & 18.2 & 0.9 & -21.0 + 23 & se11 & 06:30:06.84 & 68:52:22.20 & 0.1001 & 12.7 & 1.407 & 16.5 & 18.1 & 1.2 & -21.9 + 24 & nw7 & 06:26:43.70 & 69:14:06.91 & 0.1009 & 19.9 & 2.215 & 16.7 & 18.2 & 1.4 & -21.6 + 25 & nw9 & 06:29:03.89 & 69:17:33.40 & 0.1108 & 13.5 & 1.639 & 16.4 & 18.2 & 1.3 & -22.2 + 26 & nw4a & 06:29:35.43 & 69:07:25.80 & 0.1125 & 3.4 & 0.415 & 18.5 & 20.2 & 1.2 & -20.1 + 27 & nw8 & 06:28:29.39 & 69:17:28.89 & 0.1126 & 14.9 & 1.832 & 16.7 & 18.5 & 1.3 & -21.9 + 28 & nw10 & 06:29:05.25 & 69:17:46.69 & 0.1129 & 13.7 & 1.685 & 18.1 & 19.9 & 1.3 & -20.5 + 29 & nw6 & 06:29:35.30 & 69:09:44.99 & 0.1429 & 5.3 & 0.794 & 19.4 & 20.4 & 0.8 & -19.8 + 30 & se2 & 06:30:44.35 & 69:01:08.09 & 0.1664 & 5.4 & 0.927 & 19.3 & 20.8 & 1.0 & -20.2 + 31 & nw5 & 06:29:43.52 & 69:09:19.01 & 0.2061 & 4.6 & 0.927 & 18.4 & 20.3 & 1.4 & -21.7 + 32 & nw4b & 06:29:35.62 & 69:07:37.91 & 0.3008 & 3.5 & 0.940 & 21.0 & 22.2 & 0.4 & -19.9 the redshift measurements were made following the procedure described by jenkins et al . we used the iraf routine fxcor to cross - correlate the galaxy spectra with that of the radial velocity standard hd 182572 . in general we only used the blue channel dis data for the cross - correlation , where the 4000 break and stellar absorption lines were most apparent . red channel data were usually used to identify and measure the wavelengths of redshifted emission lines ( [ o iii ] , h@xmath39 , h@xmath16 , etc . ) when present . the galaxy redshifts obtained in this way are summarized in table [ tab : spec_red ] and are accurate to between 70 and 170 ( which corresponds to a sight line distance displacement uncertainty of 1.0 to 2.4 mpc for an unperturbed hubble flow ) . we also observed three galaxies with the echellette spectrometer and imager ( esi ; sheinis et al . 2001 ) on the 10 m keck ii telescope on the nights of 2004 september 10 and 11 during morning twilight . we observed galaxy ne3 ( see table [ tab : spec_red ] ) in echellette mode with the 0.5@xmath40 slit which provides @xmath41 spectral resolution ( fwhm ) . the fainter se13 and sw3 galaxies were observed in low dispersion mode using a 1@xmath40 slit which affords @xmath42 at @xmath43 . the exposures were flat fielded and wavelength calibrated with the esiredux package ( prochaska et al . 2003xavier / esi / index.html ] ) . the ne3 redshift was derived from the centroids of the high - resolution na@xmath4roman1 and h@xmath16 absorption lines , and the redshift uncertainty is @xmath4430 km s@xmath28 . for se13 and sw3 , redshifts were measured by fitting na@xmath4roman1 , h@xmath39 , and ca@xmath4roman2 h and k , and the uncertainties are @xmath44150 km s@xmath28 . the completeness of our galaxy redshift survey ( i.e. , the percentage of targets brighter than a given magnitude in the sextractor galaxy catalog with good spectroscopic redshifts ) is graphically summarized in figure [ fig : completeness ] as a function of limiting @xmath45 magnitude and angular separation from the sight line ( @xmath46 ) . in the @xmath47 region centered on 0624 , we have measured spectroscopic redshifts for all galaxies brighter than @xmath45 = 19.0 , and the survey is @xmath48 per cent complete for @xmath49 . as we shall see , there is a prominent cluster of absorption lines in the 0624 spectrum at @xmath50 0.0635 ; at this redshift , 5@xmath51 corresponds to a projected distance of 367 @xmath7 kpc , and @xmath45 = 19.0 corresponds to @xmath52 or @xmath53 ( taking @xmath54 from lin et al . 1996 ) . for comparison , the large magellanic cloud has a magnitude equal to @xmath55 or @xmath56 . at this redshift , we have good completeness even for low luminosity galaxies . at larger radii , a substantial number of bright galaxies are found , and our redshift survey is shallower . nevertheless , within a 10 radius circle , our survey is still 60 per cent complete for galaxies brighter than @xmath57 . using information gleaned from the literature in combination with our galaxy redshift survey , we can identify several large - scale structures that are pierced by the 0624 sight line . in this section we comment on these structures including nearby abell clusters ( [ sec : abell_clusters ] ) as well as smaller ( and closer ) galaxy groups ( [ sec : close_groups ] ) . clusters are clustered and often reveal even larger cosmic structures , i.e. , superclusters ( einasto et al . 2001 and references therein ) . in cosmological simulations , clusters are found at the nodes where large - scale filaments connect . to test the fidelity of cosmological simulations , which are now being used in a wide variety of astrophysical analyses , it is important to search for observational evidence of the expected _ gaseous _ filaments feeding into clusters and to measure the properties of the filaments . the sight line to 0624 passes through a region of relatively high abell cluster density and is well - suited for investigation of this topic . shows the positions of abell clusters around the sight line to 0624 , including the cluster richness class and spectroscopic redshift ( when available from the literature ) . the density of abell clusters in this region is relatively high compared to the vicinity of the other clusters in the abell catalog : the number of abell clusters within 2@xmath58 ( 3@xmath59 ) of a557 ( the cluster closest to the 0624 sight line ) is 2 ( 1.6 ) times larger than the average number within 2@xmath60 ( 3@xmath59 ) of all abell clusters . einasto et al . ( 2001 ) have identified two superclusters in the direction of 0624 . their supercluster scl71 ( at @xmath61 ) includes a554,a557 , a561 , a562 , and a565 while scl72 ( at @xmath62 ) includes a559 and a564 . however , a557 and a561 do not have spectroscopic redshifts , and based on our spectroscopic redshifts in the field of hs0624 + 6907/abell557 ( see table [ tab : spec_red ] ) , it appears likely that the visually identified a557 is a false cluster due to the superposition of galaxy groups at several different redshifts . to be conservative , we only use clusters with spectroscopic redshifts to identify large - scale structures . the clusters at ( a564 and a559 ) are separated by 4.7 from each other , and the clusters at ( a565 , a562 and a554 ) are separated by 3.9 and 8.7 . according to colberg et al . ( 2004 ) , in cosmological simulations , more than 85 per cent of the clusters with a separation lower than 10 are connected with a filament . we will show in subsequent sections that both absorption lines in the spectrum of 0624 and galaxies close to the sight line are detected at the redshifts of both of these abell cluster structures , which indicates that gaseous filaments connect into the clusters . in this section we offer some brief comments about specific galaxies and galaxy groups close to the sight line of 0624 , as revealed by our optical spectroscopy . we place these observations in the context of the abell clusters described in the previous section . we plot in the redshift distribution of the galaxies from table [ tab : spec_red ] . from this figure we can identify three galaxy groups : two galaxy groups appear to be present at redshifts similar to those of the abell clusters , i.e , at @xmath63 ( 7 galaxies ) and @xmath64 ( 6 galaxies ) . this indicates that the filament of galaxies connecting a559 to a564 must extend more then 3 degrees ( 15 ) west from a559 and structure linking a562 and a554 likely extends by at least 3 degrees ( 22 ) in order to cross the qso sight line . however , the most prominent group in includes 10 galaxies at , which does not match up with any abell cluster with a known spectroscopic redshift . to show the spatial distribution of galaxies in the three prominent groups in the mosa field , provides a colour - coded map of projected spatial coordinates of the galaxies . we see that the group at @xmath65 is mostly southeast of the sight line while the galaxies associated with abell 554/562/565 at are predominantly northwest of 0624 . the galaxies associated with the abell 559/564 supercluster appear to extend from the southwest across the sight line to the northeast . we note that spectroscopic redshifts are not available for several of the clusters shown in fig . [ fig : abell_map ] , including the one that is closest to the sight line , a557 . however , abell clusters are visually identified without the aid of spectroscopy , and it can be seen from figures [ fig : field ] and [ fig : gal_colour ] that several discrete groups are found at the location of a557 . it is likely that a557 is not a true cluster but rather is the superposition of several groups in projection . is gas also present in these large - scale cosmic filaments ? to address this question , we searched the spectrum of 0624 for any absorption counterparts at the redshifts of galaxies and galaxy structures near the qso sight line . the redshifts of the h@xmath4roman1 systems that we have identified and measured ( see [ sec : absline ] ) in the spectrum of 0624 are plotted at the top of ; the length of the line is proportional to the rest equivalent width of the line . this search has revealed three interesting results : first , when a galaxy is located at an impact parameter @xmath66 from the sight line , absorption is almost always found within a few hundred of the galaxy redshift ( compare table [ tab : spec_red ] to table [ tab : lyalist ] ) , consistent with the findings of previous studies ( e.g. , lanzetta et al . 1995 ; tripp et al . 1998 ; impey et al . 1999 ; chen et al . 2001 ; bowen et al . 2002 ; penton et al . 2002 ) . second , strong absorption is clearly detected at the redshift of the @xmath67 abell 564/559 supercluster . this absorption system is detected in , , and the c@xmath4roman4 @xmath68 1548.20 , 1550.78 doublet ( [ ss : z076 ] ) , and the absorption redshift is very similar to that of a559 . weak absorption is also detected at @xmath13 = 0.10822 , which is within 500 of the abell 554/562/565 filament . evidently , and not surprisingly , gas is also found in the filaments that feed into the clusters near 0624 . third , figure [ fig : zgaldist ] qualitatively indicates that the strongest lines are situated in the regions of highest galaxy density , which is similar to the conclusions of bowen et al . ( 2002 ) and c^ ot et al . ( 2005 ) . could these absorbers simply arise in the halos of individual galaxies ? as we show in the next section , the strong h@xmath4roman1 system at is comprised of a large number of components spread over 1000 . such kinematics are unprecedented in single galaxy halos . in addition , we find no obvious pattern that shows a connection between individual lines and individual galaxies in this complex . in the case of this strong h@xmath4roman1 system at , the closest observed galaxy to the sight line around this redshift has @xmath69 ( ne3 in table [ tab : spec_red ] ) . however , a closer and fainter galaxy could have been missed by the spectroscopic survey . using photometric redshifts ( measured as described in chen et al . 2003 ) to cull the distant background galaxies with photometric redshifts @xmath70 , we find that there are only four galaxies closer to the sight line than ne3 that could be near . these four objects are only candidates since photometric redshifts have substantial uncertainties . however , if we assume the redshift of these candidates to be @xmath71 , the closest one to 0624 has still a large impact parameter @xmath72 kpc . in the case of at @xmath13 = 0.07573 , the closest galaxy in projection is nw3 at @xmath73 kpc . it is conceivable that this absorption originates in the large halo of this particular galaxy , but we note that three galaxies are found at @xmath74 kpc at this @xmath75 , and many other origins are possible ( e.g. , intragroup gas or tidally stripped debris ) . finally , the absorption at @xmath76 is at a substantial distance ( 415 ) from the nearest known galaxy ( nw4a ) and is unlikely to be halo gas associated with that object . we now turn to the absorption - line measurements . as discussed in the previous section , figure [ fig : zgaldist ] compares the distributions of galaxies and ly@xmath16 lines in the direction of 0624 . we have measured the column densities and doppler parameters of the ly@xmath16 lines in the spectrum of 0624 using the voigt profile decomposition software vpfit ( see webb 1987rfc / vpfit.html ] ) . table [ tab : lyalist ] summarizes the h@xmath4roman1 equivalent widths , column densities , and @xmath77values measured in this fashion ( some of the lines are strongly saturated and consequently voigt profile fitting does not provide reliable measurements ; we discuss our treatment of these cases below ) . a particularly dramatic cluster of ly@xmath16 lines is evident at @xmath78 0.0635 in figure [ fig : zgaldist ] . the portion of the stis spectrum of hs0624 + 6907 covering this ly@xmath16 cluster is shown in figure [ fig : hi00635 ] . to avoid confusion with galaxy clusters , we will hereafter refer to this group of lines as a `` complex '' . this complex contains at least 13 h@xmath4roman1 components spread over a velocity range of 1000 km s@xmath28 . we will refer to the strongest ly@xmath16 absorption in figure [ fig : hi00635 ] at @xmath13 = 0.06352 as `` component a '' . component a is detected in absorption in the h@xmath4roman1 ly@xmath16 , ly@xmath39 , and ly@xmath79 transitions as well as the si@xmath4roman3 @xmath291206.50 , si@xmath4roman4 @xmath681393.76 , 1402.77 , and c@xmath4roman4 @xmath681548.20 , 1550.78 lines . low ionisation stages such as o@xmath4roman1 , c@xmath4roman2 , and si@xmath4roman2 are not detected at the redshift of component a or at the redshifts of any of the other components evident in figure [ fig : hi00635 ] . the o@xmath4roman6 doublet at the redshifts of the ly@xmath16 cluster in falls in a region that is partially blocked by galactic h@xmath80 and fe@xmath4roman2 lines . nevertheless , much of the region is free from blending , and we find no evidence for o@xmath4roman6 absorption . we also do not see the n@xmath4roman5 doublet . lccl|lccl & @xmath81 & @xmath82 & @xmath83 & & @xmath81 & @xmath82 & @xmath83 + @xmath75 & ( m ) & ( ) & ( ) & @xmath75 & ( m ) & ( ) & ( ) 0.017553@xmath84 1.0e-5 & 45@xmath8410 & [email protected] & 29@xmath84 4.3 & 0.207540@xmath84 0.5e-5 & 150@xmath84 9 & [email protected] & 27@xmath84 1.5 + 0.030651@xmath84 0.4e-5 & 99@xmath84 9 & [email protected] & 22@xmath84 1.7 & 0.213232@xmath84 1.6e-5 & 98@xmath8414 & [email protected] & 45@xmath84 5.6 + 0.041156@xmath84 0.8e-5 & 104@xmath8411 & [email protected] & 41@xmath84 3.0 & 0.219900@xmath84 2.3e-5 & 143@xmath8415 & [email protected] & 60@xmath84 8.6 + 0.053942@xmath84 0.6e-5 & 85@xmath84 7 & [email protected] & 24@xmath84 2.3 & 0.223290@xmath84 0.3e-5 & 256@xmath8412 & [email protected] & 25@xmath84 0.9 + 0.054367@xmath84 4.1e-5@xmath85 & 65@xmath8413 & [email protected] & [email protected]@xmath86 & 0.232305@xmath84 2.8e-5@xmath87 & 125@xmath8413 & [email protected] & 44@xmath84 7.7@xmath86 + 0.054829@xmath88 & 458@xmath8410 & @xmath4414.5@xmath89 & @xmath4435@xmath89 & 0.232547@xmath84 2.3e-5@xmath87 & 44@xmath8410 & [email protected] & 24@xmath84 7.3 + 0.055153@xmath84 7.8e-5@xmath87 & 237@xmath8414 & [email protected] & [email protected]@xmath86 & 0.240599@xmath84 0.6e-5 & 110@xmath8410 & [email protected] & 20@xmath84 2.0 + 0.061879@xmath84 0.4e-5 & 184@xmath84 6 & [email protected] & 21@xmath84 1.4 & 0.252251@xmath84 1.2e-5 & 55@xmath8411 & [email protected] & 24@xmath84 4.2 + 0.062014@xmath84 1.0e-5 & 21@xmath84 4 & [email protected] & 8@xmath84 4.7 & 0.268559@xmath84 2.1e-5 & 68@xmath8414 & [email protected] & 51@xmath84 7.2 + 0.062150@xmath84 1.5e-5@xmath87 & 13@xmath84 4 & [email protected] & 10@xmath84 7.9 & 0.272240@xmath84 0.6e-5 & 37@xmath84 8 & [email protected] & 12@xmath84 2.2 + 0.062343@xmath84 0.8e-5@xmath87 & 128@xmath84 7 & [email protected] & 30@xmath84 4.0 & 0.279771@xmath84 1.7e-5@xmath87 & 174@xmath8413 & [email protected] & 34@xmath84 4.9 + 0.062647@xmath84 2.5e-5@xmath87 & 101@xmath84 7 & [email protected] & [email protected] & 0.280171@xmath84 0.7e-5@xmath87 & 576@xmath8415 & [email protected] & 43@xmath84 1.9@xmath86 + 0.062762@xmath84 0.7e-5@xmath87 & 39@xmath84 3 & [email protected] & 8@xmath84 3.7 & 0.295307@xmath84 0.7e-5 & 309@xmath8415 & [email protected] & 42@xmath84 2.0 + 0.062850@xmath84 1.2e-5@xmath87 & 110@xmath84 5 & [email protected] & 20@xmath84 7.0 & 0.296607@xmath84 0.9e-5 & 203@xmath8418 & [email protected] & 52@xmath84 2.9 + 0.063037@xmath84 1.4e-5@xmath87 & 101@xmath84 6 & [email protected] & 27@xmath84 8.8 & 0.308991@xmath84 0.6e-5 & 167@xmath8412 & [email protected] & 28@xmath84 1.8 + 0.063456@xmath84 1.6e-5@xmath87 & 569@xmath84 9 & [email protected] & 48@xmath84 8.4@xmath86 & 0.309909@xmath84 5.5e-5@xmath87 & 246@xmath8418 & [email protected] & [email protected]@xmath86 + 0.063481@xmath84 1.6e-5@xmath87 & 443@xmath84 6 & [email protected] & 24@xmath84 5.5 & 0.310454@xmath84 8.0e-5@xmath87 & 170@xmath8417 & [email protected] & [email protected]@xmath86 + 0.063620@xmath84 2.7e-5@xmath87 & 153@xmath84 4 & [email protected] & 10@xmath84 5.6 & [email protected] & 88@xmath8416 & [email protected] & [email protected] + 0.064753@xmath84 0.9e-5@xmath87 & 257@xmath84 9 & [email protected] & 33@xmath84 3.0 & 0.312802@xmath84 4.4e-5 & 257@xmath8418 & [email protected] & 54@xmath84 9.3 + 0.065016@xmath84 0.8e-5@xmath87 & 282@xmath84 8 & [email protected] & 31@xmath84 2.7 & 0.313028@xmath84 1.6e-5 & 72@xmath8410 & [email protected] & 17@xmath84 6.8 + 0.075731@xmath84 0.2e-5 & 292@xmath84 8 & [email protected] & 24@xmath84 0.8 & 0.313261@xmath84 4.7e-5 & 244@xmath8419 & [email protected] & [email protected] + 0.090228@xmath84 4.2e-5 & 106@xmath8412 & [email protected] & [email protected] & 0.317901@xmath84 1.2e-5 & 139@xmath8418 & [email protected] & 34@xmath84 3.6 + 0.130757@xmath84 1.0e-5 & 114@xmath84 9 & [email protected] & 34@xmath84 3.6 & 0.320889@xmath84 0.4e-5 & 349@xmath8414 & [email protected] & 31@xmath84 1.2 + 0.135966@xmath84 3.9e-5 & 119@xmath8411 & [email protected] & [email protected] & 0.327245@xmath84 5.0e-5@xmath87 & 316@xmath8421 & [email protected] & [email protected]@xmath86 + 0.160541@xmath84 5.0e-5@xmath87 & 69@xmath84 7 & [email protected] & [email protected] & [email protected]@xmath87 & 264@xmath8426 & [email protected] & [email protected]@xmath86 + 0.160744@xmath84 1.0e-5@xmath87 & 200@xmath84 7 & [email protected] & 30@xmath84 2.4 & 0.332674@xmath84 1.1e-5 & 202@xmath8418 & [email protected] & 38@xmath84 3.4 + 0.199750@xmath84 0.6e-5 & 87@xmath84 9 & [email protected] & 17@xmath84 2.0 & 0.339759@xmath84 0.3e-5 & 647@xmath8413 & [email protected] & 42@xmath84 1.3 + 0.199946@xmath84 1.2e-5@xmath87 & 83@xmath8411 & [email protected] & 26@xmath84 4.6 & 0.346824@xmath84 0.6e-5 & 221@xmath8416 & [email protected] & 39@xmath84 1.9 + 0.204831@xmath84 0.3e-5 & 208@xmath84 9 & [email protected] & 24@xmath84 1.0 & 0.348645@xmath84 0.9e-5 & 40@xmath8410 & [email protected] & 18@xmath84 3.0 + 0.205326@xmath84 0.2e-5 & 322@xmath84 8 & [email protected] & 25@xmath84 0.8 figure [ fig : syst0063 ] compares the absorption profiles of the ly@xmath16 , ly@xmath39 , ly@xmath79 , si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 lines at @xmath13 = 0.06352 , and table [ tab : cldnlist0064 ] lists the equivalent widths , column densities , and @xmath77values of the metals detected at this redshift as well as upper limits on undetected species of interest . both voigt - profile fitting and direct integration of the `` apparent column density '' profile ( see savage & sembach 1991 ; jenkins 1996 ) were used to estimate the metal column densities . these independent methods are generally in good agreement for the metal lines , which do not appear to be strongly affected by unresolved saturation . the si@xmath4roman3 @xmath90 absorption at @xmath91 0.06352 is slightly blended with a strong h@xmath4roman1 line at @xmath91 0.05486 ( see figure [ fig : syst0063 ] ) . however , the metal lines at this redshift have a distinctive two - component profile ( see figures [ fig : syst0063 ] and [ fig : civ064 ] ) , and the si@xmath4roman3 profile shape is in good agreement with those of the c@xmath4roman4 and si@xmath4roman4 lines . this indicates that ( unrelated ) blended from @xmath13 = 0.05486 contributes little optical depth to the wavelength range where the si@xmath4roman3 absorption occurs . we fitted the and h@xmath4roman1 at @xmath920.05486 simultaneously , assuming all of the components are centered shortward of the si@xmath4roman3 line . the resulting joint fit is shown in figure [ fig : syst0063 ] . the profile parameters derived for most of the lines in the @xmath93 cluster are reasonably well - constrained . some of the components are strongly blended and are consequently more uncertain than the formal profile - fitting error bars indicate ; these are marked in table [ tab : lyalist ] . component a was also difficult to measure for a different reason : the three usable lyman series lines ( ly@xmath16 , ly@xmath39 and ly@xmath79 ) are all saturated , and consequently voigt profile fitting did not provide good constraints for the determination of the component a h@xmath4roman1 column densities . as shown in figure [ fig : cog ] , using a single - component curve of growth and the observed equivalent widths of ly@xmath94 , and @xmath79 , we find that the component a h@xmath4roman1 absorption lines can be reproduced by two distinct sets of values for the h@xmath4roman1 column density and the doppler parameter . one of the two sets implies n(h@xmath4roman1)@xmath95 , which should produce a strong absorption edge characteristic of a lyman limit system ( lls ) at an observed wavelength of 970 . this wavelength region is covered by the spectrum in the sic2a channel and is shown in figure [ fig : lls_rutr ] . the s / n of the sic2a channel is low but is adequate to constrain @xmath96(h@xmath4roman1 ) . the optical depth @xmath97 of the lyman limit absorption and the h@xmath4roman1 column density are approximately related by @xmath98 where @xmath99 ) is the absorption cross section and @xmath29 is the rest - frame wavelength . the sic2a spectrum does not show any compelling evidence of a lyman limit edge at the expected wavelength , but the continuum placement is somewhat uncertain and because of this , a small lyman limit decrement could be present . based on the small depth of the flux decrement at @xmath100 = 970 , we derive a @xmath101 upper limit of @xmath96(h@xmath4roman1 ) @xmath102 @xmath103 ( upper black solid curve in figure [ fig : lls_rutr ] ) . we also show in figure [ fig : lls_rutr ] the lyman limit absorption expected for @xmath96(h@xmath4roman1 ) = @xmath104 @xmath103 ( lower black curve ) , which is too strong with our adopted continuum placement . the absence of a strong lyman limit edge rules out the higher h@xmath4roman1 column density of 10@xmath105 @xmath103 predicted by the curve of growth shown in figure [ fig : cog ] . the lower @xmath96(h@xmath4roman1 ) derived from the curve of growth ( 10@xmath106 @xmath103 ) is consistent with the lack of a lyman limit edge . to be conservative , we present below the metallicities derived both from the upper limit [ @xmath96(h@xmath4roman1 ) @xmath107 and from the somewhat lower best value from the curve of growth shown in figure [ fig : cog ] . we next examine the physical conditions and metal enrichment of the absorption systems implied by the column densities and the doppler parameters obtained from voigt profile fitting . we concentrate on the absorbers at @xmath13=0.06352 and 0.0757 because these systems show metal absorption and can be associated with nearby galaxies / structures as discussed in [ sec : environment ] . to derive abundances from the detected metals in these systems ( si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 ) , we must apply ionisation corrections , which depend on the ionisation mechanism and physical conditions of the gas . we will show that the gas is predominantly photoionised , and that the implied metallicities are relatively high . to investigate the absorber ionisation corrections and metallicities , we employ cloudy photoionisation models ( v96 , ferland et al . 1998 ) as described in tripp et al . ( 2003 ) . in these models , the absorbers are approximated as constant density , plane - parallel gas slabs with a thickness that reproduces the observed h@xmath4roman1 column density . the gas in the cloud is photoionised by the uv background from quasars at @xmath108 . we used the uv background spectrum shape computed by haardt & madau ( 1996 ) with the intensity normalized to @xmath109 at 1 rydberg . this value is consistent with theoretical and observational constraints ( shull et al . 1999 ; weymann et al . 2001 ; dav & tripp 2001 , and references therein ) . with the models required to match the observed @xmath96(h@xmath4roman1 ) , the predicted metal column densities depend mainly on the ionisation parameter @xmath110 ( @xmath111 ionising photon density / total hydrogen number density ) , the overall metallicity , = log ( @xmath96(x)/@xmath96(y ) ) - log ( x / y)@xmath112 . ] and the assumed relative abundances of the metals . we assume solar relative abundances , and we adopt recent revisions reported by allende prieto et al . ( 2001,2002 ) and holweger ( 2001 ) for oxygen , carbon , and silicon , respectively . the high - ion column densities predicted by the photoionisation models depend on the assumed uv background shape . in this paper , we primarily use the uv background shape computed by haardt & madau ( 1996 ) , but we note that other assessments of the uv background ( e.g. , madau , haardt , & rees 1999 ; shull et al . 1999 ) adopt a somewhat steeper euv spectral index for quasars , which changes the cloudy ionisation patterns for a given metallicity and ionisation parameter . we investigate how these uv background uncertainties affect our results by modeling our systems with both uv background shapes ( see below ) . @l@ c@ c@ ccc@ & @xmath114 & & @xmath81 & & + species & ( ) & @xmath115 & ( m ) & @xmath116 & @xmath117h i & 1215.670 & 2.704 & 603@xmath84 9 & 15.37@xmath118 & @xmath119 + & 1025.722 & 1.909 & 349@xmath8421 & & @xmath119 + & 972.537 & 1.450 & 302@xmath8422 & & @xmath119 + o i & 1302.168 & 1.804 & 6@xmath84 9 & & @xmath120 + o vi & 1037.617 & 1.836 & 28@xmath8417 & & @xmath121 + c ii & 1334.532 & 2.232 & 104@xmath84 9 & & @xmath122 + c iv & 1548.204 & 2.470 & 143@xmath84 9 & 13.67@xmath123 & [email protected] + & 1550.781 & 2.169 & 90@xmath8410 & & [email protected] + si ii & 1260.422 & 3.104 & 23@xmath8412 & & @xmath124 + si iii & 1206.500 & 3.304 & 151@xmath8410 & 13.02@xmath125 & [email protected] + si iv & 1393.760 & 2.855 & 76@xmath8410 & 13.05@xmath126 & [email protected] + & 1402.773 & 2.554 & 49@xmath8411 & & [email protected] + n v & 1242.804 & 1.988 & 23@xmath8410 & & @xmath127 as discussed above , the h@xmath4roman1 absorption lines detected at @xmath128 are spread over a wide velocity range of @xmath129 ( see figure [ fig : hi00635 ] ) . from the velocity centroids of the 13 fitted voigt profiles and by using the biweight statistic as described in beers et al . ( 1990 ) , we estimate that the line - of - sight velocity dispersion of this h@xmath4roman1 absorption complex is @xmath130 . the velocity dispersion of galaxies near this redshift is comparable to this value though substantially more uncertain ( due to the larger uncertainties in the galaxy redshifts ) . this velocity dispersion is comparable to those observed in elliptical - rich galaxy groups ( e.g. zabludoff & mulchaey 1998 ) , which is interesting because elliptical - rich groups often show diffuse x - ray emission ( mulchaey & zabludoff 1998 ) indicative of hot gas in the intragroup medium . however , we will show in [ sec : discussion ] that the available information suggests that this group is predominantly composed of late - type spiral and s0 galaxies . spiral - rich groups are much fainter in x - rays but could still contain hot intragroup gas if the gas is somewhat cooler ( @xmath131 k ) or has a much lower density than that found in elliptical - rich groups ( mulchaey 2000 ) . however , we argue that most of the gas in the complex at @xmath132 is unlikely to be hot , collisionally ionised gas for several reasons : first , the h@xmath4roman1 lines in the complex are generally too narrow . if the line broadening is dominated by thermal motions , then the doppler parameter is directly related to the gas temperature , @xmath133 , where @xmath134 is the mass , @xmath135 is the atomic mass number , and the numerical coefficient is for @xmath83 in km s@xmath28 and @xmath136 in k. since other factors such as turbulence and multiple components can contribute to the line broadening , @xmath77values provide only upper limits on the temperature . applying this equation to the line @xmath77values from table [ tab : lyalist ] , we find that most of the h@xmath4roman1 lines in the @xmath137 complex indicate that @xmath8 k , which is colder than expected for the diffuse intragroup medium based on observed group velocity dispersions , even in spiral - rich groups ( e.g. , mulchaey et al . 1996 ) . in a complex cluster of lines , it is easy to hide a broad component indicative of hot gas ( see , e.g. , figure 6 in tripp & savage 2000 ) , so the narrow lines do not preclude the presence of hot gas , but they do indicate that many cool clouds are present in the intragroup medium . second , the metal line profiles in component a favor cool , photoionised gas . if component a metal lines were to arise in gas in collisional ionisation equilibrium ( cie ) , the @xmath96(c@xmath4roman4)/@xmath96(si@xmath4roman4 ) and @xmath96(si@xmath4roman4)/@xmath96(si@xmath4roman3 ) column density ratios ( integrated across both components seen in these species , see table [ tab : cldnlist0064 ] ) would require a gas temperature @xmath138 k ( sutherland & dopita 1993 ) . however the c@xmath4roman4 component at @xmath139 km s@xmath28 is rather narrow . to show this , figure [ fig : civ064 ] plots an expanded view of the c@xmath4roman4 doublet . we see that the @xmath139 km s@xmath28 is marginally resolved at the stis e140 m resolution of @xmath140 km s@xmath28 . voigt profile fitting for this component formally yields @xmath141 , which is significantly lower than the @xmath77value implied by the cie temperature , i.e. , @xmath142 km s@xmath28 . the stronger component at @xmath143 km s@xmath28 is broader ( see figure [ fig : civ064 ] ) , but the @xmath96(c@xmath4roman4)/@xmath96(si@xmath4roman4 ) and @xmath96(si@xmath4roman4)/@xmath96(si@xmath4roman3 ) ratios are similar in the components at @xmath144 and 0 km s@xmath28 , and we expect the ionisation mechanism and physical conditions to be similar in both components . third , cloudy models photoionised by the uv background from qsos are fully consistent with the measured si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 column densities ( and upper limits on undetected species ) at @xmath13 = 0.06352 . figure [ fig : cloudym ] shows the relevant metal column densities predicted by cloudy models ( small symbols connected with solid lines ) with log @xmath96(h@xmath4roman1 ) = 15.37 and [ m / h ] = @xmath145 compared to the observed column densities ( large symbols ) . we can see that the metal column densities are in agreement ( within the 1@xmath146 observational uncertainties ) with this model at log @xmath147 ( log @xmath148 ) . the narrow h@xmath4roman1 and c@xmath4roman4 components at this redshift could still arise in shock - heated material if they originate in gas that is not in ionisation equilibrium . many papers have considered the properties of gas that is initially shock - heated to some high temperature and then cools more rapidly than it can recombine ( e.g. , shapiro & moore 1976 ; edgar & chevalier 1986 ) . however , if component a metal lines were to arise in gas in such a state , according to both computations from shapiro & moore ( 1996 ) and schmutzler & tscharnuter ( 1992 ) , the @xmath96(si@xmath4roman3)/@xmath96(si@xmath4roman4 ) column density ratio would require a gas temperature similar ( @xmath138 k ) to the one found for the collisional ionisation equilibrium hypothesis . moreover , assuming solar abundances , the predicted o@xmath4roman6 column density at this temperature is always higher than our observed upper limit ( @xmath445 times higher in the schmutzler model ) . finally , the @xmath96(c@xmath4roman4)/@xmath96(si@xmath4roman4 ) column density ratio implies an even higher temperature than 10@xmath149 k ( 2.5 times higherr in the shapiro model ) . because of these points , this non - equilibrium cooling gas scenario seems unlikely to apply to the @xmath113 absorber toward 0624 . the cloudy modeling has some other interesting implications in addition to the basic conclusion that the gas is photoionised . for example , the photoionisation model indicates that the absorber has a relatively high metallicity of @xmath150 even though we have found no luminous galaxies within @xmath151 . a similarly high metallicity ( [ o / h]@xmath152 ) was recently reported by jenkins et al . ( 2005 ) for a lls in the spectrum of phl1811 , but that system is much closer in projection to a luminous galaxy . if we adopt the more conservative upper limit on @xmath96(h@xmath4roman1 ) from the absence of a lyman limit edge ( @xmath96(h@xmath4roman1)@xmath153 , see figure [ fig : lls_rutr ] ) instead of the curve - of - growth h@xmath4roman1 column , we obtain [ m / h ] @xmath154 . this lower limit is still substantially higher than metallicities typically observed in analogous absorbers at higher redshifts ( e.g. , schaye et al . 2003 ) and is comparable to abundances seen in high - velocity clouds near the milky way ( e.g. , sembach et al . 2001,2004 ; collins , shull , & giroux 2003 ; tripp et al . 2003 ; ganguly et al . 2005 ; fox et al . 2005 ) . to derive confidence limits on parameters extracted from our cloudy models , for combinations of metallicity @xmath155 and ionisation parameter @xmath110 we calculated the @xmath156 statistic , @xmath157 where @xmath96 indicates column density and the sum is over the three ions si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 . with the minimum @xmath156 obtained at [ m / h ] = @xmath145 and log @xmath158 , we evaluated confidence limits by finding parameters that increased @xmath159 by the amount appropriate for a given confidence level ( see lampton , margon , & bowyer 1976 ; press et al . 1992 ) . in this way , we find [ m / h ] = @xmath160 at the 2@xmath146 confidence level . of course , these confidence limits do not fully reflect potential sources of systematic error such as uncertainties in the shape of the ionising flux field or accuracy of the atomic data incorporated into cloudy . when we used the steeper uv background shape ( e.g. madau , haardt , & rees 1999 ; shull et al . 1999 ) , the observations are still consistent with the cloudy model for a lower metallicity of [ m / h ] = @xmath161 and a larger ionisation parameter log @xmath162 . we can also place constraints on physical quantities such as the absorber size ( the length of the path through the absorbing region ) and the thermal gas pressure ( but see the caveats discussed in 5 of tripp et al . figure [ fig : absorbersize ] shows confidence interval contours for the absorber size @xmath163 and thermal pressure @xmath164 implied by the photoionisation model . the best fit implies that @xmath165 3.5 kpc . if spherical , the baryonic mass of this cloud would be @xmath166 . however , we can see from figure [ fig : absorbersize ] that the model allows a large range for @xmath163 at the 2@xmath146 level . the low thermal pressure implied is also notable . when the steeper uv background shape is used for the cloudy model , the predicted pressure is lower by a factor 1.5 and the absorber size increases to @xmath1678 kpc.the range of pressures within the contours in figure [ fig : absorbersize ] is several orders of magnitude lower than the gas pressure measured in the disk of the milky way ( see jenkins & tripp 2001 ) and is even lower than pressures measured in hvcs in the milky way halo ( e.g. wakker , oosterloo , & putman 2002 ; fox et al . . however , sembach et al . ( 1995 , 1999 ) found similar pressure for civ hvcs surrounding the milky - way with somewhat lower metallicity . moreover , pressures this low are predicted in some theoretical models of galactic halos ( wolfire et al . finally , the derived pressure depends on the intensity used to normalize the ionising flux field ( see tripp et al . 2005 ) and both the particle density and the pressure could be higher if the radiation field is brighter than we assumed . as discussed in [ sec : environment ] , the absorption lines detected at @xmath13 = 0.07573 occur in a large - scale structure connected to a559 and a564 . only h@xmath4roman1 ly@xmath16 , ly@xmath39 and the c@xmath4roman4 @xmath681548.20,1550.78 doublet are clearly detected at this redshift ; the profiles of these absorption lines are shown in figure [ fig : syst0076 ] . the weakness of ly@xmath39 and the absence of ly@xmath79 suggest that the h@xmath4roman1 lines are not badly affected by unresolved saturation , and profile fitting measurements are robust . likewise , comparison of the c@xmath4roman4 apparent column density profiles shows no evidence of unresolved saturation . we have fitted these lines with only one component . the results of the fit are listed in table [ tab : cldnlist076 ] . the width of the h@xmath4roman1 line ( @xmath169 ) indicates a temperature for the gas lower than @xmath170 k , which again favors a photoionisation process . no o@xmath4roman6 is evident at this redshift , but several strong unrelated lines of various elements are found close to the expected wavelength of the o@xmath4roman6 doublet , and these lines might mask weak o@xmath4roman6 absorption . despite the fact that c@xmath4roman4 is the only metal detected in this system , we can nevertheless place an interesting lower limit on the absorber metallicity . the carbon abundance can be expressed as [ c / h]= log[/ ] + log[/ ] - log ( c / h)@xmath171 , where and are the ion fractions of h@xmath4roman1 and c@xmath4roman4 , respectively combination . the maximum is not evident in figure [ fig : cloudym ] because it occurs at a higher value of @xmath110 than the range shown . ] . with the h@xmath4roman1 column from the ly@xmath16+ly@xmath39 fit ( table [ tab : cldnlist076 ] ) , and again assuming that the gas is photoionised by the uv background from qsos ( haardt & madau 1996 ) , we find that log[/]@xmath172 , and therefore @xmath173>-0.6 $ ] in the absorber . once again , this metallicity lower limit is relatively high despite the fact that no luminous galaxies have been found near the sight line ( the closest galaxy is nw3 at @xmath174 kpc , see table [ tab : spec_red ] ) . @l@c@c@c@c@c@ & & & & & + & & & & & h i & 1215.670 & 309@xmath8410 & [email protected] & [email protected] & [email protected] + & 1025.722 & 108@xmath8421 & & & [email protected] + o vi & 1037.617 & 0@xmath8419 & & & @xmath175 + c iv & 1548.204 & 51@xmath84 7 & [email protected] & [email protected] & [email protected] + & 1550.781 & 27@xmath84 8 & & & [email protected] + si iii & 1206.500 & 0@xmath8410 & & & @xmath176 + si iv & 1393.760 & 17@xmath8414 & & & @xmath124 + n v & 1238.821 & 25@xmath84 9 & & & @xmath177 in [ sec : environment ] we noted that the clusters a554 , a562 , and a565 indicate that a large - scale filament / supercluster is foreground to 0624 . our galaxy redshift survey has also revealed four galaxies close to 0624 at the redshift of the abell 554/562/565 structure ( see figure [ fig : field ] and table [ tab : spec_red ] ) , which suggests that the large - scale filament extends across the 0624 field . however , we only find a couple of weak ly@xmath16 lines near the redshift of this structure at redshifts substantially offset from those of the galaxies and abell clusters . it is possible that the ly@xmath16 lines are weak / absent because the gas in the filament is so hot that the h@xmath4roman1 ion fraction makes the ly@xmath16 line undetectable , but we also note that the nearest galaxies are farther from the sight line in this case ( @xmath179 mpc ) than in the structures at @xmath180 and discussed above , so it is also possible that the sight line does not penetrate the part of the dark matter filament where the potential is deep enough to accumulate gas and galaxies ( see discussion in bowen et al . how do the properties of the ly@xmath16 absorbers at , , and compare to the other ly@xmath16 lines in the 0624 spectrum ( and in other sight lines ) ? we have found that the systems at and arise in photoionised cool gas ; is this true of the majority of the ly@xmath16 lines in the spectrum ? in particular , do we find ly@xmath16 lines that arise in hot gas ? richter et al . ( 2004 , 2005 ) and sembach et al . ( 2004 ) have recently identified a population of broad ly@xmath16 lines ( blas ) with @xmath181 40 km s@xmath28 in the spectra of several low@xmath0 qsos ( pg0953 + 415 , pg1116 + 215 , pg1259 + 593 , and h1821 + 643 ) . williger et al . ( 2005 ) similarly find a substantial number of blas in the spectrum of pks0405 - 123 . bowen et al . ( 2002 ) have also identified some bla candidates using somewhat lower resolution data . if these lines are mainly broadened by thermal motions , then they trace the warm - hot igm , and moreover , they in this case would contain a substantial portion of the baryons in the universe at the present epoch ( see richter et al . 2004 , 2005 ; sembach et al . 2004 ) . based on simulations , richter et al . ( 2005 ) and williger et al . ( 2005 ) find that some of the blas are not predominantly thermally broadened but instead are due to line blends that are difficult to recognize at the s / n afforded by typical stis echelle spectra . however , richter et al . ( 2005 ) conclude that approximately 50 per cent of the blas are mainly thermally broadened , and some high s / n examples in the above papers are remarkably smooth and broad and appear to entirely consistent with a single broad gaussian ( see , e.g. , figures 4 and 5 in richter et al . 2005 ) . in this paper , in addition to using a different sight line , we have employed methods that are independent from ( e.g. , using different software ) the techniques used in the papers above for continuum normalization , line detection , and profile fitting . consequently , we have an opportunity to independently check the bla findings reported in these papers . for the 0624 sight line , we find that the mean @xmath77value for all ly@xmath16 lines is @xmath182 = 37 km s@xmath28 , and the median @xmath183 km s@xmath28 . however , as noted in table [ tab : lyalist ] , some of the ly@xmath16 lines are significantly blended , and the line parameters are accordingly uncertain . if we exclude these uncertain blended cases , we find @xmath184 km s@xmath28 and @xmath185 km s@xmath28 . these ensemble @xmath77values are in reasonable agreement with the previous high - resolution ly@xmath16 studies . in figure [ fig : larg ] we compare our measurements of the h@xmath4roman1 @xmath77values and column densities from table [ tab : lyalist ] to the measurements reported by richter et al . ( 2004 ) and sembach et al . the solid line indicates @xmath83 vs. @xmath96(h@xmath4roman1 ) for a gaussian line with central optical depth = 0.1 . this is effectively a detection threshold ; lines that have ( @xmath186 ) combinations to the left of this line are not likely to be detected . the dotted line in figure [ fig : larg ] shows the minimum @xmath77value as a function of @xmath96(h@xmath4roman1 ) predicted by dav & tripp ( 2001 , see their equation 5 ) from the hydrodynamic cosmological simulations of dav et al . this predicted lower envelope appears to be in reasonable agreement with the observed lower envelope for the three sight lines shown in the figure . from figure [ fig : larg ] , we see that the ( @xmath186 ) distribution that we have obtained from the 0624 sight line appears to be generally similar to those obtained by sembach et al . ( 2004 ) and richter et al . sembach et al . and richter et al . did find more extremely broad ly@xmath16 lines ( @xmath181 80 km s@xmath28 ) than we have been able to positively identify in the 0624 spectrum . this may be partly due to signal - to - noise differences the data employed by richter et al . and sembach et al . have higher s / n because such broad and shallow lines are difficult to detect in the 0624 data . for @xmath187 km s@xmath28 , the different sight lines appear to be in broad agreement . excluding lines within 5000 km s@xmath28 of the qso redshift , of the qso redshift can arise in intrinsic gas ejected by the qso , and these lines can be rather broad ( see , e.g. , yuan et al . 2002 ) , even if not part of a full - blown broad absorption line outflow . ] the 0624 spectrum can be used to search for blas between @xmath188 and @xmath189 . accounting for regions in which broad lines could have been masked by igm or ism lines , we obtain a blocking - corrected total redshift path @xmath190 . with 21 ly@xmath16 lines in the sample with @xmath181 40 km s@xmath28 , we thus obtain @xmath191(bla ) = 64@xmath8416 . this is somewhat larger than the values reported by sembach et al . ( 2004 ) and richter et al . ( 2004,2005 ) . however , richter et al . have excluded blas that are located in complex blends on the grounds that these cases are more likely to be affected by non - thermal broadening . if we follow the same procedure , we must reject 10 blas ( see table [ tab : lyalist ] ) ; the remaining 11 blas would then imply @xmath191(bla ) = 33@xmath8410 . using equations 1 , 5 , and 6 from sembach et al . ( 2004 ) , but adjusted for the somewhat different cosmological parameters assumed in this paper , we find that our full sample implies that the bla baryonic content is @xmath192(bla ) = 0.017 @xmath7 ( in the usual notation , i.e. , @xmath193 ) . this high value probably substantially overestimates the bla baryonic content , largely because of false blas that arise from blends . if we exclude blas that are located in complex blends , this drops to @xmath192(bla ) = 0.0036 @xmath7 , which is similar to values obtained by richter et al . and sembach et al . the uncertainties in @xmath192(bla ) due to , e.g. , lines that are broad due to blends or other non - thermal broadening mechanisms , are large and currently difficult to assess ( see discussion in richter et al . however , these initial calculations suggest that blas may harbor an important quantity of baryons . with future uv spectrographs , it would be valuable to obtain high - resolution spectra with substantially better s / n in order to accurately assess the baryonic content of blas as part of the general census of ordinary matter in the nearby universe . we have acquired detailed information about the abundances , physical conditions , and galaxy proximity of absorption systems in the direction of 0624 . what are the implications of these measurements for broader questions of galaxy evolution and cosmology ? the processes that add gas to galaxies ( e.g. accretion ) and remove gas from galaxies ( e.g. , winds , dynamical stripping ) can have profound effects on galaxy evolution , and the `` feedback '' of matter and energy from galaxies into the igm is now believed to play an important role in shaping structures that subsequently grow out of the igm ( voit g. m. , 2005 ) . the quantity and implications of the @xmath194 k whim gas is a topic of particular interest currently . the galaxies and absorption systems in the direction of 0624 , particularly the galaxy group and ly@xmath16 complex at @xmath75 = , have some interesting , and perhaps surprising , implications regarding these questions , which we now discuss . rosat observations of diffuse x - ray emission have established that galaxy groups that are dominated by early - type galaxies often contain diffuse , hot intragroup gas ( mulchaey 2000 , and references therein ) . based on the observed relation between intragroup gas temperature and velocity dispersion @xmath146 in x - ray bright groups ( @xmath195 ) and the fact that spiral - rich groups have lower velocity dispersions than elliptical - rich groups , mulchaey et al . ( 1996 ) have hypothesized that spiral - rich groups might have somewhat cooler intragroup media that could give rise to qso absorption lines at whim temperatures ( e.g. , o@xmath4roman6 ) . however , the galaxy group at @xmath75 = appears to have properties that are not consistent with the elliptical - rich groups detected with rosat nor with the idea that spiral - rich groups contain warm - hot intragroup gas . it is unclear if the galaxy group at @xmath75 = is a spiral - rich group . figure [ fig : galpics ] shows @xmath45 images from the mosa data of the 10 galaxies that we have found in this group . most of the galaxies in the group show evidence of disks and bulges ( both in the direct images and in radial brightness profiles ) . we find from visual inspection that 4 - 5 of the 10 galaxies have indications of spiral structure ( sw3 , se1 , se8 , se6 , and possibly se4 ) . however , the remaining galaxies could be early - type s0 galaxies , and therefore the early - type fraction might be comparable to groups that show diffuse x - ray emission ( see , e.g , figure 7 in zabludoff & mulchaey 1998 ) . the ne3 , se13 , and se5 galaxies , which morphologically appear to be early - type galaxies , have colours and magnitudes consistent with the `` red sequence '' colour - magnitude relation observed in clusters ( e.g. , bower , lucey , & ellis 1992 ; mcintosh et al . 2005 ) ; these galaxies are likely s0s ( the other 7 galaxies have blue colours characteristic of late types ) . the velocity dispersion of the group at @xmath75 = , albeit uncertain , is more comparable to those of elliptical - dominated groups than spiral - rich groups ( mulchaey et al . 1996 ; zabludoff & mulchaey 1998 ) . regardless of whether the group is elliptical- or spiral - rich , it is surprising that we find a large number of cool , photoionised clouds in the intragroup medium ( [ ss : z064 ] ) . in the hot intragroup medium of an elliptical - rich group , h@xmath4roman1 lines should be extremely broad and weak , but instead we find strong , narrow lines ( see figure [ fig : hi00635 ] ) . even in the cooler gas predicted to be found in late - type dominated groups , the h@xmath4roman1 lines should be broader . we could entertain models of cooling intragroup gas , but in such models o@xmath4roman6 is expected to be stronger . likewise , if the intragroup gas is a multiphase medium with cooler clouds ( which cause the h@xmath4roman1 absorption lines ) embedded in a hotter phase , then we might expect to detect o@xmath4roman6 from the interface between the phases ( fox et al . 2005 ) , unless conduction is somehow suppressed . the lack of evidence of hot gas leads us to question whether this group is a bound , virialized system . an alternative possibility is that our sight line passes along the long axis of a large - scale filamentary structure in the cosmic web . in this case , the projection of the galaxies and ly@xmath16 clouds along the sight line could give a false impression of a group in which hot gas would be expected . however , in cosmological simulations of large - scale filaments , whim gas is expected to be widespread at the present epoch , even in modest - overdensity regions ( see , e.g. , figure 4 in cen & ostriker 1999b ) , so it is interesting that we find a substantial number of cool clouds at @xmath12 , somewhat contrary to theoretical expectations . as noted above , our data do not preclude the presence of whim gas at @xmath12 , but we find no clear evidence of it . other sight lines show similar clusters of ly@xmath16 lines , e.g. , the ly@xmath16 complex at @xmath196 0.057 toward pks2155 - 304 ( shull et al . 1998 ; shull , tumlinson , & giroux 2003 ) or the ly@xmath16 lines at @xmath196 0.121 toward h1821 + 643 ( tripp et al . however , unlike the 0624 ly@xmath16 complex , the pks2155 - 304 and h1821 + 643 examples both show evidence of warm - hot intragroup gas . to test whether the observations and simulations are in accord , it would be useful to assess the frequency and physical properties of these ly@xmath16 complexes in cosmological simulations for comparison with the observations . it is also interesting that the two systems for which we have obtained abundance constraints ( at @xmath75 = and ) both indicate relatively high metallicities , but both of these systems are at least 100 kpc away ( in projection ) from the nearest known galaxy . this naturally raises a question : how did gas that is so far from a galaxy attain such a high metallicity ? the gas could have been driven out of a galaxy by a galactic wind ; some wind models predict that the outflowing material will have a high metallicity ( mac low & ferrara 1999 ) , the difficulty with this interpretation is that winds from nearby galaxies are usually observed to contain substantial amounts of hot gas ( e.g. , strickland et al . 2004 ) , which seems to be inconsistent with the absorption line properties as we have discussed . a more likely explanation is that the high - metallicity gas we have detected in absorption has been tidally stripped out of one of the nearby galaxies . there are indications that tidal stripping could be a more gentle process for removing gas from galaxies , and a tidally stripped origin can therefore more easily accommodate the observed low - ionisation state of the gas . for example , in the direction of ngc3783 , the galactic high - velocity cloud ( hvc ) at @xmath197 247 is now recognized to be tidally stripped debris from the smc . while this tidally stripped material shows a wide array of low - ionisation absorption lines , it has little or no associated high - ion absorption ( lu et al . 1994 ; sembach et al . moreover , the tidally - stripped hvc contains molecular hydrogen , which sembach et al . argue formed in the smc and survived the rigors of tidal stripping ( as opposed to forming in situ in the stream ) . both the absence of high ions and the survival of h@xmath80 suggest that the stripping process did not substantially ionise and heat this hvc . several galaxies are close enough to the 0624 sight line to be plausible sources of the gas in a tidal stripping scenario . one of the nearby galaxies , se1 , has a distorted spiral morphology . this galaxy is a plausible source of tidally stripped matter . we have presented a study of absorption - line systems in the direction of 0624 using a combination of high - resolution uv spectra obtained with _ hst_/stis and plus ground - based imaging and spectroscopy of galaxies within @xmath4430 of the sight line . in addition to presented the basic measurements and ancillary information , we have reported the following findings : \1 . there are several abell galaxy clusters in the foreground of 0624 , including two clusters at @xmath12 0.077 ( a559 and a564 ) and three at @xmath12 0.110 ( a554 , a562 , and a565 ) . these clusters trace large - scale dark matter structures , i.e. , superclusters or filaments of the `` cosmic web '' . our galaxy redshift survey has revealed galaxies at these supercluster redshifts in the immediate vicinity of the 0624 sight line , and therefore our qso spectra provide an opportunity to study the gas in large - scale intergalactic filaments . the most prominent group of galaxies found in our galaxy redshift survey is at @xmath12 and is not associated with an abell cluster with a spectroscopic redshift from the literature . however , a557 , for which no spectroscopic redshift has been reported , is at least partly due to the galaxy group at @xmath12 . the two strongest ly@xmath16 absorption systems at @xmath198 arise in galaxy groups at @xmath199 and . the ly@xmath16 absorption at is particularly dramatic : at this redshift , we find 13 ly@xmath16 lines spread over a velocity range of 1000 km s@xmath28 with a line - of - sight velocity dispersion of 265 km s@xmath28 . the second - strongest system at is associated with the abell 559/564 large - scale structure , and this indicates that a filament containing gas and galaxies feeds into the abell 559/564 supercluster . analysis of the ly@xmath16 absorption - line complex at @xmath200 provides strong evidence that the gas is photoionised and relatively cool ; we find no compelling evidence of warm - hot gas in this large - scale filament . we detect si@xmath4roman3 , si@xmath4roman4 , and c@xmath4roman4 in the strongest component in this ly@xmath16 complex , and photoionisation models indicate that the gas metallicity is high , [ m / h ] = @xmath201 . this is somewhat surprising because we do not find any luminous galaxies close to the sight line ; the closest galaxy is at a projected distance @xmath202 kpc . the ly@xmath16 system at is only detected in the c@xmath4roman4 doublet , but nevertheless we find a similar result : the lower limit on the metallicity is relatively high ( [ c / h ] @xmath14 ) while the nearest galaxy is at @xmath203 kpc . we have compared the distribution of ly@xmath16 doppler parameters and h@xmath4roman1 column densities to high - resolution measurements obtained from other sight lines , and we find good agreement . we find that the number of broad ly@xmath16 absorbers with @xmath181 40 km s@xmath28 per unit redshift is in agreement with results recently reported by richter et al . ( 2004,2005 ) and sembach et al . the baryonic content of these broad h@xmath4roman1 lines is still highly uncertain and requires confirmation with higher s / n data , but it is probable that some of these broad line arise in warm - hot gas and contain an important portion of the baryons in the nearby universe . we also find that the lower bound on @xmath83 vs. @xmath96(h@xmath4roman1 ) is in agreement with predictions from cosmological simulations . the absence of warm - hot gas in the galaxy group / ly@xmath16 complex at @xmath200 is difficult to reconcile with x - ray observations of bound galaxy groups . it seems more likely that this in this case we are viewing a large - scale cosmic web filament along its long axis . the filament contains a mix of early- and late - type galaxies and many cool , photoionised clouds . the high - metallicity , cool cloud at @xmath13 = 0.06352 is probably tidally stripped material . this origin can explain the high metallicity and the lack of hot gas . one of the nearby galaxies has a disturbed morphology consistent with this hypothesis . similar clusters of ly@xmath16 lines have been observed in other sight lines , and additional examples are likely to be found as we continue to analyse stis data . we look forward to comparisons of these observations to predictions from cosmological simulations and other theoretical work in order to better understand the processes that affect the evolution of galaxies and the intergalactic medium . we thank dan mcintosh and neal katz for useful discussions . the stis observations of hs0624 + 6907 were obtained for _ hst _ program 9184 with financial support through nasa grant hst go-9184.08-a from the space telescope science institute . this research was also supported in part by nasa through long - term space astrophysics grant nng 04gg73 g . the data were obtained by the pi team of the nasa - cnes - csa project , which is operated by johns hopkins university with financial support through nasa contract nas 5 - 32985 . this research has made use of the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , california institute of technology , under contract with nasa . allende prieto , c. , lambert , d. l. , & asplund , a. 2001 , [ @]jnlapj , 556 , l63 allende prieto , c. , lambert , d. l. , & asplund , a. 2002 , [ @]jnlapj , 573 , l137 , c. m. , croton , d. j. , gazta~ naga , e. , norberg , p. , colless , m. , baldry , i. k. , et al . 2004 , [ @]jnlmnras , 351 , l44 beers , t. c. , flynn , k. , & gebhardt , k. 1990 , [ @]jnlaj , 100 , 32 bergeron , j. , aracil , b. , petitjean , p. & pichon , c. 2002 , a&a , 396 , 11 bertin , e. , & arnouts , s. 1996 , a&as , 117 , 393 bowen , d. v. , pettini , m. , & blades , j. c. 2002 , [ @]jnlapj , 580 , 169 bower , r. g. , lucey , j. r. , & ellis , r. s. 1992 , mnras , 254 , 601 bregman , j. n. , dupke , r. a. , & miller , e. d. 2004 , [ @]jnlapj , 614 , 31 cen , r. , & ostriker , j. p. 1999a , [ @]jnlapj , 514 , 1 , r. & ostriker , j. p. 1999 , [ @]jnlapj , 519 , l109 chen , h .- w . , lanzetta , k. m. , webb , j. k. , & barcons , x. 2001 , [ @]jnlapj , 559 , 654 chen , h .- w . , marzke , r. o. , mccarthy , p. j. , martini , p. , carlberg , r. g. , persson , s. e. , bunker , a. , bridge , c. r. , abraham , r. g. 2003 , [ @]jnlapj , 586 , 745 , j. m. , krughoff , k. s. & connolly , a. j. 2005 , [ @]jnlmnras , 359 , 272 , m. , dalton , g. , maddox , s. , sutherland , w. , norberg , p. , cole , s. , et al . 2001 , [ @]jnlmnras , 328 , 1039 collins , j. a. , shull , j. m. , & giroux , m. l. 2003 , [ @]jnlapj , 585 , 336 danforth , c. w. , & shull , j. m. 2005 , [ @]jnlapj , 624 , 555 dav , r. , hernquist , l. , katz , n. , & weinberg , d. h. 1999 , [ @]jnlapj , 511 , 521 , r. , cen , r. , ostriker , j. p. , bryan , g. l. , hernquist , l. , katz , n. , weinberg , d. h. , norman , m. l. & oshea , b. 2001 , [ @]jnlapj , 552 , 473 , r. & tripp , t. m. 2001 , [ @]jnlapj , 553 , 528 , j. p. , clowe , d. , schneider , p. , kerp , j. & romano - daz , e. 2004 , iau symposium , 34 , a. , tucker , d. l. , allam , s. & way , m. j. 2004 , [ @]jnla&a , 418 , 7 edgar , r. j. , & chevalier , r. a. 1986 , [ @]jnlapj , 310 , l27 , m. , einasto , j. , tago , e. , mller , v. , & andernach , h. 2001 , aj , 122 , 2222 , g. j. , korista , k. t. , verner , d. a. , ferguson , j. w. , kingdon , j. b. & verner , e. m. 1998 , [ @]jnlpasp , 110 , 761 fox , a. j. , wakker , b. p. , savage , b. d. , tripp , t. m. , sembach , k. r. , & bland - hawthorn , j. 2005 , [ @]jnlapj , 630 , 332 futamoto , k. , mitsuda , k. , takei , y. , fujimoto , r. , & yamasaki , n. y. 2004 , [ @]jnlapj , 605 , 793 ganguly , r. , sembach , k. r. , tripp , t. m. , & savage , b. d. 2005 , [ @]jnlapjs , 157 , 251 , m. e. , taylor , a. n. , meisenheimer , k. , dye , s. , wolf , c. & thommes , e. 2002 , [ @]jnlapj , 568 , 141 , f. & madau , p. 1996 , [ @]jnlapj , 461 , 20 heckman , t. m. , norman , c. a. , strickland , d. k. , & sembach , k. r. 2002 , [ @]jnlapj , 577 , 691 holweger , h. 2001 , in aip conf . proc . 598 , solar and galactic composition , ed . r. f. wimmer - schweingruber ( new york : aip ) , 23 impey , c. d. , petry , c. e. , & flint , k. p. 1999 , [ @]jnlapj , 524 , 536 , a. , frenk , c. s. , pearce , f. r. , thomas , p. a. , colberg , j. m. , white , s. d. m. , couchman , h. m. p. , peacock , j. a. , efstathiou , g. & nelson , a. h. 1998 , [ @]jnlapj , 499 , 20 , e. b. 1996 , [ @]jnlapj , 471 , 292 jenkins , e. b. , bowen , d. v. , tripp , t. m. , & sembach , k. r. 2005 , [ @]jnlapj , 623 , 767 jenkins , e. b. , & tripp , t. m. 2001 , [ @]jnlapjs , 137 , 297 , r. , colless , m. , raychaudhury , s. & peterson , b. a. 2003 , [ @]jnlmnras , 339 , 652 katz , n. , weinberg , d. h. , hernquist , l. , & miralda - escud , j. 1996 , [ @]jnlapj , 457 , l57 , r. a. , woodgate , b. e. , bowers , c. w. , kraemer , s. b. , kaiser , m. e. , gull , t. r. , et al . 1998 , [ @]jnlapj , 492 , l83 lampton , m. , margon , b. , & bowyer , s. 1976 , [ @]jnlapj , 208 , 177 landolt , a. u. 1992 , aj , 104 , 340 lanzetta , k. m. , bowen , d. v. , tytler , d. , & webb , j. k. 1995 , [ @]jnlapj , 442 , 538 lin , h. , kirshner , r. p. , shectman , s. a. , landy , s. d. , oemler , a. , tucker , d. l. , & schechter , p. l. 1996 , [ @]jnlapj , 464 , 60 lu , l. , savage , b. d. , & sembach , k. r. 1994 , apj , 426 , 563 mac low , m .- m . , & ferrara , a. 1999 , [ @]jnlapj , 513 , 142 maller , a. h. , mcintosh , d. h. , katz . n. , & weinberg , m. d. 2003 , [ @]jnlapj , 598 , l1 mcintosh , d. h. , zabludoff , a. i. , rix , h .- w . , & caldwell , n. 2005 , [ @]jnlapj , 619 , 193 mckernan , b. , yaqoob , t. , & reynolds , c. s. 2004 , [ @]jnlapj , 617 , 232 miralda - escud , j. , cen , r. , ostriker , j. p. , & rauch , m. 1996 , [ @]jnlapj , 471 , 582 , h. w. , cash , w. c. , cowie , l. l. , davidsen , a. f. , dupree , a. k. , feldman , p. d. , et al . 2000 , [ @]jnlapj , 538 , l1 , h. w. , sembach , k. r. , vidal - madjar , a. , york , d. g. , friedman , s. d. , h ' ebrard , g. , kruk , j. w. , et al . 2002 , [ @]jnlapjs , 140 , 3 mulchaey , j. s. 2000 , ara&a , 38 , 289 mulchaey , j. s. , mushotzky , r. f. , burstein , d. , & davis , d. s. 1996 , [ @]jnlapj , 456 , l5 mulchaey , j. s. , & zabludoff , a. i. 1998 , [ @]jnlapj , 496 , 73 muller , g. p. , reed , r. , armandroff , t. , boroson , t. , & jacoby , g. 1998 , spie , 3355 , 721 nicastro , f. et al . 2002 , [ @]jnlapj , 573 , 157 penton , s. v. , stocke , j. t. , & shull , j. m. 2002 , [ @]jnlapj , 565 , 720 press , w. h. , teukolsky , s. a. , vetterling , w. t. , & flannery , b. p. 1992 , numerical recipes in fortran ( 2nd ed . ; cambridge : cambridge univ . press ) prochaska , j. x. , chen , h .- w . , howk , j. c. , weiner , b. j. , & mulchaey , j. 2004 , [ @]jnlapj , 617 , 718 prochaska , j. x. , gawiser , e. , wolfe , a. m. , cooke , j. , & gelino , d. 2003 , apjs , 147 , 227 richter , p. , savage , b. d. , sembach , k. r. , & tripp , t. m. 2005 , a&a , submitted , p. , savage , b. d. , tripp , t. m. & sembach , k. r. 2004 , [ @]jnlapjs , 153 , 165 , k. , mahdavi , a. , geller , m. j. , diaferio , a. , mohr , j. j. & wegner , g. 2001 , [ @]jnlapj , 555 , 558 savage , b. d. , lehner , n. , wakker , b. p. , sembach , k. r. , & tripp , t. m. 2005 , [ @]jnlapj , 626 , 776 , b. d. & sembach , k. r. 1991 , [ @]jnlapj , 379 , 245 savage , b. d. , sembach , k. r. , tripp , t. m. , & richter , p. 2002 , [ @]jnlapj , 564 , 631 , c. , donahue , m. , voit , g. m. , rosati , p. & postman , m. 2000 , [ @]jnlapj , 528 , l73 schaye , j. , aguirre , a. , kim , t .- s . , theuns , t. , rauch , m. , & sargent , w. l. w. 2003 , [ @]jnlapj , 596 , 768 schmutzler , t. , tscharnuter , w.m . 1993 , a&a , 273 , 318 sembach , k. r. , savage , b. d. , lu , l. , murphy , e. m. 1995 , [ @]jnlapj , 451 , 616 sembach , k. r. , savage , b. d. , lu , l. , murphy , e. m. 1999 , [ @]jnlapj , 515 , 108 sembach , k. r. , howk , j. c. , savage , b. d. , & shull , j. m. 2001 , aj , 121 , 992 , k. r. , tripp , t. m. , savage , b. d. & richter , p. 2004a , [ @]jnlapjs , 155 , 351 sembach , k. r. , wakker , b. p. , tripp , t. m. , et al . 2004b , [ @]jnlapjs , 150 , 387 shapiro , p. r. , & moore , r. t. 1976 , [ @]jnlapj , 207 , 460 sheinis , a. i. , bolte , m. , epps , h. w. , kibrick , r. i. , miller , j. s. , radovan , m. v. , bigelow , b. c. , & sutin , b. m. 2002 , pasp , 114 , 851 shull , j. m. , penton , s. v. , stocke , j. t. , giroux , m. l. , van gorkom , j. h. , lee , y. h. , & carilli , c. 1998 , [ @]jnlaj , 116 , 2094 shull , j. m. , roberts , d. , giroux , m. l. , penton , s. v. , & fardal , m. a. 1999 , aj , 118 , 1450 shull , j. m. , tumlinson , j. , & giroux , m. l. 2003 , [ @]jnlapj , 594 , 107 stocke , j. t. , shull , j. m. , penton , s. , donahue , m. , & carilli , c. 1995 , [ @]jnlapj , 451 , l75 , c. , lupton , r. h. , bernardi , m. , blanton , m. r. , burles , s. , castander , f. j. , et al . 2002 , [ @]jnlaj , 123 , 485 strickland , d. k. , heckman , t. m. , colbert , e. j. m. , hoopes , c. g. , & weaver , k. a. 2004 , [ @]jnlapjs , 151 , 193 , r. s. & dopita , m. a. 1993 , [ @]jnlapjs , 88 , 253 , e. r. & henriksen , m. 2001 , [ @]jnlapj , 563 , 673 tripp , t. m. , bowen , d. v. , sembach , k. r. , jenkins , e. b. , savage , b. d. , & richter , p. 2004 , in astrophysics in the far ultraviolet : five years of discovery with , eds . g. sonneborn , h. w. moos , & b .- andersson ( san francisco : asp ) , in press ( astro - ph/0411151 ) , t. m. , giroux , m. l. , stocke , j. t. , tumlinson , j. & oegerle , w. r. 2001 , [ @]jnlapj , 563 , 724 , t. m. , jenkins , e. b. , bowen , d. v. , prochaska , j. x. , aracil , b. & ganguly , r. 2005 , [ @]jnlapj , 619 , 714 tripp , t. m. , lu , l. , & savage , b. d. 1998 , apj , 508 , 200 tripp , t. m. , & savage , b. d. 2000 , [ @]jnlapj , 542 , 42 tripp , t. m. , savage , b. d. , & jenkins , e. b. 2000 , [ @]jnlapj , 534 , l1 , t. m. , wakker , b. p. , jenkins , e. b. , bowers , c. w. , danks , a. c. , green , r. f. , heap , s. r. , joseph , c. l. , kaiser , m. e. , linsky , j. l. & woodgate , b. e. 2003 , [ @]jnlaj , 125 , 3122 voit , g. m. 2005 , rvmp , 77 , 207 wakker , b. p. , oosterloo , t. a. , & putman , m. e. 2002 , [ @]jnlaj , 123 , 1953 , q. d. , connolly , a. & brunner , r. 1997 , [ @]jnlapj , 487 , l13 , r. j. , vogel , s. n. , veilleux , s. & epps , h. w. 2001 , [ @]jnlapj , 561 , 559 webb , j. k. 1987 , phd thesis , univ . cambridge williger , g. m. , heap , s. r. , weymann , r. j. , dav , r. , ellingson , e. , carswell , r. f. , tripp , t. m. , & jenkins , e. b. 2005 , [ @]jnlapj , in press ( astro - ph/0505586 ) wolfire , m. g. , mckee , c. f. , hollenbach , d. , & tielens , a. g. g. m. 1995 , [ @]jnlapj , 453 , 673 , b. e. , kimble , r. a. , bowers , c. w. , kraemer , s. , kaiser , m. e. , danks , a. c. , et al . 1998 , [ @]jnlpasp , 110 , 1183 yuan , q. , green , r. f. , brotherton , m. , tripp , t. m. , kaiser , m. e. , & kriss , g. a. 2002 , [ @]jnlapj , 575 , 687 zabludoff , a. i. , & mulchaey , j. s. 1998 , apj , 496 , 39

we present high - resolution ultraviolet spectra of absorption - line systems toward the low@xmath0 qso 0624 ( @xmath1 ) . coupled with ground - based imaging and spectroscopic galaxy redshifts , we find evidence that many of these absorbers do not arise in galaxy halos but rather are truly integalactic gas clouds distributed within large - scale structures , and moreover , the gas is cool ( @xmath2 k ) and has relatively high metallicity ( @xmath3 ) . _ hst _ space telescope imaging spectrograph ( stis ) data reveal a dramatic cluster of 13 h@xmath4roman1 lines within a 1000 interval at @xmath5 . we find 10 galaxies at this redshift with impact parameters ranging from @xmath6 kpc to 1.37 @xmath7 mpc . the velocities and velocity spread of the lines in this complex are unlikely to arise in the individual halos of the nearby galaxies ; instead , we attribute the absorption to intragroup medium gas , possibly from a large - scale filament viewed along its long axis . contrary to theoretical expectations , this gas is not the shock - heated warm - hot intergalactic medium ( whim ) ; the width of the lines all indicate a gas temperature @xmath8 k , and metal lines detected in the complex also favor photoionised , cool gas . no o@xmath4roman6 absorption lines are evident , which is consistent with photoionisation models . remarkably , the metallicity is near - solar , [ m / h ] @xmath9 ( @xmath10 uncertainty ) , yet the nearest galaxy which might pollute the igm is at least 135 @xmath7 kpc away . tidal stripping from nearby galaxies appears to be the most likely origin of this highly enriched , cool gas . more than six abell galaxy clusters are found within @xmath11 of the sight line suggesting that the qso line of sight passes near a node in the cosmic web . at @xmath12 0.077 , we find absorption systems as well as galaxies at the redshift of the nearby clusters abell 564 and abell 559 . we conclude that the sight line pierces a filament of gas and galaxies feeding into these clusters . the absorber at @xmath13 = 0.07573 associated with abell 564/559 also has a high metallicity with [ c / h ] @xmath14 , but again the closest galaxy is relatively far from the sight line ( @xmath15 kpc ) . the doppler parameters and h@xmath4roman1 column densities of the ly@xmath16 lines observed along the entire sight line are consistent with those measured toward other low@xmath0 qsos , including a number of broad ( @xmath17 ) lines . intergalactic medium galaxies : abundances large - scale structure of the universe quasars : individual ( hs0624@xmath186907 )

astro-ph0512163

about a decade ago , galactic plane surveys revealed large numbers of infrared dark clouds ( irdcs , @xcite ; @xcite ) . these are identified as dark patches against the diffuse galactic mid - infrared background . first studies of very opaque irdcs suggested that these have very high densities , column densities , and masses ( @xmath2\gtrsim{}10 ^ 5~\rm{}cm^{-3}$ ] , @xmath3\gtrsim{}10^{23}~\rm{}cm^{-2}$ ] , @xmath4 ; @xcite ) . since they are dark , they are likely to be in an early evolutionary phase . embedded in irdcs are `` cores '' of a few dozen solar masses @xcite . it has therefore been suggested that many irdcs are the long - sought examples of clouds just at the onset of the formation of massive stars and ( proto-)clusters . this notion was corroborated by observations of young massive stars in a few individual irdcs ( @xcite ; @xcite ; @xcite ) . such views also form the framework of schemes for irdc evolution ( e.g. , @xcite , @xcite ) and reviews ( e.g. , @xcite , @xcite ) . irdc samples are usually compared to regions of massive star formation ( msf ) , such as orion and m17 ( e.g. , @xcite).[sec : introduction ] this picture can not be complete , though . the above studies ( and @xcite ) acknowledge that regions forming low and intermediate mass stars can also appear as shadows in images at mid - infrared wavelength @xcite . such irdcs will not form massive stars . unfortunately , the number of irdcs evolving towards msf is presently not known . fractions up to 100% have been considered in the past ( section [ sec : most - stars - irdc ] ) . in this letter , we thus use a novel criterion to provide the first conclusive quantitative demonstration that only few irdcs are headed towards msf . this aids identifying pre - msf irdcs as targets for alma and herschel . as a bonus , the msf threshold identified below the first observational limit of this kind informs theory . in papers i and ii @xcite , we show that solar neighborhood clouds devoid of msf ( specifically : perseus , ophiuchus , taurus , and pipe nebula ) generally obey @xmath5 irdcs submitting to eq . ( [ eq : mass - size - limit ] ) would resemble , e.g. , ophiuchus and perseus , but not orion ( which violates eq.[eq : mass - size - limit ] ) . figure [ fig : msf - limit ] illustrates why clouds bound for msf must exceed eq.([eq : mass - size - limit ] ) . since star formation necessitates an appropriate mass reservoir , msf requires that a large mass is concentrated in a relatively small volume . based on more detailed theoretical considerations , section [ sec : msf - limits ] puts quantitative limits on this intuitively evident reasoning . as seen in fig . [ fig : msf - limit ] , the masses in this msf region are well above the mass size range bound by eq.([eq : mass - size - limit ] ) . observations of msf clouds confirm eq.([eq : mass - size - limit ] ) as a true msf limit ( section [ sec : msf - threshold ] ) . this suggests to use eq.([eq : mass - size - limit ] ) to roughly separate irdcs with ( future ) msf from those without . this letter is organized as follows . based on data from section [ sec : method ] , section [ sec : msf - threshold ] confirms ( using known msf clouds ) that eq . ( [ eq : mass - size - limit ] ) approximates an msf limit . many well - studied irdcs ( 25%50% ) fall short of this threshold ( section [ sec : well - studied - irdcs ] ) . less certain data for complete irdc samples suggests that most irdcs obey eq.([eq : mass - size - limit ] ) , and will thus not form massive stars ( section [ sec : typical - irdcs ] ) . still , most of the mass contained by irdcs might be in clouds forming massive stars ( i.e. , those violating eq . [ eq : mass - size - limit ] ) . ; _ green and blue shading _ ) in relation to mass - size laws ( e.g. , @xmath6=m_0\cdot{}r^b$ ] ) observed for non - msf clouds ( eq . [ eq : mass - size - limit ] , fig . [ fig : mass - size - comparison ] ; _ yellow shading _ ) . at small radii , msf clouds ( _ highlighted in red _ ) must contain fragments bound by the theoretical msf - limits . depending on the interplay of slope , @xmath7 , and intercept , @xmath8 , such clouds must also at radii @xmath9 be more massive than fragments in non - msf clouds.[fig : msf - limit ] ] data for solar neighbourhood clouds not forming massive stars ( here : taurus , perseus , ophiuchus , pipe nebula ) are taken from paper ii ( and references therein ) . we rely on bolometer surveys to characterize msf sites : @xcite study fir color - selected msf candidates with cs - detected dense gas but no radio continuum ; @xcite map water masers embedded in cs clumps of high bolometric luminosity ( @xmath10 ) ; @xcite explore methanol masers and ultra - compact hii regions ; @xcite study the nearby cygnus - x msf site ( we use their ` clumps ' ) . to exclude fragments not forming massive stars , we only use the ` type 1 ' sources ( @xmath11 and/or @xmath12 emission , no resolved radio continuum ) from the @xcite survey , and ignore the secondary ` mm - only ' cores ( without masers and hii regions ) in the @xcite study . the irdc samples were created using msx and spitzer images . @xcite ( @xcite ; using bolometers ) and @xcite ( @xcite ; using @xmath13 extinction ) focus on clouds with stark @xmath13 contrast . @xcite report @xmath14-based results for all irdcs evident in their @xmath14 galactic plane survey . @xcite catalogue extinction properties for @xmath15 spitzer @xmath13 irdcs with unknown distances . the mass - size data for solar neighborhood clouds are derived in paper ii ( using methods summarized in section 2.1 and fig . 1 of paper i ) . they are based on column density maps derived from dust emission ( mambo and bolocam ) and extinction ( 2mass ) data . using a dendogram method introduced by @xcite , starting from a set of local column density maxima , a given column density map is contoured with infinitesimal level spacing . every contour defines the boundary of a cloud fragment . we derive the contour - enclosed mass and the effective radius , @xmath16 . subsequent contours / fragments are usually nested . this defines relationships between cloud fragments , essentially yielding series of mass - size measurements . in fig . [ fig : mass - size - comparison ] , such series are drawn using continuous lines . to derive column densities from the extinction maps , we assume that column density and visual extinction are related by @xmath17 @xcite . to combine dust emission and extinction observations , they must be calibrated with respect to one another . in practice , we use @xcite dust opacities ( decreased by a factor 1.5 , to match observed opacity laws ; section 4.2 of paper i ) for emission - based masses . for comparisons , we must scale all masses to the column density laws from paper ii . also , it is necessary to harmonize the different definitions of mass and size . the scaled data are shown in fig.[fig : mass - size - comparison ] . where relevant , we use dust temperatures suggested by the original studies . however , we substitute our choice of dust opacities and the aforementioned 1.5 scaling factor . @xmath14 masses are directly taken from @xcite , since their @xmath14-to - mass conversion law is in rough agreement with ( i.e. , by factors of 1.12.0 larger than ) the extinction - calibrated ones derived by @xcite . we assume that dust emission at @xmath18 wavelength and optical depth at @xmath19 wavelength are related by @xmath20 ( eq . 4 of @xcite ) , and derive column densities from these intensities ( assuming dust at @xmath21 , and using the 1.5 scaling factor ) . we thus increase the @xcite masses ( from their case ` a ' ) by a factor 1.47 ( to account for their choice of opacities and molecular weights)^2/ 1\farcs2 ^ 2)$ ] , where @xmath22 is distance , since pixels per beam ( as erroneously adopted ) have to be replaced by pixels per clump in eq . ( 5 ) of @xcite . ] . in many cases ( @xcite , @xcite , @xcite , @xcite ) , the size listed in the original publication refers to the contour at half peak intensity , while the mass measurement includes emission at much lower levels . in these cases , we assume that the sources have a near - gaussian shape ( just as explicitly assumed in many of the original papers ) . for such sources , the mass contained in the half peak column density contour is just a fraction @xmath23 of the total mass ( eq . a.23 of @xcite ; the area at half peak intensity is @xmath24 ^ 2 $ ] ) . thus we reduce the mass to a fraction @xmath25 , and use half of the published fwhm size as the effective radius . @xcite list masses for a sphere , not an aperture , and so the mass ( taken for the smaller of their radii ) has to be scaled up by a factor of order @xmath26 ( @xcite , eq.13 ) . if more than one distance is listed for a given object , we adopt the smaller one ( yielding a lower limit to @xmath6/m_{\rm{}lim}[r]$ ] derived below ) . lc ) approximates a mass - size limit for msf ( section [ sec : msf - threshold ] ) . only a fraction of the irdcs exceed this msf limit ( fig . [ fig : compactness ] , section [ sec : well - studied - irdcs ] ) . if a star - forming region contains more than one fragment ( i.e. , clump , core , etc . ) , the most compact fragment ( i.e. , with maximum @xmath6/m_{\rm{}lim}[r]$ ] ) is highlighted by a circle.[fig : mass - size - comparison],title="fig : " ] & + ) approximates a mass - size limit for msf ( section [ sec : msf - threshold ] ) . only a fraction of the irdcs exceed this msf limit ( fig . [ fig : compactness ] , section [ sec : well - studied - irdcs ] ) . if a star - forming region contains more than one fragment ( i.e. , clump , core , etc . ) , the most compact fragment ( i.e. , with maximum @xmath6/m_{\rm{}lim}[r]$ ] ) is highlighted by a circle.[fig : mass - size - comparison],title="fig : " ] & + ) approximates a mass - size limit for msf ( section [ sec : msf - threshold ] ) . only a fraction of the irdcs exceed this msf limit ( fig . [ fig : compactness ] , section [ sec : well - studied - irdcs ] ) . if a star - forming region contains more than one fragment ( i.e. , clump , core , etc . ) , the most compact fragment ( i.e. , with maximum @xmath6/m_{\rm{}lim}[r]$ ] ) is highlighted by a circle.[fig : mass - size - comparison],title="fig : " ] & at given radius , a cloud fragment ( i.e. , clump , core , etc . ) can be compared against eq . ( [ eq : mass - size - limit ] ) by deriving the mass ratio @xmath27 ( where @xmath28=870\,m_{\sun}\,[r/{\rm{}pc}]^{1.33}$ ] ) , to which we refer as the ` compactness ' . `` secondary cores '' ( only listed by @xcite and @xcite ) are suppressed by characterizing star - forming regions ( i.e. , a given massive star , or an entire irdc ) by their most compact fragment , @xmath29 $ ] . $ ] , for various cloud samples . for a given sample , the ratios below which a certain fraction ( e.g. , 25% ) of the sample members resides are indicated by bars . local non - msf clouds ( fig . [ fig : mass - size - comparison][a ] ) have a compactness @xmath30 ( eq . [ eq : mass - size - limit ] ) . the bars for the @xcite sample include ( _ left _ ) , respectively exclude ( _ right _ ) , their ` em ' cores . clearly , the irdcs do not reside in the mass - size space unambiguously associated with msf.[fig : compactness ] ] figure [ fig : compactness ] gives @xmath29 $ ] as derived for the samples examined here . this is based on the mass - size data presented in fig . [ fig : mass - size - comparison ] . the compactness assumes a range of values in every sample . this spread is captured by plotting several percentiles . as suggested by fig . [ fig : mass - size - comparison ] , we can clearly see in fig . [ fig : compactness ] that regions forming massive stars are , at given radius , more massive than the limiting mass , @xmath31 . in all surveys of msf regions , @xmath32 of the clouds have a maximum compactness @xmath33 . one survey @xcite contains a very small number of regions ( @xmath34 ) less compact than required by eq . ( [ eq : mass - size - limit ] ) . these regions might be interesting targets for follow - up studies . in general , though , this analysis corroborates the hypothesis that eq . ( [ eq : mass - size - limit ] ) approximates a threshold for msf . figure [ fig : compactness ] provides a compactness analysis for irdcs . we separately characterize the @xcite sample including and excluding their ` em ' cores with associated @xmath13 sources ( which are not dark ) . `` true '' irdcs will have properties in between these extremes . two interesting trends manifest in these @xmath27 data . first , irdcs have masses which are , for given size , comparable to those of solar neighborhood clouds not forming massive stars ( e.g. , ophiuchus and perseus ) . in all samples , @xmath35 of all clouds have a compactness @xmath36 . except for the @xcite clouds , @xmath37 of all targets exceed eq . ( [ eq : mass - size - limit ] ) by a factor @xmath38 . second , irdcs are less compact than regions forming massive stars . for example , excluding the @xcite targets , @xmath32 of all irdcs are less compact than most ( @xmath32 ) of the msf regions . in summary , the irdcs studied here have ( for given size ) masses in between those of regions with and without msf ( where `` true '' @xcite irdcs have properties in between the two extremes shown ) . very clearly , they do not reside in the mass - size space unambiguously associated with the formation of massive stars . however , before drawing final conclusions , let us consider some biases affecting our analysis . first , @xcite derive masses using clumpfind , while @xcite use gaussclumps . for the former , paper i showed explicitly that the derived masses are , for given radius , just @xmath39 of those derived using our dendrogram approach . for the latter , the same is expected , since the gaussian fits only describe a fraction of the emission . in a given map , our characterization scheme from papers i and ii would thus find larger masses . these biases are countered by other factors , though . we use the ` case a ' masses ( assuming bright ir foregrounds ) provided by @xcite . following @xcite , their ` case b ' ( fainter foregrounds ) appears to be more realistic . the masses could thus be lower by a factor @xmath40 @xcite . similarly , @xcite suggests @xmath14-to - mass conversion factors lower than used by @xcite . in any case , similar biases affect the data for msf regions . differences between these and irdcs are not likely to only come from observational uncertainties . finally , none of the irdcs in the @xcite , @xcite , and @xcite samples are `` typical '' for the general galactic population . @xcite and @xcite select clouds which are unusually dark in @xmath19 images . @xcite only characterize irdcs which are relatively large and dark , and are clearly detected in @xmath14 emission . all this excludes irdcs of low mass and density from the samples . less biased irdc samples should thus be less compact than derived here . the @xcite catalogue lists irdc angular sizes and column densities for the entire galactic plane covered by spitzer . it thus provides an ideal tool to derive a first idea of typical irdc properties . since they likely constitute ( to our present knowledge ) the typical reservoir of galactic star - forming gas , irdc characteristics probably gauge the early state of galactic star forming regions . ( c ) , but for the @xcite sample ( projected out to various distances).[fig : peretto - fuller ] ] cccccccccccccc ' '' '' distance@xmath41&number@xmath42&fraction@xmath43 & mass@xmath42&mass fraction@xmath43 + ' '' '' kpc&&%&@xmath44&% + ' '' '' 2&831&7&2.0&71 + 4&2218&20&9.8&87 + 6&3639&32&23.6&93 + ' '' '' 8&4778&42&43.2&96 + ' '' '' @xmath41distance to which the sample is projected @xmath42number of clouds with @xmath45 , and their total mass @xmath43mass and number fraction of compact clouds since no distances are known for the @xcite irdcs , we constrain their masses and sizes assuming a reasonable range of distances . analysis by @xcite and @xcite suggests that most irdcs have distances of @xmath46@xmath47 . figure [ fig : peretto - fuller ] illustrates the derived masses and sizes , and table [ tab : peretto - fuller ] characterizes the irdcs found to be compact ( i.e. , @xmath6/m_{\rm{}lim}[r]>1 $ ] ) . this analysis has two interesting results . first , by number , most of the @xcite irdcs have masses and sizes comparable to those of solar neighborhood clouds devoid of massive stars ( i.e. , they are not compact ) . this holds even when adopting the largest reasonable distance . second , the compact clouds contain most of the mass ( more accurately : most of the area - integrated column density ) seen in these irdcs , even for small irdc distances . unfortunately , the @xcite survey is ( like most extinction studies ) uncertain in the sense that it assumes that the diffuse galactic emission can be reliably modelled in its spatial distribution . this may not be true . in this spirit , the results from this section should be taken as an indication , not as a final result . consider the following toy model to understand the expected mass - size properties of msf clouds . stars probably form on a timescale @xmath48 slower ( @xmath49 ) than the free - fall timescale)^{1/2}$ ] , where @xmath50 is the constant of gravity and @xmath51 is the volume - averaged density ] , @xmath52 . in spherical symmetry , mass , size , and density are related by @xmath53 , where @xmath54 takes line - of - sight material not associated with the sphere into account . a specific star formation timescale then requires that @xmath55 further , to form a star of certain mass , @xmath56 , a mass reservoir larger than @xmath56 is necessary : @xmath57 figure [ fig : msf - limit ] evaluates these limits for a star of @xmath58 , based on efficiencies @xcite . typical free - fall timescales of their natal cores @xmath59 @xcite then imply @xmath60 . further , @xmath61 ( @xcite , eq . 13 ) and @xmath62 @xcite . since massive stars might form faster , and the star formation efficiency is not constrained well , we explore @xmath63 and @xmath64 in fig . [ fig : msf - limit ] ] and timescales from spitzer observations of solar neighborhood clouds . within the model , cloud collapse will only yield a massive star if initiated inside the boundaries set by eqs.([eq : msf - ff][eq : msf - efficiency ] ) . @xcite provide a similar limit , derived assuming that the collapsing region is heated by a cluster of low - mass stars ( in our terminology , they use @xmath65 ) . in order to sustain msf , at least a few cloud fragments in msf clouds must reside within the theoretical msf boundaries mentioned above ( fig . [ fig : msf - limit ] ) . the global structure of these clouds can usually be described by power laws , @xmath66 , with @xmath67 ( paper ii ) . such power laws imply that msf clouds violate @xmath68 ( eq . [ eq : mass - size - limit ] ) . depending on slope ( @xmath7 ) , intercept ( @xmath8 ) , and their interplay , such excesses are expected for radii @xmath69 ( fig.[fig : msf - limit ] ) . this is just what we find for msf clouds ( fig.[fig : mass - size - comparison][b ] ) . msf is thus only possible if a clouds slopes are shallow , intercepts are large , or both , when compared to eq.([eq : mass - size - limit ] ) . this permits a new way to quantitatively compare the structure of clouds with and without msf . pure differences in @xmath8 imply that msf and non - msf clouds only differ in their absolute properties . differences in slopes @xmath7 , however , imply _ relative _ differences in the structure , such as deviations in the hierarchical cloud structure . the irdc properties mentioned in the introduction ( @xmath2\gtrsim{}10 ^ 5~\rm{}cm^{-3}$ ] , @xmath3\gtrsim{}10^{23}~\rm{}cm^{-2}$ ] , @xmath4 ) only seem to characterize the densest patches in very large and massive irdcs . they are not well suited to describe irdcs on average . some irdcs with @xmath68 might further evolve and eventually undergo msf . and particular dust properties could , in principle , erroneously indicate @xmath68 where the reverse is true . however , such caveats are not usually considered when using irdc data to constrain msf . thus we abstain from such considerations . our study suggests that many irdcs , if not most , are not related to msf . one thus has to be prudent when using irdc properties to constrain msf initial conditions . most studies discussing irdcs as pre - msf sites concentrated on very opaque irdcs of large angular size . these clouds often violate eq . ( [ eq : mass - size - limit ] ) , and many of them are good msf candidates . @xcite suggest that most of the galactic star formation might come from irdcs . the absence of other likely reservoirs of star - forming gas evinces this too . by number , most irdcs are likely to form stars and clusters of low and intermediate mass , just as ophiuchus and perseus do . still , many irdcs will turn towards msf . interestingly , table [ tab : peretto - fuller ] suggests that most of the mass located in irdcs is in clouds that will form massive stars . for example , the 250 most compact clouds from the @xcite sample ( identified assuming a common distance ) contain more than 50% of the area - integrated column density of all irdcs . this suggests that they also contain a major fraction of the mass seen in irdcs . if this reasoning is correct , just few @xmath1 irdcs ( and not all @xmath70 : @xcite ) might contain most of the galaxy s star - forming gas . given the uncertain nature of the properties derived from the @xcite data ( section [ sec : typical - irdcs ] ) , this conclusion is far from certain , though . this letter studies whether infrared dark clouds ( irdcs ) are able to form massive stars . our main conclusions are as follows . * observations of regions with and without massive star formation ( msf ) suggest that the condition @xmath71 ( eq . [ eq : mass - size - limit ] ) approximates a threshold for msf ( section [ sec : msf - threshold ] ) . msf clouds differ from those obeying eq . ( [ eq : mass - size - limit ] ) in mass - size slope or intercept ( fig . [ fig : msf - limit ] , section [ sec : msf - limits ] ) . * many irdcs ( section [ sec : well - studied - irdcs ] ) , if not most ( section [ sec : typical - irdcs ] ) , do not exceed eq . ( [ eq : mass - size - limit ] ) . without significant further evolution , such clouds are unlikely candidates for msf , but they might well form stars and clusters of up to intermediate mass ( like perseus and ophiuchus ) . very opaque irdcs of large angular size constitute good msf candidates . * provided extinction - based masses can be trusted , just few @xmath1 irdcs might contain a major fraction of the galaxy s star - forming gas ( section [ sec : most - stars - irdc ] ) . these irdcs would be dense and massive enough to host msf . abergel , a. , bernard , j. p. , boulanger , f. , cesarsky , c. , desert , f. x. , falgarone , e. , lagache , g. , perault , m. , puget , j .- l . , reach , w. t. , nordh , l. , olofsson , g. , huldtgren , m. , kaas , a. a. , andre , p. , bontemps , s. , burgdorf , m. , copet , e. , davies , j. , montmerle , t. , persi , p. , & sibille , f. 1996 , astronomy and astrophysics evans , n. j. , dunham , m. m. , j rgensen , j. k. , enoch , m. l. , mern , b. , van dishoeck , e. f. , alcal , j. m. , myers , p. c. , stapelfeldt , k. r. , huard , t. l. , allen , l. e. , harvey , p. m. , van kempen , t. , blake , g. a. , koerner , d. w. , mundy , l. g. , padgett , d. l. , & sargent , a. i. 2009 , , 181 , 321 perault , m. , omont , a. , simon , g. , seguin , p. , ojha , d. , blommaert , j. , felli , m. , gilmore , g. , guglielmo , f. , habing , h. , price , s. , robin , a. , de batz , b. , cesarsky , c. , elbaz , d. , epchtein , n. , fouque , p. , guest , s. , levine , d. , pollock , a. , prusti , t. , siebenmorgen , r. , testi , l. , & tiphene , d. 1996 , astronomy and astrophysics

we present a new assessment of the ability of infrared dark clouds ( irdcs ) to form massive stars and clusters . this is done by comparison with an empirical mass - size threshold for massive star formation ( msf ) . we establish @xmath0 as a novel approximate msf limit , based on clouds with and without msf . many irdcs , if not most , fall short of this threshold . without significant evolution , such clouds are unlikely msf candidates . this provides a first quantitative assessment of the small number of irdcs evolving towards msf . irdcs below this limit might still form stars and clusters of up to intermediate mass , though ( like , e.g. , the ophiuchus and perseus molecular clouds ) . nevertheless , a major fraction of the mass contained in irdcs might reside in few @xmath1 clouds sustaining msf .

1009.1617

the manipulation of ultracold matter waves can now , somewhat routinely , be performed above microchip or magnetized surfaces @xcite . in such experiments , the quantum nature of the dilute atomic gases dominates over the classical , enabling precision matter wave control @xcite . one of the key requirements in using an `` atom chip '' to perform atom optics is the ability to transport atoms from one atom optical component to another . here , we present calculations of wave propagation through waveguides with idealized perturbations consisting of sudden changes to the transverse confining potential . an increase ( decrease ) in the tranverse confining potential results in a decrease ( increase ) in the kinetic energy along the direction of wave propagation , providing effective step potentials along the waveguide . the present study was motivated in three ways . firstly , recent experiments have demonstrated a significant fragmentation of a bose - einstein condensate ( bec ) in a waveguide located close to the surfaces @xcite , attributed , at least in part , to imperfections created during the wire fabrication processes @xcite . secondly , to further understand some of the limitations to designing atom optics devices that are based on variations of the waveguide potentials , for example , the smoothly varying wide narrow wide wire geometry has been proposed as a quantum - point - contact type device for atoms @xcite . thirdly , the literature has been lacking a multimode analysis of many of the simplest waveguide geometries , as it has been experimentally shown that introducing a perturbation in a waveguide can result in the transverse excitation of a bec @xcite . to characterize the impact of transverse discontinuities , here we explore the schrdinger wave mechanics of waveguides with step - like , barrier - like and well - like potentials along the direction of propagation . there have already been some theoretical investigations using time - dependent calculations of wave propagation through smooth potentials such as a bottleneck ( step - up ) and a smooth termination ( extreme step - down ) in the limit of non - interacting atoms @xcite , while non - linear ( atom - atom ) effects in the bottleneck - type geometry have also been examined @xcite . the advantages in using abrupt potentials whilst neglecting atom - atom interactions is that simple time - independent calculations can be used to characterize the transmission and reflection probabilities . under these conditions , we have previously investigated a circular bend @xcite , which consists of an abrupt transition from the lead waveguides into the bend and at low - energies behaves like a potential well . our multimode analysis , restricted as it is to the the linear regime , provides a baseline for comparison of bec propagation through quasi - one - dimensional ( 1-d ) waveguides including the transverse degrees of freedom . for example , previous time - independent studies have investigated non - linear wave propagation through shallow - well , step and gaussian shaped 1-d potentials @xcite . such simple waveguide potentials could be generated by modifying the transverse confinement , where knowledge of the transverse excitation probabilities , in the abrupt and linear limits , should be useful . for ground mode matter waves propagating at low energies through the various perturbations , the present results show that the spectra rapidly depart from single - moded , with significant transmission and reflection of excited modes . the high - energy limit sees @xmath0 transmission , and we present a simple model to determine the distribution of the transmitted modes that combines the overlap of the mode wave functions with the multi - path interference of the modes . there are a number of atom chip wire configurations that can create waveguides @xcite , but we follow the theoretical ansatz adopted in refs . that is , we assume that the waveguides consist of an idealized trapping potential that is quadratic near the minimum and operate at low matter - wave densities such that atom - atom interactions can be neglected . furthermore , so that the waveguide potentials reduce to an effective 2d problem , the waveguides are assumed to be created by multiple wire configurations with abrupt changes in the spacing between the wires , such that the height and transverse position of the potential minima remains constant along the waveguide . the out - of - plane quantum number is then a conserved quantity . we consider simple harmonic oscillator ( sho ) confining potentials , which , for barrier - like or well - like effective potentials , are given by @xmath1 the barrier - like potential has @xmath2 ; the well - like potential has @xmath3 . the step - like potential consists of only one change in frequency . oscillator units are used throughout this paper , where energies are in units of @xmath4 , while lengths are given in units of @xmath5 . an example barrier - like potential is shown in fig . [ fig : abruptpot](a ) , where the reference frequency is @xmath6 and @xmath7 . ( a ) potential energy surface of a barrier - like waveguide with @xmath6 and @xmath7 and barrier length @xmath8 . ( b ) energy levels of the leads and barrier transverse sho potential along @xmath9 . the solid lines at @xmath10 and @xmath11 should be vertical , but instead are drawn on an angle to highlight the lead barrier lead transition . the dot - dashed lines correspond to the parity - forbidden levels ( assuming an even incoming mode ) . all the energies and coordinates are given in terms of oscillator units for the leads . , width=336 ] the corresponding energy levels of eq . ( [ eqn : pots2d ] ) are shown in fig . [ fig : abruptpot](b ) . these energy levels behave as effective potentials for the longitudinal motion since we expand the total wavefunction in each region on the transverse oscillator states . in this model , all coupling between modes occurs through the matching between regions . the potentials in eq . ( [ eqn : pots2d ] ) are symmetric in @xmath12 so that parity in @xmath12 is conserved , simplifying the present analysis and discussion considerably [ the dot - dashed lines in fig . [ fig : abruptpot](b ) are not coupled to the solid lines ] . in experiments , imperfections would as likely be off - center , resulting in populating of all modes ( as was possible in our previous study of the circular bend @xcite ) . the fundamental physics , however , remains much the same , so we chose to adopt parity - conserving perturbations with the incoming waves restricted to the ground ( even ) oscillator mode . to perform the time - independent scattering calculations we initially adopted the transfer matrix method @xcite , although most of the calculations reported in this paper use the interface matching method @xcite . the two methods are similar , however , and since the transfer matrix approach facilitates the discussion of our results , we outline it here . the extension of the transfer matrix method from one - dimension to include transverse degrees of freedom is trivial @xcite , so only a short summary is given here as it applies to a barrier / well - like geometry of length @xmath13 . firstly , the time - independent wavefunction is expanded on transverse sho states , @xmath14 for frequencies @xmath15 , and @xmath16 for @xmath17 : @xmath18 \quad z \le z_0 \ ; , \\ \psi_\mathrm{ii}(x , z ) & = \sum_m \chi_m(x ) \ ; \big[c_m e^{i \kappa_m ( z - z_0 ) } + d_m e^{-i \kappa_m ( z - z_0)}\big ] \quad z_0 \le z \le z_1 \ ; , \\ \psi_\mathrm{iii}(x , z ) & = \sum_n \varphi_n(x ) \ ; \big[g_n e^{i k_n ( z - z_1 ) } + h_n e^{-i k_n ( z - z_1)}\big ] \quad z_1 \le z \ ; . \end{split}\ ] ] where the momenta are @xmath19 and @xmath20 . matching the wavefunctions and their first derivatives across each interface and then projecting out the modes gives the following sets of equations : @xmath21 the matrix elements of each submatrix are @xmath22 using the notation @xmath23 . while generating functions are known for the overlaps of sho - functions with different frequencies @xcite ( such overlaps are also found in calculations of transitions between molecular vibrational modes @xcite ) , we performed the transverse integrations numerically using a b - spline basis . the scattering solution is obtained by constructing the transfer matrix @xmath24 , which relates one lead s coefficients , @xmath25 and @xmath26 , to the other s , @xmath27 and @xmath28 . given that the wave is restricted to incoming from @xmath29 in the ground mode ( @xmath30 , @xmath31 , and @xmath32 ) , the linear equation @xmath33 is solved , and then @xmath34 . the transmission and reflection probabilities for each mode are then given by : @xmath35 there are convergence difficulties with the transfer matrix approach , the demonstration and discussion of which is mostly relegated to the appendix . in brief , the problems are related to the slow decay of the sho overlaps @xmath36 with @xmath37 , which requires the inclusion of strongly closed channels , leading to the appearance of large exponentials in the transfer matrix . while the transfer matrix method eliminates the need to find intermediate coefficients ( ie . @xmath38 and @xmath39 ) , it was generally found to be unstable beyond the smallest of the perturbations considered in this paper . the interface matching method , in contrast , explicitly solves for the intermediate coefficients and is able to include enough closed channels to ensure near machine - precision convergence for the range of geometries and energies given in this paper . the transfer matrix method works well , however , for step - like geometries with a single interface ( ie . @xmath40 ) as there are no exponentials in the matrix elements , and furthermore , all the coefficients are explicitly solved for . the multimoded transmission and reflection probabilities for three basic geometries are given in this section : the step - like potential , the barrier - like potential and the well - like potential . the calculations for the step - like potentials use the transfer matrix method , while interface matching is used for the barrier - like and the well - like potentials . before presenting these calculations , it is instructive to discuss how the interface overlaps @xmath36 scale with frequency . the @xmath36 dependence on the frequency ratio @xmath41 is discussed here since it strongly influences the amount of mode excitation caused by the different geometries . using the orthonormality of the sho eigenstates , along with the recursion relations of the hermite polynomials , it can be shown that the overlap integrals of the @xmath42 ground mode with the @xmath17 even modes reduce to the particularly simple form : @xmath43 for all @xmath44 . due to symmetry , @xmath45 for @xmath46 . ( [ eqn : overlaps ] ) was also obtained by aslangul @xcite . the dependence of the first three overlaps ( @xmath47 ) on frequency is shown in fig . [ fig : omegadependence](a ) . sho wavefunction overlaps @xmath48 between the ground mode with fixed frequency @xmath42 , and another mode with variable frequency @xmath17 . ( a ) shows the @xmath47 modes as a function of @xmath17 . the arrows labelled by @xmath49 indicate the maxima . ( b ) shows the overlaps as a function of @xmath37 for four frequencies : @xmath50 ( squares ) , @xmath51 ( filled - triangle ) , @xmath52 ( hollow - triangle ) and @xmath53 ( circles ) . the lines between dots in ( b ) are added to guide the eye . the frequencies shown in both ( a ) and ( b ) are given in oscillator units relative to @xmath15 . , width=336 ] at @xmath54 , there is a perfect waveguide match , and we must have @xmath55 and @xmath56 for @xmath57 . as @xmath17 increases , @xmath58 monotonically decreases towards zero . at the same time , the overlap with each excited mode increases and reaches a maximum when the characteristic equation , @xmath59 , is satisfied . as a function of frequency , these maxima occur at @xmath60 , which , for @xmath61 , is @xmath62 and for @xmath63 , is @xmath64 . comparing the width ( @xmath65 ) of the sho functions at these frequencies against the ground mode s reveals that @xmath66 . this connection , while natural , is not particularly illuminating and is not pursued here any further . past the maxima , the overlaps slowly decrease as @xmath67 to zero . due to symmetry of the ratio @xmath41 , a second maxima of @xmath48 also exists at @xmath68 . the slow decay of the overlaps with the higher modes can be seen in fig . [ fig : omegadependence](b ) . this implies that , even at low energies , many closed channels must be included in the following calculations to achieve computational convergence . the transmission and reflection probabilities , @xmath69 and @xmath70 , of ground mode plane - waves traversing four step - like waveguides are shown in fig . [ fig : stepwithenergy ] . at incident energies below the lowest reflection threshold ( @xmath71 , the lowest excited mode energy ) , the system behaves like the familiar 1-d step potential . ground state transmission , @xmath72 , remains the dominant channel across the range of energies shown in fig . [ fig : stepwithenergy](a ) ( for the range of @xmath17 examined here ) , although excited mode transmission can also be seen in fig . [ fig : stepwithenergy](a ) as each mode opens . significant reflection is seen into the ground @xmath73 mode in fig . [ fig : stepwithenergy](b ) , which rapidly drops off from threshold . as each reflection threshold opens , the reflection into the excited modes @xmath74 , seen in fig . [ fig : stepwithenergy](c ) , firstly increases then is seen to experience an overall decrease . all of which are consistent with the wigner threshold laws for multichannel systems @xcite . multimoded scattering probabilities of step - like potentials which consist of a single abrupt change in the transverse confinement potential from @xmath42 to @xmath50,@xmath75,@xmath76 , and @xmath77 . the incoming waves are in the ground state @xmath78 . ( a ) shows @xmath72 and @xmath79 , the transmission probabilities into the ground mode and the sum of the transmission probabilities into the excited modes , respectively . ( b ) gives @xmath80 , the reflection probabilities into the ground mode , while ( c ) gives @xmath81 , the sum of the reflection probabilities into the excited modes . the total energy @xmath82 is given in oscillator units relative to @xmath15 . the crosses in ( a ) and ( b ) are the analytic transmission and reflection probabilities for a 1-d step potential for the case of @xmath52 ( ie . of height @xmath83 with a background potential @xmath84 added to correct the reflection ground mode threshold ) . , width=336 ] for incidence in the ground channel , the total transmission approaches @xmath0 in the high energy limit . in this limit the transmission probability is given simply by @xmath85 with the sho overlaps of eq . ( [ eqn : overlaps ] ) . such projections were introduced as part of the waveguide calculations of jskelinen and stenholm @xcite , in which the transmission excitation probabilities generated by both bottleneck and split - potential waveguides were briefly discussed as the potentials tended towards abrupt . similar multimode projections have also been theoretically examined during expansion of a bec from a microtrap into a waveguide @xcite , with an emphasis on the effects of atom - atom interactions . equation ( [ eqn : project ] ) can be seen as the high - energy limit of the matrix elements of @xmath86 , given by eq . ( [ eqn : mes ] ) . at energies high compared to the step height , the momenta of the lowest few modes are approximately the same on both sides of the step , @xmath87 . given that the overlaps limit the number of channels involved , the matrix elements that dominate the @xmath40 transfer matrix are then @xmath88 , while @xmath89 . for an incoming wave in the ground mode with @xmath30 , @xmath31 , and @xmath90 , then the outgoing waves have @xmath91 while @xmath92 . to more clearly show this limit , the results of fig . [ fig : stepwithenergy ] are replotted in fig . [ fig : stepwithenergyscaled ] on an energy axis scaled by @xmath17 instead of @xmath15 so that the transmission channels for each waveguide open at the same scaled energy . the scattering probabilities of the same four step - like potentials as fig . [ fig : stepwithenergy ] ( @xmath50,@xmath75,@xmath76 , and @xmath77 ) plotted here as a function of @xmath93 , where @xmath94 . for each waveguide , ( a ) shows the transmission probabilities of the ground mode , ( b ) the sum of the excited mode transmission probabilities @xmath79 , and ( c ) the total reflection probabilities @xmath95 . the arrows at @xmath96 correspond to the @xmath97 interface overlaps . , width=336 ] at high energies , the transmission probabilities shown in fig . [ fig : stepwithenergyscaled](a ) increase towards asymptotes of @xmath98 , in agreement with eq . ( [ eqn : project ] ) . for the four different waveguides shown here , the limits are @xmath99 , @xmath100 , @xmath101 , and @xmath102 . figures [ fig : stepwithenergyscaled](b ) and ( c ) show that at the onset of each transmitted mode ( ie . at @xmath103 etc . ) the transmission probability into that mode increases , taking flux from reflection . the mismatch in mode wavefunctions for @xmath52 , for example , is particularly severe , with @xmath104 , @xmath105 , @xmath106 , and @xmath107 [ see fig . [ fig : omegadependence](b ) ] . in this case , these four transmission modes must be open before the high energy limit [ @xmath108 , as per eq . ( [ eqn : project ] ) ] is reached to within @xmath109 . whether the wave is incident from the left or the right , these results apply . the ground mode transmission @xmath110 is _ absolutely identical _ as a function of total energy @xmath82 for both the waveguide constriction ( ie . step - up from @xmath42 to @xmath111 ) , and the waveguide expansion ( ie . step - down from @xmath112 to @xmath54 ) this results also holds for the familiar 1-d step potential , and , although we do not show it here , the transmission and reflection mode mixing conspires to ensure this is also the case in the multichannel system . the mode mixing as a function of @xmath82 is different for either geometry , however , since for the step - up case there can be many reflection channels open at the lowest transmission threshold , @xmath113 , while for the step - down , there can be many transmission channels open at @xmath113 . at high energies , neither of the step geometries generates reflection , and for incidence in the ground mode the limits from eq . ( [ eqn : project ] ) apply . to demonstrate the characteristics of a single barrier - like potential , we consider the case shown in fig . [ fig : abruptpot ] , for a fixed length @xmath114 and frequencies that change from @xmath42 to @xmath2 and back to @xmath42 . we also present the high - energy characteristics for scattering from this potential . the transmission and reflection probabilities for four waveguide constrictions @xmath50,@xmath75,@xmath76,@xmath77 and length @xmath115 are shown in fig . [ fig : barrierwithenergy ] . multimoded scattering probabilities of four barrier - like potentials with @xmath50,@xmath75,@xmath76 , and @xmath77 for a fixed length @xmath115 as a function of @xmath82 ( in oscillator units relative to @xmath15 ) . the legend for the different frequencies is the same as figs . ( [ fig : stepwithenergy ] ) and ( [ fig : stepwithenergyscaled ] ) . the organisation of the probabilities is also the same as fig . ( [ fig : stepwithenergy ] ) . the crosses in ( a ) and ( b ) are the analytic transmission and reflection probabilities for a 1-d barrier potential for the case of @xmath116 ( ie . of height @xmath117 with a background potential @xmath84 ) . , width=336 ] resonances appear in all of the spectra at low energies due to the wavelength matching condition @xmath118 . the 1-d analytic results for @xmath116 in fig . [ fig : barrierwithenergy](a ) highlight the transmission resonances as the result of low - energy ground - mode propagation over the barrier . as soon as the @xmath42 , @xmath119 mode opens at @xmath120 , however , multichannel physics takes over ( @xmath121 is not allowed due to symmetry ) . above @xmath120 , there is significant excited mode transmission [ fig . [ fig : barrierwithenergy](a ) ] , while the amount of reflection into the ground mode [ fig . [ fig : barrierwithenergy](b ) ] and the excited modes [ fig . [ fig : barrierwithenergy](c ) ] is significant across the energy range . to observe the high - energy limit , the results for the @xmath52 barrier of fig . [ fig : barrierwithenergy ] were extended to higher energies , and are shown in fig . [ fig : barrierwithenergyscaled ] with the energy scaled by @xmath122 . the scattering probabilities of the @xmath52 , @xmath115 , barrier - like potential seen in fig . [ fig : barrierwithenergy ] plotted here as a function of @xmath93 where @xmath122 . ( a ) shows the individual transmission probabilities of the @xmath123 modes alongside the sum of the transmission probabilities for the @xmath124 modes . ( b ) shows the ground - mode reflection probabilities as well as the sum of the excited mode reflection probabilities . the crosses in ( a ) correspond to the @xmath125 two - mode interference model of eq . ( [ eqn : interf ] ) . the first peak of @xmath126 reaches nearly up to 1 , which is not shown due to the limited energy resolution of the figure . , width=336 ] figure [ fig : barrierwithenergyscaled](b ) shows that while there is more structure in the reflection probabilities than for the step - like potential in fig . [ fig : stepwithenergyscaled](c ) , the total amount of reflection still tends towards zero as more barrier modes become open at @xmath127 . to obtain an expression analogous to eq . ( [ eqn : project ] ) , we must take into account the fact that the transmitted waves going through a barrier experience at least two interface projections as in eq . ( [ eqn : project ] ) . the phases accumulated while propagating the length @xmath114 of the barrier must also be included in such a prescription , which suggests that at high energies ( relative to the barrier height ) @xmath128 where @xmath37 is only summed over the propagating barrier modes . the crosses shown in fig . [ fig : barrierwithenergyscaled](a ) demonstrate that the two - mode version of this model does a remarkable job in describing the transmission probability above the @xmath61 barrier threshold ( at @xmath129 ) . depending on the phase differences , the modes that are excited at the first interface can be converted back to the ground mode by the second interface . we also noted this behavior in circular waveguide bends @xcite , where the amount of excitation could be suppressed by changing the angle swept out by the bend to the point where the accumulated phase difference between the @xmath130 and @xmath121 modes was a multiple of @xmath131 . a similar design consideration could perhaps be useful for atom optics devices such as the quantum point contacts @xcite , where any unavoidable yet unwanted mode excitations could be minimised by varying the length between the changes in waveguide potentials . the transmission and reflection probabilities for well - like waveguides due to a potential bulge ( @xmath132 ) are the focus in this last section . the scattering behavior for a well is complicated by the presence of bound states which translate into the presence of feshbach resonances in a multichannel problem . much of the resonance physics seen here has been extensively discussed as part of our studies of the circular waveguide bend @xcite . the propagation thresholds in a bend lie slightly lower than the connecting leads @xcite , resulting in very weakly bound states and energetically narrow resonances . the present well - like waveguides can provide extreme differences between the lead and bulge energy thresholds , and thus the possibility of multiple narrow resonances located below the thresholds . two bulges are considered here , the first from @xmath133 to @xmath54 and back to @xmath133 , and the second with @xmath134 to @xmath54 and back to @xmath134 . in both cases , we choose @xmath54 to be the reference oscillator frequency , to simplify the comparison with the barriers in the previous section . the transmission and reflection probabilities of both of these potentials with well length @xmath115 are shown in fig . [ fig : wellwithenergy ] . multimoded scattering probabilities of two well - like potentials with @xmath133 and @xmath75 for a fixed length @xmath115 as a function of @xmath82 ( in oscillator units relative to @xmath17 ) . the organisation of the probabilities is the same as fig . [ fig : stepwithenergyscaled ] . the crosses in ( a ) correspond to the @xmath125 two - mode interference model of eq . ( [ eqn : interf ] ) for @xmath134 . , width=336 ] multiple @xmath0 reflection resonances exist at energies below the first excited lead mode energy , ie . below @xmath135 for @xmath133 and @xmath136 for @xmath134 . as was noted in our circular bend studies @xcite , the reflection resonances below the second excited mode ( ie . below @xmath137 for @xmath133 , and at @xmath138 for @xmath134 ) do not result in complete reflection due to the reduced coupling between the ground and second excited mode and due to the existence of alternate pathways to reflection . the high - energy transmission probability asymptotes for the well - like potential are again given by eq . ( [ eqn : interf ] ) , although the energy of the first excited mode within the well means that the two - mode model starts at @xmath120 ( in other words , at a lower energy than for the equivalent barrier - like potential ) . the two - mode model is shown in fig . [ fig : wellwithenergy](a ) for the @xmath134 well , and is seen to be a bad approximation near the @xmath120 threshold due to the significant reflection there . the two - mode model generally provides a reasonable approximation at higher energies once the total reflection probability has dropped below the total excited mode transmission probability , and also when the many narrow reflection resonances no longer play a role . waveguides with more extreme discontinuities such as @xmath139 and @xmath77 were also explored . they exhibit so many resonances across the range of energies shown in fig . [ fig : wellwithenergy ] , however , that the transmission and reflection probabilities essentially become a dense series of closely - spaced vertical spikes . at energies above the second excited mode threshold ( @xmath140 ) , there are not so many resonances , although there remains significant reflection probability . as an example , the @xmath141 , @xmath115 , well - like waveguide has a total reflection probability at @xmath142 ( @xmath143 ) that has only dropped down to about @xmath144 . this reflection probability can be compared with the @xmath52 , @xmath115 barrier - like geometry in fig . [ fig : barrierwithenergyscaled](b ) at @xmath142 ( @xmath96 ) , where the total reflection probability was only about @xmath145 . in other words , reflections play a far more dominant role for the well - like potentials than the barrier - like . using time - independent solutions of the schrdinger equation , we have explored the propagation of dilute , ultracold atomic gases through abruptly changing waveguide potentials . previous studies have discussed the conditions for `` adiabatic '' waveguide propagation through microstructures ( eg . refs . in contrast , the interest here was on the consequences of sudden potential variations for mode excitation with a view towards modelling waveguide imperfections , examining the effects of using abrupt potentials in atom optical devices , and simply to explore the behavior of some simple geometries . three idealized geometries with changes in the transverse guiding potential have been the focus of study : step - like , barrier - like and well - like . the low - energy behavior of all the geometries departed from single - moded , with the exception of the mildest perturbations at energies below the lowest excitation threshold . significant generation of both transmission and reflection excited modes was caused by the mismatch of the modes at the interfaces between the waveguide sections . the strong coupling to excited modes is due to the significant overlap of the ground sho function of one frequency with the excited sho functions with a different frequency . certainly , care should be taken during wire fabrication of atom optical elements to ensure that any deliberate ( or not ) changes in a waveguides transverse frequency are not abrupt . high energy wave propagation through abrupt potentials amounts to @xmath0 wave transmission via projections across each interface , along with multimode interference . the present sho - based waveguides behave somewhat differently than the hard - walled models for ballistic electron propagation through waveguides with a wide - narrow junction ( see ref . @xcite , and the references thereafter that cite it ) . in that case an impedence mismatch occurs , where there is always some amount of reflection at high energies due to the fact that the narrow guide modes can never represent the wide guide modes over the whole width of the lead . although , to the best of our knowledge , calculations for such electron waveguides have not discussed high - energy / abrupt potential transmission and reflection limits . a further condition was suggested for high energy transmission through microstructures with multiple interfaces to account for the interference between modes . it was shown that a simple two - mode model can give a reasonable approximation to the amount of ground mode transmission , and provides an additional consideration for the design of `` atom - chip '' waveguides to control single - moded wave - transmission through potentials that generate multimoded excitations . this research was supported by the department of the navy , office of naval research , and also by the research corporation . our transfer matrix program was validated by reproducing the single - mode calculations of electron propagation through a linear array of 1-d potentials @xcite , and secondly by comparing with multi - moded 2-d results from an interface matching program @xcite . the multimode transfer matrix method , however , has numerical convergence difficulties as the number of modes included in a calculation increases . this behavior can be seen in table [ tab : transconv ] , for scattering with different incident energies @xmath82 off a barrier - like potential with @xmath146 of length @xmath147 . the unitarity of the transfer matrix results deviate significantly from @xmath148 , even as @xmath149 is increased . the interface matching calculations are shown for comparison , and barely suffer from the same problems . lcccc & & + @xmath149 & @xmath150 & unitarity & @xmath150 & unitarity + + 1 & 0.00339154 & & 0.999976 & + + 1 & 0.00381548 & 4.2394@xmath151 & 1.124973 & 1.2500@xmath152 + 2 & 0.00267029 & 3.0328@xmath153 & 0.772832 & 5.2616@xmath154 + 4 & 0.00274164 & 2.8303@xmath155 & 0.804769 & 8.1844@xmath151 + 8 & 0.00274223 & 4.1060@xmath155 & 0.807601 & 3.4533@xmath156 + 16 & 0.00264476 & 9.4352@xmath157 & 0.849707 & 3.4386@xmath154 + + 1 & 0.00339154 & 3.3307@xmath158 & 0.999976 & 2.0912@xmath158 + 2 & 0.00270062 & 1.4129@xmath159 & 0.795176 & 3.3826@xmath154 + 4 & 0.00274192 & 1.7813@xmath160 & 0.805169 & 3.1198@xmath157 + 8 & 0.00274223 & 4.5390@xmath161 & 0.807546 & 1.5052@xmath153 + 16 & 0.00277009 & 4.0555@xmath157 & 0.916579 & 5.6275@xmath154 + + 1 & 0.00339154 & 2.2204@xmath158 & 0.999976 & 0 + 2 & 0.00270062 & 1.1102@xmath158 & 0.795176 & 3.3826@xmath154 + 4 & 0.00274192 & 3.3307@xmath158 & 0.805169 & 3.1198@xmath157 + 8 & 0.00274223 & 0 & 0.807605 & 3.8357@xmath161 + 16 & 0.00274223 & 4.4409@xmath158 & 0.807607 & 7.6739@xmath162 + 32 & 0.00274223 & 1.1102@xmath158 & 0.807607 & 2.2205@xmath158 + in an attempt to rescue the transfer matrix approach , a second method for constructing the transfer matrix was investigated . this second attempt involved inverting a matrix @xmath163 , @xmath164 with matrix elements , @xmath165 , and @xmath166 while the matrix elements in @xmath86 of eq . ( [ eqn : mes ] ) were modified to include the @xmath167 terms . the numerical results of solving @xmath168 are also shown in table [ tab : transconv ] , and while unitarity results for the single - channel calculation , it is as unstable , if not worse that @xmath169 . for the geometries considered in this paper the transfer matrix approach was not able to adequately converge over the required range of energies . the exception to this was for the step - like geometry , where for the most extreme calculations in this paper ( fig . [ fig : stepwithenergyscaled ] : @xmath170 and @xmath142 ) @xmath171 modes could easily be included ( since the transfer matrix contains no exponential terms , and all coefficients are explicitly solved for ) . we also find that , since the @xmath172 overlaps are energy independent , obtaining convergence is much easier for the present interface matching calculations than for previous circular waveguide bend calculations using the same method @xcite . for the most extreme barrier - like calculation ( fig . [ fig : barrierwithenergyscaled ] : @xmath170 , @xmath115 and @xmath142 ) , the interface matching method achieved unitarity to better than @xmath173 with @xmath174 modes . some of the transfer matrix convergence issues were passingly mentioned by wu _ et.al _ @xcite , where the transfer matrix method was initially applied to a system of multiple circular bends ( the transfer matrix is simply built as a number of matrix products ) . in the end , however , the calculations shown in ref . @xcite were performed using multiple interface matching . we labor on this point since no such unitarity problems were noted in the multichannel transfer matrix calculations of ref . @xcite , where the examples given were for 2 or 3 channel calculations with delta potentials . on the other hand , the advantage of the transfer matrix approach is that , in the single - mode approximation , one can easily obtain the transmission characteristics of both finite and infinite length periodic potentials @xcite .

the propagation of ultracold atomic gases through abruptly changing waveguide potentials is examined in the limit of non - interacting atoms . time - independent scattering calculations of microstructured waveguides with discontinuous changes in the transverse harmonic binding potentials are used to mimic waveguide perturbations and imperfections . three basic configurations are examined : step - like , barrier - like and well - like with waves incident in the ground mode . at low energies , the spectra rapidly depart from single - moded , with significant transmission and reflection of excited modes . the high - energy limit sees @xmath0 transmission , with the distribution of the transmitted modes determined simply by the overlap of the mode wave functions and interference .

physics0411176

the first observational evidence that the universe had entered a period of accelerated expansion was obtained when supernovae type ia ( snia ) were found to be fainter than expected @xcite . this fact has been confirmed by many independent observations such as temperature anisotropies of the cosmic microwave background ( cmb ) @xcite , inhomogeneities in the matter distribution @xcite , the integrated sachs wolfe ( isw ) effect @xcite , baryon acoustic oscillations ( bao ) @xcite , weak lensing ( wl ) @xcite , and gamma - ray bursts @xcite . within the framework of general relativity ( gr ) , the accelerated expansion is driven by a new energy density component with negative pressure , termed dark energy ( de ) . the nature of this unknown matter field has given rise to a great scientific effort in order to understand its properties . the observational evidence is consistent with a cosmological constant @xmath0 being the possible origin of the dark energy ( de ) driving the present epoch of the accelerated expansion of our universe and a dark matter ( dm ) component giving rise to galaxies and their large scale structures distributions @xcite . the dm is assumed to have negligible pressure and temperature and is termed cold . thanks to the agreement with observations the model is commonly known as @xmath0cdm , to indicate the nature of its main components . while favored by the observations , the model is not satisfactory from the theoretical point of view : the value of the cosmological constant is many orders of magnitude smaller than what it was estimated in particle physics @xcite . it was suggested soon that de could be dynamic , evolving with time @xcite . this new cosmological model also suffers from a severe fine - tune problem known as _ coincidence problem _ @xcite that can be expressed with the following simple terms : if the time variation of matter and de are very different why are their current values so similar ? cosmological models where dm and de do not evolve separately but interact with each other were first introduced to justify the currently small value of the cosmological constant @xcite but they were found to be very useful to alleviate the coincidence problem . in this review we will summarize the theoretical developments of this field and the observational evidence that constrains the dm / de interaction and could , eventually , lead to its detection . the emergence of galaxies and large scale structure is driven by the growth of matter density perturbations which themselves are connected to the anisotropies of the cmb @xcite . an interaction between the components of the dark sector will affect the overall evolution of the universe and its expansion history , the growth matter and baryon density perturbations , the pattern of temperature anisotropies of the cmb and the evolution of the gravitational potential at late times would be different than in the concordance model . these observables are directly linked to the underlying theory of gravity @xcite and , consequently , the interaction could be constrained with observations of the background evolution and the emergence of large scale structure . this review is organized as follows : in this introduction we describe the concordance model and we discuss some of its shortcomings that motivates considering interactions within the dark sector . since the nature of de and dm are currently unknown , in sec . [ sec : sec2 ] we introduce two possible and different approaches to describe the de and the dm : fluids and scalar fields . based on general considerations like the holographic principle , we discuss why the interaction within the dark sector is to be expected . in sec . [ sec : sec3 ] we review the influence of the interaction on the background dynamics . we find that a dm / de interaction could solve the coincidence problem and satisfy the second law of thermodynamics . in sec . [ sec : sec4 ] the evolution of matter density perturbations is described for the phenomenological fluid interacting models . in sec . [ sec : sec5 ] we discuss how the interaction modifies the non - linear evolution and the subsequent collapse of density perturbations . in sec . [ sec : sec6 ] we describe the main observables that are used in sec . [ sec : sec7 ] to constrain the interaction . finally , in sec . [ sec : sec8 ] we describe the present and future observational facilities and their prospects to measure or constrain the interaction . in table [ table : acronyms ] we list the acronyms commonly used in this review . @llll acronym & meaning & acronym & meaning + a - p & alcock - packzynki & ksz & kinematic sunyaev - zeldovich + bao & baryon accoustic oscillations & lbg & lyman break galaxies + cdm & cold dark matter & lhs & left hand side ( of an equation ) + cl & confidence level & lisw & late integrated sachs - wolfe + cmb & cosmic microwave background & lss & large scale structure + de & dark energy & mcmc & monte carlo markov chain + detf & dark energy task force & rhs & right hand side ( of an equation ) + dm & dark matter & rsd & redshift space distortions + eos & equation of state & sl & strong lensing + eisw & early integrated sachs - wolfe & snia & supernova type ia + frw & friedman - robertson - walker & tsz & thermal sunyaev - zeldovich + isw & integrated sachs - wolfe & wl & weak lensing + the current cosmological model is described by the friedmann - robertson - walker ( frw ) metric , valid for a homogeneous and isotropic universe @xcite @xmath1 , \label{eq : frw - canonical}\ ] ] where @xmath2 is the scale factor at time @xmath3 , the present time is denoted by @xmath4 and the scale factor is normalized to @xmath5 ; @xmath6 is the gaussian curvature of the space - time . we have chosen units @xmath7 but we will reintroduce the speed of light when needed . a commonly used reparametrization is the _ conformal time _ , defined implicitly as @xmath8 . in terms of this coordinate , the line element is @xmath9 . \label{eq : frw - conformal}\ ] ] if we describe the matter content of the universe as a perfect fluid with mean energy density @xmath10 and pressure @xmath11 , friedmann s equations are @xcite @xmath12 where @xmath13 is the hubble function and @xmath14 are the energy density and pressure of the different matter components , related by an equation of state ( eos ) parameter @xmath15 . in terms of the conformal time , the expression @xmath16 is used . usually densities are measured in units of the critical density : @xmath17 with @xmath18 . the curvature term can be brought to the right hand side ( rhs ) by defining @xmath19 . as a matter of convention , a sub - index `` 0 '' denotes the current value of any given quantity . due to the historically uncertain value of the hubble constant , its value is usually quoted as @xmath20kms@xmath21mpc@xmath21 so the parameter @xmath22 encloses the observational uncertainty on the value of the hubble constant . [ cols="^,^ " , ] the phenomenological description of the interaction between dark sectors was introduced in sec . [ sec : sec2 ] and the linear perturbation theory of the model was discussed in sec . [ sec : sec4 ] . for the sake of simplicity in our subsequent discussion , we will review only models with a de eos parameter @xmath23 . the results for a variable eos have been reported in @xcite . we restrict our analysis to the models that satisfy the stability condition ( sec . [ sec : sec4.stability ] ) . the interaction kernels were summarized in table [ table : models ] . the results of the mcmc analysis using different data sets are shown in fig . [ figure2_sec8 ] . when the coupling is proportional to the energy density of the de , the data constraints the value of the interaction parameter to be in the range @xmath24 . when the coupling is proportional to the energy density of the dm or the total energy density of the dark sector , the constraint is much tighter and the coupling are positive at the 68% confidence level ( cl ) . including additional data tightens the constraints on the cosmological parameters compared with the cmb data alone . results for different models can be found in @xcite . the conclusion of these studies shows that interacting dm / de models are compatible with observations . in @xcite it was found evidence of the existence of interaction but with a low cl . more recently the planck collaboration also found that coupled dm / de were in agreement with observations @xcite . in the next section we will further discuss the evidence in favor of an interactions . as a function of the interaction parameters @xmath25 for models iii and i , ii , respectively . in the left panel we used the cosmological parameters from table v , xi and xii of @xcite . in the right panel , labeled boss , the cosmological parameters are those of table [ table : boss_parameters ] . the horizontal line corresponds to the boss measured value @xmath26 with the shaded areas representing @xmath27 and @xmath28 cl . ] together with cmb data , low redshift observations like luminosity distances from snia have been used to test the dm / de interaction . at low redshifts the deviations from @xmath0cdm are not very pronounced and it is easier to establish the existence of a dynamical de component or even an interacting one using high redshift observations . recently , the analysis of boss data presented evidence against the concordance model by measuring the bao in the redshift range @xmath29 from the correlation function of the ly@xmath30 forest from high redshift quasars @xcite . their result indicates a @xmath31 deviation ( from _ _ planck__+wmap ) and @xmath32 deviation ( from act / spt ) from @xmath0cdm at @xmath33 . while the statistical evidence is still not significant , if confirmed , this result can not be explained by a dynamical de component , and it suggests a more exotic type of de . an interacting de appears as a simple and efficient solution to explain the boss result . if de and dm interact and the former transfers energy to the latter , as required to alleviate the coincidence problem ( see sec [ sec : sec2.direction ] ) and indicated by the data @xcite , it would explain the value of the hubble parameter , @xmath34 , obtained by the boss collaboration , value that is smaller than the expectation from @xmath0cdm @xcite . it would explain the discrepancy of the angular diameter distance at high redshifts . let us now briefly summarize which of the models given in table [ table : models ] can explain the boss results . let us consider a universe filled only with dm , de and baryons . we can use the hubble parameter obtained from the friedmann equation and compare it with the value obtained by the boss collaboration for different sets of adjusted cosmological parameters . we can also compare the constraints for @xmath35 and @xmath36 given by the boss experiment with constraints from cmb adjusted data using @xmath0cdm and the interaction model . to carry out this analysis we need to establish first the evolution with redshift of the energy densities of each component , specially de and dm since due to the interaction they are not independently conserved . for the models i and ii , they behave as @xcite @xmath37 \rho_d^0}{\xi_2+\omega_d}+\rho_c^0\right\ } . \label{int_de}\end{aligned}\ ] ] the baryonic density is given by the usual expression @xmath38 . for the model iii , the evolution is @xmath39 from these solutions , it is easy to establish that when the energy is transferred from the de to the dm , the energy density of the dm is always smaller than what it would have been in the standard @xmath0cdm model . since @xmath40 is the dominant component at redshifts @xmath41 and it is smaller than in the concordance model , so it would be the hubble parameter , as indicated by the boss data . ccc parameter & bestfit & @xmath42 + @xmath22 & @xmath43 & @xmath44 + @xmath45 & @xmath46 & @xmath47 + @xmath48 & @xmath49 & @xmath50 + @xmath51 & @xmath52 & @xmath53 + [ table : boss_parameters ] to make the previous statement more quantitative we took two sets of values for the cosmological parameters @xmath54 and @xmath55 : ( 1 ) the values used by boss collaboration , obtained from the planck collaboration analysis of the @xmath0cdm model and listed in table [ table : boss_parameters ] , and ( 2 ) the values derived by @xcite by fitting dm / de interacting fluid models to the _ planck _ , bao , snia and @xmath55 indicated above . using both data sets , the hubble parameter at @xmath33 has been computed using eqs . ( [ int_dm],[int_de ] ) for the cosmological models listed in table [ table : models ] . the results are shown in fig . ( [ fig : all_models ] ) . the right panel corresponds to the cosmological parameters of the boss collaboration and the left panel to those of @xcite . the figure shows the measured value @xmath35 and its @xmath27 and @xmath28 contours . in both panels , the @xmath0cdm model that corresponds to the case of no interaction is always outside the @xmath27cl . while still not significant , it does show that the data prefers an dm / de interacting model with positive interaction . further improvements on the data could help to establish the existence of an interaction . is the critical energy density today . the attractor solutions of @xmath56 does not depend on the initial conditions at the early time of the universe . the purple lines represent the density evolution of cosmological model with different initial conditions . solid circles represent the density contrast @xmath56 today . values change with the initial conditions but they are bounded in two attractor solutions @xmath57 in @xmath58 plane.,width=480,height=432 ] one motivation to study interacting dm / de models is to alleviate the coincidence problem . only at the present time the dm / de ratio in the @xmath59 model is of order unity , demanding a fine - tuning on the initial conditions at the planck scale of 90 orders of magnitude ( see sec . [ sec : sec1.coincidence ] ) . let us examine if once interacting de models are confronted with observations , the goal can be satisfied . we will concentrate on the analysis of the phenomenological fluid model . the results shown in fig . [ figure2_sec8 ] suggest that positive coupling parameters are compatible with the data and positive values work in the direction of solving the cosmological coincidence problem @xcite and similar conclusions have also been reached for the field description of the de @xcite . let us particularize our analysis for fluid models , and in particular for model iv . in sec . [ sec : sec2.solution ] we have demonstrated how this model solves the coincidence problem . as illustrated in fig . [ fig : ratios](a ) the ratio has two attractor solutions @xmath60 ; the past solution and future ratios are given in eq . ( [ r+- ] ) . when the coupling is @xmath61 , the ratios behave asymptotically as @xmath62 i.e. , the behavior of the attractor solutions of the ratio @xmath56 only depends on the coupling constant @xmath63 and does not depend on the initial conditions at the planck scale . this solution of the coincidence problem is illustrated in fig . [ figure5_sec7 ] . purple solid lines represent the evolution of the energy densities in units of the critical energy density today , @xmath64 , with different initial conditions . the density contrast @xmath56 at present is different for different initial conditions but all the curves are bounded by the two attractors solutions @xmath65 and @xmath66 . in this particular case we fixed @xmath67 and @xmath68 . during the whole thermal history of the universe , the dm to de ratio takes values within the range @xmath69 ; it changes much less than that in the @xmath59 model , thus the cosmological coincidence problem is greatly alleviated . the discovery that the expansion of the universe is accelerating has led to large observational programs being carried out to understand its origin . new facilities are being designed and built aiming to measure the expansion history and the growth of structure in the universe with increasing precision out to greater redshifts . since the interaction in the dark sectors changes the expansion history of the universe and the evolution of matter and radiation density perturbations , peculiar velocities and gravitational fields , these new facilities will not only test the current period of accelerated expansion but also explain the nature on the interaction between dark sectors . thus , current and future observations could and will be used to set up constraints on interactions between dark sectors and clarify the nature of dm / de interactions . observations of type ia supernova , baryon acoustic oscillations ( bao ) , gravitational lensing , redshift - space distortions and the growth of cosmic structure probe the evolution of the universe at @xmath70 . in parallel , the physics of dm / de interactions at recombination can be probed by the cmb radiation power spectrum while the isw effect and lensing pattern of the cmb sky are sensitive to the growth of matter at lower redshifts . the de task force ( detf ) was established to advise the different u.s . funding agencies on future de research . their report categorized different experimental approaches by introducing a quantitative `` figure of merit '' that is sensitive to the properties of de , including its evolution with time @xcite . using this figure of merit , they evaluated ongoing and future de studies based on observations of baryon acoustic oscillations , galaxy clusters , supernova and weak lensing . the detf categorized the different experiments by their different degree of development . stage i referred to the discovery experiments , stage ii to the on - going experiments at that time when the report was elaborated ( circa 2006 ) , stage iii was defined as the next generation that are currently in full operation . they also looked forward to a stage iv generation of more capable experiments . examples of stage ii surveys are the canada - france - hawaii telescope ( cfht ) legacy survey , with observations of snia @xcite and weak lensing @xcite and that ended in 2009 , the essence @xcite and sdss - ii @xcite supernova surveys and bao measurements from the sdss @xcite . while new observations continue to be expanded and improved with more recent instruments , the chft lensing survey remains the largest weak lensing survey to date . in this section we will briefly review projects that are currently operating or under construction ( stage iii and iv ) . all of these facilities share the common feature of surveying wide areas to collect large samples of galaxies , clusters , and/or supernovae and they will help clarify the nature of the interaction between dark sectors . more details can be found in @xcite . the existing and planned ground based de experiments collect data on snia , galaxy clustering and gravitational lensing . wide - field imaging is used to measure weak gravitational lensing and clustering of galaxies in bins of photometrically estimated redshifts and wide - field spectroscopy , to map the clustering of galaxies , quasars and the ly-@xmath30 forest and measure distances and expansion rates with bao and the history of structure growth with redshift - space distortions ( rsd ) . type ia supernovae are searched to determine the distance - redshift relation . the 6-degree field galaxy survey ( 6dfgs ) has mapped the nearby universe over @xmath71 deg@xmath72 of the southern sky with galactic latitude @xmath73 . the median redshift of the survey is @xmath74 . it is the largest redshift survey of the nearby universe , reaching out to @xmath75 . the survey data includes images , spectra , photometry , redshifts and a peculiar velocity survey of a subsample of 15,000 galaxies . the final release of redshift data is given in @xcite . the baryon oscillation spectroscopic survey ( boss ) is currently the largest spectroscopic redshift survey worldwide , mapping @xmath76 deg@xmath72 up to @xmath77 . boss is the largest of the four surveys that comprise sdss - iii and has been in operation for 5 years since 2009 . its goals are to measure angular diameter distance and expansion rate using bao , using 1.5 million galaxies @xcite . using ly-@xmath30 lines towards a dense grid of high - redshift quasars , it has pioneered a method to measure bao at redshifts @xmath78 $ ] . the analysis of sdss data release 9 has provided a measurement of the bao scale at @xmath79 with a precision of 2 - 3% @xcite . this survey will be followed by the extended boss ( eboss ) that will be operating for six years and will extend the boss survey to higher redshifts . similar to boss , the hobby - eberly telescope de experiment ( hetdex ) at the austin mcdonald observatory has the goal of providing percent - level constraints on the hubble parameter and angular diameter distance on the redshift range @xmath80 $ ] by using a combination of bao and power spectrum shape information . it will be achieved by surveying 0.8 million @xmath81 emitting galaxies on a field of view of 420deg@xmath72 @xcite . the javalambre physics of the accelerating universe astronomical survey ( j - pas ) is a new astronomical facility dedicated to mapping the observable universe in 56 colors @xcite . the starting date for this multi - purpose astrophysical survey is 2015 . in five years , j - pas will cover @xmath82deg@xmath72 using a system of 54 narrow band and 2 broad - band filters in the range @xmath83 nm . the filter system was optimized to accurately measure photometric redshifts for galaxies up to @xmath84 . the main instruments are a 2.5 m telescope located at el pico del buitre ( teruel , spain ) and a 1.2 giga - pixel camera . the main goals of the survey are to measure angular and radial components of bao from the galaxy clustering , determine the evolution of the cosmic volume from cluster counts and luminosity distances from snia . the filter system will permit to determine the redshifts of the observed supernovae . the camera is not optimized to measure galaxy elipticities so weak lensing studies would require ellipticity measurements obtained from other surveys . the jpas telescope will measure bao from high redshift quasars to achieve a better precision than boss @xcite , open the possibility of using the test described in sec . [ sec : sec7.evidence ] to disproof the concordance model . the panoramic survey telescope and rapid response system ( pan - starrs ) describes a facility with a cosmological survey among its major goals . the final goal is to use four coordinated telescopes to carry out survey of the full sky above dec=@xmath85 @xcite that will go a factor @xmath86 deeper than the sdss imaging survey . the survey will provide data on high redshift sn , galaxy clustering and gravitational lensing . for that purpose , in addition to the wide survey , an ultra - deep field of @xmath87deg@xmath72 will be observed down to magnitude 27 in the @xmath88 band with photometric redshifts to measure the growth galaxy clustering . data from this facility has already been used to constrain the equation of state parameter @xcite . the wigglez de survey is a large - scale redshift survey carried out at the anglo - australian telescope and is now complete . it has measured redshifts for @xmath89 galaxies over 1000deg@xmath72 in the sky . it combines measurements of cosmic distance using bao with measurements of the growth of structure from redshift - space distortions out to redshift @xmath90 @xcite . the atacama cosmology telescope ( act ) operates at 148 , 218 and 277 ghz with full - width at half maximum angular resolutions of @xmath91 @xcite . act observes the sky by scanning the telescope in azimuth at a constant elevation of @xmath92 as the sky moves across the field of view in time , resulting in a stripe - shaped observation area . the collaboration has released two observed areas of @xmath93deg@xmath72 and @xmath94deg@xmath72 @xcite . sky maps , analysis software , data products and model templates are available through nasa legacy archive for microwave background data analysis ( lambda ) . the south pole telescope ( spt ) is a 10 m telescope designed to map primary and secondary anisotropies in the cmb , currently operating at 95 , 150 , 220 ghz with a resolution with resolution ( 1.7 , 1.2 , 1.0)@xmath95 . the noise levels are 18@xmath96k at 150ghz and @xmath97 larger for the other two channels @xcite . it has observed a region of @xmath98deg@xmath72 . data in the three frequencies were used to produce a radiation power spectra covering the multipole range @xmath99 . at present is the most precise measurement of the radiation power spectrum at @xmath100 at those frequencies ; at those angular scales the signal is dominated by the sz effect and is not so relevant to constrain models of dm / de interaction . a polarization - sensitive receiver have been installed on the spt ; data at 95 and 150 ghz has provided a measurement of the @xmath101-mode polarization power spectrum from an area of @xmath102deg@xmath72 , spanning the range @xmath103 . the resulting power spectra was consistent with predictions for the spectrum arising from the gravitational lensing of @xmath104-mode polarization @xcite . the de survey ( des ) is a wide - field imaging and supernova survey on the blanco 4 m telecope at cerro tololo ( chile ) using the de camera . it has started operations and it will continue for five years . the de spectroscopic survey instrument ( desi ) is a wide field spectroscopic instrument intended to start in 2018 and operate also for five years in the nearly twin mayall telescope at kitt peak ( arizona ) . desi will obtain spectra and redshifts for at least 18 million emission - line galaxies , 4 million luminous red galaxies and 3 million quasi - stellar objects , to probe the effects of de on the expansion history bao and measure the gravitational growth history through rsd . the resulting 3-d galaxy maps at redshift @xmath105 and ly-@xmath30 forest at @xmath106 are expected to provide the distance scale in 35 redshift bins with a one - percent precision @xcite . the imaging survey will detect 300 million galaxies , with approximately 200 million wl shape measurements , almost a two - order of magnitude improvement over the cfhtlens weak lensing survey . approved as a major cosmology survey in sdss - iv ( 2014 - 2020 ) , eboss will capitalize on this premier facility with spectroscopy on a massive sample of galaxies and quasars in the relatively uncharted redshift range that lies between the boss galaxy sample and the boss ly-@xmath30 sample . compared with boss , this new survey will focus on a smaller patch of 7500 deg@xmath72 but it will reach higher magnitudes . it will measure both the distance - redshift relation and the evolution of the hubble parameter using different density tracers ; the clustering of luminous red galaxies ( lgrs ) and emission line galaxies ( elgs ) , quasars and ly-@xmath30 systems to probe the bao scale in the redshift ranges [ 0.6,0.8 ] , [ 1,2.2 ] and [ 2.2,3.5 ] respectively and it will achieve 1 - 2% accuracy in distance measurements from baos between @xmath107 . the javalambre physics of the accelerating universe astronomical survey ( jpas ) is a new astronomical facility dedicated to mapping the observable universe in 56 colors @xcite . the starting date for this multi - purpose astrophysical survey is 2015 . in five years , jpas will cover @xmath82deg@xmath72 using a system of 54 narrow band and 2 broad - band filters in the range @xmath83 nm . the filter system was optimized to accurately measure photometric redshifts for galaxies up to @xmath84 . the main instruments are a 2.5 m telescope located at el pico del buitre ( teruel , spain ) and a 1.2 giga - pixel camera . the main goals of the survey are to measure angular and radial components of bao from the galaxy clustering , determine the evolution of the cosmic volume from cluster counts and luminosity distances from snia . the filter system will permit to determine the redshifts of the observed supernovae . the camera is not optimized to measure galaxy elipticities so weak lensing studies would require ellipticity measurements obtained from other surveys . the jpas telescope will measure bao from high redshift quasars to achieve a better precision than boss @xcite , open the possibility of using the test described in sec . [ sec : sec7.evidence ] to disproof the concordance model . the large synoptic survey telescope ( lsst ) is a wide - field , ground - based telescope , designed to image @xmath108 deg@xmath72 in six optical bands from 320 nm to 1050 nm . the telescope will be located on cerro pachn ( chile ) and it will operate for a decade allowing to detect galaxies to redshifts well beyond unity . its science goals are to measure weak and strong gravitational lensing , bao , snia and the spatial density , distribution , and masses of galaxy clusters as a function of redshift . its first light is expected on 2019 . the square kilometre array ( ska ) is a radio - facility which is scheduled to begin construction in 2018 . the hi galaxy redshift survey can provide us with accurate redshifts ( using the 21 cm line ) of millions of sources over a wide range of redshifts , making it an ideal redshift survey for cosmological studies @xcite . although technically challenging , the ska could measure the expansion rate of the universe in real time by observing the neutral hydrogen ( hi ) signal of galaxies at two different epochs @xcite . wide - field multi - object spectrograph ( wfmos ) is a camera specially devoted to galaxy surveys . it will be mounted atop the 8.2 m subaru telescope on mauna kea ( hawaii ) . one of the science goals of the wfmos camera is high precision measurements of bao . the wfmos de survey comprises two parts : a 2,000 deg@xmath72 survey of two million galaxies at redshifts @xmath109 and a high redshift survey of about half a million lyman break galaxies ( lbgs ) at redshifts @xmath110 that would probe distances and the hubble rate beyond @xmath111 ( see @xcite for more details ) . bingo is a radio telescope designed to detect bao at radio frequencies by measuring the distribution of neutral hydrogen at cosmological distances using a technique called intensity mapping . the telescope will be located in a disused , open cast , gold mine in northern uruguay . it will operate in the range [ 0.96,1.26 ] ghz to observe the redshifted 21 cm hydrogen line . it will consist of a two - mirror compact range design with a 40 m diameter primary and it will have no moving parts to provide an excellent polarization performance and very low side - lobe levels required for intensity mapping . currently , the interest on cmb ground experiments is centered on polarization . for a cosmic variance limited experiment polarization alone places stronger constraints on cosmological parameters than cmb temperature @xcite . experiments like sptpol @xcite and quixote @xcite are currently taken data aiming to characterize the polarization of the cmb and of the galactic and extragalactic sources . cmb experiments devoted to measuring polarization from the ground are also being proposed ; the scientific capabilities of a cmb polarization experiment like cmb - s4 have been considered that in combination with low redshift data would be able to constrain , among other parameters , the de equation of state and dark matter annihilation @xcite . satellite surveys usually require a dedicated facility and , consequently , are more expensive than those carried out from the ground . their significant advantage is that , by observing outside the atmosphere , the data usually contains a lower level of systematic errors . the wilkinson microwave anisotropy probe ( wmap ) was a satellite mission devoted to measure cmb temperature fluctuations at frequencies operating between 23 and 94ghz . launched on june 30 , 2001 and operated for 9 years up to the end of september 2010 . the main results and data products of the nine years of operation are described in @xcite . the final 9yr data released was soon followed by those of the planck collaboration . the planck satellite observed the microwave and sub - millimeter sky from august 12th , 2009 to oct 23rd , 2013 in nine frequencies between 30 and 857 ghz , with angular resolution between 33 and 5. its goal was to produce cmb maps both in temperature and polarization . the planck collaboration has released data on cmb temperature anisotropies , thermal sunyaev - zledovich ( tsz ) effect . the measured temperature and polarization , a catalog of sunyaev - zeldovich ( sz ) clusters and likelihood codes to assess cosmological models against the planck data @xcite and other data products can be downloaded from the planck legacy archive @xcite . the temperature - temperature , temperature - e mode and e mode - e mode power spectra are measured up to @xmath112 @xcite , allowing the cmb lensing potential @xcite and the constraint on cosmological models beyond the @xmath0cdm model @xcite . erosita will be a x - ray satellite that will be launched in 2016 . it will perform the first imaging all - sky survey in energy range 0.3 - 10 kev @xcite . the goal of erosita is the detection of @xmath113 galaxy clusters out to redshifts @xmath114 , in order to study the large scale structure in the universe and test and characterize cosmological models including de . in the soft x - ray band ( 0.5 - 2 kev ) , it will be about 20 times more sensitive than the rosat all sky survey , while in the hard band ( 2 - 10 kev ) it will provide the first ever true imaging survey of the sky at those energies . euclid is a european space agency de satellite mission scheduled for launch in 2020 . this mission is designed to perform two surveys : a wide 15,000 deg@xmath72 survey in the optical and near - infrared and a deep survey on 40 deg@xmath72 two magnitudes deeper . these facilities are not independent between each other . euclid will map the extra - galactic sky with the resolution of the hubble space telescope , with optical and near - infrared ( nir ) imaging and nir spectroscopy . photometric redshifts for the galaxies in the wide survey will be provided from ground photometry and from the nir survey . in addition , 50 million spectroscopic redshifts will be obtained . euclid data will allow to measure the expansion history and the growth of structure with great precision . a detailed quantitative forecast of euclid performance has been discussed in @xcite . the data will allow to constrain many different cosmological models ; when the growth factor is parametrized as @xmath115 the value @xmath116 corresponds to the @xmath0cdm model and euclid will measure this parameter with a precision of @xmath117 @xcite . forecasts for other parametrizations of the growth factor and for other magnitudes such as the bias , de sound speed , redshift space distortions are given in @xcite . the wide field infrared survey telescope ( wfirst ) is an american satellite mission that is currently being reviewed and expected to be launch in 2023 . this mission updates and expands earlier proposed missions like the super nova acceleration probe ( snap ) and the joint de mission ( jdem ) . like euclid , one of its primary science goals is to determine the properties of de and in many respects complements euclid . wfirst strategy is to construct a narrow and deep galaxy redshift survey of 2000 deg@xmath72 . both satellites will measure the redshift for a similar number of galaxies and will obtain a comparable precision for the baryon acoustic oscillations derived angular diameter distances and hubble constant redshift evolution @xcite . nevertheless , due to their different observing strategy will allow cross - checks that will help to identify and eliminate systematics . the combination of both data sets will significantly improve the constraints on the dark energy parameters . many synergies will come from cross - correlating data from different observations for instance , euclid , wfirst and ska have similar scientific aims but will carry observations at different wavelengths . euclid and wfirst probe the low redshift universe , through weak lensing and galaxy clustering measurements . the ska has the potential to probe a higher redshift regime and a different range in scales of the matter power spectrum , which are linear scales rather than the quasi - non - linear scales to which euclid and wfirst will be sensitive . the combination of different observations will particularly sensitive to signatures of modified gravity . cross - correlation of different data sets will help to control systematics for the primary science . the ska , wfirst and euclid will be commissioned on similar timescales offering an exciting opportunity to exploit synergies between these facilities . @xcite the cosmic origins explorer ( core ) is a stage iv full - sky , microwave - band satellite proposed to esa within cosmic vision 2015 - 2025 . core will provide maps of the microwave sky in polarization and temperature in 15 frequency bands , ranging from 45 ghz to 795 ghz , with angular resolutions from @xmath118 at 45 ghz and @xmath119 at 795 ghz , with sensitivities roughly 10 to 30 times better than planck @xcite . the polarized radiation imaging and spectroscopy mission ( prism ) is a large - class mission proposed to esa in may 2013 within the framework of the esa cosmic vision program . its main goal is to survey the cmb sky both intensity and polarization in order to precisely measure the absolute sky brightness and polarization . the mission will detect approximately @xmath120 clusters using the thermal sz effect and a peculiar velocity survey using the kinetic sz effect that comprises our entire hubble volume @xcite . nasa is carrying similar efforts through the primordial polarization program definition team ( pppdt ) that converge towards a satellite dedicated to the study of cmb polarization ( cmbpol ) @xcite . combing these complementary ground based and space based observations , we would hopefully achieve a better understanding of the nature of dm , de and the interaction within the dark sectors . we thank s. tsujikawa for comments and suggestions . e. a. wishes to thank fapesp and cnpq ( brazil ) for support and a. a. costa , e. ferreira and r. landim for discussions and suggestions . f. a. b. acknowledges financial support from the ministerio de ciencia e innovacin , grant fis2012 - 30926 and the `` programa de profesores visitantes severo ochoa '' of the instituto de astrofsica de canarias . b. w. would like to acknowledge the support by national basic research program of china ( 973 program 2013cb834900 ) and national natural science foundation of china and he wishes to thank j. h. he and x. d. xu for helpful discussions .

models where dark matter and dark energy interact with each other have been proposed to solve the coincidence problem . we review the motivations underlying the need to introduce such interaction , its influence on the background dynamics and how it modifies the evolution of linear perturbations . we test models using the most recent observational data and we find that the interaction is compatible with the current astronomical and cosmological data . finally , we describe the forthcoming data sets from current and future facilities that are being constructed or designed that will allow a clearer understanding of the physics of the dark sector .

1603.08299

one of the long - standing problems in nuclear many - body theory has been the convergence of the perturbative expansion for the effective interaction @xmath3 ( or equally well that of effective operators ) derived from realistic nuclear forces to be used in nuclear structure calculations . conventionally , the various terms which appear in a perturbative expansion for @xmath3 are displayed by way of feynman - goldstone diagrams , examples of such diagrams are shown in fig . [ fig : fig1 ] . it is well known that a realistic nucleon - nucleon interaction @xmath4 contains strong components of short range , which renders a perturbative expansion in terms of @xmath4 meaningless . to overcome this problem , one takes into account the short - range correlations through the solution of the bethe - brueckner - goldstone equation and considers a perturbation in terms of the nuclear reaction matrix @xmath5 . the wavy lines in fig . [ fig : fig1 ] represent such @xmath5 interactions . however , higher - order perturbative contributions in terms of the @xmath5-matrix , may be large , and the convergence of the perturbative expansion slow or not convergent at all . actually , barrett and kirson @xcite showed that third - order contributions to the effective interaction were substantial , and raised the question whether the perturbative rayleigh - schrdinger ( rs ) expansion in terms of the nuclear @xmath5-matrix did converge at all . schucan and weidenmller @xcite even pointed out that the order - by - order expansion of the effective interaction will ultimately diverge when so - called intruder states are present . typical intruder states for nuclei like @xmath6o and @xmath7ca are four - particle - two - hole core - deformed states . it ought however to be mentioned that for nuclei with more valence nucleons in e.g. the oxygen mass area , such intruder state configurations may not be important , and a two - body ( or many - body ) effective interaction defined within the @xmath8-shell only , may represent the relevant degrees of freedom . most microscopic investigations of @xmath3 have been performed for nuclei in the @xmath8-shell , with few valence nucleons outside a @xmath9o core . however , when one extends the area of investigation to nuclei in the mass regions of calcium , tin or lead , one has to face the problem that for diagrams like those displayed in fig . [ fig : fig1 ] , there are more intermediate states which contribute to each diagram of the effective interaction in e.g. the @xmath10-shell than in the @xmath8-shell . moreover , the energy spacing between the various major shells is also smaller for nuclei in the @xmath10-shell than for those around @xmath9o . this leads to smaller energy denominators which should enhance third - order or higher - order contributions . thus , the combined action of the above effects could seriously deteriorate the order - by - order convergence ( if it does converge ) of the effective interaction . the only mechanism which could quench these effects , is the fact that the matrix elements of @xmath5 calculated in the @xmath10-shell should in general be weaker than those in the @xmath8-shell . the single - particle wave functions for the states around the fermi energy exhibit larger radii and , as the nucleon - nucleon interaction is of short range , the matrix elements of @xmath5 should be weaker for the heavier nuclei . the same arguments apply of course as well for the tin and lead regions . it is then the scope of this work to study the convergence of the effective interaction in terms of the mass number @xmath0 , in order to assess whether higher - order contributions to the two - body effective interaction decrease or increase as @xmath0 increases . to achieve this , we calculate all non - folded valence linked diagrams through third - order in the interaction @xmath5 , and sum higher - order folded diagrams to infinite order for the mass regions beyond closed - shell cores with @xmath1 , @xmath11 , @xmath12 , @xmath13 and @xmath2 . the details on how to obtain these effective interactions are briefly sketched in the next section , together with our results and discussions . some concluding remarks are given in section three . there are basically two main approaches in perturbation theory used to define an effective operator and effective interaction , each with its hierarchy of sub - approaches . one of these main approaches is an energy - dependent approach , known as brillouin - wigner perturbation theory , while the rayleigh - schrdinger ( rs ) perturbation expansion stands for the energy independent approach . the latter is the most commonly used approach in the literature @xcite , an approach which we will also employ here . it is then common practice in perturbation theory to reduce the infinitely many degrees of freedom of the hilbert space to those represented by a physically motivated subspace , the shell - model valence space . in such truncations of the hilbert space , the notions of a projection operator @xmath14 on the model space and its complement @xmath15 are introduced . the projection operators defining the model and excluded spaces are defined by @xmath16 and @xmath17 with @xmath18 being the dimension of the model space , and @xmath19 , @xmath20 , @xmath21 and @xmath22 . the wave functions @xmath23 are eigenfunctions of the unperturbed hamiltonian @xmath24 ( with eigenvalues @xmath25 ) , where @xmath26 is the kinetic energy and @xmath27 an appropriately chosen one - body potential , in this work that of the harmonic oscillator ( h.o . ) . the oscillator energies @xmath28 will be derived from @xmath29 , @xmath0 being the mass number . this yields @xmath30 , @xmath31 , @xmath32 , @xmath33 and @xmath34 mev for @xmath1 , @xmath11 , @xmath12 , @xmath13 and @xmath2 , respectively . the full hamiltonian is then rewritten as @xmath35 with @xmath36 , @xmath4 being the nucleon - nucleon ( nn ) interaction . below we will replace @xmath4 by the @xmath5-matrix , which will be used as the starting point for our perturbative treatment . following the above philosophy , we choose the model spaces which are believed , from both experiment and theoretical calculations , to be relevant for calculations of particle - particle effective interactions in the mass areas from @xmath1 to @xmath2 . these are the @xmath37 and @xmath38 orbits for @xmath3 in the mass area of @xmath1 , the @xmath39 , @xmath40 and @xmath41 orbits for @xmath11 , the @xmath42 , @xmath43 , @xmath44 and @xmath45 orbits for nuclei in the mass region of @xmath12 and the @xmath46 , @xmath47 , @xmath48 , @xmath49 and @xmath50 orbits for @xmath13 . for these systems , the closed - shell cores ( @xmath51he , @xmath9o , @xmath52ca and @xmath53sn ) have equal numbers of protons and neutrons , and the model spaces are the same for both protons and neutrons . for lead however , with @xmath54 and @xmath55 , the proton and neutron model spaces are different , i.e. the orbits @xmath56 , @xmath57 , @xmath58 , @xmath59 , @xmath60 and @xmath61 for the proton model space and @xmath62 , @xmath63 , @xmath64 , @xmath65 , @xmath66 , @xmath67 and @xmath68 for the neutron model space . since the effective interaction theory we will employ is tailored to degenerate model spaces , we will make no attempt to derive for lead an effective proton - neutron interaction for these two model spaces . moreover , as discussed in ref . @xcite , a multishell effective interaction may show strong non - hermiticities , or even divergencies if a h.o . basis is used . thus , for @xmath1 to @xmath13 we will discuss both isospin @xmath69 and @xmath70 effective interactions , whereas for lead we restrict the attention to @xmath71 and @xmath72 , where @xmath73 is the projection of the total isospin . for the above model spaces , there are in total 15 matrix elements for the effective interaction of @xmath1 , 63 for @xmath11 , 195 for @xmath12 , 353 for @xmath13 , 711 for the neutron model space of @xmath2 and 353 for the proton model space of @xmath2 . the effective interactions for @xmath11 , @xmath12 and @xmath13 are listed in ref.@xcite , and have been tested in nuclear structure calculations and a good agreement with the experimental data obtained for several isotopes in these mass areas . the spectra for isotopes in the lead region will be published elsewhere @xcite . having defined the various model spaces , the next step in our calculation is to obtain the nuclear reaction matrix @xmath5 , given by @xmath74 where @xmath75 is the unperturbed energy of the interacting nucleons , and @xmath76 is the unperturbed hamiltonian . for the bare nn interaction we use the one - boson - exchange potential bonn a defined in table a.1 of ref . the operator @xmath77 is a projection operator which prevents the interacting nucleons from scattering into states occupied by other nucleons . note that the exclusion operator used in the calculation of the @xmath5-matrix in this work is different from the @xmath15 operator used in the evaluation of the effective interaction . the definition of the pauli operator for the @xmath5-matrix can be found in refs . @xcite , where the so - called double - partitioned scheme has been used . this means that low - lying two - particle states are excluded by @xmath78 from the intermediate states in the bethe - goldstone eq . ( [ eq : betheg ] ) . for the example of the @xmath8-shell this exclusion refers to states with two nucleons in the @xmath10-shell . as a consequence , we have to include in our perturbation expansion ladder type diagrams , such as ( 2 - 3 ) in fig . [ fig : fig1 ] , where the allowed intermediate states are those of the @xmath10-shell or corresponding ones for the other model - spaces . the next step is to define the so - called @xmath79-box of the effective interaction and the @xmath5-matrix @xmath78 . ] given by @xmath80 where we will replace @xmath81 with @xmath5 ( @xmath5 replaces the free nn interaction @xmath4 ) . the @xmath79-box is made up of non - folded diagrams which are irreducible and valence linked . a diagram is said to be irreducible if between each pair of vertices there is at least one hole state or a particle state outside the model space . in a valence - linked diagram the interactions are linked ( via fermion lines ) to at least one valence line . note that a valence - linked diagram can be either connected ( consisting of a single piece ) or disconnected . in the final expansion including folded diagrams as well , the disconnected diagrams are found to cancel out @xcite . this corresponds to the cancellation of unlinked diagrams in the goldstone expansion @xcite . we can then obtain an effective interaction @xmath82 in terms of the @xmath79-box , with @xcite @xmath83 observe also that the effective interaction @xmath84 is evaluated at a given model space energy @xmath75 , as is the case for the @xmath5-matrix as well . for all mass areas , we fix @xmath85 mev . the first iteration is then given by @xmath86 in this work we define the @xmath79-box to consist of all diagrams through third order in the @xmath5-matrix , as discussed in ref . less than ten iterations were needed in order to obtain a converged effective interaction for the various values of @xmath0 . for further details , see ref . @xcite . in the calculation of the various diagrams , we limit the intermediate state excitations to @xmath87 in oscillator energy , an approximation which is viable if one employs an nn potential with a weak tensor force ( such as the bonn potential used here ) , as discussed by sommermann _ it is the aim of this study to explore the effects of the various contributions to @xmath88 . as it will be rather confusing to discuss the effects for individual matrix elements ( recall that depending on the model - space there are up to few hundred matrix elements ) , we define averages of matrix elements by @xmath89 where the summation index @xmath90 refers to all two - particle states of the model - space under consideration , coupled to angular momentum @xmath91 and isospin @xmath26 and [ tab : tab2 ] we omit to divide with the number of configurations , as this gives rather small numbers for the heavier nuclei . ] . in the averaging procedure defined in this equation we have weighted the matrix elements by the factor @xmath92 since this factor accounts for the degeneracy of two - particle states with respect to the projection quantum numbers and occurs e.g. in the calculation of the energy if all valence states are occupied . it turned out , however , that the main features of the results discussed below are obtained as well , if this weighting factor is dropped . for the operator @xmath93 we will consider @xmath94 , which corresponds to the bare @xmath5 matrix , @xmath95 , the @xmath96-box including terms up to second order in @xmath5 without folded diagrams , and @xmath97 ( @xmath98 ) the effective interaction including all @xmath96-box diagrams up to second ( third ) order plus all folded diagrams derived from these @xmath96-boxes . note , that the average defined in eq . ( [ eq : aver1 ] ) includes only diagonal matrix elements . in order to study if the conclusions remain valid for all matrix elements we also define a mean value including all matrix elements by @xmath99 where the summation indices @xmath100 and @xmath101 include again all two - particle states of the model - space considered . beside these averages , which include matrix elements of isospin @xmath69 and @xmath70 , we will also report on results where the averaging is restricted to one of these isospins only . results for the mean values of diagonal matrix elements ( see eq . ( [ eq : aver1 ] ) ) are listed in table [ tab : tab1 ] , while averages including the non - diagonal matrix elements as well ( see eq . ( [ eq : aver2 ] ) ) are presented in table [ tab : tab2 ] for the various model - spaces considered . inspecting these tables one observes very clearly that the mean values for the matrix elements are getting less attractive for the model spaces referring to heavy nuclei . this trend can be observed independent on the approximation used to calculate @xmath88 . this behavior reflects the fact that also the effective interaction , calculated with inclusion of higher order terms , is of short range and therefore , as we discussed already above , yield weaker matrix elements for the valence nucleons in heavy nuclei as compared to the light systems . furthermore , we observe some features which are valid independent on the mass number and model space considered : * the inclusion of second - order @xmath96-box diagrams in @xmath102 yields a substantial attraction for the @xmath69 matrix elements and a repulsion for @xmath70 . this difference may be understood by the following argument : for the @xmath70 channel , the major mechanism which accounts for the difference between first and second order , is provided by the core - polarization diagram in ( 2 - 2 ) of fig . [ fig : fig1 ] . moreover , in the @xmath70 channel , the tensor force component of the nucleon - nucleon interaction is not so important , whereas in the @xmath69 channel the contribution from the @xmath103-@xmath104 partial wave plays an important role in ladder - type diagrams , such as several of the folded diagrams , or the particle - particle ladder diagram in ( 2 - 3 ) of fig . [ fig : fig1 ] . typically , for many @xmath105 and @xmath69 particle - particle effective interactions , the particle - particle ladder is of the size of or larger than the core - polarization diagram , while for @xmath106 and @xmath70 , the core - polarization diagram and the @xmath5-matrix yield the largest contribution to the effective interaction . * the inclusion of folded diagrams yields a repulsive trend going form @xmath102 to @xmath97 . the effect is again much larger in the @xmath69 than in the @xmath70 matrix elements , which can as well be understood from the importance of the particle - particle ladder diagrams in the @xmath69 states . comparing the results of @xmath107 and @xmath97 one observes a repulsion for both isospins . * contrary to this repulsion due to the second - order terms in the folded - diagram expansion , the additional inclusion of terms of third order in @xmath5 yields some attraction in @xmath98 as compared to @xmath97 . except for the case of @xmath51he , the effect of third - order terms is very weak for the @xmath70 states . this was also observed in ref.@xcite in the study of the spectra of nuclei with valence particles being only neutrons or protons . there the authors noted that the spectra of e.g. @xmath6o or @xmath7ca obtained with either a second - order or third - order effective interaction were quite similar . for calculations of the effective interaction for lead or tin , this is a gratifying property since it means that one needs only to evaluate the @xmath79-box to second order and sum all folded diagrams . finally , in order to discuss the convergence of the perturbation expansion , we compare in table [ tab : tab1a ] the ratios evaluated from the mean values defined in eq . ( [ eq : aver1 ] ) . these ratios reflect of course the same features which we already discussed above . they emphasize , however , in a much better way that the different ratios are rather insensitive on the mass number which is considered . this means that one can expect the convergence of the perturbation expansion for the residual interaction to be as good ( or bad ) for heavy nuclei as for the light nuclei around @xmath9o , which are usually studied . for nuclear structure studies of heavy nuclei with neutron numbers quite different from the proton number one typically considers model - spaces , which are separate for protons and neutrons , ignoring the residual interaction beyond the mean - field approximation . for these cases ( isospin @xmath70 ) , the effects of terms of second order in @xmath5 seem to be rather important with @xmath97 containing a correction of around 50 percent of the average of @xmath107 . however , it is encouraging to note that the inclusion of third order terms yields a correction of only 5 percent or even below . we have studied the behavior of the perturbation expansion for the effective interaction to be used in shell - model studies of nuclei with various mass numbers . inspecting appropriate mean values of matrix elements , we have found that the fact that the @xmath5-matrix becomes smaller in absolute value with increasing mass numbers , counterbalances the effects that there are more intermediate states to sum over and that the energy denominators become smaller in each individual diagram of the effective interaction . therefore , the convergence of the perturbation expansion seems to be rather insensitive to the nuclear mass number . we observe that various features of the folded - diagram expansion , which had been discussed for the mass region @xmath108 , can also be found in heavy nuclei . the nuclear structure calculations for heavy nuclei are mainly sensitive to the proton - proton and neutron - neutron residual interactions . for these @xmath70 matrix elements the third - order and second - order averages are very close , indicating that for this isospin channel one can approximate the effective interaction by including all diagrams to second order plus folded diagrams to all orders . for @xmath69 , one still needs to account for third - order contributions . the fact that third - order contributions seem to stabilize for heavier nuclei , has also important consequences for nuclear structure calculations in nuclei in the mass regions of e.g. @xmath109sn and @xmath110pb . this means that the methods used to calculate the effective interaction for valence nucleons , applied mainly in the mass regions of @xmath9o and @xmath52ca , can be applied to the mass regions of @xmath109sn and @xmath110pb , as done recently in refs . @xcite . we gratefully acknowledge the financial support of the norfa ( nordic academy for advanced study ) , grant 93.40.018/00 . one of us , mhj , thanks the istituto trentino di cultura , italy and the research council of norway for their support . barrett b r and kirson m w 1970 _ nucl . phys . _ a * 148 * 145 schucan t h and weidenmller h a 1972 _ ann . * 73 * 108 kuo t t s and osnes e 1990 _ folded - diagram theory of the effective interaction in atomic nuclei ( springer lecture notes in physics * 364 * ) _ lindgren i and morrison j 1985 _ atomic many - body theory _ ( springer : berlin ) hjorth - jensen m , kuo t t s and osnes e 1995 , phys . reports , in press hjorth - jensen m , engeland t , holt a and osnes e 1995 , in preparation krenciglowa e m , kung c l , kuo t t s and osnes e 1976 _ ann . ny _ * 101 * 154 machleidt r 1989 _ adv . nucl . _ * 19 * 189 sommerman h m , mther h , tam k c , kuo t t s and faessler a 1981 _ phys . _ c * 23 * 1765 engeland t , hjorth - jensen m , holt a , kristiansen t and osnes e 1995 , in preparation .the mean values for diagonal matrix elements calculated according to eq . ( 7 ) in model - spaces with cores as indicated in the first row assuming various approximations for the effective interaction . averages are listed for all isospins ( @xmath111 ) as well as for @xmath69 ( @xmath112 ) and @xmath70 ( @xmath113 ) . for lead , results for averages in the proton - proton model - space ( @xmath114 ) and the neutron - neutron model - space ( @xmath115 ) are listed . all entries in mev . [ cols="<,^,^,^ " , ]

the convergence of the perturbation expansion for the effective interaction to be used in shell - model calculations is investigated as function of the mass number @xmath0 , from @xmath1 to @xmath2 . as the mass number increases , there are more intermediate states to sum over in each higher - order diagram which contributes to the effective interaction . together with the fact that the energy denominators in each diagram are smaller for larger mass numbers , these two effects could largely enhance higher - order contributions to the effective interaction , thereby deteriorating the order - by - order convergence of the effective interaction . this effect is counterbalanced by the short range of the nucleon - nucleon interaction , which implies that its matrix elements are weaker for valence single - particle states in `` large '' nuclei with large mass number as compared to those in light nuclei . these effects are examined by comparing various mean values of the matrix elements . it turns out that the contributions from higher - order terms remain fairly stable as the mass number increases from @xmath1 to @xmath2 . the implications for nuclear structure calculations are discussed .

nucl-th9509006

neutrino magnetic moments are no doubt among the most well theoretically understood and experimentally studied neutrino electromagnetic properties . @xcite as it was shown long ago @xcite , in a wide set of theoretical frameworks neutrino magnetic moment is proportional to the neutrino mass and in general very small . for instance , for the minimally extended standard model the dirac neutrino magnetic moment is given by @xcite : @xmath0 at the same time , the magnetic moment of hypothetical heavy neutrino ( with mass @xmath1 ) is @xmath2 @xcite . it should be noted here that much larger values for the neutrino magnetic moments are possible in various extensions of the standard model ( see , for instance , in @xcite ) constraints on the neutrino magnetic moment can be obtained in @xmath3 scattering experiments from the observed lack of distortions of the recoil electron energy spectra . recent reactor experiments provides us with the following upper bounds on the neutrino magnetic moment : @xmath4 ( munu collaboration @xcite ) , @xmath5 ( texono collaboration @xcite ) . the gemma collaboration has obtain the world best limit @xmath6 @xcite . another kind of neutrino experiment borexino ( solar neutrino scattering ) has obtained rather strong bound : @xmath7 @xcite . the best astrophysical constraint on the neutrino magnetic moment has been obtained from observation of the red giants cooling @xmath8 @xcite . as it was pointed out above the most stringent terrestrial constraints on a neutrino effective magnetic moments have been obtained in ( anti)neutrino - electron scattering experiments and the work to attain further improvements of the limits is in process . in particular , it is expected that the new bound on the level of @xmath9 can be reached by the gemma collaboration in a new series of measurements at the kalinin nuclear power plant with much closer displacements of the detector to the reactor that can significantly enhanced the neutrino flux(see @xcite ) . an attempt to reasonably improve the experimental bound on a neutrino magnetic moment was undertaken in @xcite where it was claimed that the account for the electron binding effect in atom can significantly increase the electromagnetic contribution to the differential cross section in respect to the case when the free electron approximation is used in calculations of the cross section . however , as it was shown in a series of papers @xcite the neutrino reactor experiments on measurements of neutrino magnetic moment are not sensitive to the electron binding effect , so that the free electron approximation can be used for them . one may expect that neutrino electromagnetic properties can be much easier visualized when neutrino is propagating in external magnetic fields and dense matter . also , neutrino propagation in matter is a rather longstanding research field nevertheless still having advances and obtaining a lot of interesting predictions for various phenomena . the convenient and elegant way for description of neutrino interaction processes in matter has been recently offered in a series of papers @xcite . the developed method is based on the use of solutions of the modified dirac equation for neutrino in matter in feynman diagrams . the method was developed before for studies of different processes in quantum electrodynamics and was called as `` the method of exact solutions '' @xcite the gain from the introduction of the method was sustained by prediction and detailed quantum description of the new phenomenon of the spin light of neutrino in matter ( the @xmath10 ) , first predicted in @xcite within the quasi - classical treatment of neutrino spin evolution . the essence of the @xmath10 is the electromagnetic radiation in neutrino transition between two different helicity states in matter . the simplification of the process framework , such as use of the uniform , unpolarized and non - moving matter , neglect of the matter influence on the radiated photon , makes the estimate of real process relevance in astrophysical settings far from the practical scope . in this short paper we should like to make a step towards the completeness of the physical picture and to consider the incomprehensible at first glance question of the plasmon mass influence on the @xmath10 . the importance of plasma effects for the @xmath10 in matter was first pointed out in @xcite . the investigations already carried out in this area @xcite indicated that the plasmon emitted in the @xmath10 has a considerable mass that can affect the physics of the process . to see how the plasmon mass enters the @xmath10 quantities we appeal to the method of exact solutions and carry out all the computations relevant to the @xmath10 . in this respect , in order to have the conformity we also set all the conditions for the task the same as for corresponding studies on the @xmath10 . in particular , we consider only the standard model neutrino interactions and take matter composed of electrons . in the exact solutions method , one starts with the modified dirac equation for the neutrino in matter in order to have initial and final neutrino states , which would enter the process amplitude . the equation reads as follows @xcite : @xmath11 where in the case of neutrino motion through the non - moving and unpolarized matter @xmath12 with @xmath13 being matter ( electrons ) number density . under this conditions the equation ( [ eq : dirac ] ) has plane - wave solution determined by 4-momentum @xmath14 and quantum numbers of helicity @xmath15 and sign of energy @xmath16 . for the details of equation solving and exact form of the wave functions @xmath17 the reader is referred to @xcite and @xcite , here we cite only the expression for the neutrino energy spectrum : @xmath18 the s - matrix of the process involves the usual dipole electromagnetic vertex @xmath19+i\gamma^{5}{\bf \sigma}\big\}$ ] and for given spinors for the initial and final neutrino states @xmath20 can be written as @xmath21 here @xmath22 is the photon polarization vector , @xmath23 is the transitional magnetic moment and @xmath24 is the normalization length . the delta - functions before spinors convolution part lead to the conservation laws @xmath25 with energies for the initial and final neutrinos @xmath26 taken in accordance to ( [ eq : dispersion ] ) . for the photon dispersion , for the purpose of our study it is sufficient to use the simplest expression @xmath27 as it was discussed in our previous studies on the @xmath10 @xcite the most appropriate conditions for the radiation to manifest its properties are met in dense astrophysical objects . this is the setting we will use further for the process and in the case of cold plasma the plasmon mass should be taken as @xmath28 the numerical evaluation at typical density gives @xmath29 , while the density parameter @xmath30 . let us now consider the influence of dense plasma on the process of spin light of neutrino . similarly to the original spin light calculation we consider the case of initial neutrino possessing the helicity quantum number @xmath31 and the corresponding final neutrino helicity is @xmath32 . using the neutrino energies ( [ eq : dispersion ] ) with corresponding helicities one can resolve the equations ( [ eq : conservation ] ) in relation to plasmon momentum which is not equal to its energy since we take into account the dispersion of the emitted photon in plasma ( [ photon dispersion ] ) . for convenience of calculations it is possible to use the following simplification . in most cases the neutrino mass appeared to be the smallest parameter in the considered problem and it is several orders smaller then any other parameter in the system . so we could first examine our process in approximation of zero neutrino mass , though we should not forget that only neutrino with non - zero mass could naturally possess the magnetic moment . this our simplification should be considered only as a technical one . it should be pointed here that in order to obtain the consistent description of the @xmath10 one should account for the effects of the neutrino mass in the dispersion relation and the neutrino wave functions . from the energy - momentum conservation it follows @xcite that the process is kinematically possible only under the condition ( taking account of the above - mentioned simplification ) : @xmath33 provided with the plasmon momentum we proceed with calculation of the @xmath10 radiation rate and total power . the exact calculation of total rate is an intricate problem and the final expression is too large to be presented here . however one can consider the most notable ranges of parameters to investigate some peculiarities of the rate behavior . first of all we calculate the rate for the case of the @xmath10 without plasma influence . this can be done by choosing the limit @xmath34 and the obtained result is in full agreement with @xcite : @xmath35 from ( [ gammasl ] ) one easily derives the @xmath10 rate for two important cases , _ i.e. _ high and ultra - high densities of matter just by choosing correspondingly @xmath14 or @xmath36 as the leading parameter in the brackets . while neutrino mass is the smallest quantity , our system fall within the range of relativistic initial neutrino energies . the corresponding expression for the total power also covers high and ultra - high density cases @xcite as well as the intermediate area where the density parameter and the neutrino momentum are comparable : @xmath37 if we account for the plasma influence ( thus , @xmath38 ) on the @xmath10 we can discuss two important situations . one is the area of parameters near the threshold , and the other is connected with direct contribution of @xmath39 into the radiation rate expression . the later case is particularly important for this study , because it fulfill the aim of the present research in finding the conditions under which the plasmon mass can not be neglected . for physically reliable conditions the density parameter usually appears to be less then the plasmon mass , which in its turn is less then the neutrino momentum : @xmath40 . obviously the threshold condition ( [ threshold ] ) should be satisfied . as we consider the conditions similar to different astrophysical objects it is natural to use high - energy neutrino . using the series expansion of the total rate one could obtain the rate of the process in the following form : @xmath41 where @xmath42 . approaching the threshold ( @xmath43 ) , the expansion ( [ gammaslseries ] ) becomes inapplicable , however it is correct in rather wide range of parameters with @xmath44 and @xmath45 . near the threshold the the total rate can be presented in the form @xmath46 but the exact coefficient is too unwieldy to be presented here . concerning the power of the @xmath10 with plasmon , one can use the expansion : @xmath47 the expression ( [ intensslseries ] ) is correct only if the system meets the requirement @xmath48 . otherwise one should use higher orders of quantity @xmath49 in the expansion to achieve a reliable value of intensity . near the threshold the power has the same dependence on the `` distance '' from the threshold @xmath50 as the rate of the process . there is an increasing interest to neutrino electromagnetic properties and neutrino magnetic moments in particular . this interest is stimulated , first by the progress in experimental bounds on magnetic moments which have been recently achieved , as well as theoretical predictions of new processes emerging due to neutrino magnetic moment , such as the @xmath10 and a believe in its importance for possible astrophysical applications . further developing the theory of the spin light of neutrino , we have explicitly shown that the influence of plasmon mass becomes significant ( see ( [ gammaslseries ] ) and ( [ intensslseries ] ) ) when the parameter @xmath51 is comparable with @xmath52 , this corresponds to the system near the threshold . as soon as the quantity @xmath48 ( so the system is far from the threshold ) one can use either @xmath10 radiation rate and total power from @xcite or their rather compact generalizations ( [ gammaslseries ] ) and ( [ intensslseries ] ) where the plasmon mass is accounted for as a minor adjustment . since high energy neutrinos propagating in matter could be rather typical situation in astrophysics , for instance in neutron stars , the influence of photon dispersion in plasma on the @xmath10 process can be neglected and the threshold generated by the non - zero plasmon mass should not be taken into account . however , the method of exact solutions of modified dirac equation provides us with analytical expressions for probability and intensity in the whole range of possible parameters . one of the authors ( a.s . ) is thankful to giorgio bellettini , giorgio chiarelli , mario greco and gino isidori for the invitation to participate in les rencontres de physique de la vallee daoste on results and perspectives in particle physics .

recent discussion on the possibility to obtain more stringent bounds on neutrino magnetic moment has stimulated new interest to possible effects induced by neutrino magnetic moment . in particular , in this note after a short review on neutrino magnetic moment we re - examine the effect of plasmon mass on neutrino spin light radiation in dense matter . we track the entry of the plasmon mass quantity in process characteristics and found out that the most substantial role it plays is the formation of the process threshold . it is shown that far from this point the plasmon mass can be omitted in all the corresponding physical quantities and one can rely on the results of massless photon spin light radiation theory in matter . [ 1999/12/01 v1.4c il nuovo cimento ]

1112.5263

for many physical cases in qcd , an observable quantity is usually expressed in terms of truncated series in the coupling constant @xmath0 with given coefficients , so that in the next - to - next - to - next - to - leading order ( n@xmath4lo ) we get @xmath5 where @xmath6 are some numbers , and @xmath7 is a fixed scale . so , the value @xmath8 is the single - scale quantity . the exhausted examples are the followings : 1 . the hadronic fraction of @xmath1-decay width @xcite @xmath9 } { \gamma[\tau\to \nu_{\tau}e^+\nu_e ] } = \nonumber\\ & & { \cal r}_{\tau}^{[0 ] } \left ( 1 + c_1^{\tau}\,\frac{\alpha_s(m_{\tau})}{\pi}+ c_2^{\tau}\,\left(\frac{\alpha_s(m_{\tau})}{\pi}\right)^2 + c_3^{\tau}\,\left(\frac{\alpha_s(m_{\tau})}{\pi}\right)^3 + \delta r_{\mbox{\small np}}\right ) , \label{eq:2}\end{aligned}\ ] ] where @xmath10}=3.058 $ ] , the coefficients are given by @xmath11 and @xmath12 is a nonperturbative contribution . the hadronic fraction of @xmath2-decay width @xcite @xmath13 } { \gamma[\eta_c\to \gamma\gamma ] } = { \cal r}_{\eta_c}^{[0 ] } \left ( 1 + d_1\,\frac{\alpha_s(2 m_c)}{\pi}\right),\end{aligned}\ ] ] where @xmath14 } = \frac{c_f}{2n_c}\,\frac{1}{e_c^4}\ , \frac{\alpha_s^2(2m_c)}{\alpha^2_{\rm em}}\ ] ] with @xmath15 , @xmath16 is the number of colors , @xmath17 is the electric charge of charmed quark , and the coefficient @xmath18 is given by @xmath19 where @xmath20 is the number of ` active ' flavors , and @xmath21 is the pole mass of charmed quark . the above formulae can be used for the extraction of @xmath0 at the appropriate scale . the value of @xmath0-corrections is numerically significant . so , the problem is how the truncated series can be improved . the well - established approach to the solution of such the problem is a resummation of some significant terms . we mention two of such techniques . the first is the summation of @xmath22 contributions , where @xmath23 is the first coefficient of @xmath24-function in qcd @xcite . the second procedure is based on an appropriate change of renormalization scheme by @xmath25 to the given order in the coupling constant , which allows one to decrease a role of higher - order corrections or even to minimize it with the modification of @xmath26-function resulting in a different running of @xmath27 @xcite . the disadvantage of above methods is twofold . first , the next - order correction while computed exactly can essentially differ from the approximation of @xmath28-dominance . second , the redefinition of renormalization scheme leads to the scale or normalization - point dependence of matching procedure . in this paper we present a procedure to improve the truncated series in the framework of renormalization group by introducing an auxiliary scale and taking a single - scale limit . a general formalism is given in section [ sec:2 ] . the numerical estimates are presented in section [ sec:3 ] . the analysis of scale dependence for the @xmath2-decay rate is performed , since the normalization at the pole mass involves the additional problem caused by the residual change of @xmath21 by the variation of normalization point in the @xmath3-mass @xmath29 @xcite . our results are summarized in conclusion . for the sake of clarity , let us start with the consideration of first - order correction . @xmath30 } = 1+c_1 \frac{\alpha_s(\lambda)}{\pi}.\ ] ] introduce an auxiliary scale @xmath31 , so that @xmath32 making use of the renormalization group relation to the first order in @xmath0 , @xmath33 we clearly get @xmath34^{\displaystyle \frac{2 c_1}{\beta_0\ln\kappa}},\ ] ] which gives the ordinary presentation improved by the renormalization group . note , that one finds the limit @xmath35 which will be correct for the further consideration at a fixed order in @xmath0 . the single - scale limit of @xmath36 can be easily evaluated @xmath37,\ ] ] which is our result for the case of first - order correction . in order to proceed with the higher - order corrections , let me perform the derivation in another way . so , the @xmath24-function has the form @xmath38 with @xmath39 . to the first order it gives @xmath40,\ ] ] at @xmath36 . then , @xmath41^{\displaystyle \frac{2 c_1}{\beta_0\ln\kappa}}\approx \exp\left[c_1 \frac{\alpha_s(\lambda)}{\pi } \right],\ ] ] and expanding in @xmath0 , we rederive the renormalization group improvement ( rgi ) for the first - order correction . further , we can easily find the rgi for the third order in @xmath0 ( n@xmath4lo ) . indeed , since @xmath42,\ ] ] we get @xmath43^{\displaystyle \frac{c_1 + 4\bar c_2\,{\mathfrak a}+16 \bar c_3\,{\mathfrak a}^2}{\beta_0+\beta_1\,{\mathfrak a}+ \beta_2\,{\mathfrak a}^2}\frac{4}{\ln\kappa^2 } } = \exp\left[c_1 \frac{\alpha_s(\lambda)}{\pi}+\bar c_2 \left(\frac{\alpha_s(\lambda)}{\pi}\right)+\bar c_3 \left(\frac{\alpha_s(\lambda)}{\pi}\right ) \right],\ ] ] where we put @xmath44 expanding in @xmath0 at @xmath36 , we find @xmath45^{\displaystyle \frac{c_1 + 4\bar c_2\,{\mathfrak a}+16 \bar c_3\,{\mathfrak a}^2}{\beta_0+\beta_1\,{\mathfrak a}+ \beta_2\,{\mathfrak a}^2}\frac{4}{\ln\kappa^2}}\approx 1 + c_1\,\frac{\alpha_s(\lambda)}{\pi}+ c_2\,\left(\frac{\alpha_s(\lambda)}{\pi}\right)^2 + c_3\,\left(\frac{\alpha_s(\lambda)}{\pi}\right)^3.\ ] ] thus , the third - order improved expression has the form @xmath46\ ] ] we stress the renormalization group motivation used in contrast to _ ad hoc _ method of pad approximants . let us show how the improvement works in a simple example . so , we consider a rather oscillating sum , @xmath47 which reveals a ` slow ' convergency , since @xmath48}=1,\quad { \cal e}^{[1]}=0.5,\quad { \cal e}^{[1]}=0.8,\ ] ] while @xmath49\ ] ] results in @xmath50 } = 1,\quad { \cal e}^{\mbox{\sc rgi}}_{[1 ] } = 0.61,\quad { \cal e}^{\mbox{\sc rgi}}_{[2 ] } = 0.72,\ ] ] which is ` more stable ' . thus , we expect that @xmath51 possesses a more numerical stability in the truncated series . of course , if a series is essentially asymptotic , the improvement can not cancel a ` bad ' convergency . next , we have to mention the numerical problem often appearing with the @xmath0-corrections to the amplitudes and the amplitudes squared if those corrections are significantly large . indeed , the correction to the amplitude @xmath52}(1+c_1\alpha_s)\ ] ] should lead to @xmath53}\right)^2(1 + 2\,c_1\alpha_s),\ ] ] so that the ratio @xmath54 numerically deviates from unit . the rgi has no such the problem , since the exponent does not involve the above mismatching . finally , we stress that the rgi does not present some kind of resummation of higher orders . in the resummation technique one certainly suggests a form of higher - order terms . in contrast , we give the exact expression produced by the renormalization group . at small @xmath0 as dictated by the perturbative paradigm , the expression can be expanded till the appropriate order . thus , one could claim that the rgi procedure looks like overflying the accuracy . to my opinion , one should use the rgi point as a central value of the calculated quantity , while the expansion truncated to the given order would indicate a systematic error of numerical estimate . the rgi formula for the @xmath1-lepton decays into hadrons reads off @xmath55 } \left\{\exp\left [ c_1^{\tau}\,\frac{\alpha_s(m_{\tau})}{\pi}+ \bar c_2^{\tau}\,\left(\frac{\alpha_s(m_{\tau})}{\pi}\right)^2 + \bar c_3^{\tau}\,\left(\frac{\alpha_s(m_{\tau})}{\pi}\right)^3\right ] + \delta r_{\mbox{\small np}}\right\ } , \label{eq:17a } \end{aligned}\ ] ] where @xmath56 implementing @xmath57 we find @xmath58 which results in @xmath59 where we include the experimental uncertainty , only . for the sake of comparison , the pdg value extracted by the same measurement of @xmath1 rate reads off @xmath60 which respectively gives @xmath61 . we point out that the theoretical uncertainty in pdg is slightly overestimated , to our opinion , since the displacement of central value extracted in two ways equals @xmath62 . thus , the preferable value of coupling constant following from the @xmath1-lepton hadronic width is given by @xmath63 with the central point closer to the ` world average ' . the problem with the estimate of hadronic width of @xmath2-charmonium is twofold . first , the scale setting in the @xmath0-correction is beyond the accuracy , since its variation contributes to @xmath65 . so , we should put the arbitrary scale by @xmath66 } \left ( 1 + d_1\,\frac{\alpha_s(\mu)}{\pi}\right ) . \label{eq:4a}\ ] ] the second point is the prescription for the pole mass of charmed quark . in the perturbative qcd , the pole mass is strictly defined . the relation between the @xmath3-running mass @xmath29 and the pole mass is known to the @xmath67-order @xcite . explicitly , to the @xmath65-terms @xcite we put @xmath68 with @xmath69 where @xmath70 , and @xmath71 . th evalue of pole mass is the renormalization invariant . however , at reasonable scales @xmath72 , the residual dependence due to the truncation of perturbative series is numerically significant . the reason of such the dependence is a growth of coefficients in series as caused by the renormalon . in fact , the pole mass becomes a scale - dependent quantity . to avoid this problem , the operative procedure is to fix a short - distance mass @xmath73 free off the renormalon and to perform the calculations with the series expressed in terms of @xmath73 . we exploit two schemes , which lead to results close enough to each other . the first scheme is given by the @xmath3-running mass @xmath74 . taking @xmath75 we calculate the pole mass shown in fig . [ fig:1 ] . we have checked that the implication of rgi procedure to the relation between the pole and running masses is consistent with the above result , and the effect of rgi can be absorbed into the decrease of @xmath76-value by about @xmath77 mev , which below the systematic accuracy of matching procedure as discussed below . ( 100,70 ) ( 3,3)=100 ( 90,0)@xmath72 , gev ( 0,67)@xmath21 , gev the second is the potential scheme described in ref @xcite . in this case , we calculate the scale - dependent matching of perturbative 2-loop scatic potential @xmath78 involving the 3-loop running @xmath0 with the phenomenological qcd - motivated static potential @xmath79 containing both the 2-loop short - distance coulomb - like contribution as well as the long - distance linear confining term preserving the infrared stability . then , the potential and , hence , the @xmath80-masses are free off the renormalon . the heavy quark masses are fixed by the measured spin - average mass - spectra of heavy quarkonia . so , @xmath81 the matching of scale - dependent perturbative potential @xmath82 is extracted numerically as described in ref @xcite . thus , the cancellation of renormalon in the sum of @xmath83 gives @xmath84 up to a constant shift @xmath85 independent of the scale . the matching with the perturbative pole mass in ( [ pole ] ) gives @xmath86 mev , depending on the variation of coupling constant @xmath87 in the limits of @xmath88 . the value of @xmath85 indicates the accuracy of matching procedure . the result is presented in fig.[fig:1 ] , which reveals a good agreement of two schemes used . ( 100,70 ) ( 3,3)=100 ( 90,0)@xmath72 , gev ( 0,67)@xmath89 then , the perturbative formula ( [ eq:4a ] ) with ( [ eq:23 ] ) results in the @xmath89 shown in fig . [ fig:2 ] , wherefrom we get @xmath90 at @xmath91 gev with @xmath92 the estimate in ( [ eq:24 ] ) is slightly greater than the value @xmath93 given by bodwin and chen @xcite . we stress the scale - stability of our result . further , at the same scale we find @xmath94 then , comparing ( [ eq:25 ] ) with ( [ eq:24 ] ) we obtain the final estimate including the theoretical uncertainty due to possible contributions of higher orders and , hence , the induced scale - dependence by the variation of central values as @xmath95 which is in agreement with the experimental value @xmath96 to be compared with @xmath97 obtained in @xcite under the resummation of @xmath22-terms . we point out that the improvement of the experimental accuracy combined with the calculation of @xmath65-correction would give a good opportunity to extract the mass of charmed quark . in this respect , we refer to ref . @xcite , where the @xmath65-corrections were taken into account in the ratio of widths for the decays of @xmath98 and @xmath99 , so that the analysis suffers from the uncertainties related with the relativistic corrections entering the ratio for the different initial states . the advantage of @xmath89 is the cancellation of such the initial state corrections . we have developed a general scheme to improve the estimate of truncated perturbative series in qcd by the tool of renormalization group for the single - scale quantities . the method allows one to get more realistic central values of the quantities as well as to estimate the theoretical uncertainty of results by comparison of rgi values with the perturbatively expanded ones . the rgi receipt for the calculation of quantity ( [ eq:1 ] ) , ( [ eq:6 ] ) is given by ( [ eq:16 ] ) and ( [ eq:17 ] ) . we have applied the approach to the fractions of hadronic widths for the @xmath1-lepton and @xmath2-charmonium , which allows us to get realistic estimates of @xmath100}/{\gamma[\eta_c\to \gamma\gamma]}\ ] ] in a reasonable agreement with the appropriately measured values . the author thanks prof.g.bodwin for an exciting presentation of his results on the resummation technique for the hadronic fraction of @xmath2 width as he gave at the heavy quarkonium workshop held in cern , nov . 8 - 11 , 2002 . a special gratitude goes to the organizing committee of the workshop , and personally to antonio vairo and nora brambilla for the invitation and a kind hospitality . i also thank prof.r.dzhelyadin for the possibility to visit cern in collaboration with the lhcb group , to which members i express my gratitude for a hospitatily . i thank prof.a.k.likhoded , who asked me for the meaning of resummation technique , which initiated this work . * * s. narison and a. pich , phys . b * 211 * , 183 ( 1988 ) ; + e. braaten , s. narison and a. pich , nucl . b * 373 * , 581 ( 1992 ) ; + a. l. kataev and v. v. starshenko , mod . lett . a * 10 * , 235 ( 1995 ) [ arxiv : hep - ph/9502348 ] . k. hagiwara _ et al . _ [ particle data group collaboration ] , phys . d * 66 * , 010001 ( 2002 ) . r. barbieri , e. demilio , g. curci and e. remiddi , nucl . b * 154 * , 535 ( 1979 ) ; + k. hagiwara , c. b. kim and t. yoshino , nucl . b * 177 * , 461 ( 1981 ) . m. beneke and v. m. braun , phys . b * 348 * , 513 ( 1995 ) [ arxiv : hep - ph/9411229 ] , nucl . b * 426 * , 301 ( 1994 ) [ arxiv : hep - ph/9402364 ] . g. t. bodwin and y. q. chen , phys . d * 64 * , 114008 ( 2001 ) [ arxiv : hep - ph/0106095 ] . j. g. korner , f. krajewski and a. a. pivovarov , phys . d * 63 * , 036001 ( 2001 ) [ arxiv : hep - ph/0002166 ] . m. beneke , phys . rept . * 317 * , 1 ( 1999 ) [ arxiv : hep - ph/9807443 ] . k. melnikov and t. v. ritbergen , phys . b * 482 * , 99 ( 2000 ) [ arxiv : hep - ph/9912391 ] ; + k. g. chetyrkin and m. steinhauser , nucl . phys . b * 573 * , 617 ( 2000 ) [ arxiv : hep - ph/9911434 ] . n. gray , d. j. broadhurst , w. grafe and k. schilcher , z. phys . c * 48 * , 673 ( 1990 ) ; + d. j. broadhurst , n. gray and k. schilcher , z. phys . c * 52 * , 111 ( 1991 ) . v. v. kiselev , a. e. kovalsky and a. i. onishchenko , phys . d * 64 * , 054009 ( 2001 ) [ arxiv : hep - ph/0005020 ] ; + v. v. kiselev , a. k. likhoded , o. n. pakhomova and v. a. saleev , phys . d * 65 * , 034013 ( 2002 ) [ arxiv : hep - ph/0105190 ] . a. czarnecki and k. melnikov , phys . b * 519 * , 212 ( 2001 ) [ arxiv : hep - ph/0109054 ] .

we formulate a general scheme to improve the truncated perturbative expansion in @xmath0 by means of the renormalization group in qcd for the single - scale quantities . the procedure is used for the evaluation of hadronic decay rates of @xmath1-lepton and @xmath2-charmonium . the scale dependence of result for @xmath2 is studied in the scheme of fixed value for the @xmath3-mass of charmed quark . = -1 cm = -2 cm

hep-ph0211261

a two - dimensional electron gas ( 2deg ) reacts sensitively to perpendicular magnetic fields . making the magnetic field inhomogeneous opens the door to a wide variety of fascinating effects and applications , ranging from magnetic superlattices @xcite and magnetic waveguides @xcite to hall sensors for magnetic nanostructures . @xcite one particularly simple magnetic field structure is the _ magnetic barrier _ , namely a perpendicular magnetic field configuration strongly localized along one in - plane direction and homogeneous in the second one . @xcite in a classical picture , magnetic barriers can be considered as selective transmitters that filter the electrons according to their angle of incidence . @xcite in a ballistic sample without edges , one would therefore expect that above a critical barrier height the barrier _ closes _ , all electrons are reflected and the resistance approaches infinity . recently , magnetic barriers have received renewed interest due to their potential applications as tunable spin filters and detectors , both of which are highly desirable spintronics devices . @xcite these theoretical works suggest that a high degree of spin polarization may be achievable with magnetic barriers in suitable materials . considering the elementary character and the simplicity of a magnetic barrier , it is surprising that only a few transport experiments on such structures have been reported . in @xcite , a magnetic barrier with a square profile has been experimentally realized in a highly sophisticated sample , namely a 2deg containing a graded step . it was observed that even for strong magnetic fields , the barrier resistance remains finite . the results of these experiments have been subsequently interpreted within a classical model @xcite , which shows that @xmath0 drift effects at the edge of the 2deg , as well as elastic scattering , limit the resistance to finite values . in all other experiments reported so far except ref . @xcite , the magnetic barrier has been generated in conventional ga[al]as heterostructures by magnetizing a ferromagnetic platelet , located on top of the sample , by an in - plane magnetic field . @xcite in such a setup , the magnetic barrier originates from the z - component of the stray field of the ferromagnet , see fig . this experimental implementation is also the basis for a significant fraction of the theoretical studies . @xcite + for an experimental implementation of the theoretical concepts , a detailed and quantitative understanding of the measured transmission properties of tunable magnetic barriers is needed . previous studies have already shown that both edge transmission and scattering in the barrier region are relevant . @xcite here , we build on these results and discuss in detail how the resistance of tunable magnetic barriers depends upon the @xmath1 drift at the edges , on the elastic scattering and on thermal smearing . in order to magnify these influences , we have prepared our ferromagnetic films from dysprosium which has a particularly large bulk saturation magnetization of @xmath2 . @xcite this allows us to drive the barriers well into the closed regime , where the transport through the structure is exclusively determined by the effects of interest here . in addition , a top gate was used to tune the electron density . these measurements are interpreted in a semi - classical picture based on the billiard model for ballistic conductors . @xcite we find that ( i ) the combination of both @xmath1 type edge drifts and elastic scattering in the barrier determines the barrier resistance , ( ii ) reasonable assumptions regarding the distribution of scattering angles for the elastic electron scattering lead to excellent agreement of the experimental data with the model , and ( iii ) thermal smearing has a marginal influence at liquid helium temperatures . the outline of the paper is as follows : in section ii , we describe the sample preparation , the experimental setup and the measurement methodology . the experimental results are presented in section iii , while the semi - classical model and its application to our measurements is the topic of section iv . the paper concludes with a summary and a discussion ( section v ) . a commercially available @xmath3 - heterostructure @xcite with a 2deg @xmath4 below the surface was laterally patterned by using optical lithography and subsequent processing steps . a hall bar geometry ( fig . [ mb1 ] ) was defined by wet chemical etching . au / ge ohmic contacts were defined at source and drain contacts and at the voltage probes 1 to 8 . a dysprosium ( dy ) platelet with a thickness of @xmath5 was defined at the heterostructure surface by dy thermal evaporation at a base pressure of @xmath6 . a cr / au gate layer of @xmath7 thickness was deposited on top to prevent the dy from oxidizing under ambient conditions and to allow the carrier density to be tuned . six samples were measured , and all showed qualitatively identical behavior . here , we discuss data taken from one representative sample . the samples were inserted in a liquid helium cryostat with a variable temperature insert that permits variation of the temperature between @xmath8 and room temperature . the sample stage is equipped with a rotatable sample holder , such that the magnetic field could be oriented within the x - z plane with an accuracy better than @xmath9 degrees . is highly localized in the x - direction . also shown is the x - component of the fringe field . ] the carrier densities and the electron mobility were determined from conventional four - probe measurements of the components of the resistance tensor , @xmath10 and @xmath11 in perpendicular magnetic fields . the ungated electron density is @xmath12 , and the mobility at a temperature of @xmath8 is @xmath13 , corresponding to a drude scattering time of @xmath14 and an elastic mean free path of @xmath15 . the quantum scattering time was determined from the envelope of the shubnikov - de haas oscillations @xcite as @xmath16 . the vanishing of the hall voltage was furthermore used to detect the parallel magnetic field configuration . at the maximum magnetic field @xmath17 of @xmath18 used in our experiments , we estimate that the maximum external perpendicular magnetic field component is below @xmath19 . strong parallel magnetic fields are well known to affect the transport properties of 2degs by modifying the density of states and the interactions . @xcite in addition , the electron effective mass becomes slightly anisotropic . @xcite these effects show up as a weak and approximately parabolic magnetoresistivity . increasing b also magnetizes the dy film along the x - direction . the z - component of the fringe field at the 2deg is strongly localized at the edge of the dy in the x - direction and forms the magnetic barrier . @xcite the x- and z- components of the fringe field are shown in fig . [ mb1](c ) for the literature value of the saturation magnetization of dy . assuming that the fringe field follows the corresponding analytic expressions @xcite , @xmath20 is given by @xmath21 where @xmath22 is the distance of the 2deg from the surface and @xmath23 denotes the thickness of the dy film , see fig . [ mb1](c ) . this relation neglects the second magnetic barrier residing at contacts 4 and 8 , which is justified since it is sufficiently far away from the region probed between contacts 2 and 3 . edge roughness of the magnetic film may also lead to deviations from eq . . we characterized the dy edge by atomic force microscope measurements and found an edge roughness ( single standard deviation ) of @xmath24 , which is smeared out to a large extent at the 2deg . we therefore neglect the edge roughness in the following . the magnetization in the x - direction as a function of @xmath17 is denoted by @xmath25 , which can be estimated from hall measurements on a magnetic barrier @xcite as described below . the x - component of the fringe field has a much smaller effect on the 2deg and is moreover small compared with the b field required to establish saturation ( i.e. @xmath26 , see the inset in fig . [ mb2 ] ) . it is therefore neglected in the following . a current of @xmath27 at a frequency of @xmath28 is passed from source to drain . the barrier resistance is obtained from the voltage measured between contacts 2 and 3 ( fig . 1 ) with a lock - in amplifier . the hall voltage measured between contacts 4 and 8 is used to determine @xmath25 . we assume in the discussion below that the magnetic barriers at both edges of the dy platelet differ only by their sign . figure 2 shows a typical magnetoresistance trace @xmath29 of our samples , measured on sample a. the traces are hysteretic , reflecting the magnetization characteristics of the dy film ( inset ) . at the coercive magnetic fields at @xmath30 , @xmath29 has minima which equal to high accuracy the magnetoresistance outside the dy film measured over an identical distance . this shows that the micromagnetic structure in the dy film which becomes most relevant around the coercive magnetic field @xcitehas no noticeable effect in our experiments . @xmath29 increases as one goes away from the coercive field , but neither saturates nor approaches infinity , even well above the saturation magnetic field . the slope @xmath31 above @xmath32 is only partly due to the parabolic background . as discussed in more detail below , if our barrier were fully ballistic and without edges , it would close at @xmath33 away from the minimum of @xmath10 . thus , in 88% of the magnetic field interval scanned , the transmission is governed by the edge and scattering effects of interest . note that the @xmath29 traces are slightly asymmetric around their minimum . we attribute this effect to the proximity of the voltage probes ( @xmath34 ) to the magnetic barrier , over which the electrons ejected from the barrier may not yet form a fermi sphere , even though the probes are about 8 elastic mean free paths away from the barrier . similar effects have been observed by leadbeater et al . @xcite and subsequently been explained in detail by ibrahim et al . @xcite . the hall resistance of the magnetic barrier ( inset in fig . [ mb1 ] ) measures @xmath20 , averaged over the spatial extension @xmath35 in the x - direction of the hall probe contacts 4 and 8 , i.e. over @xmath36 , according to @xmath37 here , @xmath38 represents a hall factor which may deviate from 1 , depending on the sample geometry and the mean free path . @xcite for our structure , @xmath39 is expected @xcite , and we have no reason to assume otherwise , in contrast to the findings reported in ref . @xcite . here , we have assumed that the magnetic barrier on top of our hall cross is adequately described by the ballistic model , even though the mean free path is smaller that the width of the voltage probes . since , however , the fwhm of our barrier in the closed regime is no larger than @xmath40 and thus much smaller than the mean free path , and the maximum magnetic field multiplied by the electron mobility @xmath41 for our system in the closed regime , the diffusive model developed in ref . @xcite does not strictly apply as well , @xcite while to the best of our knowledge , a model for the intermediate regime is not available . for @xmath42 , we determine from the measured @xmath43 a saturation magnetization of @xmath44 for our dy films . this is significantly below the literature value for bulk dysprosium . we attribute this reduction to the embedding of oxygen into the dy film during metallization , as well as to the granularity of the film.@xcite this interpretation is supported by our observation of @xmath45 dropping over time in samples where the dy films is not covered by a cr / au layer . figure [ mb3 ] reproduces the behavior of the barrier magnetoresistance as the temperature is changed . as the temperature is increased , the magnetic barrier resistance is reduced . at the same time , the shape of the barrier becomes more nearly triangular and the hysteresis decreases . simultaneously , the resistance minima increase and are shifted to smaller absolute values of @xmath17 . as will be explained in more detail below , this behavior can be understood in terms of a combination of increased scattering and a reduced coercivity of the dy film as the temperature is increased , while the thermal smearing of the fermi function plays a marginal role . a slight hysteresis is observed even above the literature value for the curie temperature of bulk dy , @xmath46 . @xcite the enhancement of the curie temperature is a second indication of crystal imperfections in our dy films . our numerical approximation of the magnetic barrier resistance is exemplified using the data of fig . the analysis is based upon the billiard model for quasi - ballistic conductors @xcite and the landauer - bttiker formalism @xcite . electrons are injected into the barrier region starting from a fixed x - position @xmath47 to the left of the barrier at random positions in the y - direction across the hall bar of width @xmath48 . they start out with the fermi velocity , while their directions are arbitrarily distributed over all angles with positive x - component . we solve the differential equations describing the classical motion of the electrons to obtain their trajectories until either it is rejected by the barrier and passes the starting line in the opposite direction , or until it travels through the barrier and reaches the x position @xmath47 to its right . at this distance , @xmath49 is negligible . injecting the electrons at larger distances does not modify the results . the edges of the hall bar have been incorporated by constant , reflective electric fields in the regions @xmath50 . strong electric edge fields generate specular reflection . the magnetic barrier is incorporated as given by eq . in the x - direction and homogeneous in the y - direction . we have introduced scattering in the 2deg by assuming scattering after time of flights which obey a poisson distribution with a time constant of @xmath51 . the electrons in a 2deg in a modulation - doped ga[al]as heterostructure are predominantly scattered at the ionized donors in the doping layer , and the corresponding _ screened coulomb scattering _ is known to form an approximately gaussian distribution of scattering angles.@xcite we therefore assume a gaussian distribution of scattering angles @xmath52 with a standard deviation of @xmath53 , centered at @xmath54 and limited to @xmath55 . within our model , the times of flight between two subsequent scattering events form a poisson distribution with an expectation value of @xmath56 . at a scattering event , the angles between the initial and the final electron velocity vector direction are changed according to the distribution function described above . these two distributions reproduce the experimentally determined values for @xmath51 and @xmath57 . + we remark that our simulation results are rather insensitive to the chosen distribution function of scattering angles . we have also used a rectangular distribution , namely a constant probability for scattering angles @xmath58 and a probability of zero for larger angles . even though we find slightly higher values for the resistance ( about 2.5 % ) at small fields , , the same values are found in the closed regime . for each magnetic field , 40000 electrons of fermi energy @xmath59 are injected . the transmission is determined by @xmath60 here , @xmath38 is the angle between the x - direction and the direction in which the electron is injected and @xmath61 is either @xmath62 or @xmath63 depending on whether the electron with the corresponding initial conditions is transmitted or not . we note that the carrier density under the dy film may differ from that underneath the cr / au gate , as a consequence of different schottky barriers . hall measurements at voltage probes 1 and 5 , as well as at 3 and 7 , respectively , indicate that the electron density under the dy is roughly 5% larger as compared with the density measured outside , but this value is ambiguous since the z - component of the dy fringe field superimposes on the homogeneous magnetic field in the z - direction and we were unable to separate these two contributions to the modified hall resistance.@xcite we have neglected this density step in our simulations . the conductance is given by @xmath64 where @xmath65 is the number of modes in our hall bar and @xmath66 is the fermi wavelength . we note that according to eq . [ eq4 ] , the contact resistance ( @xmath67 ) between an infinitely extended 2deg and the hall bar does not contribute to @xmath10.@xcite from eq . , the longitudinal resistance @xmath10 across the barrier for a given carrier density is readily obtained from @xmath68 . as a test simulation , we have turned off all scattering and set the electric field at the edges to zero , thereby simulating a ballistic magnetic barrier which extends to infinity in the y - direction . in this case , the numerical results closely match the corresponding analytical expression @xcite and reproduce the critical angle of incidence for which the magnetic barrier closes to an accuracy of 1 degree . + with the experimental trace of fig . 2 ( full line ) , corrected for the magnetoresistivity of the 2deg which is set to its constant value at @xmath69 . full circles : @xmath10 for a barrier with edges ( edge electric field @xmath70 ) and no scattering ; triangles : @xmath10 for the barrier with no edges but scattering according to the experimentally determined scattering times ; open circles : barrier resistance with both scattering and edge electric field ; open squares : @xmath10 for the structure with edges and a small quantum scattering time of @xmath71 . inset : typical calculated trajectories in the closed regime : scattering inside the magnetic barrier ( trace 1 ) as well as @xmath0 - drifts by edge electric fields ( trace 3 ) are responsible for a finite resistance . ( b ) conductance of the magnetic barrier as a function of electron energy for several values of @xmath72 including edges and scattering . large symbols in the figure correspond to the critical energy for which an infinitely extended barrier in a ballistic system will close ( the corresponding energy dependent conductance for @xmath73 is shown in the inset ) . ] for a comparison of the simulations with the experiments , the magnetization trace @xmath25 shown in the inset of fig . [ mb2 ] is used to map the height of the magnetic barrier onto the experimental value of @xmath17 . results of the simulations are represented in fig . [ mb4](a ) . most significantly , the addition of the edge electric fields to the ballistic system limits the resistance in the closed regime to finite values . some electrons that would be reflected at the barrier away from the edges are pulled through the barrier at the edges due to the @xmath1 drift , see the inset in fig . [ mb4](a ) . this effect is the sole reason for a finite resistance in the closed regime , as long as scattering is disregarded . the barrier resistance decreases as the edge electric field is increased . in our simulations , we have assumed an electric field of @xmath70 . this value can be considered an upper limit , based upon measurements of the steepness of the confining walls @xcite and in agreement with the consideration that a potential change of the order of @xmath74 can not occur over a length smaller than the screening length . @xcite we note that the simulated barrier resistance is only weakly dependent on the strength of the edge electric field . + for an infinitely extended closed magnetic barrier in a disordered system , the resistance is also kept finite by scattering events in the barrier region , see fig . [ mb4](a ) . this is again illustrated by the characteristic trajectories , see the inset in fig . [ mb4](a ) : a scattering event may redirect an electron which in the absence of scattering would be rejected by the barrier . for elastic mean free paths comparable to or below the spatial extension of the barrier , the barrier becomes unimportant ( open squares in fig . [ mb4](a ) , where the simulated elastic mean free path was @xmath75 ) . + in comparison with the experimental data in fig . [ mb2 ] ( a ) , we observe that the numerical trace for @xmath29 that takes only the scattering into account but disregards edge effects disagrees significantly with the experiment . here , the measured background magnetoresistance ( fig . [ mb2 ] ) has been replaced by its value at @xmath69 . the shape of the measurement trace is reproduced , but its absolute value differs by up to 30% . we point out that in this simulation , the only adjustable parameter is the distribution of scattering angles under the constraints set by the measured values for @xmath57 and @xmath51 , which is determined by the details of the disorder potential landscape . from this separate discussion of the two mechanisms , it emerges that a combination of both edge transmission and scattering - induced transmission determines @xmath29 . in fact , inclusion of both elastic scattering in accordance with the measured scattering times , as well as an electrostatic edge field of @xmath76 , gives a very good reproduction of the measured trace , see fig . [ mb4 ] ( a ) . we refrain from fitting the experimental data since further uncertainties may have an influence on this level of accuracy . first , there is a slight asymmetry of the measured traces , which we attribute to asymmetries in the voltage probe geometry.@xcite second , the shape of the magnetic barrier may deviate from eq . , and it may be inhomogeneous in the y - direction . furthermore , the electron density below the dy deviates from that below the cr . this effect could in principle be avoided by preparing a thin , homogeneous metal electrode between the semiconductor surface and the ferromagnet . + in order to investigate the influence of thermal smearing , we have calculated the energy dependent conductance @xmath77 by varying the energy in eq . , from which the conductance at non - zero temperatures is obtained via @xmath78 where @xmath79 denotes the fermi - dirac distribution function . [ mb4 ] ( b ) shows that the simulated @xmath77 is a nearly linear function of the electron energy . according to eq . , @xmath80 becomes independent of temperature for @xmath81 . a similar relation is also approximately found within an analytic treatment of the infinitely extended , open magnetic barrier in a ballistic 2deg @xcite , see the inset in fig . [ mb4 ] ( b ) . hence , the inclusion of both edge effects and scattering does not change this insensitivity of the magnetic barrier resistance to thermal smearing . we conclude , therefore , that the changes of @xmath29 with temperature are , besides the temperature dependence of @xmath82 , mainly due to the temperature dependence of the scattering times . in our experiments , we find that both scattering times are constant up to @xmath83 , and in addition , we do not see significant changes in @xmath29 , see fig . [ mb3 ] , while for larger temperatures , the observed shubnikov - de haas oscillations no longer allow a meaningful determination of @xmath51 . hence , a reasonable approximation of the measurements at higher temperatures requires more detailed information regarding the scattering times and the distribution of scattering angles than available from our experiments . we have furthermore studied numerically the effect of zeeman splitting on @xmath29 ( using an effective g - factor of -0.44 ) due to which the two spin directions acquire different fermi energies and therefore different partial conductances , resulting in a spin polarization of the current . our simulations suggest that the influence of the spin splitting on @xmath29 is marginal . also , it is found numerically that the spin polarization of the current increases with increasing barrier height in the closed regime , but is below @xmath84 for all magnetic fields . however , this value is about a factor of 5 larger than the simulated values for magnetic barriers without edges , and we conclude that edge transmission tends to increase the spin polarization . even accounting for this increase , the effect for a 2deg in ga[al]as remains very small . finally , we have also incorporated magnetic mass effects induced by the strong parallel magnetic field @xcite and find that they are negligible in our parameter range . we have studied the resistance of magnetic barriers defined in ga[al]as heterostructures in the quasi - ballistic regime as a function of in - plane magnetic fields . we have also described the system numerically within a semiclassical model , and we find that the finite resistance observed in the closed regime originates from both elastic scattering in the barrier region and from transmission via @xmath0 drifts at the edges of the hall bar . by using the scattering times as extracted from the experiment , a very good agreement between measurement and simulation is obtained , especially given the uncertainties involved regarding the exact shape and homogeneity of the magnetic barrier . furthermore , the barrier magnetoresistance is insensitive to thermal smearing , spin polarization and magnetic mass effects . the results also show how the resistance change induced by the magnetic barrier can be increased , which may be of importance if one wishes to observe quantum effects such as resonant tunneling @xcite or spin polarization . first of all , both larger mobilities and hall bars of reduced width will reduce the scattering in the barrier region and thereby increase the barrier resistance . also , defining soft edges reduces in principle the transmission via @xmath0 drift effects ; our simulations however suggest that very soft edges with edge fields in the range of @xmath85 are required to obtain a noticeable effect . finally , the deposition of clean ferromagnetic films under ultra high vacuum conditions should enhance the saturation magnetization almost up to a factor of two in our samples . our model can easily be extended to describe more complicated magnetic barrier structures , for example those suggested recently for use as tunable spin filters.@xcite + the authors acknowledge the careful reading of the manuscript by matthew jenkins , stimulating discussions with hengyi xu as well as financial support by the _ heinrich - heine - universitt dsseldorf _ .

strong magnetic barriers are defined in two - dimensional electron gases by magnetizing dysprosium ferromagnetic platelets on top of a ga[al]as heterostructure . a small resistance across the barrier is observed even deep inside the closed regime . we have used semiclassical simulations to explain this behavior quantitatively in terms of a combined effect of elastic electron scattering inside the barrier region and e x b drift at the intersection of the magnetic barrier with the edge of the hall bar .

cond-mat0701109

spectra of type 1 agn show a diversity of broad and narrow emission lines that provide direct insights into the structure and kinematics of photoionized , and otherwise excited , gas in the vicinity of the putative central massive object . broad emission lines , like much studied h@xmath0 ( e.g. , * ? ? ? * hereafter z10 ) , are thought to arise in or near an accretion disk acting as the fuel reservoir for the central supermassive black hole ( log m@xmath1 m@xmath2 ) . h@xmath0 shows a diversity of line widths as well as profile shifts and asymmetries @xcite . despite this diversity some systematics have emerged and are best highlighted via the concept of two type 1 agn populations @xcite . population a show the smallest broad - line widths fwhm h@xmath0=1000 - 4000 and includes the narrow line seyfert 1 ( nlsy1 ) sources ( fwhm @xmath3 2000 ) . a h@xmath0 profiles are currently best fit by a single lorentz function . population b sources show fwhm h@xmath0=4000 - 12000 and require two gaussians ( one unshifted and one redshifted ) for a reasonable profile description . `` broad - line '' h@xmath0 profiles as narrow as fwhm = 500 @xcite and as broad as fwhm = 40000 @xcite have been found . a is predominantly radio - quiet while pop . b involves a mix of radio - quiet and the majority of radio - loud quasars . broad- and narrow - line profile shifts are known and the phenomenology can be confusing . narrow emission lines like [ oiii]5007 are regarded as a reliable measure of the local quasar rest frame except in the case of `` blue outliers '' , usually found in sources with fwhm h@xmath0= 1500 - 3500 and weak [ oiii ] @xcite . blue outliers show [ oiii ] blueshifts as large as @xmath41000 . no pop . b sources with blueshifted [ oiii ] are known at low z ( or luminosity ) . careful use of [ oiii ] and h@xmath0 narrow line as rest frame measures suggests that broad h@xmath0 in pop . a sources rarely shows a systematic red or blue shift above the fwhm profile level . a blueshifted component or asymmetry is observed in some extreme feii strong pop . a sources @xcite . b sources show more complex line shift properties . the h@xmath0 profile usually shows two components : 1 ) a `` classical '' broad component ( bc ; fwhm = 4000 5000 ) with zero or small ( red or blue ) shift , and 2 ) a very broad ( vbc ; 10000 ) and redshifted ( @xmath51000 ) component . composites involving the 469 brightest sdss - dr5 quasars suggest that these two components represent the underlying stable structure of h@xmath0 in pop . b sources . broad feii emission has been found in type 1 quasars since the era of photographic spectroscopy in the 60s . feii emission blends are almost ubiquitous in a sample of the brightest ( usually highest s / n ) sdss quasars ( z10 ) . circumstantial evidence has accumulated supporting the assumption that feii emission arises in or near the emitting clouds that produce other low ionization lines like h@xmath0 ( see e.g. , @xcite ) . fwhm feii appears to correlate with fwhm h@xmath0 over the full range where feii can be detected ( fwhm=1000 - 12000 ) . this can be clearly seen at low @xmath6 by observing the shape ( e.g. , smoothness ) of the feii 4450 - 4700 blue blend ( and the feii multiplet 42 line at 5018 ) near [ oiii]5007 . in pop . a sources the blend resolves into individual lines while it becomes much smoother in pop . b sources . sources with the strongest feii emission also show a weakening of h@xmath0 emission as expected if the latter is collisionally quenched in the same dense medium where strong feii emission can be produced @xcite . obviously systematic line shifts place important constraints on models for the geometry and kinematics of the broad line region . the most famous example involves a systematic blueshift of high ionization lines ( e.g. , civ 1549 ) relative to low ionization lines ( e.g. , balmer ) especially in pop . a sources ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? evidence was recently advanced ( * ? ? ? * hereafter h08 ) for the existence of a _ systematic _ redshift of feii relative to [ oiii]5007 ( and hence the balmer lines ) in a majority of type 1 quasars . this result , along with a narrower estimated feii line width , has been ascribed to feii emission arising in a region with dynamics dominated by infall and located at larger radius than the region producing the bulk of h@xmath0 . h08 argue that the amplitude of the shifts correlates inversely with source eddington ratio ( l / l@xmath7@xmath8@xmath9 ) . interpretations for such an feii redshift have already appeared @xcite reflecting the potential importance of such a first - order kinematic signature . having worked on line spectra and profile shifts for many years we were surprised by the h08 claims and decided to test the hypothesis of a systematic feii redshift . could we have missed it ? first let us consider what we know . a quasars show relatively symmetric unshifted lorentz - like h@xmath0 profiles with fwhm@xmath34000 . in our work using the brightest ( @xmath10 17.5 or @xmath11 17.5 ; @xcite ) sdss dr5 quasars we processed spectra for @xmath4260 pop . a sources ( from a sample of 469 quasars ; z10 ) and we found no evidence for a systematic shift of feii lines relative to h@xmath0 or . such an feii shift should be easiest to detect in the brightest pop . a sdss spectra with narrowest broad - line profiles and strongest feii emission . it is immediately suspicious that more and larger feii redshifts are claimed for pop . b sources . in only one pop . a source in our sample sdss j0946 + 0139 do we find a large h@xmath0 peak ( 90@xmath12 intensity level ) redshift of 1830 . this source is similar to oq208 ( @xcite and discussed in h08 ) which shows @xmath13 @xmath142000 . sdss j0946 is the only pop . a source with a large feii redshift in our z10 sample ( 1/260 ) . z10 found 19 quasars with an h@xmath0 peak ( 9/10 fractional intensity ) blueshifted more than -320 and 4 sources with the peak redshift more than + 320 . the remaining 241 pop . a sources showed no significant h@xmath0 peak shift ( figure 8 of z10 ) . best feii template fits to these sources show no significant difference in centroid redshift between feii and h@xmath0 . there are two possible causes of small and spurious h@xmath0 ( or feii ) shifts : 1 ) host galaxy contamination and 2 ) blue outliers.except in rare cases host galaxy contamination is unlikely to induce systematic redshifts with the amplitudes reported by h08 . extreme blue outliers with [ oiii ] blueshifts in the range 400 - 1000 are rare and therefore can not be the cause of the large and systematic shifts reported in h08 . in fact h08 selection criteria rejected sources likely to be seriously affected by 1 ) or 2 ) . h08 chose 4000 + sources from sdss dr5 with computed s / n @xmath15 10 . z10 also used dr5 where @xmath494% of sources show s / n @xmath15 10 . our sample was magnitude - limited with a slightly shallower redshift upper limit ( z=0.7 instead of 0.8 ) . why do we reach different conclusions about feii shifts ? a big part of the answer could involve how s / n was computed . compute s / n over the range 44305500@xmath16 . this procedure overestimates the quality of the data because it includes major emission lines in the computation . we compute s / n in the range 5600 - 5800@xmath16 , which is free of strong lines and represents as close as one can approach to an estimate of continuum s / n near h@xmath0 . using our range the h08 sample shows mean and median s / n values of 10.6 and 7.4 , respectively ; approx . 65% show s / n @xmath3 10 . we find that only 182 spectra of our bright sample are included in the h08 s . the majority of the h08 s sources are lower s / n than those in our sample . one can not estimate reliable feii line shifts using individual sdss spectra for sources fainter than about g@xmath417 - 17.5 . in rough order of importance our studies indicate that the accuracy of feii shift measures depends on : 1 ) feii strength and feii / h@xmath0 profile widths , 2 ) spectral s / n and 3 ) if estimates depend heavily on fits to the 4430 - 4680 blend , strength of heii 4686 emission . typical individual spectra used by h08 _ show _ too low s / n to allow convincing conclusions about feii shift and width typical parameter uncertainties for individual sources are much larger than the ones connected with our high s / n composites ( for a typical a2 source with s / n@xmath1720 uncertainties of shift estimates are larger than @xmath181500 ) . individual source spectra with large quoted feii redshift and s / n near the sample median were extracted from the h08 sample and specfit modelled . using an feii template with fixed shifts ranging from zero up to the largest values quoted by h08 , @xmath19 can not distinguish between zero and e.g. , 1000 redshift in the majority of the sources . the best recourse is to generate high s / n composite spectra . h08 argue that one can not confirm or refute the existence of a systematic feii redshift using composite spectra because of the large dispersion of fwhm , shifts and flux values for both h@xmath0 and feii . this is likely true for composites generated from random subsamples of sources but not true if one generates composites over more limited ranges of parameter values . one can generate binned composites over limited ranges of fwhm h@xmath0 and feii strength using the 4de1 formalism ( @xcite ; z10 ) . 4de1 bins a2 ( fwhm h@xmath0= 1000 - 4000 , 0.5 @xmath20 @xmath20 1.0 ) and b1 ( fwhm h@xmath0=4000 8000 , @xmath20 0.5 ) are of particular interest because they include the largest numbers of sources in random samples . specfit analysis ( @xcite ; details in @xcite ) of an a2 median composite involving n = 130 z10 sources ( s / n 90 ) gives a best - fit consistent with zero feii redshift . the situation for the b1 composite ( n=131 sources from z10 ; s / n 110 ) is less constraining because lines are broader and feii weaker . table 1 reports feii template shifts and 2@xmath21 uncertainties for specfit tests discussed in this paper . we also report peak shifts of bc extracted from the best specfit solutions along with `` core '' shifts measured at the centroid of the line peak after nc subtraction . in no case do we find a significant shift between feii and the rest frame or between feii and . we also do not find any significant feii shifts if we restrict to sources with l / l@xmath7 ratio @xmath22 ( h08 suggested the shifts might be largest for low l / l@xmath7 sources ) . since we find no evidence for systematic feii redshifts in our z10 bright quasar sample composites it is useful to generate feii shift composites using the h08 sample . we generate them within the 4de1 context thereby restricting the ranges of fwhm h@xmath0 , feii relative strength ( and likely also fwhm feii ) for each composite . since the distribution of feii shifts shown in h08 is continuous we focus on the sources with largest quoted shift values . if these shifts are not confirmed then smaller shifts are even less likely to be real . we therefore focus on constructing median composites for all h08 sources falling in 4de1 bins a2 and b1 with h08 feii redshift estimates @xmath23 ( figure 1 ) . two composites were constructed for each spectra bin : 1 ) one with no restriction on feii width ( h08 do not constrain fwhm feii in their template fits so it is sometimes very different from fwhm h@xmath0 ) and 2 ) one with feii width constrained to the fwhm range of h@xmath0 in a particular bin ( i.e. , @xmath24 for a2 and @xmath25 for b1 ) . upper and lower panels of figure 1 show bins a2 ( fwhm h@xmath0 @xmath3 4000 ) and b1 ( fwhm h@xmath0=4000 - 8000 ) , respectively ( n=156 for bin a2 and n=240 for bin b1 ) . the s / n @xmath4 55 - 60 for both composites . spectra show best - fit specfit models superimposed . the left and center panels involved feii templates fixed to the best fit and 1500 shifts , respectively . our template prescription is described in @xcite graphical results for the best - fits are shown in the right panels of figure 1 . fits were performed over the range @xmath17 4470 5450 , where feii and continuum emission account for the total flux making it the safest region for normalized @xmath26 computations . @xmath26 values are shown for the range of adopted feii shifts . in order to estimate confidence intervals we considered a set of fits with displacements @xmath27= + @xmath28 , for integer @xmath29 , along with the best fit and a few additional @xmath27 cases in proximity to the minimum @xmath26 . one can see a clear preference for zero or near - zero fits . the significance of @xmath26 variations is described by @xmath30 statistics appropriate for ratios of @xmath26 values @xcite . given the large number of degrees of freedoms in the sampling range ( 4500 4630 , 5040 5090 , 5310 5360 ) any @xmath26 differences between two fits become significant at a 95% confidence level if @xmath31 . the @xmath26 values indicate that zero shift and `` best shift '' values in table 1 are not significantly different . all fits involving shifts @xmath5 500 are statistically significant . the middle panel of figure 1 upper row demonstrates visually that the fit with @xmath13 = + @xmath32 ( and even more the fits with larger displacement ) do not reproduce the observed feii emission . both the residuals and @xmath19 results rule out any systematic redshift for at least half of the h08 sample ( pop . note especially the fits to the two relatively isolated multiplet 42 feii lines between h@xmath0 and [ oiii]4959 and on the red wing of [ oiii]5007 . the redshifted fit fails to include the blue side of the 4450 4700 blend and the red side is confused by the frequent presence of heii 4686 . the latter line is not mentioned in the h08 study leaving us to conclude that it was not included in their fits . it can certainly give the impression of a redshift of the feii blue blend , which is the most useful feii diagnostic in the optical spectra of low redshift quasars ( the red feii blend is frequently affected by coronal lines as well as mgii host galaxy absorption in lower redshift sources ) . pioneering principal component analysis of the bqs survey @xcite found that heii4686 equivalent width anticorrelates with sources luminosity ( it is eigenvector 2 ) . there is a tendency for the h08 sources with largest feii redshifts to favor a smaller and lower ( @xmath33 ) range of source luminosity than those with near zero shifts ( @xmath34 ) . thus the effect of heii will tend to play a larger role in the sources where the largest feii redshifts have been found . show composite spectra for five bins of feii redshift in their figure 12 . the three bins involving largest feii redshift sources show a prominent heii signature that , if not subtracted , will increase the apparent significance of any assumed feii redshift . only the bin involving sources with no feii redshift ( within the uncertainties ) shows no evidence of heii emission . figure 9a of h08 suggests that a larger fraction of quasars with fwhm h@xmath35 4000 ( population b ) show large feii redshifts . the lower panels of our figure 1 show specfit models superimposed on a b1 median composite constructed from all h08 sources with quoted feii redshift greater than 1000 . the situation is certainly more ambiguous than for the a2 composite . it is hard to identify individual feii features . lines are broad , feii is weak and under these conditions there are serious limitations on the reliability of fwhm and shift estimates for feii ( cf . fig . 3 of @xcite ) . the same analysis as done for a2 composite shows much poorer constraints on the feii shift . the best fit yields @xmath13 @xmath36 but is not distinguishable from a zero shift solution . if one actually computes @xmath26 values over the ranges 4474 4640 , 5040 5105 , 5320 5400 , the @xmath26 monotonically increases from 0 shift ( figure 1 , lower rightmost panel ) , although the increase remains insignificant until @xmath17 1100 , where @xmath37 . b1 feii is too faint and the lines are too broad to make inferences about line shifts and widths . the claim of large feii shifts are not , and can not be , confirmed . @xcite recently report an feii study of sdss quasars and any feii redshifts they measure ( their figure 16 ) are much smaller than those reported by h08 ( the average feii shift relative to the narrow lines is 100 @xmath18 240 ) . returning to our previous list of major sources of uncertainty for feii shift and fwhm estimates leads us to suggest that low spectral s / n and above average heii strength are the culprits . the fit to the 4430 4680 blue blend drives the best fit @xmath19 results . the exclusion of heii4686 from the h08 fits likely results in a tendency for heii to `` redshift '' the blue feii blend . this effect in a typically low luminosity sample , where heii is stronger than average , likely drove the conclusion that feii was systematically redshifted . we tested this conclusion omitting the heii line from our fits to the bin a2 and b1 composites generated from the h08 sample . feii shifts in lines 2 and 5 of table 1 increase from -60 to + 770km / s and from 730 to 1570km / s , respectively . the more constraining a2 results suggest that heii can produce the entire systematic redshift claimed by h08 . we _ do not _ confirm large feii redshifts relative to narrow [ oiii ] and broad h@xmath0 emission in type 1 agn but can not rule out the existence of small red ( or blue ) shifts in particular subsamples . fitting median composites built from spectra with large claimed feii shifts ( @xmath23 ) indicates small shifts with an upper limit @xmath17 300 for bin a2 . in the case of b1 the best fit suggests @xmath17700 but the shift is very poorly constrained . in both cases the shifts are not significantly different from 0 . these results do not support the origin of feii emission from a dynamical disjoint region from the one emitting the broad core of . our result also challenges the usefulness of feii shift as orientation parameter . small systematic shifts of feii with respect to the rest frame seem plausible but a reliable analysis is possible only on spectra of high s / n ratio . lcccccc a2 - z10 & -40 @xmath18 90 & 10 & 0 & @xmath38 & 130 + a2 - h08 , @xmath39 & -60 @xmath18 400 & 80 & 90 & 70 & 156 + a2 - h08 , @xmath40 & -160 @xmath18 375 & 100 & 10 & 45 & 194 + b1 - z10 & -340 @xmath18 400 & -150 & -80 & @xmath38 & 131 + b1 - h08 , @xmath39 & + 730@xmath41 & 0 & 100 & @xmath38 & 240 + b1 - h08 , @xmath40 & + 180 @xmath18 450 & -80 & 40 & @xmath38 & 410 + b1 - h08 @xmath42 z10 , @xmath40 & -150 @xmath18 470 & -170 & -140 & 45 & 22 +

we test the recent claim by hu et al . ( 2008 ) that feii emission in type 1 agn shows a systematic redshift relative to the local source rest frame and broad - line h@xmath0 . we compile high s / n median composites using sdss spectra from both the hu et al . sample and our own sample of the 469 brightest dr5 spectra . our composites are generated in bins of fwhm h@xmath0 and feii strength as defined in our 4d eigenvector 1 ( 4de1 ) formalism . we find no evidence for a systematic feii redshift and consistency with previous assumptions that feii shift and width ( fwhm ) follow h@xmath0 shift and fwhm in virtually all sources . this result is consistent with the hypothesis that feii emission ( quasi - ubiquitous in type 1 sources ) arises from a broad - line region with geometry and kinematics the same as that producing the balmer lines .

1203.5992

recent developments in photoelectron spectroscopy have challenged the apparent simple truth that the fermi surface of cuprate superconductors is simply the one corresponding to lda band structures with the only effect of the closeness to the mott - hubbard insulator being a moderate correlation narrowing of the band width . the discovery of the ` shadow bands'@xcite , the temperature dependent pseudogap in the underdoped state@xcite and the substantial doping dependence of the quasiparticle band structure@xcite leave little doubt that a simple single - particle description is quite fundamentally inadequate for these materials . moreover , photoemission experiments on one - dimensional ( 1d ) copper oxides@xcite have shown very clear signatures of spin charge separation . the equally clear nonobservation of these signatures in the cuprate superconductors at any doping level advises against another apparent simple truth , namely that the fermi surface seen in the cuprates is simply that of the ` spinons ' in a 2d version of the tomonaga - luttinger liquid ( tll ) realized in 1d . motivated by these developments , we have performed a detailed exact diagonalization study of the electron removal spectrum in the 1d and 2d @xmath0@xmath1@xmath2 model . this model reads @xmath7 there by the ` constrained ' fermion operators are written as @xmath8 and @xmath9 denotes the spin operator on site @xmath10 . the summation @xmath11 extends over all pairs of nearest neighbors in a 1d or 2d square lattice . + the electron removal spectrum is defined as @xmath12 denote the ground state energy and wave function . for small finite clusters , this function can be evaluated numerically by means of the lanczos algoritm@xcite . + in 1d the @xmath0@xmath1@xmath2 model is solvable by bethe ansatz in the case @xmath2@xmath13@xmath14@xcite , but even for this limit the complexity of the bethe ansatz equations precludes an evaluation of dynamical correlation functions . for the closely related hubbard model in the limit @xmath15@xmath16@xmath17 the bethe - ansatz equations simplify@xcite , and an actual calculation of the spectral function becomes possible@xcite . in all other cases lanczos diagonalization is the only way to obtain accurate results for @xmath18@xcite . + in order to analyze our numerical results , we first want to develop an intuitive picture of the scaling properties of the elementary excitations in 1d , which will turn out to be useful also in 2d . it has been shown by ogata and shiba@xcite that for @xmath15@xmath16@xmath17 the wave functions can be constructed as products of a spinless fermion wave function , which depends only on the positions of the holes , and a spin wave function , which depends only on the sequence of spins . a naive explanation for this remarkable property is the ` decay ' of a hole created in a nel ordered spin background into an uncharged spin - like domain wall , and a charged spinless domain wall . then , since it is the kinetic energy @xmath19@xmath0 which propagates the charge - like domain walls , whereas the exchange energy @xmath19@xmath2 moves the spin - like domain walls , one may expect that the two types of domain walls have different energy scales . namely the excitations of the charge part of the wave function ( i.e. , the ` holons ' ) have @xmath0 as their energy scale , whereas those of the spin part ( i.e. , the ` spinons ' ) have @xmath2 as their energy scale . scanning the low energy excitation spectrum of 1d @xmath0@xmath1@xmath2 rings then shows that indeed most of the excited states have excitation energies of the form @xmath20@xcite , which indicates the presence of two different elementary excitations with different energy scales . + surprisingly enough the low energy spectrum of the 2d model shows the same scaling behavior of the excitation energies as in 1d@xcite , which seems to indicate the existence of two types of spin and charge excitations if very different nature also in this case . other cluster results indicate , however , that these two types of excitations do not exist as ` free particles ' : the dynamical density correlation function , which corresponds to the ` particle - hole excitations ' of holons and shows sharp low energy peaks in 1d@xcite is essentially incoherent in 2d and has practically no sharp low energy excitations@xcite . the optical conductivity in 2d shows an incoherent high energy part with energy scale @xmath2@xcite - which is completely unexpected for the correlation function of the current operator which acts only on the charge degrees of freedom . there is moreover rather clear numerical evidence@xcite that the hole - like low energy excitations can be described to very good approximation as spin @xmath21 ` spin bags'@xcite - i.e. , holes dressed heavily by a local cloud of spin excitations . + to obtain further information about similarities and differences between 1d and 2d , also in comparison to the spectroscopic results , we have performed a systematic comparison of the electron removal spectra in both cases . as will become apparent , there are some similarities , but also clear differences . we suggest that the main difference between 1d and 2d is a strong attractive interaction between ` spinon ' and ` holon ' in 2d , which leads to a band of bound states being pulled out of the continuum of free spinon and holon states . this band of bound states - which are nothing but simple spin @xmath21 fermions corresponding to the doped holes - then sets the stage for the low energy physics of the system , i.e. , true spin - charge separation as in 1d never occurs . we begin with a discussion of the 1d model at half - filling . figure [ fig1 ] shows the electron removal spectra for the @xmath22-site ring . let us first consider the left panel , where energies are measured in units of @xmath2 . then , one can distinguish different types of states according to their scaling behavior with @xmath0 : there is one ` band ' of peaks ( connected by the thin full line ) whose energies relative to the single - hole ground state at @xmath23@xmath13@xmath24 remains practically unchanged under a variation of @xmath0 , i.e. , these states have @xmath2 as their energy scale . as a remarkable fact , this ` band ' abruptly disappears half - way in the brillouin zone , i.e. , there are no peaks whose energy scales with @xmath2 beyond @xmath23@xmath13@xmath24 . this looks like a half - filled free - electron band with a fermi level crossing at @xmath24 , which however is quite remarkable because inverse photoemission is not possible at half - filling . next , in addition to this ` @xmath2-band ' , there are several groups of peaks whose excitation energy shows a very systematic progression with @xmath0 . indeed , when plotting the same spectra but measuring energies in units of @xmath0 ( right panel of fig . [ fig1 ] ) these peaks coalesce , i.e. , to excellent approximation the energy scale of these states is @xmath0 . this coexistence of states with different energy scales can be nicely seen in the ` double peak ' for @xmath25@xmath13@xmath26 and momentum @xmath27 : the peak with lower binding energy falls into the @xmath2-band , the one with the higher binding energy belongs to the @xmath0-band . the dispersion of the @xmath0-band resembles a slightly asymmetric parabola with minimum near @xmath24 for the low excitation energies that we are considering . the states that fall onto this parabola correspond to the creation of a spinon with momentum @xmath4@xmath13@xmath28 , and a holon of momentum @xmath29 . since the spinon momentum is fixed , this group of states then simply traces out the holon dispersion . on the other hand , the ` @xmath2-branch ' corresponds to the holon momentum being fixed at the minimum of the holon dispersion , and thus traces out the spinon dispersion . + this building principle for the spectra can be pushed further . namely , one might expect that not only @xmath4 but any spinon momentum may serve as the starting point for a complete branch of peaks which trace out the full holon dispersion . that this is indeed the case is shown in fig . [ fig2 ] . there , the entire width of the spectra is shown and we have chosen the zero of energy at the excitation energy of either the topmost ` @xmath2-peak ' at @xmath30 ( left panel ) or the topmost ` @xmath2-peak ' at @xmath31 ( right panel ) . due to this choice of the zero of energy , the energy @xmath6 of the spinon with the respective momentum drops out . then , when measuring energies in units of @xmath0 different holon bands ` become sharp ' , i.e. , their energy _ relative to the respective spinon energy _ scales accurately with @xmath0 . moreover , these different groups of peaks to good approximation all trace out the same simple backfolded nearest neighbor hopping dispersion , i.e. , the dispersion of the holon is simply @xmath32 . as discussed above , the first holon band is shifted by the spinons fermi momentum , @xmath30 , so that its dispersion near the band minimum at @xmath33 could be seen in fig . we have also verified that by alligning the spinon peaks at @xmath34 yet another complete holon band can be identified . + we can thus infer the following building principle for the spectral function : the basis for the whole construction is the ` half - filled ' spinon band , with dispersion @xmath35 ; this is indicated by the thick dashed line in fig . then , each @xmath36-point of this band provides the ` basis ' for a complete holon band @xmath37 , which is ` hooked on ' to the spinon band at its band maximum ; these holon bands are indicated by the thin full lines in fig . [ fig3](a ) . comparison with the numerical results ( in this case for the @xmath38-site ring ) in fig . [ fig3](b ) shows that indeed to excellent approximation the poles of the single particle spectral function fall onto these bands . there are some deviations at high binding energies , which however are most probably a deficiency of the lanczos spectra , which are highly accurate only at low excitation energy . moreover , the holon bands in fig . [ fig3](b ) have been slightly shifted , i.e. , they are ` hooked on ' to the spinon band not precisely at their maximum - we have verified that this shift has oscillating sign for different chain lengths , so that it probably is a finite size effect . as an interesting feature , the pole strength seems to be constant along each of these holon bands , i.e. , the weight is a function only of the spinon momentum ( this seems not to be correct for @xmath23@xmath13@xmath17 and @xmath23@xmath13@xmath39 ; here it should be noted that for these momenta the holon band intersects itself , which leads to a doubling of the peak weight ) . in the thermodynamic limit , the density of bands increases , while simultaneously their spectral weight decreases , resulting in incoherent continua . comparing with the exact results of sorella and parola@xcite for the case @xmath15@xmath16@xmath17 it is obvious that the outermost holon band in our calculation , originating from the spinon fermi momentum , develops into a cusp - like singularity of the spectral weight . the spinon band itself , whose energy scale is @xmath2 , turns into a second dispersionless cusp in this limit , which skims at zero excitation energy between @xmath23@xmath13@xmath17 and @xmath23@xmath13@xmath24 . sorella and parola found the excitation energy of the dispersive cusp to be @xmath40 , which corresponds to a the backfolded and shifted nearest neighbor hopping band , @xmath41 . + summarizing the data for 1d we see that the entire electron removal spectrum obeys a very simple building principle , which moreover holds for all momenta and frequencies . analyzing the scaling of the different features with @xmath2 and @xmath0 one can identify ` branches ' of states which trace out the dispersion of the true elementary excitations of the tll , namely the collective spin and charge excitations . the dispersions of the spinons and holons are both consistent with simple nearest neighbor hopping bands , the spinons moreover have a half - filled fermi surface . while these results may not be really new or surprising , we note that they demonstrate that exploiting the scaling properties of excitation energies provides a very useful method to identify the different ` subbands ' . in the following , we will make extensive use of this principle to address the far less understood problems of 2d and finite doping . we proceed to the 2d model , and also consider first the case of half filling . the spectra shown below refer to the standard @xmath38-site cluster , which is the largest cluster for which the calculation of the electron removal spectrum is feasible also in the doped case . the @xmath42-net for this cluster , which is shown in fig . [ fig3a ] , consists of the group of momenta which roughly follows the @xmath43 direction , and a second group along @xmath44@xmath16@xmath45 . we would like to stress that results for other clusters are completely consistent with those for the @xmath38-site cluster . then , the left panel of fig . [ fig4 ] shows the photoemission spectrum for this cluster at half - filling ; thereby we again focus on energies within a few @xmath2 from the top of the band and measure energies in units of @xmath2 . when the spectra are aligned at the top of the band , the positions of the other dominant low energy peaks do not show a strong variation with @xmath0 . some peaks do show a slight but systematic drift with @xmath0 , which however is much weaker than in 1d . a peculiar feature is the peak at @xmath46 , whose relative excitation energy decreases rather than increases with @xmath0 . inspection shows , however , that the ( very weak ) dispersion along the line @xmath44@xmath16@xmath45 ( i.e. , the lowest three momenta in fig . [ fig4 ] ) scales with @xmath0 to good approximation . a possible explanation is the fact that a hole in a 2d system has two distinct mechanisms for propagation , firstly by ` string truncation ' , which gives effective hopping integrals @xmath19@xmath2 , and secondly by hopping along spiral paths@xcite , which gives ( smaller ) effective hopping integrals @xmath19@xmath0@xcite . it can be shown@xcite that the dispersion relation for a single hole to good approximation can be written as @xmath47 where @xmath48 are numerical constants . the first term , which originates from the string truncation mechanism , gives a dispersion which is degenerate along @xmath44@xmath16@xmath45 and this degeneracy is lifted by the second term which is the contribution from the spiral paths ; this naturally explains the scaling of the dispersion along this line with @xmath0 . comparing with 1d we note that with the exception of @xmath49 the ` @xmath2-band ' is present in the entire brillouin zone , i.e. , the spinon fermi surface seen at half - filling in 1d does not exist . + we turn to the right panel of fig . [ fig4](b ) , which shows the entire width of the spectra , with energies measured in units of @xmath0 . it is first of all quite obvious that the spectra generally are more ` diffuse ' than in 1d , with sharp features existing only in the immediate neighborhood of the top of the band ( except for one relatively sharp high energy peak at @xmath49 ) . next , among the diffuse features at high energy there are some whose energy accurately scales with @xmath0 . although these ` peaks ' are rather broad , so that the assignment of a dispersion is not really meaningful , their centers of gravity can be roughly fitted by the expression @xmath50 , which is reminiscent of the dispersion of the ` holon - cusp ' found by sorella and parola@xcite in 1d . an important difference as compared to 1d is the fact that this @xmath0-band does not seem to reach the top of the photoemission spectrum - rather it stays an energy of @xmath19@xmath0 below the @xmath2-band , which forms the first ionization states . we believe that ` in 1d language ' the most plausible interpretation of the data is the formation of bound states of spinon and holon : assuming a strong attraction between these two excitations , which may originate , e.g. , from the well - known string mechanism for hole motion in an antiferromagnet@xcite , one may expect that a band of bound states is pulled out of the continuum of free spinons and holons . this band of bound states corresponds to the @xmath2-band ( which however has a small contribution @xmath51 in its dispersion due to the spiral path mechanism ) . such a bound state of spinon and holon should be a spin - bag like spin @xmath21 fermion , i.e. , a hole heavily dressed by spin excitations . there is strong numerical evidence@xcite , that this is indeed the character of the low energy states in 2d at low doping . one may expect , however , that such a bound state may not be stable for all momenta , and we believe that this is the reason for the absence of a @xmath2-peak at @xmath49 . in this picture , the 2d analogue of the holon is not a coherently propagating excitation , because it is bound to the much slower spinon by the linearly ascending string potential . this picture fits nicely with the diffuse character of the dynamical density correlation function in 2d@xcite : this function , which in a tll should measure basically the response of the free holons , in 2d has almost exclusively diffuse high energy ` peaks ' , with virtually no sharp low energy peaks . moreover , the unexpected ( in the framework of spin charge separation ) appearance of @xmath2 as energy scale in the optical conductivity is also readily understood in terms of the dipole - excitations of a bound spinon - holon pair@xcite . + summarizing the data for 2d , we see a band of quasiparticle peaks , which predominantly has @xmath2 as its energy scale , and some diffuse high energy ` band ' with energy scale @xmath0 . both , the absence of the ` spinon fermi surface ' , as well as the lack of sharp ` holon bands ' are in clear contrast to the situation in 1d . the formation of bound states of spinon and holon , resulting in a split - off band of spin - bag like spin @xmath21 fermions explains this in a natural way . we return to 1d and consider the doped case . figure [ fig5 ] shows the spectral function for the @xmath22-site ring with @xmath26 holes . measuring excitation energies in units of @xmath2 ( left panel ) we can again identify the spinon band . for @xmath26 holes in @xmath22 sites the nominal fermi momentum is @xmath4@xmath13@xmath52 ( i.e. , half way between @xmath31 and @xmath24 ) and the spinon band extends up to this momentum . as was the case at half - filling , some other peaks show a systematic progression of their excitation energy , and switching the unit of energy to @xmath0 ( right panel ) again makes a nearly complete ` holon band ' visible to which these peaks belong . the holon band again takes the form of a backfolded tight - binding band , but this time the top of the parabola around @xmath23@xmath13@xmath24 is missing . the holon band now seems to touch the fermi energy at @xmath4 and at @xmath5@xmath13@xmath53 ( the latter momentum is half way between @xmath54 and @xmath55 ) . this picture of the spectral function nicely fits with the recent exact calculation in the limit @xmath15@xmath16@xmath17 by penc _ et al._@xcite : on the photoemission side , this calculation showed a high intensity ` band ' which is very similar to the backfolded tight - binding dispersion of the holon band . in addition there was a dispersionless low intensity band at zero excitation energy , which corresponds to the spinon band in the limit @xmath15@xmath16@xmath17 . for both , the exact result in the limit @xmath15@xmath16@xmath56 , and our numerical data for finite @xmath2 , there are thus two branches of states which reach excitation energy zero : the ` main band ' which touches @xmath3 at @xmath4 , and the ` shadow band'@xcite , which reaches @xmath3 at @xmath5 . the ` fermi level crossings ' of these two bands may be thought of producing the well known ( marginal ) singularities in the electron momentum distribution @xmath57 at @xmath4 and @xmath5 , found by ogata and shiba@xcite . + the numerical spectra demonstrate a peculiar feature of the tll , namely a kind of pauli exclusion principle which holds for both holons and spinons : the dispersions of both types of excitations become incomplete upon doping , i.e. , the spinon fermi surface shrinks as if the spinons were spin @xmath21 particles , while simultaneously the top of the holon band is ` sawed off ' as if the holons were spinless fermions . it should be noted that this is quite naturally to be expected in that the rapidities for the different ` particles ' in the bethe ansatz solution both obey a pauli - like exclusion principle@xcite . this has negative implications for , e.g. , slave boson mean - field calculations , which necessarily have to treat one type of excitation as a boson . while spin - charge separation is often quoted as justification for the mean - field decoupling , it is obvious that this approximation must fail to reproduce the excitation spectrum even qualitatively in 1d , the only situation where spin - charge separation is really established . + for a more quantitative discussion of the fermi points , we note that the fermi momentum for hole concentration @xmath58 is @xmath59 . for this momentum the first branch of low energy excitations reaches @xmath3 . for small @xmath58 the second branch of low energy excitations comes up to @xmath3 at @xmath60 . the two marginal singularities thus enclose a hole pocket of length @xmath61 as one would expect for holes corresponding to spinless fermions . it is easy to see , that this hole pocket is nothing but the manifestation of the holon ` fermi surface ' around @xmath23@xmath13@xmath39 : the lowest charge excitations , which may be thought of as corresponding to a particle hole excitation between the two edges of the holon pocket have wave vector @xmath62 , i.e. , the holon pocket has a diameter of @xmath63 , precisely the distance between the two marginal singularities . the spectral function for the doped case thus follows the same building principle as for the case of half filling , with the sole difference being that occupied spinon or holon momenta are no longer available for the construction of final states . the singularities in @xmath57 may be thought of as enclosing a hole pocket corresponding to spinless fermions , and thus reflect the fermi surface of the holons . the two holon pockets are placed such that their inner edges at @xmath64 enclose the volume corresponding to the fermi sea of spinons of density @xmath65 . we proceed to the doped case in 2d . let us note from the very beginning that for very simple technical reasons the situation is much more unfavorable in this case . to begin with , due to the higher symmetry of 2d clusters the available @xmath42 meshes are much coarser : for example , amongst the @xmath66 allowed momenta in the @xmath66 site cluster only @xmath67 @xmath42-points are actually non symmetry - equivalent , so that the amount of nonredundant information is much smaller than in 1d . next , unlike 1d where a unique relationship exists between hole density and fermi momentum , most electron numbers in small 2d clusters correspond to open - shell configurations with highly degenerate ground states for noninteracting particles . in an open - shell situation multiplet effects are guaranteed to occur , so that it is in general unpredictable which momenta are occupied and which ones are not ( this holds for a fermi liquid , but is most probably true also for other ` effective particles ' ) . unexpected problems may arise from this . bearing this in mind , one therefore may not expect to see a similarly detailed and clear picture as in 1d . + then , figure [ fig6 ] shows the photoemission spectra for the @xmath66-site cluster , with two holes . we first consider the left hand panel , where energies are measured in units of @xmath2 . comparing with fig . [ fig5 ] , some similarities are quite obvious : the excitation energies of the topmost peaks at @xmath46 , @xmath68 are independent of @xmath0 ( although the spectra for @xmath25@xmath13@xmath26 show a slight deviation ) so that we can identify a ` band ' of states with energy scale @xmath2 . the situation actually is not entirely clear , in that the peak at @xmath69 is so close in energy to the one at @xmath68 that it is not possible to decide if their energy difference scales with @xmath2 or @xmath0 . next , the topmost peaks at @xmath70 and @xmath71 show a systematic progression with @xmath0 , which is very reminiscent of , e.g. , fig . [ fig1 ] . plotting the same spectra with energy scale @xmath0 indeed to good approximation aligns these peaks ( although the peak at @xmath70 still has a slight drift ) , i.e. , their excitation energy relative to the topmost peak at @xmath68 scales with @xmath0 . moreover one can identify a number of diffuse ` features ' at energies between @xmath72 and @xmath73 , which also are roughly aligned ; these are indicated by the dashed line . in analogy with 1d , we can thus distinguish different branches of states , with different energy scales in their excitation energies . while the coarseness of the @xmath42-meshes introduces some uncertainty , the data are consistent with a ` @xmath2-band ' dispersing upwards in the interior of the antiferromagnetic brillouin zone , and a ` @xmath0-band ' dispersing downwards in the outer part , i.e. , the same situation as seen in 1d . a major difference is the fact that the ` features ' at higher binding energies are all very diffuse , at least for @xmath15@xmath74@xmath75 . more significantly , despite the fact that its energy scale seems to be @xmath0 , the dispersion of the ` shadow band ' is much weaker than in 1d . in other words , the effective mass of that band is @xmath19@xmath76 , but with a very large prefactor . + we proceed to the @xmath38-site cluster , also doped with two holes ( see fig . [ fig7 ] ) . choosing @xmath2 as the unit of energy , we see the already familiar situation : the topmost peaks for the states at @xmath77 and @xmath78 are aligned ( although @xmath25@xmath13@xmath26 again deviates slightly ) and several other peaks show a systematic progression with @xmath0 ( an unexpected exception is @xmath46 where a well defined peak actually is not observed ) . changing to energy scale @xmath0 aligns a number of these peaks , which suggests that these peaks form a ` @xmath0-band ' which originates from the topmost peak at @xmath77 . this is a second unexpected feature of the @xmath38-site cluster , in that for the spectra in 1d ( and for those of the @xmath66-site cluster in 2d ) the most intense @xmath0-band always seemed to originate from the topmost peak of the photoemission spectrum . we can only speculate that these unusual features are the consequence of , e.g. , the multiplet effects mentioned above . we also note in this context that the spectra at @xmath46 look actually quite different for @xmath66 and @xmath38 site cluster , which shows the impact of finite - size effects . + ascribing the special behavior at @xmath46 to finite size - effects , we have a quite similar picture as in the @xmath66-site cluster , i.e. , the topmost peaks for spectra inside the antiferromagnetic zone have @xmath2 as their energy scale whereas the topmost peaks in the outer part of the zone have energy scale @xmath0 ( this also holds for @xmath44 which is on the boundary of the antiferromagnetic zone ) . as was the case in the @xmath66 site cluster the ` shadow band ' , while having @xmath0 as its energy scale , has a much weaker dispersion than in 1d . indeed , fitting the @xmath0-bands in both @xmath66 and @xmath38 site cluster by an expression of the form @xmath79 requires to choose @xmath80 - it is tempting to speculate that this may actually be @xmath81 , as one would expect e.g. in the gutzwiller picture . another notable feature is that the @xmath0-band is restricted to the outer part of the brillouin zone . only the diffuse high - energy ` band ' indicated by the dashed line in figure [ fig7 ] seems to scale with @xmath0 . + for completeness we would like to mention that a similar analysis was not possible for the @xmath82-site cluster with @xmath26 holes . the reason is essentially that for some momenta there are no more sharp ` peaks ' , but rather a multitude of densely spaced small peaks . due to this , we were not able to assign any defined ` bands ' , or groups of peaks which showed a systematic scaling of their excitation energy . we have also performed this kind of analysis for the @xmath82 site cluster with @xmath83 holes and found no more indication of the energy scale @xmath2 : at this somewhat higher concentration the entire spectra scale with @xmath0 . + summarizing the data for 2d , hole doping seems to lead to behavior which is more reminiscent of 1d than for half filling , in that the @xmath2-band dispersing upwards in the inner part of the brillouin zone and the @xmath0-band dispersing downwards in the outer part seem to exist also in this case . much unlike 1d , however , the shadow band , while in principle having @xmath0 as its energy scale , still has a very weak dispersion , so that the band structure in the doped case is practically identical to that in the undoped system@xcite . we note however , that the fact that the shadow band has @xmath0 as energy scale has profound implications for its explanation : there have been attempts to interpret the shadow band in bi2212 as a ` dynamical replica ' of the main band , created by scattering of quasiparticles in the standard tight - binding band from antiferromagnetic spin fluctuations@xcite . experimentally , however , the fact that the shadow bands are observed also in the overdoped compounds@xcite , where antiferromagnetism is very weak , as well as the fact that they do not seem to become more pronounced in the underdoped compounds@xcite , where antiferromagnetism is strong , both suggest otherwise . on the theoretical side , we believe that our data very clearly rule out this interpretation : both the ` main band ' and the spin correlation function@xcite have @xmath2 as their relevant energy scale , and it would be very hard to understand how the energy scale of @xmath0 for the shadow band should emerge from a combination of these two types of excitations . in fact , the relatively accurate scaling with physically very different parameters suggests completely different propagation mechanisms for the two types of excitations . we therefore believe that the shadow band is a separate branch of excitations , probably best comparable to the states which produce the @xmath5 singularity in the 1d systems . in the previous sections we have investigated the photoemission spectrum for the one and two dimensional @xmath0@xmath1@xmath2 model . by studying the parameter dependence of the spectra , we could in 1d identify ` branches ' of states which trace out the dispersions of the elementary excitations of the tll , the spinons and holons . both elementary excitations have a simple nearest neighbor hopping dispersion , but with different band width : that of the spinons is @xmath19@xmath2 , that of the holons @xmath19@xmath0 . + in the doped case there are two groups of states which touch the fermi energy ( see figure [ figx ] ) . ` inside ' the noninteracting fermi surface , there is a whole continuum of bands dispersing upwards to @xmath3 . the uppermost of these bands traces out the spinon dispersion and has @xmath2 as its energy scale , the lowermost band traces out the holon dispersion and has @xmath0 as its energy scale . in the thermodynamic limit these bands degenerate into ` cusps ' and merge at @xmath3 . in the outer half of the brillouin zone , there are only states which have @xmath0 as their energy scale . these reach the fermi energy at @xmath5 , giving rise to a second fermi point . while the resolution in @xmath23 and @xmath84 available in our finite clusters is not sufficient to make statements about extreme low energy excitations , the positions of the singularities in the electron momentum distribution as determined from exact solutions clearly shows that both branches of states indeed do touch @xmath3 . the two singularities may be thought of enclosing a hole pocket of extent @xmath85 , which is essentially the image of the holon fermi surface . + in 2d at half - filling , the situation is quite different : while it is still possible to distinguish bands with different scaling behavior with @xmath2 and @xmath0 , the spinon fermi surface present in 1d does not exist and the ` holons ' seem to correspond to overdamped resonances rather than sharp excitations as in 1d . we propose that strong attraction between spin and charge excitations , most probably due to the well - known string mechanism , pulls a band of bound states out of the continuum of ` free ' holon and spinon states . the relevant physics thus is that of spin - bag like spin 1/2 quasiparticles , as suggested by a considerable amount of numerical evidence . + for the doped case in 2d , the situation is less clear and actually somewhat ambiguous . the numerical photoemission spectra show some analogy with 1d , in that there seems to be a high intensity ` main band ' with energy scale @xmath2 dispersing upwards in the inner part of the brillouin zone , and a low intensity ` shadow band ' with energy scale @xmath0 dispersing downwards in the outer part of the brillouin zone ( see figure [ figx ] ) . in contrast to 1d the dispersion of the shadow band is much weaker , i.e. , while the energy scale of the dispersion is @xmath0 , it has an additional very small prefactor ( of the order of the hole concentration ) . moreover the @xmath0-band seems limited to the outer part of the brillouin zone , i.e. there are no indications for a holon band with energy scale @xmath0 dispersing upwards in the inner part of the brillouin zone . only in the @xmath66-site cluster a diffuse ` band ' with energy scale @xmath0 can be roughly identified at higher binding energies . the different energy scales of main band and shadow band suggest that these are excitations of quite different nature , and in particular rule out the explanation that that the shadow band is created by scattering from antiferromagnetic spin fluctuations . + turning to experiment , the results for 2d immediately suggests a comparison with the data of aebi _ et al._@xcite . these authors found that in addition to the ` bright ' part of the band structure , which seems to be consistent with the noninteracting one , there is also a low intensity ` replica ' , shifted approximately by @xmath49 , which had been consistently overlooked in all previous studies . if one wants to make a correspondence to the situation for the @xmath0@xmath1@xmath2 model , one thus should identify this low intensity part with the @xmath0-band dispersing downwards in the @xmath0@xmath1@xmath2 model . our data imply that the shadow band should have a slightly different dispersion than the main band . the limitations of the cluster method probably preclude any meaningful quantitative statements , but it might be interesting to see if this difference in dispersion can be resolved experimentally . + we conclude by outlining a somewhat speculative scenario , based on the assumption that the two bands represent indeed different excitations , which persist at all temperatures and independent of antiferromagnetic correlations . in this case , the topology in 2d opens an interesting possibility : whereas in 1d the two classes of low energy excitations forming the @xmath4 and @xmath5 singularities in @xmath86 are well separated in @xmath42 and @xmath84 space for simple topological reasons , the experimental data of aebi _ et al . _ indicate that the main and shadow band intersect at certain points in the brillouin zone ( see fig . [ fig9 ] , left panel ) . neglecting the small difference in dispersion between main and shadow band , we might therefore model the low energy excitation spectrum by the effective hamiltonian @xmath87 where @xmath88 is the dispersion of the main band , @xmath89 , and the @xmath90 and @xmath91 operators refer to the main and shadow band , respectively . choosing a dispersion of the form @xmath92 this hamiltonian reproduces the fermi surface topology found by aebi _ _ quite well ( see fig . [ fig9 ] ) . however , as mentioned above the two branches of excitations intersect at some points of the brillouin zone , so that already a small mixing between the two bands , which in turn may originate from the spinon - holon interaction , has a dramatic effect on the topology of the low energy excitation spectrum . namely adding a term of the form @xmath93 i.e. , a hybridization between the two types of bands , even relatively small values of @xmath94 open up a gap around @xmath44 and transform the fermi surface transformed into a hole pocket ( see left panel of fig . [ fig9 ] ) . thereby we fix the chemical potential by requiring that the number of @xmath90 and @xmath91 particles remains unchanged ; it is easy to see that the area covered by the pockets then equals the hole concentration @xmath58 , precisely as it was the case in 1d . thereby the fermi surface has predominant main band character at its inner edge , and shadow band character at the outer edge , implying a very different ` visibility ' in photoemission . finally , it is tempting to speculate that the ` pseudo gap order parameter ' @xmath94 decreases with increasing temperature / hole concentration . its vanishing at a certain temperature @xmath95 then could produce a crossover from the hole pockets to the ` large ' fermi surface at @xmath95 , which picture would nicely reproduce the pseudogap phenomenology observed@xcite in cuprate superconductors . + financial support of r. e. by the european community and of y. o. by the saneyoshi foundation and a grant - in - aid for scientific reserach from the ministry of education , science and culture of japan is most gratefully acknowledged . et al . _ , 72 * , 2757 ( 1995 ) . s. la rosa _ et al . _ , preprint . a. g. loeser _ et al . _ , science * 273 * , 325 ( 1996 ) . d. s. marshall _ et al . _ , phys . lett . * 76 * , 4841 ( 1996 ) . c. kim _ et al . _ , phys . rev . lett . * 77 * , 4054 ( 1996 ) . e. dagotto , rev . phys . * 66 * , 763 ( 1994 ) . . bares and m. blatter , phys . lett . * 64 * , 2567 ( 1990 ) . m. ogata and h. shiba , phys . b * 41 * , 2326 ( 1990 ) . s. sorella and a. parola , j. phys . cond . mat . * 4 * , 3589 ( 1992 ) . k. penc , k. hallberg , and h. shiba , phys . lett . * 77 * , 1390 ( 1996 ) . j. favand _ et al . _ , e - print cond - mat 9611223 . r. eder _ et al . _ , unpublished . r. eder , y. ohta , and s. maekawa , phys . lett . * 74 * , 5124 ( 1995 ) . r. eder , p. wrobel , and y. ohta , phys . b * 54 * , r11034 ( 1996 ) . e. dagotto and j. r. schrieffer , phys . b * 43 * , 8705 ( 1990 ) . r. eder and y. ohta , phys . b * 50 * , 10043 ( 1994 ) . j. riera and e. dagotto , unpublished . j. r. schrieffer , x. g. wen , and s. c. zhang , phys . b * 39 * , 11663 ( 1989 ) . s. a. trugman , phys . b * 37 * , 1597 ( 1987 ) . r. eder and k. w. becker , z. phys . b. 78 , 219 ( 1990 ) ; see also m. vojta and k. w. becker , phys . b * 54 * , 15483 ( 1996 ) . l. n. bulaevskii , e. l. nagaev , and d. i. khomskii , sov . jetp * 27 * , 836 ( 1968 ) . r. eder , y. ohta , and t. shimozato , phys . b * 50 * , 3350 ( 1994 ) . a. chubukov , phys . b * 52 * , r3840 ( 1995 ) .

we present an exact diagonalization study of the single particle spectral function in the 1d and 2d @xmath0@xmath1@xmath2 model . by studying the scaling properties with @xmath2 and @xmath0 we find a simple building pattern in 1d and show that every spectral feature can be uniquely assigned by a spinon and holon momentum . we find two types of low energy excitations : a band with energy scale @xmath2 and high spectral weight disperses upwards in the interior part of the brillouin zone and reaches @xmath3 at @xmath4 , and a band with energy scale @xmath0 and low spectral weight disperses downwards in the outer part of the zone , touching @xmath3 at @xmath5 . an analogous analysis of the 2d case at half filling shows that the @xmath0-band exists also in this case , but is diffuse and never reaches the fermi energy . for the doped case in 2d the picture is more reminiscent of 1d , in particular the ` main - band ' with a dispersion @xmath6 and the ` shadow band ' with energy scale @xmath0 can be identified also in this case . this leads us to propose that the shadow bands discovered by aebi _ et al . _ in bi2212 are the 2d analogue of the @xmath5 singularity in 1d systems and unrelated to antiferromagnetic spin fluctuations . 2

cond-mat9702221

in this talk i will describe some recent work together with michael dinsdale concerning the relative size of non - perturbative power corrections for qcd event shape observables @xcite . for @xmath1 event shape _ means _ the delphi collaboration have found in a recent analysis that , if the next - to - leading order ( nlo ) perturbative corrections are evaluated using the method of effective charges @xcite , then one can obtain excellent fits to data without includingany power corrections @xcite . in contrast fits based on the use of standard fixed - order perturbation theory in the @xmath2 scheme with a physical choice of renormalization scale equal to the c.m . energy , require additional power corrections @xmath3 with @xmath4 . power corrections of this size are also predicted in a model based on an infrared finite coupling @xcite , which is able to fit the data reasonably well in terms of a single parameter . given the delphi result it is interesting to consider how to extend the method of effective charges to event shape _ distributions _ rather than means . consider an @xmath1 observable @xmath5 , e.g. an event shape observable- thrust or heavy - jet mass , @xmath6 being the c.m . @xmath7 here @xmath8 . normalised with the leading coefficient unity , such an observable is called an _ effective charge_. the couplant @xmath9 satisfies the beta - function equation @xmath10 here @xmath11 and @xmath12 are universal , the higher coefficients @xmath13 , @xmath14 , are rs - dependent and may be used to label the scheme , together with dimensional transmutation parameter @xmath0 @xcite . the _ effective charge _ @xmath15 satisfies the equation @xmath16 this corresponds to the beta - function equation in an rs where the higher - order corrections vanish and @xmath17 , the beta - function coefficients in this scheme are the rs - invariant combinations @xmath18 eq.(3 ) for @xmath19 can be integrated to give @xmath20 + \int_{0}^{{\cal{r}}(q)}{dx}\left[\frac{b}{\rho(x)}+\frac{1}{{x}^{2}(1+cx)}\right]\;.\ ] ] the dimensionful constant @xmath21 arises as a constant of integration . it is related to the dimensional transmutation parameter @xmath22 by the exact relation , @xmath23 here @xmath24 with @xmath25 , is the nlo perturbative coefficient . eq.(5 ) can be recast in the form @xmath26 the final factor converts to the standard convention for @xmath0 . here @xmath27 is the _ universal _ function @xmath28 and @xmath29 is @xmath30 here @xmath31 is the nnlo ech rs - invariant . if only a nlo calculation is available , as is the case for @xmath1 jet observables , then @xmath32 , and @xmath33 eq.(10 ) can be used to convert the measured data for the observable @xmath15 into a value of @xmath34 bin - by - bin . such an analysis was carried out in ref . @xcite for a number of @xmath1 event shape observables , including thrust and heavy jet mass which we shall focus on here . it was found that the fitted @xmath0 values exhibited a clear plateau region , away from the two - jet region , and the region approaching @xmath35 where the nlo thrust distribution vanishes . the result for 1-thrust corrected for hadronization effects is shown in fig . 1 . another way of motivating the effective charge approach is the idea of `` complete renormalization group improvement '' ( corgi ) @xcite . one can write the nlo coefficient @xmath36 as @xmath37 hence one can identify scale - dependent @xmath38-logs and rs - invariant `` physical '' uv @xmath6-logs . higher coefficients are polynomials in @xmath39 . @xmath40 given a nlo calculation of @xmath39 , parts of @xmath41 are `` rg - predictable '' . one usually chooses @xmath42 then @xmath43 is @xmath6-independent , and so are all the @xmath44 . the @xmath6-dependence of @xmath5 then comes entirely from the rs - dependent coupling @xmath45 . however , if we insist that @xmath38 is held constant _ independent of @xmath6 _ the only @xmath6-dependence resides in the `` physical '' uv @xmath6-logs in @xmath43 . asymptotic freedom then arises only if we resum these @xmath6-logs to _ all - orders_. given only a nlo calculation , and assuming for simplicity that that we have a trivial one loop beta - function @xmath46 so that @xmath47 the rg - predictable terms will be @xmath48 summing the geometric progression one obtains @xmath49 \nonumber \\ & = & 1/b{\ln}(q/{\lambda}_{\cal{r}}).\end{aligned}\ ] ] the @xmath38-logs `` eat themselves '' and one arrives at the nlo ech result @xmath50 . + as we noted earlier , @xcite , use of nlo effective charge perturbation theory ( renormalization group invariant ( rgi ) perturbation theory ) leads to excellent fits for @xmath1 event shape _ means _ consistent with zero power corrections , as illustrated in figure 2 . taken from ref.@xcite . given this result it would seem worthwhile to extend the effective charge approach to event shape _ distributions_. it is commonly stated that the method of effective charges is inapplicable to exclusive quantities which depend on multiple scales . however given an observable @xmath51 depending on @xmath52 scales it can always be written as @xmath53 here the @xmath54 are _ dimensionless _ quantities that can be held fixed , allowing the @xmath55 evolution of @xmath15 to be obtained as before . in the 2-jet region for @xmath1 observables large logarithms @xmath56 arise and need to be resummed to all - orders . event shape distributions for thrust ( @xmath57 ) or heavy - jet mass ( @xmath58 ) contain large kinematical logarithms , @xmath59 , where @xmath60 . @xmath61 here @xmath62 , @xmath63 , denote leading logarithms , next - to - leading logarithms , etc . for thrust and heavy - jet mass the distributions _ exponentiate _ @xcite @xmath64 here @xmath65 contains the ll and @xmath66 the nll . @xmath67 is independent of @xmath68 , and @xmath69 contains terms that vanish as @xmath70 . it is natural to define an effective charge @xmath71 so that @xmath72 this effective charge will have the expansion @xmath73 here @xmath74 , and the higher coefficients @xmath75 have the structure @xmath76 usually one resums these logarithms to all - orders using the known closed - form expressions for @xmath77 and @xmath78 , where @xmath79 is taken to be the @xmath80 coupling with a `` physical '' scale choice @xmath25 ( @xmath80ps ) . instead we want to resum logarithms to all - orders in the @xmath81 function ( ech ) . the form of the @xmath82 rs - invariants ( eq.(4 ) ) means that the @xmath82 have the structure @xmath83 one can then define all - orders rs - invariant @xmath62 and @xmath63 approximations to @xmath81 , @xmath84 the resummed @xmath85 can then be used to solve for @xmath86 by inserting it in eq.(5 ) . notice that since @xmath21 involves the _ exact _ value of @xmath87 there is no matching problem as in the standard @xmath2ps approach . the resummed @xmath88 can be straightforwardly numerically computed using @xmath89 with @xmath79 chosen so that @xmath90 . the same relation with @xmath91 suffices for @xmath92 , although in this case one needs to remove @xmath93 terms , e.g. an @xmath94 term which would otherwise be included in @xmath31 . this can be accomplished by numerically taking limits @xmath95 with @xmath96 fixed . + as we have noted a crucial feature of the effective charge approach is that it resums to all - orders _ rg - predictable _ pieces of the higher - order coefficients , thus the nlo ech result ( assuming @xmath97 for simplicity ) corresponds to an rs - invariant resummation ( c.f . eq.(13 ) . ) @xmath98 thus even at fixed - order without any resummation of large logs in @xmath81 a _ partial _ resummation of large logs is automatically performed . furthermore one might expect that the ll ech result contains already nll pieces of the standard @xmath80ps result . + in figure 3 we show various nlo approximations . notice that the solid curve , which corresponds to the exponentiated nlo ech result , is a surprisingly good fit even in the 2-jet region , whereas the dashed curve which is the nlo @xmath2ps result , has a badly misplaced peak . the all - orders partial resummation of large logs in eq.(15 ) gives a reasonable 2-jet peak . figure 4 shows that the nll @xmath80ps coefficients `` predicted '' from the ll ech result by re - expanding it in the @xmath2ps coupling are in good agreement with the exact coeffiecients out to o(@xmath99 ) . we now turn to fits simultaneously extracting @xmath34 and the size of power corrections @xmath3 from the data . to facilitate this we use the result that inclusion of power corrections effectively shifts the event shape distributions , which can be motivated by considering simple models of hadronization , or through a renormalon analysis @xcite . thus we define @xmath100 this shifted result is then fitted to the data for 1-thrust and heavy jet mass . @xmath1 data spanning the c.m . energy range from @xmath101 gev was used ( see @xcite for the complete list of references ) . the resulting fits for 1-thrust and heavy - jet mass are shown in figures 5 . and 6 .. + the ech fits for thrust and heavy jet mass show great stability going from nlo to ll to nll , presumably because at each stage a partial resummation of higher logs is automatically performed . the power corrections required with ech are somewhat smaller than those found with @xmath80ps , but we do not find as dramatic a reduction as delphi find for the means . this may be because their analysis corrects the data for bottom quark mass effects which we have ignored . the fitted value of @xmath34 for ech is much smaller than that found with @xmath80ps , ( @xmath102 ( thrust ) and @xmath103 ( heavy - jet mass ) ) . similarly small values are found with the dressed gluon exponentiation ( dge ) approach @xcite . a problem with the effective charge resummations is that the @xmath81 function contains a branch cut which limits how far into the 2-jet region one can go . we are limited to @xmath104 in the fits we have performed . this branch cut mirrors a corresponding branch cut in the resummed @xmath105 function . similarly as @xmath106 approaches @xmath107 the leading coefficient @xmath108 vanishes and the effective charge formalism breaks down . we need to restrict the fits to @xmath109 . from the `` rg - predictability '' arguments we might expect that these difficulties would also become apparent for a nnll @xmath80ps resummation . one will be able to check this expectation when a result for @xmath110 becomes available . event shape means have also been studied in dis at hera @xcite . for such processes one has a convolution of proton pdf s and hard scattering cross - sections , @xmath111 there is no way to directly relate such quantities to effective charges . the dis cross - sections will depend on a _ factorization scale _ @xmath112 , and a renormalization scale @xmath38 at nlo . in principle one could identify unphysical scheme - dependent @xmath113 and @xmath114 , and physical uv @xmath6-logs , and then by all - orders resummation get the @xmath112 and @xmath38-dependence to `` eat itself '' . the pattern of logs is far more complicated than the geometrical progression in the effective charge case , and a corgi result for dis has not been derived so far . instead one can use the principle of minimal sensitivity ( pms ) @xcite , and for an event shape mean @xmath115 look for a stationary saddle point in the @xmath116 plane @xcite . it turns that there are large cancellations between the nlo corrections for quark and gluon initiated subprocesses . one can distinguish between two approaches , @xmath117 where one seeks a saddle point in the @xmath116 plane for the sum of parton subprocesses , and @xmath118 where one introduces two separate scales @xmath119 and @xmath120 and finds a saddle point in @xmath121 . @xmath117 gives power corrections fits comparable to @xmath80ps with @xmath122 . @xmath118 in contrast gives substantially reduced power corrections . this is shown in figure 7 for a selection of hera event shape means . given large cancellations of nlo corrections rg - improvement should be performed _ separately _ for the @xmath123 and @xmath124-initiated subprocesses , and so @xmath118 which indeed fits the data best , is to be preferred . event shape means in @xmath1 annihilation are well - fitted by nlo perturbation theory in the effective charge approach , without any power corrections being required . with the usual @xmath80ps approach power corrections @xmath125 are required with @xmath126 gev . similarly sized power corrections are predicted in the model of ref.@xcite . it would be interesting to modify this model so that its perturbative component matched the effective charge prediction , but this has not been done . we showed how resummation of large logarithms in the effective charge beta - function @xmath127 could be carried out for @xmath1 event shape distrtibutions . if the distributions are represented by an exponentiated effective charge then even at nlo a partial resummation of large logarithms is performed . as shown in figure 3 this results in good fits to the 1-thrust distribution , with the peak in the 2-jet region in rough agreement with the data . in contrast the @xmath80ps prediction has a badly misplaced peak in the 2-jet region , and is well below the data for the realistic value of @xmath128 mev assumed . we further showed in figure 4 that the ll ech result contains already a large part of the nll @xmath80ps result . we found unfortunately that @xmath127 contains a branch point mirroring that in the resummed @xmath77 function . this limited the fit range we could consider . we fitted for power corrections and @xmath34 to the 1-thrust distribution and heavy - jet mass distributions , finding somewhat reduced power corrections for the ech fits compared to @xmath80ps , with good stability going from nlo to ll to nll . the suggestion of the `` rg - predictability '' manifested in figure 4 would be that the nll ech result contains a large part of the nnll @xmath80ps result . this suggests that the branch point problem which limits the ability to describe the 2-jet peak , would also show up given a nnll analysis . this can be checked once the @xmath129 function becomes available . recent work on event shape means in dis was briefly mentioned and seemed to indicate that greatly reduced power corrections are found when a correctly optimised pms approach is used . yasaman farzan and the rest of the organising committee of the ipm lph-06 meeting are thanked for their painstaking organisation of this stimulating and productive school and conference . many thanks are also due to abolfazl mirjalili for organising my wonderful post - conference visits to esfahan , yazd and shiraz , and to all those whose welcoming hospitality made my first visit to iran so extremely enjoyable . dinsdale and c.j . maxwell , nucl . b713 * ( 2005 ) 465 . maxwell , talk delivered at the frif workshop on first principles qcd of hadron jets . [ hep - ph/0607039 ] . g. grunberg , phys . b95 * ( 1980 ) 70 ; phys . rev . * d29 * ( 1984 ) 2315 . delphi collaboration ( j. abdallah et . j. * c29 * ( 2003 ) 285 . k. hamacher , talk delivered at the frif workshop on first principles qcd of hadron jets . [ hep - ex/0605123 ] . y.l . dokshitzer and b.r . webber , phys . * b404 * ( 1997 ) 321 . p.m. stevenson , phys . rev . * d23 * ( 1981 ) 2916 . burby and c.j . maxwell , nucl . b609 * ( 2001 ) 193 . maxwell [ hep - ph/9908463 ] ; c.j . maxwell and a. mirjalili , nucl . b611 * , 423 ( 2001 ) . s. catani , g. turnock , b.r . webber and l. trentadue , phys . b263 * ( 1991 ) 491 . webber [ hep - ph/9411384 ] . e. gardi and j. rathsman , nucl . phys . * b638 * ( 2002 ) 243 . c. adloff _ et al . _ [ h1 collaboration ] , eur . j. c * 14 * ( 2000 ) 255 [ erratum - ibid . c * 18 * ( 2000 ) 417 ] [ arxiv : hep - ex/9912052 ] . dinsdale [ arxiv : hep - ph/0512069 ] .

we introduce and motivate the method of effective charges , and consider how to implement an all - orders resummation of large kinematical logarithms in this formalism . fits for qcd @xmath0 and power corrections are performed for the @xmath1 event shape obesrvables 1-thrust and heavy - jet mass , and somewhat smaller power corrections found than in the usual approach employing the `` physical scale '' choice .

hep-ph0611169

precision experiments in the last two decades have elevated the standard model ( sm ) of particle physics from a promising description to a provisional law of nature , tested as a quantum field theory at the level of one percent or better . despite its triumphs the sm is not an entirely satisfactory theory , however , because it has various theoretical shortcomings . in particular , the gauge hierarchy problem , _ i.e. _ , the instability of the electroweak scale @xmath7 under radiative corrections , has spurred the imagination of many theorists and led to the development of a plethora of models of physics beyond the sm that envision new phenomena at or not far above the tev scale . a particularly appealing proposal for stabilizing the electroweak scale , featuring one compact extra dimension with a non - factorizable anti - de sitter ( ads@xmath8 ) metric , is the randall - sundrum ( rs ) model @xcite , which by virtue of the ads / cft correspondence @xcite can be thought of as dual to a strongly coupled four - dimensional ( 4d ) cft . with two three - branes acting as the boundaries of the warped extra dimension , the ads@xmath8 background generates an exponential hierarchy of energy scales , so that the natural scale at one orbifold fixed point ( the ultra - violet ( uv ) brane ) is much larger than at the other ( the infra - red ( ir ) brane ) , @xmath9 . in the rs framework the gauge hierarchy problem is thus solved by gravitational red - shifting . there are numerous possibilities for building models of electroweak symmetry breaking in ads@xmath8 . the basic building blocks for the construction of a viable theory include , among others , the choice of the bulk gauge group , the zero - mode fermion localization , and the dynamical mechanism for localizing the higgs field on ( or near ) the ir brane . while in the original rs proposal all sm fields were constrained to reside on the ir boundary and the gauge group was taken to be @xmath10 , it was soon realized that allowing gauge @xcite and matter fields @xcite to spread in the ads@xmath8 bulk not only avoids dangerous higher - dimensional operators suppressed only by powers of @xmath11 , but also admits a natural explanation of the flavor structure of the sm @xcite via geometrical sequestering @xcite . this way of generating fermion hierarchies also implies a certain amount of suppression of dangerous flavor - changing neutral currents ( fcncs ) @xcite , a scheme referred to as the rs - gim mechanism @xcite . harmful contributions to the @xmath1 parameter can be cured in an elegant way by extending the bulk hypercharge group to @xmath12 and breaking it to @xmath13 on the uv brane @xcite . an appropriate embedding of the down - type sm quarks into the custodial rs model further furnishes the possibility to reduce the tree - level corrections to the @xmath14 vertex @xcite and its flavor - changing counterparts @xcite . as a result , all existing electroweak precision and cp - conserving fcnc constraints are typically satisfied for the mass of the lightest kaluza - klein ( kk ) gauge boson below a few tev . however , in spite of the rs - gim mechanism , cp - violating effects in the neutral kaon system @xcite and corrections to the neutron electric dipole moment @xcite tend to be too large in models with flavor anarchy , pushing the new - physics scale to at least 10 tev and thus beyond the reach of the cern large hadron collider ( lhc ) . relaxing the latter bounds seems to require an additional flavor alignment in warped models and has triggered a lot of model - building activity . the purpose of this article is to perform a thorough analysis of the structure of the rs variant with extended gauge symmetry @xmath15 in the bulk , where @xmath3 interchanges the two @xmath16 groups and is responsible for the protection of the @xmath14 vertex . while in the existing literature on the custodial rs model the couplings of the bulk fields to the higgs sector are treated as a perturbation , we instead construct the exact solutions to the bulk equations of motion ( eoms ) subject to appropriate boundary conditions ( bcs ) . in that way we obtain exact results for the profiles and masses of the various sm particles and their kk excitations . this approach is not only more elegant but also offers several advantages over the perturbative approach . in particular , it facilitates the analytic calculation of all terms of order @xmath17 , including those arising from the breaking of the @xmath3 symmetry by the bcs and possibly the bulk masses . the physical interpretation of the obtained results in terms of ( ir)reducible sources of symmetry breaking is thus evident in our approach , while it remains somewhat hidden if the couplings of the bulk fields to the higgs sector are treated as a perturbation from the very beginning . the exact approach also permits to include the mixing of fermions between different generations in a completely general way , making the dependence on the exact realization of the matter sector explicit . in turn , it is straightforward to address questions about the model - dependence of the resulting gauge- and higgs - boson interactions with the sm fermions . in summary , our work puts the theory of custodial warped extra dimensions on a more sound basis , both at the field theoretical and phenomenological level . in a forthcoming paper we will apply the derived results to tree - level flavor - violating @xmath18 and @xmath19 processes in the quark sector . this article is organized as follows . after recalling important definitions and notations , we discuss in section [ sec : gauge ] the kk decomposition of the bulk gauge fields in the presence of the brane - localized higgs sector , working in a covariant @xmath20 gauge . we also show how to compute sums over kk towers of gauge bosons in closed form . the analogous discussion for bulk fermions is presented in section [ sec : fermions ] . special attention is devoted to the correct implementation of yukawa couplings containing @xmath21-odd fermion profiles . in sections [ sec : custodialprotection ] and [ sec : higgscouplings ] we present the main results of our work . we first analyze the structure of gauge - boson interactions with sm fermions and then study the couplings of the higgs boson to matter . in the first case , we give analytic formulas that expose , on one hand , the prerequisites for achieving a custodial protection of the left - handed @xmath2-boson couplings and , on the other , which are the terms that necessarily escape protection . in addition , we show explicitly that no protection mechanism is present in the charged - current sector . in the second case , the exact dependence on the realization of the fermion sector of the higgs - fermion couplings is worked out . in our article we concentrate on the leading contributions to the observables of interest , ignoring possible effects of brane - localized kinetic terms @xcite . although the uv dynamics is not specified , it is natural to assume that these terms are loop suppressed , so that they can be neglected to first order . the most important phenomenological implications of our findings are discussed in section [ sec : pheno ] . we begin by studying the constraints imposed by the precision measurements of the bottom - quark pseudo observables , including all tree - level corrections that avoid protection . we further discuss the phenomenology of rare top decays in the extended rs model and compare it to the one of the minimal formulation . finally , we explore the possible changes of the higgs production cross section and branching fractions at the lhc , including all leading - order quantum corrections stemming from the extended electroweak gauge - boson and fermion sectors . in a series of appendices we collect details on the derivation of the ir bcs and higgs - boson fcncs in the presence of both @xmath21-even and -odd yukawa couplings , our input values for the sm parameters , and the explicit expressions for the form factors needed to calculate the production cross section and the branching ratios of the higgs boson in the rs model . we work with the non - factorizable rs geometry @xmath22 where @xmath23 denote the coordinates on the 4d hyper - surfaces of constant @xmath24 with metric @xmath25 . the fifth dimension is an @xmath26 orbifold of size @xmath27 labeled by @xmath28 $ ] . the extra dimension has orbifold fixed points at @xmath29 ( the uv brane ) and @xmath30 ( the ir brane ) . since the ratio of the warp factor and the curvature , @xmath31 , corresponds to an inverse energy scale in the 4d theory , the gauge hierarchy problem can be tamed by an appropriate choice of the product @xmath32 . in order to address the hierarchy between the electroweak scale , @xmath33 , and the fundamental planck scale , @xmath34 , one has to choose @xmath35 below we will sometimes refer to @xmath36 as the `` volume '' of the extra dimension . the quantity @xmath37 also sets the mass scale for the low - lying kk excitations of the sm fields to be of order of the `` kk scale '' @xmath38 for instance , the masses of the first kk photon and gluon are approximately @xmath39 . it will often be convenient to introduce a coordinate @xmath40 , which equals @xmath41 on the uv brane and 1 on the ir brane @xcite . integrals over the orbifold are then obtained using the following replacements @xmath42 we now have enough definitions and notations in place to start our discussion . in this section we construct the kk decomposition in the gauge sector and derive exact solutions for the bulk fields , including the effects of an ir brane - localized higgs sector . in all previous works on the rs model with custodial protection , the couplings of the higgs sector to bulk fields were treated as a perturbation , expanding the theory in powers of @xmath43 . this leads to the necessity of diagonalizing infinite - dimensional matrices which have to be truncated , including only one ( or a few ) kk excitations . while such an approach should in general lead to sensible results @xcite , it is worthwhile to study the set - up within the basis of mass eigenstates , thereby obtaining exact results @xcite . indeed , we will see that the summation over the entire kk tower receives additional contributions , which are lost through truncation @xcite . as the sum can be evaluated in closed form , it is convenient to work with the complete sum and afterwards expand the obtained expressions in powers of @xmath43 . proceeding in this way one can clearly distinguish between leading and subleading terms . alternatively one could use five - dimensional ( 5d ) propagators @xcite , which would be equivalent to our method . an exhaustive treatment of the perturbative approach featuring truncation after the first mode can be found in @xcite . we consider the rs model with custodial protection as proposed in @xcite , with the bulk gauge symmetry @xmath44 . on the ir brane , the symmetry - breaking pattern @xmath45 provides a custodial symmetry , which protects the @xmath1 parameter . the additional @xmath3 symmetry prevents the left - handed @xmath46 coupling from receiving excessively large corrections @xcite . on the uv brane , the symmetry breaking @xmath47 generates the sm gauge group . the symmetry breaking down to @xmath48 is related to the interplay of uv and ir bcs and will become clear later on . the 5d action of the gauge sector takes the form @xmath49 with the gauge - kinetic terms @xmath50 where @xmath51 denotes the 5d metric . as it is not needed for our analysis , we ignore the faddeev - popov lagrangian . the higgs lagrangian @xmath52 is localized on the ir brane . a simple prescription of how to deal with @xmath53 has already been presented in @xcite and we postpone a refined treatment to section [ sec : fermkk ] . the gauge - fixing lagrangian , @xmath54 , will be given in the next section . we choose the four - vector components of the gauge fields to be even under the @xmath21 parity , while the scalar fifth components are odd , in order to arrive at a low - energy spectrum that is consistent with observation . the higgs bi - doublet , responsible for breaking @xmath55 to the diagonal subgroup @xmath56 on the ir brane , transforms as @xmath57 and explicitly reads @xmath58 where @xmath59 are real scalar fields , @xmath60 , and @xmath61 is the vacuum expectation value ( vev ) of the higgs field . @xmath62 transformations act from the left on the bi - doublet , while the @xmath63 transformations act from the right . the covariant derivative in the higgs sector reads @xmath64 with @xmath65 . an explicit calculation leads to @xmath66 \\[-3 mm ] & \phantom{xx } + \text{terms bi - linear in fields } \ , , \nonumber \end{aligned}\ ] ] where we have introduced @xmath67 the structure of ( [ eq : cohigg ] ) motivates us to define the new fields @xcite @xmath68 which lead to a diagonal mass matrix . we have introduced the 4d gauge couplings @xmath69 . the rotations are analogous to the usual definitions of the @xmath2 boson and photon fields in the sm , which are themselves postponed to ( [ eq : za ] ) . finally , the mass term adopts the form @xmath70 and reveals the breaking pattern @xmath71 induced by the higgs vev @xmath72 . appropriate bcs break the extended electroweak gauge group down to the sm gauge group on the uv boundary @xmath73 explicitly , this is done by introducing the new fields @xmath74 and giving dirichlet bcs to @xmath75 and @xmath76 on the uv brane . the @xmath13 hypercharge coupling is related to the @xmath77 couplings by @xmath78 the sm - like neutral electroweak gauge bosons are defined in the standard way through @xmath79 it follows that the definitions of the sine and cosine of the weak - mixing angle , @xmath80 agree with the ones in the sm . furthermore , the fields @xmath81 and @xmath82 can be rotated to the photon field @xmath83 and a state @xmath84 via @xmath85 where @xmath86 and we write @xmath87 , as it is a linear combination of @xmath88 and @xmath89 , which is orthogonal to @xmath84 as we will see below . @xmath90 in table [ tab : bcs ] we summarize the bcs that we choose for the fields in order to obtain the correct mass spectrum for the sm gauge bosons . they are given in terms of fields with individual bcs at the two different branes . in the following we will refer to these sets of fields as the uv and ir basis , respectively . the situation is summarized in figure [ fig : bases ] , where we also recall the symmetry - breaking patterns on the different branes . the bcs can easily be transformed to another basis at the expense of obtaining expressions that mix different fields . the photon @xmath91 has individual and source - free neumann bcs at both branes , and therefore its zero mode remains massless . note that there is just one mass parameter @xmath92 entering the ir bcs , in contrast to the two parameters @xmath93 and @xmath94 appearing in the minimal model . in the custodial model , the different masses for the lightest electroweak gauge bosons are accomplished through the mixed uv bcs of the gauge fields in the ir basis ( see ( [ eq : rots ] ) below ) . the fact that there is just one fundamental mass parameter is crucial for the custodial protection of the @xmath1 parameter . we will elaborate on this in section [ sec : mwmzstu ] . ( 0,0)(0,0 ) ( -285,-7.5 ) ( -330,170 ) ( -282.5,45 ) ( -292.5,120 ) ( -110,-7.5 ) ( -165,170 ) ( -108.5,45 ) ( -117.5,120 ) [ fig : bases ] the action of the theory still contains mixing terms between gauge fields and scalars , which can be removed by an appropriate gauge - fixing lagrangian . as the higgs sector is localized on the ir brane , it is natural to work in the ir basis for that purpose . for this reason , we define the 5d theory in the ir basis . the concrete form of the gauge fixing will be given below in ( [ eq : sgf ] ) . before discussing the kk decomposition , we summarize the relations between the uv ( right ) and ir basis ( left ) . they read @xmath95 \left ( \begin{array}{c } \tilde a_m^\pm\\ v_m^\pm \end{array } \right ) & = \left ( \begin{array}{cr } \cos\theta_w & -\sin\theta_w \\ \sin\theta_w & \cos\theta_w \end{array } \right ) \left ( \begin{array}{c } l_m^\pm\\ r_m^\pm \end{array } \right ) \equiv { \bm r}_w \left ( \begin{array}{c } l_m^\pm\\ r_m^\pm \end{array } \right ) , \end{split}\ ] ] where @xmath96 \sin\theta_w & = \frac{g_r}{\sqrt{g_l^2+g_r^2 } } \ , , & \qquad \cos\theta_w & = \frac{g_l}{\sqrt{g_l^2+g_r^2 } } \,,\end{aligned}\ ] ] and @xmath97 has been defined in ( [ eq : glrx2 ] ) . in order to shorten the notation we will hereafter employ the abbreviations @xmath98 and @xmath99 for @xmath100 . we now perform the kk decomposition of the 5d fields . it is convenient to work with profiles that obey definite neumann ( @xmath101 ) or dirichlet ( @xmath102 ) bcs at the uv brane . therefore we include a rotation to the uv basis , _ i.e. _ , the basis in which the uv bcs decouple , in our decomposition . furthermore , as different uv fields get mixed by the ir bcs , these fields should be expressed through the same 4d basis . we consequently introduce the vectors @xmath103 and @xmath104 and write @xmath105 where the sums run over @xmath106 . note that @xmath107 _ etc . _ are 4d mass eigenstates and the lightest modes are identified with the sm gauge bosons . the matrices @xmath108 are defined in ( [ eq : rots ] ) and we have introduced the diagonal matrix @xmath109 as well as two - component vectors @xmath110 , with @xmath111 , representing the mixings between the different gauge fields and their kk excitations . these vectors are normalized according to @xmath112 notice that the matrix @xmath113 should in principle also carry a superscript @xmath114 , indicating the field to which it belongs , but we will not show it , as the correct index should be always clear from the context . the superscripts @xmath115 and @xmath116 label the type of bc we impose on the profiles at the uv brane , _ i.e. _ , they indicate untwisted and twisted even functions - parity , which obey dirichlet bc on the uv brane and are thus not smooth at this orbifold fix point . these fields are sometimes called odd , as they look like an odd function if one just considers half of the orbifold . untwisted even functions correspond to ordinary profiles with neumann uv bcs . ] on the orbifold . remember from table [ tab : bcs ] that both profiles satisfy a neumann bc at the ir boundary , which we do not indicate explicitly by a superscript @xmath115 to avoid unnecessary clutter of notation . let us also introduce the shorthand notations @xmath117 for the profiles of the uv fields . in analogy to the fermion decomposition in @xcite , the profiles @xmath118 do not obey exact orthonormality conditions . this fact is related to the decomposition of fields with neumann and dirichlet bcs into the same 4d gauge - boson basis . the complete vectors @xmath119 with @xmath120 are however orthonormal on each other , @xmath121 note also that the photon obeys a standard orthonormality condition . we also expand the 4d goldstone bosons in the basis of mass eigenstates @xmath122 and @xmath123 by writing @xcite @xmath124 employing the notation introduced in this section , the gauge - fixing lagrangian takes the form @xmath125 \right)^2\\ & \quad\mbox{}- \frac{1}{2\xi } \left ( \partial^\mu \vec z_\mu - \xi \left [ \frac{\delta(|\phi|-\pi)}{r}\,m_{\tilde a}\,\vec\varphi^{\ , 3 } + \frac{\partial_\phi\,e^{-2\sigma(\phi)}}{r^2}\ , \vec z_\phi \right ] \right)^2 \\ & \quad\mbox{}- \frac{1}{\xi } \left ( \partial^\mu \vec w_\mu^+ - \xi \left [ \frac{\delta(|\phi|-\pi)}{r}\,m_{\tilde a}\,\vec\varphi^{\ , + } + \frac{\partial_\phi\,e^{-2\sigma(\phi)}}{r^2 } \ , \vec w_\phi^+ \right ] \right)^t \\ & \qquad\times \left ( \partial^\mu \vec w_\mu^- - \xi \left [ \frac{\delta(|\phi|-\pi)}{r}\,m_{\tilde a}\,\vec\varphi^{\ , - } + \frac{\partial_\phi\,e^{-2\sigma(\phi)}}{r^2 } \ , \vec w_\phi^- \right ] \right ) . \end{split}\end{aligned}\ ] ] inserting the decomposition ( [ eq : kkdec ] ) into the action and defining the projectors @xmath126 and @xmath127 , we derive the eoms @xcite @xmath128 where @xmath129 with @xmath130 and @xmath131 , as well as @xmath132 and @xmath133 . in order to avoid boundary terms due to integration by parts , we move the @xmath134-distribution by an infinitesimal amount into the bulk @xcite . we will indicate this limiting procedure by a superscript in the argument of the profiles , _ e.g. _ , by writing @xmath135 . the appropriate ir bcs for the profiles can be obtained by integrating the eoms ( [ eq : gaugeeom ] ) over an infinitesimal interval around @xmath136 . at the 5d level they have already been given in table [ tab : bcs ] . however , since the profiles of the scalar components are taken to be proportional to the @xmath24-derivative of the vector profiles , they develop discontinuities at the ir brane @xcite . we arrive at @xmath137 where @xmath138 notice that for the photon the right - hand side in ( [ eq : irbc2 ] ) is equal to zero . after applying the eoms and the orthonormality condition ( [ eq : ortho ] ) , we observe that the 4d action takes the desired canonical form , if @xmath139 the spectrum of the theory is determined by the ir bcs ( [ eq : irbc2 ] ) . the eigenvalues @xmath140 are thus solutions of @xmath141=0\,,\ ] ] with @xmath142 once the eigenvalues are known , the eigenvectors @xmath143 are determined by ( [ eq : irbc2 ] ) . we now derive expressions for the profiles @xmath144 . in order to get the eoms for the uv basis we multiply ( [ eq : gaugeeom ] ) by @xmath145 from the left . introducing the coordinate @xmath146 , we then write the solutions as @xcite @xmath147 with @xmath148 c_n^{(-)+}(t ) & = y_1(x_n \epsilon ) j_1(x_n t ) -j_1(x_n \epsilon ) \,y_1(x_n t ) \ , , \\[1 mm ] c_n^{(+)-}(t ) & = \frac{1}{x_n t}\ , \frac{d}{dt } \big ( t\,c_n^{(+)+}(t ) \big ) = y_0(x_n \epsilon ) \,j_0(x_n t ) -j_0(x_n \epsilon ) \,y_0(x_n t ) \ , , \\ c_n^{(-)-}(t ) & = \frac{1}{x_n t}\ , \frac{d}{dt } \big ( t\,c_n^{(-)+}(t ) \big ) = y_1(x_n \epsilon ) \,j_0(x_n t ) -j_1(x_n \epsilon ) \,y_0(x_n t ) \ , . \end{split}\ ] ] the masses of the kk states normalized to the kk scale , @xmath149 , are determined by the ir bcs as explained above . from the latter expressions , it is obvious that the profiles fulfill the uv bcs , since @xmath150 . the normalization constants @xmath151 are determined from the orthonormality condition ( [ eq : ortho ] ) . with respect to the formula given in @xcite , they contain additional terms due to the different uv bcs . we obtain @xmath152 ^ 2 + \left [ c_n^{(\pm)-}(1 ^ - ) \right]^2 - \frac{2}{x_n } \big ( \,c_n^{(\pm)+}(1)\,c_n^{(\pm)-}(1 ^ -)- \epsilon \,c_n^{(\pm)+}(\epsilon)\,c_n^{(\pm)-}(\epsilon^+ ) \big)\\ & - \epsilon^2 \left ( \left [ c_n^{(\pm)+}(\epsilon ) \right]^2 + \left [ c_n^{(\pm)-}(\epsilon^+ ) \right]^2\right ) . \end{split}\ ] ] note that , depending on the type of the uv bcs , some of the terms in ( [ eq : nngauge ] ) vanish identically . it will turn out to be useful to have simple analytical expressions for the masses and profiles of the lightest modes . expanding ( [ eq : irbc3 ] ) in powers of @xmath43 and inserting the definitions of the mixing angles ( [ eq : thetas ] ) , which connect the uv and ir bases , we arrive at analytic expressions for the masses of the @xmath153 and @xmath2 bosons . they read @xmath154 , \\ m_z^2 & = \frac{(g_l^2+g_y^2 ) \ , v^2}{4 } \left [ 1 - \frac{(g_l^2+g_y^2 ) \ , v^2}{8\mkk^2 } \left ( l - 1 + \frac{1}{2l}\right ) - \frac{(g_r^2-g_y^2 ) \ , v^2}{8\mkk^2 } \ , l + { { \cal o}}\left ( \frac{v^4}{\mkk^4 } \right ) \right ] , \quad \end{split}\end{aligned}\ ] ] where the last terms inside the square brackets are new compared to the minimal model studied in @xcite . interestingly , the latter terms are responsible for the custodial protection of the peskin - takeuchi @xcite parameter @xmath1 , which is sensitive to the difference between the corrections to the @xmath153- and @xmath2-boson vacuum - polarization functions and thus measures isospin violation . the set of oblique corrections , which are defined as @xmath155 , \\ t & = \frac{4 \hspace{0.5 mm } \pi}{e^2 c^2_w m_z^2 } \ , \big [ \pi_{ww}(0)-c^2_w \ , \pi_{zz}(0 ) -2 \ , s_w c_w \ , \pi_{za}(0)- s^2_w \ , \pi_{aa}(0 ) \big]\,,\\ u & = \frac{16 \hspace{0.5 mm } \pi s^2_w}{e^2 } \ , \big [ \pi_{ww}^{\hspace{0.25 mm } \prime}(0)-c^2_w \ , \pi_{zz}^{\hspace{0.25 mm } \prime}(0 ) -2 \ , s_w c_w \ , \pi_{za}^{\hspace{0.25 mm } \prime}(0 ) -s^2_w \ , \pi_{aa}^{\hspace{0.25 mm } \prime}(0 ) \big ] \ , , \end{split}\ ] ] can be computed in an effective lagrangian approach @xcite . gauge invariance guarantees that @xmath156 to all orders in perturbation theory , and one further has @xmath157 as long as one works at tree level . the non - zero tree - level correlators @xmath158 with @xmath159 are calculated from the corrections to the zero - mode masses ( [ eq : mwmz ] ) and profiles ( [ eq : expprof ] ) , where the latter also give rise to non - zero derivatives @xmath160 of the correlators at zero momentum . we find @xmath161 , \\ \pi^{\hspace{0.25 mm } \prime}_{ww}(0)&=\frac{g_l^2v^2}{8 \mkk^2 } \left(1-\frac 1 l\right ) , \\ \pi_{zz}(0)&=-\frac{(g_l^2+g_y^2 ) \ , v^4}{32 \mkk^2}\left[\left(g_l^2+g_y^2\right)\left(l-\frac 1{2l}\right)+\left(g_r^2-g_y^2\right ) l\right ] , \\ \pi^{\hspace{0.25 mm } \prime}_{zz}(0)&=\frac{(g_l^2+g_y^2 ) \ , v^2}{8 \mkk^2 } \left(1-\frac 1 l\right ) . \end{split}\ ] ] inserting these expressions into ( [ eq : studef ] ) yields @xmath162 in agreement with @xcite . in contrast to the minimal model @xcite there is no @xmath36-enhanced term in the @xmath1 parameter . it has been cancelled by the additional corrections appearing in the contributions to the mass formula ( [ eq : mwmz ] ) , which introduces extra terms in the correlators @xmath163 and @xmath164 . a related discussion including estimates of loop effects on the @xmath1 parameter has been given in @xcite . the one - loop corrections to the @xmath165 parameter in the custodial rs model arising from higgs loops have been calculated in @xcite and found to be logarithmically uv divergent . this could result in a large and positive @xmath165 parameter , which is rather problematic @xcite in view of the consistency of the global fit of the oblique electroweak precision observables . the zero - mode profiles , which were used for the above derivations , read @xmath166 , \\[1 mm ] \chi_0^{(-)}(t ) & = \sqrt{\frac{l}{2\pi}}\ ; t^2 \left [ \ , -2 + \frac{x_{a}^2}{4 } \left(t^2-\frac 2 3\right ) + { { \cal o}}\left ( x_a^4 \right ) \right ] , \end{split}\ ] ] for @xmath159 . here @xmath167 denotes the relevant zero - mode solution in ( [ eq : mwmz ] ) . the profiles @xmath168 with neumann ir bc are identical to those appearing in the minimal model , while the profiles @xmath169 satisfying dirichlet ir bc scale like @xmath170 , reflecting the localization of kk modes close to the ir boundary . notice that ( [ eq : expprof ] ) contains , besides the @xmath171-independent terms that are included in ( [ eq : stures ] ) , also @xmath171-dependent contributions that will in general lead to non - universal vertex corrections . while these corrections modify the interactions of the sm fermions with the @xmath153 and @xmath2 bosons , they turn out to be negligibly small for light fermions localized near the uv brane . this is the case for the first two generations of sm fermions , and it helps to avoid excessive contributions to fcncs . in such a case the oblique corrections are adequately parametrized by the @xmath165 , @xmath1 , @xmath172 parameters as given in ( [ eq : stures ] ) . finally , we can also expand @xmath173 . including corrections up to @xmath43 , we find @xmath174 where the second component parametrizes the admixture of @xmath175 in the zero mode . as we will see below in section [ sec : custodialprotection ] , the results ( [ eq : expprof ] ) and ( [ eq : veca0a ] ) play a crucial role in the custodial protection mechanism of the @xmath14 vertex and its flavor - changing counterparts . in this section we will show how to compute the following sum over gauge - boson profiles @xmath176 weighted by inverse powers of the normalized kk mode masses @xmath177 , @xmath178 which arises when one attempts to calculate the tree - level exchange of a sm electroweak gauge boson accompanied by its kk excitations in the limit of zero ( or negligibly small ) momentum transfer . the infinite sum in ( [ eq : kksum1 ] ) can be calculated by employing the methods developed in @xcite . we first integrate the eoms ( [ eq : gaugeeom ] ) twice , accounting for the bcs on both the uv and ir brane . after switching to @xmath171 coordinates this yields @xmath179 \ ! { \bm p}_{(+ ) } \ , \frac{\vec{\chi}_n^{\ , a } ( \epsilon)}{\big ( x_n^a \big)^2 } \,,\ ] ] where we have defined @xmath180 using the completeness relation @xmath181 for the gauge - boson profiles , it is then easy to prove that @xmath182 where @xmath183 . with these results at hand it is now a matter of simple algebraic manipulations to arrive at @xmath184 \\[1 mm ] & \phantom{xx } + \left [ \,\bm{1 } - \left ( t^2 - \epsilon^2 \right ) { \bm x}_a \ , \right ] { \bm p}_{(+ ) } \ , { \bm \sigma}_a ( \epsilon , \epsilon ) \ , { \bm p}_{(+ ) } \left [ \ , \bm{1 } - \left ( t^{\prime \ , 2 } - \epsilon^2 \right ) { \bm x}_a \ , \right]^t , \end{split}\ ] ] which is exact to all orders in @xmath185 . with the help of the orthonormality relation ( [ eq : ortho ] ) , the remaining sum over gauge profiles evaluated at the uv brane can be brought into the form @xmath186 \ , \bigg ] ^{-1 } { \bm p}_{(+)}\ , , \end{split}\ ] ] where @xmath187 denotes the @xmath188 component of the corresponding zero - mode vector . this formula has the advantage that it can be easily expanded in powers of @xmath43 using ( [ eq : expprof ] ) and ( [ eq : veca0a ] ) . retaining the first two terms in the expansion leads to @xmath189 + { \cal o } ( x_a^2 ) \right ) { \bm p}_{(+)}\,.\ ] ] keeping in mind that @xmath190 and dropping phenomenologically irrelevant terms of second order in @xmath191 , we finally arrive at @xmath192 + \left [ \frac{1}{2 \pi x_a^2 } + \frac{1}{4 \pi } \left ( 1 - \frac{1}{2 l } \right ) \right ] { \bm p}_{(+ ) } + { \cal o } ( x_a^2 ) \,,\ ] ] where @xmath193 having at hand an analytic expression for the zero - mode contribution to ( [ eq : kksum1 ] ) alone , @xmath194 will also prove useful later in our discussion . employing the results ( [ eq : expprof ] ) and ( [ eq : veca0a ] ) , a straightforward calculation leads to @xmath195 \\[1 mm ] & \phantom{xx } + \left [ \ , \frac{1}{2 \pi x_a^2 } + \frac{1}{4 \pi } \left ( 1 - \frac{1}{l } + t^2 \left ( \frac{1}{2 } - \ln t \right ) + t^{\prime \ , 2 } \left ( \frac{1}{2 } - \ln t^\prime \right ) \right ) \right ] { \bm p}_{(+ ) } + { \cal o } ( x_a^2 ) \ , . \hspace{4 mm } \end{split}\end{aligned}\ ] ] comparing ( [ eq : sigmafinal ] ) to ( [ eq : pifinal ] ) we see that all @xmath36-enhanced terms in @xmath196 besides the one proportional to the non - factorizable term @xmath197 arise from the zero - mode contribution @xmath198 . factorizable contributions due to the ground - state @xmath153 and @xmath2 bosons are thus enhanced by the logarithm of the warp factor with respect to the contributions from the tower of kk excitations @xcite . we also recall that the term @xmath197 reflects the full 5d structure of the rs model , which is lost when one considers only a few low - lying kk modes @xcite . our analytic results for @xmath199 and @xmath200 will turn out to be phenomenologically quite important , as they allow for a clear understanding of the cancellation of certain terms in @xmath18 and @xmath19 fcnc interactions . in particular , the exact form of the matrix @xmath201 and its interplay with the terms proportional to the @xmath202 unit matrix @xmath203 are key ingredients for the custodial protection of the flavor - conserving @xmath14 coupling as well as of the flavor - violating @xmath204 vertices . we will for the moment no further dwell on this issue , but will return to it in detail in section [ sec : custodialprotection ] . before moving on , let us remark that the kk sums involving photon and gluon excitations do not depend on whether the electroweak gauge group is minimal or extended , so that the results for the corresponding sums ( excluding the zero modes ) can be taken over from @xcite . we will now present the explicit realization of the quark sector in the model under consideration . then we will turn to the kk decomposition and derive the bulk profiles for the corresponding fields . as we want to have a custodial protection of the @xmath205 vertex @xcite , we impose a discrete @xmath3 symmetry that interchanges the two @xmath16 groups . as a consequence , the left - handed bottom quark has to be part of a @xmath206 bi - doublet with isospin quantum numbers @xmath207 ( see section [ sec : custodialprotection ] ) . this fixes the quantum numbers of the other fields uniquely and implies the following multiplet structure for the quark fields with even @xmath21 parity : @xmath208 { \cal t}_r & \equiv { \cal t}_{1r}\oplus{\cal t}_{2r}\equiv\left(\begin{array}{c } { \lambda_r^{\prime\ , ( -)}}_{\frac 53}\\ { u_r^{\prime\ , ( -)}}_{\frac 23}\\ { d_r^{\prime\ , ( -)}}_{-\frac 13 } \end{array}\right)_{\frac 23 } \oplus\left(\begin{array}{ccc } { d_r^{(+)}}_{-\frac 13}\ { u_r^{(-)}}_{\frac 23}\ { \lambda_r^{(-)}}_{\frac 53 } \end{array}\right)_{\frac 23}\ , . \end{split}\ ] ] here the superscripts @xmath115 and @xmath116 of the chiral fields specify the type of bc on the uv boundary , and as before we have not explicitly shown the bcs at the ir brane , which are understood to be of neumann type in all cases . the choice of the parities is motivated by the constraint to arrive at a low - energy spectrum of the theory that is consistent with observations . the subscripts correspond to the @xmath48 and @xmath209 charges , respectively , which are connected through the relations @xmath210 and @xmath211 . for completeness and future reference , we summarize the quantum numbers of the quark fields in table [ tab : charges ] . the right - handed down - type quarks have to be embedded in a @xmath63 triplet in order to arrive at an @xmath209-invariant yukawa coupling . note that we have chosen the same @xmath55 representations for all three generations , which is necessary if one wants to consistently incorporate quark mixing in the fully anarchic approach to flavor in warped extra dimensions . the chosen representations also play a crucial role in the suppression of flavor - changing left - handed @xmath2-boson couplings @xcite . altogether they feature 15 different quark fields in the up - type and nine in the down - type sector . due to the bcs , there will be three light modes in each sector to be identified with the sm quarks . these are accompanied by kk towers which consist of groups of 15 and nine modes of similar masses in the up- and down - type quark sector , respectively . moreover one also faces a kk tower of exotic fermion fields of electric charge @xmath212 , which exhibits nine excitations with small mass splitting in each level . in addition to ( [ eq : multiplets ] ) we have a second set of multiplets , belonging to the components of opposite chirality . the corresponding states have opposite bcs . in particular , they all obey dirichlet bcs at the ir brane . remember that the @xmath62 transformations act vertically , while the @xmath63 transformations act horizontally on the multiplets . [ cols="^,^,^,^,^,^",options="header " , ] compared to gluon - gluon fusion , higgs - boson production through weak gauge - boson fusion , @xmath213 with @xmath214 , which is known to be extremely useful for discovery at the lhc , receives only moderate corrections of around @xmath215 ( @xmath216 ) for @xmath217 ( @xmath218 ) . the same reduction will affect associated @xmath153-boson production , @xmath219 , which is the only channel that in principle would allow for a higgs discovery at the tevatron . the rs predictions for the production cross section for @xmath220 at the tevatron and for @xmath221 at the lhc are illustrated by the solid blue lines in the left and right panels of figure [ fig : prodplots ] , respectively . the corresponding sm predictions are taken from @xcite and represented by the blue dashed lines . finally , the cross section of associated top - quark pair production , @xmath222 , will also experience a reduction . for values of the kk scale in the ballpark of @xmath223 , this suppression can amount up to @xmath224 . since @xmath225 , @xmath226 , and @xmath227 are tree - level processes , their rs predictions have all been obtained by a simple rescaling of the corresponding sm results . in summary , we find that the main higgs - boson production modes at hadron colliders are suppressed in the custodial rs model relative to the sm . suppression effects in @xmath228 were also reported in @xcite . a direct numerical comparison with our findings is however not possible , since @xcite only included zero - mode corrections , while @xcite studied rs variants that differ from the specific set - up considered here . in @xcite the authors studied corrections to gluon - gluon fusion arising from virtual exchange of light fermionic kk modes . there it has been claimed that for a heavy bottom - quark partner with a mass @xmath229 of a few hundred gev the higgs - boson production cross section via @xmath228 can be significantly enhanced . we would like to point out in this context that in order to achieve @xmath230 with the embedding of quarks as chosen in ( [ eq : multiplets ] ) , the @xmath3 symmetry has to be broken strongly via the bulk mass parameters of the @xmath231 multiplets by choosing @xmath232 rather far away from @xmath233 . while for @xmath234 it is possible to achieve @xmath235 and thus an enhancement of the @xmath236 cross section , such choices of parameters need to be fine - tuned to reproduce the measured mass spectrum of the sm quarks for anarchic yukawa couplings . if , on the other hand , @xmath237 , we find that @xmath238 remains strictly negative , and as a result the @xmath228 channel experiences a reduction . we furthermore add that choices of @xmath232 corresponding to a strong breaking of the @xmath3 symmetry lead , barring an accidental cancellation , to a sizable negative shift in the @xmath14 coupling through ( [ deltad2 ] ) , which is problematic in view of the stringent constraints arising from the @xmath239 pseudo observables . to which extent electroweak precision measurements constrain the masses of light fermionic kk partners deserves further study . we now move on to study the decay modes of the higgs boson . in this context , we will consider all processes with quarks and gauge bosons in the final state that can receive important rs corrections and have a branching fraction larger than @xmath240 . as we have not explicitly specified the embedding of the fermions in the lepton sector , we ignore decays into taus and muons . due to the uv localization of the leptonic fields , we however expect that the decay widths of the higgs into charged leptons are all sm - like . furthermore , we will not include loop contributions of kk leptons in our analysis of the @xmath241 and @xmath242 decay channels . we will comment on the potential impact of this omission below . in order to be able to calculate the decay rates of the higgs boson into massive gauge bosons , we still need to evaluate the rs corrections to the @xmath243 , @xmath244 , and @xmath245 tree - level vertices . due to the unbroken @xmath48 gauge group , the @xmath246 coupling is unchanged with respect to the sm to all orders in @xmath43 . the weak couplings involving the higgs boson are derived from the cubic and quartic interactions due to ( [ eq : d ] ) . in unitary gauge , the relevant terms in the lagrangian read @xmath247 , \ ] ] where @xmath248+{{\cal o}}\left(x_v^4\right ) , \end{split}\end{aligned}\ ] ] and @xmath249 for @xmath214 . in the case of the @xmath3 symmetry ( [ plr ] ) , one has @xmath250 and @xmath251 , which implies that the leading correction due to @xmath252 takes the form @xmath253 . for @xmath217 ( @xmath254 ) these terms lead to a suppression of the @xmath243 and @xmath244 couplings by about @xmath255 ( @xmath256 ) compared to the sm . notice that in the minimal rs model the expressions ( [ eq : hbosoncoupl ] ) hold in the limit @xmath257 , and consequently the corrections to the couplings of the higgs to massive gauge bosons are smaller by about a factor of 2 . our finding that the couplings @xmath243 and @xmath244 experience a reduction from their sm expectations confirms the model - independent statements made in @xcite . the partial decay widths @xmath258 of the higgs boson decaying to a final state @xmath259 are again obtained by rescaling the sm decay widths . we use @xmath260 with @xmath261 in the case of the decay of the higgs boson into a pair of @xmath153 and @xmath2 bosons , respectively . the relevant @xmath262 parameters for decays into two gluons , top or bottom quarks have already been given in ( [ eq : kappagg ] ) and ( [ tbratios ] ) . in figure [ fig : hprodec ] the diagrams inducing the decay into a pair of heavy quarks and massive gauge bosons are shown on the right in the top row . apart from the change in the @xmath263 coupling , we neglect rs corrections to the three - body decay @xmath264 , which relative to the two - body mode @xmath265 amounts to a correction of ( far below ) 1% in the sm . given the smallness of this effect , the omission of possible new - physics effects in the @xmath266 coupling that would affect the @xmath267 channel is for all practical purposes irrelevant . in the case of the final state with two photons , we employ @xmath268 in ( [ eq : gammahtof ] ) , where @xmath269 , @xmath270 , @xmath271 , @xmath272 , @xmath273 , and the explicit expression for the form factor @xmath274 , encoding the @xmath153-boson contribution , can be found in appendix [ app : formfactors ] . the first , second , and third terms in the numerator describe the effects of virtual heavy - quark , @xmath153-boson , and kk - quark exchange , respectively . the corresponding one - loop graphs are shown on the left in the bottom row of figure [ fig : hprodec ] and in the center plot of figure [ fig : hkkcontr ] . the amplitude @xmath275 interferes destructively with the quark contribution @xmath276 , falling from @xmath277 for @xmath278 to @xmath279 at the @xmath280 threshold @xmath281 and finally approaching @xmath282 in the limit @xmath283 . comparing these numbers with the ones for @xmath276 quoted earlier , one observes that within the sm the @xmath153-boson contribution to the @xmath284 decay amplitude is always dominant below threshold . we emphasize that in ( [ kappagamma ] ) contributions from leptonic kk modes are not included . while the precise impact of these effects depends on the exact realization of the lepton sector ( which we have not specified ) , it is possible to predict their relative sign as well as estimate their size . generalizing the result ( [ kappagamma ] ) to include contributions from triangle diagrams with kk leptons only requires to perform the replacement @xmath285 where @xmath286 and the parameter @xmath287 encodes the effects due to kk - lepton loops . under the reasonable assumption that @xmath288 , we conclude from ( [ kkleptonsinkappaa ] ) that the kk lepton contribution to the @xmath289 amplitude amounts to approximately @xmath290 of the kk quark corrections and interferes constructively with the latter . based on this estimate we expect that an omission of kk lepton effects in the calculation of @xmath291 has only a minor numerical impact on the obtained higgs - boson branching fractions . the quantity @xmath292 representing the one - loop contribution of the @xmath153-boson kk modes can be calculated analytically in the decoupling limit . the corresponding feynman diagram is displayed on the very right in figure [ fig : hkkcontr ] . employing the results for the kk sums derived in section [ sec : kksum ] , we obtain @xmath293 \hspace{0.5 mm } \vec{d}_w \ , \left ( -\frac{21}{4 } + { \cal o } \left ( 1/\tau_n^w \right ) \right ) \\ & = -\,\frac{21}8\ , \delta g_h^w \left(1+{\cal o } \left ( 1/\tau_n^w \right ) \right ) , \end{split}\ ] ] where @xmath294 and @xmath295 . since already @xmath296 , the terms suppressed by powers of @xmath297 in ( [ eq : nugammaw ] ) can be ignored in practice . the result for @xmath298 can be found in ( [ eq : hbosoncoupl ] ) . in order to compute the last missing decay channel , namely @xmath299 , we use @xmath300 in ( [ eq : gammahtof ] ) . here @xmath301 , and @xmath302 for @xmath303 . the amplitudes @xmath304 encoding the effects of virtual quarks and @xmath153 bosons in @xmath242 are collected in appendix [ app : formfactors ] . the corresponding feynman diagrams are shown on the right in the bottom row of figure [ fig : hprodec ] . like in the case of @xmath289 , the sm decay rate for @xmath299 is in large parts of the parameter space dominated by the @xmath153-boson loop contribution . one has @xmath305 for @xmath306 and @xmath307 for @xmath308 . on the other hand , the function @xmath309 rises from around @xmath310 to @xmath311 between @xmath312 and @xmath281 , and then falls to approximately @xmath313 in the limit @xmath283 . the first term in the numerator of ( [ kappagammaz ] ) depends on the ratios @xmath314 which quantify the relative shift in the vector coupling of the @xmath2 boson to top and bottom quarks . in the left panel of figure [ fig : vectorzplot ] we show the predictions for @xmath315 versus @xmath316 for 150 randomly chosen model parameter points . it is evident from the plot that the vector coupling of the @xmath2 boson to top quarks is always reduced in the custodial rs model relative to the sm . numerically , the suppression amounts to a moderate effect of @xmath256 ( @xmath317 ) for @xmath217 ( @xmath318 ) . in contrast , the @xmath2-boson coupling to bottom - quark pairs is larger than its sm value , but numerically the resulting effects turn out to be negligibly small due to the custodial protection mechanism . consequently , we will set @xmath319 to 1 in our numerical analysis . parameterizing the average value of the relative shift @xmath320 by @xmath321 the coefficient @xmath322 can again be determined through a fit . employing the shown set of parameter points , we obtain the value for @xmath322 given in table [ tab : kappas ] . the second term in the numerator of ( [ kappagammaz ] ) encodes the contribution to the @xmath242 transition arising from the @xmath153-boson triangle graph . the calculation of this zero - mode contribution is greatly simplified by the following two observations . first , one has @xmath323 and second @xmath324 ^ 2 = { \cal o } ( v^4/\mkk^4)$ ] . the expressions for @xmath325 and @xmath326 necessary to derive these results can be found in ( [ eq : expprof ] ) and ( [ eq : veca0a ] ) . in combination these two relations imply that the triple gauge - boson vertex involving two @xmath153- and one @xmath2-boson fields does not receive corrections at @xmath327 in the rs model , regardless of the specific gauge group . by the same line of reasoning , it is also readily seen that all quartic gauge - boson vertices first differ at order @xmath328 from the corresponding sm expressions . in view of this extra suppression , we will set the triple gauge - boson couplings of the zero modes to their sm values when evaluating the higgs - boson branching fractions . in this approximation the effect of virtual @xmath153-boson exchange to ( [ kappagammaz ] ) is simply given by the combination @xmath329 , which up to the different form factor resembles the form of the corresponding term in ( [ kappagamma ] ) . the third term in the numerator of ( [ kappagammaz ] ) describes the contribution to the @xmath242 amplitude stemming from the virtual exchange of kk quarks . the corresponding one - loop diagram is displayed in the middle of figure [ fig : hkkcontr ] . in the up - type quark sector we find @xmath330 where @xmath331 denotes the relative strength of vector coupling of the @xmath2 boson to the @xmath332 up - type quark kk mode defined in analogy to ( [ eq : kappaiv ] ) , and @xmath333 . analog expressions apply in the case of down- and @xmath334-type quark kk modes . since an analytic calculation of ( [ eq : nugammazu ] ) turns out to be impractical , we resort to a numerical evaluation of the kk sum employing the method described in section [ sec : higgsproduction ] . the predictions for the real parts of @xmath335 , @xmath336 , and @xmath337 corresponding to a set of 150 random model parameter points are depicted in the right panel of figure [ fig : vectorzplot ] . the solid lines displayed there indicate the best fit of the form @xmath338 to the sample of points with kk scales in the range @xmath339 \ , { \rm tev}$ ] . as before , points with @xmath340 have been excluded in the fit , since they are subject to significant higher - order corrections . the corresponding coefficients @xmath341 can be found in table [ tab : kappas ] . the average values of the real parts of @xmath335 , @xmath342 , and @xmath337 obtained from the fit formulas are @xmath343 ( @xmath344 ) , @xmath345 ( @xmath346 ) , and @xmath347 ( @xmath348 ) for @xmath217 ( @xmath218 ) , respectively . the imaginary parts @xmath349 turn out to be tiny . the reason for this feature has already been discussed in section [ sec : higgsproduction ] . contributions from kk - lepton triangle graphs have again not been incorporated in ( [ kappagammaz ] ) . denoting these corrections by @xmath350 , they can be included via the simple replacement @xmath351 in order to estimate the typical size of @xmath350 we need an analytic formula for the relative strength of the vector coupling between the @xmath2 boson and fermionic kk modes appearing in ( [ eq : nugammazu ] ) . we find @xmath352 where the expressions for @xmath353 can be found in ( [ eq : delta2 ] ) . in the case of extended @xmath3 symmetry ( [ eq : extendedplr ] ) , it turns out that for down- and @xmath334-type kk quarks the result for @xmath354 can be expressed in terms of the electric charge and the third component of the weak isospin of the involved fermion , while no such formula can be derived for up - type quark kk modes . we obtain to excellent approximation ( @xmath355 ) @xmath356 which implies that all down - type ( @xmath334-type ) kk - quark modes couple with universal strength to the vector part of the @xmath2-boson coupling . it follows that in the decoupling limit , @xmath357 , one has @xmath358 from the numbers of the fit coefficients given in table [ tab : kappas ] , we see that this relation is satisfied to an accuracy of around @xmath359 . the kk - fermion effects in the down- and @xmath334-type quark sectors that contribute to @xmath360 , and @xmath361 are thus universal , in the sense that they can be simply obtained from each other by an appropriate replacement of the vector couplings of the external fields . making now the plausible assumption that in the decoupling limit the sums @xmath336 and @xmath362 differ only by the presence of the vector couplings @xmath363 and @xmath364 , we obtain the following estimate for the contribution to ( [ kkleptonsinkappaaz ] ) from leptonic relative to down - type quark kk modes : @xmath365 as a result , the sum ( [ kkleptonsinkappaaz ] ) can be approximated as @xmath366 where the last term on the right - hand side encodes the effects due to kk leptons , and in order to obtain the numerical values we have inserted the relevant electroweak quantum numbers and used @xmath367 . for @xmath368 , the real part of the relation ( [ eq : sumlkk ] ) evaluates to @xmath369 ( @xmath370 ) if effects due to kk leptons are excluded ( included ) . while these numbers imply that an omission of kk lepton effects can change the numerical value of the kk fermion contribution notably , it is not difficult to see that the impact on ( [ kappagammaz ] ) itself is limited , since the coefficient @xmath371 is dominated by the @xmath153-boson triangle contribution . we thus conclude that the absence of kk - lepton contributions in our prediction for @xmath242 ( which is anyhow difficult to study at the lhc ) will not change any of the conclusions drawn below . the coefficient @xmath372 in ( [ kappagammaz ] ) incorporates the effects in @xmath242 due to charged kk - boson excitations in the loop . the associated feynman graph is displayed on the very right in figure [ fig : hkkcontr ] . this contribution can be written as @xmath373 with @xmath374}\ , , \end{split}\ ] ] and @xmath375 . notice that the prefactor in the second line of the above formula corresponds to the choice @xmath376 . since the first term in the sum of ( [ eq : nugammaz ] ) is already suppressed by a factor of @xmath43 , the computation of @xmath372 to this order only requires the knowledge of the overlap integral ( [ eq : iwwz ] ) to zeroth order in the ratio of the weak over the kk scale . we obtain @xmath377 it is again an excellent approximation to evaluate the loop function @xmath378 in the infinite mass limit @xmath379 , in which the form factor approaches @xmath380 . we perform the sum in ( [ eq : nugammaz ] ) numerically , including sufficiently many kk levels until the series converges . in this way , we find @xmath381 @xmath382 for @xmath383 ( @xmath384 ) . values for @xmath372 corresponding to different kk scales can be obtained by means of the fit formula @xmath385 with the coefficient @xmath386 given in table [ tab : kappas ] . in the two panels of figure [ fig : kappas ] we display the relative corrections @xmath387 , @xmath388 , and @xmath389 for @xmath217 ( left ) and @xmath390 ( right ) . the depicted curves represent the rs results obtained from ( [ eq : hbosoncoupl ] ) and ( [ eq : nugammaw ] ) as well as the relevant fit formulas with the values of the coefficients collected in table [ tab : kappas ] . while the behavior of @xmath387 has already been explained in section [ sec : higgsproduction ] , we see that @xmath391 is close to 1 and independent of the value of the higgs - boson mass . this implies that the partial decay width @xmath392 in the custodial rs model is essentially unchanged with respect to the sm . the relative correction @xmath388 is , on the other hand , a non - trivial function of @xmath393 . below the @xmath280 threshold , the @xmath153-boson amplitude dominates the sm @xmath289 decay rate and the contributions due to kk quarks and @xmath153 bosons both interfere constructively with the sm gauge - boson triangle graph . for @xmath394 , the new - physics contributions amount to around @xmath395 ( @xmath396 ) of the total sm amplitude for @xmath368 ( @xmath397 ) , resulting in values @xmath398 ( @xmath399 ) . for @xmath400 , the higgs - mass dependence of the sm amplitude becomes less pronounced and the rs prediction stays almost constant . the strong rise of @xmath388 , visible at higher values of the higgs mass , results from the fact that for @xmath401 the top - quark loop nearly cancels the @xmath153-boson contribution in the sm . in consequence , for @xmath402 the partial width @xmath403 is almost entirely due to loops involving heavy kk modes , with the contribution from kk quarks being the dominant correction . the various higgs - boson branching ratios obtained using the above results are shown in figure [ fig : hxx ] . the dashed lines illustrate the sm expectations calculated with the help of hdecay @xcite , , the original input file of hdecay version 3.51 is used . ] while the solid lines represent the rs predictions based on the results for @xmath404 quoted above and the curves for @xmath405 displayed in figure [ fig : kappas ] . it is evident that in the custodial rs model the branching ratios @xmath406 , @xmath407 , and @xmath408 receive only insignificant corrections , not exceeding the level of @xmath409 . for @xmath410 the experimentally cleanest signature for the discovery of the higgs boson at the lhc is its `` golden '' decay to four leptons , @xmath411 . since the @xmath408 branching fraction is essential sm - like , the reduction in the @xmath228 production cross section will make an observation of the higgs boson in the golden channel more difficult . moderate effects occur in the non - discovery channels @xmath299 and @xmath412 . in the relevant ranges for the higgs mass , the modifications in the branching ratios amount to around @xmath413 ( @xmath414 ) and @xmath415 ( @xmath255 ) for @xmath217 ( @xmath218 ) . the most pronounced effects are found for @xmath416 and @xmath289 . for higgs masses below the @xmath280 threshold , the branching fraction of the former mode is reduced by a factor of almost 4 ( 8) , while the branching ratio of the latter transition is enhanced by a factor of around 4 ( 2 ) . the corresponding maximal values of @xmath417 are @xmath418 ( @xmath419 ) for @xmath217 ( @xmath254 ) and arise at @xmath420 . calculating the rescaling factor @xmath421 for @xmath422 and the quoted maximal branching fractions , we obtain the values @xmath423 ( @xmath424 ) . these numbers suggest that the statistical significance for a lhc discovery of the higgs boson in @xmath289 can be enhanced in the custodial rs model for low kk scales . a detailed study of how the deviations found in the rs framework affect the searches for the higgs boson at the lhc will be presented elsewhere . we add that if the kk scale is lowered to @xmath425 , the branching ratio of @xmath426 can reach values above @xmath240 for higgs masses above @xmath427 . , the corresponding decay rate is simply obtained from ( [ brthc ] ) by multiplying the branching fraction for @xmath428 with @xmath429 and replacing @xmath430 through @xmath431 . of course , an analogous formula applies in the case of @xmath432 . ] for such a low kk scale , also the decay channel @xmath433 can open up below the @xmath280 threshold , but typically stays below the level of @xmath434 . note that our results for the higgs - boson branching fractions depend primarily on the value of the kk scale , and are rather insensitive to the other free parameters present in the model . for example , the final results do not strongly depend on the precise localization pattern of the fermionic bulk fields . we also verified that the omission of kk - lepton effects does not have a pronounced effect . rs predictions for the various branching fractions of the higgs boson have been presented previously in @xcite . yet a direct comparison with our results is difficult , as the latter work only includes rs corrections affecting the tree - level couplings of the higgs boson to fermions . we have performed a thorough analysis of the structure of tree - level effects in the rs model with enlarged bulk gauge symmetry @xmath0 and an ir brane - localized higgs sector . in contrast to the existing literature , where the yukawa couplings have always been treated as a perturbation , we have performed the kk decomposition of the gauge fields in a covariant @xmath20 gauge within the basis of mass eigenstates , by constructing the exact solutions to the bulk equations of motion augmented with appropriate boundary conditions . the kk decomposition in the matter sector has been performed employing the same formalism and including the mixing of fermionic fields between different representations and generations in a completely general way . by expanding the exact results , we have derived simple analytic expressions for the profiles and masses of the various sm particles as well as for the sums over kk towers of gauge bosons , which include all terms up to second order in the ratio of the higgs vacuum expectation value @xmath435 over the kk mass scale @xmath316 . we have demonstrated that our exact approach is not only more elegant , but also offers some distinct advantages over treating the couplings of the bulk fields to the higgs sector perturbatively . by expanding the low - energy spectrum as well as the gauge couplings in powers of @xmath43 , we have obtained analytic formulas which allow not only for a numerical treatment , but for a transparent and explicit understanding of the model - specific protection mechanisms of the peskin - takeuchi parameter @xmath1 and the left - handed @xmath2-boson vertices involving down - type quarks . in the case of the gauge - boson corrections to the @xmath436 couplings , we have pointed out all terms that escape the custodial protection and identified them with the irreducible sources of @xmath3-symmetry breaking , originating from the different boundary conditions of untwisted and twisted gauge - boson profiles on the uv brane . unlike in the perturbative approach , which in general requires diagonalizing high - dimensional matrices numerically , the interpretation of our results in physical terms is thus very clear . by making the dependence on the implementation of the matter sector explicit , we were also able to address the important question about the model - dependence of the resulting gauge - boson interactions with sm fermions . we have shown in this context , that the @xmath3 symmetry is explicitly broken by the bulk mass parameters of the @xmath21-odd @xmath62 singlet fields if their values differ from the ones of their @xmath21-even counterparts . turning our attention to the charged - current interactions , we have then demonstrated that a custodial protection in not at work in this case . we have finally revisited the issue of the flavor - misalignment between fermion zero - mode masses and yukawa couplings , extending existing analyses of the structure of the flavor - changing higgs - boson couplings to the case of the rs scenario with custodial protection . subsequently we have considered some simple applications of our general results . a thorough discussion of the constraints imposed by the precision measurements of the bottom - quark pseudo observables opened our phenomenological survey . we found that , contrary to the minimal case , the prediction for the correction to the @xmath205 vertex in the rs model with extended @xmath3 symmetry is essentially independent of the left - handed bulk mass parameter of the third - generation quarks . this feature relaxes the bounds that originate from the precision measurements of the left - handed @xmath2-boson coupling significantly , giving a strong motivation to protect the latter vertex through a suitable embedding of the bottom quarks . we have furthermore pointed out that , irrespectively of the bulk gauge group and barring an unnatural large value of the bulk mass parameter of the right - handed top quark , the requirement to obtain the correct top- and bottom - quark masses excludes large corrections to the @xmath437 coupling . a direct explanation of the anomaly in the forward - backward asymmetry for bottom quarks seems therefore generically challenging in warped extra - dimension models in which the left - handed bottom and top quark are part of the same multiplet . allowing for a heavy higgs boson with a mass in the ballpark of 0.5 tev ( which is the naturally expected mass range for @xmath393 in models with a brane - localized higgs sector ) leads however to a good agreement between @xmath438 data and theory . yet , a heavy higgs boson would need tuning in models with custodial symmetry , since the shifts induced by @xmath439 in the parameters @xmath165 and @xmath1 can not be compensated by rs tree - level effects , and thus would require the presence of sizable oblique loop corrections in order not to spoil the global electroweak fit . detailed numerical analyses of the new - physics effects in rare top - quark decays as well as of the changes in the production cross section and branching fractions of the higgs boson completed our phenomenological investigations . in the former case , we found that due to the protection of the @xmath14 vertex , the experimental prospects for observing @xmath440 and @xmath441 are more favorable in the extended than in the minimal rs scenario . in particular , for kk gauge - boson masses below @xmath442 the branching fractions of both @xmath443 and @xmath428 can be within the reach of the lhc . in the latter case , our study revealed that due to the composite nature of the higgs boson , the top quark , and the kk modes , observable effects in higgs physics can naturally occur in the scenario under consideration . in order to arrive at this conclusion , we have performed the first complete one - loop calculation of all higgs - boson production and decay channels relevant at hadron colliders , incorparating all effects stemming from the extended electroweak gauge boson and fermion sectors . concerning the main higgs - boson production modes at the tevatron and the lhc , proceeding through @xmath228 , @xmath220 , and @xmath221 , we found that they are all suppressed in the custodial rs model relative to the sm . since the shifts in the production cross sections can exceed the combined experimental and theoretical uncertainties , the reduction in higgs events predicted in the rs framework might be observable at the lhc . on the other hand , the reduced @xmath228 production cross section should make an observation of the higgs boson with a mass above the @xmath444 threshold via the `` golden '' four - lepton channel more difficult , because the @xmath408 branching fraction remains essential sm - like in the custodial rs model . the possible enhancement of the branching ratio for @xmath289 might however lead to a higher statistical significance and a faster lhc discovery of the higgs boson , if its mass is below the @xmath280 threshold . we emphasize that our findings concerning higgs physics have to be considered robust predictions , since they depend rather weakly on the details of the spectrum ( and thus the specific rs parameter values ) once the contributions of the entire kk towers have been included . the analytical and numerical results obtained in this article form the basis for general calculations of flavor - changing processes in the custodial rs model . a detailed phenomenological analysis of the potential new - physics effects in neutral - meson mixing and in rare decays of kaons and @xmath445 mesons , including both inclusive and exclusive processes , is left for future work . it is a pleasure to thank a. azatov , m. toharia , and l. zhu for helpful correspondence concerning flavor - changing higgs - boson couplings . we are also grateful to v. ahrens , m. benzke , and d. dolce for useful discussions . the feynman diagrams shown in this work are drawn using feynarts @xcite . the research of s.c . is supported by the dfg cluster of excellence `` origin and structure of the universe '' . the research of f.g . , m.n . , and t.p . is supported in part by the german federal ministry for education and research grant 05h09ume ( `` precision calculations for collider and flavour physics at the lhc '' ) , and by the research centre `` elementary forces and mathematical foundations '' funded by the excellence initiative of the state of rhineland - palatinate . u.h . thanks the galilo galilei institute for theoretical physics for the hospitality and the infn for partial support during the final stage of this work . in this appendix we rederive ( [ eq : bcirrescaled ] ) and ( [ eq : gtil1 ] ) to ( [ eq : bmh ] ) , using the rectangular function @xmath446 \ , , \\[4 mm ] \ , 0 \ , , & { \rm otherwise } \ , , \end{cases}\ ] ] to regularize the @xmath134-functions appearing in the eoms ( [ eq : eom ] ) . keeping only terms relevant in the range @xmath447 $ ] , the eoms ( [ eq : eom ] ) close to the ir brane take the simpler form @xmath448 combining the first ( second ) with the fourth ( third ) relation and using ( [ eq : rect ] ) , we obtain @xmath449 \bm{s}_n^q(t ) & = 0\ , , \qquad \left [ \partial_t^2 - \bigg ( \frac{\bm{\tilde x}_{\vec q}}{\eta } \bigg ) ^2 \right ] \bm{s}_n^{\hspace{0.25 mm } q}(t ) = 0 \ , , \end{split}\ ] ] where @xmath450 imposing now the bcs @xmath451 and matching @xmath452 onto the solutions of ( [ eq : eom ] ) evaluated in the limit @xmath453 , we find that the differential equations ( [ eq : sdgl ] ) are solved by @xmath454 this implies that in the interval @xmath455 $ ] the @xmath21-even fermion profiles take the form @xmath456 reinserting the solutions ( [ eq : ssol ] ) and ( [ eq : csol ] ) into ( [ eq : eom_close_to_brane ] ) , allows us to determine the ir bcs which relate the @xmath21-even profiles with the -odd ones at @xmath457 . the resulting expressions read @xmath458 -{\bm s}_n^{\hspace{0.25 mm } q } ( 1 ^ - ) \ , \vec a_n^{\hspace{0.25 mm } q } & = \frac{v}{\sqrt{2 } m_{\rm kk } } \ , { \bm y}_{\vec q}^\dagger \ , \big ( \bm{x}_{\vec q } \big ) ^{-1 } \ , \tanh \big ( \bm{x}_{\vec q } \big ) \ , { \bm c}_n^{q } ( 1 ^ - ) \ , \vec a_n^{q } \ , , \end{split}\ ] ] which , after introducing the rescaled yukawa couplings @xmath459 , resembles ( [ eq : bcirrescaled ] ) employing the regularization ( [ eq : rect ] ) for the @xmath134-function , the flavor - changing higgs - boson couplings ( [ eq : deltagtilde ] ) become @xmath460 combining ( [ eq : ssol ] ) , ( [ eq : csol ] ) , and ( [ eq : irbcsagain ] ) and using @xmath461 valid for any arbitrary invertible matrix @xmath462 , we then obtain ( [ eq : gtil1 ] ) to ( [ eq : bmh ] ) . the central values and errors of the quark masses used in our analysis are @xmath463 they correspond to @xmath464 masses evaluated at the scale @xmath465tev , obtained by using the low - energy values as compiled in @xcite . the central values and errors of the wolfenstein parameters are taken from @xcite and read @xmath466 the central values and errors for the parameters entering our analysis of the bottom - quark pseudo observables are @xcite @xmath467 we refer to the central values for these quantities as sm reference values . unless noted otherwise , the reference value for the higgs - boson mass is @xmath468gev . the form factors @xmath469 and @xmath470 describing the effects of quark and @xmath153-boson loops in the production and the decay of the higgs boson are given by @xcite @xmath471 \ , , \\ a_{w}^h ( \tau ) & = -\frac{3}{4 } \left [ \hspace{0.25 mm } 2 + 3 \tau + 3 \tau \left ( 2 - \tau \right ) f ( \tau ) \hspace{0.25 mm } \right ] \ , , \\[2 mm ] a_{q}^h ( \tau , \lambda ) & = - i ( \tau , \lambda ) + j ( \tau , \lambda ) \ , , \\[1 mm ] a_{w}^h ( \tau , \lambda ) & = c_w \hspace{0.5 mm } \left \ { 4 \left ( 3 - \frac{s_w^2}{c_w^2 } \right ) i ( \tau , \lambda ) + \left [ \left ( 1 + \frac{2}{\tau } \right ) \frac{s_w^2}{c_w^2 } - \left ( 5 + \frac{2}{\tau } \right ) \right ] j ( \tau , \lambda ) \right \ } \ , . \end{split}\ ] ] the functions @xmath472 and @xmath473 take the form @xmath474 \ , , \\ j ( \tau , \lambda ) & = \frac{\tau \lambda}{2 \left ( \tau - \lambda \right ) } + \frac{\tau^2 \lambda^2}{2 \left ( \tau - \lambda \right ) ^2 } \ , \big [ f(\tau ) - f(\lambda ) \big ] + \frac{\tau^2 \lambda}{(\tau - \lambda)^2 } \ , \big [ g(\tau ) - g(\lambda ) \big ] \ , , \end{split}\ ] ] while the functions @xmath475 and @xmath476 read @xmath477 ^ 2 \ , , & \tau \leq 1 \ , , \\[4 mm ] \arcsin^2 \left ( \displaystyle \frac{1}{\sqrt{\tau } } \right ) \ , , & \tau > 1 \ , , \end{cases } \\[2 mm ] g(\tau ) & = \begin{cases } \sqrt{\tau - 1 } \hspace{0.5 mm } \arcsin \left ( \displaystyle \frac{1}{\sqrt{\tau } } \right ) \ , , & \tau \leq 1 \,,\\[6 mm ] \displaystyle \frac{1}{2 } \ , \sqrt{1 - \tau } \ , \left [ \ , \ln \left ( \displaystyle \frac{1 + \sqrt{1 - \tau}}{1 - \sqrt{1 - \tau } } \right ) - i \pi \ , \right ] \ , , & \tau > 1 \ , . \end{cases}\end{aligned}\ ] ] l. randall and r. sundrum , phys . lett . * 83 * , 3370 ( 1999 ) [ arxiv : hep - ph/9905221 ] . j. m. maldacena , adv . * 2 * , 231 ( 1998 ) [ int . j. theor . phys . * 38 * , 1113 ( 1999 ) ] [ arxiv : hep - th/9711200 ] . s. s. gubser , i. r. klebanov and a. m. polyakov , phys . b * 428 * , 105 ( 1998 ) [ arxiv : hep - th/9802109 ] . e. witten , adv . * 2 * , 253 ( 1998 ) [ arxiv : hep - th/9802150 ] . h. davoudiasl , j. l. hewett and t. g. rizzo , phys . b * 473 * , 43 ( 2000 ) [ arxiv : hep - ph/9911262 ] . a. pomarol , phys . b * 486 * , 153 ( 2000 ) [ arxiv : hep - ph/9911294 ] . s. chang , j. hisano , h. nakano , n. okada and m. yamaguchi , phys . 084025 ( 2000 ) [ arxiv : hep - ph/9912498 ] . y. grossman and m. neubert , phys . b * 474 * , 361 ( 2000 ) [ arxiv : hep - ph/9912408 ] . t. gherghetta and a. pomarol , nucl . b * 586 * , 141 ( 2000 ) [ arxiv : hep - ph/0003129 ] . s. j. huber and q. shafi , phys . b * 498 * , 256 ( 2001 ) [ arxiv : hep - ph/0010195 ] . s. j. huber , nucl . b * 666 * , 269 ( 2003 ) [ arxiv : hep - ph/0303183 ] . n. arkani - hamed and m. schmaltz , phys . d * 61 * , 033005 ( 2000 ) [ arxiv : hep - ph/9903417 ] . k. agashe , g. perez and a. soni , phys . lett . * 93 * , 201804 ( 2004 ) [ arxiv : hep - ph/0406101 ] . k. agashe , g. perez and a. soni , phys . d * 71 * , 016002 ( 2005 ) [ arxiv : hep - ph/0408134 ] . k. agashe , a. delgado , m. j. may and r. sundrum , jhep * 0308 * , 050 ( 2003 ) [ arxiv : hep - ph/0308036 ] . k. agashe , r. contino , l. da rold and a. pomarol , phys . b * 641 * , 62 ( 2006 ) [ arxiv : hep - ph/0605341 ] . m. blanke , a. j. buras , b. duling , s. gori and a. weiler , jhep * 0903 * , 001 ( 2009 ) [ arxiv:0809.1073 [ hep - ph ] ] . c. csaki , a. falkowski and a. weiler , jhep * 0809 * , 008 ( 2008 ) [ arxiv:0804.1954 [ hep - ph ] ] . o. gedalia , g. isidori and g. perez , phys . b * 682 * , 200 ( 2009 ) [ arxiv:0905.3264 [ hep - ph ] ] . m. bauer , s. casagrande , u. haisch and m. neubert , arxiv:0912.1625 [ hep - ph ] . h. davoudiasl , j. l. hewett and t. g. rizzo , phys . d * 68 * , 045002 ( 2003 ) [ arxiv : hep - ph/0212279 ] . m. s. carena , e. ponton , t. m. p. tait and c. e. m. wagner , phys . d * 67 * , 096006 ( 2003 ) [ arxiv : hep - ph/0212307 ] . m. s. carena , a. delgado , e. ponton , t. m. p. tait and c. e. m. wagner , phys . d * 68 * , 035010 ( 2003 ) [ arxiv : hep - ph/0305188 ] . s. casagrande , f. goertz , u. haisch , m. neubert and t. pfoh , jhep * 0810 * , 094 ( 2008 ) [ arxiv:0807.4937 [ hep - ph ] ] . f. goertz and t. pfoh , jhep * 0810 * ( 2008 ) 035 [ arxiv:0809.1378 [ hep - ph ] ] . m. bauer , s. casagrande , l. grnder , u. haisch and m. neubert , phys . d * 79 * ( 2009 ) 076001 [ arxiv:0811.3678 [ hep - ph ] ] . l. randall and m. d. schwartz , jhep * 0111 * ( 2001 ) 003 [ arxiv : hep - th/0108114 ] . m. e. albrecht , m. blanke , a. j. buras , b. duling and k. gemmler , jhep * 0909 * , 064 ( 2009 ) [ arxiv:0903.2415 [ hep - ph ] ] . g. burdman and l. da rold , jhep * 0811 * ( 2008 ) 025 [ arxiv:0809.4009 [ hep - ph ] ] . m. e. peskin and t. takeuchi , phys . * 65 * , 964 ( 1990 ) . m. e. peskin and t. takeuchi , phys . rev . d * 46 * ( 1992 ) 381 . c. csaki , j. erlich and j. terning , phys . d * 66 * ( 2002 ) 064021 [ arxiv : hep - ph/0203034 ] . a. delgado and a. falkowski , jhep * 0705 * , 097 ( 2007 ) [ arxiv : hep - ph/0702234 ] . m. s. carena , e. ponton , j. santiago and c. e. m. wagner , nucl . b * 759 * ( 2006 ) 202 [ arxiv : hep - ph/0607106 ] . m. s. carena , e. ponton , j. santiago and c. e. m. wagner , phys . d * 76 * ( 2007 ) 035006 [ arxiv : hep - ph/0701055 ] . j. hirn and v. sanz , phys . d * 76 * , 044022 ( 2007 ) [ arxiv : hep - ph/0702005 ] . a. azatov , m. toharia and l. zhu , phys . d * 80 * , 035016 ( 2009 ) [ arxiv:0906.1990 [ hep - ph ] ] . j. a. bagger , f. feruglio and f. zwirner , phys . * 88 * ( 2002 ) 101601 [ arxiv : hep - th/0107128 ] . a. j. buras , b. duling and s. gori , jhep * 0909 * , 076 ( 2009 ) [ arxiv:0905.2318 [ hep - ph ] ] . k. agashe , g. perez and a. soni , phys . d * 75 * ( 2007 ) 015002 [ arxiv : hep - ph/0606293 ] . k. agashe and r. contino , phys . d * 80 * , 075016 ( 2009 ) [ arxiv:0906.1542 [ hep - ph ] ] . b. duling , arxiv:0912.4208 [ hep - ph ] . j. h. field , mod . a * 13 * , 1937 ( 1998 ) [ arxiv : hep - ph/9801355 ] . s. schael _ et al . _ [ aleph collaboration ] , phys . rept . * 427 * , 257 ( 2006 ) [ arxiv : hep - ex/0509008 ] . a. b. arbuzov _ et al . _ , comput . commun . * 174 * , 728 ( 2006 ) [ arxiv : hep - ph/0507146 ] . a. djouadi , g. moreau and f. richard , nucl . b * 773 * , 43 ( 2007 ) [ arxiv : hep - ph/0610173 ] . c. bouchart and g. moreau , nucl . b * 810 * , 66 ( 2009 ) [ arxiv:0807.4461 [ hep - ph ] ] . the tevatron electroweak working group , arxiv:0803.1683 [ hep - ex ] . t. aaltonen _ et al . _ [ cdf collaboration ] , arxiv:0805.2109 [ hep - ex ] . j. carvalho _ et al . _ [ atlas collaboration ] , eur . j. c * 52 * , 999 ( 2007 ) [ arxiv:0712.1127 [ hep - ex ] ] . g. l. bayatian _ et al . _ [ cms collaboration ] , j. phys . g * 34 * , 995 ( 2007 ) . j. a. aguilar - saavedra and g. c. branco , phys . b * 495 * , 347 ( 2000 ) [ arxiv : hep - ph/0004190 ] . m. spira , arxiv : hep - ph/9510347 and references therein . a. falkowski , phys . d * 77 * , 055018 ( 2008 ) [ arxiv:0711.0828 [ hep - ph ] ] . i. low , r. rattazzi and a. vichi , jhep * 1004 * , 126 ( 2010 ) [ arxiv:0907.5413 [ hep - ph ] ] . j. r. ellis , m. k. gaillard and d. v. nanopoulos , nucl . b * 106 * , 292 ( 1976 ) . m. a. shifman , a. i. vainshtein , m. b. voloshin and v. i. zakharov , sov . j. nucl . * 30 * , 711 ( 1979 ) [ yad . * 30 * , 1368 ( 1979 ) ] . b. a. kniehl and m. spira , z. phys . c * 69 * , 77 ( 1995 ) [ arxiv : hep - ph/9505225 ] . a. pomarol , phys . * 85 * , 4004 ( 2000 ) [ arxiv : hep - ph/0005293 ] . w. d. goldberger and i. z. rothstein , phys . rev . lett . * 89 * , 131601 ( 2002 ) [ arxiv : hep - th/0204160 ] . k. agashe , a. delgado and r. sundrum , nucl . b * 643 * , 172 ( 2002 ) [ arxiv : hep - ph/0206099 ] . w. d. goldberger and i. z. rothstein , phys . d * 68 * , 125011 ( 2003 ) [ arxiv : hep - th/0208060 ] . r. contino , p. creminelli and e. trincherini , jhep * 0210 * , 029 ( 2002 ) [ arxiv : hep - th/0208002 ] . k. w. choi and i. w. kim , phys . d * 67 * , 045005 ( 2003 ) [ arxiv : hep - th/0208071 ] . w. d. goldberger and i. z. rothstein , phys . d * 68 * , 125012 ( 2003 ) [ arxiv : hep - ph/0303158 ] . f. del aguila and j. santiago , phys . b * 493 * ( 2000 ) 175 [ arxiv : hep - ph/0008143 ] . v. ahrens , t. becher , m. neubert and l. l. yang , eur . j. c * 62 * , 333 ( 2009 ) [ arxiv:0809.4283 [ hep - ph ] ] . r. v. harlander and w. b. kilgore , phys . lett . * 88 * , 201801 ( 2002 ) [ arxiv : hep - ph/0201206 ] . c. anastasiou and k. melnikov , nucl . b * 646 * , 220 ( 2002 ) [ arxiv : hep - ph/0207004 ] . v. ravindran , j. smith and w. l. van neerven , nucl . b * 665 * , 325 ( 2003 ) [ arxiv : hep - ph/0302135 ] . s. moch and a. vogt , phys . b * 631 * , 48 ( 2005 ) [ arxiv : hep - ph/0508265 ] . e. laenen and l. magnea , phys . b * 632 * , 270 ( 2006 ) [ arxiv : hep - ph/0508284 ] . a. idilbi , x. d. ji , j. p. ma and f. yuan , phys . d * 73 * , 077501 ( 2006 ) [ arxiv : hep - ph/0509294 ] . v. ravindran , nucl . b * 752 * , 173 ( 2006 ) [ arxiv : hep - ph/0603041 ] . a. idilbi , x. d. ji and f. yuan , nucl . b * 753 * , 42 ( 2006 ) [ arxiv : hep - ph/0605068 ] . v. ahrens , t. becher , m. neubert and l. l. yang , phys . d * 79 * , 033013 ( 2009 ) [ arxiv:0808.3008 [ hep - ph ] ] . a. d. martin , w. j. stirling , r. s. thorne and g. watt , phys . b * 652 * , 292 ( 2007 ) [ arxiv:0706.0459 [ hep - ph ] ] . g. aad _ et al . _ [ the atlas collaboration ] , arxiv:0901.0512 [ hep - ex ] . u. aglietti _ et al . _ , arxiv : hep - ph/0612172 and http://maltoni.home.cern.ch/maltoni/ tev4lhc / sm.html . g. cacciapaglia , a. deandrea and j. llodra - perez , jhep * 0906 * , 054 ( 2009 ) [ arxiv:0901.0927 [ hep - ph ] ] . j. r. espinosa , c. grojean and m. mhlleitner , arxiv:1003.3251 [ hep - ph ] . a. djouadi and g. moreau , phys . b * 660 * , 67 ( 2008 ) [ arxiv:0707.3800 [ hep - ph ] ] . a. djouadi , j. kalinowski and m. spira , comput . commun . * 108 * , 56 ( 1998 ) [ arxiv : hep - ph/9704448 ] . t. hahn , comput . commun . * 140 * , 418 ( 2001 ) [ arxiv : hep - ph/0012260 ] . c. amsler _ et al . _ [ particle data group ] , phys . b * 667 * , 1 ( 2008 ) and updated results available at : http://pdglive.lbl.gov/ j. charles _ et al . _ [ ckmfitter group ] , eur . j. c * 41 * , 1 ( 2005 ) [ arxiv : hep - ph/0406184 ] and updated results available at : http://ckmfitter.in2p3.fr/ a. djouadi , phys . rept . * 457 * , 1 ( 2008 ) [ arxiv : hep - ph/0503172 ] and references therein .

we reexamine the randall - sundrum ( rs ) model with enlarged gauge symmetry @xmath0 in the presence of a brane - localized higgs sector . in contrast to the existing literature , we perform the kaluza - klein ( kk ) decomposition within the mass basis , which avoids the truncation of the kk towers . expanding the low - energy spectrum as well as the gauge couplings in powers of the higgs vacuum expectation value , we obtain analytic formulas which allow for a deep understanding of the model - specific protection mechanisms of the @xmath1 parameter and the left - handed @xmath2-boson couplings . in particular , in the latter case we explain which contributions escape protection and identify them with the irreducible sources of @xmath3 symmetry breaking . we furthermore show explicitly that no protection mechanism is present in the charged - current sector confirming existing model - independent findings . the main focus of the phenomenological part of our work is a detailed discussion of higgs - boson couplings and their impact on physics at the cern large hadron collider . for the first time , a complete one - loop calculation of all relevant higgs - boson production and decay channels is presented , incorporating the effects stemming from the extended electroweak gauge - boson and fermion sectors . mz - th/10 - 18 + may 23 , 2010 + * the custodial randall - sundrum model : + from precision tests to higgs physics * s. casagrande@xmath4 , f. goertz@xmath5 , u. haisch@xmath5 , m. neubert@xmath6 and t. pfoh@xmath5 +

1005.4315

we present a detailed study on the nature of biases in network sampling strategies to shed light on how best to sample from networks . a _ network _ is a system of interconnected entities typically represented mathematically as a graph : a set of vertices and a set of edges among the vertices . networks are ubiquitous and arise across numerous and diverse domains . for instance , many web - based social media , such as online social networks , produce large amounts of data on interactions and associations among individuals . mobile phones and location - aware devices produce copious amounts of data on both communication patterns and physical proximity between people . in the domain of biology also , from neurons to proteins to food webs , there is now access to large networks of associations among various entities and a need to analyze and understand these data . with advances in technology , pervasive use of the internet , and the proliferation of mobile phones and location - aware devices , networks under study today are not only substantially larger than those in the past , but sometimes exist in a decentralized form ( e.g. the network of blogs or the web itself ) . for many networks , their global structure is not fully visible to the public and can only be accessed through `` crawls '' ( e.g. online social networks ) . these factors can make it prohibitive to analyze or even access these networks in their entirety . how , then , should one proceed in analyzing and mining these network data ? one approach to addressing these issues is _ sampling _ : inference using small subsets of nodes and links from a network . from epidemiological applications @xcite to web crawling @xcite and p2p search @xcite , network sampling arises across many different settings . in the present work , we focus on a particular line of investigation that is concerned with constructing samples that match critical structural properties of the original network . such samples have numerous applications in data mining and information retrieval . in @xcite , for example , structurally - representative samples were shown to be effective in inferring network protocol performance in the larger network and significantly improving the efficiency of protocol simulations . in section [ sec : applications ] , we discuss several additional applications . although there have been a number of recent strides in work on network sampling ( e.g. @xcite ) , there is still very much that requires better and deeper understanding . moreover , many networks under analysis , although treated as complete , are , in fact , _ samples _ due to limitations in data collection processes . thus , a more refined understanding of network sampling is of general importance to network science . towards this end , we conduct a detailed study on _ network sampling biases_. there has been a recent spate of work focusing on _ problems _ that arise from network sampling biases including how and why biases should be avoided @xcite . our work differs from much of this existing literature in that , for the first time in a comprehensive manner , we examine network sampling bias as an _ asset to be exploited_. we argue that biases of certain sampling strategies can be advantageous if they `` push '' the sampling process towards inclusion of specific properties of interest . our main aim in the present work is to identify and understand the connections between specific sampling biases and specific definitions of structural representativeness , so that these biases can be leveraged in practical applications . * summary of findings . * we conduct a detailed investigation of network sampling biases . we find that bias towards high _ expansion _ ( a concept from expander graphs ) offers several unique advantages over other biases such as those toward high degree nodes . we show both empirically and analytically that such an expansion bias `` pushes '' the sampling process towards new , undiscovered clusters and the discovery of wider portions of the network . in other analyses , we show that a simple sampling process that selects nodes with many connections from those already sampled is often a reasonably good approximation to directly sampling high degree nodes and locates well - connected ( i.e. high degree ) nodes significantly faster than most other methods . we also find that the breadth - first search , a widely - used sampling and search strategy , is surprisingly among the most dismal performers in terms of both discovering the network and accumulating critical , well - connected nodes . finally , we describe ways in which some of our findings can be exploited in several important applications including disease outbreak detection and market research . a number of these aforementioned findings are surprising in that they are in stark contrast to conventional wisdom followed in much of the existing literature ( e.g. @xcite ) . not surprisingly , network sampling arises across many diverse areas . here , we briefly describe some of these different lines of research . * network sampling in classical statistics . * the concept of sampling networks first arose to address scenarios where one needed to study hidden or difficult - to - access populations ( e.g. illegal drug users , prostitutes ) . for recent surveys , one might refer to @xcite . the work in this area focuses almost exclusively on acquiring unbiased estimates related to variables of interest attached to each network node . the present work , however , focuses on inferring properties related to the _ network itself _ ( many of which are not amenable to being fully captured by simple attribute frequencies ) . our work , then , is much more closely related to _ representative subgraph sampling_. * representative subgraph sampling . * in recent years , a number of works have focused on _ representative subgraph sampling _ : constructing samples in such a way that they are condensed representations of the original network ( e.g. @xcite ) . much of this work focuses on how best to produce a `` universal '' sample representative of _ all _ structural properties in the original network . by contrast , we subscribe to the view that no single sampling strategy may be appropriate for all applications . thus , our aim , then , is to better understand the _ biases _ in specific sampling strategies to shed light on how best to leverage them in practical applications . * unbiased sampling . * there has been a relatively recent spate of work ( e.g. @xcite ) that focuses on constructing uniform random samples in scenarios where nodes can not be easily drawn randomly ( e.g. settings such as the web where nodes can only be accessed through crawls ) . these strategies , often based on modified random walks , have been shown to be effective for various frequency estimation problems ( e.g. inferring the proportion of pages of a certain language in a web graph @xcite ) . however , as mentioned above , the present work focuses on using samples to infer structural ( and functional ) properties of the _ network itself_. in this regard , we found these unbiased methods to be less effective during preliminary testing . thus , we do not consider them and instead focus our attention on other more appropriate sampling strategies ( such as those mentioned in _ representative subgraph sampling _ ) . * studies on sampling bias . * several studies have investigated _ biases _ that arise from various sampling strategies ( e.g. @xcite ) . for instance , @xcite showed that , under the simple sampling strategy of picking nodes at random from a scale - free network ( i.e. a network whose degree distribution follows the power law ) , the resultant subgraph sample will _ not _ be scale - free . the authors of @xcite showed the converse is true under traceroute sampling . virtually all existing results on network sampling bias focus on its negative aspects . by contrast , we focus on the _ advantages _ of certain biases and ways in which they can be exploited in network analysis . * property testing . * work on sampling exists in the fields of combinatorics and graph theory and is centered on the notion of _ property testing _ in graphs @xcite . properties such as those typically studied in graph theory , however , may be less useful for the analysis of _ real - world _ networks ( e.g. the exact meaning of , say , @xmath0-colorability @xcite within the context of a social network is unclear ) . nevertheless , theoretical work on property testing in graphs is excellently surveyed in @xcite . * other areas . * decentralized search ( e.g. searching unstructured p2p networks ) and web crawling can both be framed as network sampling problems , as both involve making decisions from subsets of nodes and links from a larger network . indeed , network sampling itself can be viewed as a problem of information retrieval , as the aim is to seek out a subset of nodes that either individually or collectively match some criteria of interest . several of the sampling strategies we study in the present work , in fact , are graph search algorithms ( e.g. breadth - first search ) . thus , a number of our findings discussed later have implications for these research areas ( e.g. see @xcite ) . for reviews on decentralized search both in the contexts of complex networks and p2p systems , one may refer to @xcite and @xcite , respectively . for examples of connections between web crawling and network sampling , see @xcite . we now briefly describe some notations and definitions used throughout this paper . [ defn : network ] @xmath1 is a _ network _ or _ graph _ where @xmath2 is set of vertices and @xmath3 is a set of edges . [ defn : sample ] a _ sample _ @xmath4 is a subset of vertices , @xmath5 . [ defn : neighborhood ] @xmath6 is the _ neighborhood _ of @xmath4 if @xmath7 . [ defn : inducedsubgraph ] @xmath8 is the _ induced subgraph _ of @xmath9 based on the sample @xmath4 if @xmath10 where the vertex set is @xmath5 and the edge set is @xmath11 . the induced subgraph of a sample may also be referred to as a _ subgraph sample_. we study sampling biases in a total of twelve different networks : a power grid ( powergrid @xcite ) , a wikipedia voting network ( wikivote @xcite ) , a pgp trust network ( pgp @xcite ) , a citation network ( hepth @xcite ) , an email network ( enron @xcite ) , two co - authorship networks ( condmat @xcite and astroph @xcite ) , two p2p file - sharing networks ( gnutella04 @xcite and gnutella31 @xcite ) , two online social networks ( epinions @xcite and slashdot @xcite ) , and a product co - purchasing network ( amazon @xcite ) . these datasets were chosen to represent a rich set of diverse networks from different domains . this diversity allows a more comprehensive study of network sampling and thorough assessment of the performance of various sampling strategies in the face of varying network topologies . table [ tab : datasets ] shows characteristics of each dataset . all networks are treated as undirected and unweighted . [ th ] .network properties . * key : * _ n= # of nodes , d= density , pl = characteristic path length , cc = local clustering coefficient , ad = average degree . _ [ cols="<,^,^,^,^,^",options="header " , ] -0.15 in in the present work , we focus on a particular class of sampling strategies , which we refer to as _ link - trace sampling_. in _ link - trace sampling _ , the next node selected for inclusion into the sample is always chosen from among the set of nodes directly connected to those already sampled . in this way , sampling proceeds by tracing or following links in the network . this concept can be defined formally . [ defn : linktracesampling ] given an integer @xmath0 and an initial node ( or seed ) @xmath12 to which @xmath4 is initialized ( i.e. @xmath13 ) , a _ link - trace sampling _ algorithm , @xmath14 , is a process by which nodes are iteratively selected from among the current neighborhood @xmath6 and added to @xmath4 until @xmath15 . _ link - trace sampling _ may also be referred to as _ crawling _ ( since links are `` crawled '' to access nodes ) or viewed as _ online _ sampling ( since the network @xmath9 reveals itself iteratively during the course of the sampling process ) . the key advantage of sampling through link - tracing , then , is that complete access to the network in its entirety is _ not _ required . this is beneficial for scenarios where the network is either large ( e.g. an online social network ) , decentralized ( e.g. an unstructured p2p network ) , or both ( e.g. the web ) . as an aside , notice from definition [ defn : linktracesampling ] that we have implicitly assumed that the neighbors of a given node can be obtained by visiting that node during the sampling process ( i.e. @xmath6 is known ) . this , of course , accurately characterizes most real scenarios . for instance , neighbors of a web page can be gleaned from the hyperlinks on a visited page and neighbors of an individual in an online social network can be acquired by viewing ( or `` scraping '' ) the friends list . having provided a general definition of _ link - trace sampling _ , we must now address _ which _ nodes in @xmath6 should be preferentially selected at each iteration of the sampling process . this choice will obviously directly affect the properties of the sample being constructed . we study seven different approaches - all of which are quite simple yet , at the same time , ill - understood in the context of real - world networks . * breadth - first search ( bfs ) . * starting with a single seed node , the bfs explores the neighbors of visited nodes . at each iteration , it traverses an unvisited neighbor of the _ earliest _ visited node @xcite . in both @xcite and @xcite , it was empirically shown that bfs is biased towards high - degree and high - pagerank nodes . bfs is used prevalently to crawl and collect networks ( e.g. @xcite ) . * depth - first search ( dfs ) . * dfs is similar to bfs , except that , at each iteration , it visits an unvisited neighbor of the most _ recently _ visited node @xcite . * random walk ( rw ) . * a random walk simply selects the next hop uniformly at random from among the neighbors of the current node @xcite . * forest fire sampling ( ffs ) . * ffs , proposed in @xcite , is essentially a probabilistic version of bfs . at each iteration of a bfs - like process , a neighbor @xmath16 is only explored according to some `` burning '' probability @xmath17 . at @xmath18 , ffs is identical to bfs . we use @xmath19 , as recommended in @xcite . * degree sampling ( ds ) . * the ds strategy involves greedily selecting the node @xmath20 with the highest degree ( i.e. number of neighbors ) . a variation of ds was analytically and empirically studied as a p2p search algorithm in @xcite . notice that , in order to select the node @xmath21 with the highest degree , the process must know @xmath22 for each @xmath20 . that is , knowledge of @xmath23 is required at each iteration . as noted in @xcite , this requirement is acceptable for some domains such as p2p networks and certain social networks . the ds method is also feasible in scenarios where 1 ) one is interested in efficiently `` downsampling '' a network to a connected subgraph , 2 ) a crawl is repeated and history of the last crawl is available , or 3 ) the proportion of the network accessed to construct a sample is less important . * sec ( sample edge count ) . * given the currently constructed sample @xmath4 , how can we select a node @xmath20 with the highest degree _ without _ having knowledge of @xmath23 ? the sec strategy tracks the links from the currently constructed sample @xmath4 to each node @xmath20 and selects the node @xmath16 with the most links from @xmath4 . in other words , we use the degree of @xmath16 in the induced subgraph of @xmath24 as an approximation of the degree of @xmath16 in the original network @xmath9 . similar approaches have been employed as part of web crawling strategies with some success ( e.g. @xcite ) . * xs ( expansion sampling ) . * the xs strategy is based on the concept of expansion from work on expander graphs and seeks to greedily construct the sample with the maximal expansion : @xmath25 , where @xmath0 is the desired sample size @xcite . at each iteration , the next node @xmath16 selected for inclusion in the sample is chosen based on the expression : @xmath26 like the ds strategy , this approach utilizes knowledge of @xmath23 . in sections [ sec : rep.reach ] and [ sec : biases.xs ] , we will investigate in detail the effect of this expansion bias on various properties of constructed samples . what makes one sampling strategy `` better '' than another ? in computer science , `` better '' is typically taken to be structural _ representativeness _ ( e.g. see @xcite ) . that is , samples are considered better if they are more representative of structural properties in the original network . there are , of course , numerous structural properties from which to choose , and , as correctly observed by ahmed et al . @xcite , it is not always clear which should be chosen . rather than choosing arbitrary structural properties as measures of representativeness , we select specific measures of representativeness that we view as being potentially useful for real applications . we divide these measures ( described below ) into three categories : degree , clustering , and reach . for each sampling strategy , we generate 100 samples using randomly selected seeds , compute our measures of representativeness on each sample , and plot the average value as sample size grows . ( standard deviations of computed measures are discussed in section [ sec : rep.seedsensitivity ] . applications for these measures of representativeness are discussed later in section [ sec : applications ] . ) due to space limitations and the large number of networks evaluated , for each evaluation measure , we only show results for two datasets that are illustrative of general trends observed in all datasets . however , full results are available as supplementary material . the degrees ( numbers of neighbors ) of nodes in a network is a fundamental and well - studied property . in fact , other graph - theoretic properties such as the average path length between nodes can , in some cases , be viewed as byproducts of degree ( e.g. short paths arising from a small number of highly - connected hubs that act as conduits @xcite ) . we study two different aspects of degree ( with an eye towards real - world applications , discussed in section [ sec : applications ] ) . * degree distribution similarity ( distsim ) . * we take the degree sequence of the sample and compare it to that of the original network using the two - sample kolmogorov - smirnov ( k - s ) d - statistic @xcite , a distance measure . our objective here is to measure the agreement between the two degree distributions in terms of both shape and location . specifically , the d - statistic is defined as @xmath27 , where @xmath28 is the range of node degrees , and @xmath29 and @xmath30 are the cumulative degree distributions for @xmath9 and @xmath8 , respectively @xcite . we compute the distribution similarity by subtracting the k - s distance from one . * hub inclusion ( hubs ) . * in several applications , one cares less about matching the _ overall _ degree distribution and more about accumulating the highest degree nodes into the sample quickly ( e.g. immunization strategies @xcite ) . for these scenarios , sampling is used as a tool for information retrieval . here , we evaluate the extent to which sampling strategies accumulate hubs ( i.e. high degree nodes ) quickly into the sample . as sample size grows , we track the proportion of the top @xmath31 nodes accumulated by the sample . for our tests , we use @xmath32 . figure [ fig : rep.degree ] shows the _ degree distribution similarity _ ( distsim ) and _ hub inclusion _ ( hubs ) for the slashdot and enron datasets . note that the sec and ds strategies , both of which are biased to high degree nodes , perform best on _ hub inclusion _ ( as expected ) , but are the _ worst _ performers on the distsim measure ( which is also a direct result of this bias ) . ( the xs strategy exhibits a similar trend but to a slightly lesser extent . ) on the other hand , strategies such as bfs , ffs , and rw tend to perform better on distsim , but worse on hubs . for instance , the ds and sec strategies locate the majority of the top 100 hubs with sample sizes less than @xmath33 in some cases . bfs and ffs require sample sizes of over @xmath34 ( and the performance differential is larger when locating hubs ranked higher than @xmath35 ) . more importantly , no strategy performs best on _ both _ measures . this , then , suggests a tension between goals : constructing small samples of the most well - connected nodes is in conflict with producing small samples exhibiting representative degree distributions . more generally , when selecting sample elements , choices resulting in gains for one area can result in losses for another . thus , these choices must be made in light of how samples will be used - a subject we discuss in greater depth in section [ sec : applications ] . we conclude this section by briefly noting that the trend observed for sec seems to be somewhat dependent upon the quality and number of hubs actually present in a network ( relative to the size of the network , of course ) . that is , sec matches ds more closely as degree distributions exhibit longer and denser tails ( as shown in figure [ fig : rep.dd ] ) . we will revisit this in section [ sec : biases.sec ] . ( other strategies are sometimes affected similarly , but the trend is much less consistent . ) in general , we find sec best matches ds performance on many of the social networks ( as opposed to technological networks such as the powergrid with few `` good '' hubs , lower average degree , and longer path lengths ) . however , further investigation is required to draw firm conclusions on this last point . + -0.01 in -0.15 in + -0.01 in -0.15 in many real - world networks , such as social networks , exhibit a much higher level clustering than what one would expect at random @xcite . thus , clustering has been another graph property of interest for some time . here , we are interested in evaluating the extent to which samples exhibit the level of clustering present in the original network . we employ two notions of clustering , which we now describe . * local clustering coefficient ( ccloc ) . * the local clustering coefficient @xcite of a node captures the extent to which the node s neighbors are also neighbors of each other . formally , the local clustering coefficient of a node is defined as @xmath36 where @xmath37 is the degree of node @xmath16 and @xmath38 is the number of links among the neighbors of @xmath16 . the average local clustering coefficient for a network is simply @xmath39 . * global clustering coefficient ( ccglb ) . * the global clustering coefficient @xcite is a function of the number of triangles in a network . it is measured as the number of closed triplets divided by the number of connected triples of nodes . results for clustering measures are less consistent than for other measures . overall , dfs and rw strategies appear to fare relatively better than others . we do observe that , for many strategies and networks , estimates of clustering are initially higher - than - actual and then gradually decline ( see figure [ fig : rep.clustering ] ) . this agrees with intuition . nodes in clusters should intuitively have more paths leading to them and will , thus , be encountered earlier in a sampling process ( as opposed to nodes not embedded in clusters and located in the periphery of a network ) . this , then , should be taken into consideration in applications where accurately matching clustering levels is important . + -0.01 in -0.15 in we propose a new measure of representativeness called _ network reach_. as a newer measure , _ network reach _ has obviously received considerably less attention than degree and clustering within the existing literature , but it is , nevertheless , a vital measure for a number of important applications ( as we will see in section [ sec : applications ] ) . _ network reach _ captures the extent to which a sample _ covers _ a network . intuitively , for a sample to be truly representative of a large network , it should consist of nodes from diverse portions of the network , as opposed to being relegated to a small `` corner '' of the graph . this concept will be made more concrete by discussing in detail the two measures of _ network reach _ we employ : _ community reach _ and the _ discovery quotient_. * community reach ( cnm and rak ) . * many real - world networks exhibit what is known as _ community structure_. a _ community _ can be loosely defined as a set of nodes more densely connected among themselves than to other nodes in the network . although there are many ways to represent community structure depending on various factors such as whether or not overlapping is allowed , in this work , we represent community structure as a _ partition _ : a collection of disjoint subsets whose union is the vertex set @xmath2 @xcite . under this representation , each subset in the partition represents a community . the task of a community detection algorithm is to identify a partition such that vertices within the same subset in the partition are more densely connected to each other than to vertices in other subsets @xcite . for the criterion of _ community reach _ , a sample is more representative of the network if it consists of nodes from more of the communities in the network . we measure _ community reach _ by taking the number of communities represented in the sample and dividing by the total number of communities present in the original network . since a community is essentially a cluster of nodes , one might wonder why we have included _ community reach _ as a measure of _ network reach _ , rather than as a measure of _ clustering_. the reason is that we are slightly less interested in the structural details of communities detected here . rather , our aim is to assess how `` spread out '' a sample is across the network . since community detection is somewhat of an inexact science ( e.g. see @xcite ) , we measure _ community reach _ with respect to two separate algorithms . we employ both the method proposed by clauset et al . in @xcite ( denoted as cnm ) and the approach proposed by raghavan et al . in @xcite ( denoted as rak ) . essentially , for our purposes , we are defining communities simply as the output of a community detection algorithm . * discovery quotient ( dq ) . * an alternative view of _ network reach _ is to measure the proportion of the network that is _ discovered _ by a sampling strategy . the number of nodes discovered by a strategy is defined as @xmath40 . the _ discovery quotient _ is this value normalized by the total number of nodes in a network : @xmath41 . intuitively , we are defining the _ reach _ of a sample here by measuring the extent to which it is one hop away from the rest of the network . as we will discuss in section [ sec : applications ] , samples with high _ discovery quotients _ have several important applications . note that a simple greedy algorithm for coverage problems such as this has a well - known sharp approximation bound of @xmath42 @xcite . however , link - trace sampling is restricted to selecting subsequent sample elements from the current neighborhood @xmath6 at each iteration , which results in a much smaller search space . thus , this approximation guarantee can be shown not to hold within the context of link - trace sampling . as shown in figure [ fig : rep.reach ] , the xs strategy displays the overwhelmingly best performance on all three measures of _ network reach_. we highlight several observations here . first , the extent to which the xs strategy outperforms all others on the rak and cnm measures is quite striking . we posit that the expansion bias of the xs strategy `` pushes '' the sampling process towards the inclusion of new communities not already seen ( see also @xcite ) . in section [ sec : biases.xs ] , we will analytically examine this connection between expansion bias and _ community reach_. on the other hand , the sec method appears to be among the least effective in reaching different communities or clusters . we attribute this to the fact that sec preferentially selects nodes with many connections to nodes already sampled . such nodes are likely to be members of clusters already represented in the sample . second , on the dq measure , it is surprising that the ds strategy , which explicitly selects high degree nodes , often fails to even come close to the xs strategy . we partly attribute this to an overlap in the neighborhoods of well - connected nodes . by explicitly selecting nodes that contribute to _ expansion _ , the xs strategy is able to discover a much larger proportion of the network in the same number of steps - in some cases , by actively sampling comparatively _ lower _ degree nodes . finally , it is also surprising that the bfs strategy , widely used to crawl and explore online social networks ( e.g @xcite ) and other graphs ( e.g. @xcite ) , performs quite dismally on all three measures . in short , we find that nodes contributing most to the expansion of the sample are unique in that they provide specific and significant advantages over and above those provided by nodes that are simply well - connected and those accumulated through standard bfs - based crawls . these and previously mentioned results are in contrast to the conventional wisdom followed in much of the existing literature ( e.g. @xcite ) . + -0.01 in + -0.01 in -0.15 in as described , link - trace sampling methods are initiated from randomly selected seeds . this begs the question : how sensitive are these results to the seed supplied to a strategy ? figure [ fig : std ] shows the standard deviation of each sampling strategy for both _ hub inclusion _ and _ network reach _ as sample size grows . we generally find that methods with the most explicit biases ( xs , sec , ds ) tend to exhibit the least seed sensitivity and variability , while the remaining methods ( bfs , dfs , ffs , rw ) exhibit the most . this trend is exhibited across all measures and all datasets . let us briefly summarize two main observations from section [ sec : rep ] . we saw that the xs strategy dramatically outperformed all others in accumulating nodes from many different communities . we also saw that the sec strategy was often a reasonably good approximation to directly sampling high degree nodes and locates the set of most well - connected nodes significantly faster than most other methods . here , we turn our attention to analytically examining these observed connections . we begin by briefly summarizing some existing analytical results . * random walks ( rw ) . * there is a fairly large body of research on random walks and markov chains ( see @xcite for an excellent survey ) . a well - known analytical result states that the probability ( or _ stationary _ probability ) of residing at any node @xmath16 during a random walk on a connected , undirected graph converges with time to @xmath43 , where @xmath44 is the degree of node @xmath16 @xcite . in fact , the _ hitting time _ of a random walk ( i.e. the expected number of steps required to reach a node beginning from any node ) has been analytically shown to be directly related to this stationary probability @xcite . random walks , then , are naturally biased towards high degree ( and high pagerank ) nodes , which provides some theoretical explanation as to why rw performs slightly better than other strategies ( e.g. bfs ) on measures such as _ hub inclusion_. however , as shown in figure [ fig : rep.degree ] , it is nowhere near the best performers . thus , these analytical results appear only to hold in the limit and fail to predict actual sampling performance . * degree sampling ( ds ) . * in studying the problem of searching peer - to - peer networks , adamic et al . @xcite proposed and analyzed a greedy search strategy very similar to the ds sampling method . this strategy , which we refer to as a degree - based walk , was analytically shown to quickly find the highest - degree nodes and quickly cover large portions of scale - free networks . thus , these results provide a theoretical explanation for performance of the ds strategy on measures such as _ hub inclusion _ and the _ discovery quotient_. * other results . * as mentioned in section [ sec : relatedwork ] , to the best of our knowledge , much of the other analytical results on sampling bias focus on _ negative _ results @xcite . thus , these works , although intriguing , may not provide much help in the way of explaining _ positive _ results shown in section [ sec : rep ] . + we now analyze two methods for which there are little or no existing analytical results : xs and sec . a widely used measure for the `` goodness '' or the strength of a community in graph clustering and community detection is _ conductance _ @xcite , which is a function of the fraction of total edges emanating from a sample ( lower values mean stronger communities ) : @xmath45 where @xmath46 are entries of the adjacency matrix representing the graph and @xmath47 , which is the total number of edges incident to the node set @xmath4 . it can be shown that , provided the conductance of communities is sufficiently low , sample expansion is directly affected by community structure . consider a simple random graph model with vertex set @xmath2 and a community structure represented by partition @xmath48 where @xmath49 . let @xmath50 and @xmath51 be the number of each node s edges pointing within and outside the node s community , respectively . these edges are connected uniformly at random to nodes either within or outside a node s community , similar to a configuration model ( e.g. , @xcite ) . note that both @xmath50 and @xmath51 are related directly to conductance . when conductance is lower , @xmath51 is smaller is @xmath52 , the total number of edges incident to @xmath53 is @xmath54 , and @xmath50 and @xmath51 are random variables denoting the inward and outward edges , respectively , of each node ( as opposed to constant values ) . then , @xmath55 and @xmath56 . if @xmath57 , then @xmath58 . ( in this example , the expectations are over nodes in @xmath53 only . ) ] as compared to @xmath50 . the following theorem expresses the link between expansion and _ community reach _ in terms of these inward and outward edges . [ thm : xsbias ] let @xmath4 be the current sample , @xmath16 be a new node to be added to @xmath4 , and @xmath59 be the size of @xmath16 s community . if @xmath60 , then the expected expansion of @xmath24 is higher when @xmath16 is in a new community than when @xmath16 is in a current community . let @xmath61 be the expected value for @xmath62 when @xmath16 is in a new community and let @xmath63 be the expected value when not . we compute an upper bound on @xmath63 and a lower bound on @xmath61 . + deriving @xmath63 : assume @xmath16 is affiliated with a current community already represented by at least one node in @xmath4 . since we are computing an upper bound on @xmath63 , we assume there is exactly one node from @xmath4 within @xmath16 s community , as this is the minimum for @xmath16 s community to be a _ current _ community . by the linearity of expectations , the upper bound on @xmath63 is @xmath64 , where the term @xmath65 is the expected number of nodes in @xmath16 s community that are both linked to @xmath16 _ and _ in the set @xmath66 . + deriving @xmath61 : assume @xmath16 belongs to a new community not already represented in @xmath4 . ( by definition , no nodes in @xmath4 will be in @xmath16 s community . ) applying the linearity of expectations once again , the lower bound on @xmath61 is @xmath67 , where the term @xmath68 is the expected number of nodes in @xmath16 s community that are both linked to @xmath16 _ and _ already in @xmath6 . + solving for @xmath51 , if @xmath60 , then @xmath69 . theorem [ thm : xsbias ] shows analytically the link between expansion and community structure - a connection that , until now , has only been empirically demonstrated @xcite . thus , a theoretical basis for performance of the xs strategy on _ community reach _ is revealed . recall that the sec method uses the degree of a node @xmath16 in the induced subgraph @xmath70 as an estimation for the degree of @xmath16 in @xmath9 . in section [ sec : rep ] , we saw that this choice performs quite well in practice . here , we provide theoretical justification for the sec heuristic . consider a random network @xmath9 with some arbitrary expected degree sequence ( e.g. a power law random graph under the so - called @xmath71 model @xcite ) and a sample @xmath5 . let @xmath72 be a function that returns the expected degree of a given node in a given random network ( see @xcite for more information on _ expected _ degree sequences ) . then , it is fairly straightforward to show the following holds . [ prop : secbias ] for any two nodes @xmath73 , + if @xmath74 , then @xmath75 . the probability of an edge between any two nodes @xmath76 and @xmath77 in g is @xmath78 where @xmath79 . let @xmath80 . then , @xmath81 since @xmath82 only when @xmath83 , the proposition holds . combining proposition [ prop : secbias ] with analytical results from @xcite ( described in section [ sec : biases.existing ] ) provides a theoretical basis for observed performance of the sec strategy on measures such as _ hub inclusion_. finally , recall from section [ sec : rep.degree.results ] that the extent to which sec matched the performance of ds on hubs seemed to partly depend on the tail of degree distributions . proposition [ prop : secbias ] also yields insights into this phenomenon . longer and denser tails allow for more `` slack '' when deviating from these expectations of random variables ( as in real - world link patterns that are not purely random ) . we now briefly describe ways in which some of our findings may be exploited in important , real - world applications . although numerous potential applications exist , we focus here on three areas : 1 ) outbreak detection 2 ) landmarks and graph exploration 3 ) marketing . what is the most effective and efficient way to predict and prevent a disease outbreak in a social network ? in a recent paper , christakis and fowler studied outbreak detection of the h1n1 flu among college students at harvard university @xcite . previous research has shown that well - connected ( i.e. high degree ) people in a network catch infectious diseases earlier than those with fewer connections @xcite . thus , _ monitoring _ these individuals allows forecasting the progression of the disease ( a boon to public health officials ) and _ immunizing _ these well - connected individuals ( when immunization is possible ) can prevent or slow further spread . unfortunately , identifying well - connected individuals in a population is non - trivial , as access to their friendships and connections is typically not fully available . and , collecting this information is time - consuming , prohibitively expensive , and often impossible for large networks . matters are made worse when realizing that most existing network - based techniques for immunization selection and outbreak detection assume full knowledge of the global network structure ( e.g. @xcite ) . this , then , presents a prime opportunity to exploit the power of _ sampling_. to identify well - connected students and predict the outbreak , christakis and fowler @xcite employed a sampling technique called _ acquaintance sampling _ ( acq ) based on the so - called friendship paradox @xcite . the idea is that random neighbors of randomly selected nodes in a network will tend to be highly - connected @xcite . christakis and fowler @xcite , therefore , sampled random friends of randomly selected students with the objective of constructing a sample of highly - connected individuals . based on our aforementioned results , we ask : can we do better than this acq strategy ? in previous sections , we showed empirically and analytically that the sec method performs exceedingly well in accumulating hubs . ( it also happens to require less information than ds and xs , the other top performers . ) figure [ fig : outdet ] shows the sample size required to locate the top - ranked well - connected individuals for both sec and acq . the performance differential is quite remarkable , with the sec method faring overwhelmingly better in quickly zeroing in on the set of most well - connected nodes . aside from its superior performance , sec has one additional advantage over the acq method employed by christakis and fowler . the acq method assumes that nodes in @xmath2 can be selected uniformly at random . it is , in fact , dependent on this @xcite . ( acq , then , is _ not _ a link - trace sampling method . ) by contrast , sec , as a pure link - trace sampling strategy , has no such requirement and , thus , can be applied in realistic scenarios for which acq is unworkable . -0.15 in recall from section [ sec : rep.reach ] that a community in a network is a cluster of nodes more densely connected among themselves than to others . identifying communities is important , as they often correspond to real social groups , functional groups , or similarity ( both demographic and not ) @xcite . the ability to easily construct a sample consisting of members from diverse groups has several important applications in marketing . marketing surveys often seek to construct stratified samples that collectively represent the diversity of the population @xcite . if the attributes of nodes are not known in advance , this can be challenging . the xs strategy , which exhibited the best _ community reach _ , can potentially be very useful here . moreover , it has the added power of being able to locate members from diverse groups with absolutely no _ a priori _ knowledge of demographics attributes , social variables , or the overall community structure present in the network . there is also recent evidence to suggest that being able to construct a sample from many different communities can be an asset in effective word - of - mouth marketing @xcite . this , then , represents yet another potential marketing application for the xs strategy . _ landmark - based methods _ represent a general class of algorithms to compute distance - based metrics in large networks quickly @xcite . the basic idea is to select a small sample of nodes ( i.e. the landmarks ) , compute offline the distances from these landmarks to every other node in the network , and use these pre - computed distances at runtime to approximate distances between pairs of nodes . as noted in @xcite , for this approach to be effective , landmarks should be selected so that they _ cover _ significant portions of the network . based on our findings for _ network reach _ in section [ sec : rep.reach ] , the xs strategy overwhelmingly yields the best _ discovery quotient _ and covers the network significantly better than any other strategy . thus , it represents a promising landmark selection strategy . our results for the _ discovery quotient _ and other measures of _ network reach _ also yield important insights into how graphs should best be explored , crawled , and searched . as shown in figure [ fig : rep.reach ] , the most prevalently used method for exploring networks , bfs , ranks low on measures of _ network reach_. this suggests that the bfs and its pervasive use in social network data acquisition and exploration ( e.g. see @xcite ) should possibly be examined more closely . we have conducted a detailed study on sampling biases in real - world networks . in our investigation , we found the bfs , a widely - used method for sampling and crawling networks , to be among the worst performers in both discovering the network and accumulating critical , well - connected hubs . we also found that sampling biases towards high expansion tend to accumulate nodes that are uniquely different from those that are simply well - connected or traversed during a bfs - based strategy . these high - expansion nodes tend to be in newer and different portions of the network not already encountered by the sampling process . we further demonstrated that sampling nodes with many connections from those already sampled is a reasonably good approximation to sampling high degree nodes . finally , we demonstrated several ways in which these findings can be exploited in real - world application such as disease outbreak detection and marketing . for future work , we intend to investigate ways in which the top - performing sampling strategies can be enhanced for even wider applicability . one such direction is to investigate the effects of alternating or combining different biases into a single sampling strategy .

from social networks to p2p systems , network sampling arises in many settings . we present a detailed study on the nature of biases in network sampling strategies to shed light on how best to sample from networks . we investigate connections between specific biases and various measures of structural representativeness . we show that certain biases are , in fact , beneficial for many applications , as they `` push '' the sampling process towards inclusion of desired properties . finally , we describe how these sampling biases can be exploited in several , real - world applications including disease outbreak detection and market research . [ data mining ]

1109.3911

it has long been thought that electro - magnetic probes _ i.e. _ real or virtual photons would provide a way to detect the formation of a quark - gluon plasma in ultra - relativistic heavy ion collisions . the energy distribution of the photons would allow to measure the temperature of the plasma provided the rate of production in the plasma exceeds that of various backgrounds . it is expected that this will occur in a small window in the gev range for the energy of the photon . at lower values of the energy the rate is dominated by various hadron decay processes while at higher values the usual hard processes ( those occurring in the very early stage of the collision before the plasma is formed ) , calculable by standard perturbative qcd methods , would dominate . in contrast to hadronic observables ( or heavy quarkonia ) which are sensitive to the late evolution of the plasma as well as to the re - hadronisation phase , the photons in the gev range are produced soon after the plasma is formed and then they escape the plasma without further interaction . we assume the plasma in thermal equilibrium ( temperature t ) with vanishing chemical potential . the rate of production , per unit time and volume , of a real photon of momentum @xmath0 is @xmath1 while for a lepton pair of mass @xmath2 it is @xmath3 where @xmath4 is the retarded photon polarisation tensor . the pre - factor @xmath5 provides the expected exponential damping @xmath6 when @xmath7 . this report is devoted to the study of @xmath8 which contains the strong interaction dynamics of quarks and gluons in the plasma . the theoretical framework is that of the effective theory with re - summed hard thermal loops ( htl ) @xcite . we briefly review the status of @xmath9 calculated up to the two - loop approximation . some phenomenological consequences are mentioned . then we turn to a discussion of higher loop corrections . following the htl approach @xcite one distinguishes two scales : the hard " scale , typically of order @xmath10 or larger ( the energy of quarks and gluons in the plasma ) and the soft " scale of order @xmath11 where @xmath12 , the strong coupling , is assumed to be small . collective effects in the plasma modify the physics at scale @xmath11 _ i.e. _ over long distances of @xmath13 . these effects lead to a modification of the propagators and vertices of the theory and one is led to introduce effective ( re - summed ) propagators and vertices . this is easily illustrated with the example of the fermion propagator , @xmath14 , which in the bare " theory is simply @xmath15 ( we neglect spin complications and make only a dimensional analysis ) . the thermal contribution to the one loop correction @xmath16 is found to be @xmath17 which is of the same order as the inverse propagator when @xmath18 is of order @xmath19 . the re - summed propagator @xmath20 is then deeply modified for momenta of @xmath21 whereas the thermal corrections appear essentially as higher order effects for hard momenta . likewise , the gluon propagator and vertices are modified by hard thermal loops when the external momenta are soft @xcite . one can construct an effective lagrangian @xcite in terms of effective propagators and vertices and calculate observables in perturbation theory . in the one - loop approximation , the photon production rate is given by the diagram shown in fig . [ fig:1loop ] where the symbol @xmath22 means that effective propagators and vertices are used . the result has been known for some time and can be expressed , in simplified notation , as @xcite @xmath23 where @xmath24 is related to the thermal mass of the quark . one notes the presence of a large " logarithmic term @xmath25 dominating over a constant term " @xmath26 . the two - loop diagrams are displayed in fig . [ fig:2loop ] . in principle , there are more diagrams in the effective theory but only those leading to the dominant contribution are shown . all propagators and vertices should be effective but since the largest contribution arises from hard fermions it is enough , following the htl strategy , to keep bare fermion propagators and -15pt vertices as indicated . only the gluon line needs to be effective since soft momentum @xmath27 through the gluon line dominates the integrals . to evaluate these diagrams it is convenient to distinguish between the contribution arising from a time - like gluon ( @xmath28 ) and a space like gluon ( @xmath29 ) . the first type leads to a contribution similar to eq . ( [ eq:1loop ] ) and requires some care as counter - terms ( not shown ) eliminate the parts of the two - loop diagrams already contained in the one - loop diagrams @xcite . we concentrate on the second case which in terms of physical processes corresponds to bremsstrahlung production of a photon or production in a quark - antiquark annihilation process where one of the quark is put off - shell by scattering in the plasma ( see fig . [ fig : processes ] ) . the result for hard photons is @xcite @xmath30 the reason why these two - loop contributions have the same order as the one - loop one is due to the presence of strong collinear singularities . to calculate @xmath9 one has to cut the propagators as indicated by the dash - dotted lines in fig . [ fig:2loop ] . in the integration over the loop hard momentum @xmath31 ( with @xmath32 , @xmath33 on shell ) the denominators @xmath34 and @xmath35 of the un - cut fermion propagators simultaneously almost vanish when @xmath36 is parallel to @xmath37 _ i.e. _ in the collinear configuration . this leads to an enhancement factor of type @xmath38 where the cut - off @xmath39 emerges from the calculation . for the kinematic range of concern to us here , @xmath40 so that the two - loop diagram is enhanced by a factor @xmath41 which compensates the @xmath42 factor associated to the coupling of the gluon to the quarks . an interesting result of the calculation is the importance of process ii of fig . [ fig : processes ] which grows with the energy of the photon and dominates over the other contributions when @xmath43 as shown in fig . [ fig : compar ] . phenomenological applications of these results have been carried out and the two - loop processes have been included in hydrodynamic evolution codes to predict the rate of real photon production at rhic or lhc @xcite . it is found that the two - loop processes ( especially the annihilation with scattering ) lead to an increase by an order of magnitude compared to the one - loop processes . this may even have consequences for heavy ion collisions at sps energies @xcite . several effects may reduce these over - optimistic predictions : lack of chemical equilibrium and more importantly higher order corrections as discussed next . since the one - loop and two - loop results are of the same order it is reasonable to worry about the convergence of the perturbative expansion in the effective theory ! the enhancement mechanism operative at two - loop could also be at work at the multi - loop level especially in ladder diagrams , an example of which is shown in fig . [ fig : ladder ] : indeed many small " fermion denominators appear in such diagrams which can produce a pile - up of collinear singularities . a recent study of the three - loop ladder diagram shows that @xcite @xmath44 where @xmath45 is the largest of the cut - offs : + @xmath46 , which is the collinear cut - off encountered above : it depends on the thermal quark mass and momentum ( @xmath47 ) as well as on the external variables ; + @xmath48 , the debye mass if the added gluon is longitudinal , or @xmath49 if it is transverse . + for the kinematic configuration of interest , in the case of an extra longitudinal gluon one can check that @xmath50 and the debye mass acts as a cut - off with the result that the three - loop contribution is suppressed by a factor @xmath12 compared to the two - loop . on the contrary , for a transverse gluon , both regulators are of order @xmath51 ( as long as @xmath52 ) and the three - loop diagram is of the same order as the two - loop one . one is therefore in a non - perturbative regime . the problem is similar to the magnetic mass problem pointed out by linde in the perturbative calculation of the free energy @xcite , except that here it appears at leading order . another effect which can modify the collinear enhancement mechanism is related to the fermion damping rate . indeed , including the damping rate on the fermion lines , will shift the pole of the propagators away from the real axis : this affects the enhancement mechanism based on the near - vanishing of the denominators . ignoring the requirement of gauge invariance and concentrating only on the mathematical effect of shifting the poles to the complex plane one can do again the two - loop calculation with fermion propagators including the damping rate @xmath53 . the result is intuitively simple as a regulator of the form @xcite @xmath54 comes out , with @xmath55 defined above . the effect of @xmath56 on @xmath57 is shown on fig . [ fig : damping ] for the case of a real photon ( @xmath58 ) . the region @xmath59 is dominated by bremsstrahlung emission while the region @xmath60 receives a contribution mainly from the annihilation with scattering process ( see fig . [ fig : compar ] ) . the top curve is the result obtained with a vanishingly small width . one notes the change in the @xmath61 behaviour of @xmath62 as @xmath56 increases : this is due to the different @xmath61 dependences of the real and imaginary parts of @xmath63 . for virtual photon production , one notes that the quantity @xmath64 increases with @xmath65 at fixed @xmath61 so that the ratio @xmath66 , which controls the relative importance of the width , decreases . for @xmath65 large enough the effect of @xmath56 will become negligible and the two - loop calculation should be adequate . this is illustrated in fig . [ fig:2 ] . equation ( [ eq : damping ] ) lends itself to a simple interpretation . it can be written as @xmath67 where @xmath68 is the mean free path of the quark in the plasma and @xmath69 can be shown to be the formation length of the photon . then , if @xmath70 the effect of the damping rate can be ignored and the corresponding higher order diagrams are suppressed . in the opposite case , re - scattering in the plasma modifies the two - loop result . this is equivalent to say that the landau - pomeranchuk - migdal ( lpm ) effect @xcite has to be taken into account in the calculation . two interesting features emerge from the above discussion : 1 ) the lpm effect not only modifies the production of bremsstrahlung photon but also that of very hard photons emitted in the annihilation with scattering " process as illustrated in fig . [ fig : damping ] ; 2 ) if the virtuality @xmath71 of the hard lepton pair is large enough then one falls in the domain @xmath72 and the perturbative calculation at two - loop is sufficient . the problems discussed above are an illustration of a more general situation concerning thermal green s function with external momenta close to the light - cone @xcite . the production mechanism of hard photons in the plasma is very complex . new processes appear at two - loop which considerably increase the rate of photon production calculated at one - loop . however , for real or small mass virtual photons the higher loop diagrams become important and the rate turns out to be non - perturbative . taking into account higher order effects to obtain a quantitative estimate remains to be done . i thank f. gelis , r. kobes and h. zaraket for a fruitful collaboration on the work discussed above .

we discuss the production of real or virtual photons in a quark - gluon plasma . laboratoire de physique thorique lapth , + bp110 , f-74941 , annecy le vieux cedex , france lapth - conf-810/2000

hep-ph0009015

i am grateful to alekos kechris for informing me of t.dyck/ ; the proof given seems to be due to alain louveau . i thank norm levenberg for references . hough , j.b . , krishnapur , m. , peres , y. , and virg , b. , _ zeros of gaussian analytic functions and determinantal point processes_. university lecture series , * 51*. american mathematical society , providence , ri , 2009 . mester , p. , invariant monotone coupling need not exist . * 41 * ( 2013 ) , 3a , 11801190 . morris , b. , the components of the wired spanning forest are recurrent . _ probab . theory related fields _ * 125 * ( 2003 ) , 259265 .

we describe the fundamental constructions and properties of determinantal probability measures and point processes , giving streamlined proofs . we illustrate these with some important examples . we pose several general questions and conjectures . primary 60k99 , 60g55 ; secondary 42c30 , 37a15 , 37a35 , 37a50 , 68u99 . random matrices , eigenvalues , orthogonal projections , positive contractions , exterior algebra , stochastic domination , negative association , point processes , mixtures , spanning trees , orthogonal polynomials , completeness , bernoulli processes . determinantal point processes were originally defined by macchi @xcite in physics . starting in the 1990s , determinantal probability began to flourish as examples appeared in numerous parts of mathematics @xcite . recently , applications to machine learning have appeared @xcite . a discrete determinantal probability measure is one whose elementary cylinder probabilities are given by determinants . more specifically , suppose that @xmath0 is a finite or countable set and that @xmath1 is an @xmath2 matrix . for a subset @xmath3 , let @xmath4 denote the submatrix of @xmath1 whose rows and columns are indexed by @xmath5 . if @xmath6 is a random subset of @xmath0 with the property that for all finite @xmath7 , we have e.dpm = ( qa ) , then we call @xmath8 a . the inclusion - exclusion principle in combination with yields the probability of each elementary cylinder event . therefore , for every @xmath1 , there is at most one probability measure , to be denoted @xmath9 , on subsets of @xmath0 that satisfies . conversely , it is known ( see , e.g. , b.lyons:det/ ) that there is a determinantal probability measure corresponding to @xmath1 if @xmath1 is the matrix of a positive contraction on @xmath10 ( in the standard orthonormal basis ) . technicalities are required even to define the corresponding concept of determinantal point process for @xmath0 being euclidean space or a more general space . we present a virtually complete development of their basic properties in a way that minimizes such technicalities by adapting the approach of b.lyons:det/ from the discrete case . in addition , we use an idea of goldman b.goldman/ to deduce properties of the general case from corresponding properties in the discrete case . space limitations prevent mention of most of what is known in determinantal probability theory , which pertains largely to the analysis of specific examples . we focus instead on some of the basic properties that hold for all determinantal processes and on some intriguing open questions . let @xmath0 be a denumerable set . we identify a subset of @xmath0 with an element of @xmath11 in the usual way . there are several approaches to prove the basic existence results and identities for determinantal probability measures . we sketch the one used by b.lyons : det/. this depends on understanding first the case where @xmath1 is the matrix of an orthogonal projection . it also relies on exterior algebra so that the existence becomes immediate . any unit vector @xmath12 in a hilbert space with orthonormal basis @xmath0 gives a probability measure @xmath13 on @xmath0 , namely , @xmath14 associated to orthogonal projections @xmath15 . we refer to b.lyons:det/ for details not given here . identify @xmath0 with the standard orthonormal basis of the real or complex hilbert space @xmath10 . for @xmath16 , let @xmath17 denote a collection of ordered @xmath18-element subsets of @xmath0 such that each @xmath18-element subset of @xmath0 appears exactly once in @xmath17 in some ordering . define @xmath19 if @xmath20 , then @xmath21 and @xmath22 . we also define @xmath23 to be the scalar field , @xmath24 or @xmath25 . the elements of @xmath26 are called of @xmath18 , or for short . we then define the ( or ) of multivectors in the usual alternating multilinear way : @xmath27 for any permutation @xmath28 , and @xmath29 for any scalars @xmath30 ( @xmath31,\ ; e \in e'$ ] ) and any finite @xmath32 . ( thus , @xmath33 unless all @xmath34 are distinct . ) the inner product on @xmath26 satisfies e.ipdet = _ i , j when @xmath35 and @xmath36 are 1-vectors . ( this also shows that the inner product on @xmath26 does not depend on the choice of orthonormal basis of @xmath37 . ) we then define the ( or ) @xmath38 , where the summands are declared orthogonal , making it into a hilbert space . ( throughout the paper , @xmath39 is used to indicate the sum of orthogonal summands , or , if there are an infinite number of orthogonal summands , the closure of their sum . ) vectors @xmath40 are linearly independent iff @xmath41 . for a @xmath18-element subset @xmath3 with ordering @xmath42 in @xmath17 , write @xmath43 . we also write @xmath44 for any function @xmath45 . although there is an isometric isomorphism @xmath46 for @xmath47 , this does not simplify matters in the discrete case . it will be very useful in the continuous case later , however . if @xmath48 is a closed linear subspace of @xmath37 , written @xmath49 , then we identify @xmath50 with its inclusion in @xmath51 . that is , @xmath52 is the closure of the linear span of the @xmath18-vectors @xmath53 . in particular , if @xmath54 , then @xmath55 is a 1-dimensional subspace of @xmath51 ; denote by @xmath56 a unit multivector in this subspace . note that @xmath56 is unique up to a scalar factor of modulus 1 ; which scalar is chosen will not affect the definitions below . we denote by @xmath15 the orthogonal projection onto @xmath48 for any @xmath49 or , more generally , @xmath57 . l.projection for every closed subspace @xmath49 , every @xmath16 , and every @xmath58 , we have @xmath59 write @xmath60 and expand the product . all terms but @xmath61 have a factor of @xmath62 in them , making them orthogonal to @xmath50 by e.ipdet/. a multivector is called or if it is the wedge product of 1-vectors . b.whitney:book/ , p. 49 , shows that e.whitney . we shall use the defined by duality : @xmath63 in particular , if @xmath64 and @xmath65 is a multivector that does not contain any term with @xmath66 in it ( that is , @xmath67 ) , then @xmath68 and @xmath69 . more generally , if @xmath70 with @xmath71 and @xmath72 , then @xmath73 and @xmath74 . note that the interior product is sesquilinear , not bilinear , over @xmath25 . for @xmath70 , write @xmath75 $ ] for the subspace of scalar multiples of @xmath12 in @xmath37 . if @xmath48 is a finite - dimensional subspace of @xmath37 and @xmath76 , then e.hwedge _ h e = p_h^e_h + [ e ] ( up to signum ) . to see this , let @xmath77 be an orthonormal basis of @xmath48 , where @xmath78 . put @xmath79 . then @xmath80 is an orthonormal basis of @xmath81 $ ] , whence @xmath82 } = u_1 \wedge u_2 \wedge \cdots \wedge u_r \wedge v = \mv_h \wedge v = \mv_h \wedge e/\|p_h^\perp e\|\ ] ] since @xmath83 . this shows e.hwedge/. similarly , if @xmath84 , then e.hvee _ h e = p_h e_h e^ ( up to signum ) . indeed , put @xmath85 . let @xmath86 be an orthonormal basis of @xmath48 with @xmath87 . then @xmath88 ( up to signum ) , as desired . finally , we claim that e.reverse = . indeed , @xmath89 , so this is equivalent to @xmath90 thus , it suffices to show that @xmath91 by sesquilinearity , it suffices to show this for @xmath92 members of an orthonormal basis of @xmath48 . but then it is obvious . for a more detailed presentation of exterior algebra , see b.whitney : book/. let @xmath48 be a subspace of @xmath37 of dimension @xmath93 . define the probability measure @xmath94 on subsets @xmath95 by e.xihpr ^h(\{b } ) : = ||^2 . note that this is non-0 only for @xmath96 . also , by l.projection/ , @xmath97 for @xmath96 , which is non-0 iff @xmath98 are linearly independent . that is , @xmath99 iff the projections of the elements of @xmath100 form a basis of @xmath48 . let @xmath101 be any basis of @xmath48 . if we use e.ipdet/ and the fact that @xmath102 for some scalar @xmath103 , then we obtain another formula for @xmath94 : we use @xmath104 to denote a random subset of @xmath0 arising from a probability measure @xmath94 . to see that e.dpm/ holds for the matrix of @xmath15 , observe that for @xmath96 , @xmath105 = \bigip{p_{\ext(h ) } \theta_b , \theta_b } = \bigip{\bigwedge_{e \in b } p_h e , \bigwedge_{e \in b } e } = \det [ \ip{p_h e , f}]_{e , f \in b}\ ] ] by e.ipdet/. this shows that e.dpm/ holds for @xmath106 since @xmath107 @xmath94-a.s . the general case is a consequence of multilinearity , which gives the following extension of e.dpm/. we use the convention that @xmath108 and @xmath109 for any multivector @xmath65 . t.genprs if @xmath110 and @xmath111 are ( possibly empty ) subsets of a finite set @xmath0 , then e.genprs ^h[a_1 , a_2 = ] = . in particular , for every @xmath3 , we have e.included ^h[a ] = p_(h ) _ a^2 . c.dualrep if @xmath0 is finite , then for every subspace @xmath49 , we have e.dualrep ^h^(\{e b } ) = ^h(\{b } ) . these extend to infinite @xmath0 . in order to define @xmath94 when @xmath48 is infinite dimensional , we proceed by finite approximation . let @xmath112 be infinite . consider first a finite - dimensional subspace @xmath48 of @xmath37 . define @xmath113 as the image of the orthogonal projection of @xmath48 onto the span of @xmath114 . by considering a basis of @xmath48 , we see that @xmath115 in the weak operator topology ( wot ) , i.e. , matrix - entrywise , as @xmath116 . it is also easy to see that if @xmath117 , then @xmath118 for all large @xmath18 and , in fact , @xmath119 in the usual norm topology . it follows that e.genprs/ holds for this subspace @xmath48 and for every finite @xmath120 . now let @xmath48 be an infinite - dimensional closed subspace of @xmath37 . choose finite - dimensional subspaces @xmath121 . it is well known that @xmath115 ( wot ) . then e.detgenprs a ( p_h_k a ) ( p_h a ) , whence @xmath122 has a weak@xmath123 limit that we denote @xmath94 and that satisfies e.genprs/. we also note that for _ any _ sequence of subspaces @xmath113 , if @xmath124 ( wot ) , then @xmath125 weak@xmath123 because e.detgenprs/ then holds . we call @xmath1 a if @xmath1 is a self - adjoint operator on @xmath37 such that for all @xmath126 , we have @xmath127 . a of @xmath1 is an orthogonal projection @xmath15 onto a closed subspace @xmath128 for some @xmath129 such that for all @xmath126 , we have @xmath130 , where we regard @xmath131 as the orthogonal sum @xmath132 . in this case , @xmath1 is also called the of @xmath15 to @xmath37 . choose such a dilation ( see e.vecdilate/ or e.dilate/ ) and define @xmath9 as the law of @xmath133 when @xmath104 has the law @xmath94 . then e.dpm/ for @xmath1 is a special case of e.dpm/ for @xmath15 . of course , when @xmath1 is the orthogonal projection onto a subspace @xmath48 , then @xmath134 . basic properties of @xmath9 follow from those for orthogonal projections , such as : t.q if @xmath1 is a positive contraction , then for all finite @xmath135 , e.qgenprs ^q= . if e.dpm/ is given , then e.qgenprs/ can be deduced from e.dpm/ without using our general theory and , in fact , without assuming that the matrix @xmath1 is self - adjoint . indeed , suppose that @xmath136 is any diagonal matrix . denote its @xmath137-entry by @xmath138 . comparing coefficients of @xmath138 shows that e.dpm/ implies , for finite @xmath3 , e.xe = ( ( q + x ) a ) . replacing @xmath5 by @xmath139 and choosing @xmath140 gives e.qgenprs/. on the other hand , if we substitute @xmath141 , then we may rewrite e.xe/ as e.ze = ( ( q z + i - q ) a ) , where @xmath142 is the diagonal matrix of the variables @xmath143 . let @xmath0 be finite . write @xmath144 for @xmath3 . then e.ze/ is equivalent to e.affine _ a e ^q[= a ] z^a = ( i - q+qz ) . this is the same as the laplace transform of @xmath9 after a trivial change of variables . when @xmath145 , we can write @xmath146 with @xmath147 . thus , for all @xmath3 , we have a probability measure @xmath8 on @xmath148 is called if its generating polynomial @xmath149 z^a$ ] satisfies the inequality for all @xmath150 and all real @xmath151 . this property is satisfied by every determinantal probability measure , as was shown by b.bbl:rayleigh/ , who demonstrated its usefulness in showing other properties , such as negative associations and preservation under symmetric exclusion processes . for a set @xmath152 , denote by @xmath153 the @xmath154-field of events that are measurable with respect to the events @xmath155 for @xmath156 . define the @xmath154-field to be the intersection of @xmath157 over all finite @xmath158 . we say that a measure @xmath8 on @xmath148 has if every event in the tail @xmath154-field has measure either 0 or 1 . t.tail b.lyons:det/ if @xmath1 is a positive contraction , then @xmath9 has trivial tail . for finite @xmath0 and a positive contraction @xmath1 , define the of @xmath9 to be @xmath159 numerical calculation supports the following conjecture b.lyons:det/ : g.concave for all positive contractions @xmath160 and @xmath161 , we have e.concave ( ( q_1+q_2)/2 ) ( ( q_1 ) + ( q_2))/2 . let @xmath0 be denumerable . a function @xmath162 is called if for all @xmath163 and all @xmath164 , we have @xmath165 . an event is called increasing or if its indicator is increasing . given two probability measures @xmath166 , @xmath167 on @xmath148 , we say that and write @xmath168 if for all increasing events @xmath169 , we have @xmath170 . this is equivalent to @xmath171 for all bounded increasing @xmath172 . a of two probability measures @xmath166 , @xmath167 on @xmath148 is a probability measure @xmath173 on @xmath174 whose coordinate projections are @xmath166 , @xmath167 ; it is if @xmath175 by strassen s theorem @xcite , stochastic domination @xmath176 is equivalent to the existence of a monotone coupling of @xmath166 and @xmath167 . t.dominate-infinite b.lyons:det/ if @xmath177 , then @xmath178 . it would be very interesting to find a natural or explicit monotone coupling . a coupling @xmath173 has @xmath8 if for all events @xmath179 , we have @xmath180 . q.unioncoupling @xcite given @xmath181 , is there a coupling of @xmath182 and @xmath183 with union marginal @xmath94 ? a positive answer is supported by some numerical calculation . it is easily seen to hold when @xmath184 by c.dualrep/. in the sequel , we write @xmath185 if @xmath186 for all @xmath126 . t.dominate @xcite if @xmath187 , then @xmath188 . by t.dominate-infinite/ , it suffices that there exist orthogonal projections @xmath189 and @xmath190 that are dilations of @xmath160 and @xmath161 such that @xmath191 . this follows from namark s dilation theorem @xcite , which says that any measure whose values are positive operators , whose total mass is @xmath192 , and which is countably additive in the weak operator topology dilates to a spectral measure . the measure in our case is defined on a 3-point space , with masses @xmath160 , @xmath193 , and @xmath194 , respectively . if we denote the respective dilations by @xmath195 , @xmath196 , and @xmath197 , then we set @xmath198 and @xmath199 . a positive answer in general to q.unioncoupling/ would give the following more general result by compression : if @xmath160 , @xmath161 and @xmath200 are positive contractions on @xmath37 , then there is a coupling of @xmath201 and @xmath202 with union marginal @xmath203 . it would be very useful to have additional sufficient conditions for stochastic domination : see the end of s.orthogpoly/ and g.fkdom/. for examples where more is known , see t.gmdom/. we shall say that the events in @xmath153 are @xmath158 and likewise for functions that are measurable with respect to @xmath153 . we say that @xmath8 has if for every pair @xmath204 , @xmath205 of increasing functions that are measurable with respect to complementary subsets of @xmath0 , e.negass . @xcite if @xmath206 , then @xmath9 has negative associations . the details for finite @xmath0 were given in b.lyons : det/. for infinite @xmath0 , let @xmath204 and @xmath205 be increasing bounded functions measurable with respect to @xmath207 and @xmath208 , respectively . choose finite @xmath209 . the conditional expectations @xmath210 $ ] and @xmath211 $ ] are increasing functions to which e.negass/ applies ( because restriction to @xmath212 corresponds to a compression of @xmath1 , which is a positive contraction ) and which , being martingales , converge to @xmath204 and @xmath205 in @xmath213 . write @xmath214 for the distribution of a bernoulli random variable with expectation @xmath215 . for @xmath216 $ ] , let @xmath217 be the distribution of a sum of independent @xmath218 random variables . recall that @xmath75 $ ] is the set of scalar multiples of @xmath12 . t.eigmix @xcite ; lemma 3.4 of @xcite ; ( 2.38 ) of @xcite ; @xcite let @xmath1 be a positive contraction with spectral decomposition @xmath219}$ ] , where @xmath220 are orthonormal . let @xmath221 be independent . let @xmath222 $ ] ; thus , @xmath223 . then @xmath224 . hence , if @xmath225 , then @xmath226 . by t.dominate/ , it suffices to prove it when only finitely many @xmath227 . then by t.q/ , we have @xmath228 = \bigip{\bigwedge_{e \in a } q e , \theta_a } $ ] for all @xmath3 . now @xmath229 } e & = \sum_{j \colon a \to \bbn } \prod_{e \in a } \lambda_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e \\ & = \sum_{j \colon a \rightarrowtail \bbn } \prod_{e \in a } \lambda_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e\end{aligned}\ ] ] because @xmath230 and @xmath231 } e$ ] is a multiple of @xmath12 , so none of the terms where @xmath232 is not injective contribute . thus , @xmath233 } e = \ebig{\sum_{j \colon a \rightarrowtail \bbn } \prod_{e \in a } i_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e } \\ & = \ebig{\sum_{j \colon a \to \bbn } \prod_{e \in a } i_{j(e ) } \bigwedge_{e \in a } p_{[v_{j(e ) } ] } e } = \be\bigwedge_{e \in a } \sum_k i_k p_{[v_k ] } e = \be \bigwedge_{e \in a } p_{\rh } e \,.\end{aligned}\ ] ] we conclude that @xmath234 = \be \leftip{\bigwedge_{e \in a } p_{\rh } e , \theta_a } = \ebig { \bp^{\rh}\left [ a \subseteq \ba \right ] } $ ] by e.qgenprs/. we sketch another proof : let @xmath235 be disjoint from @xmath0 with the same cardinality . choose an orthonormal sequence @xmath236 in @xmath131 . define then @xmath1 is the compression of @xmath15 to @xmath37 . expanding @xmath237 in the obvious way into orthogonal pieces and restricting to @xmath0 , we obtain the desired equation from e.xihpr/. the first proof shows more generally the following : let @xmath238 be a positive contraction . let @xmath220 be ( not necessarily orthogonal ) vectors such that @xmath239 } \lloew i$ ] . let @xmath240 be independent bernoulli random variables with @xmath241 . write @xmath242}$ ] . then @xmath243 . this was observed by ghosh and krishnapur ( personal communication , 2014 ) . note that in the mixture of t.eigmix/ , the distribution of @xmath244 is determinantal corresponding to the diagonal matrix with diagonal @xmath245 . thus , it is natural to wonder whether @xmath246 can be taken to be a general determinantal measure . if such a mixture is not necessarily determinantal , must it be strongly rayleigh or at least have negative correlations ? here , we say that a probability measure @xmath8 on @xmath148 has if for every pair @xmath5 , @xmath100 of finite disjoint subsets of @xmath0 , we have @xmath247 \le \bp [ a \subseteq \qba ] \bp [ b \subseteq \qba ] $ ] . note that negative associations is stronger than negative correlations . the most well - known example of a ( nontrivial discrete ) determinantal probability measure is that where @xmath6 is a uniformly chosen random spanning tree of a finite connected graph @xmath248 with @xmath249 . here , we regard a spanning tree as a set of edges . the fact that holds for the uniform spanning tree is due to b.burpem/ and is called the transfer current theorem . the case with @xmath250 was shown much earlier by b.kirchhoff/ , while the case with @xmath251 was first shown by b.bsst/. write @xmath252 for the uniform spanning tree measure on @xmath253 . to see that @xmath252 is indeed determinantal , consider the vertex - edge incidence matrix @xmath254 of @xmath253 , where each edge is oriented ( arbitrarily ) and the @xmath255-entry of @xmath254 equals 1 if @xmath256 is the head of @xmath66 , @xmath257 if @xmath256 is the tail of @xmath66 , and 0 otherwise . identifying an edge with its corresponding column of @xmath254 , we find that a spanning tree is the same as a basis of the column space of @xmath254 . given @xmath258 , define the at @xmath256 to be the @xmath256-row of @xmath254 , regarded as a vector @xmath259 in the row space , @xmath260 . it is easy that the row - rank of @xmath254 is @xmath261 . let @xmath262 and let @xmath65 be the wedge product ( in some order ) of the stars at all the vertices other than @xmath263 . thus , @xmath264 for some @xmath265 . since spanning trees are bases of the column space of @xmath254 , we have @xmath266 iff @xmath5 is a spanning tree . that is , the only non - zero coefficients of @xmath65 are those in which choosing one edge in each @xmath259 for @xmath267 yields a spanning tree ; moreover , each spanning tree occurs exactly once since there is exactly one way to choose an edge incident to each @xmath267 to get a given spanning tree . this means that its coefficient is @xmath268 . hence , @xmath269 is indeed uniform on spanning trees . simultaneously , this proves the matrix tree theorem that the number of spanning trees equals @xmath270_{x , y \ne x_0}$ ] , since this determinant is @xmath271 . one can define analogues of @xmath252 on infinite connected graphs @xcite by weak limits . for brevity , we simply define them here as determinantal probability measures . again , all edges of @xmath253 are oriented arbitrarily . we define @xmath272 as the closure of the linear span of the stars . an element of @xmath273 that is finitely supported and orthogonal to @xmath272 is called a ; the closed linear span of the cycles is @xmath274 . the is @xmath275 , while the is @xmath276 . our discussion of the continuous " case includes the discrete case , but the discrete case has the more elementary formulations given earlier . let @xmath0 be a measurable space . as before , @xmath0 will play the role of the underlying set on which a point process forms a counting measure . while before we implicitly used counting measure on @xmath0 itself , now we shall have an arbitrary measure @xmath173 ; it need not be a probability measure . the case of lebesgue measure on euclidean space is a common one . the hilbert spaces of interest will be @xmath277 . there may be no natural order in @xmath0 , so to define , e.g. , a probability measure on @xmath278 points of @xmath0 , it is natural to use a probability measure on @xmath279 that is symmetric under coordinate changes and that vanishes on the diagonal @xmath280 . likewise , for exterior algebra , it is more convenient to identify @xmath281 with @xmath282 for @xmath283 . thus , @xmath284 is identified with the function @xmath285_{i , j \in \{1 , \ldots , n\}}/\sqrt{n ! } $ ] . note that @xmath286 \det [ { v_i(x_j ) } ] = \det [ u_i(x_j ) ] \det [ { v_i(x_j)}]^t \nonumber \\ & = \det [ u_i(x_j)][{v_i(x_j)}]^t = \det [ k(x_i , x_j)]_{i , j \in \{1 , \ldots , n\}}\end{aligned}\ ] ] with @xmath287 . here , @xmath288 denotes transpose . suppose from now on that @xmath0 is a locally compact polish space ( equivalently , a locally compact second countable hausdorff space ) . let @xmath173 be a radon measure on @xmath0 , i.e. , a borel measure that is finite on compact sets . let @xmath289 be the set of radon measures on @xmath0 with values in @xmath290 . we give @xmath289 the vague topology generated by the maps @xmath291 for continuous @xmath172 with compact support ; then @xmath289 is polish . the corresponding borel @xmath154-field of @xmath289 is generated by the maps @xmath292 for borel @xmath3 . let @xmath293 be a simple point process on @xmath0 , i.e. , a random variable with values in @xmath289 such that @xmath294 for all @xmath295 . the power @xmath296 lies in @xmath297 . thus , @xmath298 $ ] is a borel measure on @xmath299 ; the part of it that is concentrated on @xmath300 is called the of @xmath293 . if the intensity measure is absolutely continuous with respect to @xmath301 , then its radon - nikodym derivative @xmath302 is called the or the : since the intensity measure vanishes on the diagonal @xmath303 , we take @xmath302 to vanish on @xmath303 . we also take @xmath302 to be symmetric under permutations of coordinates . intensity functions are the continuous analogue of the elementary probabilities e.dpm/. since the sets @xmath304 generate the @xmath154-field on @xmath300 for pairwise disjoint borel @xmath305 , a measurable function @xmath306 is the " @xmath18-point intensity function iff since @xmath293 is simple , @xmath307 , where @xmath308 . since @xmath302 vanishes on the diagonal , it follows from e.rn/ that for disjoint @xmath309 and non - negative @xmath310 summing to @xmath18 , again , this characterizes @xmath302 , even if we use only @xmath311 . in the special case that @xmath312 a.s . for some @xmath313 , then the definition e.rn/ shows that a random ordering of the @xmath278 points of @xmath293 has density @xmath314 . more generally , e.rn/ shows that for all @xmath315 , whence in this case , we call @xmath293 if for some measurable @xmath316 and all @xmath16 , @xmath317 @xmath301-a.e . here , @xmath318 is the matrix @xmath319_{i , j \le k}$ ] . in this case , we denote the law of @xmath293 by @xmath320 . we consider only @xmath158 that are locally square integrable ( i.e. , @xmath321 is radon ) , are hermitian ( i.e. , @xmath322 for all @xmath323 ) , and are positive semidefinite ( i.e. , @xmath324 is positive semidefinite for all finite @xmath325 , written @xmath326 ) . in this case , @xmath158 defines a positive semidefinite integral operator @xmath327 on functions @xmath328 with compact support . for every borel @xmath3 , we denote by @xmath329 the measure @xmath173 restricted to borel subsets of @xmath5 and by @xmath330 the compression of @xmath158 to @xmath5 , i.e. , @xmath331 for @xmath332 . the operator @xmath158 is locally trace - class , i.e. , for every compact @xmath3 , the compression @xmath330 is trace class , having a spectral decomposition @xmath333 , where @xmath334 are orthonormal eigenfunctions of @xmath330 with positive summable eigenvalues @xmath335 . if @xmath110 is the set where @xmath336 , then @xmath337 and @xmath338 converges on @xmath339 , with sum @xmath340-a.e . equal to @xmath158 . we normally redefine @xmath158 on a set of measure 0 to equal this sum . such a @xmath158 defines a determinantal point process iff the integral operator @xmath158 extends to all of @xmath341 as a positive contraction @xcite . the joint intensities determine uniquely the law of the point process ( * ? ? ? * lemma 4.2.6 ) . poisson processes are not determinantal processes , but when @xmath173 is continuous , they are distributional limits of determinantal processes . to see that a positive contraction defines a determinantal point process , we first consider @xmath158 that defines an orthogonal projection onto a finite - dimensional subspace , @xmath48 . then @xmath342 for every orthonormal basis @xmath343 of @xmath48 and @xmath344 is a unit multivector in the notation of s.ext/. because of e.prodtensor/ , we have i.e. , @xmath345/n!$ ] is a density with respect to @xmath346 . although in the discrete case , the absolute squared coefficients of @xmath347 give the elementary probabilities , now coefficients are replaced by a function whose absolute square gives a probability density . as noted already , e.firstdensity/ means that @xmath348 is the @xmath278-point intensity function . in order to show that this density gives a determinantal process with kernel @xmath158 , we use the cauchy - binet formula , which may be stated as follows : for @xmath349 matrices @xmath350 $ ] and @xmath351 $ ] with @xmath352_{\substack{i \le k \\ j \in j}}$ ] , we have @xmath353 [ b_{i , j}]^t\big ) = \sum_{|j| = k } \det a^j \cdot \det b^j = \sum _ { \substack{\sigma , \tau \in \sym(k , n ) \\ \operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{i=1}^k a_{i , \sigma(i ) } b_{i , \tau(i ) } \,,\ ] ] where @xmath354 denotes the image of @xmath154 and the sums extend over all pairs of injections @xmath355 here , the sign @xmath356 of @xmath154 is defined in the usual way by the parity of the number of pairs @xmath357 for which @xmath358 . we have @xmath359 \,d\mu^{n - k}(x_{k+1 } , \ldots , x_n ) \nonumber \\ & = \frac{1}{(n - k ) ! } \int_{e^{n - k } } \sum_{\sigma \in \sym(n ) } ( -1)^\sigma \prod_{i=1}^n \phi_{\sigma(i)}(x_i ) \cdot { } \nonumber \\ \noalign{$\displaystyle \hfill \cdot \sum_{\tau \in \sym(n ) } ( -1)^\tau \prod_{i=1}^n \overline{\phi_{\tau(i)}(x_i ) } \,d\mu^{n - k}(x_{k+1 } , \ldots , x_n ) $ } & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\ \operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{i=1}^k \phi_{\sigma(i)}(x_i ) \overline{\phi_{\tau(i)}(x_i ) } \nonumber \\ & = \det \big(k \restrict ( x_1 , \ldots , x_k)\big ) \,.\end{aligned}\ ] ] here , the first equality uses e.integrate/ , the second equality uses e.prodtensor/ , the third equality uses the fact that @xmath360 is 1 or 0 according as @xmath361 or not , and the fourth equality uses cauchy - binet . note that a factor of @xmath362 arises because for every pair of injections @xmath363 with equal image , there are @xmath362 extensions of them to permutations @xmath364 with @xmath361 for all @xmath365 ; in this case , @xmath366 . we write @xmath94 for the law of the associated point process on @xmath0 . l.weaklimit let @xmath367 with @xmath368 for some @xmath369 . then @xmath370 is tight and every weak limit point of @xmath371 is simple . by using the kernel @xmath372 with respect to the measure @xmath373 , we may assume that @xmath374 . tightness follows from @xmath375 \le \be[\sx_n(a ) ] = \int_a k_n(x , x ) \,d\mu(x)\,.\ ] ] for the rest , we may assume that @xmath0 is compact and @xmath376 . let @xmath293 be a limit point of @xmath371 . let @xmath377 be the atomic part of @xmath173 and @xmath378 . choose @xmath379 and partition @xmath0 into sets @xmath380 with @xmath381 . let @xmath5 be such that @xmath382 and @xmath383 . let @xmath384 be open such that @xmath385 and @xmath386 . then @xmath387 & \le \limsup_n \big(\bp[\sx_n(u \setminus a ) \ge 1 ] + \bp[\texists i \sx_n(a_i ) \ge 2]\big ) \\ & \le \limsup_n \big(\be[\sx_n(u \setminus a ) ] + \sum_i \be[(\sx_n(a_i))_2]\big ) \\ & \le \muc(u ) + \sum_i \mu(a_i)^2 < 2/m \ , . \tag*{\qedhere}\end{aligned}\ ] ] now , given any locally trace - class orthogonal projection @xmath158 onto @xmath48 , choose finite - dimensional subspaces @xmath388 with corresponding projections @xmath389 . clearly @xmath390 @xmath391-a.e.and @xmath392 @xmath173-a.e . thus , the joint intensity functions converge a.e . by dominated convergence , if @xmath393 is relatively compact and borel , then @xmath394 \to \int_a \det ( k \restrict f ) \,d\mu^k(f)$ ] . by uniform exponential moments of @xmath395 ( * ? ? ? * proof of lemma 4.2.6 ) , it follows that all weak limit points of @xmath396 are equal , and hence , by l.weaklimit/ , define @xmath94 with kernel @xmath158 . ( in s.cinequalities/ , we shall see that @xmath397 is stochastically increasing . ) finally , let @xmath158 be any locally trace - class positive contraction . define the orthogonal projection on @xmath398 whose block matrix is take an isometric isomorphism of @xmath277 to @xmath131 for some denumerable set @xmath235 and interpret the above as an orthogonal projection @xmath399 on @xmath400 . then @xmath399 is clearly locally trace - class and @xmath158 is the compression of @xmath399 to @xmath0 . thus , we define @xmath320 by intersecting samples of @xmath401 with @xmath0 . we remark that by writing @xmath399 as a limit of increasing finite - rank projections that we then compress , we see that @xmath320 may be defined as a limit of determinantal processes corresponding to increasing finite - rank positive contractions . g.ctail if @xmath158 is a locally trace - class positive contraction , then @xmath320 has trivial tail in that every event in @xmath402 is trivial . rather than using compressions as in the last paragraph above , an alternative approach to defining @xmath320 uses mixtures and starts from finite - rank projections , as in s.mix/. this approach is due to b.hkpv : survey/. consider first a finite - rank @xmath403 . let @xmath404 be independent . let @xmath405 $ ] ; thus , @xmath406 . we claim that @xmath407 is determinantal with kernel @xmath158 . indeed , it is clearly a simple point process . write @xmath408 , @xmath409 , and @xmath410 . let @xmath411 . combining cauchy - binet with e.prodtensor/ yields @xmath412 . similarly , the joint intensities of @xmath413 are the expectations of the joint intensities of @xmath414 , which equal @xmath415 essentially the same works for trace - class @xmath416 ; we need merely take , in the last step , a limit in the above equation as @xmath417 for @xmath418 , since all terms are non - negative and @xmath419 a.e . given this construction of @xmath320 for trace - class @xmath158 , one can then construct @xmath320 for a general locally trace - class positive contraction by defining its restriction to each relatively compact set @xmath5 via the trace - class compression @xmath330 . as noted by b.hkpv:survey/ , a consequence of the mixture representation is a clt due originally to b.soshnikov:gauss/ : t.clt let @xmath389 be trace - class positive contractions on spaces @xmath420 . let @xmath367 and write @xmath421 . if @xmath422 as @xmath417 , then @xmath423 obeys a clt . in order to simulate @xmath320 when @xmath158 is a trace - class positive contraction , it suffices , by taking a mixture as above , to see how to simulate @xmath424 when @xmath425 . the following algorithm ( * ? ? ? * algo . 18 ) gives a uniform random ordering of @xmath293 as @xmath426 . since @xmath427 , the measure @xmath428/n = n^{-1 } k(x , x)\,d\mu(x)$ ] is a probability measure on @xmath0 . select a point @xmath429 at random from that measure . if @xmath430 , then we are done . if not , then let @xmath431 be the orthogonal complement in @xmath48 of the function @xmath432 , where @xmath343 is an orthonormal basis for @xmath48 . then @xmath433 and we may repeat the above for @xmath431 to get the next point , @xmath434 , then @xmath435 , etc . the conditional density of @xmath436 given @xmath437 is @xmath438 by e.densityktuple/ , i.e. , @xmath439 times the squared distance from @xmath440 to the linear span of @xmath441 . it can help for rejection sampling to note that this is at most @xmath442 . one can also sample faster by noting that the conditional distribution of @xmath436 is the same as that of @xmath443 , where @xmath444 is a uniformly random vector on the unit sphere of @xmath113 . note that if @xmath445 are bounded @xmath446-valued random variables , then the function @xmath447 determines the joint distribution of @xmath448 since it gives the derivatives at @xmath449 of the probability generating function @xmath450 . let us re - examine e.falling/ in the context of a finite - rank @xmath451 . given disjoint @xmath452 and non - negative @xmath310 summing to @xmath18 , it will be convenient to write @xmath453 for @xmath454 . we have by cauchy - binet @xmath455 = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \rho_k \,d\mu^k = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \det ( k \restrict ( x_1 , \ldots , x_k ) ) \,\prod_{j=1}^k d\mu(x_j ) \\ & = \int_{\prod_{\ell=1}^r a_\ell^{k_\ell } } \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\ \operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{j=1}^k \lambda_{\sigma(j ) } \phi_{\sigma(j)}(x_j ) \overline{\phi_{\tau(j)}(x_j ) } \,\prod_{j=1}^k d\mu(x_j ) \\ & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\ \operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \prod_{j=1}^k \int_{a_{\kappa(j ) } } \lambda_{\sigma(j ) } \phi_{\sigma(j)}(x_j ) \overline{\phi_{\tau(j)}(x_j ) } \,d\mu(x_j ) \\ & = \sum_{\substack{\sigma , \tau \in \sym(k , n ) \\ \operatorname{im}(\sigma ) = \operatorname{im}(\tau ) } } ( -1)^\sigma ( -1)^\tau \lambda^{\operatorname{im}(\sigma ) } \prod_{j=1}^k \bigip{\boi{a_{\kappa(j ) } } \phi_{\sigma(j ) } , \overline{\phi_{\tau(j ) } } } \\ & = \sum_{\sigma \in \sym(k , n ) } ( -1)^\sigma \lambda^{\operatorname{im}(\sigma ) } \det \big [ \bigip{\boi{a_{\kappa(j ) } } \phi_{\sigma(j ) } , \overline{\phi_\ell } } \big]_{\substack{j \le k \hfill \\ \ell \in \operatorname{im}(\sigma ) } } \,.\end{aligned}\ ] ] as an immediate consequence of this formula , we obtain the following important principle of goldman ( * ? ? ? * proposition 12 ) that allows one to infer properties of continuous determinantal point processes from corresponding properties of discrete determinantal probability measures : t.transfer let @xmath456 and @xmath457 be two radon measure spaces on locally compact polish sets . let @xmath458 be pairwise disjoint borel subsets of @xmath0 and @xmath459 be pairwise disjoint borel subsets of @xmath325 . let @xmath460 $ ] with @xmath461 . let @xmath462 be orthonormal in @xmath277 and @xmath463 be orthonormal in @xmath464 . let @xmath465 and @xmath466 . if @xmath467 for all @xmath468 , then the @xmath320-distribution of @xmath469 equals the @xmath470-distribution of @xmath471 . when only finitely many @xmath227 , this follows from our previous calculation . the general case follows from weak convergence of the processes corresponding to the partial sums , as in the paragraph following l.weaklimit/. this permits us to compare to discrete measures via ( * ? ? ? * lemma 16 ) : l.compare let @xmath173 be a radon measure on a locally compact polish space , @xmath0 . let @xmath458 be pairwise disjoint borel subsets of @xmath0 . let @xmath472 for @xmath16 . then there exists a denumerable set @xmath325 , pairwise disjoint subsets @xmath459 of @xmath325 , and @xmath473 such that @xmath474 and @xmath475 for all @xmath468 . without loss of generality , we may assume that @xmath476 . for each @xmath477 , fix an orthonormal basis @xmath478 for the subspace of @xmath277 spanned by @xmath479 . here , @xmath480 . define @xmath481 and @xmath482 . let @xmath483 be the isometric isomorphism from the span of @xmath484 to @xmath485 that sends @xmath486 to @xmath487 . defining @xmath488 yields the desired vectors . we now show how the discrete models of s.transf/ allow us to obtain the analogues of the stochastic inequalities known to hold for discrete determinantal probability measures . for a borel set @xmath7 , let @xmath207 denote the @xmath154-field on @xmath289 generated by the functions @xmath489 for borel @xmath490 . we say that a function that is measurable with respect to @xmath207 is , more simply , measurable with respect to @xmath5 . the obvious partial order on @xmath289 allows us to define what it means for a function @xmath491 to be . as in the discrete case , we say that @xmath8 has if @xmath492 \le \be[f_1 ] \be[f_2 ] $ ] for every pair @xmath204 , @xmath205 of bounded increasing functions that are measurable with respect to complementary subsets of @xmath0 . an event is increasing if its indicator is increasing . then @xmath8 has negative associations iff for every pair @xmath493 , @xmath494 of increasing events that are measurable with respect to complementary subsets of @xmath0 . we also say that and write @xmath495 if @xmath496 for every increasing event @xmath169 . call an event if it has the form @xmath497 , where @xmath100 is a relatively compact borel set and @xmath498 . write @xmath499 for the closure under finite unions and intersections of the collection of elementary increasing events with @xmath500 ; the notation @xmath501 is chosen for upwardly closed " . note that every event in @xmath499 is measurable with respect to some finite collection of functions @xmath502 for pairwise _ disjoint _ relatively compact borel @xmath503 . write @xmath504 for the closure of @xmath499 under monotone limits , i.e. , under unions of increasing sequences and under intersections of decreasing sequences ; these events are also increasing . this is the same as the closure of @xmath499 under countable unions and intersections . l.approxincr let @xmath5 be a borel subset of a locally compact polish space , @xmath0 . then @xmath504 is exactly the class of increasing borel sets in @xmath207 . we give a proof at the end of this subsection . first , we derive two consequences . a weaker version ( negative correlations of elementary increasing events ) of the initial one is due to b.ghosh/. t.cfm let @xmath173 be a radon measure on a locally compact polish space , @xmath0 . let @xmath158 be a locally trace - class positive contraction on @xmath277 . then @xmath320 has negative associations . let @xmath505 be borel . let @xmath506 and @xmath507 . then @xmath508 for some compact @xmath100 by definition of @xmath509 . we claim that e.cnegass/ holds for @xmath493 , @xmath494 , and @xmath510 , i.e. , for @xmath511 . now @xmath493 is measurable with respect to a finite number of count functions @xmath502 for some disjoint @xmath512 ( @xmath513 ) and likewise @xmath494 is measurable with respect to a finite number of functions @xmath514 for some disjoint @xmath515 ( @xmath513 ) . thus , there are functions @xmath516 and @xmath517 such that @xmath518 and @xmath519 . by t.transfer/ and l.compare/ , there is some discrete determinantal probability measure @xmath9 on some denumerable set @xmath325 and pairwise disjoint sets @xmath520 such that the joint @xmath521-distribution of all @xmath522 and @xmath523 is equal to the joint @xmath9-distribution of all @xmath524 and @xmath525 . define the corresponding events @xmath526 by @xmath527 and @xmath528 . since @xmath526 depend on disjoint subsets of @xmath325 , t.fm/ gives that @xmath529 . this is the same as e.cnegass/ by t.transfer/. the same e.cnegass/ clearly then holds in the less restrictive setting @xmath530 by taking monotone limits . l.approxincr/ completes the proof . t.cdom theorem 3 of b.goldman/ suppose that @xmath531 and @xmath532 are two locally trace - class positive contractions such that @xmath533 . then @xmath534 . it suffices to show that @xmath535 for every @xmath536 . again , it suffices to assume that @xmath537 are trace class . l.compare/ applied to all eigenfunctions of @xmath531 and @xmath532 yields a denumerable @xmath325 and two positive contractions @xmath538 on @xmath485 , together with an event @xmath539 , such that @xmath540 for @xmath541 . furthermore , by construction , every function in @xmath485 is the image of a function in @xmath542 under the isometric isomorphism @xmath483 used to prove l.compare/ , whence @xmath543 . therefore t.dominate/ yields @xmath544 , as desired . again , it would be very interesting to have a natural monotone coupling of @xmath545 with @xmath546 . for some examples where this would be desirable , see s.orthogpoly/. l.approxincr/ will follow from this folklore variant of a theorem of dyck b.dyck/ : t.dyck let @xmath136 be a polish space on which @xmath547 is a partial ordering that is closed in @xmath548 . let @xmath501 be a collection of open increasing sets that generates the borel subsets of @xmath136 . let @xmath549 be the closure of @xmath501 under countable intersections and countable unions . suppose that for all @xmath550 , either @xmath551 or there is @xmath552 and an open set @xmath553 such that @xmath554 , @xmath555 , and @xmath556 . then @xmath549 equals the class of increasing borel sets . obviously every set in @xmath549 is borel and increasing . to show the converse , we prove a variant of lusin s separation theorem . namely , we show that if @xmath557 is increasing and analytic ( with respect to the paving of closed sets , as usual ) and if @xmath558 is analytic with @xmath559 , then there exists @xmath560 such that @xmath561 and @xmath562 . taking @xmath563 to be borel and @xmath564 forces @xmath565 and gives the desired conclusion . to prove this separation property , we first show a stronger conclusion in a special case : suppose that @xmath566 are compact such that @xmath110 is contained in an increasing set @xmath563 that is disjoint from @xmath111 ; then there exists an open @xmath560 and an open @xmath567 such that @xmath568 , @xmath569 , and @xmath556 . indeed , since @xmath563 is increasing , for every @xmath570 , we do _ not _ have that @xmath551 , whence by hypothesis , there exist @xmath571 and an open @xmath572 with @xmath573 , @xmath574 , and @xmath575 . because @xmath111 is compact , for each @xmath576 , we may choose @xmath577 such that @xmath578 . define @xmath579 . then @xmath580 is open , contains @xmath256 , and is disjoint from @xmath581 , whence compactness of @xmath110 ensures the existence of @xmath582 with @xmath583 . then @xmath584 is open , contains @xmath111 , and is disjoint from @xmath384 , as desired . to prove the general case , let @xmath585 and @xmath586 be the two coordinate projections on @xmath587 . define @xmath588 for @xmath589 to be 0 if there exists @xmath560 such that @xmath590 and @xmath591 ; and to be 1 otherwise . we claim that @xmath192 is a capacity in the sense of ( * ? ? ? * ( 30.1 ) ) . it is obvious that @xmath592 if @xmath593 and it is simple to check that if @xmath594 , then @xmath595 . suppose for the final property that @xmath5 is compact and @xmath596 ; we must find an open @xmath597 for which @xmath598 . there exists some @xmath599 with @xmath600 and @xmath601 . then the result of the second paragraph yields sets @xmath384 and @xmath567 that give @xmath602 as desired . now let @xmath563 and @xmath603 be as in the first paragraph . if @xmath604 is compact , then setting @xmath605 and applying the second paragraph shows that @xmath596 . thus , by the choquet capacitability theorem ( * ? ? ? * ( 30.13 ) ) , @xmath606 . l.approxincr/ clearly every set in @xmath504 is increasing and in @xmath207 . for the converse , endow @xmath5 with a metric so that it becomes locally compact polish while preserving its class of relatively compact sets and its borel @xmath154-field : choose a denumerable partition of @xmath5 into relatively compact sets @xmath607 and make each one compact and of diameter at most 1 ; make the distance between @xmath256 and @xmath608 be 1 if @xmath256 and @xmath608 belong to different @xmath607 . let @xmath609 with the vague topology and let @xmath501 be the class of elementary increasing events defined with respect to ( relatively compact ) sets @xmath500 that are open for this new metric . apply t.dyck/. since @xmath610 , the result follows . natural examples of determinantal point processes arise from orthogonal polynomials with respect to a probability measure @xmath173 on @xmath25 . assume that @xmath173 has infinite support and finite moments of all orders . let @xmath389 denote the orthogonal projection of @xmath611 onto the linear span @xmath612 of the functions @xmath613 . there exist unique ( up to signum ) polynomials @xmath614 of degree @xmath18 such that for every @xmath278 , @xmath615 is an orthonormal basis of @xmath612 . by elementary row operations , we see that for variables @xmath616 , the map @xmath617_{i , j \le n}$ ] is a vandermonde polynomial up to a constant factor , whence @xmath618 [ \phi_i(z_j)]^ * = c_n \prod_{1 \le i < j \le n } |z_i - z_j|^2\ ] ] for some constant @xmath619 . therefore , the density of @xmath620 ( with points randomly ordered ) with respect to @xmath346 is given by @xmath621 times the square of a vandermonde determinant . classical examples include the following : 1 . if @xmath173 is gaussian measure on @xmath24 , i.e. , @xmath622 , then @xmath614 are the hermite polynomials , @xmath623 , and @xmath620 is the law of the , which is the set of eigenvalues of @xmath624 , where @xmath625 is an @xmath626 matrix whose entries are independent standard complex gaussian . ( a standard complex gaussian random variable is the same as a standard gaussian vector in @xmath627 divided by @xmath628 in order that the complex variance equal 1 . its density is @xmath629 with respect to lebesgue measure on @xmath25 . ) this is due to wigner ; see b.mehta/. 2 . if @xmath173 is unit lebesgue measure on the unit circle @xmath630 , then @xmath631 , so @xmath632 , and @xmath620 is the law of the , which is the set of eigenvalues of a random matrix whose distribution is haar measure on the set of @xmath626 unitary matrices . this ensemble was introduced by dyson , but the law of the eigenvalues is due to weyl ; see b.hkpv : book/. 3 . if @xmath173 is standard gaussian measure on @xmath25 , then @xmath633 , @xmath623 , and @xmath620 is the law of the , which is the set of eigenvalues of an @xmath626 matrix whose entries are independent standard complex gaussian . this is due to ginibre ; see b.hkpv : book/. 4 . if @xmath173 is unit lebesgue measure on the unit disk @xmath634 , then @xmath635 , so @xmath636 , and the limit of @xmath620 is the law of the zero set of the random power series whose coefficients are independent standard complex gaussian , which converges in the unit disk a.s . this is due to peres and virg b.peresvirag/. 5 . if @xmath173 has density @xmath637 with respect to lebesgue measure on @xmath25 , then @xmath638 for @xmath315 , so @xmath639 , and @xmath620 is the law of the , which is the set of eigenvalues of @xmath640 when @xmath641 are independent @xmath626 matrices whose entries are independent standard complex gaussian . ( here , we are limited to @xmath612 since the larger spaces do not lie in @xmath341 . ) this is due to krishnapur b.krishnapur:thesis/ ; see b.hkpv : book/. the process was studied earlier by b.caillol/ and b.fjm/ , but without observing the connection to eigenvalues . inverting stereographic projection , we identify this process with one whose density with respect to lebesgue measure on the unit sphere in @xmath642 is proportional to @xmath643 . for additional information on such processes , see @xcite . for an extension to complex manifolds , see @xcite . by t.cdom/ , the processes @xmath620 stochastically increase in @xmath278 for each of the examples above except the last . it would be interesting to see natural monotone couplings . perhaps the last example also increases stochastically in @xmath278 . the is the limit of the @xmath278th ginibre processes as @xmath417 ; it has the kernel @xmath644 with respect to standard gaussian measure on @xmath25 . this process is invariant under all isometries of @xmath25 . for each of the plane , sphere , and hyperbolic disk , there is only a 1-parameter family of determinantal point processes having a kernel @xmath645 that is holomorphic in @xmath646 and in @xmath647 and whose law is isometry invariant ( * ? ? ? * theorem 3.0.5 ) . for the sphere , that family has already been given above ; the parameter is a positive integer . for the other two families , the parameter is a positive real number , @xmath648 . in the case of the plane , the processes are related simply by homotheties , @xmath649 . the push - forward of the ginibre process with respect to @xmath650 has kernel @xmath651 with respect to the measure @xmath652 , where @xmath173 is lebesgue measure on @xmath25 . do these processes increase stochastically in @xmath648 , like poisson processes do ? in the hyperbolic disk , the processes have kernel @xmath653 with respect to the measure @xmath654 , where @xmath173 is lebesgue measure on @xmath655 . ( we fix a branch of @xmath656 for @xmath657 . ) these give orthogonal projections onto the generalized bergman spaces . the case @xmath658 is that of the limiting ope4 above . do these processes stochastically increase in @xmath648 ? recall that when @xmath48 is a finite - dimensional subspace of @xmath37 , the measure @xmath94 is supported by those subsets @xmath95 that project to a basis of @xmath48 under @xmath15 . similarly , when @xmath158 is the kernel of a finite - rank orthogonal projection onto @xmath659 , define the functions @xmath660 . then the measure @xmath320 is supported by those @xmath661 such that @xmath662 is a basis of @xmath48 , since @xmath663 . here , @xmath664 means that @xmath665 . the question of extending this to infinite - dimensional @xmath48 turns out to be very interesting . a basis of a finite - dimensional vector space is a minimal spanning set . although @xmath666 is @xmath94-a.s . linearly independent , minimality does not hold in general , even for the wired spanning forest of a tree , as shown by the examples in b.heicklenlyons/. see also c.ell2min/. however , the other half of being a basis does hold in the discrete case and is open in the continuous case . let @xmath667 $ ] be the closed linear span of @xmath668 . t.basis b.lyons:det/ for every @xmath49 , we have @xmath669 = h$ ] @xmath94-a.s . we give an application of t.basis/ for @xmath670 , but it has an analogous statement for every countable abelian group . let @xmath671 be the unit circle equipped with unit lebesgue measure . for a measurable function @xmath672 and @xmath673 , the of @xmath172 at @xmath278 is @xmath674 . let @xmath675 denote the restriction of @xmath676 to @xmath677 . if @xmath678 is measurable , we say @xmath679 is @xmath5 if the set @xmath680 is dense in @xmath681 , where we identify @xmath681 with the set of functions in @xmath682 that vanish outside @xmath5 . the case where @xmath5 is an interval is quite classical ; see b.redheffer/ for a review . a crucial role in that case is played by the following notion of density of @xmath677 . d.bm for an interval @xmath683 \subset \bbr \setminus \ { 0 \}$ ] , define its @xmath684\big ) : = \max \big\ { |a| , |b| \big\}/ \min \big\ { |a| , |b| \big\ } \,.\ ] ] for a discrete @xmath685 , the of @xmath677 , denoted @xmath686 , is the supremum of those @xmath687 for which there exist disjoint nonempty intervals @xmath688 with @xmath689 for all @xmath278 and @xmath690 ^ 2 = \infty$ ] . a simpler form of the beurling - malliavin density was provided by b.red:two/ , who showed that e.bmred ( s ) = \ { c : s _ k s | - | < } . c.seqdual b.lyons:det/ let @xmath691 be lebesgue measurable with measure @xmath692 . then there is a set of beurling - malliavin density @xmath692 in @xmath693 that is complete for @xmath5 . indeed , let @xmath694 be the determinantal probability measure on @xmath695 corresponding to the toeplitz matrix @xmath696 . then @xmath694-a.e . @xmath697 is complete for @xmath5 and has @xmath698 . when @xmath5 is an interval , the celebrated theorem of beurling and malliavin b.bm/ says that if @xmath677 is complete for @xmath5 , then @xmath699 , and that if @xmath700 , then @xmath677 is complete for @xmath5 . ( this holds for @xmath677 that are not necessarily sets of integers , but we are concerned in this subsection only with @xmath679 . ) c.seqdual/ can be compared ( take @xmath701 and @xmath702 ) to a theorem of bourgain and tzafriri b.btz/ , according to which there is a set @xmath697 of ( schnirelman ) density at least @xmath703 such that if @xmath704 and @xmath676 vanishes off @xmath677 , then @xmath705 it would be interesting to find a quantitative strengthening of c.seqdual/ that would encompass this theorem of @xcite . the following theorem is equivalent to t.basis/ by duality : t.morris b.lyons:det/ for every @xmath49 , we have @xmath706 } h } = [ \ba]$ ] @xmath94-a.s . as an example , consider the wired spanning forest of a graph , @xmath253 . here , @xmath707 . in this case , @xmath708 } \star(g ) } = \star(b)$ ] for @xmath709 . thus , the conclusion of t.morris/ is that @xmath710 , which equals @xmath711 , is concentrated on the singleton @xmath712 for @xmath713-a.e . @xmath714 . this was a conjecture of , established by b.morris/. c.ell2min for every @xmath49 , @xmath94-a.s . the maps @xmath715 \to h$ ] and @xmath716 } \colon h \to [ \ba]$ ] are injective with dense image . both statements are equivalent to @xmath717 \cap h^\perp = \{0\ } = h \cap \ba^\perp$ ] , and these are the contents of theorems [ thm : basis ] and [ thm : morris ] . proved that on any network @xmath718 ( where @xmath253 is the underlying graph and @xmath719 is the function assigning conductances , or weights , to the edges ) , for @xmath720-a.e . forest @xmath714 and for every component tree @xmath483 of @xmath714 , the @xmath721 of @xmath722 equals @xmath483 a.s . this suggested b.lyons:det/ the following extension . given a subspace @xmath48 of @xmath37 and a set @xmath95 , the subspace of @xmath723 $ ] most like " or closest to " @xmath48 is the closure of the image of @xmath48 under the orthogonal projection @xmath724}$ ] ; we denote this subspace by @xmath725 . for example , if @xmath726 , then @xmath727 since for each @xmath728 , we have @xmath724 } ( \star_x^g ) = \star_x^b$ ] . to say that @xmath729 is concentrated on @xmath730 is the same as to say that @xmath731 $ ] . this motivated the following theorem and shows how it is an extension of morris s theorem . if @xmath158 is a locally trace - class orthogonal projection onto @xmath48 , then for @xmath732 , we have @xmath733 in other words , @xmath158 is a reproducing kernel for @xmath48 . a subset @xmath677 of @xmath48 is called if the closed linear span of @xmath677 equals @xmath48 ; equivalently , the only element of @xmath48 that is orthogonal to @xmath677 is 0 . an analogue of t.basis/ was conjectured by lyons and peres in 2010 : g.cbasis if @xmath158 is a locally trace - class orthogonal projection onto @xmath48 , then for @xmath320-a.e . @xmath293 , @xmath734 = h$ ] , i.e. , if @xmath735 and @xmath736 , then @xmath737 . just as in the discrete case , this appears to be on the critical border for many special instances , as we illustrate for several processes where @xmath738 : 1 . let @xmath173 be lebesgue measure on @xmath24 and @xmath739 , the . denote the fourier transform on @xmath24 by @xmath740 for @xmath741 , and , by isometric extension , for @xmath742 . write @xmath743}$ ] . since @xmath744 , we have @xmath745 , where @xmath746 is the inverse fourier transform of @xmath172 . therefore , the induced operator @xmath158 arises from the orthogonal projection onto the paley - wiener space @xmath747 . the sine - kernel process arises frequently ; e.g. , it is various scaling limits of the @xmath278th gaussian unitary ensemble in the bulk " as @xmath417 . ( a related scaling limit of the gue is wigner s semicircle distribution . ) we may more easily interpret g.cbasis/ for fourier transforms of functions in @xmath748 $ ] : it says that for @xmath320-a.e . @xmath293 , the only @xmath749 $ ] such that @xmath750 is @xmath751 . although the beurling - malliavin theorem applies , no information can be deduced because @xmath752 a.s . however , ghosh b.ghosh/ has proved this case . 2 . let @xmath173 be standard gaussian measure on @xmath25 and @xmath753 . this is the ginibre process . it corresponds to orthogonal projection onto the @xmath754 consisting of the entire functions that lie in @xmath611 ; this is the space of power series @xmath755 such that @xmath756 . completeness of a set of elements @xmath757 in @xmath754 is equivalent to completeness in @xmath758 ( with lebesgue measure ) of the gabor system of windowed complex exponentials @xmath759 \st \lambda \in \sqrt{2}\lambda\big\ } \,,\ ] ] which is used in time - frequency analysis of non - band - limited signals . the equivalence is proved using the bargmann transform @xmath760 \,dt \big ) \,,\ ] ] which is an isometry from @xmath758 to @xmath754 . that the critical density is 1 was shown in various senses going back to von neumann ; see b.clp/. this case has also been proved by ghosh b.ghosh/. 3 . let @xmath173 be unit lebesgue measure on the unit disk @xmath761 and @xmath762 . this process is the limiting ope4 in s.orthogpoly/. it corresponds to orthogonal projection onto the @xmath763 consisting of the analytic functions that lie in @xmath764 . what is known about the zero sets of functions in the bergman space b.duren/ is insufficient to settle g.cbasis/ in this case and it remains open . the two instances above that have been proved by ghosh b.ghosh/ follow from his more general result that g.cbasis/ holds whenever @xmath173 is continuous and @xmath320 is , which means that @xmath765 is measurable with respect to the @xmath320-completion of @xmath766 for every ball @xmath767 . the limiting process ope4 is not rigid b.hs : tolerance/. ghosh and krishnapur ( personal communication , 2014 ) have shown that @xmath320 is rigid only if @xmath158 is an orthogonal projection . it is not sufficient that @xmath158 be a projection , as the example of the bergman space shows . a necessary and sufficient condition to be rigid is not known . let @xmath158 be a locally trace - class orthogonal projection onto @xmath768 . for a function @xmath172 , write @xmath769 for the function @xmath770 . let @xmath771 . clearly @xmath772 for a.e . @xmath293 . also , for @xmath773 , the function @xmath774 is bounded . a conjecture analogous to c.ell2min/ is that @xmath293 is a sort of set of interpolation for @xmath48 in the sense that given any countable dense set @xmath775 , for a.e . @xmath293 , the set @xmath776 is dense in @xmath777 . one may also ask about completeness for appropriate poisson point processes . suppose @xmath778 is a group that acts on @xmath0 and that @xmath158 is @xmath778-invariant , i.e. , @xmath779 for all @xmath780 , @xmath295 , and @xmath781 . ( this is equivalent to the operator @xmath158 being @xmath778-equivariant . ) then the probability measure @xmath320 is @xmath778-invariant . this contact with ergodic theory and other areas of mathematics suggests many interesting questions . lack of space prevents us from considering more than just a few aspects of the case where @xmath0 is discrete and from giving all definitions . let @xmath782 . in this case , @xmath158 is invariant iff @xmath783 for some @xmath784 $ ] , where @xmath785 . we write @xmath786 in place of @xmath320 . some results and questions from b.ls:dyn/ follow . t.bern for all @xmath172 , the process @xmath786 is isomorphic to a bernoulli process . this was shown in dimension 1 by b.shitak:ii/ for those @xmath172 such that @xmath787 by showing that those @xmath786 are weak bernoulli ( wb ) , also called @xmath788-mixing " and absolutely regular " . despite its name , it is known that wb is strictly stronger than bernoullicity . the precise class of @xmath172 for which @xmath786 is wb is not known . as usual , the of a nonnegative function @xmath172 is @xmath789 . t.gmdom for all @xmath172 , the process @xmath786 stochastically dominates product measure @xmath790 and is stochastically dominated by product measure @xmath791 . these bounds are optimal . we conjecture that ( kolmogorov - sinai ) entropy is concave , as would follow from g.concave/. g.invconcave for all @xmath172 and @xmath792 , we have @xmath793 . q.block let @xmath794 $ ] be a trigonometric polynomial of degree @xmath795 . then @xmath786 is @xmath795-dependent , as are all @xmath796-block factors of independent processes . is @xmath797 an @xmath796-block factor of an i.i.d . process ? this is known when @xmath798 b.broman/. let @xmath778 be a sofic group , a class of groups that includes all finitely generated amenable groups and all finitely generated residually amenable groups . no finitely generated group is known not to be sofic . let @xmath0 be @xmath778 or , more generally , a set acted on by @xmath778 with finitely many orbits , such as the edges of a cayley graph of @xmath778 . the following theorems are from b.lyonsthom/. t.sbern for every @xmath778-equivariant positive contraction @xmath1 on @xmath37 , the process @xmath9 is a @xmath799-limit of finitely dependent ( invariant ) processes . if @xmath778 is amenable and @xmath800 , then @xmath9 is isomorphic to a bernoulli process . even if @xmath166 and @xmath167 are @xmath778-invariant probability measures on @xmath801 with @xmath168 , there need not be a @xmath778-invariant monotone coupling of @xmath166 and @xmath167 b.mester : mono/. the proof of the preceding theorem depends on the next one : t.monojoin if @xmath160 and @xmath161 are two @xmath778-equivariant positive contractions on @xmath37 with @xmath185 , then there exists a @xmath778-invariant monotone coupling of @xmath201 and @xmath202 . the proof of t.sbern/ also uses the inequality @xmath802 for equivariant positive contractions , @xmath1 and @xmath803 , where @xmath804 is the schatten 1-norm . when @xmath1 and @xmath803 commute , one can improve this bound to @xmath805 we do not know whether this inequality always holds . write @xmath806 for the fuglede - kadison determinant of @xmath1 when @xmath1 is a @xmath778-equivariant operator . the following would extend t.gmdom/. it is open even for finite groups . g.fkdom for all @xmath778-equivariant positive contractions @xmath1 on @xmath807 , the process @xmath9 stochastically dominates product measure @xmath808 and is stochastically dominated by product measure @xmath809 , and these bounds are optimal . it turns out that the expected degree of a vertex in the free uniform spanning forest of a cayley graph depends only on the group , via its first @xmath810-betti number , @xmath811 , and not on the generating set used to define the cayley graph b.lyons:betti/ : t.betti in every cayley graph @xmath253 of a group @xmath778 , we have @xmath812 = 2 \beta_1(\gp ) + 2 \,.\ ] ] this is proved using the representation of @xmath813 as a determinantal probability measure . it can be used to give a uniform bound on expansion constants b.lpv/ : t.lpv for every finite symmetric generating set @xmath677 of a group @xmath778 , we have @xmath814 for all finite non - empty @xmath815 . there are extensions of these results to higher - dimensional cw - complexes and higher @xmath810-betti numbers b.lyons : betti/. in unpublished work with d. gaboriau @xcite , we have shown the following : t.damien let @xmath253 be a cayley graph of a finitely generated group @xmath778 and @xmath816 . then there exists a @xmath778-invariant finitely dependent determinantal probability measure @xmath9 on @xmath817 that stochastically dominates @xmath818 and such that @xmath819 \le \be_\fsf\big[\deg_\fo(\bp)\big ] + \epsilon \,.\ ] ] in addition , if @xmath778 is sofic , then @xmath820 . if it could be shown that @xmath9 , or indeed every invariant finitely dependent probability measure that dominates @xmath813 , yields a connected subgraph a.s . , then it would follow that @xmath821 is equal to the cost of @xmath778 , a major open problem of b.gaboriau : invar/.

1406.2707

set theory was proposed with the intended use to the fields of pattern classification and information processing [ 1 ] . indeed , it has attracted many researchers , and their applications to real - life problems are of a great significance . simpson [ 2 ] presented the fuzzy min max neural network ( fmm ) , which makes the soft decisions to organize hyperboxes by its degree of belongingness to a particular class , which is known as a membership function . hyperbox is a convex box , completely represented by min and max points . fmm classification results are completely characterized with the help of a membership function . along with this elegant proposal , [ 2 ] also presented the characteristics for a good classifier , among which , nonlinear separability , overlapping classes and tuning parameters have proved to be of a great interest to a research community . simpson also presented a clustering approach using fmm in [ 3 ] . but many problems in real - life require both classification and clustering . to address this issue , gfmm [ 4 ] brought this generality . besides generality , the more significant contribution has proved to be modification to the membership function . the presented membership function computes the belongingness to the hyperbox so that the membership value decreases uniformly as we move away from the hyperbox . another weakness of fmm was the patterns belonging to overlapped region , where the rate of misclassification is considerably high . the tuning parameter , theta ( @xmath0 ) , which controls the size of a hyperbox , has a great impact on this overlapped region . smaller theta values produce less overlaps producing high training accuracy , but the efficacy of the network gets compromised , and for larger theta values , accuracy gets decreased . multiple approaches were presented to tackle this problem . earlier , the process of contraction [ 1][4 ] was employed , which used to eliminate all the overlapping regions . this method had the intrinsic problem of representing patterns not belonging to any of the hyperbox , in turn lessening the accuracy . exclusion / inclusion fuzzy classification ( hefc ) network was introduced in [ 5 ] , which further reduced the number of hyperboxes and increased the accuracy . inclusion hyperboxes were used to represent patterns belonging to the same class , while exclusion hyperboxes were used to denote the overlapped region , treated as if it is a hyperbox . this notion is used as it is in almost all the newly introduced models [ 6][7][8][9 ] . fuzzy min - max neural network classifier with compensatory neurons ( fmcn ) was acquainted in [ 7 ] . authors categorized the overlap into three parts , namely , full containment , partial overlap and no overlap , and then a new membership function to accommodate belongingness based on the compensation value . authors also analyzed that neatly taking care of overlapped region automatically brings the insensitivity to the hyperbox size parameter , @xmath0 . data core based fuzzy min - max neural network ( dcfmn ) [ 8 ] further improved upon fmcn . authors eliminated the need of overlap categorization . they also suggest a new membership function based on noise , geometric center and data cores of the hyperbox . wherein dcfmn improved the accuracy in few cases , there are some serious drawbacks . * * dcfmn introduces two new user controlled variables , @xmath1 and @xmath2 . @xmath1 is used to suppress the influence of the noise and @xmath2 is used to control the descending speed of the membership function . these two variables greatly impact the performance of the model and naturally , defining their values is a tedious job . * there exists an underlying assumption that noise within all the hyperboxes is similar , which may not be true . moreover , the sequence of the training exemplars plays a role as well . * mlf conveys that this membership function is not always preferred , in that , it does not work well for high percentage of samples belonging to overlapped area . multi - level fuzzy min max neural network ( mlf ) [ 9 ] addresses the problem of overlapped region with an elegant approach . it uses separate levels for overlapping regions , and monotonically decreases the hyperbox size ( @xmath0 ) . for most cases , mlf produces 100% training accuracy . though mlf achieves a significant milestone , entertaining testing accuracy is rather more important than training accuracy , as it greatly sways the usage of the algorithm in practical scenarios . in this brief , we identify and define a new boundary region , where misclassification rate is substantial . to the best of our knowledge , this kind of approach is presented for the first time , at least we did not come across any similar published work . hence we propose a method , based on data centroids , to evidentially prove that handling this newly introduced area of confusion between hyperboxes of different classes significantly increases the testing accuracy . the paper is organized as follows . mlf is reviewed in section ii . we introduced d - mlf algorithm in section iii . an illustrative example and comparative results of d - mlf with mlf model are presented in section iv and v , respectively . finally , conclusion is given in section vi . multi - level fuzzy min max neural network ( mlf ) is a classifier which efficiently caters misclassification of patterns belonging to overlapped region by maintaining a tree structure , which is a homogeneous tree [ 9 ] . in mlf training phase , exemplars are continuously recurred to form the hyperboxes and overlaps , each recursion resulting in one level . this recursive procedure is carried till the predefined maximum depth or till overlap exists . hyperbox expansion , based on hyperbox size controlling parameter ( @xmath0 ) , is validated using equation ( 1 ) and expansion is carried out by equation ( 2 ) . @xmath3 @xmath4 where , @xmath5 and @xmath6 are min point and max point of hyperbox _ b _ respectively , @xmath7 is the @xmath8 dimension of pattern _ a _ and _ d _ is the number of dimensions . also , prior to each recursion , @xmath0 is updated using equation ( 3 ) @xmath9 where , @xmath10 and @xmath11 thetas for next level and previous level , respectively and @xmath12 , being the value between 0 and 1 , ensures that size of hyperbox in overlapped region is less than its previous level . in the testing phase , overlap regions are first traversed recursively , to discover appropriate subnet to which a test pattern belongs to . thence , in that level , a class of hyperbox having highest membership value with the hyperboxes in the discovered subnet , is selected as a predicted class . mlf is able to achieve higher accuracy rates than previous fmm methods . this is due to an elegant treatment to the boundary region a confusion area . but , after training , there exists a room for yet another boundary . the region where membership function generates very close by values , it becomes difficult to assign a class with high degree of assurance . as per our experiments , mlf , and all the previous classifiers , do not perform well in this area . hence , a definition of this new region , and a methodology to solve it is proposed . in this section , we give details about a newly proposed algorithm , specifically , we define a new boundary region generated due to trained network and propose a solution to correctly classify test patterns belonging to it . _ figure 1 _ describes the d - mlf structure , each node in s@xmath13net contains two segments , hyperboxes segment ( hbs ) and overlapped segment ( ols ) . hbs represents hyperboxes generated in that level , whereas ols represents overlaps in that level . along with hyperbox information , data centroid ( dc ) . _ figure 2 _ shows the area of confusion considered by mlf and d - mlf . we introduce a boundary region that exists between any two hyperboxes , where , according to our experiments , the rate of misclassification is comparatively high . in the proposed method , the recommendations of mlf are intact , in addition to it , we use distance with the data centroids to improve a classification rate in the anew boundary region . similar to the mlf learning procedure , d - mlf maintains @xmath14 using hbs and ols structures . first , all the patterns are passed through , resulting in creation and expansion of hyperboxes using equations ( 1 ) and ( 2 ) . then each hyperbox is checked with the rest of hyperboxes to detect the overlap using equation ( 4 ) . @xmath15 where @xmath16 and @xmath17 are the max points and @xmath18 and @xmath19 are the min points of the two hyperboxes , among which overlap is tested . moreover , d - mlf adds a new step at the learning phase , known as data centroid ( dc ) computation , where dc of all input patterns belonging to each hyperbox is maintained in the hbs . dc is computed as follows : @xmath20 where @xmath21 is the data centroid of the @xmath22 hyperbox , @xmath23 is number of patterns belonging to @xmath22 hyperbox and @xmath24 is the @xmath8 pattern in @xmath22 hyperbox . if there exists an overlap , patterns belonging to the overlapped region are again sent to training procedure , where hbs and ols creation takes place for the next level . this process of recursion is followed afterward to train all the patterns . due to computation of ols and process of finding patterns belonging to ols , d - mlf and mlf are not single pass algorithms . in general , given the n overlaps in the first level , entire training data has to be traversed n times . thereafter , in the subsequent stages , data belonging to overlapped region is traversed in order of magnitude of number of overlaps in that region . this is a novel finding , and contradictory to what mlf authors have mentioned [ 9 ] . note that , the patterns belonging to overlapped region are not part of the dc computation . this step makes sure that training patterns balloting for more than one class are omitted in the final decision making . + net = d - mlf - train(net , @xmath0 ) + @xmath25 @xmath26 = h.centroid / h.membercount return null h.centroid + = sample ; h.membercount + = 1 ; create new hyperbox h ; h.centroid = sample ; h.membercount = 1 ; sdata = samples which inhabit in i region ; hi.centroid -= s ; hi.membercount -= 1 ; create an overlap - box as @xmath27 and add to ols @xmath28 = d - mlf - train ( sdata , @xmath29 ) ; link @xmath27 to @xmath28 with link @xmath30 ; @xmath31 the original mlf used a decision making based on the subnets decision . the selected subnet need not be a leaf node in the tree . we do not alter this model , rather enhance the process of how subnet marks the choice . membership function mentioned in the equation ( 11 ) is used against overlapped boxes . after recursively traversing the ols an appropriate subnet is discovered , to which test pattern belongs to . a membership function explained in the equation ( 6 ) is used , this time , to compute the membership with the hyperboxes within the selected subnet . @xmath32 \\ [ 1- f(v_i^j - a_h^i,\gamma_i ) ] ) ) \\ f(x,\gamma)=\begin{cases}{1}\;\;\;\ ; if \ ; x\gamma\;>\;1 \\ { x}\;\;\;\ ; if \ ; 0\;\leq\;x\gamma\;\leq\;1 \\ { 0}\;\;\;\ ; if \ ; x\gamma\;<\;0 \end{cases } \end{split}\ ] ] where @xmath33 represents belongingness of sample @xmath34 with @xmath35 hyperbox . @xmath36 is a difference between min and max point with sample @xmath34 and @xmath37 is a tuning parameter to control fuzziness . within these membership values , hyperboxes with highest two values are selected to define a boundary . medial region of these hyperboxes , controlled by @xmath38 , is treated as a boundary region . @xmath38 is a user controlled variable , mentioned in the percentage value . at this point , it is necessary to check if test pattern belongs to the boundary region . we define @xmath391 and @xmath392 as incident angles between test pattern and two hyperboxes , respectively . inclusion value is evaluated as follows : @xmath40 further , based on the inclusion value , output class is chosen . if pattern exists in the area outside of the defined boundary , we simply follow a path of mlf , and classify the pattern based on the maximum membership value , which is already computed . if the pattern belongs to the boundary region , euclidean distance [ 10 ] between test pattern and the data centroids of the selected hyperboxes is computed . hence , centered on the inclusion value , the output of the network is denoted as either the class of maximum @xmath41 among all the hyperboxes , or as a minimum of the distances of the topmost two hyperboxes @xmath42 where @xmath43 is given by ; @xmath44 where @xmath45 is the @xmath8 class membership for the test sample in @xmath46 subnet , @xmath47 is edge between subnet @xmath46 and the corresponding overlap box that enables the subnet if test sample is in this overlap box . and @xmath48 is the output of ols , which is given by equation ( 10 ) @xmath49 where @xmath50 is number of overlap boxes in ols and @xmath51 is membership function of the @xmath35 overlap box for test sample @xmath34 , given by equation ( 11 ) @xmath52 and @xmath53 is given by equation(12 ) @xmath54 where @xmath55 is the euclidean distance computed amongst sample @xmath46 and the data centroid of the topmost @xmath8 hyperbox using equation ( 13 ) @xmath56 out = d - mlf - test(net , sample ) + @xmath25 out = d - mlf - test ( @xmath28 , sample ) ; return null ; mv = [ ] ; mv + = membership ( sample , @xmath57 ) ; = [ max(mv ) , max(mv ( mv @xmath58 max(mv ) ) ) ] d = eudistance(sample , h1.dc , h2.dc ) ; out = min(d).class ; out = max ( mv).class ; @xmath31 in this illustration , we describe the effectiveness of the proposed model , clearly pointing out the identification and handling of the stated area of confusion . _ figure 3 _ illustrates the 2-diamentional data space . we consider 14 data samples for training and 6 data samples for testing . hyperbox size parameter ( @xmath0 ) is fixed at 0.3 and a boundary parameter ( @xmath38 ) is fixed at 5% . both mlf and d - mlf create two hyperboxes at @xmath59 layer . d - mlf also computes data centroids ( dc ) for each hyperbox , @xmath60 and @xmath61 . here , data centroids of @xmath60 and @xmath61 are @xmath62 and @xmath63 , respectively . patterns which do not belong to boundary region are classified correctly by mlf . but when it comes to boundary region , it fails to correctly classify the patterns . whereas the proposed d - mlf works better in the boundary region as well , as its decision making is not completely based on the membership value , but it also considers data centroids . it can be noted that the patterns in the above example are not uniformly spread out . which is a very common scenario in real - world examples . it occurs because of the dominance of the parameters such as outliers , temporal nature of the variables , etc . due to them , most of the times , the patterns within the overall data , and in case of fuzzy min max hierarchy , within hyperboxes , will not be steadily spread across all the dimensions . as demonstrated above , our proposed method treats them elegantly , without many of the modifications to the state of the art . performance of proposed method ( d - mlf ) is studied on the basis of the classification rate . various experiments were carried out to test d - mlf on different standard datasets . standard datasets such as iris , glass , wine , wisconsin breast cancer ( wbc ) , wisconsin diagnostic breast cancer ( wdbc ) and ionosphere were used . these datasets were obtained from the uci repository of machine learning databases [ 11 ] . in these experiments , hyperbox size parameter ( @xmath0 ) was chosen as 0.2 , 0.5 and 0.9 . this was to perform the measurements across the spectrum . as we increase the size of the hyperbox , the number of overlaps increase , and so does the misclassification rate . we split the data evenly for training and testing . the average results are shown over 100 experiments . for each iteration , training and testing data is chosen randomly . _ table 1 _ shows results , we compare our results to mlf method , as it has been already proven to perform better than the previously proposed fmm methods [ 9 ] . .results [ cols="^,^,^,^",options="header " , ] in this brief , we introduced a new boundary region and distance based mlf classification method to handle patterns belonging to that boundary region . a data centroid based method , d - mlf , minimizes significance of outliers and similar errors in decision making . it has been evidentially proven that the proposal outperforms all the previously proposed fmm methods . more importantly , we have proposed a model suited for data in the real world , extending the state of the art . d - mlf will help humongous application areas such as security , natural language processing , biomedical reasoning , etc . l. a. zadeh , fuzzy sets , information and control , vol . 3 , pp . 338 - 353 , 1965 . p. k. simpson , fuzzy min - max neural networks . classification , ieee trans . neural network , vol . 5 , pp . 776786 , sep . 1992 . simpson , p. k. , fuzzy min - max neural networks - part 2 : clustering , ieee trans fuzzy systems 1 , 3245 1993 . b. gabrys and a. bargiela , general fuzzy min - max neural network for clustering and classification , ieee trans . neural networks , vol . 11 , pp . 769783 , 2000 . bargiela , w. pedrycz , and m. tanaka , an inclusion / exclusion fuzzy hyperbox classifier , int . based intell . , vol . 8 , no . 2 , pp . 9198 , 2004 . a. rizzi , m. panella , and f. m. f. mascioli , adaptive resolution min - max classifiers , ieee trans . neural netw . 2 , pp . 402414 , mar . a. v. nandedkar and p. k. biswas , a fuzzy min - max neural network classifier with compensatory neuron architecture , ieee trans . neural netw . 1 , pp . 4254 , jan . 2007 . h. zhang , j. liu , d. ma , and z.wang , data - core - based fuzzy min max neural network for pattern classification , ieee trans . neural netw . 12 , pp . 23392352 , dec . r. davtalab , m. h. dezfoulian and m. mansourizade , multi - level fuzzy min - max neural network classifier , ieee trans . neural netw . 3 , pp.470 - 481 , mar . w. bezdel and h. j. chandler , results of an analysis and recognition of vowels by computer using zero - crossing data , proc . 2060 - 2066 , nov . k. bache and m. lichman . , uci machine learning repository , school inf . california , irvine , ca , usa . , 2013.[online available ] http://archive.ics.uci.edu/ml

recently , a multi - level fuzzy min max neural network ( mlf ) was proposed , which improves the classification accuracy by handling an overlapped region ( area of confusion ) with the help of a tree structure . in this brief , an extension of mlf is proposed which defines a new boundary region , where the previously proposed methods mark decisions with less confidence and hence misclassification is more frequent . a methodology to classify patterns more accurately is presented . our work enhances the testing procedure by means of data centroids . we exhibit an illustrative example , clearly highlighting the advantage of our approach . results on standard datasets are also presented to evidentially prove a consistent improvement in the classification rate . hyperbox , fuzzy min - max , data centroids , neural networks , neurofuzzy , classification , machine learning .

1608.05513

we only consider finite and simple graphs . let @xmath1 be a graph with vertex set @xmath4 and edge set @xmath5 . a perfect matching or 1-factor @xmath0 of a graph @xmath1 is a set of edges of @xmath1 such that each vertex of @xmath1 is incident with exactly one edge in @xmath0 . a kekul structure of some molecular graph ( for example , benzenoid and fullerene ) coincides with a perfect matching of a graph . randi and klein @xcite proposed the _ innate degree of freedom _ of a kekul structure , i.e. the least number of double bonds can determine this entire kekule structure , nowadays it is called the forcing number by harary et al . @xcite . a _ forcing set _ @xmath6 of a perfect matching @xmath0 of @xmath1 is a subset of @xmath0 such that @xmath6 is contained in no other perfect matchings of @xmath1 . the _ forcing number _ of @xmath0 is the smallest cardinality over all forcing sets of @xmath0 , denoted by @xmath7 . an edge of @xmath1 is called a _ forcing edge _ if it is contained in exactly one perfect matching of @xmath1 . the _ minimum _ _ maximum _ ) _ forcing number _ of @xmath1 is the minimum ( resp . maximum ) value of forcing numbers of all perfect matchings of @xmath1 , denoted by @xmath8 ( resp . @xmath9 ) . in general to compute the minimum forcing number of a graph with the maximum degree 3 is an np - complete problem @xcite . let @xmath0 be a perfect matching of a graph @xmath1 . a cycle @xmath10 of @xmath1 is called an _ @xmath0-alternating cycle _ if the edges of @xmath10 appear alternately in @xmath0 and @xmath11 . @xcite[forcing ] a subset @xmath12 is a forcing set of @xmath0 if and only if each @xmath0-alternating cycle of @xmath1 contains at least one edge of @xmath6 . for planar bipartite graphs , pachter and kim obtained the following minimax theorem by using lucchesi and younger s result in digraphs @xcite . [ cycle]@xcite let @xmath0 be a perfect matching in a planar bipartite graph @xmath1 . then @xmath13 , where @xmath14 is the maximum number of disjoint @xmath0-alternating cycles of @xmath1 . a hexagonal system ( or benzenoid ) is a 2-connected finite plane graph such that every interior face is a regular hexagon of side length one . it can also be formed by a cycle with its interior in the infinite hexagonal lattice on the plane ( graphene ) . a hexagonal system with a perfect matching is viewed as the carbon - skeleton of a benzenoid hydrocarbon . let @xmath3 be a hexagonal system with a perfect matching @xmath0 . a set of _ disjoint _ @xmath0-alternating hexagons of @xmath3 is called an @xmath0-_resonant set_. a set of @xmath0-alternating hexagons of @xmath3 ( the intersection is allowed ) is called an @xmath0-_alternating set_. a maximum resonant set of @xmath3 over all perfect matchings is a _ clar structure _ or _ clar set _ , and its size is the _ clar number _ of @xmath3 , denoted by @xmath15 ( cf . a fries set of @xmath3 is a maximum alternating set of @xmath3 over all perfect matchings and the fries number of @xmath3 , denoted by @xmath16 , is the size of a fries set of @xmath3 . both clar number and fries number can measure the stability of polycyclic benzenoid hydrocarbons @xcite . @xcite[clar ] let @xmath3 be a hexagonal system . then @xmath17 . in this paper we consider the anti - forcing number of a graph , which was previously defined by vukievi and trinajsti @xcite as the smallest number of edges whose removal results in a subgraph with a single perfect matching ( see refs @xcite for some researches on this topic ) . by an analogous manner as the forcing number we define the anti - forcing number , denoted by @xmath2 , of a perfect matching @xmath0 of a graph @xmath1 as the minimal number of edges not in @xmath0 whose removal to fix a single perfect matching @xmath0 of @xmath1 . we can see that the anti - forcing number of a graph @xmath1 is the minimum anti - forcing number of all perfect matchings of @xmath1 . we also show that the anti - forcing number has a close relation with forcing number : for any perfect matching @xmath0 of @xmath1 , @xmath18 , where @xmath19 denotes the maximum degree of @xmath1 . for plane bipartite graph @xmath1 , we obtain a minimax result : for any perfect matching @xmath0 of @xmath1 , the anti - forcing number of @xmath0 equals the maximal number of @xmath0-alternating cycles of @xmath1 any two members of which intersect only at edges in @xmath0 . for a hexagonal system @xmath3 , we show that the maximum anti - forcing number of @xmath3 equals the fries number of @xmath3 . as a consequence , we have that the fries number of @xmath3 is between the clar number of @xmath3 and twice . discussions for some extremal graphs about the anti - forcing numbers show the anti - forcing number of a graph @xmath1 with the maximum degree three can achieve the minimum forcing number or twice . an anti - forcing set @xmath6 of a graph @xmath1 is a set of edges of @xmath1 such that @xmath20 has a unique perfect matching . the smallest cardinality of anti - forcing sets of @xmath1 is called the _ anti - forcing number _ of @xmath1 and denoted by @xmath21 . given a perfect matching @xmath0 of a graph @xmath1 . if @xmath10 is an @xmath0-alternating cycle of @xmath1 , then the symmetric difference @xmath22 is another perfect matching of @xmath1 . here @xmath10 may be viewed as its edge - set . a subset @xmath23 is called an anti - forcing set of @xmath0 if @xmath20 has a unique perfect matching , that is , @xmath0 . [ anti]a set @xmath6 of edges of @xmath1 not in @xmath0 is an anti - forcing set of @xmath0 if and only if @xmath6 contains at least one edge of every @xmath0-alternating cycle of @xmath1 . if @xmath6 is an anti - forcing set of @xmath0 , then @xmath20 has a unique perfect matching , i.e. @xmath0 . so @xmath20 has no @xmath0-alternating cycles . otherwise , if @xmath20 has an @xmath0-alternating cycle @xmath10 , then the symmetric difference @xmath22 is another perfect matching of @xmath20 different from @xmath0 , a contradiction . hence each @xmath0-alternating cycle of @xmath1 contains at least one edge of @xmath6 . conversely , suppose that @xmath6 contains at least one edge of every @xmath0-alternating cycle of @xmath1 . that is , @xmath20 has no @xmath0-alternating cycles , so @xmath20 has a unique perfect matching . the smallest cardinality of anti - forcing sets of @xmath0 is called the anti - forcing number of @xmath0 and denoted by @xmath2 . so we have the following relations between the forcing number and anti - forcing number . [ bound]let @xmath1 be a graph with the maximum degree @xmath19 . for any perfect matching @xmath0 of @xmath1 , we have @xmath24 given any anti - forcing set @xmath6 of @xmath0 . for each edge @xmath25 in @xmath6 , let @xmath26 and @xmath27 be the edges in @xmath0 adjacent to @xmath25 . all such edges @xmath25 in @xmath6 are replaced with one of @xmath26 and @xmath27 to get another set @xmath28 of edges in @xmath0 . it is obvious that @xmath29 . further we claim that @xmath28 is a forcing set of @xmath0 . for any @xmath0-alternating cycle @xmath10 of @xmath1 , by lemma [ anti ] @xmath10 must contain an edge @xmath25 in @xmath6 . then @xmath10 must pass through both @xmath26 and @xmath27 . by the definition for @xmath28 , @xmath10 contains at least one edge of @xmath28 . so lemma [ forcing ] implies that @xmath28 is a forcing set of @xmath0 . hence the claim holds . so @xmath30 , and the first inequality is proved . now we consider the second inequality . let @xmath31 be a minimum forcing set of @xmath0 . then @xmath32 . for each edge @xmath25 in @xmath31 , we choose all the edges not in @xmath0 incident with one end of @xmath25 . all such edges form a set @xmath33 of size no larger than @xmath34 , which is disjoint with @xmath0 . we claim that @xmath33 is an anti - forcing set of @xmath0 . otherwise , lemma [ anti ] implies that @xmath35 contains an @xmath0-alternating cycle @xmath10 . since each edge in @xmath31 is a pendant edge of @xmath35 , @xmath10 does not pass through an edge of @xmath31 . this contradicts that @xmath31 is a forcing set of @xmath0 by lemma [ forcing ] . hence @xmath36 . @xmath37 . by the definitions the above result is immediate . hence we may say , @xmath21 is the _ minimum anti - forcing number _ of @xmath1 . whereas , @xmath38 is the _ maximum anti - forcing number _ of @xmath1 . the following is an immediate consequence of theorem [ bound ] . [ bounds]let @xmath1 be a graph with a perfect matching and the maximum degree @xmath19 . then @xmath39 further , @xmath40 and @xmath41 are called the _ forcing spectrum _ @xcite and the _ anti - forcing spectrum _ of @xmath1 respectively . for example , @xmath42 and @xmath43 ( see fig . [ spec](a ) ) , @xmath44 @xcite(see fig . [ spec](b ) ) . randi and vukievi @xcite computed the distributions of forcing numbers of kekul structures of @xmath45 and @xmath46 respectively . for any given graph @xmath1 with a perfect matching @xmath0 , we now consider the anti - forcing number @xmath2 . if @xmath1 has two @xmath0-alternating cycles that either are disjoint or intersect only at edges in @xmath0 , then by lemma [ anti ] any anti - forcing set of @xmath0 contains an edge of each one of such @xmath0-alternating cycles . thus it naturally motivates us to propose a novel concept : a collection @xmath47 of @xmath0-alternating cycles of @xmath1 is called a _ compatible @xmath0-alternating set _ if any two members of @xmath47 either are disjoint or intersect only at edges in @xmath0 . let @xmath48 denote the maximum cardinality of compatible @xmath0-alternating sets of @xmath1 . by the above discussion we have the following immediate result . [ ineqality]for any perfect matching @xmath0 of a graph @xmath1 , we have @xmath49 . for plane bipartite graphs @xmath1 we can show that the equality in the above lemma always holds . the vertices of @xmath1 are colored with white and black such that any pair of adjacent vertices receive different colors . such two color classes form a bipartition of @xmath1 . [ minimax]let @xmath1 be a planar bipartite graph with a perfect matching @xmath0 . then @xmath50 to obtain such a minimax result we need a classical result of lucchesi and younger @xcite about directed graphs ; its shorter proof was ever given by lovsz @xcite . let @xmath51 be a finite directed graph . feedback set _ of @xmath51 is a set of arcs that contains at least one arc of each directed cycle of @xmath51 . @xcite[ly]for a finite planar digraph , a minimum feedback set has cardinality equal to that of a maximum collection of arc - disjoint directed cycles . _ proof of theorem [ minimax ] . _ first assign a specific orientation of @xmath1 concerning @xmath0 to obtain a digraph @xmath52 : any edge in @xmath0 is directed from white end to black end , and the edges not in @xmath0 are directed from black ends to white ends . obviously the @xmath0-alternating cycles of @xmath1 corresponds naturally to directed cycles of its orientation . then contract each edge of @xmath0 in @xmath52 to a vertex ( i.e. delete the edge and identify its ends ) to get a new digraph , denoted by @xmath53 . we can see that there is a one - to - one correspondence between the @xmath0-alternating cycles of @xmath1 and directed cycles of @xmath53 . that is , an @xmath0-alternating cycle of @xmath1 becomes a directed cycle @xmath53 , and a directed cycle of @xmath53 can produce an @xmath0-alternating cycle of @xmath1 when each vertex is restored to an edge of @xmath0 . so by lemma [ anti ] a subset @xmath23 is an anti - forcing set of @xmath0 if and only if @xmath6 is a feedback set of @xmath53 . hence @xmath2 equals the smallest cardinality of feedback sets of @xmath53 . on the other hand , a compatible @xmath0-alternating set of @xmath1 corresponds to a set of arc - disjoint directed cycles of @xmath53 . that implies that @xmath48 equals the maximum number of arc - disjoint directed cycles of @xmath53 . note that @xmath53 is a planar digraph . so theorem [ ly ] implies @xmath54 . @xmath55 0.2 cm however , the equality in lemma [ ineqality ] does not necessarily hold in general . a counterexample is dodecahedron ( see fig . [ spec](b ) ) ; for this specific perfect matching marked by bold lines , it can be confirmed that there are at most three compatible alternating cycles , but its anti - forcing number is at least four . in this section we restrict our consideration to a hexagonal system @xmath3 with a perfect matching @xmath0 . without loss of generality , @xmath3 is placed in the plane such that an edge - direction is vertical and the peaks ( i.e. those vertices of @xmath3 that just have two low neighbors , but no high neighbors ) are black . an @xmath0-alternating cycle @xmath10 of @xmath3 is said to be _ proper _ ( resp . _ improper _ ) if each edge of @xmath10 in @xmath0 goes from white end to black end ( resp . from black end to white end ) along the clockwise direction of @xmath10 . the boundary of @xmath3 means the boundary of the outer face . an edge on the boundary is a boundary edge . the following main result shows that the maximum anti - forcing number equals the fries number in a hexagonal system . [ fries]let @xmath3 be a hexagonal system with a perfect matching . then @xmath56 . since any fries set of @xmath3 is a compatible @xmath0-alternating set @xmath47 for some perfect matching @xmath0 of @xmath3 , we have that @xmath57 from theorem [ minimax ] . so we now prove that @xmath58 . it suffices to prove that for a compatible alternating set @xmath47 of @xmath3 with @xmath59 , we can find a fries set @xmath31 of @xmath3 such that @xmath60 . given any compatible @xmath0-alternating set @xmath47 of @xmath3 with a perfect matching @xmath0 . two cycles @xmath61 and @xmath62 in @xmath47 are _ crossing _ if they share an edge @xmath25 in @xmath0 and the four edges adjacent to @xmath25 alternate in @xmath61 and @xmath62 ( i.e. @xmath61 enters into @xmath62 from one side and leaves from the other side via @xmath25 ) . such an edge @xmath25 is said to be a crossing . for example , see fig . we say @xmath47 is _ non - crossing _ if any two cycles in @xmath47 are not crossing . -alternating cycles @xmath61 and @xmath62 ( bold lines are edges in @xmath0 ) . ] * claim 1 . * for any compatible @xmath0-alternating set @xmath47 of @xmath3 , we can find the corresponding non - crossing compatible @xmath0-alternating set @xmath63 of @xmath3 such that @xmath64 . suppose @xmath47 has a pair of crossing members @xmath61 and @xmath62 . in fact @xmath61 and @xmath62 have even number of crossings . let @xmath26 and @xmath27 be two consecutive crossings , which are edges in @xmath0 . so we may suppose along the counterclockwise direction @xmath62 from edge @xmath65 enters into the interior of @xmath61 , then reaches the crossing @xmath66 . note that @xmath67 is the first vertex of @xmath62 entering in @xmath61 and @xmath68 the first vertex of @xmath62 leaving from @xmath61 after @xmath67 . for convenience , if a cycle @xmath10 in @xmath3 has two vertices @xmath69 and @xmath70 , we always denote by @xmath71 the path from @xmath69 to @xmath70 along @xmath10 clockwise . if @xmath61 is a proper @xmath0-alternating cycle and @xmath62 is an improper @xmath0-alternating cycle , let @xmath72 and @xmath73 ( see fig . 2(left ) ) . if @xmath61 and @xmath62 both are proper ( resp . improper ) @xmath0-alternating cycles , let @xmath74 and @xmath75 ( see fig . 2(right ) ) . in all such cases @xmath61 and @xmath62 in @xmath47 can be replaced with @xmath76 and @xmath77 to get a new compatible @xmath0-alternating set of @xmath3 and such a pair of crossings @xmath26 and @xmath27 disappeared . since such a change can not produce any new crossings , by repeating the above process we finally get a compatible @xmath0-alternating set @xmath63 of @xmath3 that is non - crossing . it is obvious that @xmath64 . for a cycle @xmath10 of @xmath3 , let @xmath78 denote the number of hexagons in the interior of @xmath10 . by claim 1 we can choose a perfect matching @xmath0 of @xmath3 and a maximum compatible @xmath0-alternating set @xmath47 satisfying that ( i ) @xmath59 and ( ii ) @xmath47 is non - crossing , and @xmath79 is as minimal as possible subject to ( i ) and ( ii ) . we call @xmath80 the @xmath81-_index _ of @xmath47 . by the above choice we know that for any two cycles in @xmath47 their interiors either are disjoint or one contains the other one . hence the cycles in @xmath47 form a _ poset _ according to the containment relation of their interiors . since each @xmath0-alternating cycle has an @xmath0-alternating hexagon in its interior ( cf . @xcite ) , we immediately obtain the following claim . * claim 2 . * every minimal member of @xmath47 is a hexagon . it suffices to prove that all members of @xmath47 are hexagons . suppose to the contrary that @xmath47 has at least one non - hexagon member . let @xmath10 be a minimal non - hexagon member in @xmath47 . then @xmath10 is an @xmath0-alternating cycle . we consider a new hexagonal system @xmath82 formed by @xmath10 and its interior as a subgraph of @xmath3 . without loss of generality , suppose that @xmath10 is a proper @xmath0-alternating cycle ( otherwise , analogous arguments are implemented on right - top corner of @xmath82 ) . so we can find a substructure of @xmath82 in its left - top corner as follows . , @xmath83 . ] we follow the notations of zheng and chen @xcite . let @xmath84 and @xmath85 , be a series of hexagons on the boundary of @xmath82 as fig . 3 that form a hexagonal chain and satisfy that neither @xmath86 nor @xmath87 is contained in @xmath82 . we denote edges , if any , by @xmath88 and @xmath89 , and by @xmath90 and @xmath85 ; and denote the hexagons ( not necessarily contained in @xmath82 ) with both edges @xmath91 and @xmath92 , by @xmath93 and @xmath85 ( see fig . * claim 3 . * ( a ) @xmath94 , and @xmath95 , + ( b ) @xmath96 or 2 for all @xmath97 , + ( c ) for all @xmath97 , @xmath98 , and + ( d ) if @xmath99 , @xmath100 , then @xmath101 . we now prove the claim by induction on @xmath102 . we first consider @xmath103 . if @xmath104 , then @xmath105 is a proper @xmath0-alternating hexagon . so @xmath10 in @xmath47 can be replaced with @xmath105 to produce a new compatible @xmath0-alternating set @xmath63 . that is , @xmath106 , but @xmath107 , a contradiction . so @xmath108 , which implies that @xmath109 and all edges @xmath110 belong to @xmath0 . hence @xmath111 is a hexagon of @xmath82 and @xmath95 . if @xmath112 , since the boundary @xmath10 of @xmath82 is a proper @xmath0-alternating cycle , none of the edges @xmath113 is a boundary edge of @xmath82 . in this case the cycle @xmath10 can be replaced with @xmath114 to get another compatible @xmath0-alternating set with less index @xmath81-index than @xmath47 , also a contradiction . hence @xmath94 . so the claim holds for @xmath103 . suppose @xmath115 and claim 3 holds for any integer @xmath116 . we want to show that it holds for @xmath117 . there are two cases to be considered . * case 1 . * @xmath96 . suppose that @xmath118 . if @xmath119 , then @xmath120 all belong to @xmath0 . by an analogous argument as above , we have that @xmath121 are hexagons of @xmath82 , and @xmath10 can be replaced with @xmath122 to get another @xmath0-compatible alternating set with less @xmath81-index than @xmath81 , also a contradiction . hence @xmath123 . by the induction hypothesis we have @xmath124 , and @xmath125 is an @xmath0-alternating hexagon . if @xmath126 , the similar contradiction occurs so @xmath127 . we can see that none of members of @xmath47 but @xmath10 intersect @xmath125 . then @xmath128 is a compatible @xmath129-alternating set , which is larger than @xmath47 , contradicting the choice of @xmath47 . hence @xmath130 . if @xmath131 , then @xmath132 otherwise , @xmath10 in @xmath47 would be replaced with @xmath125 to obtain a similar contradiction . if @xmath133 , by the similar arguments we have that @xmath123 and @xmath134 . so @xmath135 . * case 2 . * @xmath99 . choose an integer @xmath136 with @xmath137 such that @xmath138 , and @xmath139 . by the induction hypothesis , we have that the right vertical edge of hexagon @xmath140 belongs to @xmath0 , the hexagons @xmath141 are all proper @xmath0-alternating hexagons , which all belong to @xmath47 , and @xmath142 . if @xmath143 , then @xmath132 . we have that @xmath131 ; otherwise , @xmath144 and @xmath10 would be replaced with @xmath122 to get another @xmath0-compatible alternating set with less @xmath81-index than @xmath47 , also a contradiction . so suppose that @xmath145 . then @xmath125 is a proper @xmath0-alternating hexagon . we claim that @xmath133 and @xmath146 . if @xmath131 , then @xmath147 belongs to @xmath10 . so @xmath10 can be replaced with @xmath125 also to get a contradiction . hence @xmath144 . suppose @xmath127 . let @xmath148 . then @xmath149 is a perfect matching of @xmath3 so that @xmath150 are @xmath0-alternating hexagons . let @xmath151 . then @xmath63 is a compatible @xmath149-alternating set of @xmath3 with @xmath152 , contradicting the choice for @xmath47 . hence @xmath126 and @xmath134 . if @xmath118 , then @xmath153 all belong to @xmath0 , so @xmath10 can be replaced with @xmath122 to get a similar contradiction . hence @xmath133 and the claim holds . further we have that @xmath135 now we have completed the proof of claim 3 . by claim 3 we have that @xmath154 . that implies that @xmath155 . so @xmath156 exists in @xmath82 , a contradiction . hence each member of @xmath47 is a hexagon . combining theorems [ clar ] and [ fries ] with corollary [ bounds ] , we immediately obtain the following relations between the clar number and fries number . let @xmath3 be a hexagonal system . then @xmath157 . let @xmath3 be a hexagonal system . the inner dual @xmath158 of @xmath3 is a plane graph : the center of each hexagon @xmath81 of @xmath3 is placed a vertex @xmath159 of @xmath158 , and if two hexagons of @xmath3 share an edge , then the corresponding vertices are joined by an edge . @xmath3 is called catacondensed if its inner dual is a tree . further @xmath3 is called _ all - kink catahex _ @xcite if it is catacondensed and no two hexagons share a pair of parallel edges of a hexagons . the following result due to harary et al . gives a characterization for a hexagonal system to have the fries number ( or the maximum anti - forcing number ) achieving the number of hexagons . an independent ( or stable ) set of a graph @xmath1 is a set of vertices no two of which are adjacent . the independence number of @xmath1 , denoted by @xmath162 , is the largest cardinality of independent sets of @xmath1 . by theorem [ harary ] , @xmath164 equals the number @xmath160 of vertices of @xmath158 . note that any set of disjoint hexagons of @xmath3 is a resonant set . by theorem [ clar ] , @xmath165 . since @xmath158 is a bipartite graph , @xmath166 , where @xmath167 denotes the matching number of @xmath158 , the size of a maximum matching of @xmath158 . so this equality implies the result . for a hexagonal system @xmath3 with a perfect matching @xmath0 , let @xmath168 be the number of @xmath0-alternating hexagons of @xmath3 . then @xmath16 is the maximal value of @xmath168 over all perfect matchings . the minimal value of @xmath168 over all perfect matchings @xmath0 is called the _ minimum fries number _ , denoted by @xmath169 . for an all - kink catahex , each hexagon has two choices for three disjoint edges , and just one s edges can be glued with other hexagons , so these three edges are called _ fusing edges_. if a fusing edge is on the boundary , then an additive hexagon is glued along it to get a larger all - kink catahex . a _ dominating set _ of a graph @xmath1 is a set @xmath6 of vertices of @xmath1 such that every vertex not in @xmath6 has a neighbor in @xmath6 . an independent dominating set of @xmath1 is a set of vertices of @xmath1 that is both dominating and independent in @xmath1 @xcite . the independent domination number of @xmath1 , denoted by @xmath170 , is the minimum size of independent dominating sets of @xmath1 . ( for a survey on independent domination , see @xcite ) for any perfect matching @xmath0 of @xmath3 , by theorem [ cycle ] we have that @xmath172 . note that @xmath3 has no interior vertices . since each @xmath0-alternating cycle of @xmath3 contains an @xmath0-alternating hexagon in its interior , @xmath14 equals the maximum number of disjoint @xmath0-alternating hexagons of @xmath3 . it is obvious that for a hexagon of @xmath3 a non - fusing edge belongs to @xmath0 if and only if the three non - fusing edges belong to @xmath0 . choose a perfect matching @xmath0 of @xmath3 such that @xmath173 . let @xmath6 be a maximum set of disjoint @xmath0-alternating hexagons of @xmath3 and @xmath174 . then @xmath175 . we claim that @xmath176 is an independent dominating set of @xmath158 . let @xmath81 be any hexagon of @xmath3 not in @xmath6 . if some hexagon @xmath177 of @xmath3 adjacent to @xmath81 has the three non - fusing edges in @xmath0 , then @xmath178 . otherwise , @xmath81 is an @xmath0-alternating hexagon . since @xmath179 and @xmath6 is maximum , some hexagon of @xmath3 adjacent to @xmath81 must belong to @xmath6 . so the claim holds , and @xmath180 . conversely , given a minimum independent dominating set @xmath176 of @xmath158 . construct a perfect matching @xmath181 of @xmath3 as follows . the three non - fusing edges of each hexagon in @xmath6 are chosen as edges of @xmath181 . for any hexagon of @xmath3 not in @xmath6 , a fusing edge that is a boundary edge or shared by the other hexagon not in @xmath6 is also an edge of @xmath181 . so we can see that @xmath181 is a perfect matching of @xmath3 and any hexagon of @xmath3 not in @xmath6 is not @xmath181-alternating . hence @xmath6 is the maximum set of @xmath181-alternating hexagons of @xmath3 . so @xmath182 . hence @xmath183 . according to the above construction , @xmath6 is the set of all @xmath181-alternating hexagons of @xmath3 . hence @xmath184 . on the other hand , for any perfect matching @xmath0 of @xmath3 , @xmath185 , and thus @xmath186 . both inequalities imply the second equality . beyer et al . @xcite observed an algorithm of linear time to compute the independent domination number of a tree . so the minimum forcing number of all - kink catahexes can be computed in linear time . for example , fig . [ cata ] gives the minimum forcing numbers of two all - kink catahexes . but the anti - forcing number of an all - kink catahex may be larger than its minimum forcing number ; for example , the triphenylene has the minimum forcing number 1 and the anti - forcing number 2 ( see fig . [ spec](a ) ) . li @xcite gave the structure of hexagonal systems with an anti - forcing edge ( i.e. an edge that itself forms an anti - forcing set ) . for integers @xmath188 , let @xmath189 be a hexagonal system with @xmath190 horizontal rows of @xmath188 hexagons and last hexagon of each row being immediately below and to the right of the last one in the previous row , and we call it _ truncated parallelogram _ @xcite ; for example , see fig . [ para ] . in particular , @xmath191 with @xmath192 and @xmath193 and @xmath194 with @xmath193 are parallelogram and linear chain respectively . note that a truncated parallelogram can be placed and represented in other ways . precisely , a single hexagon has six anti - forcing edges , a linear chain has four anti - forcing edges , and a parallelogram has two anti - forcing edges . a true truncated parallelogram has just one anti - forcing edge ( see fig . [ para ] ) . in the following we will give a construction for hexagonal systems with the anti - forcing number 2 . some necessary preliminary is needed . let @xmath1 be a connected plane bipartite graph . an edge of @xmath1 is said to be _ fixed single _ _ double _ ) if it belongs to no ( resp . all ) perfect matchings of @xmath1 . @xmath1 is _ normal _ or _ elementary _ if @xmath1 has no fixed single edges . the non - fixed edges of @xmath1 form a subgraph whose components are normal and thus 2-connected graphs , which are called _ normal components _ of @xmath1 . further , a normal component of @xmath1 is called a _ normal block _ if it is formed by a cycle of @xmath1 with its interior . a pendant vertex of a graph is a vertex of degree one , and its incident edge is a pendant edge . @xcite[component ] let @xmath3 be a connected plane bipartite graph with a perfect matching . if all pendant vertices of @xmath1 are of the same color and lie on the boundary , then @xmath1 has at least one normal block . if @xmath1 has a fixed single edge and @xmath197 , then @xmath1 has at least two normal blocks . let @xmath3 be a hexagonal system with a perfect matching . let @xmath198 be a set of parallel edges of @xmath3 such that @xmath199 and @xmath200 belong to the same hexagon and the @xmath26 and @xmath201 are boundary edges . then @xmath202 is an edge - cut of @xmath3 and @xmath203 is invariant for all perfect matchings @xmath0 of @xmath3 . by lemma [ component ] @xmath3 has at least two normal components . such normal component is a hexagonal system with the anti - forcing number at least one . note that the anti - forcing number of @xmath3 equals the sum of the anti - forcing numbers of such normal components . hence @xmath204 if and only if @xmath3 has exactly two normal components , which are truncated parallelograms by theorem [ li ] . [ af2]let @xmath3 be a normal hexagonal system . then @xmath204 if and only if @xmath3 is not truncated parallelogram and @xmath3 can be obtained by gluing two truncated parallelograms @xmath205 and @xmath206 along their boundary parts as a fused path @xmath207 of odd length such that + ( i ) an anti - forcing edge of @xmath205 remains on the boundary , + ( ii ) the hexagons of each @xmath208 with an edge of @xmath207 form a linear chain or a chain with one kink ( i.e. the inner dual is a path with exactly one turning vertex ) , and + ( iii ) when the fused path @xmath207 passes through edge @xmath209 ( or @xmath210 ) of @xmath205 , the hexagons of @xmath205 ( resp . @xmath206 ) with an edge of @xmath207 form a linear chain that is the last ( or first ) row of @xmath205 ( resp . a chain with one kink ) . ( see fig . [ construction ] ) suppose that @xmath204 . then @xmath3 has distinct edges @xmath25 and @xmath211 such that @xmath212 has a unique perfect matching @xmath0 . so by lemma [ unique ] @xmath82 has two pendant vertices with different colors . then one of @xmath25 and @xmath211 , say @xmath25 , must be a boundary edge of @xmath3 ; otherwise @xmath82 has at most one pendant vertex , a contradiction . otherwise , suppose that @xmath25 has both ends with degree three . then @xmath213 has the minimum degree two . if @xmath213 is 2-connected , it must be a hexagonal system other than truncated parallelogram , contradicting that @xmath213 has an anti - forcing edge @xmath211 . if @xmath213 has a cut edge , by lemma [ component ] @xmath213 has at least two normal components . so @xmath214 , also a contradiction , and claim 1 holds . so @xmath213 has a pendant vertex @xmath67 . the edge @xmath215 between @xmath67 and its neighbor belongs to all perfect matchings of @xmath213 , and is thus anti - forced by @xmath25 . deleting the ends of this edge and incident edges , any pendant edges of the resulting graph also belong to all perfect matchings of @xmath213 , such pendant edges are anti - forced by @xmath25 . repeating the above process , until to get a graph without pendant vertices , denoted by @xmath216 . if @xmath216 is empty , then @xmath25 is an anti - forcing edge of @xmath3 , a contradiction . otherwise , @xmath216 has a perfect matching and the minimum degree two . note that the interior faces of @xmath216 are hexagons . by the similar arguments as the proof of claim 1 , we have that @xmath216 is a hexagonal system with an anti - forcing edge @xmath211 . hence claim 2 holds . without loss of generality , suppose that edge @xmath25 is from the left - up end @xmath67 to the right - low end . then @xmath215 is a slant edge . let @xmath69 be the hexagon with edge @xmath25 , @xmath217 the vertical edge of @xmath69 adjacent to @xmath25 , @xmath218 the other edge of @xmath69 parallel to @xmath25 . from the center @xmath219 of @xmath69 draw a ray perpendicular to and away from @xmath217 ( resp . @xmath215 ) intersecting a boundary edge @xmath210 at @xmath47 ( resp . edge @xmath209 at @xmath86 ) such that @xmath220 ( resp . @xmath221 ) only passes through hexagons of @xmath3 . let @xmath222 and @xmath223 be the linear chains of @xmath3 consisting of hexagons intersecting @xmath220 and @xmath221 ; see fig . [ case1 ] . by the similar reasons as claim 1 , we have the following claim . by claim 3 we may suppose that @xmath3 has no hexagons adjacent above to @xmath222 . let @xmath225 denote a series of edges in @xmath222 parallel to @xmath215 and above @xmath220 , @xmath226 denote a series of edges in @xmath222 parallel to @xmath218 and below @xmath220 ( see fig . [ case1 ] ) . hence @xmath227 are anti - forced by @xmath25 in turn and thus belong to @xmath0 . let @xmath228 be the graph consisting of the hexagons adjacent to @xmath222 and below it . if @xmath229 is a boundary edge of @xmath3 , then @xmath230 are further anti - forced by @xmath25 and thus belong to @xmath0 . so @xmath228 is a linear chain with an end hexagon in @xmath223 , and thus @xmath228 has at most many hexagons as @xmath222 . otherwise , by lemma [ constant ] we have that some vertical edges in @xmath228 are fixed single edges , contradicting that @xmath3 is normal . in general , for @xmath231 let @xmath232 be the graph consisting of the hexagons adjacent to @xmath233 and below it . if @xmath232 has no hexagon adjacent left to the left end hexagon of @xmath233 , by the same reasons as above we have that @xmath232 is a linear chain with an end hexagon in @xmath223 and the edges in @xmath232 parallel to @xmath218 are anti - forced by @xmath25 and thus belong to @xmath0 . there are two cases to be considered . in this case there must be an integer @xmath234 such that for each @xmath97 , @xmath235 is a linear chain with an end hexagon in @xmath223 and @xmath235 has at most many hexagons as @xmath236 , but @xmath232 has a hexagon adjacent left to the left end hexagon of @xmath233 . otherwise @xmath3 is a truncated parallelogram , a contradiction . along chain @xmath223 , similarly as rows @xmath235 we can define @xmath237 in turn and have the similar fact : there must be an integer @xmath238 such that for each @xmath239 , @xmath240 is a linear chain with an end hexagon in @xmath222 and @xmath240 has at most many hexagons as @xmath241 , but @xmath242 has a hexagon adjacent below to the lowest hexagon of @xmath243 ( see fig . 7 ) . then @xmath233 and @xmath243 have exactly one hexagon @xmath244 in common . let @xmath245 be the center of @xmath244 , @xmath63 the center of the most - left vertical edge of @xmath233 and @xmath87 the center of the lowest right edge of @xmath246 . hence @xmath247 is just a subhexagonal system lying in left - low side of the line @xmath248 . let @xmath205 be the graph consisting of @xmath249 and @xmath250 . it is obvious that @xmath205 is a truncated parallelogram , @xmath205 and @xmath206 intersect at a path of odd length , and statements ( i ) and ( ii ) holds . * case 2*. @xmath3 has hexagons adjacent right to @xmath223 . let @xmath252 be the graph consisting of hexagons of @xmath3 adjacent right to @xmath223 . let @xmath234 be the least integer such that @xmath253 has the right end hexagon adjacent to a hexagon of @xmath252 . note that @xmath234 may be zero . let @xmath254 be a series of vertical edges of @xmath255 on its right side ( see fig . [ case2 ] ) . then the edges @xmath256 are anti - forced by @xmath25 and thus belong to @xmath0 . if every @xmath235 is a linear chain and @xmath257 has no hexagons adjacent left to the left - end hexagon of @xmath236 , then @xmath252 is a linear chain that intersect @xmath223 at a path of odd length , so @xmath258 must be a truncated parallelogram consisting of @xmath252 and its right side . otherwise , by analogous arguments we have that for each @xmath97 , @xmath235 is a linear chain with an end hexagon in @xmath223 and @xmath235 has at most many hexagons as @xmath236 , but @xmath232 has a hexagon adjacent left to the left end hexagon of @xmath233 . let @xmath244 be the right end hexagon of @xmath233 , @xmath245 the center of @xmath244 , @xmath63 the center of the most - left vertical edge of @xmath233 and @xmath87 the center of the edge of @xmath244 adjacent above to @xmath259 . then @xmath247 just lies below @xmath248 , and @xmath205 consists of @xmath260 ( see fig . [ case2 ] ) . so the necessity is proved . conversely , suppose that @xmath3 is obtained from the construction that the theorem states . we can see that the anti - forcing edge @xmath25 of @xmath205 can anti - forces all double and single edges of @xmath205 except for the path @xmath207 . that is , @xmath261 . hence @xmath262 . since @xmath3 is not truncated parallelogram , @xmath204 . finally we give some examples of applying the construction of theorem [ af2 ] as shown in fig . [ examples ] . the last graph has the minimum forcing number one . in fact , zhang and li @xcite , and hansen and zheng @xcite determined hexagonal systems with a forcing edge . in hexagonal systems @xmath3 with @xmath262 , we can see that in addition to such kind of graphs , we always have that @xmath263 . + * acknowledgments * 99 h. abeledo , g.w . atkinson , unimodularity of the clar number problem , linear algebra appl . 420 ( 2007 ) 441448 p. adams , m. mahdian , e. s. mahmoodian , on the forced matching numbers of bipartite graphs , discrete math . 281 ( 2004 ) 112 . p. afshani , h. hatami , e.s . mahmoodian , on the spectrum of the forced matching number of graphs , australas . j. combin . 30 ( 2004 ) 147160 . t. beyer , a. proskurowski , s. hedetniemi , s. mitchell , independent domination in trees , congr . ( 1977 ) 321328 . z. che , z. chen , forcing on perfect matchings a survey , match commun . 66 ( 2011 ) 93136 . e. clar , the aromatic sextet , wiley , london , 1972 . cyvin , i. gutman , kekul structures in benzenoid hydrocarbons ( lecture notes in chemistry 46 ) , springer verlag , berlin , 1988 . h. deng , the anti - forcing number of hexagonal chains , match commun . comput . chem . 58 ( 2007 ) 675682 . h. deng , the anti - forcing number of double hexagonal chains , match commun . comput . chem . 60 ( 2008 ) 183192 . w. goddard , m.a . henning , independent domination in graphs : a survey and recent results , discrete math . 313 ( 2013 ) 839854 . p. hansen , m. zheng , bonds fixed by fixing bonds , j. chem . 34 ( 1994 ) 297304 . p. hansen , m. zheng , upper bounds for the clar number of benzenoid hydrocarbons , j. chem . faraday trans . 88 ( 1992 ) 16211625 . f. harary , d. klein , t. ivkovi , graphical properties of polyhexes : perfect matching vector and forcing , j. math . ( 1991 ) 295306 . d. klein , m. randi , innate degree of freedom of a graph , j. comput . ( 1987 ) 516521 . x. li , hexagonal systems with forcing single edges , discrete appl . 72 ( 1997 ) 295301 . l. lovsz , on two minimax theorems in graph , j. combin . theory ser . b 21 ( 1976 ) 96 - 103 . l. lovsz , m.d . plummer , matching theory , annals of discrete math . 29 , north - holland , amsterdam , 1986 . lucchesi , d.h . younger , a minimax theorem for directed graphs , j. london math . soc . 17 ( 1978 ) 369374 . l. pachter , p. kim , forcing matchings on square grids , discrete math . 190 ( 1998 ) 287294 . m. randi , d. klein , mathematical and computational concepts in chemistry , n. trinajsti , ed . , john wiley & sons , new york , 1985 , pp . m. randi , d. vukievi , kekul structures of fullerene c@xmath264 , croat . acta 79 ( 2006 ) 471481 . riddle , the minimum forcing number for the torus and hypercube , discrete math . 245 ( 2002 ) 283292 . h. sachs , perfect matchings in hexagonal systems , combinatorica 4 ( 1984 ) 8999 . shiu , p.c.b . lam , f. zhang , h. zhang , normal components , kekul patterns , and clar patterns in plane bipartite graphs , j. math . 31 ( 2002 ) 405420 . d. vukievi and m. randi , on kekukl structures of buckminsterfullerene , chem . 401 ( 2005 ) 446450 . d. vukievi , n. trinajsti , on the anti - forcing number of benzenoids . 42 ( 2007 ) 575583 . d. vukievi , n. trinajsti , on the anti - kekul number and anti - forcing number of cata - condensed benzenoids , j. math . 43 ( 2008 ) 719726 . l. xu , h. bian , f. zhang , maximum forcing number of hexagonal systems , match commun . 70 ( 2013 ) 493500 . f. zhang , x. li , hexagonal systems with forcing edges , discrete math . 140 ( 1995 ) 253263 . q. zhang , h. bian , e. vumar , on the anti - kekul and anti - forcing number of cata - condensed phenylenes , match commun . 65 ( 2011 ) 799806 . h. zhang , d. ye , w.c . shiu , forcing matching numbers of fullerene graphs , discrete appl . ( 2010 ) 573582 . h. zhang , f. zhang , plane elementary bipartite graphs , discrete appl . math . 105 ( 2000 ) 291311 . m. zheng , r. chen , a maximal cover of hexagonal systems , graphs combin . 1 ( 1985 ) 295 - 298 .

we define the anti - forcing number of a perfect matching @xmath0 of a graph @xmath1 as the minimal number of edges of @xmath1 whose deletion results in a subgraph with a unique perfect matching @xmath0 , denoted by @xmath2 . the anti - forcing number of a graph proposed by vukievi and trinajsti in kekul structures of molecular graphs is in fact the minimum anti - forcing number of perfect matchings . for plane bipartite graph @xmath1 with a perfect matching @xmath0 , we obtain a minimax result : @xmath2 equals the maximal number of @xmath0-alternating cycles of @xmath1 where any two either are disjoint or intersect only at edges in @xmath0 . for a hexagonal system @xmath3 , we show that the maximum anti - forcing number of @xmath3 equals the fries number of @xmath3 . as a consequence , we have that the fries number of @xmath3 is between the clar number of @xmath3 and twice . further , some extremal graphs are discussed . * keywords : * graph ; hexagonal system ; perfect matching ; forcing number ; anti - forcing number ; fries number .

1406.3796

during recent years there has been increasing evidence for a connection between active galactic nuclei ( agn ) and star formation in the vicinity of the central black holes . this subject forms the central topic of this paper , and is discussed in sections [ sec : prop ] and [ sec : staragn ] . a large number of studies have addressed the issue of star formation around agn . those which have probed closest to the nucleus , typically on scales of a few hundred parsecs , have tended to focus on seyferts notably seyfert2 galaxies since these are the closest examples @xcite . the overall conclusion of these studies is that in 3050% cases the agn is associated with young ( i.e. age less than a few 100myr ) star formation . while this certainly implies a link , it does not necessarily imply any _ causal _ link between the two phenomena . instead , it could more simply be a natural consequence of the fact that both agn and starburst require gas to fuel them . and that in some galaxies this gas has fallen towards the nucleus , either due to an interaction or secular evolution such as bar driven inflow . one aspect which must be borne in mind when interpreting such results , and which has been pointed out by @xcite , is the discrepancy in the scales involved . agn and starburst phemonena occur on different temporal and spatial scales ; and observations are sensitive to scales that are different again . for example , star formation has typically been studied on scales of several kiloparsecs down to a few hundred parsecs . in contrast , accretion of gas onto an agn will occur on scales much less than 1pc . similarly , the shortest star formation timescales that most observations are sensitive to are of order 100myr to 1gyr . on the other hand , in this paper we show that the active phase of star formation close around a black hole is typically rather less than 100myr . correspondingly short accretion timescales for black holes are reflected in the ages of jets which , for a sample of radio galaxies measured by @xcite , span a range from a few to 100myr . in seyfert galaxies the timescales are even shorter , as typified by ngc1068 for which @xcite estimate the age of the jets to be only @xmath4myr . that the putative causal connection between agn and starbursts might occur on relatively small spatial scales and short timescales can help us to understand why no correlation has been found between agn and ( circum-)nuclear starbursts in general . it is simply that the circumnuclear activity on scales greater than a few hundred parsecs is , in most cases , too far from the agn to influence it , or be strongly influenced by it ( cf * ? ? ? in this paper we redress this imbalance . while the optical spectroscopy pursued by many authors allows a detailed fitting of templates and models to the stellar features , we also make use of established star formation diagnostics and interpret them using starburst population synthesis models . observing at near infrared wavelengths has brought two important advantages . the optical depth is 10 times less than at optical wavelengths , and thus our data are less prone to the effects of extinction which can be significant in agn . and we have employed adaptive optics to reach spatial resolutions of 0.10.2 , bringing us closer to the nucleus . applying these techniques , we have already analysed the properties of the nuclear star formation in a few objects @xcite . here we bring those data together with new data on 5 additional objects . our sample enables us to probe star formation in agn from radii of 1kpc down to less than 10pc . our aim is to ascertain whether there is evidence for star formation on the smallest scales we can reach ; and if so , to constrain its star formation history . ultimately , we look at whether there are indications that the nuclear starburst and agn are mutually influencing each other . in [ sec : obs ] we describe the sample selection , observations , data reduction , psf estimation , and extraction of the emission and absorption line morphologies and kinematics . in [ sec : diag ] we discuss the observational diagnostics and modelling tools . brief analyses of the relevant facets of our new data for the individual objects are provided in appendix [ sec : obj ] , where we also summarise results of our previously published data , re - assessing them where necessary to ensure that all objects are analysed in a consistent manner . the primary aims of our paper are addressed in [ sec : prop ] and [ sec : staragn ] . in [ sec : prop ] we discuss global results concerning the existence and recent history of nuclear star formation for our whole sample . in [ sec : staragn ] we discuss the implications of nuclear starbursts on the starburst - agn connection . finally , we present our conclusions in [ sec : conc ] . the agn discussed in this paper form a rather heterogeneous group . they include type 1 and type 2 seyferts , ulirgs , and even a qso , and do not constitute a complete sample . in order to maximise the size of the sample , we have combined objects on which we have already published adaptive optics near infrared spectra with new observations of additional targets . source selection was driven largely by technical considerations for the adaptive optics ( ao ) system , namely having a nucleus bright and compact enough to allow a good ao correction . this is actually a strength since it means that 7 of the 9 agn are in fact broad line objects as given either by the standard type 1 classification or because there is clear broad ( fwhm @xmath5kms@xmath6 ) br@xmath7 emission in our spectra . [ fig : bbrg ] shows broad br@xmath7 in k - band spectra of 3 agn that are not usually classified as broad line galaxies . this is in contrast to most other samples of agn for which star formation has been studied in detail , and avoids any bias that might arise from selecting only type 2 syeferts . that there may be a bias arises from the increasing evidence that the obscuration in perhaps half of type 2 agn lies at kpc scales rather than in the nucleus , which may be caused by spatially extended star formation in the galaxy disk @xcite . such agn do not fit easily into the standard unification scheme ( and perhaps should not really be considered type 2 objects ) . because broad lines can be seen in the infrared , we know that we are seeing down to the nuclear region and hence our results are not subject to any effects that this might otherwise introduce . it is exactly broad line agn for which little is known about the nuclear star formation , because the glare of the agn swamps any surrounding stellar light in the central arcsec . as a result , most studies addressing star formation close to agn have focussed on type 2 seyferts . adaptive optics makes it possible to confine much of the agn s light into a very compact region , and to resolve the stellar continuum around it . the use of adaptive optics does give rise to one difficulty when attempting to quantify the results in a uniform way , due to the different resolutions achieved which is a combination of both the distance to each object ( i.e. target selection ) and the ao performance . as a result , the standard deviation around the logarithmic mean resolution of our sample ( excluding ngc2992 , see section [ sec : obj ] ) of 22pc is a factor of 3 . however , this has enabled us to study the centers of agn across nearly 3 orders of magnitude in spatial scale , from 1kpc in the more distant objects to only a few parsecs in the nearby objects with the best ao correction . a summary of our observations is given in table [ tab : obs ] . a description of observations and processing of the new data is given below . data for iras05189 - 2524 and ngc1068 were taken in december 2002 at the vlt with naco , an adaptive optics near infrared camera and long slit spectrograph @xcite . since iras05189 - 2524 is nearly face on , there is no strongly preferred axis and the slit was oriented north - south ; for ngc1068 two orientations were used , north - south and east - west . in all cases the slit width was 0.086 , yielding a nominal resolution of @xmath8 with the wide - field camera ( pixel scale 0.054 ) and medium resolution grism . the galaxy was nodded back and forth along the slit by 10 to allow sky subtraction . for iras05189 - 2524 , 12 integrations of 300sec were made ; for ngc1068 12 integrations of 300sec were made at one position angle , and 14 frames of 200sec at the other . all data were reduced and combined , using standard longslit techniques in iraf , to make the final h - band spectrum . data for ngc7469 , ngc2992 , ngc1097 , ngc1068 , and ngc3783 were taken during 20042005 at the vlt with sinfoni , an adaptive optics near infrared integral field spectrograph @xcite . data were taken with various gratings covering the h and k bands either separately ( @xmath9 ) or together ( @xmath8 ) . the pixel scales were [email protected] or [email protected] , depending on the trade - offs between field of view , spatial resolution , and signal - to - noise ratio . individual exposure times are in the range 50300sec depending on the object brightness . object frames were interspersed with sky frames , usually using the sequence o - s - o - o - s - o , to facilitate background subtraction . the data were processed using the dedicated _ spred _ software package @xcite , which provides similar processing to that for longslit data but with the added ability to reconstruct the datacube . the data processing steps are as follows . the object frames are pre - processed by subtracting sky frames , flatfielding , and correcting bad pixels ( which are identified from dark frames and the flatfield ) . the wavemap is generated , and edges and curvature of the slitlets are traced , all from the arclamp frame . the arclamp frame is then reconstructed into a cube , which is checked to ensure that the calibration is good . the pre - processed object frames are then also reconstructed into cubes , spatially shifted to align them using the bright nucleus as a reference , and combined . in some cases the final cube was spatially smoothed using a @xmath11 median filter . estimation of the spatial resolution ( see below ) was always performed after this stage . in some cases , the strong near - infrared oh lines did not subtract well . with longer exposure times this is to be expected since the timescale for variation of the oh is only 12mins . if visual inspection of the reconstructed cubes showed signs of over- or under - subtraction of the oh lines , these cubes were reprocessed using the method described in @xcite . standard star frames are similarly reconstructed into cubes . telluric correction and flux calibration were performed using b stars ( k - band ) or g2v stars ( h - band ) . in addition , flux calibration was cross - checked in 3 apertures using 2mass data , and in smaller 13 apertures using broad - band imaging from naco or hst nicmos . agreement between cubes with different pixel scales , and also with the external data , was consistent to typically 20% . there are a multitude of ways to derive the point spread function ( psf ) from adaptive optics data , five of which are described in @xcite . with agn , it is usually possible to estimate the psf from the science data itself , removing any uncertainty about spatial and temporal variations of the psf due to atmospheric effects . typically one or both of the following methods are employed on the new data presented here . if a broad emission line is detected , this will always yield a measure of the psf since the blr of seyfert galaxies has a diameter that can be measured in light days . alternatively , the non - stellar continuum will provide a sufficiently good approximation in all but the nearest agn since at near infrared wavelengths it is expected to originate from a region no more than 12pc across . in every case we have fit an analytical function to the psf . since the strehl ratio achieved is relatively low , even a gaussian is a good representation . we have used a moffat function , which achieves a better fit because it also matches the rather broad wings that are a characteristic of partial adaptive optics correction . the psf measured for ngc3227 , which is shown in fig . 1 of @xcite , can be considered typical . if one applies the concept of ` core plus halo ' to this psf , then the gaussian fit would represent just the core while the moffat fit the entire ` core plus halo ' . integrating both of these functions indicates that about 75% of the flux is within the ` core ' , and it is thus this component which dominates the psf . in this paper , a more exact representation of the psf is not needed since we have not performed a detailed kinematic analysis , and we have simply used the moffat to derive a fwhm for the spatial resolution . the resolutions achieved are listed in table [ tab : obs ] . the 2d distribution of emission and absorption features has been found by fitting a function to the continuum - subtracted spectral profile at each spatial position in the datacube . the function was a convolution of a gaussian with a spectrally unresolved template profile in the case of emission lines it was an oh sky emission line , and for stellar absorption features we made use of template stars observed in the same configuration ( pixel scale and grism ) . a minimisation was performed in which the parameters of the gaussian were adjusted until the convolved profile best matched the data . during the minimisation , pixels in the data that consistently deviated strongly from the data were rejected . the uncertainties were boot - strapped using monte carlo techniques , assuming that the noise is uncorrelated and the intrinsic profile is well represented by a gaussian . the method involves adding a gaussian with the derived properties to a spectral segment that exhibits the same noise statistics as the data , and refitting the result to yield a new set of gaussian parameters . after repeating this 100 times , the standard deviation of the center and dispersion were used as the uncertainites for the velocity and line width . the kinematics were further processed using kinemetry @xcite . this is a parameterisation ( i.e. a mathematical rather than a physical model ) of the 2d field . as such , beam smearing is not a relevant issue to kinemetry , which yields an analytical expression for the observed data . of course , when the coefficients of this expression are interpreted or used to constrain a physical model , then beam smearing should be considered . mathematically , the kinemetry procedure fits the data with a linear sum of sines and cosines with various angular scalings around ellipses at each radius . we have used it for 3 purposes : to determine the best position angle and axis ratio for the velocity field , to remove high order noise from the raw kinematic extraction , and to recover the velocity and dispersion radial profiles . in all of the cases considered here , the kinematic centre of the velocity field was assumed to be coincident with the peak of the non - stellar continuum . in addition , the uniformity of the velocity field permitted us to make the simplifying assumption of a single position angle and axis ratio i.e. there is no evidence for warps or twisted velocity contours . we then derived the position angle and inclination of the disk by minimising the _ a1 _ and _ b3 _ parameters respectively ( see @xcite for a description of these ) . the rotation curves were recovered by correcting the measured velocity profile for inclination . we have assumed throughout the paper that the dispersion is isotropic , and hence no inclination correction was applied to the dispersion that was measured . the innermost parts of the kinematics derived as above are of course still affected by beam smearing . in general , the central dispersion can not necessarily be taken at face value since it may either be artificially increased by any component of rotation included within the beam size , or decreased if neighbouring regions within the beam have a lower dispersion . in the galaxies we have studied , there are two aspects which mitigate this uncertainty : the rotation speed in the central region is much less than the dispersion and so will not significantly alter it ; and when estimating the central value we consider the trend of the dispersion from large radii , where the effect of the beam is small , to the center . for the basic analyses performed here , we have therefore adopted the central dispersion at face value . more detailed physical models for the nuclear disks , which properly account for the effects of beam smearing , will be presented in future publications . lastly , we emphasize that the impact of the finite beam size on the derived rotation curve does not affect our measurement of the dynamical mass . the reason is that , for all the dynamical mass estimates we make , the mass is estimated at a radius much large than the fwhm of the psf as can be seen in the relevant figures . in this section we describe the tools of the trade used to analyse the data , and which lead us to the global results presented in section [ sec : prop ] . specific details and analyses for individual objects can be found in appendix [ sec : obj ] . we use the same methods and tools for all the objects to ensure that all the data are analysed in a consistent manner . perhaps the most important issue is how to isolate the stellar continuum , which is itself a powerful diagnostic . in addition , we use three standard and independent diagnostics to quantify the star formation history and intensity in the nuclei of these agn . these are the br@xmath7 equivalent width , supernova rate , and mass - to - light ratio . much of the discussion concerns how we take into account the contribution of the agn when quantifying these parameters . we also consider what impact an incorrect compensation could have on interpretation of the diagnostics . we model these observational diagnostics using the stellar population and spectral synthesis code stars ( e.g. * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? * ; * ? ? ? this code calculates the distribution of stars in the hertzsprung - russell diagram as a function of age for an assumed star formation history . we usually assume an exponentially decaying star formation rate , which has an associated timescale @xmath12 . spectral properties of the cluster are then computed given the stellar population present at any time . we note that the model output of stars is quantitatively similar to that from version 5.1 of starburst99 , which unlike earlier versions does include agb tracks @xcite . as discussed in detail below , particular predictions of stars include , for ages greater than 10myr : equivalent widths of @xmath13 and @xmath14 , and h - k color of 0.15mag . the equivalent quantities predicted by starburst99 v5.1 are @xmath15 and @xmath16 , and h - k color of 0.2mag . for small observational apertures a significant fraction of the k - band ( and even h - band ) continuum can be associated with non - stellar agn continuum . the agn contribution can be estimated from a simple measurement of the equivalent width of a stellar absorption feature . we use co2 - 0 2.29 in the k - band or co6 - 3 1.62 in the h - band . although the equivalent widths @xmath17 and @xmath18 vary considerably for individual stars , the integrated values for stellar clusters span only a rather limited range . this was shown by @xcite who measured these values for elliptical , spiral , and star - forming ( hii ) galaxies . we have plotted their measurements of these two absorption features in the left - hand panel of fig . [ fig : equiv ] , together with the equivalent widths of giant and supergiant stars from @xcite . in stars we use empirically determined equivalent widths from library spectra @xcite to compute the time - dependent equivalent width for an entire cluster of stars . results for various star formation histories are shown in the centre and right panels of fig . [ fig : equiv ] , for @xmath18 and @xmath17 respectively . typical values are @xmath19 and @xmath20 . the dashed box in the left panel shows that the locus of 20% deviation from each of these computed values is consistent with observations . that the hii galaxies have slightly higher @xmath17 can be understood because these are selected to have bright emission lines and hence are strongly biassed towards young stellar ages often corresponding to the maximum depth of the stellar features that occurs at 10myr due to the late - type supergiant population . it may be this bias for galaxies selected as ` starbursts ' , and the similarity of the co depth for starbursts of all other ages , that led @xcite to conclude that there is no evidence for strong starbursts in seyfert2 galaxies . similarly , an estimate of the dilution can be found from the nai2.206 line . fig . 7 of @xcite shows that for nearly all star formation histories the value @xmath21 remains in the range 23 . our conclusion here is that within a reasonable uncertainty of @xmath2220% ( see fig . [ fig : equiv ] ) , one can assume that the intrinsic equivalent width of the absorption most notably co features of any stellar population that contains late - type stars is independent of the star formation history and age . for a stellar continuum diluted by additional non - stellar emission , the fraction of stellar light is @xmath23 where @xmath24 and @xmath25 are the observed and intrinsic equivalent widths of the co features discussed above . thus , we are able to correct the observed continuum magnitude for the contribution associated with the agn . our data cover both the h and k - bands hence the reason for using both @xmath18 and @xmath17 . in order to homogenize the dataset , we need to convert h - band stellar magnitudes to k - band . the stars computation in fig . [ fig : col_lum ] shows that this conversion is also independent of the star formation history , being close to @xmath26mag ( no extinction ) for all timescales and ages . this result is supported empirically by photometry of elliptical and spiral galaxies performed by @xcite . for ellipticals @xmath270.25 , and for spirals @xmath270.3 . some of the difference between the data and models could be due to extinction since @xmath28 ; and for @xmath29 , @xmath30 . however , at the level of precision required here , the 510% difference between model and data can be considered negligible . to convert from absolute magnitude to luminosity we use the relation @xmath31 where @xmath32 is the total luminosity in the 1.92.5 band in units of bolometric solar luminosity ( @xmath33w ) , and as such different from the other frequently used monochromatic definition with units of the solar k - band luminosity density ( @xmath34w@xmath35m@xmath6 ) . we then use stars to estimate the bolometric stellar luminosity @xmath36 . the relation between @xmath36 and @xmath32 is shown in the right panel of fig . [ fig : col_lum ] . the dimensionless ratio @xmath37 depends on the age and the exponential decay timescale of the star formation . however , the range spanned is only 20200 for ages greater than 10myr . thus even if the star formation history can not be constrained , a conversion ratio of @xmath38 will have an associated uncertainty of only 0.3dex . in general we are able to apply constraints on the star formation age , and so our errors will be accordingly smaller . graphs showing how the diagnostics vary with age and star formation timescale are shown in fig . [ fig : stars ] . once the stellar continuum luminosity is known , an upper limit to the equivalent width of br@xmath7 asssociated with star formation can be found from the narrow br@xmath7 line flux . in some cases it is possible to estimate what fraction of the narrow br@xmath7 might be associated with the agn . this can be done both morphologically , for example if the line emission is extended along the galaxy s minor axis ; and/or kinematically , for example if the line shows regions that are broader , perhaps with fwhm a few hundred kms@xmath6 , suggestive of outflow . even if acounting for the agn contribution is not possible , one may be able to set interesting upper limits or even rule out continuous star formation scenarios , and put a constraint on the time since the star formation was active . this can be seen in the lefthand panel of fig . [ fig : stars ] , which shows for example that for ages less than @xmath39yrs , continuous star formation scenarios will always have @xmath40 . we estimate the type ii ( core collapse ) supernova rate @xmath41 from the radio continuum using the relation @xcite : @xmath42 where @xmath43 is the non - thermal radio continuum luminosity , @xmath44 is frequency of the observation and @xmath45 the spectral index of the non - thermal continuum . this relation was derived for galactic supernova remnants ; but a similar one , differing only in having a coefficient of @xmath46 , was derived by @xcite for m82 . for the 5ghz non - thermal radio continuum luminosity of arp220 ( 176mjy , * ? ? ? * ) it would lead to a supernova rate of 2.9yr@xmath6 , comfortably within the 1.753.5yr@xmath6 range estimated by @xcite based on the detection of individual luminous radio supernovae . this , therefore , seems a reasonable relation to apply to starbursts . we have to be careful , however , to take into account any contribution from the agn to the radio continuum . our premise for the nuclei of seyfert galaxies is that if the nuclear radio continuum is spatially resolved ( i.e. it has a low brightness temperature ) and does not have the morphology of a jet , it is likely to originate in extended star formation . at the spatial scales of a few parsec or more that we can resolve , emission from the agn will be very compact . as a result , we can use the peak surface brightness to estimate the maximum ( unresolved ) contribution from an agn . wherever possible , we use radio continuum observations at a comparable resolution to our data to derive the extended emission ; and observations at higher resolution to estimate the agn contribution . details of the data used in each case are given in the relevant sub - sections for each object in appendix [ sec : obj ] . in addition , we exclude any emission obviously associated with jets , for example as in ngc1068 . to use @xmath41 as a diagnostic , we normalise it with respect to the stellar k - band luminosity . this gives the ratio @xmath47 , for which stars output is drawn in fig . [ fig : stars ] . models indicate that the ratio @xmath48 of the stellar mass to k - band luminosity should be an excellent diagnostic since , for ages greater than 10myr , it increases monotonically with age as shown in fig . [ fig : stars ] . however , in practice estimating the stellar mass is not entirely straightforward . in many cases it is only practicable to derive the dynamical mass . it may be possible to estimate and hence correct for the molecular gas mass based on millimetre co maps , but these are scarce at sufficiently high spatial resolution and are associated with their own co - to - h@xmath49 conversion uncertainties . we also note that it is often not possible to separate the ` old ' and ` young ' stellar populations . the best one can do is estimate the overall mass - to - light ratio , and argue that this is an upper limit to the true ratio for the young population . while there inevitably remains uncertainty on the true ratio , the limit is often sufficient to apply useful constraints on the age of the ` young ' population . our estimates of the dynamical mass are based wherever possible on the stellar kinematics , since the gas kinematics can be perturbed by warps , shocks , and outflows . we begin by estimating the simple keplerian mass assuming that the stars are supported by ordered rotation at velocity @xmath50 in a thin plane . however , the stellar kinematics in all the galaxies exhibit a significant velocity dispersion indicating that a considerable mass is supported by random rather than ordered motions . thus the simple keplerian mass is very much an underestimate , and any estimate of the actual mass is associated with large uncertainties see for example @xcite , who derive masses of spheroidal systems . as stated in section [ sec : obs ] , we assume that the random motions are isotropic . our relation for estimating the mass enclosed within a radius @xmath51 is then @xmath52 where @xmath53 is the observed 1-dimensional velocity dispersion . we note that when taking rotation into account in estimating the masses of spheroids with various density profiles , @xcite also use a factor 3 between the @xmath54 and @xmath53 terms in their appendix b. despite the complexities involved , within the unavoidable uncertainties ( a factor 23 ) , their relation gives the same mass as that above . although this uncertainty appears to be quite large , it does not impact the results and conclusions in this paper since we are concerned primarily with order - of - magnitude estimates when considering mass surface densities . in the following section we bring to together the individual results ( detailed in appendix [ sec : obj ] ) to form a global picture . it is possible to do this because all the data have been analysed in a consistent manner , using the tools described in section [ sec : diag ] to compare in each object the same diagnostics to the same set of stellar evolutionary synthesis models . we note that the discussion that follows is based on results for 8 of the agn we have observed . as explained in appendix [ sec : obj ] we exclude ngc2992 because we are not able to put reliable constraints on the properties of the nuclear star formation . despite this , there are indications that at higher spatial resolution one should expect to find a distinct nuclear stellar population as has been seen in other agn . [ [ size - scale ] ] size scale + + + + + + + + + + tracing the stellar features rather than the broad - band continuum , we have in all cases resolved a stellar population in the nucleus close around the agn . while this should not be unexpected if the stellar distribution follows a smooth @xmath55 or exponential profile , we have in several cases been able to show that on scales of @xmath56pc there is in fact an excess above what one would expect from these profiles . this suggests that in general we are probing an inner star forming component . [ fig : sizes ] shows normalised azimuthally averaged stellar luminosity profiles for the agn . these have not been corrected for a possible old underlying population , nor has any deconvolution with the psf been performed . nevertheless , it is still clear that the stellar intensity increases very steeply towards the nucleus . in 6 of the 8 galaxies shown , the half - width at half - maximum is less than 50pc . the remaining 2 galaxies are the most distant in the sample , and the spatial resolution achieved does not permit a size measurement on these scales . we may conclude that the physical radial size scale of the nuclear star forming regions in seyfert galaxies does not typically exceed 50pc . [ [ stellar - age ] ] stellar age + + + + + + + + + + + for 8 of the agn studied here , we have been able to use classical star formation diagnostics based on line and continuum fluxes as well as kinematics to constrain the ages of the inner star forming regions . the resulting ages should be considered ` characteristic ' , since in many cases there may simultaneously be two or more stellar populations that are not co - eval . for example , if a bulge population exists on these small spatial scales , it was not usually possible to account for the contamination it would introduce . while this would have little effect on @xmath57 , it could impact on @xmath58 more strongly , increasing the inferred age . the ages we find lie in the range 10300myr , compelling evidence that it is common for there to be relatively young star clusters close around agn . intriguingly , we also find rather low values of @xmath57 : typically @xmath59 ( see table [ tab : derprop ] ) . this indicates directly that there is currently little or no on - going star formation . coupled with the relatively young ages , we conclude that the star formation episodes are short - lived . one may speculate then that the star formation is episodic , recurring in short bursts . the scale of the bursts and time interval between them would certainly have an impact on the fraction of seyfert nuclei in which observational programmes are able to find evidence for recent star formation . [ [ nuclear - stellar - disks ] ] nuclear stellar disks + + + + + + + + + + + + + + + + + + + + + the first evidence for nuclear stellar disks came from seeing limited optical spectroscopy , for which a slight reduction in @xmath60 was seen for some spiral galaxies @xcite . and there is now a growing number of spiral galaxies more than 30 in which the phenomenon has been observed , suggesting that they might occur in 30% or more of disk galaxies @xcite . the @xmath60-drop has been interpreted by @xcite as arising from a young stellar population that is born from a dynamically cold gas component , and which makes a significant contribution to the total luminosity . this appears to be borne out by n - body and sph simulations of isolated galaxies @xcite , which suggest that although the entire central system will slowly heat up with time , the @xmath60-drop can last for at least several hundred myr . indeed , preliminary analysis of optical integral field data for ngc3623 suggest that the stellar population responsible for the @xmath60-drop can not be younger than 1gyr @xcite . our results provide strong support for the nuclear disk interpretation . in previous work @xcite , we had argued that in both circinus and ngc3227 the inner distributions were disk - like , albeit thickened . we have now found much more direct evidence for this phenomenon in ngc1097 and ngc1068 . in both of these galaxies , we have spatially resolved a @xmath60-drop and an excess stellar continuum over the same size scales . in ngc1097 this size was @xmath610.5 , corresponding to about 40pc . for ngc1068 these effects were measured out to @xmath611 , equivalent to 70pc . these are not the scale lengths of the disks , but simply the maximum radius to which we can detect them . in both cases the mean mass surface densities are of order @xmath62(1 - 3)@xmath63m@xmath1pc@xmath2 . for an infinitely large thin self gravitating stellar disk , one can use the expression @xmath64 to estimate the scale height . although this may not be entirely appropriate , we use it here to obtain a rough approximation to the scale heights , which are 520pc . thus while the disks appear to be flattened , they should still be considered thick since the radial extent is only a few times the scale height . the impact of nuclear starbursts on the central light profile of galaxies was considered theoretically more than a decade ago by @xcite . they performed numerical simulations of galaxy mergers to study the mass and luminosity profiles of the remnants , taking gas into account , and estimating the star formation rate using a modified schmidt law . they found that there should be a starburst in the nucleus which would give rise to an excess stellar continuum above the @xmath55 profile of the older stars in the merged system . several years ago , compact nuclei were found to be present in a significant fraction of spiral galaxies @xcite as well as coma cluster dwarf ellipticals @xcite . more recently , nuclei with a median half - light radius of 4.2pc have been found in the majority of early - type members of the virgo cluster @xcite ; and traced out to @xmath611 , equivalent to @xmath65pc , in some of the ` wet ' merger remnants in that cluster @xcite . while the nuclear starbursts in these latter cases are caused by a merger event , whereas those we are studying arise from secular evolution as gas from the galaxy disk accretes in the nucleus , there appear to be many parallels in the phenomenology of the resulting starbursts . [ [ star - formation - rate ] ] star formation rate + + + + + + + + + + + + + + + + + + + it is possible to estimate the bolometric luminosity @xmath66 of the stars from their k - band luminosity @xmath67 even if one knows nothing about the star formation history . as discussed in section [ sec : diag ] , this would result in an uncertainty of about a factor 3 . the diagnostics in table [ tab : derprop ] and discussions in appendix [ sec : obj ] enable us to apply some constraints to the characteristic age of the star formation . because continuous star formation is ruled out by the low @xmath57 , we have assumed exponential decay timescales of @xmath68100myr . we have then used stars to estimate the average star formation rates . in order to allow a meaningful comparison between the objects , the rates have been normalised to the same area of 1kpc@xmath69 . these are the rates given in table [ tab : derprop ] . they are calculated simply as the mass of stars produced divided by the entire time since the star forming episode began . because @xmath12 is shorter than the age , the average includes both active and non - active phases of the starburst . indeed , for @xmath68myr one would expect the star formation rate during the active phases to be at least a factor of a few , and perhaps an order of magnitude , greater . the table shows that on scales of a few hundred parsecs one might expect a few @xmath70m@xmath1yr@xmath6kpc@xmath2 , while on scales of a few tens of parsecs mean rates reaching @xmath65m@xmath1yr@xmath6kpc@xmath2 should not be unexpected ; and correspondingly higher up to an order of magnitude , see fig . [ fig : starstoy ] during active phases . an obvious question is why there should be such vigorous star formation in these regions . star formation rates of 10100m@xmath1yr@xmath6kpc@xmath2 are orders of magnitude above those in normal galaxies and comparable to starburst galaxies . the answer may lie in the schmidt law and the mass surface densities we have estimated in table [ tab : derprop ] . [ fig : msurfden ] shows these surface densities at the radii over which they were estimated , revealing a trend towards higher densities on smaller scales and values of a few times @xmath71m@xmath1pc@xmath2 in the central few tens of parsecs . the global schmidt law , as formulated by @xcite , states that the star formation rate depends on the gas surface density as @xmath72 . if one assumes that 1030% of the mass in our agn is gas , then this relation would predict time - averaged star formation rates in the range 10100m@xmath1yr@xmath6kpc@xmath2 , as have been observed . that the high star formation rates may simply be a consequence of the high mass surface densities is explored futher by hicks et al . ( in prep . ) . [ [ stellar - luminosity ] ] stellar luminosity + + + + + + + + + + + + + + + + + + as a consequence of the high star formation rates , the stellar luminosity per unit area close around the agn is very high in these objects . despite this , because the star formation is occurring only in very small regions , the absolute luminosities are rather modest . this can be seen in fig . [ fig : mag_agn ] which shows the bolometric luminosity of the stars as a fraction of the entire bolometric luminosity of the galaxy . we have calculated a range for the ratio @xmath73 appropriate for each galaxy based on the ages in table [ tab : derprop ] for different @xmath12 . because we assume that all the k - band stellar continuum is associated with the young stars , we have adopted the lower end of each range in an attempt to minimise possible overestimation of @xmath66 . the resulting values for the ratio used span 30130 , within a factor of 2 of the ` baseline ' value of 60 given in section [ sec : diag ] . in the central few tens of parsecs , young stars contribute a few percent of the total . but integrated over size scales of a few hundred parsecs , this fraction can increase to more than 20% . on these scales , the star formation is energetically significant when compared to the agn . such high fractions imply that on the larger scales the extinction to the young stars must be relatively low . on the other hand , on the smallest scales where in absolute terms the stellar luminosity is small , there could in general be considerable extinction even at near infrared wavelengths . in this paper we have not tried to account for extinction since it is very uncertain . the primary effect of doing so would simply be to increase the stellar luminosity above the values discussed here . [ fig : mag_bol ] shows the stellar bolometric luminosity @xmath66 integrated as a function of radius . all the curves follow approximately the same trend , with the luminosity per unit area increasing towards smaller scales and approaching @xmath74l@xmath1kpc@xmath2 in the central few parsecs . this appears to be a robust trend and will not change significantly even with large uncertainties of a factor of a few . it is remarkable that the luminosity density of @xmath74l@xmath1kpc@xmath2 is that estimated by @xcite for ulirgs , which they modelled as optically thick starburst disks . the main difference between the ulirg model and the starbursts close around agn is the spatial scales on which the starburst occurs . based on this model , they argued that ulirgs are radiating at the eddington limit for a starburst , defined as when the radiation pressure on the gas and dust begins to dominate over self - gravity . the limiting luminosity - to - mass ratio was estimated to be @xmath75l@xmath1/m@xmath1 by @xcite . he argued that in a star cluster , once the upper end of the main sequence was populated , the radiation pressure would halt further accretion on to the star cluster and hence terminate the star formation . following @xcite , we apply this definition to the entire disk rather than a single star cluster . for the @xmath74l@xmath1kpc@xmath2 , this implies a mass surface density of @xmath76m@xmath1pc@xmath2 . comparing these quantities to the agn we have observed , we find that on scales of a few tens of parsecs they are an order of magnitude below the eddington limit . on the other hand , we have already seen that the low @xmath57 indicates that there is little on - going star formation and hence that the starbursts are short - lived . this is important because short - lived starbursts fade very quickly . as shown in fig . [ fig : starstoy ] , for a decay timescale of @xmath68myr , @xmath66 will have decreased from its peak value by more than an order of magnitude at an age of 100myr . thus it is plausible and probably likely that while the star formation was active , the stellar luminosity was an order of magnitude higher . in this case the starbursts would have been at , or close to , their eddington limit at that time . the luminosity - to - mass ratio of @xmath77l@xmath1/m@xmath1 associated with the eddington limit is in fact one that all young starbursts would exceed if , beginning with nothing , gas was accreted at the same rate that it was converted into stars . that , however , is not a realistic situation . a more likely scenario , shown in fig . [ fig : ratio_bol ] , is that the gas is already there in the disk . in this case , a starburst with a star - forming timescale of 100myr could never exceed @xmath78l@xmath1/m@xmath1 . to reach @xmath77l@xmath1/m@xmath1 , the gas would need to be converted into stars on a timescale @xmath79myr . this timescale is independent of how much gas there is . thus , for a starburst to reach its eddington limit , it must be very efficient , converting a significant fraction of its gas into stars on very short @xmath80myr timescales . this result is consistent with the prediction of the schmidt law , which states that disks with a higher gas surface density will form stars more efficiently . the reason is that the star formation efficiency is simply @xmath81 . thus , from arguments based solely on the schmidt law and mass surface density , one reaches the same conclusion that the gas supply would be used rather quickly and the lifetime of the starburst would be relatively short . summarising the results above , a plausible scenario could be as follows . the high gas density leads to a high star formation rate , producing a starburst that reaches its eddington limit for a short time . because the efficiency is high , the starburst can only be active for a short time and then begins to fade . inevitably , one would expect that the starburst is then dormant until the gas supply is replenished by inflow . this picture appears to be borne out by the observations presented here . in the previous sections we have presented and discussed evidence that in general there appears to have been moderately recent star formation on small spatial scales around all the agn we have observed . [ fig : age ] shows the first empirical indication of a deeper relationship between the star formation and the agn . in this figure we show the luminosity of the agn , both in absolute units of solar luminosity and also in relative units of its eddington luminosity @xmath82 , against the age of the most recent known nuclear star forming episode . since the agn luminosity is not well known , we have made the conservative assumption that it is equal to half the bolometric luminosity of the galaxy as may be the case for ngc1068 ( @xcite , but see also @xcite ) . to indicate the expected degree of uncertainty in this assertion we have imposed errorbars of a factor 2 in either direction , equivalent to stating that the agn luminosity in these specific objects is likely to be in the range 25100% of the total luminosity of the galaxy . the eddington luminosity is calculated directly from the black hole mass , for which estimates exist for these galaxies from reverberation mapping , the m@xmath83 relation , maser kinematics , etc . these are listed in table [ tab : basicdata ] . for the age of the star formation , we have plotted the time since the most recent known episode of star formation began , as given in table [ tab : derprop ] . for galaxies where a range of ages is given , we have adopted these to indicate the uncertainty ; the mean of these , @xmath8430% , has been used to estimate the uncertainty in the age for the rest of the galaxies . we note that these errorbars reflect uncertainties in characterising the age of the star formation from the available diagnostics and also in the star formation timescale @xmath12 . however , there are still many implicit assumptions in this process , and we therefore caution that the actual errors in our estimation of the starburst ages may be larger than that shown . conceding this , we do not wish to over - interpret the figure . keeping the uncertainties in mind , fig . [ fig : age ] shows the remarkable result that agn which are radiating at lower efficiency @xmath85l / l@xmath86 are associated with younger @xmath87100myr starbursts ; while those which are more efficient @xmath88l / l@xmath86 have older @xmath89100myr starbursts . if one were to add to this figure the galactic centre which is known to have an extremely low luminosity ( l / l@xmath90 ; @xcite ) and to have experienced a starburst @xmath91myr ago @xcite it would be consistent with the categories above . the inference is that either there is a delay between the onset of starburst activity and the onset of agn activity , or star formation is quenched once the black hole has become active . in section [ sec : prop ] we argued that the starbursts are to some extent self - quenching : that very high star formation efficiencies are not sustainable over long periods . in addition , an intense starburst will provide significant heat input to the gas , which is perhaps partially responsible for the typically high gas velocity dispersions in these regions ( hicks et al . , in prep . ) . this itself could help suppress further star formation . heating by the agn could also contribute to this process , and has been proposed as the reason why the molecular torus is geometrically thick @xcite . it is also used to modulate star formation ( at least on global scales ) in semi - analytic models of galaxy evolution @xcite . while this is certainly plausible , it does not explain either why the star formation in some galaxies with a lower luminosity agn has already ceased , nor why none of the agn associated with younger starbursts are accreting efficiently . instead we argue for the former case above , that efficient fuelling of a black hole is associated with a starburst that is at least 50100myr old . it may be because of such a delay between agn and starburst activity that recent star formation is often hard to detect close to agn : the starburst has passed its most luminous ( very young ) age , and is in decline while the agn is in its most active phase ( see fig . [ fig : starstoy ] ) . this does not necessarily imply that the _ a priori _ presence of a starburst is required before an agn can accrete gas although it seems inevitable that one will occur as gas accumulates in the nucleus . nor does it imply that all starbursts will result in fuelling a black hole ; indeed it is clear that there are many starbursts not associated with agn . as we argue below , the crucial aspect may be the stellar ejecta associated with the starburst ; and in particular , not just the mass loss rate , but the speed with which the mass is ejected . [ [ winds - from - ob - stars ] ] winds from ob stars + + + + + + + + + + + + + + + + + + + in the galactic center , @xcite proposed that it is the recent starburst there that is limiting the luminosity of the black hole . in this scenario , mechanical winds from young stars both the outflow and the angular momentum of the gas ( which is a consequence of the angular momentum of the stars themselves ) hinder further inflow . the authors argued that almost none of the gas flowing into the central parsec reached the black hole because of outflowing winds from irs16 and hei stars in that region . detailed modelling of the galactic center region as a 2-phase medium was recently performed by @xcite . they included both the fast young stellar winds with velocities of 700kms@xmath6 @xcite and the slower winds of @xmath92kms@xmath6 @xcite ; and also took into account the orbital angular momentum of the stars @xcite , which had a strong influence on reducing the accretion rate . they found that the average accretion rate onto the black hole was only @xmath93m@xmath1yr@xmath6 , although an intermittent cold flow superimposed considerable variability onto this . in contrast , the hypothetical luminosity @xcite estimate that sgra@xmath94 would have if it could accrete all the inflowing gas , would be @xmath95ergs@xmath6 , typical of seyfert galaxies . in principle this process could be operating in other galaxy nuclei where there has been a starburst which extends to less than 1pc from the central black hole . however , it can not explain the timescale of the delay we have observed , which is an order of magnitude greater than the main sequence lifetime of ob and wolf - rayet stars . [ [ winds - from - agb - stars ] ] winds from agb stars + + + + + + + + + + + + + + + + + + + + stars of a few ( 18m@xmath1 ) solar masses will evolve on to the asymptotic giant branch ( agb ) at the end of their main sequence lifetimes . the timescale for stars at the upper end of this range to reach this phase is @xmath96myr , comparable to the delay apparent in fig . [ fig : mag_bol ] . since agb stars are known to have high mass - loss rates , of order @xmath97@xmath98m@xmath1yr@xmath6 at velocities of 1030kms@xmath6 @xcite , they may be prime candidates for explaining the delay between starburst and agn activity . to quantify this , we consider how much of the mass in the wind could be accreted by the central supermassive black hole . the bondi parameterisation of the accretion rate onto a point particle for a uniform spherically symmetric geometry is given by @xcite @xmath99 where @xmath100 is the mass of the point particle moving through a gas cloud , @xmath54 is the velocity of the particle with respect to the cloud , @xmath101 is the density of the cloud far from the point particle , and @xmath102 is the sound speed . this approximation is still used to quantify accretion on to supermassive black holes in models of galaxy evolution @xcite , even though it may be significantly inaccurate for realistic ( e.g. turbulent ) media @xcite . here , it is sufficient to provide an indication of the role that stellar winds may play in accretion onto a central black hole . the density of the stellar wind at a distance @xmath51 from the parent star is given by @xmath103 in our case , @xmath51 is the distance from the star to the black hole . one would therefore expect that the accretion rate on to the black hole could be written as ( see also @xcite ) @xmath104 this equation shows that @xmath105 . we have implicitly assumed that @xmath106 is greater than the orbital velocity @xmath107 of the star from which it originates . this is not the case for agb winds , and so one reaches the limiting case of @xmath108 , where for the galaxies we have observed @xmath109100kms@xmath6 . this is still at least an order of magnitude less than the winds from ob and wolf - rayet stars . thus , even though the mass loss rates from individual ob and wolf - rayet stars are similar to those of agb stars , the agb winds will fuel a black hole much more efficiently . however , for slow stellar winds that originate close to a @xmath110m@xmath1 black hole , the equation breaks down because the conditions of uniformity and spherically symmetry are strongly violated . indeed , the apparent accretion rate exceeds the outflow rate implying that essentially the entire wind can be accreted . for agb wind velocities of 1030kms@xmath6 , the maximum radius at which the entire wind from a star in keplerian orbit around a @xmath110m@xmath1 black hole will not exceed the escape velocity from that orbit ( i.e. @xmath111 ) is around 1070pc . we adopt the middle of this range , 40pc , as the characteristic radius within which it is likely that a significant fraction , and perhaps most , of the agb winds are accreted onto the black hole . [ fig : mag_bol ] indicates that the stellar luminosity within this radius is @xmath112l@xmath1 . it is this luminosity that has been used to scale the stars model ( for @xmath68myr and an age of 100myr ) in fig . [ fig : starstoy ] , and so one can also simply read off the mass loss from the figure . the mass loss rate for such winds peaks at about 0.1m@xmath1yr@xmath6 and then tails off proportionally to the k - band luminosity , leading to a cumulative mass lost of @xmath113m@xmath1 after 1gyr ( although most of the loss occurs actually occurs within half of this timespan ) . this mass loss rate is sufficient to power a seyfert nucleus for a short time . a typical seyfert with m@xmath114m@xmath1 requires 0.02m@xmath1yr@xmath6 to radiate at the eddington limit . even for the short bursts we have modelled , fig . [ fig : starstoy ] shows that this can be supplied by agb winds for starburst ages in the range 50200myrs . we note that taking an agb star luminosity of @xmath71l@xmath1 ( which is at the high end of the likely average , @xcite ) we then find that there are @xmath115 agb stars close enough to the black hole to contribute to accretion . in order to provide at least 0.02m@xmath1yr@xmath6 , the typical mass loss rate per star must exceed @xmath97m@xmath1yr@xmath6 , which is the lower limit of the range measured for galactic agb stars given above . thus the mass losses and rates estimated here appear to be plausible . the low speed of these winds means they will not create much turbulence . we quantify this by considering their total mechanical energy @xmath116 integrated over the same timespan , which is @xmath117j . these two quantities gas mass ejected and mechanical energy are compared to those for supernovae below . [ [ supernovae ] ] supernovae + + + + + + + + + + type ii supernovae are the stellar outflows most able to create turbulence in the interstellar medium , since they typically eject masses of @xmath118m@xmath1 at velocities of @xmath119kms@xmath6 @xcite . each supernova therefore represents a considerable injection of mechanical momentum and energy into the local environment . a large number of compact supernova remnants are known , for example in m82 and arp220 , and are believed to have expanded into dense regions with @xmath120@xmath71@xmath121 @xcite . these authors argue that such remnants become radiative when they reach sizes of @xmath122pc , at which point the predicted expansion velocity will have slowed to @xmath75kms@xmath6 . by this time , the shock front will have driven across @xmath123m@xmath1 of gas . when integrated over the age of the starburst , even for low supernova rates e.g. the current rate within 30pc of the nucleus of ngc3227 is @xmath124yr@xmath6 @xcite this represents a substantial mass of gas that has been affected by supernova remnants . the stars model we have constructed in fig . [ fig : starstoy ] indicates that typically one could expect @xmath125 supernovae to occur as a result of one of the short - lived starbursts ; and that most of these will occur around 1050myr after the beginning of the starburst . for a decay timescale of the star formation rate that is longer than @xmath68myr , this timespan will increase . hence , supernovae may also play a role in causing the observed delay between starburst and agn activity . stars calculates the mass loss and mass loss rates using a very simple scheme , assuming that a star ejects all of its lost mass at the end of its life on a stellar track . thus , it does not calculate the mass lost from supernovae explicitly , rather the combined mass lost from ob winds and supernovae which is much higher . we therefore adopt the @xmath118m@xmath1 per supernova given above , which yields a total ejected mass of @xmath126m@xmath1 . this is about 40% of that released by agb winds . however , since this gas is ejected at high speed and @xmath105 , the efficiency with which it can be accreted onto the black hole is extremely low . this can also be seen in the total mechanical energy of @xmath127j , which is several orders of magnitude greater than for agb winds . in fact the total mechanical energy exceeds the binding energy of the nuclear region , which is of order @xmath128j ( assuming @xmath129m@xmath1 within 40pc ) . as a result , it is highly likely that supernova cause some fraction of the gas to be permanently expelled . indeed , superwinds driven by starbursts are well known in many galaxies . this is not important as long as sufficient gas either remains to fuel the agn , or more is produced by stellar winds which , as we have argued above , appears to be the case for agb stars . we have obtained near infrared spectra of 9 nearby active galactic nuclei using adaptive optics to achive high spatial resolution ( in several cases better than 10pc ) . for 7 of these , integral field spectroscopy with sinfoni allows us to reconstruct the full 2-dimensional distributions and kinematics of the stars and gas . although the individual agn are very varied , we have analysed them in a consistent fashion to derive : the stellar k - band luminosity , the dynamical mass , and the equivalent width of the br@xmath7 line . we have combined these with radio continuum data from the literature , which has been used to estimate the supernova rate . we have used these diagnostics to constrain stars evolutionary synthesis models and hence characterize the star formation timescales and ages of the starbursts close around agn . our main conclusions can be summarised as follows : * the stellar light profiles show a bright nuclear component with a half - width at half - maximum of less than 50pc . in a number of cases these nuclear components clearly stand out above an inward extrapolation of the profile measured on larger scales . in addition , there are 2 cases which show kinematical evidence for a distinct stellar component , indicating that the nuclear stellar populations most probably exist in thick nuclear disks . the mean mass surface densities of these disks exceeds @xmath71m@xmath1pc@xmath2 . * there is abundant evidence for recent star formation in the last 10300myr . but the starbursts are no longer active , implying that the star formation timescale is short , of order a few tens of myr . while the starbursts were active , the star formation rates would have been much higher than the current rates , reaching as high as 1000m@xmath1kpc@xmath2 in the central few tens of parsecs ( comparable to ulirgs , but on smaller spatial scales ) . these starbursts would have been eddington limited . due to the very high star forming efficiency , the starbursts would have also exhausted their fuel supply on a short timescale and hence have been short - lived . it therefore seems likely that nuclear starbursts are episodic in nature . * there appears to be a delay of 50100myr ( and in some cases perhaps more ) between the onset of star formation and the onset of agn activity . we have interpreted this as indicating that the starburst has a significant impact on fuelling the central black hole , and have considered whether outflows from stars might be responsible . while supernovae and winds from ob stars eject a large mass of gas , the high velocity of this gas means that its accretion efficiency is extremely low . on the other hand , winds from agb stars ultimately dominate the total mass ejected in a starburst ; and the very slow velocities of these winds mean they can be accreted onto the black hole very efficiently . the authors thank all those who assisted in the observations , and also the referee for a thorough review of the paper . this work was started at the kavli institute for theoretical physics at santa barabara and as a result was supported in part by the national science foundation under grant no . phy05 - 51164 . rd aknowledges the interesting and useful discussions he had there with eliot quartaert , norm murray , julian krolik and todd thompson . _ facilities : _ , . this appendix contains specific details on the individual objects . we summarize our published results from near infrared adaptive optics spectroscopy of individual objects , and present a brief analysis of the new data for several other objects . the aim of re - assessing the data for mkn231 that has already been published is to ensure that it is analysed using stars in a manner that is consistent with the new data . for ngc7469 , we make a significant update of the analysis using new data from integral field spectroscopy . in general , for objects with new data , we provide only the part of the analysis relevant to understanding star formation around the agn . our intention is that a complete analysis for each object will be presented in future publications . our analyses are restricted to the nuclear region . since there is no strict universal definition of what comprises the ` nuclear region ' , we explicitly state in table [ tab : derprop ] the size of the region we study in each galaxy . the table also presents a summary of the primary diagnostics . the way in which these have been derived , and their likely uncertainties , has been discussed in some detail already in section [ sec : diag ] . as such , the description of these methods is not repeated , and in this section we discuss only issues that require special attention . a detailed analysis of the star formation in the nucleus of mkn231 at a resolution of about 0.18 ( 150pc ) was given in @xcite . here we summarize only the main points ; no new data is presented , but the analysis is updated using stars to make it consistent with the other objects studied in this paper . the presence of stellar absorption features across the nucleus demonstrates the existence of a significant population of stars . the radial distribution and kinematics indicate they lie , like the gas @xcite , in a nearly face - on disk . @xcite found that the dynamical mass imposed a strong constraint on the range of acceptable starburst models , yielding an upper limit to the age of the stars of around 120myr . re - assessing the mass - to - light ratio using stars models suggests that for the increased mass required by a more face - on orientation ( @xmath130 ) an upper age of 250myr is also possible , depending on the star formation timescale . however , either a small change of only a few degrees to the inclination ( to @xmath131 ) , or a relatively short star formation timescale of 10myr would reduce the limit to the @xmath61100myr previously estimated . this is more consistent with the extremely high supernova rate . the stellar luminosity , found from the dilution of the co absorption @xcite , indicates that stars within 1 ( 800pc ) of the nucleus contribute 2540% of the bolometric luminosity of the galaxy . similarly , within 200pc , stars comprise 1015% of l@xmath132 . the age , star formation rate , and size scale ( disk scale length of 0.180.2 ) are all consistent with high resolution radio continuum imaging @xcite . star formation in the central 16pc of circinus was addressed by @xcite . the diagnostics given in table [ tab : derprop ] are taken from this reference . we used the depth of the co2 - 0 bandhead to estimate the stellar luminosity , combined with the narrow br@xmath7 flux ( which we argued originated in star forming regions rather than the agn narrow line region ) and the radio continuum , to constrain starburst models . the conclusion was that the starburst was less than 80myr old and was already decaying . on these scales it contributes 1.4% of l@xmath132 , or more if extinction is considered . a similar nuclear star formation intensity was estimated by @xcite , who were also able to study circinus on larger scales . they found that the luminosity of young stars within 200pc of the agn was of order @xmath133@xmath134 , and hence comparable to the agn . an analysis similar to that for circinus was performed on ngc3227 by @xcite , and the diagnostics given in table [ tab : derprop ] are taken from this reference . in this case we were able to make estimates of and correct for contributions of : ( 1 ) the narrow line region to br@xmath7 , because there were clear regions along the minor axis that had higher dispersion ; ( 2 ) the agn to the radio continuum , by estimating the maximum contribution from an unresolved source ; and ( 3 ) the bulge stars to the stellar luminosity , by extrapolating the radial profile of the bulge to the inner regions . the stars models yielded the result that in the nucleus , star formation began approximately 40myr ago and must have already ceased . at the resolution of 0.085 , the most compact component of stellar continuum had a measured fwhm of 0.17 , suggesting an intrinsic size scale of @xmath135pc . young stars within 30pc of the agn ( i.e. more than just the most compact region ) have a luminosity of @xmath136@xmath134 , which is @xmath137% of the entire galaxy . star formation on large scales in ngc7469 has been studied by @xcite . they found that within 800pc of the nucleus , a region that includes the circumnuclear ring , the luminosity from young stars was @xmath138@xmath134 , about 70% of the galaxy s bolometric luminosity . this situation is similar to that in mkn231 . on smaller scales , the nuclear star formation in ngc7469 was directly resolved by @xcite on a size scale of 0.150.20 ( 5065pc ) fwhm . an analysis of the longslit data , similar to that for mkn231 , was made making use of stellar absorption features , kinematics , and starburst models . we estimated that the age of this region was no more than 60myr under the assumption that the fraction of stellar light in the k - band in the central 0.2 was 2030% . our new integral field sinfoni observations of ngc7469 at a spatial resolution of 0.15 ( measured from both the broad br@xmath7 and the non - stellar continuum profiles , see section [ sec : obs ] ) are used here to make a more accurate estimate of the nuclear k - band luminosity . they enable us to provide a short update to the detailed analysis in @xcite . the sinfoni data show that the equivalent width of the 2.3 co2 - 0 is @xmath139 in a 0.8 aperture and 0.9 in a 0.2 aperture . the corresponding k - band magnitudes are @xmath140 and @xmath141 respectively . if one takes the intrinsic equivalent width of the 2.3 co2 - 0 bandhead to be 12 ( see section [ sec : diag ] ) , one arrives at a more modest value of 8% for the stellar fraction of k - band continuum in the 0.2aperture . the stellar k - band luminosity in this region is then @xmath142l@xmath1 . comparing this to the dynamical mass in @xcite yields a mass - to - light ratio of m / l@xmath143m@xmath1/l@xmath1 . previously , extrapolation from a 37mas slit to a filled aperture had led to an underestimation of the total magnitude but an overestimation of the stellar contribution . fortuitously , these uncertainties had compensated each other . the same analysis for the 0.8 aperture yields a k - band stellar luminosity of @xmath144l@xmath1 and hence m / l@xmath145m@xmath1/l@xmath1 . the k - band datacube yields estimates of the upper limit to @xmath57 of 17 and 11 in 0.2 and 0.8apertures respectively . this has been corrected for dilution of the stellar continuum ( as described in section [ sec : diag ] ) but not for a possible contribution to the narrow br@xmath7 from the agn . hence the actual @xmath57 corresponding to only the stellar line and continuum emission will be less than these values indicating that the star formation is unlikely still to be on - going . we estimate the age of the star formation using the stars models in fig . [ fig : stars ] . within the 0.2 aperture this gives 100myr , comparable to our original estimate . such a young age is supported by radio continuum measurements . with a 0.2 beam , @xcite reported that the unresolved core flux in ngc7469 was 12mjy at 8.4ghz . with much higher spatial resolution of 0.03 , @xcite reported an upper limit to the unresolved 8.4ghz continuum of 7mjy . we assume that the difference of 5mjy is due to emission extended on scales of 1060pc which is resolved out of one beam but not the other . as discussed in section [ sec : diag ] , star formation is a likely candidate for such emission . in this case , we would estimate the supernova rate to be @xmath4yr@xmath6 and the ratio @xmath146 . this is likely to be a lower limit since there was only an upper limit on the core radio flux density . for a ratio of this order , even allowing for some uncertainty , fig [ fig : stars ] implies an age consistent with no more than 100myr . within the 0.8 aperture , which we adopt in table [ tab : derprop ] , continuous star formation is inconsistent with w@xmath147 . for a star formation timescale of @xmath148myr , the mass - to - light ratio implies an age of 190myr , just consistent with the measured value of w@xmath149 . if some of the narrow br@xmath7 is associated with the agn rather than star formation , then a shorter star formation timescale is required . for @xmath68myr , the ratio m / l@xmath150 yields an age of 110myr . [ fig : ir05189_spec ] shows the h - band spectrum integrated across two segments of the naco slit , located on either side of the nucleus . it shows that even away from the nucleus , the depth of the stellar absorption features is only a few percent . we have therefore decomposed the data into the stellar and non - stellar parts using both the stellar absorption features and the spectral slope of the continuum . the latter method has been shown to work for well sampled data by @xcite . the rationale is that the hot dust associated with the agn will be much redder than the stellar continuum . an agn component is also expected to be unresolved for a galaxy at the distance ( 170mpc ) of iras05189 - 2524 . the spectral slope was determined by fitting a linear function to the spectrum at each spatial position along the slit . it is plotted as a function of position in fig . [ fig : ir05189_decomp ] , showing a single narrow peak . a gaussian fit to this yields a spatial resolution of 0.12 ( 100pc ) fwhm . the stellar continuum , also shown in fig . [ fig : ir05189_decomp ] , has been determined by summing the four most prominent absorption features : co4 - 1 , sii , co5 - 2 , co6 - 3 . while a gaussian is not an optimal fit to this profile , it does yield an aproximate size scale , which we find to be 0.27 fwhm . quadrature correction with the spatial resolution yields an intrinsic size of 0.25 ( 200pc ) . as a cross - check , in the figure we have compared the sum of these two components to the full continuum profile . the good match indicates that the decomposition appears to be reasonable . remarkably , the 200pc size of the nuclear stellar light is very similar to that of the 8.44ghz radio continuum map of @xcite . with a beam size of [email protected] , they resolved the nuclear component to have an intrinsic size of [email protected] . in constrast to radio sources which are powered by agn and have brightness temperatures @xmath151k , the emission here is resolved and has a low brightness temperature of @xmath152k . this implies a star forming origin . using their scaling relations further suggests that the flux density corresponds to a supernova rate of @xmath122yr@xmath6 . as described in section [ sec : diag ] , we have estimated the stellar luminosity by comparing the h - band spectrum to a template star to correct for dilution . we used hr8465 a k1.5i star for which the equivalent width of co6 - 3 is 4.2 , within the 45 range predicted by stars in fig . [ fig : equiv ] . by extrapolating from the spatial profiles along the slit we have estimated the integrated equivalent width within a 1.1 aperture , for which @xcite gave an h - band magnitude of 11.83 . using all four features above we find for the template @xmath153 and for iras05189 - 2524 @xmath154 . this implies that in the central 1.1 , approximately 45% of the h - band continuum originates in stars . using the colour conversion @xmath26 from fig . [ fig : col_lum ] ( see section [ sec : diag ] ) we find a k - band magnitude for the stars of 12.55mag and hence a k - band stellar luminosity of @xmath155l@xmath1 . putting these results together we derive a ratio of supernova rate to k - band stellar luminosity of @xmath156 / l_{k } [ 10^{10 } l_\odot ] \sim 5 $ ] . applying corrections for extinction and an agn contribution would tend to decrease this ratio . as a second diagnostic we use @xmath57 . we estimate the dilution of the k - band continuum via two methods . firstly , we measure @xmath157 , indicating a stellar fraction of 0.100.15 . a consistency check is provided by the h - band dilution , which we extrapolate to the k - band using blackbody functions for the stars and dust assuming characteristic temperatures of 5000k and 1000k respectively . this method suggests the k - band stellar fraction is around @xmath158 . hence correcting the directly measured equivalent width of the narrow br@xmath7 for the non - stellar continuum yields @xmath1595 . since iras05189 - 2524 is close to face - on @xcite , it is not straightforward to make a reliable estimate of the dynamical mass . nevertheless , requiring @xmath160 to be high while @xmath57 is low already puts significant constraints on the star formation history . thus , although the star formation has probably ended , the age is unlikely to be greater than 100myr , and could be as low as 50myr where @xmath160 peaks . for such ages the ratio @xmath37 is in the range 100150 . hence for the young stars within 0.55 ( 450pc ) of the nucleus we find @xmath16123@xmath162l@xmath1 , about 20% of @xmath36 for the galaxy . the spatial resolution of the k - band data for ngc2992 has been estimated from both the broad br@xmath7 and the non - stellar continuum ( see section [ sec : obs ] and [ sec : diag ] ) . the two methods yield symmetric psfs , with fwhms of 0.32 and 0.29 respectively , corresponding to 50pc . unlike iras05189 - 2524 , the radio continuum in ngc2992 is quite complex . much of the extended emission on scales of a few arcsec appears to originate from a superbubble , driven either by the agn or by a nuclear starburst . on the other hand , most of the nuclear emission seems to be unresolved . with a beam size of [email protected] , @xcite measured the unresolved flux to be 7mjy at 5ghz . at a resolution better than 0.1 , @xcite reported a 2.3ghz flux of 6mjy . based on this as well as non - detections at 1.7ghz and 8.4ghz , they estimated the core flux at 5ghz to be @xmath1666mjy . taking a flat spectral index , as indicated by archival data @xcite , one might expect the 5ghz core flux to be not much less than 6mjy , leaving room for only @xmath122mjy in extended emission in the central 0.5 . if we assume this difference can be attributed to star formation , it implies a supernova rate of @xmath167yr@xmath6 and hence @xmath168 . [ fig : stars ] shows that a ratio of this order is what one might expect for ages up to 200myr . however , given the uncertainty it does not impose a significant constraint . it is also difficult to quantify what fraction of the narrow br@xmath7 is associated with star formation . this is made clear in fig . [ fig : n2992_brg ] which shows that the morphology of the line ( centre left panel ) does not follow that of the stars ( far left ) . in addition , particularly the south - west side is associated with velocities that are bluer than the surrounding emission , indicative of motion towards us . the western edge also exhibits high velocity dispersion . taken together , these suggest that we may be seeing outflow from the apex of an ionisation cone with a relatively large opening angle . this interpretation would tend to support the hypothesis that the radio bubble has been driven by the agn . the stellar continuum appears to trace an inclined disk , the north west side of which is more obscured ( fig . [ fig : n2992_brg ] ) . however , the velocity dispersion is high , exceeding 150kms@xmath6 across the whole field ( fig . [ fig : n2992_vel ] ) . this is similar to the 160kms@xmath6 reported by @xcite from optical spectroscopy , and suggests that we are seeing bulge stars . to analyse the radial luminosity profile we have fitted it with both an @xmath55 law and exponential profile . the fits in fig . [ fig : n2992_rad ] were optimised at radii @xmath169 and then extrapolated inwards , convolved with the psf . whether one could claim that there is excess continuum in the nucleus depends on the profile fitted . the @xmath55 law provides a stronger constraint since it is more cuspy , and suggests there is no excess . although this evidence is inconclusive , fig . [ fig : n2992_vel ] suggests that there is some kinematic evidence favouring the existence of a distinct nuclear stellar population . this comes in the form of a small unresolved drop in dispersion at the centre , similar to those in ngc1097 and ngc1068 . while the evidence in ngc2992 is not compelling , the dispersion is consistent with there being an equivalent but fainter nuclear disk on a scale of less than our resolution of 50pc . in general it seems that the k - band light we are seeing is dominated by the bulge , and we are therefore unable to probe in detail the inner region where it seems that more recent star formation has probably occurred . thus , although the available data suggest there has likely been recent star formation in the nucleus of ngc2992 , the only strong constraint we can apply is that continuous star formation in the central arcsec over the last gyr can be ruled out since it would require @xmath17015 . we therefore omit ngc2992 from the discussion and analysis in sections [ sec : prop ] and [ sec : staragn ] . in ngc1097 , the first evidence for recent star formation near the nucleus was in the form of a reduction in the stellar velocity dispersion . @xcite proposed this could be explained by the presence of a dynamically cold nuclear disk that had recently formed stars . direct observations of a spiral structure in the central few arcsec , from k - band imaging @xcite and [ nii ] streaming motions @xcite , have since confirmed this idea . however , some issues remain open , such as why there are three spiral arms rather than the usual two , and why gas along one of them appears to be outflowing . our data , at a resolution of 0.25 measured from the h - band non - stellar continuum , also reveal the same spiral structure . indeed , we find that it is traced by the morphology of the co bandhead absorption as well as by the 2.12 h@xmath49 1 - 0s(1 ) line . interestingly , 1 - 0s(1 ) emission is stronger where the stellar features are weaker . this suggests that obscuration by gas and dust plays an important role . [ fig : n1097_prof ] shows that an @xmath55 law , typical of stellar bulges , with effective radius @xmath171 is a good fit to the stellar radial profile at @xmath172 . it therefore seems reasonable to argue that at these radii it is only the gas that lies in a disk . in this picture the spiral structure in the stellar continuum arises solely due to extinction of the stars behind the disk . extrapolating this fit , convolved with the psf , to the nucleus indicates that at @xmath173 there is at least 25% excess stellar continuum . there could be much more , given that it coincides with a change in the dominant kinematics . for ngc1097 we parameterized the kinematics of the gas and stars quantitatively using kinemetry . based on the uniformity of the velocity field , we made the simplifying assumption that across the central 4 the gas lies in a single plane whose centre is coincident with the peak of the non - stellar emission . we were then able to derive the position angle and inclination of the disk ( see section [ sec : obs ] ) . the 2d kinematics of the stars is traced via the co2 - 0 absorption bandhead , and that of the gas through the 1 - 0s(1 ) emission line . these independently yielded similar parameters : both gave a position angle of @xmath174 and their inclinations were 43@xmath175 and 32@xmath175 respectively . these are fully consistent with values found by other authors ( @xcite ) . the resulting rotation curves and velocity dispersions are shown in fig . [ fig : n1097_dispvel ] . the residuals , which can be seen in the velocity field of the gas but not the stars , and their relation to the spiral structure described above will be discussed elsewhere ( davies et al . in prep ) . the important result here is that at our spatial resolution , we find that the central stellar dispersion is @xmath176kms@xmath6 , less than the surrounding 150kms@xmath6 and also less than that in the seeing limited spectra of @xcite . in the same region we find that the rotation velocity of the gas starts to decrease rapidly , and its dispersion increases from @xmath177kms@xmath6 to @xmath178kms@xmath6 . [ fig : n1097_dispvel ] also shows that while the kinematics of the stars and gas are rather different at large ( @xmath179 ) radii , they are remarkably similar at radii @xmath180 . this certainly provides a strong indication that in the nuclear region the stars and gas are coupled , most likely in a ( perhaps thick ) disk ; and that the stars in this disk , which are bright and hence presumably young , give rise to the excess stellar continuum observed . evidence for a recent starburst has been found by @xcite through optical and uv spectra . they argued that a number of features they observed could only arise from an @xmath181m@xmath1 instantaneous starburst , which occurred a few myr ago and is reddened by @xmath182mag of extinction . using stars we have modeled this starburst as a @xmath181m@xmath1 burst beginning 8myr ago with an exponential decay timescale of 1myr . the age we have used is a little older to keep the br@xmath7 equivalent width low ; and at this age , the model predicts @xmath159 . as fig . [ fig : n1097_maps ] shows , the observed br@xmath7 is weak , although perhaps slightly resolved . corrected for the non - stellar continuum , we measure only @xmath183 . however , the bulge population may account for a significant fraction of the k - band stellar continuum . correcting also for this could increase @xmath57 to 25 , consistent with that of the model assuming that the br@xmath7 is associated with the starburst rather than the agn . to within a factor of a few , the scale of the model starburst is also consistent with that measured : in the central 0.5 we measure a br@xmath7 flux of @xmath184wm@xmath2 , compared to that predicted by the model of @xmath185wm@xmath2 . given the uncertainties factors of a few both in the parameters of the starburst model and also in the corrections we have applied to the data , we consider this a good agreement . we can not constrain the starburst further due to its compactness . @xcite found that it was occurring in the central 0.2 , whereas our resolution is only 0.25 . the br@xmath7 emission is confined to the central 0.40.5 , although its size is hard to measure due to its weakness with respect to the stellar absorption features . in this region the k - band stellar luminosity is @xmath186l@xmath1 . to estimate the dynamical mass we use the mean kinematics of the stars and gas , i.e. v@xmath187kms@xmath6 ( corrected for inclination ) and @xmath188kms@xmath6 ( this is the central value , which is least biassed by bulge stars ) , yielding @xmath189m@xmath1 this is actually dominated by the black hole , which has a mass of @xmath190m@xmath1 @xcite . the difference between these implies a mass of gas and stars of @xmath191m@xmath1 , although with a large uncertainty . the associated mass - to - light ratio is @xmath192 . on its own , this implies that over the relatively large area that it encompasses , the maximum characteristic age for the star formation is a few hundred myr . if one speculates that star formation has been occurring sporadically for this timescale , then the starburst seen by @xcite is the most recent active episode . in order to make a rough estimate of the supernova rate in the central region we make use of measurements reported by @xcite . they find an unresolved component ( size @xmath193 ) with 5ghz flux density @xmath194mjy , but at lower resolution there is a @xmath195mjy component of size 1 . as discussed in section [ sec : diag ] we assume that the difference albeit with only marginal significance of @xmath196mjy is due to star formation in the central region , which implies a supernova rate of @xmath197yr@xmath6 and hence @xmath198 , a value consistent with rather more recent star formation . indeed , when compared to fig . [ fig : stars ] , this and the low @xmath57 imply a young age and short star formation timescale . for @xmath68myr the age is 6070myr ; for an instantaneous burst of star formation , the age would be @xmath80myr , broadly consistent with that of @xcite . thus , although our data do not uniquely constrain the age of the starburst in the nucleus of ngc1097 , they do indicate that recent star formation has occurred ; and they are consistent with a very young compact starburst similar to that derived from optical and uv data . evidence for a stellar core in ngc1068 with an intrinsic size scale of @xmath199pc was first presented by @xcite . based on kinematics measured in large ( 24 ) apertures , they assumed the core was virialized and estimated a mass - to - light ratio based on this assumption leading to an upper limit on the stellar age of 1600myr . making a reasonable correction for an assumed old component lead to a younger age of 500myr . stellar kinematics from optical integral field spectra @xcite show evidence for a drop in the stellar velocity dispersion in the central few arcsec to @xmath200kms@xmath6 , inside a region of higher 150200kms@xmath6 dispersion ( presumably the bulge ) . our near infrared adaptive optics data are able to fully resolve the inner region where @xmath60 drops , as shown in fig . [ fig : n1068_prof ] . as for ngc1097 , the velocity distribution of the stars was derived through kinemetry , again making use of the uniformity of the stellar velocity field to justify the simplifying assumption that the position angle and inclination do not change significantly in the central 4 . the derived inclination of @xmath201 and position angle of @xmath202 are quantitatively similar to those found by other authors in the central few to tens of arcseconds @xcite . the uniformity of the stellar kinematics is in contrast to molecular gas kinematics , as traced via the 1 - 0s(1 ) line , which are strongly perturbed and show several distinct structures superimposed . these are too complex to permit a comparably simple analysis and will be discussed , together with the residuals in the stellar kinematics in a future work ( mueller snchez et al . in prep ) . the crucial result relevant here is that at our h - band resolution of 0.10 we find that @xmath60 reduces from 130kms@xmath6 at 12 to only 70kms@xmath6 in the very centre . that there is in the same region an excess in the stellar continuum is demonstrated in fig . [ fig : n1068_prof_wide ] . here we show the radial profile of the stellar continuum from both sinfoni integral field spectra out to a radius of 2 and naco longslit spectra out to 5 ( 350pc ) . at radii 15 , corresponding roughly to the region of high stellar dispersion measured by @xcite , the profile is well matched by an @xmath55 law , as one might expect for a bulge . at radii @xmath203 the same radius at which we begin to see a discernable reduction in the stellar dispersion the stellar continuum increases by as much as a factor 2 above the inward extrapolation of the profile , indicating that there is extra emission . as for ngc1097 , the combined signature of dynamically cool kinematics and excess emission is strong evidence for a nuclear disk which has experienced recent star formation . we can make an estimate of the characteristic age of the star formation in the central arcsec based on the mass - to - light ratio in a similar way to @xcite . because the stars appear to lie in a disk , we estimate the dynamical mass as described in section [ sec : diag ] from the stellar kinematics , using the rotation velocity and applying a correction for the dispersion . the stellar rotation curve is essentially flat at 0.10.5 , with @xmath204kms@xmath6 ( corrected for inclination ) . we also take @xmath205kms@xmath6 , which is the central value and hence least biassed by the high dispersion bulge stars . these lead to a mass of @xmath206m@xmath1 within @xmath207 ( 35pc ) , and a mean surface density of @xmath208m@xmath1pc@xmath2 . correcting for the non - stellar continuum , the h - band magnitude ( which the behaviour of @xmath60 indicates is dominated by the disk emission ) in the same region is 11.53mag . for @xmath26mag ( fig . [ fig : stars ] ) , we find @xmath209l@xmath1 and hence @xmath210m@xmath1/l@xmath1 . if no star formation is on - going , this implies a characteristic age of 200300myr fairly independent of the timescale ( for @xmath211myr , see fig . [ fig : stars ] ) on which stars were formed . we note that this is significantly younger than the age estimated by @xcite primarily because their mass was derived using a higher @xmath60 corresponding to the bulge stars . the assumption of no current star formation is clearly demonstrated by the br@xmath7 map in fig . [ fig : n1068_maps ] . away from the knots of br@xmath7 , which are associated with the coronal lines and the jet rather than possible star formation , the equivalent width is @xmath212 . this is significantly less than that for continuous star formation of any age . thus , while it seems likely that star formation has occurred in the last few hundred myr , it also seems an unavoidable conclusion that there is no current star formation . to complete our set of diagnostics for ngc1068 , we consider also the radio continuum . this is clearly dominated by phenomena associated with the agn and jets , and our best estimate of the flux density away from these features is given by the lowest contour in maps such as figure 1 of @xcite . from this we estimate an upper limit to the 5ghz continuum associated with star formation of 128mjy within @xmath173 . however , converting to a supernova rate and comparing to the k - band stellar luminosity yields a limit that is not useful , being an order of magnitude above the largest expected values . at near infrared wavelengths , the agn in ngc3783 is remarkably bright . integrated over the central 0.5 less than 4% of the k - band continuum is stellar . in addition , the broad brackett lines are very strong and dominate the h - band . both of these phenomena are immediately clear from the h- and k - band spectra in fig . [ fig : n3783_spec ] . however , it does mean that the spatial resolution can be measured easily from both the non - stellar continuum and the broad emission lines ( see section [ sec : obs ] ) . we find the k - band psf to be symmetrical with a fwhm of 0.17 . due to the ubiquitous brackett emission in the h - band we were unable to reliably trace the stellar absorption features and map out the stellar continuum . instead we have used the co2 - 0 bandhead at 2.3 even though the dilution at the nucleus itself is extreme . the azimuthally averaged radial profile is shown in fig . [ fig : n3783_rad ] together with the psf for reference . at radii from 0.21.6 ( the maximum we can measure ) the profile is well fit by an @xmath55 de vaucouleurs law with @xmath213 ( 120pc ) . as has been the case previously , at smaller radii we find an excess that here is perhaps marginally resolved . thus a substantial fraction of the near infrared stellar continuum in the central region is likely to originate in a population of stars distinct from the bulge . we were unable to measure the stellar kinematics due to the limited signal - to - noise . instead , we used the molecular gas kinematics to estimate the dynamical mass . as before , we used kinemetry to derive the position angle of @xmath214 and the inclination in the range 3539@xmath175 . this orientation is consistent with the larger ( 20 ) scale isophotes in the j - band 2mass image and implies that in ngc3783 there is no significant warp on scales of 50pc to 4kpc . a small inclination is also consistent with its classification as a seyfert 1 . adopting these values , the resulting rotation curve is shown in fig . [ fig : n3783_rot ] . at very small radii the rising rotation curve may be the result of beam smearing across the nucleus . at @xmath215 , the falling curve suggests that the rotation is dominated by the central ( @xmath216 ) mass , perhaps the supermassive black hole . we estimate the dynamical mass within a radius of 0.3 ( 60pc ) , corresponding to the point where the excess continuum begins and also where the rotation curve appears to be unaffected by beam smearing . taking @xmath217kms@xmath6 and @xmath218kms@xmath6 we derive a dynamical mass of m@xmath219m@xmath1 . the black hole mass of @xmath220m@xmath1 ( from reverberation mapping , @xcite ) is only 30% of this , and so can not be dominating the dynamics on this scale unless its mass is underestimated . with respect to this , we note that @xcite claim the statistical uncertainty in masses derived from reverberation mapping is about a factor 3 . alternatively , there may be a compact mass of gas and stars at @xmath221 . however , including @xmath53 in the mass estimate implicitly assumes that the dispersion arises from macroscopic motions . on the other hand , because we are observing only the hot h@xmath49 , it is possible that the dispersion is dominated by turbulence arising from shocks or uv heating of clouds that generate the 1 - 0s(1 ) emission issues that are discussed in more detail by hicks et al . ( in prep . ) . in this case we will have overestimated the dynamical mass . excluding @xmath53 from the mass estimation yields @xmath222m@xmath1 . we consider these two estimates as denoting the maximum range of possible masses . subtracting m@xmath223 then gives a mass of stars and gas in the range ( 27)@xmath224m@xmath1 , implying a mass surface density of 17006000m@xmath1pc@xmath2 and @xmath2252.1m@xmath1/l@xmath1 . based on these ratios alone , fig [ fig : stars ] indicates that the characteristic age of the star formation may be as low as @xmath226myr , although it could also be an order of magnitude greater . without additional diagnostics we can not discriminate further . we are unable to use br@xmath7 as an additional constraint on the star formation history . its morphology and velocity field are similar to that of [ sivi ] , and rather different from the 1 - 0s(1 ) . it shows an extension to the north which appears to be outflowing at @xmath227kms@xmath6 ( fig [ fig : n3783_maps ] ) perhaps tracing an ionisation cone . since the br@xmath7 resembles the [ sivi ] , it is reasonable to conclude that it too is associated with the agn rather than star formation . thus the equivalent width of br@xmath7 ( with respect to the stellar continuum ) of @xmath228 represents an upper limit to that associated with star formation . the radio continuum in the nucleus of ngc3783 has been measured with several beam sizes at 8.5ghz . for a beam of @xmath229 , @xcite found it was unresolved with a flux density of @xmath230mjy . with a smaller @xmath164 beam , @xcite measured a total flux density of 8.0mjy dominated by an unresolved component of @xmath231mjy . at smaller scales still of @xmath232 corresponding to 6pc , @xcite placed an upper limit on the 8.5ghz flux density of 7mjy . taken together , these results imply that there is some modest 8.5ghz radio continuum of 0.71mjy extended on scales of 0.31 . based on this we estimate a supernova rate as described in section [ sec : diag ] of @xmath233yr@xmath6 , and hence a ratio @xmath234 . given that the unresolved radio continuum on the smallest scales is an upper limit , the extended component may be stronger and hence the true @xmath235 ratio may be greater than that estimated here . fig [ fig : stars ] then puts a relatively strong limit of @xmath96myr on the maximum age of the star formation . this age is fully consistent with that above associated with our lower mass estimate . the value of @xmath236 above does not impose additional constraints , although we note that if the br@xmath7 flux associated with star formation is only a small fraction of the total then it would imply that the timescale over which the star formation was active is no longer than a few times @xmath80myr . therefore in the nucleus ( @xmath221 ) of ngc3783 we adopt 5070myr as the age of the star formation and @xmath237m@xmath1 as the dynamical mass excluding the central supermassive black hole . knapen j. , 2004 , in _ penetrating bars through masks of cosmic dust : the hubble tuning fork strikes a new note _ , eds block d. , puerari i. , freeman k. , groess r. , block e. ( dordrecht : kluwer academic publishers ) , assl , vol . 319 , p.189 mkn231 & h & 0.176 & may 02 & keck , nirc2 + ngc7469 & k & 0.085 & nov 02 & keck , nirspao + & k & 0.15 & jul 04 & vlt , sinfoni + circinus & k & 0.22 & jul 04 & vlt , sinfoni + ngc3227 & k & 0.085 & dec 04 & vlt , sinfoni + iras05189 - 2524 & h & 0.12 & dec 02 & vlt , naco + ngc2992 & k & 0.30 & mar 05 & vlt , sinfoni + ngc1097 & h & 0.245 & oct 05 & vlt , sinfoni + ngc1068 & h & 0.10 & oct 05 & vlt , sinfoni + & h & 0.13 & dec 02 & vlt , naco + ngc3783 & k & 0.17 & mar 05 & vlt , sinfoni + mkn231 & ulirg , sy1 , qso & 170 & 12.5 & 7.2 & 1 & + ngc7469 & sy1 & 66 & 11.5 & 7.0 & 2 & + circinus & sy2 & 4 & 10.2 & 6.2 & 3 & + ngc3227 & sy1 & 17 & 10.2 & 7.3 & 4 & + iras05189 - 2524 & ulirg , sy1 & 170 & 12.1 & 7.5 & 1 & + ngc2992 & sy1 & 33 & 10.7 & 7.7 & 5 & + ngc1097 & liner , sy1 & 18 & 10.9 & 8.1 & 6 & + ngc1068 & sy2 & 14 & 11.5 & 6.9 & 7 & + ngc3783 & sy1 & 42 & 10.8 & 7.5 & 2 & + mkn231 & 0.6 & 480 & 9.3 & 9.8 & 0.9 & & 3.1 & 20 & 120250 & 2550 + ngc7469 & 0.4 & 128 & 8.5 & 8.7 & 1.0 & 11 & 1.6 & 3 & 110190 & 50100 + circinus & 0.4 & 8 & 6.2 & 7.5 & 17 & 30 & 23 & 1.5 & 80 & @xmath226 + ngc3227 & 0.4 & 32 & 7.8 & 8.0 & 3.7 & 4 & 1.9 & 2.2 & 40 & @xmath238 + iras05189 - 2524 & 0.55 & 450 & 9.3 & & & 4 & & 5 & 50100 & 3070 + ngc2992 & 0.4 & 64 & 7.5 & & & @xmath16612 & & 1 & & + ngc1097 & 0.25 & 22 & 6.7 & 8.2 & 1.3 & 1 & 4.5 & 1.4 & 8 & @xmath178 + ngc1068 & 0.5 & 35 & 7.6 & 8.1 & 3.4 & 4 & 3.0 & @xmath16620 & 200300 & 90170 + ngc3783 & 0.3 & 60 & 7.5 & 7.3 & 0.2 & @xmath16630 & 0.6 & 2 & 5070 & 3060 +

we analyse star formation in the nuclei of 9 seyfert galaxies at spatial resolutions down to 0.085 , corresponding to length scales of order 10pc in most objects . our data were taken mostly with the near infrared adaptive optics integral field spectrograph sinfoni . the stellar light profiles typically have size scales of a few tens of parsecs . in two cases there is unambiguous kinematic evidence for stellar disks on these scales . in the nuclear regions there appear to have been recent but no longer active starbursts in the last 10 - 300myr . the stellar luminosity is less than a few percent of the agn in the central 10pc , whereas on kiloparsec scales the luminosities are comparable . the surface stellar luminosity density follows a similar trend in all the objects , increasing steadily at smaller radii up to @xmath0l@xmath1kpc@xmath2 in the central few parsecs , where the mass surface density exceeds 10@xmath3m@xmath1pc@xmath2 . the intense starbursts were probably eddington limited and hence inevitably short - lived , implying that the starbursts occur in multiple short bursts . the data hint at a delay of 50100myr between the onset of star formation and subsequent fuelling of the black hole . we discuss whether this may be a consequence of the role that stellar ejecta could play in fuelling the black hole . while a significant mass is ejected by ob winds and supernovae , their high velocity means that very little of it can be accreted . on the other hand winds from agb stars ultimately dominate the total mass loss , and they can also be accreted very efficiently because of their slow speeds .

0704.1374

collisionless shocks are widely thought to be effective accelerators of energetic , nonthermal particles ( hereafter cosmic - rays or crs ) . those particles play central roles in many astrophysical problems . the physical basis of the responsible diffusive shock acceleration ( dsa ) process is now well established through in - situ measurements of heliospheric shocks @xcite and through analytic and numerical calculations @xcite . while test particle dsa model treatments are relatively well developed ; e.g. , @xcite , it has long been recognized that dsa is an integral part of collisionless shock physics and that there are substantial and highly nonlinear backreactions from the crs to the bulk flows and to the mhd wave turbulence mediating the cr diffusive transport ( see , for example , @xcite and references therein ) . most critically , the crs can capture a large fraction of the kinetic energy dissipated across such transitions . as they diffuse upstream the crs form a pressure gradient that decelerates and compresses the entering flow inside a broad shock precursor . that , in turn , can lead to greatly altered full shock jump conditions , especially if the most energetic crs , which can have very large scattering lengths , escape the system and carry significant energy with them . also in response to the momentum dependent scattering lengths and flow speed variations through the shock precursor the cr momentum distribution will take on different forms than in a simple discontinuity . effective analytic ( e.g. , @xcite ) and numerical ( e.g. , @xcite ) methods have been developed that allow one to compute steady - state modified shock properties given an assumed diffusion behavior . on the other hand , as the cr particle population evolves in time during the formation of such a shock the shock dynamics and the cr - scattering wave turbulence evolve as well . for dynamically evolving phenomena , such as supernova remnants , the time scale for shock modification can be comparable to the dynamical time scales of the problem . the above factors make it essential to be able to include both nonlinear and time dependent effects in studies of dsa . generally , numerical simulations are called for . full plasma simulations offer the most complete time dependent treatments of the associated shock microphysics @xcite , but are far too expensive to follow the shock evolution over the time , length and energy scales needed to model astrophysical cr acceleration . the most powerful alternative approach utilizes continuum methods , with a kinetic equation for each cr component combined with suitably modified compressible fluid dynamical equations for the bulk plasma ( see 2 below ) . by extending that equation set to include relevant wave action equations for the wave turbulence that mediates cr transport , a self - consistent , closed system of equations is possible ( e.g. , @xcite ) ) . continuum dsa simulations of the kind just described are still quite challenging and expensive even with only one spatial dimension . the numerical difficulty derives especially from the very large range of cr momenta that must be followed , which usually extends to hundreds of gev / c or beyond on the upper end and down to values close to those of the bulk thermal population , with nonrelativistic momenta . the latter are needed in part to account for `` injection '' of crs due to incomplete thermalization that is characteristic of collisionless shocks . one computational constraint comes from the fact that cr resonant scattering lengths from mhd turbulence , @xmath1 , are generally expected to be increasing functions of particle rigidity , @xmath2 . the characteristic length coupling the crs of a given momentum , @xmath3 , to the bulk flow and defining the width of the modified shock precursor is the so - called diffusion length , @xmath4 , where @xmath5 is the cr particle speed , and @xmath6 is the bulk flow speed into the shock . one must spatially resolve the modified shock transition for the entire range of @xmath7 in order to capture the physics of the shock formation and the spatial diffusion of the crs , in particular . the relevant @xmath7 typically spans several orders of magnitude , beginning close to the dissipation length of the thermal plasma , which defines the thickness of the classical , `` viscous '' gas shock , also called the `` subshock '' in modified structure . this resolution requirement generally leads to very fine spatial grids in comparison to the `` outer scale '' of the problem , which must exceed the largest @xmath7 . two approaches have been applied successfully so far to manage this constraint in dsa simulations . berezhko and collaborators @xcite developed a method that normalizes the spatial variable by @xmath7 at each momentum value of interest during solution of the cr kinetic equation . this approach establishes an spatial grid that varies economically in tune with @xmath7 . derived cr distribution properties at different momenta can be combined to estimate feed - back on the bulk flow at appropriate scales . the method was designed for use with cr diffusion properties known _ a priori_. it is not readily applied to cr diffusion behaviors incorporating arbitrary , nonlinear feedback between the waves and the crs . as an alternative that can accommodate those latter diffusion properties , kang @xcite have implemented diffusive cr transport into a multi - level adaptive mesh refinement ( amr ) environment . the benefit of amr in this context comes from the feature that the highest resolutions are only necessary very close to the subshock , which can still be treated as a discontinuity satisfying standard rankine - hugoniot relations . by efficient use of spatial gridding both of these computational strategies can greatly reduce the cost of time dependent dsa simulations . on the other hand , the above methods do not directly address the principal computational cost in such simulations , so they remain much more costly compared to purely hydrodynamic or mhd simulations . this is because the dependence of @xmath8 on cr momentum , @xmath3 , adds a physical dimension to the problem . in practice , the spatial evolution of the kinetic equation for each cr constituent must be updated over the entire spatial grid at multiple momentum values ; say , @xmath9 . the value of @xmath9 is usually large , since the spanned range of cr momentum is typically several orders of magnitude . physically , crs propagate in momentum space during dsa in response to adiabatic compression in the bulk flow , sometimes by momentum diffusion ( see , for example , equation [ dce ] below ) , or because of various irreversible energy loss mechanisms , such as coulomb or radiative losses . the associated evolution rates for @xmath8 depend on the process , but generally depend on @xmath10 . the conventional approach to evolving @xmath8 approximates @xmath10 through low order finite differences in @xmath11 ( e.g. , @xcite ) . experience has shown that converged solutions of @xmath12 using such methods require @xmath13 @xcite . in that case , for example , a mere five decades of momentum coverage requires more than 100 grid points in @xmath14 . since spatial update of the kinetic equation at each momentum grid point requires computational effort comparable to that for any of the accompanying hydrodynamical equations ( e.g. , the mass continuity equation ) , cr transport then dominates the computational effort by a very large factor , commonly exceeding an order of magnitude . an attractive alternative approach to evolving the kinetic equation replaces @xmath8 by its integral moments over a discrete set of finite momentum volumes , in which case @xmath10 is replaced by @xmath8 evaluated at the boundaries of those volumes . the method we outline here follows that strategy . because @xmath15 is relatively smooth , simple subvolume models can effectively be applied over moderately large momentum volumes . we have found this method to give accurate solutions to the evolution of @xmath8 with an order of magnitude fewer momentum bins than needed in our previous finite difference calculations . the computational effort to evolve the cr population is thereby reduced to a level comparable to that for the hydrodynamics . in recognition of its distinctive features we refer to the method as `` coarse - grained momentum finite volume '' or `` cgmv '' . it extends related ideas introduced in @xcite , @xcite and @xcite for test particle cr transport . those previous presentations , while satisfactorily following cr transport in many large - scale , smooth flows , did not include spatial or momentum diffusion , so could not explicitly follow evolution of @xmath8 during dsa . instead , analytic , test - particle solutions for @xmath12 were applied at shock jumps . here we extend the cgmv method so that it can be applied to the treatment of fully nonlinear cr modified shocks . we outline the basic cgmv method and its implementation in eulerian hydrodynamics codes in 2 . several tests are discussed in 3 , and our conclusions are presented in 4 . the standard diffusion - convection form of the kinetic equation describing the evolution of the isotropic cr distribution function , @xmath16 , can be written in one spatial dimension as ( e.g. , @xcite ) . @xmath17 where @xmath18 is the bulk flow speed , @xmath19 , @xmath20 is the spatial diffusion coefficient , @xmath21 is the momentum diffusion coefficient , and @xmath22 is a representative source term . we henceforth express particle momentum in units of @xmath23 , where @xmath24 is the particle mass . the first rhs term in equation ( [ dce ] ) represents `` momentum advection '' in response to adiabatic compression or expansion . for simplicity of presentation equation [ dce ] neglects for now propagation of the scattering turbulence with respect to the bulk plasma , which can be a significant influence when the sonic and alfvnic mach numbers of the flow are comparable . although it is numerically straightforward to include this effect , the details are somewhat complex , so we defer that to a follow - up work focussed on cr transport in mhd shocks . full solution of the problem at hand requires simultaneous evolution of the hydrodynamical flow , as well as the diffusion coefficients , @xmath20 and @xmath21 . again postponing full mhd , the added equations to be solved are the standard gasdynamic equations with cr pressure included . expressed in conservative , eulerian formulation for one dimensional plane - parallel geometry , they are @xmath25 @xmath26 @xmath27 where @xmath28 and @xmath29 are the isotropic gas and the cr pressure , respectively , @xmath30 is the total energy density of the gas per unit mass and the rest of the variables have their usual meanings . the injection energy loss term , @xmath31 , accounts for the energy of the suprathermal particles transferred at low energy to the crs . as usual , cr inertia is neglected in such computations , since the mass fraction of the crs is generally tiny . we note for completeness that @xmath32 can be computed from @xmath8 using the expression @xmath33 in the simulations described below we set the particle mass , @xmath34 , for convenience . as mentioned in [ intro ] , the momentum advection and diffusion terms in equation [ dce ] typically require @xmath35 when using low order finite difference methods in the momentum coordinate @xcite . the resulting large number of grid points in @xmath36 makes finding the solution of equation [ dce ] the dominate effort in simulations of dsa . on the other hand , previous studies of dsa as well as direct observations of crs in different environments have shown that @xmath12 is commonly well described by the form @xmath37 , where @xmath38 , is a slowly varying function of @xmath36 . thus , we may expect a piecewise powerlaw form to provide an efficient and accurate , two - parameter subgrid model for @xmath39 . two moments of @xmath12 are sufficient to recover the subgrid model parameters . we find it convenient to use @xmath40 and @xmath41 the first of these moments , @xmath42 , is proportional to the spatial number density of crs in the momentum bin @xmath43 $ ] , while for relativistic crs , @xmath44 is proportional to the energy density or pressure contribution of crs in the bin . then , for example , @xmath45 , \label{niq}\ ] ] where @xmath46 , and @xmath47 with obvious extension to @xmath44 . either of these moments , plus their ratio , @xmath48 , can be used in straightforward fashion ( e.g. , iteration ) to find both @xmath49 and the intrabin index , @xmath50 . to evolve @xmath42 and @xmath44 we need the associated moments of equation [ dce ] over the finite momentum volume bounded by @xmath51 . the result for @xmath42 is @xmath52 where @xmath53 is a flux in momentum space , with @xmath54 , and where @xmath55 and @xmath56 are averaged over the momentum interval , according to @xmath57 and @xmath58 extension of the momentum flux term @xmath59 to include other processes such as radiative or coulomb losses is obvious ( e.g. , @xcite ) . in practice these fluxes should be upwinded according to the signs of @xmath60 and @xmath61 . evaluation of @xmath59 and @xmath62 at the boundaries of the included momentum range requires application of suitable boundary conditions , of course . we usually have set @xmath63 . in most cases we pick a sufficiently large maximum momentum , @xmath64 , that this condition is important only late in the simulation , if at all . appropriate conditions at the lowest momenta can be more involved , depending on how one intends to connect the cr particle distribution to the thermal particle distribution , as in the injection models discussed below . the @xmath44-associated moment of equation [ dce ] is @xmath65 where @xmath66 , @xmath67 @xmath68 , and @xmath69 is given by an analogous expression to equation [ sni ] . we note that momentum binning in the cgmv scheme is quite flexible , so that it can be easily adapted to either uniform or nonuniform momentum bin sizes , or to a momentum range that evolves during the simulation . we have successfully implemented both nonuniform and evolving momentum bin structures , although , for brevity we do not illustrate them here . the coefficients in equation [ cne ] are obtained with the aid of the solutions to equations [ masscon ] - [ encon ] , which are updated prior to solution of equations [ dce - n ] and [ dce - g ] . we note again that similar methods can be applied to follow the evolution of the wave turbulence that resonantly scatters crs and that defines the spatial and momentum diffusion coefficients . in that case one begins with the wave action equation for the appropriate waves rather than the particle kinetic equation ( e.g. , @xcite ) . the solution of equation ( [ cne ] ) for either @xmath42 or @xmath44 is quite analogous to our previous fd methods . thus , since the cgmv method evolves two quantities rather than one , the relative effort required for a given @xmath9 is roughly twice in the cgmv scheme that required in the fd scheme . our tests confirm this expected scaling . on the other hand , since @xmath9 can be dramatically reduced in a cgmv simulation the method can still be more efficient by a large factor . the most common source term represented by @xmath22 in equation ( [ dce ] ) is injection at the shock of low energy crs from the thermal plasma . there is presently no generally accepted theory for that process . however , we have implemented two commonly used models successfully into the cgmv scheme . for completeness we outline those here . the simplest and one of the most frequently applied injection models assumes that a small , fixed fraction of the thermal particle flux through the gas subshock , @xmath77 , is injected at a momentum @xmath78 , where @xmath79 is a constant greater than unity and @xmath80 is the plasma sound speed immediately downstream of the subshock ( e.g. , @xcite ) . this gives @xmath81 , where @xmath82 is the plasma mass density just upstream of the subshock , @xmath83 is the plasma mean particle mass , @xmath6 is the subshock speed with respect to the plasma immediately upstream and @xmath84 is a normalized weight function that allows the injection to be distributed across the numerical shock structure . in this case @xmath85 , while @xmath86 in the momentum bin with @xmath87 . both @xmath88 and @xmath69 are zero , otherwise . the energy extracted from the thermal plasma is simply @xmath89 . for convenience we call this injection model the `` flux fraction '' or `` ff '' model . a more sophisticated approach to injection physics includes models of the physical processes moderating particle orbits in the post shock flow region in order to estimate the probability that particles of a given speed will be able to escape back upstream , across the subshock . in such `` thermal leakage '' ( tl ) models for cr injection at shocks , most of the downstream thermal protons are locally confined by nonlinear mhd waves and only particles well into the tail of the postshock maxwellian distribution can leak upstream across the subshock . in particular , `` leaking '' particles not only must have velocities large enough to swim against the downstream flow in order to return across the shock , they must also avoid being scattered during that passage by the mhd waves that mediate the plasma subshock . to model tl injection we utilize a `` transparency function '' , @xmath90 , expressing the probability that supra - thermal particles at a given velocity can leak upstream from behind the subshock ( see @xcite for details ) . in particular we set @xmath91 { ( 1-\frac{1}{\tilde{\upsilon } } ) \over ( 1-\frac{u_2}{\upsilon } ) } \exp\left\ { -\left[\tilde{\upsilon}-(1+\epsilon_b)\right]^{-2}\right\ } , \label{tau}\ ] ] where @xmath92 is the postshock flow speed in the subshock frame , @xmath93 is the heaviside step function , and the particle velocity is normalized to @xmath94 . the parameter , @xmath95 , measures the ratio of the amplitude of the postshock mhd wave turbulence @xmath96 to the general magnetic field aligned with the shock normal , @xmath97 . both hybrid simulations and theory suggest that @xmath98 @xcite , so that the model is well constrained . with this @xmath99 the shock is completely `` opaque '' to particles with momenta less than @xmath100 , _ @xmath101 for @xmath102 , where @xmath103 . so @xmath104 is the lowest momentum of the first momentum bin in the tl model and changes in time with the postshock flow speed . for @xmath105 , @xmath106 . the shock becomes virtually transparent to particles with momenta two to three times greater than @xmath100 . for strong , unmodified shocks @xmath100 in the tl model and @xmath107 in the ff model are similar when @xmath105 and @xmath108 . under those circumstances the initial injection rates will be roughly similar , although differences in model physics lead to different behaviors as such shocks become modified ( see , e.g. , @xcite , @xcite , @xcite ) . the tl model is implemented in the cgmv scheme by the following numerical approach . after solution of equations ( [ dce - n ] ) and ( [ dce - g ] ) the net changes in @xmath42 and @xmath44 are corrected ( reduced ) in the upstream region by application of the transparency function as follows : @xmath109 and @xmath110 where @xmath111 , found using equation ( [ niq ] ) , is the cr distribution updated with equations ( [ dce - n ] ) and ( [ dce - g ] ) . the energy loss rate of the bulk plasma to injection into the @xmath112-th cr momentum bin can be approximated by @xmath113 ( see @xcite ) . with the piece - wise power - law subgrid model ( @xmath114 ) the integrals in equations ( [ leakn])-([leakl ] ) can be written : @xmath115 @xmath116 and @xmath117 in practice , this leakage step is significant only for the lowest few momentum bins , so that this correction need not be applied to all bins . in order to test the performance of the new cgmv scheme we have installed it into two distinct one - dimensional eulerian hydrodynamic ( hd ) codes that we have previously applied to studies of cr - modified shocks using conventional finite difference ( fd ) methods to solve the diffusion convection equation . in this section we briefly discuss the results and compare the cgmv and fd behaviors . both of the host hd codes are constructed from high order , conservative riemann solver - based schemes designed to capture shocks sharply . first we describe results from the cgmv scheme installed in a second order `` total variation diminished '' ( tvd ) hd code based on the finite difference scheme of harten @xcite . this is the hd version of the mhd - cr code used by us in a previous study of cr modified shocks @xcite . gas subshocks in the tvd scheme typically spread over 2 - 3 numerical zones . an outline of the code mechanics and the fd cr scheme can be found there , in @xcite and references cited in those papers . the fd solver employed a crank - nicholson routine originally introduced in @xcite for evolving @xmath118 that is similar to equation ( [ cne ] ) . for the tvd tests we applied the ff injection model . in addition we present cgmv tests carried out with our cosmic ray adaptive refinement shock ( crash ) code . crash is based on the high order godunov - like shock tracking algorithm of leveque @xcite . the hydrodynamics routine in that code employs a nonlinear riemann solver to follow shock discontinuities within the zones of an initially uniform grid . thus , gas subshocks in cr - modified shocks remain discontinuous throughout a simulation , allowing cr transport to be modeled down almost to the scale of the physical shock thickness . crash also employs adaptive mesh refinement ( amr ) around shocks in order to reduce the computational effort on the spatial grid . refinement is centered on the subshock and each level spans 100 zones with a resolution twice as fine as the level above it . the number of refinement levels depends on what is required to capture diffusion of the lowest energy crs . the standard , previously documented version of crash uses the same fd methods as the tvd code to solve the diffusion convection equation for cr transport . it is described in @xcite and @xcite . for the crash tests discussed in this paper we employed the tl injection model . we first examine some results obtained using the tvd - cr code with both fd and cgmv schemes used to model the evolution of a strong cr - modified shock . fig . 1 and fig . 2 illustrate the evolution of shocks formed by the reflection of a mach 30 flow ( adiabatic index , @xmath119 ) off the left grid boundary . the resulting piston - driven shock initially has a mach number , @xmath120 . the density and sound speed of plasma entering from the right boundary were set to @xmath121 and @xmath122 , respectively , so that the inflow speed was unity . the time unit for the calculations is also set by these scalings . in order to relate hydrodynamical variables to cr momenta it is necessary to fix the unit flow speed ( the inflow speed in all the simulations discussed in this paper ) with respect to the speed of light ; i.e. , @xmath123 . we set @xmath124 in the tvd - cr simulations . time steps were fixed by a standard courant condition , @xmath125 . for the simulation illustrated in fig . 1 evolution of the cr distribution is followed over the momentum range @xmath126 = [ 2\times 10^{-4 } , 1.6\times 10 ^ 3]$ ] ( @xmath127 ) . the simulation represented in fig . 2 included the momentum range @xmath126 = [ 2\times 10^{-4 } , 2.4\times 10 ^ 5]$ ] ( @xmath128 ) . the cr diffusion coefficient , @xmath20 is spatially uniform and set to @xmath129 . in a quest for a reasonably generalized behavior that required minimal computational effort , this choice was motivated by results from malkov , who found self - similar analytic steady - state solutions for strong cr - modified shocks that apply to all powerlaw forms of @xmath130 , so long as the powerlaw index is steeper that @xmath131 @xcite . thus , our @xmath20 choice leads to fairly general shock behaviors in a way that minimizes the width of the precursor . that width , which determines the minimum space that must be simulated ahead of the subshock , is set by @xmath132 , where @xmath133 represents the maximum momentum contained . the tvd - cr simulations utilize a uniform , fixed grid , so , for example , the bohm diffusion form modeled in the crash simulations below would lead to excessive costs for the tvd - cr fd test simulations presented in this section . the spatial resolution required for the calculations is set by @xmath134 , since accurate solutions of the diffusion - convection equation require good structural information in the diffusive shock precursor upstream of the subshock . previous convergence tests have shown that @xmath135 is desirable ( e.g. , @xcite ) . for the problems illustrated in fig . 1 and fig . 2 these considerations led us to set @xmath136 for both the fd and cgmv simulations . by varying this resolution , we verified that the shock evolution is reasonably converged with respect to @xmath137 . the simulation followed in fig . 1 assumes a pre - existing cr population , @xmath138 , corresponding to an upstream cr pressure , @xmath139 . no fresh injection is included at the shock ; i.e. , @xmath140 . this test then provides a simple and direct comparison between the cgmv and fd schemes for solving the diffusion - convection equation , since it omits any complications related to the injection model . this simulation pair evolves the shocks until @xmath141 , which is sufficient to accelerate crs to ultrarelativistic momenta . the spatial grid spans the interval [ 0,16 ] , which is sufficient to contain the leading edge of the cr precursor to the end of the simulation . the cr - modified shock spatial structures and the cr momentum distributions at the subshock are shown in fig . 1 at times @xmath142 . before comparing the solutions obtained with the two different methods , it is useful to summarize briefly the physics captured during the shock evolution . all the behaviors described here have been reported previously by multiple authors . the figure shows the well - known property of strongly modified shocks that the crs extract most of the energy flux into the structure . that leads to a substantial drop in the postshock gas pressure , @xmath143 , and a large increase in the postshock density , @xmath144 . together those indicate a strongly reduced postshock gas temperature . the decreased temperature is evident in the @xmath145 plot at the subshock , which is dominated at low momenta by the maxwellian distribution of the bulk plasma . as crs diffuse upstream against the inflowing gas they compress the flow within the precursor , preheating the gas ( adiabatically in these simulations ) . initially , while the cr pressure is relatively small compared to the incoming momentum flux , the gas subshock remains strong enough to produce a full four times density compression on top of the precompression . however , the subshock weakens once @xmath146 , reducing the subshock compression in this case to a factor @xmath147 , corresponding to a subshock mach number near 2.3 . that evolution explains the well - known , transitory `` density spike '' in the shock structures seen after @xmath148 . we note that since energy extracted from the flow by crs becomes increasingly spread upstream and downstream due to cr diffusion , the total compression in such an _ evolving _ modified shock would always exceed the factor of seven one would predict for a strong , fully relativistic gas shock . for this simulation no significant cr energy escapes the spatial grid through upstream diffusion . however , at late times ( @xmath149 ) the partial pressure due to crs just below @xmath150 is sufficient that escape across the upper momentum boundary is significant . this contributes to the slow decrease in @xmath32 behind the shock and the increasing total compression through the shock transition that is visible in fig . 1 . in the early evolutionary stages of this flow , while shock modification is modest , the cr momentum distribution resembles the powerlaw form , @xmath151 , predicted by test particle theory for a strong shock ( e.g. , @xcite ) . with the spatial diffusion coefficient used in this simulation the high momentum cutoff to the distribution increases with time approximately as @xmath152 ( see , e.g. , @xcite ) . as shock modification intensifies , most of the flow compression shifts from the subshock to the precursor . then dsa of high momentum crs occurs predominantly within the precursor rather than near the less important subshock . consequently the cr distribution develops the familiar upwards - concave form resulting from the momentum dependent cr diffusion length . crs of higher momentum experience a greater velocity jump within the precursor , so gain more energy each time they are reflected within the shock structure . that flattens the distribution , @xmath12 at momenta below @xmath133 . the result is a bump in the distribution of @xmath118 . on the other hand , crs with momenta only a little above the injection range remain trapped close to the subshock . their distribution closely approaches the steady state , powerlaw , test particle form appropriate to the weakened subshock . that feature extends upwards in momentum as the bump near @xmath133 moves upwards . looking finally to compare the two methods used to evolve the shock evolution displayed in fig . 1 we see results from the fd scheme with @xmath153 and the cgmv scheme with @xmath154 . the agreement is generally very good . all the dynamical quantities , including shock jumps and the cr momentum distributions show excellent agreement . the curves representing the fd and cgmv distributions of @xmath144 , @xmath143 and @xmath32 and virtually indistinguishable in the plots . most notably , all features formed in the fd evolution of the cr momentum distribution are faithfully reproduced by the much coarser cgmv distribution . given the excellent comparison in this strongly modified flow it is satisfying to note that the execution time required for the cgmv solution was a little less than 20% of that for the fd solution , demonstrating the significantly higher efficiency of the former method . the speed - up observed in our implementations of the two methods is roughly in accordance with what we would predict for a given reduction factor in the number of momentum values used , since the cgmv method requires one to evolve two distributions @xmath42 and @xmath44 for each momentum bin . we address convergence with respect to momentum resolution in our discussion of a second shock . fig.2 shows a mach 30 flow similar to that in fig . 1 . in this case ff injection is included with commonly assumed values , @xmath155 and @xmath156 ( see 2.3 ) , the upper momentum bound is increased to @xmath157 and the spatial grid extends farther from the piston to @xmath158 . a negligible pre - existing cr population is included to avoid numerical issues coming from the fact that our cgmv scheme requires computation of the ratio @xmath159 over the entire grid . the simulations evolved the shock until @xmath160 , which leads to @xmath161 . the spatial grid , spanning the interval [ 0,25 ] , is sufficient to contain the cr leading edge of the precursor almost to the end of the simulations . however , after @xmath162 some cr energy escapes through the right boundary , due to diffusion upstream , mimicking the behavior of a `` free escape '' boundary ( feb ) ( e.g. , @xcite and references therein ) . just as for the shock simulated in fig.1 , this energy loss amounts to a cooling process , so that the total shock compression increases with time as the simulation ends . again the agreement between fd and cgmv simulations is very good . very early in the evolution of the shock , when the crs are dominated by freshly injected nonrelativistic particles , the shock evolution is slightly faster in the cgmv scheme . that influence becomes insignificant later on , so that the modified shock structures found by the two schemes are almost identical as is the distribution @xmath118 at the shock . there is a small residual effect that the position of the subshock in the cgmv simulation lags slightly behind that of the fd simulation , and that @xmath32 is slightly higher in the cgmv simulation near the piston , where the shock first formed . since the efficiency of the cgmv method comes from its ability to cover the momentum range coarsely , it is important to evaluate how broad the momentum bins can be and still faithfully model the evolution of the shock . 3 illustrates convergence of the cgmv scheme with respect to momentum bin size , @xmath163 , at @xmath164 for the flow modeled in fig . 2 . the upper panel plots the spatial @xmath32 distributions , while the lower panel shows the particle momentum distributions at the gas subshock . for reference the corresponding fd solution ( @xmath153 ) is shown by the dotted red curves . solutions from the cgmv scheme are plotted for @xmath165 , @xmath166 and @xmath167 . even the coarsest of the cgmv solutions is in basic agreement with the other cgmv solutions and with the fd solution . fine details in the momentum distribution are naturally obscured as the cgmv bin size increases . the largest bins with @xmath168 span a decade in cr momentum ( @xmath169 ) , but still capture the basic dynamical properties of the crs correctly . however , the quality of the cgmv solutions deteriorates for still larger momentum bins in these experiments , once the simple subgrid model for the momentum distribution becomes inadequate . as illustrated in the lower panel of fig . 3 already momentum bins larger than roughly @xmath170 can not closely follow sharper structures in the momentum distibution that develops at the ends of the cr distribution . that enhances @xmath32 upstream of the subshock , where the flow is both cold and strongly compressed as it approaches the subshock . when the errors become excessive , for @xmath171 , in this case , @xmath32-induced overcompression in this region can cause the riemann solver in our tvd code to perform poorly or even to fail in high mach number flows . in general the largest allowed momentum bin size , @xmath172 should depend on the strongest curvature of the cr momentum distribution function as well as the degree of shock modification . for a third test example we illustrate in figs . 4 and 5 simulation results using the crash code with the tl injection model applied to a mach 10 flow reflecting off the left computational boundary . the initial gas shock mach number is approximately 13 . as for the previous test , the upstream gas density and flow speed are set to unity , with upstream sound speed , @xmath173 , and @xmath174 . the time unit is defined accordingly . cr momenta are tracked over the range @xmath175 $ ] , where @xmath104 is the smallest momentum that can leak upstream ( see equation 15 ) . in this case a bohm - type diffusion model with @xmath176 , is adopted and the tl injection parameter , @xmath177 is used . the crash test was significantly more computationally demanding than the tvd - cr tests . note first that in the crash simulation the value of @xmath178 , while @xmath179 in the previous examples shown in figs . 1 - 3 . consequently , @xmath180 is about seven times greater in the current case , and the nominal physical scale of the precursor and its formation timescale are similarly lengthened . in addition , the stronger momentum dependence of bohm diffusion coefficient means that the precursor width expands more strongly as @xmath133 increases . the associated time rate of increase in @xmath133 is , however , slower , so that the shock must evolve longer to reach a given @xmath133 . these factors substantially increase the size of the physical domain needed to reach a given @xmath133 . 4 shows the early evolution of this cr - modified shock for @xmath181 as computed with both the fd and the cgmv methods . the spatial domain for this simulation is [ 0,20 ] . the base spatial grid included @xmath182 zones , giving @xmath183 . since it is necessary to resolve structures near the subshock on scales of the diffusion length for freshly injected , suprathermal crs , the amr feature of the crash code is utilized . the fd simulation is carried out with 7 refined grid levels ; four levels of refined grid are applied in the cgmv simulation . 240 momentum points ( @xmath184)are used in the fd simulation , while the cgmv simulation includes 20 momentum bins ( @xmath185 ) . the time step for each refinement level , @xmath186 , is determined by a standard courant condition , that is , @xmath187 . although the crank - nicholson scheme is stable with an arbitrary time step , the diffusion convection equation is solved with the time step smaller than @xmath188 ) to maintain good accuracy in the momentum space advection ( _ i.e. , _ @xmath189 ) . with @xmath184 , the required time step is smaller by a factor of three or so than the hydrodynamic time step in the fd simulation . consequently , the fd diffusion convection solver is typically subcycled about 3 times with @xmath190 for each hydrodynamic time step . because of the much larger @xmath163 , subcycling is not necessary in the cgmv simulation . that adds another relative economy to the cgmv calculation . at the end of this simulation , @xmath191 , the modified shock structure is approaching a dynamical equilibrium in the sense that the postshock values of @xmath144 , @xmath143 and @xmath32 will not change much at later times . since this shock is weaker than the mach 40 shocks examined earlier modifications are more moderate . on the other hand , as expected from the stronger momentum dependence of @xmath7 , the shock precursor broadens much more quickly in the present case . the cutoff in the cr distribution has reached roughly @xmath192 by @xmath191 . longer term evolution of this shock will be addressed below . the agreement between the fd and cgmv solutions shown in fig . 4 is good , although not as close as it was in the examples illustrated in fig . 1 and fig . 2 . the more apparent distinctions between the two solutions in the present case come from effective differences in the application of the tl injection model with bohm diffusion in fd and cgmv methods . recall that the cgmv scheme applies the diffusion coefficients averaged across the momentum bins ( see equations [ kni ] , [ kgi ] ) . the bohm diffusion model has a very steep momentum dependence for nonrelativistic particles ; namely , @xmath193 . at low momenta where injection takes place the averaging increases the effective diffusion coefficient , and , thus , the leakage flux of suprathermal particles , leading to higher injection rate compared to the fd scheme for the same tl model parameters . consequently , the distribution function in the second bin at @xmath194 is slightly higher in the cgmv scheme , as evident in figs . note that @xmath195 is anchored on the tail of maxwellian distribution . the cgmv solutions accordingly show slightly more efficient cr acceleration than the fd solutions at early times . in this test @xmath32 is about 5 % greater in the cgmv simulation at @xmath191 . since the cgmv scheme can be implemented with nonuniform momentum bins , such differences could be reduced by making the momentum bins smaller at low momentum in instances where the details relating to the injection rate were important . we show in fig . 5 the evolution of this same shock extended to @xmath196 , as computed with the cgmv method . this simulation is computed on the domain [ 0,800 ] , spanned by a base spatial grid of @xmath197 zones , giving @xmath198 . we also included 7 refined grid levels at the subshock , giving @xmath199 . this grid spacing is insufficient for convergence at the injection momentum , @xmath200 , so that the very early evolution is somewhat slower than in the simulations shown in fig . however , once shock modification becomes strong evolution becomes roughly self - similar , as pointed out previously @xcite . the time asymptotic states do not depend sensitively on the early injection history . the self - similar behavior results with bohm diffusion from a match between the upstream and downstream extensions of the cr population . one also sees from the form of the distribution function in fig . 5 that the postshock gas temperature has stabilized , while the previously - explained concave form to the cr distribution is better developed than it was at earlier times . this simulation illustrates nicely the relative efficiency of the cgmv scheme . the equivalent fd simulation would be very much more expensive , because this model requires a long execution time and a large spatial domain . with bohm diffusion @xmath201 for ultrarelativistic crs , so that the scale of the precursor , @xmath202 . at the same time the peak in the cr momentum distribution extends relatively slowly , with @xmath203 . the required spatial grid is , thus , 40 times longer than for the shorter simulation illustrated in fig . the simulated time interval in the extended simulation was 50 times longer . together those increase the total computational time by a factor 2000 . the fd calculation with @xmath204 $ ] to @xmath205 took about 2 cpu days on our fastest available processor , so the extended simulation would have been unrealistic using the fd method . the extended cgmv simulation , however , required only about 10 times the effort of the shorter fd simulation , clearly demonstrating the efficiency of the cgmv scheme . this speed - up is a result of combination of several factors : 20 times larger grid spacing , no need for subcycling for the diffusion convection solver , and , of course , a smaller number of momentum bins . detailed time dependent simulations of nonlinear cr shock evolution are very expensive if one allows for inclusion of arbitrary , self - consistent and possibly time dependent spatial diffusion , as well as various other momentum dependent transport processes . the principal computational cost in such calculations is typically the cr transport itself , and , in self - consistent calculations , the analogous transport of the mhd wave turbulence that mediates cr transport . tracking these behaviors requires adding at least one physical dimension to the simulations compared to the associated hydrodynamical calculations , since the collisionless media involved are sensitive to the phase space configurations of the particles and waves . particle kinetic equations ( commonly the so - called diffusion convection equation ) provide a straightforward approach to addressing this problem and can be coupled conveniently with hydrodynamical equations that track mass and bulk momentum and energy effectively . momentum derivatives of the cr distribution function in the diffusion convection equation are most frequently handled by finite differences . although it is simple , that approach requires moderately fine resolution in momentum space . that is a primary reason that such calculations are costly . here we introduce a new scheme to solve the diffusion convection equation based on finite volumes in momentum space with a momentum bin spacing as much as an order of magnitude larger than that of the usual finite difference scheme . we demonstrate that this coarse grained momentum finite volume ( cgmv ) method can be used successfully to model the evolution of strong , cr - modified shocks at much lower computational cost than the finite difference approach . the computation efficiency is greatly increased , not only because the number of momentum bins is smaller , but also because the required spatial grid spacing is less demanding due to the coarse - grained averaging of the diffusion coefficient used in the cgmv method . in addition , larger momentum bin size can eliminate the need of subcycling of the diffusion convection solver that can be necessary in some instances using finite differences in momentum . thus , the combination of the cgmv scheme with amr techniques as developed in our crash code , for example , should allow more detailed modeling of the diffusive shock acceleration process with a strongly momentum dependent diffusion model such as bohm diffusion , or self - consistent treatments of cr diffusion and wave turbulence transport . twj is supported by nsf grant ast03 - 07600 , by nasa grants nag5 - 10774 , nng05gf57 g and by the university of minnesota supercomputing institute . hk was supported by kosef through the astrophysical research center for the structure and evolution of cosmos ( arcsec ) . . red lines and stars were obtained using the new cgmv scheme with @xmath154 . the solutions are almost indistinguishable . a pre - existing cr population , @xmath138 , corresponding to the upstream cr pressure , @xmath139 is included , without fresh injection at the shock ( @xmath206 ) . ] at @xmath160 for the same shock as shown in fig 2 . the different curves represent results computed using the fd scheme and three different momentum resolutions with the cgmv scheme . bottom : the cr distribution function at the shock from the same simulations . ] and @xmath207 , respectively . the heavy dashed lines represent solution at @xmath205 with a conventional finite difference scheme using 240 momentum points ( @xmath184 ) . the red solid lines and x s represent cgmv solutions at @xmath208 and 1000 with 20 momentum bins ( @xmath185 ) . ]

we have developed a new , very efficient numerical scheme to solve the cr diffusion convection equation that can be applied to the study of the nonlinear time evolution of cr modified shocks for arbitrary spatial diffusion properties . the efficiency of the scheme derives from its use of coarse - grained finite momentum volumes . this approach has enabled us , using @xmath0 momentum bins spanning nine orders of magnitude in momentum , to carry out simulations that agree well with results from simulations of modified shocks carried out with our conventional finite difference scheme requiring more than an order of magnitude more momentum points . the coarse - grained , cgmv scheme reduces execution times by a factor approximately half the ratio of momentum bins used in the two methods . depending on the momentum dependence of the diffusion , additional economies in required spatial and time resolution can be utilized in the cgmv scheme , as well . these allow a computational speed - up of at least an order of magnitude in some cases . ,

astro-ph0506212

in recent years , influences of small - size and surface effects on the magnetic properties of magnetic nanoparticles have provided a conspicuous and productive field for the interaction between theoretical works @xcite and technological @xcite , as well as biomedical applications @xcite . as the physical size of a magnetic system reduces to a characteristic length , surface effects become dominant on the system , hence , some unusual and interesting magnetic phenomena can be observed , which may differ from those of bulk materials @xcite . recent developments in the experimental techniques allow the scientists to fabricate such kinds of fine nanoscaled materials @xcite , and the magnetization of certain nanomaterials such as @xmath3-@xmath4 nanoparticles has been experimentally measured @xcite . in particular , magnetic nanowires and nanotubes such as @xmath5 @xcite , @xmath6 and @xmath7 @xcite can be synthesized by various experimental techniques and they have many applications in nanotechnology @xcite . from the theoretical point of view , many studies have been performed regarding the magnetic properties of nanoparticles , such as nanowire , nanotube , and nanorod systems , and theoretical works can be classified in two basic categories based on the investigation of equilibrium or nonequilibrium phase transition properties of such nanoscaled magnetic structures . namely , in the former group , equilibrium properties of these systems have been investigated by a variety of techniques such as mean field theory ( mft ) @xcite , effective - field theory ( eft ) @xcite , green functions formalism @xcite , variational cumulant expansion ( vce ) @xcite and monte carlo ( mc ) simulations @xcite . based on mc simulations , particular attention has been paid on the exchange bias ( eb ) effect in magnetic core - shell nanoparticles where the hysteresis loop exhibits a shift below the nel temperature of the antiferromagnetic shell due to the exchange coupling on the interface region of ferromagnetic core and antiferromagnetic shell . the readers may refer to @xcite for a rigorous review about the eb phenomena . it is a well known fact that physical properties of a bulk material are independent from size ; however , below a critical size , nanoparticles often exhibit size - dependent properties , and some unique phenomena have been reported , such as superparamagnetism @xcite , quantum tunneling of the magnetization @xcite , and unusual large coercivities @xcite . as an example , it has been experimentally shown that @xmath8 ( lcmn ) nanoparticle exhibits a negative core - shell coupling , although the bulk lcmn is a ferromagnet @xcite . moreover , as a theoretical example , the total magnetizations in a nanoscaled transverse ising thin film with thickness @xmath9 are investigated by the use of both the eft with correlations and mft , and it has been shown that the magnetization may exhibit two compensation points with the increasing film thickness @xcite . the phenomena of two compensation points observed in the nanoscaled thin films has also been reported for bulk ferrimagnetic materials @xcite . however , the origin of the existence of such a phenomenon in the nanoscaled magnets is quite different from those observed in the bulk ferrimagnetic materials . namely , a compensation point originates in the bulk systems due to the different temperature dependence of the atomic moments of the sublattices @xcite . however , nanoscaled magnetic particles such as nanowires or nanotubes exhibit a compensation point , due to the presence of an antiferromagnetic interface coupling between the core and the shell , even if the lattice sites in the particle core and shell are occupied by identical atomic moments . hence , theoretical investigation of ferrimagnetism in nanoparticle systems has opened a new field in the research of the critical phenomena in nanoscaled magnetic particles @xcite . according to recent mc studies @xcite , it has been shown that the core - shell morphology can be successfully applied in equilibrium properties of nanoparticles formed by more than one compound ( i.e. ferrimagnetic nanostructures ) since the concept is capable of explaining various characteristic behaviors observed in nanoparticle magnetism . namely , we learned from these works that compensation point fairly depends on the particle size . therefore , nanoscaled magnets such as nanowires , nanotubes , etc . are currently considered as promising candidates due to their potential utilization as ultra - high density recording media . on the other hand , a magnetic system exhibits nonequilibrium phase transition properties in the presence of a driving magnetic field . namely , when a magnetic material is subject to a periodically varying time dependent magnetic field , the system may not respond to the external magnetic field instantaneously which causes interesting behaviors due to the competing time scales of the relaxation behavior of the system and periodic external magnetic field . at high temperatures and for the high amplitudes of the periodic magnetic field , the system is able to follow the external field with some delay while this is not the case for low temperatures and small magnetic field amplitudes . this spontaneous symmetry breaking indicates the presence of a dynamic phase transition ( dpt ) @xcite which shows itself in the dynamic order parameter ( dop ) which is defined as the time average of the magnetization over a full period of the oscillating field . related to this nonequilibrium phenomena , in recent years , based on glauber type of stochastic dynamics @xcite , a few theoretical studies have been devoted to the investigation of dynamical aspects of phase transition properties of cylindrical ising nanowire and nanotube systems in the presence of a time - dependent magnetic field within the eft with correlations @xcite . in those studies , the authors analyzed the temperature dependencies of the dynamic magnetization , hysteresis loop area and dynamic correlation between time dependent magnetization and magnetic field , and it has been reported that dynamic magnetization curves can be classified into well known categories , according to nel theory of ferrimagnetism @xcite . furthermore , based on mc simulations and by using unixally anisotropic heisenberg model , frequency dispersion of dynamic hysteresis in a core - shell magnetic nanoparticle system has been studied by wu et al . @xcite , in order to determine whether the dynamic hysteresis loops obey the power - law scaling or not , and they concluded that the frequency dispersion of the dynamic hysteresis shows both the spin - reversal and spin - tilting resonances , and also they found that the exchange coupling on the core - shell interface has no effect on the power - law scaling of the dynamic hysteresis dispersion . as can be seen in the previously published works mentioned above , equilibrium phase transition properties of nanoparticle systems have been almost completely understood , whereas nonequilibrium counterparts needs particular attention and the following questions need to be answered : ( i ) what is the effect of the amplitude and frequency of the oscillating magnetic field on the dynamic phase transition properties ( i.e. critical and compensation temperatures ) of the nanoparticle systems ? ( ii ) what kind of physical relationships exist between the magnetic properties ( compensation point and coercivity ) of the particle and the system size ? main motivation of the present paper is to attempt to clarify the physical facts underlying these questions . particular emphasis has also been devoted for the treatment of highly nonequilibrium situation where the applied field amplitudes are of the same order as ( or greater than ) the exchange interactions , and oscillation periods with tens of time steps ( in terms of mc steps per site ) . the physical realization of this situation can be probably achieved by applying ultrafast laser fields @xcite or by using some certain novel materials with low exchange interaction , in comparison with external field strength . outline of the paper is as follows : in section [ formulation ] we briefly present our model . the results and discussions are presented in section [ results ] , and finally section [ conclude ] contains our conclusions . we consider a cubic ferrimagnetic nanoparticle composed of a spin-3/2 ferromagnetic core which is surrounded by a spin-1 ferromagnetic shell layer . at the interface , we define an antiferromagnetic interaction between core and shell spins ( see figure 1 in ref . construction of such kind of model allows us to simulate a ferrimagnetic small particle formed by more than one compound . the particle is subjected to a periodically oscillating magnetic field . the time dependent hamiltonian describing our model of magnetic system can be written as @xmath10 where @xmath11 and @xmath12 are spin variables in the core and shell sublattices . @xmath13 , @xmath14 and @xmath15 define antiferromagnetic interface and ferromagnetic core and shell exchange interactions , respectively . @xmath16 represents the oscillating magnetic field , where @xmath17 and @xmath18 are the amplitude and the angular frequency of the applied field , respectively . period of the oscillating magnetic field is given by @xmath19 . @xmath20 denotes the nearest neighbor interactions on the lattice . we fixed the value of @xmath15 to unity throughout the simulations , and we also use normalized the exchange interactions with @xmath15 . accordingly , amplitude of the oscillating magnetic field has been normalized as @xmath21 in the calculations . in order to simulate the system , we employ metropolis mc simulation algorithm @xcite to equation ( [ eq1 ] ) on an @xmath22 simple - cubic lattice with free boundary conditions ( fbc ) which is an appropriate choice for such a finite small system . configurations were generated by selecting the sites in sequence through the lattice and making single - spin - flip attempts , which were accepted or rejected according to the metropolis algorithm , and @xmath23 sites are visited at each time step ( a time step is defined as a mc step per site or simply mcs ) . the frequency @xmath24 of the oscillating magnetic field is defined in terms of mcs in such a way that @xmath25 where @xmath26 is the number of mcss necessary for one complete cycle of the oscillating field and @xmath27 is the time interval . in our simulations , we choose @xmath28 , hence we get @xmath29 . data were generated over @xmath30 independent sample realizations by running most of the simulations for @xmath31 monte carlo steps per site after discarding the first @xmath32 steps . this amount of transient steps is found to be sufficient for thermalization for almost the whole range of the parameter sets . however , for evaluating the hysteresis loops , in order to guarantee to obtain stable loops , the first few cycles of the external field are considered as transient regime , and after this transient regime , statistical averaging has been performed ( see section [ results2 ] ) . error bars were calculated by using the jackknife method @xcite . our program calculates the instantaneous values of the core and shell layer magnetizations @xmath33 and @xmath34 , and the total magnetization @xmath35 at the time @xmath36 . these quantities are defined as @xmath37 where @xmath38 and @xmath39 denote the number of spins in core and shell layers , respectively . from the instantaneous magnetizations , we obtain the dynamic order parameters as follows @xmath40 where @xmath41 , @xmath42 and @xmath43 denote the dynamic order parameters corresponding to the core and shell layers , and to the overall lattice ( @xmath43 is assumed to represent the time averaged total magnetization over a full cycle of the oscillating field ) , respectively . we also calculate the time average of the total energy of the particle including both cooperative and field parts over a full cycle of the magnetic field as follows @xcite @xmath44dt.\\\end{aligned}\ ] ] thus , the specific heat of the system is defined as @xmath45 where @xmath46 represents the temperature . to determine the dynamic compensation temperature @xmath0 from the computed magnetization data , the intersection point of the absolute values of the dynamic core and shell magnetizations were found using @xmath47 with @xmath48 , where @xmath49 is the critical temperature i.e. nel temperature . equations ( [ eq7 ] ) and ( [ eq8 ] ) indicate that the signs of the dynamic core and shell magnetizations are different , however , absolute values of them are equal to each other at the compensation point . hence , in order to characterize the compensation points , we also define two additional order parameters belonging to core and shell layers of the particle as follows : @xmath50 in this section , we will focus our attention on the dynamic phase transition properties of the ferrimagnetic nanoparticle system . this section is divided into three parts as follows : in section [ results1 ] , we have examined the dependence of the critical temperature @xmath49 of the particle on the amplitude and frequency of the oscillating magnetic field , as well as the exchange couplings defined in equation ( [ eq1 ] ) . in this section , we have also investigated the conditions for the occurrence of a compensation point @xmath0 in the system . hysteretic response of the particle to the periodically oscillating magnetic fields have been investigated in section [ results2 ] , and size dependence of the magnetic properties have been analyzed in section [ results3 ] . in order to make a comparison with the previously published works where the equilibrium properties of the present system were discussed , we select the number of core and shell spins as @xmath51 and @xmath52 ( the same values as in @xcite where @xmath53 is the linear dimension of the lattice ) in sections [ results1 ] and [ results2 ] . plane for @xmath54 with some selected values of the external field amplitude @xmath55 . the curves are plotted for three values of oscillation period : ( a ) @xmath56 , ( b ) @xmath57 , and ( c ) @xmath58.,title="fig:",width=302 ] plane for @xmath54 with some selected values of the external field amplitude @xmath55 . the curves are plotted for three values of oscillation period : ( a ) @xmath56 , ( b ) @xmath57 , and ( c ) @xmath58.,title="fig:",width=132 ] + in order to clarify the influence of antiferromagnetic interface coupling @xmath2 between core and shell layers on the dynamic phase transition properties of the system , we represent the phase diagrams in a @xmath59 plane with three oscillation period values @xmath60 which realizes ultra - fast switching external fields , and for some selected values of the field amplitude @xmath61 in figures [ fig1]a-[fig1]c . here , we consider a weak ferromagnetic interaction , such as @xmath54 for the core spins which simulates a surface exchange enhancement in the system . one of the common findings in these figures is that transition temperature @xmath49 values gradually increase as the strength of the antiferromagnetic interface coupling @xmath2 increases . this phenomenon is independent from amplitude @xmath61 and period @xmath62 of the oscillating magnetic field . at high oscillation period values ( i.e. at relatively low frequencies ) , dynamic magnetization @xmath35 corresponding to the instantaneous ferrimagnetic order parameter of the particle can respond to the oscillating magnetic field with some delay whereas as the period of the external magnetic field gets lower , a competition occurs between the period @xmath62 of the field and the relaxation time of the system , hence the dynamic magnetization can not respond to the external field due to the increasing phase lag between the field and the magnetization @xmath35 . as a result , this makes the occurrence of the dynamic phase transition difficult . in addition , for weak @xmath63 values , core and shell layers of the particle become independent of each other . as the strength of the antiferromagnetic interface interaction gets increased then it becomes dominant against the periodic local fields , and the particle exhibits a strong ferrimagnetic order . hence , a relatively large amount of thermal energy is needed to observe a dynamic phase transition in the system , due to the response of the spins to the external magnetic field . as the value of the field amplitude increases then the antiferromagnetic exchange interaction @xmath13 loses its dominance against the external field amplitude and it becomes possible to observe a dynamic phase transition at lower temperatures . consequently , ferrimagnetically ordered phase region in the phase diagrams shown in figures [ fig1]a-[fig1]c gets narrower with increasing @xmath61 and @xmath62 values . in figures [ fig2]a and [ fig2]b , we depict the effect of the antiferromagnetic interface coupling @xmath2 on the temperature dependencies of dynamic order parameters , corresponding to the phase diagrams shown in figure [ fig1]b . as an interesting observation , we can see from figure [ fig2 ] that although the ferromagnetic exchange coupling of the particle core is relatively weaker than that of the shell layer ( i.e. @xmath54 ) , both the core and shell layers undergo a dynamic phase transition at the same critical temperature which is a result of the relatively strong interface coupling @xmath13 . as seen in the magnetization curves shown in figure [ fig2]a , magnetization of the present nanoparticle system can exhibit similar features as observed in the bulk ferrimagnetic systems . in the bulk ferrimagnetism of nel @xcite , it is possible to classify the thermal variation of the total magnetization curves in some certain categories . according to this nomenclature , the system exhibits p - type behavior at which the magnetization shows a temperature - induced maximum with increasing temperature . at this point , we should note that this result conflicts with some recent works @xcite . in particular , in these studies the existence of at least one compensation point has been predicted for equilibrium properties of the system , i.e. for @xmath64 . however , the total magnetization of the system has been defined in such a way that the inequality of the number of spins in the core and shell layers has been ignored in the aforementioned works which completely effects the results in qualitative manner . the situation can be clarified by analyzing core and shell magnetizations of the particle . these results are given in figure [ fig2]b . as seen in this figure , core ( with 1331 spins ) and shell ( with 2044 spins ) magnetizations do not cancel each other , hence we can not observe any compensation point in the system for given set of system parameters . accordingly , we can conclude that in the presence of surface exchange enhancement ( such as @xmath54 ) and at high oscillation frequencies the system does not exhibit compensation phenomena . , @xmath65 , and @xmath66 for a combination of hamiltonian parameters corresponding to phase diagrams depicted in figure [ fig1].,title="fig:",width=302 ] + on the temperature dependencies of ( a ) total magnetization @xmath43 , ( b ) dynamic order parameters @xmath65 , @xmath66 , and ( c ) dynamic heat capacity @xmath67 for @xmath68 , @xmath54 , and @xmath56 with @xmath55.,title="fig:",width=302 ] on the temperature dependencies of ( a ) total magnetization @xmath43 , ( b ) dynamic order parameters @xmath65 , @xmath66 , and ( c ) dynamic heat capacity @xmath67 for @xmath68 , @xmath54 , and @xmath56 with @xmath55.,title="fig:",width=151 ] + next , in figure [ fig3 ] , we present the influence of the amplitude @xmath61 of the external field on the temperature dependencies of total magnetization @xmath43 , dynamic order parameters @xmath65 and @xmath66 , as well as the dynamic heat capacity curves of the particle , corresponding to the phase diagrams depicted in figure [ fig1]a . in figure [ fig3]a , total magnetization curves of the overall system are plotted . as seen in figure [ fig3]a , magnetization curves exhibit p - type behavior , and @xmath49 values decrease with increasing @xmath61 values . on the other hand , dynamic heat capacity curves which are depicted in figure [ fig3]c exhibit a hump at lower temperatures and a sharp peak at the transition temperature . schottky - like rounded humps observed in the heat capacity curves get sharper with increasing @xmath61 which is a result of a sudden change in the core magnetization . behavior of the dynamic specific heat curves corresponding to highly non - monotonous magnetization profiles can be better observed in figure [ fig4 ] . namely , when the magnetization curves exhibit conventional p - type behavior , dynamic specific heat curves exhibit a hump , and a sharp peak whereas if the magnetization shows p - type behavior with two separate maxima at @xmath69 then the specific heat curves exhibit two distinct schottky - like rounded humps , and a sharp transition peak . when the amplitude @xmath61 of the external field is sufficiently large , then the first hump observed at low temperatures gets sharper whereas the shape of the other one which is observed at higher temperatures does not change , since it originates from the thermal variation of the magnetization of the shell layer which does not exhibit a sudden variation as the temperature is varied . , dynamic order parameters @xmath65 , @xmath66 , and specific heat @xmath67 of the particle for @xmath54 , @xmath70 , and @xmath56 with ( a ) @xmath68 and ( b ) @xmath71 . , title="fig:",width=302 ] + in figure [ fig5 ] , we examine the effect of the external field period @xmath62 on the dynamical phase transition characteristics of the particle . phase diagrams in this figure are plotted for a value of the field amplitude @xmath72 with relatively high frequencies ( in comparison with number of mcss ) in which nonequilibrium properties of the system have been simulated under ultrahigh fields and ultra fast speeds . from figure [ fig5]a , one can clearly observe that @xmath49 values are depressed with increasing @xmath62 . the physical facts underlying the behaviors observed in figure [ fig5]a are identical to those emphasized in figure [ fig2 ] . therefore we will not discuss these interpretations here . however , as a complementary investigation , let us represent certain magnetization profiles corresponding to the phase diagrams given in figure [ fig5]a . for instance , as seen in figure [ fig5]b , when the antiferromagnetic exchange interaction strength is selected as @xmath71 , dynamic order parameters @xmath65 and @xmath66 of the core and shell layers never intersect each other , which causes the observation of p - type characteristics in total magnetization curves . we have also performed simulations for the case of @xmath68 , but we have not observed any significant difference in characteristic behavior of magnetization curves . hence , based on the results given in figures [ fig4 ] and [ fig5 ] , we see that the existence of strong antiferromagnetic exchange interaction ( such as @xmath71 ) is not sufficient for the occurrence of compensation phenomena in the system . plane for @xmath54 , @xmath70 , and @xmath60 . ( b ) the effect of the oscillation period @xmath62 on the temperature dependencies of dynamic order parameters @xmath65 and @xmath66 of core and shell layers of the particle for @xmath71 , @xmath73 , @xmath70 , and @xmath60 . , title="fig:",width=302 ] + as a final investigation of this subsection , we will discuss the influence of the ferromagnetic exchange interaction of the core layer @xmath1 on the magnetization profiles of the particle in figure [ fig6 ] . at first sight , according to figure [ fig6 ] , we can clearly claim that as the value of @xmath1 increases then the transition temperature of the system also increases . we can also mention that the phase transition temperature of the particle shell is directly related to the value of @xmath1 , since the antiferromagnetic interface interaction is relatively large as @xmath74 . furthermore , for @xmath75 , the magnetization of the particle exhibits p - type behavior , whereas for @xmath76 , a compensation temperature appears which decreases with increasing @xmath61 , and we find an n - type dependence at which the magnetization is characterized by a compensation point at which total magnetization @xmath43 reduces to zero due to the complete cancelation of the core and shell layer magnetizations . these observations are represented in figures [ fig6]a and [ fig6]b , respectively . on the temperature dependencies of @xmath43 , @xmath65 , and @xmath66 curves of the system for @xmath74 , @xmath56 , and @xmath77 with ( a ) @xmath75 , ( b ) @xmath76 . , title="fig:",width=302 ] + hysteresis behavior in magnetic systems originates in response to varying magnetic fields , and it is one of the most important features of real magnetic materials . in dynamic systems , the phenomenon occurs as a result of a dynamic phase lag between instantaneous magnetization and periodic external magnetic field . in contrast to the behavior observed in static models where the strength of the external field does not change with time explicitly , dynamic hysteresis in nonequilibrium phase transitions is characterized by a dynamic symmetry loss at high oscillation frequencies of the external field . the shape of a hysteresis loop is determined by the coercivity and remanent magnetization of the magnetic material . in particular , coercivity -which is defined as the required amount of the external magnetic field to reduce the magnetization of a material to zero- is an essential physical property of magnetic materials which has a significant importance in technological applications . moreover , it is worth to note that hysteresis loops of equilibrium systems exhibit coercivity in ferromagnetic phase @xcite whereas coercive fields in nonequilibrium systems driven by an oscillating field are always observed in the dynamic paramagnetic phase . for several values of @xmath61 with @xmath68 , and ( a ) @xmath54 , ( b ) @xmath78 , respectively.,title="fig:",width=302 ] + oscillation period and external field amplitude dependence of the coercivity of the particle and dynamic hysteresis loops are shown in figures [ fig7]-[fig9 ] which have been calculated at a temperature @xmath79 where @xmath80 is the transition temperature of the system in the absence of the external field . this choice of the temperature allows the system to undergo a purely mechanical phase transition ( i.e. magnetic field induced transition ) . in order to acquire a stationary behavior , the first 100 cycles of the external field has been discarded and the data were collected for 400 cycles . value for several values of oscillation period @xmath62 with @xmath68 , @xmath54 , and @xmath70 . the letters accompanying each curve denote the value of @xmath62 . , title="fig:",width=302 ] + from figure [ fig7]a , for weak ferromagnetic core coupling values such as @xmath54 , coercivity curves exhibit sudden variation with increasing @xmath62 whereas for sufficiently high @xmath62 they exhibit a stable profile . moreover , it is clear from figure [ fig7]a that at low oscillation periods , higher amplitude values mean large coercive fields , however at high @xmath62 values , coercivity becomes independent from @xmath61 . however , according to figure [ fig7]b , as the strength of ferromagnetic exchange interaction @xmath1 between the core spins increases , coercivity may reduce to zero at high @xmath62 values for relatively small @xmath61 values . some typical examples of the dynamic hysteresis loops obtained from the time evolution of instantaneous magnetization @xmath35 corresponding to figure [ fig7 ] ( where there exists a relatively weak interface interaction such as @xmath68 between core and shell layers ) are depicted in figure [ fig8 ] for some selected values of oscillation period @xmath62 with @xmath70 . from figure [ fig8 ] , we see that remanent magnetization values of the system do not change as @xmath62 varies . on the other hand , width of the loops becomes narrower but does not vanish for high @xmath62 values which is only possible for enhanced ferromagnetic core coupling values such as @xmath78 . where @xmath78 , and @xmath70 . effect of antiferromagnetic interface coupling is depicted in ( a ) for @xmath68 , and in ( b ) for @xmath71 . , title="fig:",width=302 ] + in the following analysis , let us investigate the effect of antiferromagnetic @xmath2 interactions which can not be predicted from the results presented in figure [ fig7 ] . by comparing figures [ fig9]a and [ fig9]b we observe that for relatively strong antiferromagnetic interface couplings such as @xmath71 the system may exhibit an interesting phenomenon . namely , as shown in figure [ fig9]b , the system exhibits triple hysteresis loops with an apparently wide middle loop for relatively strong @xmath2 values . the width of the middle loop becomes wider as the field period @xmath62 increases . we note that observation of such hysteresis behavior is possible at sufficiently low oscillation frequency values . more clearly , we have not observed this type of behavior in the system at oscillation periods @xmath81 with the parameters given in figure [ fig9]b . origin of this phenomena can be understood by analyzing the time series of instantaneous magnetizations . according to our numerical calculations , triple hysteresis loop behavior originates from the existence of a weak ferromagnetic core coupling @xmath1 , as well as a strong antiferromagnetic interface exchange interaction @xmath2 . triple loops disappear for @xmath82 . according to figure [ fig9]a where we consider a weak interface coupling @xmath83 ; at strong fields , such as for @xmath84 both the core and shell layers can be magnetized along the field direction . as the field amplitude reaches to the value @xmath70 within a half cycle , both core and shell magnetizations easily keep their alignment with the external magnetic field . on the other hand , when @xmath71 ( see figure [ fig9]b ) , shell magnetization can align in the oscillating field direction instantaneously , however core magnetization tends to align antiferromagnetically , due to the existence of a strong @xmath13 interaction and weak ferromagnetic core interaction @xmath85 . this results in a maximum phase lag between the magnetizations of the core and shell layers , and consequently we observe triple hysteresis loops . these types of hysteresis loops have also been observed recently in cylindrical ising nanowire systems @xcite , and in molecular based magnetic materials @xcite in the presence of static magnetic fields . however , triple hysteresis loops observed in the present system may be unphysical , since the interfacial interaction strength @xmath2 is usually supposed to be within the range of @xmath86 in the exchange bias systems @xcite . with some selected values of temperature . the system parameters have been kept fixed as @xmath87 , @xmath54 , @xmath56 , and @xmath88 . corresponding hysteresis curves of the total magnetization @xmath35 have been depicted for temperature values @xmath89 and @xmath90 in ( b ) and ( c ) , respectively.,title="fig:",width=283 ] with some selected values of temperature . the system parameters have been kept fixed as @xmath87 , @xmath54 , @xmath56 , and @xmath88 . corresponding hysteresis curves of the total magnetization @xmath35 have been depicted for temperature values @xmath89 and @xmath90 in ( b ) and ( c ) , respectively.,title="fig:",width=306 ] + as a final investigation , let us represent some size dependent properties of the particle for some selected values of hamiltonian and magnetic field parameters . in figure [ fig10 ] , we show the effect of the shell thickness @xmath91 on the coercivity and hysteresis curves corresponding to high frequency regime @xmath92 . it is clear from figure [ fig10]a that as the temperature increases then coercivity of the system increases and saturates at a certain value which depends on the temperature . moreover , the curve corresponding to @xmath93 ( black full squares ) exhibits an unusual behavior . namely , coercivity of the system loses its symmetric shape with increasing @xmath91 when @xmath93 which can be regarded as a signal of a dynamic phase transition . corresponding hysteresis curves have been depicted in figures [ fig10]b and [ fig10]c for @xmath93 and @xmath94 , respectively . as seen from figure [ fig10]b , hysteresis loop loses its symmetry and the particle exhibits a dynamic phase transition from paramagnetic to a dynamically ordered phase with increasing @xmath91 values . this phase transition completely originates from the magnetization of the shell layer of the particle , since the magnetization of particle core does not change its shape as @xmath91 varies . on the other hand , according to figure [ fig10]c , for @xmath94 the particle always remains in the paramagnetic phase as @xmath91 increases , since the temperature is large enough to keep the system in a dynamically disordered state . we can also observe from figure [ fig10]c that remanent magnetization values increase and loop areas get wider with increasing @xmath91 , and after a sufficiently large value of @xmath91 , the loop areas are not affected from changing @xmath91 values . we have also investigated the situation for low frequency regime @xmath95 in figure [ fig11 ] . at low frequency values , the magnetization of the particle is able to follow the external magnetic field instantaneously , but with a phase lag . dependence of coercivity on the shell thickness @xmath91 for @xmath96 can be seen in figure [ fig11]a . by comparing figures [ fig10]a and [ fig11]a , we can clearly observe that the system exhibits large coercivities for high frequency values . hysteresis curves corresponding to figure [ fig11]a are plotted in figures [ fig11]b and [ fig11]c for temperatures @xmath93 and @xmath97 , respectively . the most remarkable observation in these figures is that the phase difference between the magnetization of the particle and oscillating external magnetic field drastically reduces , hence coercivity and remanent magnetization values become zero in low frequency and high temperature regions , and consequently loop areas reduce to zero . with some selected values of temperature . the system parameters have been kept fixed as @xmath87 , @xmath54 , @xmath96 , and @xmath88 . corresponding hysteresis curves of the total magnetization @xmath35 have been depicted for temperature values @xmath89 and @xmath98 in ( b ) and ( c ) , respectively.,title="fig:",width=283 ] with some selected values of temperature . the system parameters have been kept fixed as @xmath87 , @xmath54 , @xmath96 , and @xmath88 . corresponding hysteresis curves of the total magnetization @xmath35 have been depicted for temperature values @xmath89 and @xmath98 in ( b ) and ( c ) , respectively.,title="fig:",width=313 ] + in figures [ fig12]a-[fig12]c , in order to depict the effect of the ferromagnetic shell layer thickness @xmath91 on @xmath0 and @xmath49 values , we plot the temperature dependencies of total magnetization curves of the particle for some selected values of @xmath91 . figures [ fig12]a and [ fig12]b show the representative p - type curves with some selected values of hamiltonian parameters and both for fairly strong , and relatively moderate antiferromagnetic interface couplings such as @xmath99 and @xmath100 , respectively . it is clear from figures [ fig12]a and [ fig12]b that although the system exhibit a ferrimagnetic order , we can not observe any compensation point . moreover , it can be easily observed in figures [ fig12]a and [ fig12]b that the temperature induced maxima of the total magnetization curves get higher with increasing @xmath91 , however the magnetization curves retain their shapes even if the particle size changes and the curves approach to bulk limit with increasing @xmath91 values . on the other hand , when the surface exchange enhancement is somewhat reduced , such as @xmath76 case depicted in fig . [ fig12]c , the total magnetization curves exhibit n - type behavior with a compensation point for thinner shell layers . as the shell thickness increases , then the total magnetization curves exhibit a shape transformation from n - type characteristics to ordinary q - type shape where we observe a monotonic decrease in the magnetization with increasing temperature . dependence of total magnetization @xmath43 curves with a variety of system parameters . ( a ) @xmath99 , @xmath54 , @xmath70 , and @xmath57 . ( b ) @xmath68 , @xmath54 , @xmath70 , and @xmath56 . ( c ) @xmath74 , @xmath76 , @xmath101 , and @xmath56.,title="fig:",width=302 ] dependence of total magnetization @xmath43 curves with a variety of system parameters . ( a ) @xmath99 , @xmath54 , @xmath70 , and @xmath57 . ( b ) @xmath68 , @xmath54 , @xmath70 , and @xmath56 . ( c ) @xmath74 , @xmath76 , @xmath101 , and @xmath56.,title="fig:",width=154 ] + in conclusion , by means of mc simulations , in order to clarify how the magnetism in a nanoparticle system is affected in the presence of a periodically oscillating external perturbation , we have analyzed nonequilibrium phase transition properties and stationary - state behavior of a single domain ferrimagnetic nanoparticle which is composed of a ferromagnetic core surrounded by a ferromagnetic shell layer . by considering an antiferromagnetic exchange interaction in the interfacial region we have also investigated some of the ferrimagnetic properties of the particle . the most conspicuous observations reported in the present paper can be briefly summarized as follows : * in section [ results1 ] , a complete picture of the phase diagrams and magnetization profiles have been presented . we have observed that in the presence of surface exchange enhancement ( such as @xmath54 ) and at high oscillation frequencies the system does not exhibit compensation phenomena . moreover , we have found that the existence of strong antiferromagnetic exchange interaction ( such as @xmath71 ) is not sufficient for the occurrence of compensation phenomena in the system . @xmath1 parameter is the decisive factor for the occurrence of a compensation point . in other words , there exists a critical @xmath1 value below which the system can not exhibit compensation point . compensation points have been found to reduce as external field amplitude increases . according to nel nomenclature , the magnetization curves of the particle are found to obey p - type , n - type and q - type classification schemes under certain conditions . the numerical values of dynamic magnetizations , especially the total magnetization @xmath43 of the system is defined in such a way that different number of spins in the core and shell layers of the particle are considered . in this context , recent calculations reported in the literature should be treated carefully , since the number of spins in the core and shell are generally not equal in the core - shell nanoparticle models . * section [ results2 ] has been devoted to investigation of hysteretic response of the particle . based on the simulation results , we have reported the existence of triple hysteresis loop behavior which originates from the existence of a weak ferromagnetic core coupling @xmath1 , as well as a strong antiferromagnetic interface exchange interaction @xmath2 . triple hysteresis loops disappear under the transformation @xmath102 . however , we claim that triple hysteresis loops observed in the present system may be unphysical , since the interfacial interaction strength @xmath2 is usually supposed to be within the range of @xmath86 in the exchange bias systems @xcite * size dependent properties of the particle have been clarified in section [ results3 ] and we found that the particle may exhibit a dynamic phase transition from paramagnetic to a dynamically ordered phase with increasing ferromagnetic shell thickness @xmath91 when the oscillation frequency is sufficiently high which simulates an highly nonequilibrium scenario in the presence of ultrafast fields . we have also found that as the shell thickness increases , then the total magnetization curves exhibit a shape transformation from n - type characteristics to ordinary q - type , hence compensation point may disappear with increasing @xmath91 . all of the observations outlined above show that the shape ( amplitude and frequency ) of the driving field and particle size have an important influence on the thermal and magnetic properties , such as coercivity , remanence and compensation temperature of the particle . in fact , we note that according to our simulation results , we have not found any evidence of the first order phase transitions . the reason is most likely due to the fact that , in contrast to the conventional techniques such as mft and eft , the method we used in the present work completely takes into account the thermal fluctuations in the present system which allows us to obtain non - artificial results . moreover , it is possible to improve the proposed model to simulate more realistic systems by considering a simulation of heisenberg type of hamiltonian with an assembly of interacting nanoparticles instead of a single particle . this may be the subject of a future work . on the other hand , we should also note that the conventional mc modeling of dynamical systems is capable of explaining the time dependent properties , such as dynamic order parameters in terms of an artificial timescale . recently , in order to relate the dynamic parameters to any real time scale , a nice theoretical attempt has been introduced as a dynamical approach using the landau - lifshitz - gilbert ( llg ) equation @xcite . the method which was propounded for the first time by nowak _ @xcite is simply based on the introduction of a time quantification factor which relates mc time step to physical time used in the llg equation . this method is widely regarded to be suitable for modeling short - time scale dynamics where the time step is only of the order of several picoseconds @xcite . hence , it could also be interesting to treat the problem presented in this study within the framework of a time - quantified monte carlo technique . the authors ( y.y . and e.v . ) would like to thank the scientific and technological research council of turkey ( tbitak ) for partial financial support . this work has been completed at dokuz eyll university , graduate school of natural and applied sciences , and the numerical calculations reported in this paper were performed at tbitak ulakbim , high performance and grid computing center ( tr - grid e - infrastructure ) . 99 kodama r h 1999 _ j. magn . mater . _ * 200 * 359 and references therein kim t y , yamazaki y and hirano t 2004 _ phys . status solidi b _ * 241 * 1601 pankhurst q a , connoly j , jones s k and dobson j 2003 _ j. phys . * 36 * r167 and references therein rivas j , boobre - lpez m , pieiro - redondo y , rivas b and lpez - quintela m a 2012 _ j. magn . mater . _ doi:10.1016/j.jmmm.2012.02.075 and references therein berkowitz a e , kodama r h , makhlouf s a , parker f t , spada f e , mcniff jr e j and foner s 1999 _ j. magn . mater . _ * 196 * 591 ruhrig m , khamsehpour b , kirk k j , chapman j n , aitchison p , mcvitie s and wilkinsons c d w 1996 _ ieee transactions on magnetics _ * 32 * 4452 schrefl t , fidler j , kirk k j and chapman j n 1997 _ j. magn . * 175 * 193 martnez b , obradors x , balcells l , rouanet a and monty c 1998 _ phys . rev . lett . _ * 80 * 181 fan z and lu j g 2006 _ int . j. high speed electron . and systems _ * 16 * 883 su y c , skomski r , sorge k d and sellmyer d j 2004 _ appl . lett . _ * 84 * 1525 skomski r 2003 _ j. phys . : condens * 15 * r841 schlrb h , haehnel v , khatri m s , srivastav a , kumar a , schultz l , and fhler s 2010 _ phys . status solidi b _ * 247 * 2364 leite v s and figueiredo w 2005 _ physica a _ * 350 * 379 kaneyoshi t 2009 _ j. magn . magn . mater . _ * 321 * 3430 kaneyoshi t 2011 _ phys . status solidi b _ * 248 * 250 kaneyoshi t 2005 _ phys . status solidi b _ * 242 * 2938 kaneyoshi t 2012 _ phase transitions _ * 85 * 264 wang c d , lu z z , yuan w x , kwok s y and teng b h 2011 _ phys . lett . a _ * 375 * 3405 keskin m , sarli n and deviren b 2011 _ solid state commmun . _ * 151 * 1025 bouhou s , essaoudi i , ainane a , saber m , dujardin f and de miguel j j 2012 _ j. magn . magn . mater . _ * 324 * 2434 garanin d a and kachkachi h 2003 _ phys . lett . _ * 90 * 65504 wang h , zhou y , wang e and lin d l 2001 _ chin . _ * 39 * 85 wang h , zhou y , lin d l and wang c 2002 _ phys . status solidi b _ * 232 * 254 iglesias , batlle x and labarta a 2005 _ phys . rev . b _ * 72 * 212401 hu y and du a 2007 _ j. appl . phys . _ * 102 * 113911 iglesias , battle x and labarta a 2008 _ j. nanosci . nanotechnol . _ * 2761 vasilakaki m and trohidou k n 2009 _ phys . rev . b _ * 79 * 144402 zaim a , kerouad m and el amraoui y 2009 _ j. magn mater . _ * 321 * 1077 zaim a and kerouad m 2010 _ physica a _ * 389 * 3435 jiang l , zhang j , chen z , feng q and huang z 2010 _ physica b _ * 405 * 420 yksel y , aydiner e and polat h 2011 _ j. magn . mater . _ * 323 * 3168 kittel c 1946 _ phys . rev . _ * 70 * 965 jacobs i s , bean c p 1963 _ in magnetism _ , edited by g. t. rado and h. suhl ( academic press , new york , 1963 ) , vol . 3 , chap . 6 . chudnovsky e m and gunther l 1998 _ phys . _ * 60 * 661 kneller e f and luborsky f e 1963 _ j. appl . phys . _ * 34 * 656 bhowmik r n 2011 _ j. magn . mater . _ * 323 * 311 bhowmik r n , poddar a , ranganathan r and mazumdar c 2009 _ j. appl . * 105 * 113909 bobk a and jaur m 1995 _ phys . rev . b _ * 51 * 11533 streka j and ekiz c 2012 physica a * 391 * 4763 buenda g m and machado e 2000 _ phys . b _ * 61 * 14686 tom t and de oliveira m j 1990 _ phys . rev . a _ * 41 * 4251 lo w s and pelcovits r a 1990 _ phys . a _ * 42 * 7471 chakrabarti b k and acharyya m 1999 _ rev . * 71 * 847 and the references therein glauber r j 1963 _ j. math . phys . _ * 4 * 294 deviren b and keskin m 2012 _ phys . a _ * 376 * 1011 deviren b , kantar e and keskin m 2012 _ j. magn . * 324 * 2163 nel l 1948 _ ann . paris _ * 3 * 137 streka j 2006 _ physica a _ * 360 * 379 wu m h , dong s and liu j -m 2008 _ j. appl . phys . _ * 103 * 07b103 daniel m and sabareesan p 2009 _ j. phys . : condens . matter _ * 21 * 352001 sabareesan p and daniel m 2011 _ j. phys . : condens . matter _ * 23 * 046004 qi j , xu y , steigerwald a , liu x , furdyna j k , perakis i e and tolk n h 2009 _ phys . b _ * 79 * 085304 sukhov a and berakdar j 2009 _ phys lett . _ * 102 * 057204 stanciu c d , kimel a v , hansten f , tsukamoto a , itoh a , kirilyuk a and rasing th 2006 _ phys . b _ * 73 * 220402 choi b c , belov m , hiebert w k , ballentine g e and freeman m r 2001 _ phys . rev . _ * 86 * 728 k binder , in : k. binder ( ed . ) , monte carlo methods in statistical physics , springer , berlin , 1979 p 127 newman m e j and barkema g t 2001 _ monte carlo methods in statistical physics _ ( usa : oxford university press ) p 72 acharyya m 1997 _ phys . * 56 * 2407 sariyer o s , kabakiolu a and berker a n 2012 arxiv:1206.0230v1 jiang w , lo v c , bai b d and yang j 2010 _ physica a _ * 389 * 2227 nowak u , chantrell r w and kennedy e c 2000 _ phys . rev . * 84 * 163 cheng x z , jalil m b a and lee h k 2006 _ phys . b _ * 73 * 224438 cheng x z , jalil m b a , lee h k and okabe y 2006 _ j. appl . phys . _ * 99 * 08b901 hinzke d and nowak u 2000 _ j. magn . * 221 * 365 hinzke d and nowak u 2000 _ phys . b _ * 61 * 6734 smirnov - rueda r , chubykalo o , nowak u , chantrell r w and gonzalez j m 2000 _ j. appl . phys . _ * 87 * 4798 hinzke d and nowak u 2002 _ phys . status solidi a _ * 189 * 475 berkov d v , gorn n l and grnert p 2002 _ phys . sol a _ * 189 * 409 cheng x z , jalil m b a , lee h k and okabe y 2005 _ phys . rev . b _ * 72 * 094420

we have presented dynamic phase transition features and stationary - state behavior of a ferrimagnetic small nanoparticle system with a core - shell structure . by means of detailed monte carlo simulations , a complete picture of the phase diagrams and magnetization profiles have been presented and the conditions for the occurrence of a compensation point @xmath0 in the system have been investigated . according to nel nomenclature , the magnetization curves of the particle have been found to obey p - type , n - type and q - type classification schemes under certain conditions . much effort has been devoted to investigation of hysteretic response of the particle and we observed the existence of triple hysteresis loop behavior which originates from the existence of a weak ferromagnetic core coupling @xmath1 , as well as a strong antiferromagnetic interface exchange interaction @xmath2 . most of the calculations have been performed for a particle in the presence of oscillating fields of very high frequencies and high amplitudes in comparison with exchange interactions which resembles a magnetic system under the influence of ultrafast switching fields . particular attention has also been paid on the influence of the particle size on the thermal and magnetic properties , as well as magnetic features such as coercivity , remanence and compensation temperature of the particle . we have found that in the presence of ultrafast switching fields , the particle may exhibit a dynamic phase transition from paramagnetic to a dynamically ordered phase with increasing ferromagnetic shell thickness .

1207.2023

the study of the high - redshift progenitors of today s massive galaxies can provide us with invaluable insights into the key mechanisms that shape the evolution of galaxies in the high - mass regime . the latest generation of galaxy formation models are now able to explain the number densities and ages of massive galaxies at high redshift . however , this is only part of the challenge , as recent studies have posed new questions about how the morphologies of massive galaxies evolve with redshift . in addition to the basic question of how high - redshift galaxies evolve in size , there is also still much debate about how these massive galaxies evolve in terms of their fundamental morphological type . extensive studies of the local universe have revealed a bimodality in the colour - morphology plane , with spheroidal galaxies typically inhabiting the red sequence and disk galaxies making up the blue cloud ( e.g. ( * ? ? ? * baldry et al . 2004 ) ) . however , recent studies at both low ( e.g. ( * ? ? ? * bamford et al . 2009 ) ) and high redshift ( e.g. ( * ? ? ? * van der wel et al . 2011 ) ) have uncovered a significant population of passive disk - dominated galaxies , providing evidence that the physical processes which quench star - formation may be distinct from those responsible for driving morphological transformations . this result is particularly interesting in light of the latest morphological studies of high - redshift massive galaxies by and ( * ? ? ? * van der wel et al . ( 2011 ) ) who find that , in contrast to the local population of massive galaxies ( which is dominated by bulge morphologies ) , by @xmath6 massive galaxies are predominantly disk - dominated systems . in this work we attempt to provide significantly improved clarity on these issues . the candels ( ( * ? ? ? * grogin et al . 2011 ) , ( * ? ? ? * koekemoer et al . 2011 ) ) near - infrared f160w data provides the necessary combination of depth , angular resolution , and area to enable the most detailed study to date of the rest - frame optical morphologies of massive ( @xmath1 ) galaxies at @xmath2 in the ukidss ultra deep survey ( ( * ? ? ? * lawrence et al . 2007 ) ) . for this study we have constructed a sample based on photometric redshifts and stellar mass estimates which were determined using the stellar population synthesis models of ( * ? ? ? * bruzual & charlot ( 2003 ) ) assuming a chabrier initial mass function ( see ( * ? ? ? * bruce et al . 2012 ) for full details ) . this provides us with a total mass - complete sample of @xmath7 galaxies . we have employed the galfit ( ( * ? ? ? * peng et al . 2002 ) ) morphology fitting code to determine the morphological properties for all the objects in our sample . to conduct the double component fitting we define three components : a srsic index fixed at @xmath8 bulge , an @xmath9 fixed disk and a centrally concentrated psf component to account for any agn or nuclear starbursts within our galaxies . these three components are combined to generate six alternative multiple component model fits , of varying complexity , for every object in the sample . these models are formally nested , and thus @xmath10 statistics can be used to determine the `` best '' model given the appropriate number of model parameters . armed with this unparalleled morphological information on massive galaxies at high redshift we can consider how the relative number density of galaxies of different morphological type changes during the key epoch in cosmic history probed here . in fig . 1 we illustrate this by binning our sample into four redshift bins of width @xmath11 , and consider three alternative cuts in morphological classification as measured by @xmath12 from our bulge - disk decompositions . in the left - hand panel of fig . 1 we have simply split the sample into two categories : bulge - dominated ( @xmath13 ) and disk - dominated ( @xmath14 ) . in the central panel we have separated the sample into three categories , with any object for which @xmath15 classed as `` intermediate '' . finally , in the right - hand panel we have expanded this intermediate category to encompass all objects for which @xmath16 . and using three alternative cuts in morphological classification ( both to try to provide a complete picture , and to facilitate comparison with different categorisations in the literature).,width=528 ] from these panels it can be seen that @xmath6 marks a key transition phase , above which massive galaxies are predominantly disk - dominated systems and below which they become increasingly mixed bulge+disk systems . we also note that at the lowest redshifts probed by this study ( @xmath17 ) it is seen that , while bulge - dominated objects are on the rise , pure - bulge galaxies ( i.e. objects comparable to present - day giant ellipticals ) have yet to emerge in significant numbers , with @xmath18% of these high - mass galaxies still retaining a significant disk component . this is compared with @xmath19 of the local @xmath1 galaxy population , which would be classified as pure - bulges from our definition ( @xmath20 , corresponding to @xmath21 ) from the sample of ( * ? ? ? * buitrago et al . ( 2013 ) ) . thus , our results further challenge theoretical models of galaxy formation to account for the relatively rapid demise of massive star - forming disks , but the relatively gradual emergence of genuinely bulge - dominated morphologies . in addition to our morphological decompositions we also make use of the sed fitting already employed in the sample selection to explore the relationship between star - formation activity and morphological type . 2 shows specific star - formation rate ( @xmath22 ) versus morphological type for the massive galaxies in our sample , where morphology is quantified by single srsic index in the left - hand panel , and by bulge - to - total @xmath23-band flux ratio ( @xmath12 ) in the right - hand panel . the values of @xmath22 plotted are derived from the original optical - infrared sed fits employed in the sample selection , and include correction for dust extinction as assessed from the best fitting value of @xmath24 derived during the sed fitting . as a check of the potential failure of this approach to correctly identify reddened dusty star - forming galaxies , we have also searched for 24@xmath25 m counterparts in the _ spitzer _ spuds mips imaging of the uds , and have highlighted in blue stars those objects which yielded a mips counterpart within a search radius of @xmath26arcsec . to first order , our results show that the well - documented bimodality in the colour - morphology plane seen at low redshift , where spheroidal galaxies inhabit the red sequence , while disk galaxies occupy the blue cloud is at least partly already in place by @xmath6 . nonetheless , the sample also undoubtedly contains star - forming bulge - dominated galaxies and , perhaps more interestingly , a significant population of apparently quiescent disk - dominated objects . to highlight and quantify this population we have indicated by a box on both the panels the region occupied by objects with disk - dominated morphologies and @xmath27 . in the left - hand panel , disk - dominated is defined as @xmath28 , and @xmath29% of the quiescent galaxies lie within this box ( if we exclude the 24@xmath25 m detections ) , while in the right - hand panel , disk - dominated is defined by @xmath14 , in which case @xmath30% of the quiescent objects lie within this region . the presence of a significant population of passive disks among the massive galaxy population at these redshifts indicates that star - formation activity can cease without a disk galaxy being turned directly into a disk - free spheroid , as generally previously expected if the process that quenches star formation is a major merger . one possible mechanism for this arises from the latest generation of hydrodynamical simulations ( e.g. ( * ? ? ? * kere et al . 2005 ) , ( * ? ? ? * dekel et al . 2009a ) ) and analytic theories ( e.g. ( * ? ? ? * birnboim & dekel 2003 ) ) , which suggest a formation scenario whereby at high redshift star - formation is fed through inflows of cold gas . another scenario which can account for star - formation quenching , whilst still being consistent with the existence of passive disks , is the model of violent disk instabilities ( e.g. ( * ? ? ? * dekel et al . 2009b ) ) , coupled with morphology quenching " ( ( * ? ? ? * martig et al . 2009 ) ) .

we have used high - resolution , hst wfc3/ir , near - infrared imaging to conduct a detailed bulge - disk decomposition of the morphologies of @xmath0 of the most massive ( @xmath1 ) galaxies at @xmath2 in the candels - uds field . we find that , while such massive galaxies at low redshift are generally bulge - dominated , at redshifts @xmath3 they are predominantly mixed bulge+disk systems , and by @xmath4 they are mostly disk - dominated . interestingly , we find that while most of the quiescent galaxies are bulge - dominated , a significant fraction ( @xmath5% ) of the most quiescent galaxies , have disk - dominated morphologies . thus , our results suggest that the physical mechanisms which quench star - formation activity are not simply connected to those responsible for the morphological transformation of massive galaxies .

1301.6373

a _ @xmath0-graph _ is a graph which at each vertex has a bijection from the outgoing half - edges to the vertices of a cycle graph . half - edges which are mapped to adjacent vertices are ( formally ) _ adjacent_. half - edges are said to be _ opposite _ if they are mapped to vertices of maximal distance in the cycle graph . by an embedding of a @xmath0-graph @xmath2 into a surface @xmath3 we always mean an embedding of @xmath2 into @xmath3 such that the formal relation of being adjacent coincides with the relation of being adjacent induced by the embedding . an _ angle _ in a @xmath0-graph is a pair of adjacent half - edges at a vertex . a _ checkerboard embedding _ of a @xmath0-graph @xmath2 into @xmath3 is an embedding such that the cells of @xmath4 admit a 2-coloring under which any cells with a common edge have different colors . checkerboard embeddings are exactly those embeddings whose first @xmath1-homology class is zero . in @xcite the second named author ( v.o.m . ) gave a solution to the question of whether four - valent framed graphs are planar . in @xcite , he addressed the question of determining the genus of surfaces into which four - valent framed graphs can be embedded , in particular considering the special case of surfaces into which four - valent framed graphs may be checkerboard - embedded . in @xcite , the first named author ( t.f . ) introduced @xmath0-graphs as a generalization of four - valent framed graphs , and gave a planarity condition for @xmath0-graphs with each vertex of degree 4 or 6 . in @xcite , the authors characterized the genera of orientable surfaces into which @xmath0-graphs with each vertex of degree 4 or 6 may be checkerboard - embedded , generalizing some of the results of @xcite . in this paper , we continue the project of generalizing results about embeddability properties of framed four - valent graphs to @xmath0-graphs with each vertex of degree 4 or 6 , now considering checkerboard - embeddability into nonorientable surfaces . in theorem [ main ] we show that this is equivalent to a problem on matrices . our methods are close to those used in our previous paper @xcite , which were themselves based closely on those used by the second named author ( v.o.m . ) in @xcite for framed four - valent graphs . the goal of this paper is to provide a method for determining whether a given @xmath0-graph @xmath2 has a checkerboard embedding into a nonorientable surface of genus @xmath5 . we show that this is equivalent to a problem on matrices . to accomplish this , we fix a cycle @xmath6 in @xmath2 satisfying certain properties , called a rotating - splitting cycle . then we define a correspondence between checkerboard embeddings of @xmath2 and `` permissible separations '' of a chord diagram @xmath7 , where the result of a permissible separation is a pair of chord diagrams . we then show that the number of white ( black ) cells in the embedding is equal to the number of circles resulting from surgery of the first ( second ) of these chord diagrams . the circuit - nullity theorem allows us to calculate the number of circles resulting from surgery of each chord diagram in terms of the rank of their intersection matrices over @xmath1 . from this we have the total number of cells in the embedding , from which the genus @xmath5 of the surface can be easily calculated . the authors of this paper would like to thank victor anatolievich vassiliev and sergei vladimirovich chmutov for valuable discussions . a _ @xmath0-atom _ is a closed 2-surface @xmath3 into which a connected graph @xmath2 ( the _ skeleton _ of the @xmath0-atom ) is embedded in such a way that it divides @xmath3 into black and white cells so that cells sharing an edge have different colors . this embedding induces a @xmath0-structure on the skeleton . the @xmath0-structure at each vertex determines a set of @xmath8 angles among which we say that two angles are adjacent if they share a half - edge . two adjacent angles never have the same color . thus the angles around a vertex can be partitioned into two sets @xmath9 and @xmath10 in such a way that for any @xmath0-atom corresponding to the @xmath0-graph @xmath2 , either all angles in @xmath9 are black and all angles in @xmath10 are white , or all angles in @xmath9 are white and all angles in @xmath10 are black . thus given a connected @xmath0-graph @xmath2 , the @xmath0-atoms corresponding to @xmath2 are uniquely determined by a choice of one of the two possible colorings at each vertex . thus the main problem can be reformulated as follows : given a @xmath0-graph @xmath2 in which all vertices have degree 4 or 6 , choose a coloring for the angles around each vertex such that the genus of the resulting atom is minimal . if such a graph has @xmath11 vertices , there are @xmath12 corresponding @xmath0-atoms . an _ euler circuit _ @xmath6 of a @xmath0-graph @xmath2 is a surjective mapping @xmath13 which is one - to - one except at the vertices , and such that every vertex of degree @xmath8 has @xmath14 preimages . given an euler circuit @xmath6 of a @xmath0-graph @xmath2 , a 4-vertex @xmath15 is _ rotating _ with respect to @xmath6 if for every @xmath16 , @xmath17 and @xmath18 are on adjacent half - edges around @xmath19 . given an euler circuit @xmath6 of a @xmath0-graph @xmath2 , a 6-vertex @xmath15 is _ rotating _ with respect to @xmath6 if for every @xmath16 , @xmath17 and @xmath18 are on adjacent half - edges around @xmath19 . given an euler circuit @xmath6 of a @xmath0-graph @xmath2 , a 6-vertex @xmath15 is _ splitting _ with respect to @xmath6 if for some @xmath16 , @xmath17 and @xmath18 are on opposite half - edges around @xmath19 , and for the other two points @xmath20 , @xmath21 and @xmath22 are on adjacent half - edges , and @xmath23 and @xmath24 are on adjacent half - edges . a _ rotating - splitting circuit _ is a circuit with respect to which every vertex is rotating or splitting . a rotating - splitting circuit induces an orientation on the half - edges around a rotating 6-vertex . if the order of the edges containing @xmath17 and @xmath18 agrees with this orientation , then the angle is said be _ untwisted _ ; otherwise it is _ twisted_. see figures [ r0 t ] , [ r1 t ] , [ r2 t ] , [ r3 t ] . [ rscycle - exists ] if @xmath2 is a connected @xmath0-graph in which all vertices have degree 4 or 6 , then @xmath2 admits a rotating - splitting circuit . assign to each vertex any rotating or splitting structure . this gives a partition of the edges of @xmath2 into edgewise disjoint cycles . if there is only one such cycle , we are done . if there is more than one cycle , since the graph is connected , there must be a vertex @xmath19 shared by different cycles . if @xmath19 has degree 4 , it must be rotating , and we can join the two cycles meeting at @xmath19 by assigning to @xmath19 the other possible rotating structure . if @xmath19 has degree 6 , we consider the cycles given by starting at @xmath19 , exiting through one of its incident edges , and following the rotating - splitting structure until we come back to @xmath19 . there are three such cycles , up to a change in orientation . if each of these cycles contains a pair of adjacent edges at @xmath19 , we can assign to @xmath19 the rotating structure shown on the right side of figure [ r0t - intro ] , so that the three cycles are joined together , and @xmath19 becomes a rotating vertex with no twisted angles . note that before making the change , @xmath19 may have some structure other than that shown on the left side of figure @xmath25 ; the left side of the figure and the others referenced in this proof are merely examples . if exactly two of the three cycles contain a pair of adjacent edges around @xmath19 , then the third must contain a pair of opposite edges , and we can assign to @xmath19 the splitting structure shown on the right side of figure [ s0t - intro ] to join the cycles together . if exactly one of the three cycles contains a pair of adjacent edges around @xmath19 , then the other two must contain pairs of edges which are neither opposite nor adjacent . in this case we can join the cycles by assigning the rotating structure shown on the right side of figure [ r1t - intro ] , so that @xmath19 becomes a rotating vertex with one twisted angle . if none of the cycles contains a pair of adjacent edges , then we have two possibilities : each of the cycles contains a pair of opposite edges , or one of the cycles contains a pair of opposite edges and the other two contain a pair of edges which are neither opposite nor adjacent . if each of the cycles contains a pair of opposite edges , we can assign to @xmath19 the rotating structure shown on the right side of figure [ r3t - intro ] to join the cycles . if one of the cycles contains a pair of opposite edges and the other contains a pair of edges which are neither opposite nor adjacent , we can assign to @xmath19 the rotating structure shown on the right side of figure [ r2t - intro ] to join the cycles . . [ r0t - intro ] . [ s0t - intro ] . [ r1t - intro ] . [ r3t - intro ] . [ r2t - intro ] a _ chord diagram _ is a cubic graph @xmath26 with a distinguished hamiltonian cycle ; i.e. an embedding @xmath27 which covers all the vertices of @xmath26 . a _ signed chord diagram _ is a chord diagram in which each edge not in the distinguished cycle is assigned a positive or negative sign . a @xmath0-chord diagram is a graph @xmath26 with a distinguished simple cycle ( i.e. an embedding @xmath28 ) , such that every vertex in @xmath26 has degree 3 or 4 and for every edge @xmath29 in @xmath26 , one of the following holds : 1 . @xmath29 is in the distinguished cycle . 2 . both of the vertices on @xmath29 are in the distinguished cycle , and both have degree 3 one of the vertices on @xmath29 is in the distinguished cycle , the other is not , and both have degree 3 . both of the vertices on @xmath29 are in the distinguished cycle , one has degree 3 , and the other has degree 4 . a _ signed @xmath0-chord diagram _ is a @xmath0-chord diagram in which each edge not in the distinguished cycle is assigned a positive or negative sign . an _ arc _ of a @xmath0-chord diagram @xmath2 is an edge in the distinguished cycle of @xmath2 . a _ chord _ of a @xmath0-chord diagram @xmath2 is an edge not in the distinguished cycle of @xmath2 , connecting two vertices of degree 3 which are in the cycle . a _ triad _ of a @xmath0-chord diagram @xmath2 is a vertex @xmath19 not in the distinguished cycle of @xmath2 , together with the three edges incident to @xmath19 . the vertex @xmath19 is called a _ triad point_. a _ double chord _ of a @xmath0-chord diagram @xmath2 is a pair of edges not in the distinguished cycle of @xmath2 , which are incident to a shared vertex @xmath19 . the vertex @xmath19 is called the _ principal vertex _ of the double chord . given a @xmath0-graph @xmath2 with all vertices of degree 4 or 6 , and given a rotating - splitting circuit @xmath6 of @xmath2 , we define a @xmath0-chord diagram @xmath30 as follows : for each 4-vertex @xmath19 in @xmath2 , the two points in @xmath31 which are mapped to @xmath19 by @xmath6 are connected by a chord , whose sign is positive if and only if the two half - edges through which @xmath6 enters @xmath19 are not adjacent . for any rotating 6-vertex @xmath19 in @xmath2 , the three points in @xmath31 which are mapped to @xmath19 by @xmath6 are connected by a triad , and the edge connecting a vertex @xmath16 to the triad point has positive sign if and only the angle into which @xmath6 maps a neighborhood of @xmath32 is not twisted . for any splitting 6-vertex @xmath19 in @xmath2 , the three points in @xmath31 which are mapped to @xmath6 are connected by a double chord , whose principal vertex is @xmath16 such that @xmath18 and @xmath17 are in opposite half - edges around @xmath19 , where the sign of the edge connecting @xmath32 another vertex @xmath33 is positive if and only if @xmath21 is adjacent to @xmath18 [ figure ] . an _ expansion _ of as signed @xmath0-chord diagram @xmath30 a signed chord diagram @xmath7 such that 1 . for every chord in @xmath34 containing vertices @xmath32 and @xmath35 , there is a chord of the same sign in @xmath36 of the same sign connecting vertices @xmath32 and @xmath35 . 2 . for every triad @xmath29 in @xmath37 containing at least one edge with positive sign , for some labeling @xmath38 of the vertices of @xmath29 such that the edge connecting @xmath32 to the triad point is positive , @xmath36 contains a chord connecting @xmath39 to @xmath35 and a chord connecting @xmath40 to @xmath41 , with @xmath42 chosen in such a way that the chords are not linked . furthermore , the chords connecting @xmath32 to @xmath35 and @xmath32 to @xmath41 in @xmath36 have the same signs as the edges connecting the triad point of @xmath29 to @xmath35 and @xmath41 in @xmath37 , respectively . 3 . for every triad @xmath29 in @xmath37 in which all edges have negative sign , for some labeling @xmath38 of the vertices of @xmath29 , @xmath36 contains a chord connecting @xmath39 to @xmath35 and a chord connecting @xmath40 to @xmath41 , with @xmath42 chosen in such a way that the chords are linked . furthermore , the chords connecting @xmath32 to @xmath35 and @xmath32 to @xmath41 in @xmath36 have signs opposite to the edges connecting the triad point of @xmath29 to @xmath35 and @xmath41 in @xmath37 , respectively . 4 . for every double chord @xmath29 in @xmath37 with principal vertex @xmath32 , for some labeling @xmath43 of the nonprincipal vertices of @xmath29 , there is a chord in @xmath36 connecting @xmath44 to @xmath35 and a chord in @xmath36 connecting @xmath45 to @xmath41 . these chords are not linked . a _ permissible separation _ of a signed chord diagram @xmath7 arising as an expansion of a signed @xmath0-chord diagram @xmath30 is a pair of signed chord diagrams @xmath46 and @xmath47 such that 1 . every chord in @xmath7 is in exactly one of @xmath46 and @xmath47 . 2 . two chords in @xmath7 which come from the same triad in @xmath30 are both in @xmath46 , or both in @xmath47 . 3 . of any two chords in @xmath7 which come from the same double chord in @xmath30 , one chord is in @xmath46 and the other is in @xmath47 . suppose @xmath2 is checkerboard - embedded in a closed surface @xmath3 . then the rotating - splitting circuit @xmath6 gives a mapping from @xmath31 to @xmath3 which is one - to - one except at the preimages of vertices @xmath2 . this mapping can be smoothed to give an embedding of @xmath31 into @xmath3 , as in figure [ fig : smoothed_rscycle ] . observe that the circle @xmath48 divides the surface into a black part and a white part . we can draw the chords of @xmath7 as small edges lying in neighborhoods of vertices of @xmath2 , see figure [ fig : small_chord ] . 0.3 0.3 0.3 [ fig : small_chord ] the coloring of @xmath3 divides the chords of @xmath7 into two families : those lying in the white part and those lying in the black part . observe that the two chords in the neighborhood of a rotating vertex are in the same part , and the two chords in the neighborhood of a splitting vertex are in different parts . thus we have a permissible separation of @xmath7 . vice versa , given a @xmath0-graph @xmath2 with all vertices of degree 4 or 6 and which satisfies the source - sink condition , a rotating - splitting circuit @xmath6 of @xmath2 , and a permissible separation of @xmath7 , we can recover the coloring of the angles around each vertex of @xmath2 , and thus we can recover the surface @xmath3 . thus given a @xmath0-graph @xmath2 with all vertices of degree 4 or 6 and which satisfies the source - sink condition , and an expansion @xmath7 of its @xmath0-chord diagram , we have a one - to - one correspondence between atoms of @xmath2 and permissible separations of @xmath7 . note that the two chords to be drawn in the neighborhood of any rotating 6-vertex @xmath19 do not cross in @xmath3 , as shown in figures [ r0t - lifecycle ] , [ r1t - lifecycle ] , [ r2t - lifecycle ] , and [ r3t - lifecycle ] . thus we have an embedding of @xmath7 into @xmath3 . furthermore , since the embedding of @xmath2 divides @xmath3 into 2-cells , the embedding of @xmath7 does as well . given a chord diagram @xmath26 , _ surgery _ of @xmath26 is the following process : for each chord @xmath29 connecting points @xmath49 , delete a neighborhood of @xmath29 and connect the obtained endpoints @xmath50 to @xmath51 and @xmath52 to @xmath53 if @xmath29 is positive , and @xmath50 to @xmath53 and @xmath52 to @xmath51 if @xmath29 is negative . this produces a family of circles ; these are called the _ result of surgery _ of @xmath26 . to form the _ intersection matrix _ of a signed chord diagram @xmath26 with @xmath11 chords , first enumerate the chords @xmath54 . then the _ intersection matrix _ @xmath55 is an @xmath56 matrix over @xmath1 , such that @xmath57 if and only if the chord @xmath58 is negative , and @xmath59 for @xmath60 if and only if the chords @xmath58 and @xmath61 are linked . the number of components in the manifold obtained from a signed chord diagram @xmath26 by surgery of the circle is one plus the corank of @xmath55 . [ sep - to - genus ] given a @xmath0-graph @xmath2 in which all vertices have degree 4 or 6 and a rotating - splitting circuit @xmath6 of @xmath2 , and a checkerboard embedding of @xmath2 into a nonorientable surface @xmath3 , the nonorientable genus of @xmath3 is given by @xmath62 where @xmath46 and @xmath47 are the results of the permissible separation of @xmath7 induced by the embedding . consider the embedding @xmath63 described above . the number of 2-cells on the white side of the embedding is the number of circles resulting in surgery of @xmath46 . likewise , the number of 2-cells on the black side of the embedding is the number of circles resulting in surgery of @xmath47 . applying the circuit nullity theorem , the total number of 2-cells is @xmath64 . introducing the notation @xmath65 to represent the number of chords in @xmath7 , the number of arcs in @xmath7 is @xmath66 , so its total number of edges is @xmath67 . the number of vertices in @xmath7 is @xmath66 . thus the euler characteristic of @xmath3 is @xmath68 @xmath69 so the nonorientable genus of @xmath3 is @xmath70 . thus @xmath2 has admits an atom of genus @xmath5 if and only if some permissible separation of @xmath30 results in @xmath71 . this can be reduced to a problem on matrices , as follows . a _ permissible partition _ of the indices of @xmath72 is a partition of the indices of @xmath72 ( which are just the chords of @xmath7 ) into two parts , in such a way that chords arising from the same triad in @xmath30 are in the same part , and chords arising from the same double chord in @xmath30 are in different parts . clearly , @xmath46 and @xmath47 are a permissible separation of @xmath7 if and only if there exists a permissible partition @xmath73 of the indices of @xmath72 such that @xmath74 is the intersection matrix of @xmath46 and @xmath75 is the intersection matrix of @xmath47 . [ main ] for a @xmath0-graph @xmath2 which does not satisfy the source - target condition and which has rotating - splitting circuit @xmath6 , @xmath2 has a checkerboard embedding into a nonorientable surface of genus @xmath5 if and only if there is a permissible partition of the indices of @xmath76 into parts @xmath77 and @xmath78 such that @xmath79 . let @xmath7 be any expansion of @xmath30 . by lemma [ sep - to - genus ] , @xmath2 has a checkerboard embedding into a surface of genus @xmath5 if and only there is a permissible separation @xmath80 of @xmath7 such that @xmath81 . such a permissible separation exists if and only if there is a permissible partition of @xmath76 into parts @xmath77 and @xmath78 such that @xmath79 . thus , the problem of finding the minimal nonorientable genus into which a @xmath0-chord diagram with each vertex of degree 4 or 6 may be checkerboard - embedded , is equivalent to the problem of finding a permissible partition of the indices of a matrix @xmath82 into parts @xmath77 and @xmath78 which minimizes @xmath83 . a @xmath0-graph @xmath2 with rotating - splitting circuit @xmath6 is embeddable into the projective plane if and only if there exists a permissible separation @xmath80 of @xmath36 such that @xmath85 and @xmath86 . in other words , @xmath2 is @xmath84-embeddable if and only if there exists a permissible separation of @xmath7 into two chord diagrams , one of which consists of a family of pairwise - linked negative chords and a family of positive chords which are not linked to each other or to the negative chords , and the other of which consists of a family of pairwise - unlinked negative chords . we can test this condition by the following algorithm , which takes time quadratic in the number of chords of @xmath36 : first assign all negative chords to the same chord diagram . then for each assigned chord , assign all positive chords linked to it to the other chord diagram . if an assigned chord originates from a triad , assign the other chord coming from this triad to the same chord diagram , and if the assigned chord originates from a double chord , assign the other chord coming from this double chord to the other chord diagram . then , for each of the newly assigned chords , assign any unassigned linked chords or chords coming from the same triad or double chord , using the same rules described above . repeat this process until for every assigned chord , the linked chords and any chord coming from the same triad or double chord have been assigned . if not all chords have been assigned , take any unassigned chord and arbitrarily assign it to @xmath46 or @xmath47 , and repeat until all chords have been assigned . finally , check whether this is a permissible separation , and whether @xmath87 . @xmath2 is @xmath84-embeddable if and only if both of these conditions are true . a @xmath0-graph @xmath2 with rotating - splitting circuit @xmath6 is embeddable into the klein bottle if and only if there exists a permissible separation @xmath80 of @xmath36 such that @xmath88 . there are two possible cases in which this can occur : @xmath89 or @xmath90 and @xmath86 . we will first consider the case where @xmath89 . in this case , we have a permissible separation of @xmath7 , each of which consists of a family of pairwise - linked negative chords and a family of positive chords which are not linked to each other or to the negative chords . this condition also admits a quadratic - time test , as follows : assign one of the chords arbitrarily to @xmath46 or @xmath47 . if the chord is positive , assign all chords linked to it to the other chord diagram ; if it is negative , assign all positive linked chords and all negative unlinked chords to the other diagram . regardless of sign , if an assigned chord originates from a triad , assign the other chord coming from this triad to the same chord diagram , and if the assigned chord originates from a double chord , assign the other chord coming from this double chord to the other chord diagram . repeat this process until for every assigned chord , the linked chords and any chord coming from the same triad or double chord have been assigned . if not all chords have been assigned , take any unassigned chord and arbitrarily assign it to @xmath46 or @xmath47 , and repeat until all chords have been assigned . finally , check whether this is a permissible separation , and whether @xmath89 . these conditions are met if and only if there is an embedding of @xmath2 into the klein bottle so that a smoothing of @xmath6 divides the klein bottle into two mbius bands . if this test fails , there is still the possibility that @xmath2 has an embedding into the klein bottle where the smoothing of @xmath6 bounds a disc . to cover this possibility , we choose a negative chord @xmath41 of @xmath36 and perform surgery at that chord , in the manner shown in [ klein - surgery ] . since this reverses the orientation of part of the designated cycle , we should also change the sign of all chords which cross @xmath41 , producing a new chord diagram @xmath91 . then for any surface @xmath3 and any embedding of @xmath92 which respects the signs of the chords , there is a corresponding embedding @xmath93 , still respecting the signs of the chords . furthermore , if the distinguished cycle in the embedding @xmath92 into the klein bottle bounds a disc , then the distinguished cycle in the embedding @xmath93 bounds a mbius band . thus @xmath2 has an embedding into the klein bottle where the smoothing of @xmath6 bounds a disc if and only @xmath94 has an embedding into the klein bottle where the distinguished cycle bounds a mbius band . this condition can be checked using the algorithm in the previous paragraph .

this paper considers @xmath0-graphs in which all vertices have degree 4 or 6 , and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded . in a previous paper @xcite by the authors , the problem of calculating whether a given @xmath0-graph in which all vertices have degree 4 or 6 admits a @xmath1-homologically trivial embedding into a given orientable surface was shown to be equivalent to a problem on matrices . here we extend those results to nonorientable surfaces . the embeddability condition that we obtain yields quadratic - time algorithms to determine whether a @xmath0-graph with all vertices of degree 4 or 6 admits a @xmath1-homologically trivial embedding into the projective plane or into the klein bottle . * keywords : * graph , @xmath0-graph , surface , embedding , genus * ams subject classification : * primary 05c10 ; secondary 57c15 , 57c27

1312.6342

real time operation of the power grid and computation of electricity prices @xcite require accurate estimation of its structure and critical state variables . remote terminal units ( rtus ) transmit measurements collected from different grid components to the central control center for state estimation and subsequent use in analyzing grid stability . the collected measurements can be broadly classified into two kinds : meter readings and breaker statuses . the breaker statuses on transmission lines help create the current operational topology of the grid . the meter readings , comprising of line flow and bus power injection measurements , are then used to estimate the state variables over the estimated topology . in a practical setting , the collected measurements suffer from noise , that get added at source or during communication to the control center . the affect of such noise is minimized through placement of redundant / additional meters and use of suitable bad - data detection and correction techniques at the estimator @xcite . cyber - attacks on the power grid refer to corruption of measurements ( meter readings and breaker statuses ) by an adversary , aimed at changing the state estimation output , without getting detected by the estimator s checks . the viability of such attacks has in fact been demonstrated through controlled experiments like the aurora attack in department of energy s idaho laboratory and gps spoofing attack on phasor measurement units ( pmus ) @xcite . past literature on cyber - attacks have generally looked at adversaries that change meter data ( and not breaker statuses ) to affect state estimation . such data attacks involving injection of malicious data into meters were first analyzed in @xcite . using a dc power flow model for state estimation , the authors of @xcite provide an attack design using projection matrices . following this , several approaches have been discussed to study hidden attacks under different operating conditions . these include mixed integer programming @xcite , heuristic based detection @xcite , sparse recovery using @xmath0 relaxation @xcite , graph - cut based construction for systems with phasor measurement units ( pmus ) @xcite among others . the possible economic ill - affects of such hidden data attacks on power markets are presented in @xcite . in a recent paper @xcite , the authors investigates hidden attacks under the more general and potent regime of topology data ( breaker statuses ) and meter data corruption . all of these cited work on data alone or topology and data attacks , however , require changing floating point meter measurements in real time . the practicality of this is questionable as significant resources are required to synchronize the changes at multiple meters . in this paper , we focus on hidden attacks that primarily operate through changes in breaker statuses . here the adversary changes the statuses of a few operational breakers from @xmath1 ( closed ) to @xmath2 ( open ) , as well as jams ( blocks the communication ) of flow measurements on a subset of transmission lines in the grid . however , the adversary does not modify any meter reading to an arbitrary value . we term these attacks as breaker - jammer attacks . note that breaker statuses , unlike meter readings , are binary in nature and fluctuate with lower frequency . they are thus easier to change , even by adversaries with limited resources . jamming measurements , through jammers or by destruction of communication apparatus , is technologically less intensive than corrupting meter measurements . in fact , jamming does not raise a major alarm as measurement loss due random communication drops occurs under normal circumstances . the breaker - jammer attack model was introduced by the authors for grids with a specific meter configuration requiring sufficient line flow measurements in @xcite . this work generalizes the framework to any grid with line flow and injection meters and uses a novel graph - coloring analysis to determine the optimal hidden attack . our graph coloring based analysis is in principal similar to @xcite which studies standard data attacks as a graph partitioning problem . however , the similarly ends there as our attack model does not use corruption of meter readings . instead breaker status changes and line flow jams provide a different set of necessary and sufficient conditions for feasible attacks . the surprising revelation of our analysis is that under normal operating conditions , a single breaker status change ( with the necessary flow measurement jamming ) is sufficient to create an undetectable attack . in fact , we show that if a hidden attack can be constructed by changing the status of a set of breakers , then a hidden attack using only one break status change exists as well . this is significant as the adversary can focus on jamming the necessary flow measurements , after selecting a breaker to attack . further , our attack design does not depend on the current system state or transmission line parameter values , and has low information requirements . the rest of this paper is organized as follows . we present the system model used in generalized state estimation and describe the attack model in the next section . the graph coloring approach to determine the necessary and sufficient conditions for a hidden attack and elucidating examples are discussed in section [ sec : coloring ] . the design of the optimal hidden attack is discussed in section [ sec : design ] along with simulations on ieee test cases . finally , concluding remarks and future directions of work are presented in section [ sec : conclusion ] . first , we provide a brief description of the notation used . we represent the current operational structure of the grid by graph @xmath3 where @xmath4 denotes the set of buses / nodes of size @xmath5 and @xmath6 denotes the set of operational edges of size @xmath7 . the set of binary breakers statuses for the edges is denoted by the diagonal matrix @xmath8 of size @xmath9 . we assume that all lines to be initially operational ( @xmath8 is identity matrix ) and ignore any non - operation line for ease of notation . the edge to node incidence matrix is denoted by @xmath10 of dimension @xmath11 . each operational edge @xmath12 between nodes @xmath13 and @xmath14 has a corresponding row @xmath15 in @xmath10 , where @xmath16 . @xmath17 denotes the standard basis vector in @xmath18 with one at the @xmath19 location . the direction of flow on edge @xmath12 is taken to be from @xmath13 to @xmath14 , without any loss of generality . we consider the dc power flow model for state estimation in this paper @xcite . the state variables in this model are the bus phase angles , denoted by the @xmath20 vector @xmath21 . the set of measurements is denoted by the vector @xmath22 . here line flow measurements are included in @xmath23 and bus injection measurements are included in @xmath24 . state estimation in the power grid relies on the breaker statuses in @xmath8 for topology estimation and then uses the meter measurements @xmath25 for estimating the state vector @xmath21 . the relation between @xmath21 and @xmath25 in the dc model is given by @xmath26 where @xmath27 is the zero mean gaussian noise vector with covariance matrix @xmath28 . @xmath29 is the measurement matrix and depends on the grid structure and susceptance of transmission lines . let the @xmath30 entry in @xmath25 corresponds to the flow measurement on line @xmath12 . then @xmath31 ( the @xmath30 row in @xmath29 ) is given by @xmath32 = b_{ab}m_{ab } \label{flow}\end{aligned}\ ] ] with the non - zero values at the @xmath19 and @xmath33 locations respectively . @xmath34 is the susceptance of the line @xmath12 . on the other hand , if the @xmath35 entry corresponds to an injection measurement at node @xmath13 , we have @xmath36 . in matrix form , ignoring measurement noise , we can write equations for received measurements as @xmath37 @xmath38 is the diagonal matrix of susceptances of lines in @xmath6 . we arrange the rows in @xmath10 such that the top @xmath39 rows represent the lines with flow measurements . matrix @xmath40 , comprising of the top @xmath39 rows of a @xmath9 identity matrix , selects these measured flows . for ease of notation and analysis in later sections , we pad trailing zeros to vector @xmath23 and make it of length @xmath7 . similarly , we pad trailing all - zero rows to @xmath40 to make it a diagonal square matrix of dimension @xmath7 . @xmath41 on the other hand consists of the columns of @xmath10 that correspond to the nodes with injection measurements . the optimal state vector estimate @xmath42 is given by minimizing the residual @xmath43 . if the minimum residual does not satisfy a tolerance threshold , bad - data detection flags turn on and data correction is done by the estimator . the overall scheme of topology and state estimation processes followed by bad - data detection and correction is called generalized state estimation ( gse ) @xcite as illustrated in figure [ estimator ] . [ estimator ] * attack model : * we assume that the adversary is agnostic and has no information on the current system state @xmath21 or line susceptance matrix @xmath38 . for attack , the adversary changes the breaker statuses on some lines . the new breaker status matrix , after attack , is denoted by @xmath44 where diagonal matrix @xmath45 has a value of @xmath1 for attacked breakers . similarly , the available flow measurements after jamming are represented by @xmath46 , with diagonal matrix @xmath47 having a value of @xmath1 corresponding to jammed flows . let the new state vector estimated after the breaker - jammer attack be denoted by @xmath48 , where @xmath49 denotes the change . note that if the flow measurement on a line is not jammed , its value remains the same following the attack . using ( [ flowmat ] ) , we have @xmath50 it follows immediately that if the breaker status on the @xmath51 line with flow measurement is changed ( @xmath52 ) , to avoid detection , its flow measurement needs to be jammed as well ( @xmath53 ) . thus , @xmath54 consider the injection measurements ( @xmath24 ) now , which are not changed during the attack . the breaker attack leads to removal of lines marked as open from equation ( [ injmat ] ) , resulting in the following modification . @xmath55 equation ( [ injcond ] ) thus states that after the breaker - jammer attack , for each injection measurement , the sum of original flows contributed by lines with attacked breakers ( left side ) needs to be accommodated by changes in estimated flows on lines ( connected to the same bus ) whose breakers are intact but actual flow measurements are not received ( right side ) . finally , for unique state estimation following the adversarial attack ( with one bus considered reference bus with phase angle @xmath2 ) we need @xmath56 the necessary conditions for a successful breaker - jammer attack that results in a change in estimated state vector consists of equations ( [ flowcond ] ) , ( [ breakjam ] ) , ( [ injcond ] ) , and ( [ rank ] ) . in the next section , we describe a graph coloring based analysis of the necessary and sufficient conditions and use it to discuss design of optimal attacks of our regime . for our graph coloring based analysis , we use the following coloring scheme : _ for any change @xmath57 in the estimated state vector , neighboring buses with same value in @xmath57 are given same color . _ using this , we now discuss a permissible graph coloring corresponding to the requirements of a feasible attack discussed in the previous section . equation ( [ flowcond ] ) states that if the flow on line @xmath12 between buses @xmath13 and @xmath14 is not jammed , @xmath58 ( same color in our scheme ) . thus , _ * a set of buses connected through lines with available flow measurements ( not jammed ) has the same color . * _ this implies that the grid buses , following a feasible attack , can be divided into groups , each group having a distinct color . the lines between buses of different groups do not carry any flow measurement or are jammed by the adversary . a test example is illustrated in figure [ fig : graphcoloring ] . observe the buses with injection measurements , that are not corrupted by the adversary . for an interior bus @xmath59 , ( all neighboring nodes have the same color as itself ) , the right side of equation ( [ injcond ] ) equates to zero . the left side becomes equal to zero , under normal operating conditions , if breakers on lines connected to bus @xmath59 are not attacked . thus , we have _ * a feasible graph coloring has lines with attacked breakers connected to boundary buses . * _ a boundary bus is one that has neighboring buses of colors distinct from itself . bus system @xcite with flow measurements on all lines and injection measurements at buses @xmath13 , @xmath14 and @xmath59 . the blue , green and black buses are divided into groups and have same value of change @xmath57 in estimated state vector . the dotted red lines represent jammed lines , solid black lines represent operational lines . the grey lines with red bars represent the lines @xmath60 and @xmath61 with attacked breakers.,scaledwidth=42.0%,scaledwidth=30.0% ] [ fig : graphcoloring ] now , consider the injection meter installed on any boundary bus . such buses can exist in two configurations : a ) connected to lines with attacked breaker ( see bus @xmath13 in figure [ fig : graphcoloring ] ) or b ) connected to only lines with correct breaker statuses ( node @xmath14 if line @xmath61 did not have a breaker attack ) . in either case , using ( [ injcond ] ) , we have : _ * each injection measurement placed at a boundary bus provides one constraint relating the values of @xmath57 for neighboring differently colored buses . * _ for further analysis , we now use the coloring constraints highlighted in bold above to construct a reduced grid graph @xmath62 from @xmath63 as follows : * 1*. in each colored group , club boundary buses without injection measurements with all interior buses into one supernode of that color . make boundary buses with injection measurements into supernodes with the same color . connect supernodes of same color with artificial lines of zero susceptance . * 2*. for each line with intact breaker between two buses of different colors , create a line of same impedance between their corresponding supernodes. remove supernodes connected only to other supernodes of same color . * 3*. make injection measurements on supernodes equal to the sum of original flows on lines with attacked breakers connected to them ( positive for inflow , negative for outflow ) . if no incident line has attacked breaker , make the injection equal to @xmath2 . . the blue , green and black solid circles represent super nodes for buses @xmath13 , @xmath14 and @xmath59 respectively . the flow on the dotted red lines are not measured after attack . the grey lines with red bars represent lines with attacked breakers , that influence the injections at supernodes @xmath13 and @xmath14.,scaledwidth=33.0%,scaledwidth=22.0% ] [ fig : supernode ] figure [ fig : supernode ] illustrates the reduced graph construction for the example in figure [ fig : graphcoloring ] . note that in the reduced graph @xmath62 , original lines between buses of same color are removed . the included lines exist between buses of different colors and have jammed or unavailable flow measurements . similarly , injection measurement relation ( [ injcond ] ) at interior nodes are trivially satisfied by @xmath57 and are ignored . the reduced system , thus , only includes constraints from boundary injection measurements that are similar in form to equation ( [ injcond ] ) as shown below : @xmath64 here , @xmath13 and @xmath14 are supernodes of different colors . the numeric value for the color of supernode @xmath13 is given by @xmath65 ( not the @xmath19 entry in @xmath66 ) . @xmath67 and @xmath68 are the susceptance matrix and edge set corresponding to the reduced graph @xmath62 . @xmath69 denotes the injection measurement on supernode @xmath13 with value given by step @xmath70 in the reduced graph construction . note that equation ( [ reducedinjcond ] ) for the injection measurements involves rows of the susceptance weighted laplacian matrix for @xmath62 . a unique solution of @xmath66 for @xmath62 in turn provides a uniquely estimated @xmath57 in @xmath71 after the adversarial attack . we now look at condition ( [ rank ] ) , necessary for unique state estimation after a feasible adversarial attack in terms of graph coloring . the reduced graph @xmath62 greatly simplifies our analysis here . first , it is clear that each color must have at least one supernode or a neighboring supernode ( of different color ) with injection measurement . otherwise the value of @xmath66 for that color will not be in any injection constraint . this goes against uniqueness of state estimation . note that the number of degrees of freedom in @xmath66 ( representing distinct values in @xmath57 ) is one less than the number of colors as one color denotes the reference phase change of @xmath2 . using @xmath62 , we prove the following result regarding permissible graph coloring for unique estimation . [ oneless ] following a breaker - jammer attack , the number of injection measurements at the boundary buses should be one less than the number of distinct colors in the grid buses . let the number of colored groups be @xmath72 . then the number of independent entries in @xmath66 is @xmath73 ( one entry being @xmath2 ) . the total number of linear constraints involving the numeric values in @xmath66 is equal to the number of injection measurements at the supernodes in @xmath62 . for unique state estimation , the number of injection measurements should thus be greater than or equal to @xmath73 . we now show that exactly @xmath73 injection measurements are needed to get a solution to state estimation . consider the reduced graph @xmath62 . for real valued line susceptances and for cases where the supernodes having injection measurements do not form a closed ring with no additional branches ( see figure [ fig : supernode ] ) , the rank of @xmath73 rows is @xmath73 and we have unique state estimation . if the reduced graph @xmath62 contains a closed ring of supernodes with injection measurements , then the measurements will represent the entire susceptance weighted graph laplacian of the ring , that is rank deficient . however , the real valued entries in @xmath74 that exist on the right side of ( [ reducedinjcond ] ) and are derived from flows on lines with attacked breakers , will not cancel out under normal operating conditions . further , the adversary designing the attack is unaware of the current system state and will be able to determine if they do . hence the @xmath73 injections measurements constraints will be linearly independent ( the adversary will expect this under normal operations ) . this gives an unique @xmath66 and @xmath57 for a @xmath72 distinct colored grid graph . to summarize , the highlighted statements and theorem [ oneless ] provide the necessary and sufficient conditions for a feasible breaker - jammer attack under our graph - coloring scheme . in the next section , we show that the graph coloring approach proves a surprising result that simplifies the design of an optimal attack . we call an breaker - jammer feasible attack optimal if it requires minimum number of breaker status changes ( considering the fact that doing so is significantly more resource draining than measurement jamming ) . if multiple attacks are possible using the minimum number of breaker changes , we select as optimal the attack that requires the least number of flow measurement jams . using the reduced graph @xmath62 , we present the following result for the minimum number of breaker changes needed for a feasible attack under normal operating conditions ( non - zero real - valued bus susceptances and line flows that are distinct for different grid elements ) . [ oneenough ] if a feasible attack can be designed with @xmath72 breaker status changes , then a feasible attack exists such that all but one breaker statuses are changed back to their original operational state ( @xmath1 ) , while keeping their line flow measurements jammed . construct the reduced graph @xmath62 with its colored supernodes for the feasible attack with @xmath72 breakers and necessary flow measurement jams . let the number of colors in state estimation change @xmath57 be @xmath75 . the length of @xmath66 is then @xmath75 . by theorem [ oneless ] , there are @xmath76 injection measurements at the supernodes that provide constraint equations listed in ( [ reducedinjcond ] ) . if we revert the breaker status of an attacked line back to @xmath1 while keeping its flow measurement jammed , the only change in any constraint equation ( [ reducedinjcond ] ) involving that line will be that the injection measurement on the incident node ( entry in @xmath77 ) will become @xmath2 . since all but one breakers are brought back to the operational state , at least one injection measurement in @xmath77 will still remain non - zero and the @xmath76 constraint equations will still have linear inndependence . thus , state estimation will result in a different but non - zero @xmath66 , leading to a feasible attack . for example , consider the case in figure [ fig : graphcoloring ] where two breaker statuses are attacked . if the breaker status on line @xmath61 is changed back to @xmath1 while keeping the flow measurement jammed , the new reduced graph that will be derived is given in figure [ fig : supernode1 ] . as mentioned in theorem [ oneenough ] , the coloring scheme is still feasible and a non - zero change in state estimation results . bus case given in figure [ fig : graphcoloring ] , but with line @xmath61 being changed to a dotted red line . the blue , green and black solid circles represent super nodes for buses @xmath13 , @xmath14 and @xmath59 respectively . the flow on the dotted red lines are not received . the only grey line with red bar represents the line @xmath60 with an attacked breaker.,scaledwidth=33.0%,scaledwidth=22.0% ] [ fig : supernode1 ] this is a very significant result and simplifies the search for an optimal attack greatly . since one breaker change is sufficient , the adversary can select each line in turn ( @xmath7 iterations ) , attack its breaker ( change the corresponding entry in diagonal @xmath45 to @xmath1 ) and determine the flow measurements that need to be jammed ( given by diagonal @xmath47 ) to conduct a feasible attack . the breaker change that requires the minimum number of measurement jams ( or maximally sparse @xmath47 ) will then give the optimal attack . the selection of jammed measurements , after fixing @xmath45 , is formulated as ( [ opt_attack ] ) . @xmath78 this is simplified in formulation ( [ opt_attack1 ] ) where the jammed measurements ( with @xmath1 on diagonal of @xmath47 ) are given by the non - zero entries in @xmath79 . @xmath0 relaxation can be used to approximately solve ( [ opt_attack1 ] ) . since the adversary has no access to the actual state vector @xmath21 , a random non - zero @xmath21 similarly , unavailable line susceptance @xmath38 are replaced with distinct real values . these replacements , under normal conditions , do not affect the optimal solution as they preserve the linear independence of injection constraints given in ( [ reducedinjcond ] ) . @xmath80 the rank constraint ( [ rank ] ) is not included in the optimization framework and can be checked manually after determining @xmath47 , for consistency . + * experiments : * we simulate our attack model on ieee @xmath81 and @xmath82 bus test systems @xcite and present averaged findings in figure [ fig : topologyplot ] . for each test system considered , we place flow measurements on all lines and injection measurements on a fraction of buses , selected randomly . to design a feasible attack involving a line , we change its breaker status and solve problem ( [ opt_attack1 ] ) to jam flows measurements to prevent detection . this is repeated for each line to determine the optimal attack . in figure [ fig : topologyplot ] , note that the average number of flow measurements jammed increases with the number of injection measurements . this happens due to an increase in the number of injection constraints that require more measurement jams . [ fig : topologyplot ] in this paper , we study topology based cyber - attacks on power grids where an adversary changes the breaker statuses of operational lines and marks them as open . the adversary also jams flow measurements on certain lines to prevent detection at the state estimator . the attack framework is novel as it does not involve any injection of corrupted data into meters or knowledge of system parameters and current system state . using lesser information and resource overhead than traditional data attacks , our attack regime explores attacks on systems where all meter data are protected from external manipulation . we discuss necessary and sufficient conditions for the existence of feasible attacks through a new graph - coloring approach . the most important result arising from our analysis is that optimal topology based attacks exist that require a single breaker status change . finally , we discuss an optimization framework to select flow measurements that are jammed to prevent detection of the optimal attack . its efficacy is presented through simulations on ieee test cases . designing protection schemes for our attack model is the focus of our current work . 1 a. l. ott , experience with pjm market operation , system design , and implementation " , _ ieee trans . power syst . 18 , no . 2 , 2003 . a. abur and a. g. expsito , power system state estimation : theory and implementation " , new york : marcel dekker , 2004 . d. shepard , t. humphreys , and a. fansler , evaulation of the vulnerability of phasor measurement units to gps spoofing " , _ international conference on critical infrastructure protection _ y. liu , p. ning , and m. k. reiter , false data injection attacks against state estimation in electric power grids " , _ proc . commun . security _ o. vukovic , k. c. sou , g. dan , and h. sandberg , network - aware mitigation of data integrity attack on power system state estimation " , _ ieee journal on selected areas in communications _ , vol . 30 , no . 6 , 2012 . o. kosut , l. jia , r. j. thomas , and l. tong , limiting false data attacks on power system state estimation " , _ proc . t. kim and v. poor , strategic protection against data injection attacks on power grids " , _ ieee trans . smart grid _ , vol . 2 , no . 2 , 2011 . d. deka , r. baldick , and s. vishwanath , data attack on strategic buses in the power grid : design and protection " , _ ieee pes general meeting _ , 2014 . l. xie , y. mo , and b. sinopoli , false data injection attacks in electricity markets " , _ proc . ieee smartgridcomm _ , 2010 . j. kim and l. tong , on topology attack of a smart grid : undetectable attacks and countermeasures " , _ ieee j. select . areas commun . 31 , no . 7 , 2013 . d. deka , r. baldick , and s. vishwanath , attacking power grids with secure meters : the case for using breakers and jammers " , _ ieee infocom ccses workshop _ , 2014 . a. giani , e. bitar , m. garcia , m. mcqueen , p. khargonekar , and k. poolla , smart grid data integrity attacks " , _ ieee trans . on smart grid _ , vol . 4 , no . 3 , 2013 . a. abur and a. g. exposito , _ power system state estimation : theory and implementation _ , crc , 2000 . power system test archive " , http://www.ee.washington.edu/research/pstca .

a coordinated cyber - attack on grid meter readings and breaker statuses can lead to incorrect state estimation that can subsequently destabilize the grid . this paper studies cyber - attacks by an adversary that changes breaker statuses on transmission lines to affect the estimation of the grid topology . the adversary , however , is incapable of changing the value of any meter data and can only block recorded measurements on certain lines from being transmitted to the control center . the proposed framework , with limited resource requirements as compared to standard data attacks , thus extends the scope of cyber - attacks to grids secure from meter corruption . we discuss necessary and sufficient conditions for feasible attacks using a novel graph - coloring based analysis and show that an optimal attack requires breaker status change at only one transmission line . the potency of our attack regime is demonstrated through simulations on ieee test cases .

1506.04303

natural hamiltonian systems are the mathematical models of those physical systems for which the energy is constant , for example harmonic oscillators or the kepler system . often , as in the previous two examples , more quantities are constants of the motion ( or _ first integrals _ ) : angular momentum , laplace - runge - lentz vector , etc . usually , these constants are expressed by quadratic polynomials in the momenta or , for quantum systems , by second - order differential operators . hamiltonian systems with constants of the motion of degree higher than two are less common , nevertheless , some of them are of great interest , as for instance the three - body jacobi - calogero and wolfes systems . these systems represent the dynamics of three point - masses on a line under forces determined by the potential functions @xmath4 respectively ( we do not consider here the harmonic oscillator terms ) and they have essentially the same dynamics @xcite . both the resulting natural hamiltonians in @xmath5 admit one linear and one quadratic in the momenta constants of the motion , making the systems liouville - integrable and solvable by separation of variables ( see @xcite and references therein ) . other two independent constants of the motion do exist , one quadratic , due to the multiseparability of the hamiltonian , and one cubic . the systems are then maximally superintegrable ( ms ) , having a number of functionally independent constants of the motion equal to twice the degrees of freedom , minus one ( for quantum systems , the same number of algebraically independent symmetry operators ) . ms systems are of the greatest importance in mathematical physics , harmonic oscillators and kepler are ms and this makes them to satisfy bertrand s theorem . indeed , maximal superintegrability manifests itself , for classical systems , in the fact that all finite orbits of ms systems are closed while , for quantum systems , in the fact that the energy levels are totally degenerate @xcite . in recent years , several techniques made possible the construction of classical and quantum hamiltonian systems , ms and not , with first integrals of arbitrarily high degree @xcite whose study , still in development , produced remarkable results in special functions , quantum algebras , canonical quantization theories @xcite . in this note it is shortly introduced the work on the topic done by claudia chanu , luca degiovanni and the author ( in short cdr ) in several joint articles . in few words , the extension procedure ( theorem [ teo0 ] ) adds one degree of freedom to some suitable hamiltonian @xmath6 in such a way an extra non trivial first integral , polynomial of degree @xmath7 , of the new hamiltonian do exist . the following theorem , stated in @xcite , defines and characterizes what we intend for `` extensions '' in the particular case of natural hamiltonians on cotangent bundles of riemannian manifolds ; for a more general definition , see @xcite . given an @xmath8-dimensional natural hamiltonian @xmath6 on the cotangent bundle of a ( pseudo)-riemannian manifold @xmath9 , let be @xmath10 and @xmath11 where @xmath12 is the hamiltonian vector field of @xmath6 , then [ teo0 ] let @xmath9 be a @xmath8-dimensional ( pseudo-)riemannian manifold with metric tensor @xmath13 . the natural hamiltonian @xmath14 on @xmath15 with canonical coordinates @xmath16 admits an extension @xmath17 in the form ( [ hamext ] ) with a first integral @xmath18 with @xmath19 given by ( [ u ] ) and @xmath20 , if and only if the following conditions hold : 1 . the functions @xmath21 and @xmath22 satisfy @xmath23 @xmath24 where @xmath25 is the hessian tensor of @xmath21 . 2 . for @xmath26 the extended hamiltonian @xmath17 and the first integral @xmath27 are @xmath28 for @xmath29 the extended hamiltonian @xmath17 and the first integral @xmath27 are @xmath30 with @xmath31 , @xmath32 , @xmath33 and @xmath34 in @xcite it is proved that @xmath35 is functionally independent from @xmath17 , @xmath6 and any other first integral of @xmath6 in @xmath36 . the integrability conditions of ( [ hessteo ] ) are discussed in @xcite and it is found that their complete integrability requires @xmath37 , where @xmath31 is the constant curvature of @xmath9 . then , the function @xmath20 can depend linearly on up to @xmath38 parameters and the maximal number of parameters is attained on constant curvature manifolds only . however , non complete solutions can be found in non - constant curvature manifolds ( cdr to appear ) . from equation ( [ vteo ] ) , the expressions of the admissible potentials @xmath22 can be computed . several examples are given in @xcite . the particular form of @xmath19 makes possible to explicit any @xmath35 by expanding the @xmath39-th power of a binomial , obtaining @xcite @xmath40 with @xmath41}{\left ( \begin{matrix } m \cr 2k \end{matrix } \right ) \gamma^{2k}p_u^{m-2k}\left(-2m(cl+l_0)\right)^k},\ ] ] @xmath42}{\left ( \begin{matrix } m \cr 2k+1 \end{matrix } \right ) \gamma^{2k+1}p_u^{m-2k-1}\left(-2m(cl+l_0)\right)^k } , \quad m>1,\ ] ] where @xmath43 $ ] denotes the integer part and @xmath44 . we remark that first integrals of high degree obtained in other ways than by the extension procedure @xcite can be explicitly expressed only thanks to the fact that the dynamical equations are in these cases separated in some coordinate system . as a first example of the extension procedure we consider the one - dimensional hamiltonian @xcite @xmath45 the geodesic term of the extended hamiltonian @xmath17 is @xmath46 where @xmath31 is here the constant curvature of the extended configuration manifold . the solutions of equations ( [ hessteo ] ) and ( [ vteo ] ) are @xmath47 where @xmath48 . when @xmath49 and @xmath50 , the configuration manifold of @xmath17 is the euclidean plane , the sphere @xmath51 and the pseudosphere @xmath52 respectively , while for @xmath53 and @xmath50 , the minkowski plane , the desitter and anti - desitter manifolds , respectively . after a rescaling of the coordinate @xmath54 , the parameter @xmath39 in @xmath17 passes into @xmath6 and @xmath55 this makes evident that the extension procedure introduces some discrete symmetry into @xmath17 , in this case a dihedral symmetry of order @xmath56 , somehow connected with the extra first integral @xmath35 . in the euclidean plane ( i.e. @xmath57 ) with @xmath58 and @xmath59 , @xmath22 is associated with the jacobi - calogero potential or , equivalently , with the wolfes potential @xcite . indeed , in cylindrical coordinates of @xmath5 , @xmath60 , with axis @xmath61 parallel to @xmath62 w.r . to cartesian coordinates @xmath63 , we have @xmath64 the procedure of extension provides two new functionally independent first integrals to the extended hamiltonian @xmath17 : @xmath17 itself and @xmath27 . this fact is particularly relevant when the hamiltonian @xmath6 is ms . in this case , @xmath17 is ms too , admitting @xmath65 functionally independent first integrals . in @xcite this property of extended hamiltonians is studied in several cases . for example , let us consider @xmath66 that is a particular case of the generalized tremblay - turbiner - winternitz system ( ttw ) @xcite for @xmath67 and @xmath68 defined on constant - curvature manifolds of curvature @xmath69 . this system is ms for any rational parameter @xmath70 and admits polynomial first integrals of degree related to @xmath70 @xcite . in @xcite it is shown that @xmath6 always admits extensions of the form @xmath71 with @xmath72 , creating in this way new ms systems . similarly , harmonic oscillators in @xmath73 , isotropic or not , can be extended into harmonic oscillators in @xmath74 @xcite . the assumption @xmath20 can be generalized to @xmath75 . this leads to important generalizations , introduced in @xcite , that will be developed in a paper in preparation ( cdr ) . the extension procedure can be in this case applied with @xmath39 substituted by any positive rational @xmath76 after a suitable definition of @xmath77 , so that the generalized ttw system of above , with @xmath78 , can be written as an extension for any rational @xmath70 . to classical extended hamiltonians and their first integrals can be associated quantum hamiltonians and symmetry operators by some procedure of quantization , usually in form of laplace - beltrami operators . when the curvature of @xmath9 is not constant , the quantization requires additional terms ( quantum corrections ) in order to keep integrability or superintegrability . the quantum correction is then determined by the scalar curvature and by the weyl tensor @xcite . at least in the case @xmath79 , the simultaneous quantization of @xmath17 and @xmath80 is possible , allowing the preservation of maximal superintegrability of ms classical extended systems to the quantum limit . this will be shown in a paper in preparation ( cdr ) .

given an n - dimensional natural hamiltonian l on a riemannian or pseudo - riemannian manifold , we call `` extension '' of l the n+1 dimensional hamiltonian @xmath0 with new canonically conjugated coordinates @xmath1 . for suitable l , the functions @xmath2 and @xmath3 can be chosen depending on any natural number m such that h admits an extra polynomial first integral in the momenta of degree m , explicitly determined in the form of the m - th power of a differential operator applied to a certain function of coordinates and momenta . in particular , if l is maximally superintegrable ( ms ) then h is ms also . therefore , the extension procedure allows the creation of new superintegrable systems from old ones . for m=2 , the extra first integral generated by the extension procedure determines a second - order symmetry operator of a laplace - beltrami quantization of h , modified by taking in account the curvature of the configuration manifold . the extension procedure can be applied to several hamiltonian systems , including the three - body calogero and wolfes systems ( without harmonic term ) , the tremblay - turbiner - winternitz system and n - dimensional anisotropic harmonic oscillators . we propose here a short review of the known results of the theory and some previews of new ones .

1310.5840

nearby dark clouds like taurus and perseus contain dozens of dense molecular cores where stars like our sun are currently forming or have done so in the recent past ( myers 1995 ) . their large number , together with their proximity and simple structure , make cores unique targets to study the complex physics involved in the formation of a star . dense cores that have not yet formed stars , the so called starless or pre - stellar cores , inform us of the initial conditions of star formation , and their study can help us elucidate the process by which pockets of cloud material condense and become gravitationally unstable . cores with deeply embedded young stellar objects ( `` protostellar cores '' ) are unique targets to study the complex motions that occur during the period of accretion , when a combination of infall , outflow , and rotation is necessary to assemble the star and redistribute the gas angular momentum . finally , evolved cores are primary targets to study the interaction between the newly born star and its environment . these feedback effects are responsible for the transition of the protostar from embedded to visible , and may be important determining the final mass of the star and stabilizing the nearby gas via turbulence generation . the observational study of dense cores has advanced enormously over the last decade thanks to the increase in resolution provided by the new millimeter and submillimeter interferometers , and also due to the systematic combination of observations of dust and molecular tracers ( e.g. , bergin & tafalla 2007 ) . this brief review summarizes some new results from dense cores studies and presents a number of current issues that will greatly benefit from alma observations . the limited space of this article makes any attempt to review the field necessarily incomplete , and the reader is referred for further information to the other contributions on star formation in these proceedings , in particular to those by van dishoeck , andr , shepherd , aikawa , wilner , johnstone , and crutcher . despite significant recent progress , our understanding of the structure and evolution of dense cores is still incomplete due in part to limitations in the resolution and sensitivity of the available observations . even the highest resolution data of nearby dense cores can not discern details finer than about 100 au , which is still insufficient to disentangle the complex kinematics of infall and outflow motions in the vicinity of a protostar . probably more important , the low temperatures of the gas and the dust in cores ( @xmath0 k ) make the emission of any core tracer intrinsically weak , so any increase in the resolution needs to be accompanied by a parallel increase in the sensitivity , or the observations will not achieve enough s / n to provide useful information . this is particularly important when using weak , optically thin tracers to sample the innermost gas in the core . these tracers , in addition , often present extended emission , which poses a problem to the current generation of interferometers that cover sparsely the @xmath1 plane and therefore suffer systematically from missing flux . the high resolution and collecting area afforded by alma , combined with its great sensitivity to extended emission , promises to revolutionize the field of dense cores studies . on the one hand , alma will allow studying the dense cores of nearby clouds with the greatest detail , achieving subarcsecond resolution with high sensitivity . on the other hand , alma will permit the systematic study of dense cores in more distant clouds , enlarging the sample of available targets from the current set of the nearest clouds to cores at distances of at least 1 kpc . the earliest phase of a core , the so - called starless or pre - stellar stage , is characterized by the lack of a point - like object at its center ( e.g. , di francesco et al . 2007 ) . this characterization is of course dependent on the current sensitivity limits of the observations , and is therefore susceptible of misclassifying a core with an embedded source of very low luminosity ( see the case of vellos below ) . still , the significant number of dense cores with no pointlike source detected even after deep spitzer space telescope observations suggests that a population of truly starless cores exists in nearby clouds like taurus ( werner et al . 2006 ) . starless cores present systematically a close to constant density of @xmath2-@xmath3 @xmath4 over the central 5000 - 10000 au followed by an almost power - law drop at large distances . this central flattening of the density profile has been observed in a number of cores using different observational techniques , like millimeter dust continuum emission ( ward - thompson et al . 1999 ) , mir absorption ( bacmann et al . 2000 ) , and nir extinction ( alves et al . 2001 ) , and therefore constitutes a robust result of recent core studies . the presence of a density flattening provides further evidence that starless cores have not yet developed a central singularity , and that they are of pre - stellar nature . the physical origin of the flattening , however , is still a matter of debate , as a number of interpretations are consistent with it . the most natural one is that the profile results from an equilibrium configuration in which the pressure of an isothermal gas balances its gravitational attraction , the so called bonnor - ebert profile ( e.g. , alves et al . 2001 ) . indeed , the gas temperature in a core is typically close to constant ( @xmath0 k ) , and the associated thermal pressure dominates the turbulent component by a factor of several ( e.g. , tafalla et al . the bonnor - ebert interpretation , however , seems in conflict with the non - spherical shape of most cores ( typical axial ratio is 2:1 , myers et al . 1991 ) , and with the fact that the density contrast observed in cores often exceeds the factor of 14 limit for stability of the bonnor - ebert analysis ( bacmann et al . 2000 ) . additional magnetic field support could be responsible for these deviations from the theoretical expectation , but unfortunately , the observation of this magnetic component is extremely hard to make ( see contribution from crutcher in this volume ) . even the apparently `` simple '' structure of the cores still eludes our understanding . when the density distribution of a core , as inferred from dust measurements , is compared with the observed emission from most molecular tracers , it is commonly found that they disagree significantly . as illustrated in fig . 1 for l1498 in taurus , the dust emission of a core often appears centrally concentrated ( with of course a relative flattening at the center ) , while all molecular species but nh@xmath5 and n@xmath6h@xmath7 present ring - like distributions around the continuum peak . radiative transfer analysis of the molecular emission indicates that the abundance of most species drops by at least a factor of 10 towards the high density peak of the molecular core ( caselli et al . 1999 , bergin et al . 2002 , tafalla et al . 2002 ) . such strong abundance decrease is suffered by all the c - bearing molecules as well as other species ( like so ) , while it does not affect significantly nh@xmath5 or n@xmath6h@xmath7 ( see di francesco et al . 2007 and bergin & tafalla 2007 for reviews ) . nh@xmath5 seems in fact to be enhanced toward the center of most cores ( tafalla et al . 2002 ) , while the n@xmath6h@xmath7 abundance tends to have a constant value or may drop at the very center of some cores ( bergin et al . 2002 , pagani et al . 2005 ) . cores therefore have a differentiated ( onion - like ) molecular composition , with a center rich in nh@xmath5 and n@xmath6h@xmath7 and a series of outer layers containing c - bearing species . the inhomogeneous composition of the starless dense cores most likely results from the freeze out of the main molecular species onto the cold dust grains at the center ( bergin & langer 1997 , aikawa et al . 2005 ) . the high densities and low temperatures typical of dense core centers make the freeze out time ( @xmath8 yr ) become much shorter than the core dynamical scale ( @xmath9 1 myr ) , and as a consequence , species like co disappear rapidly from the gas phase . other molecular species suffer the same fate as co , but more importantly , the original chemical balance , characterized by a relative large co abundance ( @xmath10 ) , is changed dramatically by freeze out . a new chemical balance emerges , and it is characterized by the enhancement of certain n - bearing species , like n@xmath6h@xmath7 , which are daughter products of n@xmath6 and whose abundance is controlled by the amount of co in the gas phase ( co is the main destroyer of n@xmath6h@xmath7 ) . even as n@xmath6 freezes out on the dust grains with a similar binding energy as co ( berg et al . 2005 ) , the n@xmath6h@xmath7 abundance can increase relatively from its value in the diffuse cloud ( where co is undepleted ) and give rise to the relatively `` high '' abundances ( few 10@xmath11 ) typical of dense cores . nh@xmath5 can then form from n@xmath6h@xmath7 via dissociative recombination ( geppert et al . 2004 ) , giving rise to the observed central enhancement ( aikawa et al . 2005 ) . another effect of the co depletion in cores is the enhancement of deuterated species . deuteration at the low ( 10 k ) temperature of dense cores occurs via the enhancement of h@xmath6d@xmath7 , which then passes the deuterium atom to other species via ion - molecule reactions ( dalgarno & lepp 1984 ) . as h@xmath6d@xmath7 is mainly destroyed by co , the depletion of co further enhances the h@xmath6d@xmath7 abundance , which in turn enriches in deuterium a number of additional species . high abundance of h@xmath6d@xmath7 has in fact been observed in the heavily co - depleted dense core l1544 ( caselli et al . 2003 ) , and a correlation of co depletion and high deuteration has been reported by bacmann et al . ( 2003 ) and crapsi et al . this deuteration in the cold and dense pre - stellar phase is responsible for the extreme deuteration values of species like h@xmath6co , ch@xmath5oh , and nh@xmath5 seen toward protostellar cores ( ceccarelli et al . 1998 , roueff et al . 2000 , van der tak et al . as cores evolve , they are expected to become more and more centrally concentrated until they reach the point of gravitational instability . one of the most pressing issues in star formation studies is to understand whether this process of concentration is driven by the loss of magnetic field support via ambipolar diffusion ( e.g. , shu et al . 1987 , mouschovias & ciolek 1999 ) or by the dissipation of turbulence via shocks ( e.g. , maclow & klessen 2004 ) . observations of dense cores can not yet distinguish between these scenarios , but do show a systematic correlation between central concentration and other indicators of evolution , like co depletion and deuterium fractionation ( crapsi et al . 2005 ) . evidence for inward motions also seems correlated with central concentration , and this suggests that some cores that we see now as starless have already begun collapsing to form stars . one of the best candidates for such a collapsing system is the l1544 core in taurus , whose pattern of inward motions has been studied in a number of molecules ( tafalla et al . 1998 , williams et al . 1999 , caselli et al . the l1544 dense core is characterized by a high central density and concentration ( ward - thompson et al . 1999 , tafalla et al . 2002 ) , a high degree of co depletion and deuterium fractionation ( caselli et al . 1999 , 2002 ) , and seems starless despite deep spitzer space telescope observations in the ir ( bourke , private communication ) . clearly this core , an similar objects , will be prime targets for alma observations . cores more evolved than l1544 are expected to contain already a luminous object surrounded by an envelope of accreting material . the little observable difference between the pre and proto - stellar phases of a core is illustrated by the case of l1521f , a core initially thought from molecular data to be an almost twin of l1544 ( crapsi et al . 2004 ) and later found with spitzer observations to have a luminous central star ( bourke et al . the central object in l1521f has a luminosity close to 0.1 l@xmath12 , and is characteristic of a new group of objects identified by the spitzer telescope and usually referred as vellos ( very low luminosity objects ) . these vellos seem associated with very weak nir nebulosity and low velocity bipolar outflows ( bourke et al . 2005 ) , and their status in the evolutionary sequence of protostars is still unclear . although some vellos could represent precursors of substellar objects ( proto brown dwarfs ) , it seems more likely that in the case of l1521f we are witnessing the very first moments of accretion , when the central source has an extremely low mass . the proto brown dwarf alternative is unlikely in this case because the dense core has about 5 m@xmath12 of mass ( crapsi et al . 2004 ) , and no clear perturbation seems stopping the accretion ( the outflow has too little mechanical power ) . the pristine nature of vellos makes them ideal candidates to study star - forming infall motions . the study of these motions has a long and rich tradition , and is plagued by difficulties as illustrated by the case of b335 . this dense core harbors a very young ( class 0 ) object whose inward motions were first characterized by zhou et al . these authors found that the spectral signatures from this core are in good agreement with the expectation from the inside - out collapse model of shu ( 1977 ) . high resolution observations with the plateau de bure interferometer by wilner et al . ( 2000 ) , however , have shown that some of the signatures of `` infall '' ( like the high velocity wings in the cs lines ) arise in fact from outflow acceleration , and not from an increase in velocity of the infalling material as it approaches the central object . a revisit of b335 ( and similar objects ) making use of alma s high angular resolution and selecting appropriate ( i.e. , depletion resistant ) tracers is therefore needed to clarify the still confusing picture of star - forming infall motions . the clean appearance of some vellos , together with their weaker outflow emission , offers an interesting alternative to the more evolved ( and massive ) objects like b335 , that have fully developed outflows . because of their lower mass , vellos may present weaker signatures of infall and may be tracing the very first moments of collapse . the combined study of vellos and more luminous class 0 and class i sources should therefore allow us to reconstruct the sequence of star - forming accretion as a function of time . the presence of a protostar at the center of a core affects not only the gas kinematics but its chemistry . the newly born star heats up the nearby gas and dust introducing a temperature gradient in its vicinity . in the @xmath13 au region where the dust temperature exceeds the co evaporation temperature ( @xmath14 - 30 k ) , this molecule returns to the gas phase and undoes part of the chemical processing that occurred during the pre - stellar phase ( jrgensen et al . 2004 , jrgensen 2004 ) . closer to the protostar ( @xmath15 au ) , the dust temperature reaches the 90 - 100 k value at which water evaporates from the grains , further enriching the chemistry . observations of some very young protostellar objects , like iras 16293 - 2422 , show that these very small regions have extreme abundance of a number of complex molecules like hcooh , hcooch@xmath5 , and ch@xmath5och@xmath5 ( cazaux et al . 2003 , bottinelli et al . 2004 ) . the chemical richness of these regions rivals that of the hot cores around massive protostars , justifying their common denomination as `` hot corinos '' ( ceccarelli et al . the exact origin of the complex molecules in these regions , however , is still not fully understood . one possibility is that they result from direct evaporation of species trapped in the water ice , while an alternative is that they result from the processing of simpler evaporated molecules . even the geometry of hot corinos remains unknown , with the innermost part of the envelope or a more stable disk - like distribution as the most likely locations . despite these temporary uncertainties , hot corinos offer a unique opportunity to study the innermost vicinity of low - mass protostars . their distinctive chemical composition makes them highly selective tracers of the most complex and interesting region of the protostar , where inflow , outflow , and rotation motions play comparable roles , and angular momentum is transfered between different gas components . hot corino studies with alma will surely constitute some of the first scientific projects of the instrument . at the same time that protostars accrete material , they eject powerful bipolar outflows of supersonic speed . co observations of these outflows reveal masses that are too large to originate directly from the central protostar , and indicate that most of the moving gas is core ambient material accelerated by a collimated stellar wind ( lada 1985 ) . the lobes of bipolar outflows , in addition , commonly coincide with evacuated cavities seen via scattered light from the protostar , further illustrating how the outflow phenomenon represents a major disruption in the core internal structure ( padgett et al . 1999 ) . despite more than two decades of intense outflow research , a number of outstanding problems remain , and alma observations represent our current best hope to solve them ( see also contribution by d. shepherd in this volume ) . the properties of the underlying wind , for example , are not yet understood , and several alternative models have been proposed over the years . the two main types of models that attempt to fit the observations are the jet - driven outflow and the wind - driven shell , each of them with a number of flavors ( see bachiller 1996 for a review ) . despite significant successes , however , neither type of model can reproduce the rich variety of kinematic properties found by observations , so each of of them is necessarily incomplete ( lee et al . 2002 ) . in the jet driven model , a highly collimated agent shocks and sweeps cloud material along an almost straight line . this model succeeds in explaining the highly collimated co outflows often found toward class 0 objects , but fails to reproduce observations of less collimated flows ( usually powered by class i sources ) , where the co emission arises from gas along limb - brightened shells ( like l1551 , see moriarty - schieven et al . 1987 ) . to fit these less collimated systems , the jet models need to broaden the outflow path , and this has been done by either invoking jet precession/wandering ( masson & chernin 1993 ) or large - scale bow shocks ( raga & cabrit 1993 ) . none of these elements however seems consistent with observations ( see arce et al . 2007 for more details ) , and this leaves the jet models limited to fitting the youngest , and admittedly more spectacular , bipolar outflows . wind - driven models , on the other hand , naturally produce shell - like structures thanks to a wide - angle agent that sweeps ambient material ( shu et al . these models , unfortunately , do not reproduce the appearance of the highly collimated outflows or the mass - velocity distribution commonly observed even in the poorly collimated flows ( masson & chernin 1992 ) . a combination of high resolution observations and new developments in outflow modeling are starting to show a possible solution to the current impasse . interferometer mapping of the outflow powered by the very young source iras 04166 + 2706 in taurus shows both jet and shell features simultaneously ( see fig . 2 and poster contribution by santiago - garca et al . the jet - like feature in this outflow , seen in both co and sio emission , is extremely rectilinear , appears only at the highest velocities ( between 30 and 50 km s@xmath16 ) , and shows no evidence for precession or wandering . the shell - like part appears at low velocities ( 2 to 10 km s@xmath16 ) and seems to delineate two opposed cavities with the iras source at their vertex . this cavity interpretation is supported by the fact that the blue outflow shell coincides with the walls of a nir scattering nebula seen in spitzer images , as expected from its more favorable projection . in addition , the high velocity jet runs along the axis of the two cavities showing a remarkable degree of symmetry ( see poster contribution for further details ) . the data from iras 04166 + 2706 , therefore , leads to the inevitable conclusion that , at least in some cases , both highly collimated and wide - angle components coexist in the outflow driving agent , and that a model that considers both components simultaneously is needed to explain the observations . interestingly enough , recent realistic modeling of the interaction between the x - wind of shu et al . ( 1994 ) and a toroidal core shows that both jet and shell components should be observed simultaneously in very young outflows ( shang et al . this so - called `` unified '' model of bipolar flows shows in fact a remarkable likeness with the iras 04166 + 2706 observations , both in geometry and kinematics ( compare fig . 2 and the models in shang et al . 2006 ) . the unified outflow model not only unifies the jet and wide - angle aspects of the outflows , but also brings together the evolution of flows and the dense cores , two elements often treated separately . evidence for outflow - core interaction has been reported in a number of systems ( e.g. , tafalla & myers 1997 , arce & sargent 2006 ) , but no unified framework of how this interaction happens or how outflows and cores evolve in parallel exists yet . the beautiful simulations of shang et al . ( 2006 ) illustrate how the most important elements of this interaction occur inside the central 1000 au region , which corresponds to less than @xmath17 even towards the most nearby clouds . high angular resolution observations with alma are clearly needed to sample the complex geometry and kinematics inside this critical region , and thus compare real outflows with their simulated counterparts . producing a unified picture of the different and interacting processes occurring during the formation of a solar - type star can be one of most significant achievements of alma .

dense cores are the simplest star - forming sites that we know , but despite their simplicity , they still hold a number of mysteries that limit our understanding of how solar - type stars form . alma promises to revolutionize our knowledge of every stage in the life of a core , from the pre - stellar phase to the final disruption by the newly born star . this contribution presents a brief review of the evolution of dense cores and illustrates particular questions that will greatly benefit from the increase in resolution and sensitivity expected from alma .

astro-ph0702691

the primary aim of the efar project ( wegner 1996 ; paper 1 ) is to use the tight correlations between the global properties of early - type galaxies embodied in the fundamental plane ( fp : djorgovski & davis 1987 , dressler 1987 ) to measure relative distances to clusters of galaxies in order to investigate peculiar motions and the mass distribution on large scales . however these global relations also constrain the dynamical properties and evolutionary histories of early - type galaxies . for example , renzini & ciotti ( 1993 ) show that the tilt of the fp implies a range in mass - to - light ratio @xmath2 among ellipticals of less than a factor of three , while the low scatter about the fp implies a scatter in @xmath2 at any location in the plane of less than 12% . similar reasoning has been used to constrain the star formation history of cluster ellipticals using the colour magnitude relation ( bower 1992 , kodama & arimoto 1997 ) . recently the fp , and colour magnitude relations have been followed out to higher redshifts and used to show that the early - type galaxies seen at @xmath9@xmath101 differ from present - day early - type galaxies in a manner consistent with passive evolutionary effects ( van dokkum & franx 1996 , ziegler & bender 1997 , kelson 1997 , ellis 1997 , kodama & arimoto 1997 , stanford 1998 , kodama 1998 , bender 1998 , van dokkum 1998 ) . in this paper we consider the relation between the central velocity dispersion @xmath11 and the strength of the magnesium lines at a rest wavelength of 5174 for the early - type galaxies in the efar sample . this relation connects the dynamical properties of galaxy cores with their stellar populations . the remarkably small scatter about this relation ( burstein 1988 , guzmn 1992 , bender 1993 , jrgensen 1996 , bender 1998 ) , and its distance - independent nature , make it a potentially useful constraint on models of the star formation history of early - type galaxies and a test for environmental variations in the fp ( burstein 1988 , bender 1996 ) . there are , however , some problems with using the relation for probing galaxy formation . two of these problems are apparent from the stellar population models ( worthey 1994 , vazdekis 1996 ) : ( i ) both age and metallicity contribute to the mg linestrengths in comparable degree , so that a spread in linestrengths could be due to either a range of ages or a range of metallicities or some combination ; ( ii ) the mg linestrengths are not particularly sensitive indicators at fixed metallicity a difference in age of a factor of ten only results in a change of 0.05 - 0.1 mag , while at fixed age a difference of 1 dex in metallicity gives a change of 0.10.2 mag . thus mg linestrength measurements must be accurate in order to yield useful constraints on the ages and metallicities of stellar populations , and the relation on its own can only supply constraints on combinations of age and metallicity and not one or the other separately . recently trager ( 1997 ) has suggested that the tightness of the relation may be the result of a ` conspiracy ' , in that there appears to be an anti - correlation between the ages and metallicities of the stellar populations in early - type galaxies at fixed mass which acts to reduce the scatter in the mg linestrengths . trager takes the accurate h@xmath12 , mg and fe linestrengths from gonzlez ( 1993 ) and applies the stellar population models of worthey ( 1994 ) to derive ages and abundances from line indices with different dependences on age and metallicity . he finds that at fixed velocity dispersion the ages and abundances lie in a plane of almost constant mg linestrength , leading him to predict little scatter in the relation even for large differences in age or metallicity a factor of ten in age ( from 1.5 gyr to 15 gyr ) gives a spread in of only 0.010.02 mag . this conclusion depends on the appropriateness of the single stellar population models and requires confirmation from further high - precision linestrength measurements . it can also be tested using the high - redshift samples now becoming available . in a similar vein , a number of authors ( ferreras 1998 , shioya & bekki 1998 , bower 1998 ) have recently re - examined whether the apparent passive evolution of the colour magnitude relation out to @xmath9@xmath101 really implies a high redshift for the bulk of the star - formation in elliptical galaxies . they conclude that in fact such evolution can be consistent with a rather broad range of ages and metallicities if the galaxies assembling more recently are on average more metal - rich than older galaxies of similar luminosity . as well as studies focussing on the evolution of the galaxy population , there have also been investigations of possible variations with local environment . guzmn ( 1992 ) have suggested that there are systematic variations in the relation which affect estimates of relative distances based on the fp . they report a significant offset in the zeropoint of the relation between galaxies in the core of the coma cluster and galaxies in the cluster halo . jrgensen and co - workers ( 1996 , 1997 ) examine a sample of 11 clusters and find a weak correlation between mg linestrength and local density within the cluster which is consistent with this result . similar offsets are claimed between field and cluster ellipticals by de carvalho & djorgovski ( 1992 ) and jrgensen ( 1997 ) , although burstein ( 1990 ) find no evidence of environmental effects . such systematic differences could result from different star - formation histories in different density environments , producing variations in the mass - to - light ratio of the stellar population . fp distance measurements would then be subject to environment - dependent systematic errors leading to spurious peculiar motions . where data for field and cluster ellipticals come from different sources , however , the possibility also exists that any zeropoint differences are due to uncertainties in the relative calibrations rather than intrinsic environmental differences . the relation has thus become an important diagnostic for determinations of both the star formation history and the peculiar motions of elliptical galaxies . here we examine the relation in the efar sample , which includes more than 500 early - type galaxies drawn from 84 clusters spanning a wide range of environments . in 2 we summarise the relevant properties of the sample and the techniques used to determine the and linestrength indices , the central velocity dispersions @xmath11 , and the errors in these quantities . we present the relation in 3 and investigate how it varies from cluster to cluster within our sample , and with cluster velocity dispersion , x - ray luminosity and x - ray temperature . in 4 we compare our results with the predictions of stellar population models in order to derive constraints on the ages , metallicities and mass - to - light ratios of early - type galaxies in clusters . in particular , we consider the constraints on the dispersion in the ages and metallicities from the intrinsic scatter in the relation on its own , and in combination with the intrinsic scatter in the fp . our conclusions are given in 5 . here we give a short description of our sample and dataset , with emphasis on the velocity dispersions and line indices used in this paper . the interested reader can find more detail on the sample selection in paper 1 ( wegner 1996 ) ; on the measurement , calibration and error estimation procedures for the spectroscopic parameters in paper 2 ( wegner 1998 ) ; and on the structural and morphological properties of the galaxies in paper 3 ( saglia 1997 ) . the efar sample of galaxies comprises 736 mostly early - type galaxies in 84 clusters . these clusters span a range of richnesses and lie in two regions toward hercules corona borealis and perseus pisces cetus at distances of between 6000 and 15000 . in addition to this program sample we have also observed 52 well - known galaxies in coma , virgo and the field in order to provide a calibrating link to previous studies . the efar galaxies are listed in table 2 of paper 1 , and comprise an approximately diameter - limited sample of galaxies larger than about 20 arcsec with the visual appearance of ellipticals . photometric imaging ( paper 3 ) shows that 8% are cds , 12% are pure es and 49% are bulge - dominated e / s0s ; thus 69% of the sample are early - type galaxies , with the remaining 31% being spirals or barred galaxies . we have obtained spectroscopy for 666 program galaxies , measuring redshifts , velocity dispersions and linestrength indices ( paper 2 ) . we have used the redshifts we obtained together with literature redshifts for other galaxies in the clusters in order to assign program galaxies to physical clusters . we have used the combined redshift data for these physical clusters to estimate cluster mean redshifts and velocity dispersions . the early - type galaxies in our sample span a wide range in luminosity , size and mass : they have absolute magnitudes from @xmath13=@xmath824 to @xmath818 ( @[email protected] ; h@xmath15=50kms@xmath16mpc@xmath16 ) , effective radii from 1 to 70 ( @xmath17=9.1 ) and central velocity dispersions from less than 100 to over 400 ( @xmath18=220 ) . the sample is thus dominated by early - type galaxies with luminosities , sizes and masses typical of giant ellipticals . we summarise here the procedures used in measuring the redshifts , velocity dispersions and mg linestrengths ; full details are given in paper 2 . = redshifts and velocity dispersions were measured from each observed galaxy spectrum using the iraf task fxcor . linestrength indices on the lick system were determined using the prescription given by gonzlez ( 1993 ) . the and indices were both measured : because it is the index most commonly measured in previous work , and because it could be measured for more objects ( as it requires a narrower spectral range ) and is better - determined ( being less susceptible to variations in the non - linear continuum shape ) . we find it more convenient to express the ` atomic ' index in magnitudes like the ` molecular ' index rather than as an equivalent width in ngstroms , since this puts these two indices on similar footings . the conversion is @xmath19 where @xmath20 is the index bandpass ( 32.5 for ) . error estimates for each quantity were derived from detailed monte carlo simulations , calibrated by comparisons of the estimated errors with the results obtained from repeat measurements ( over 40% of our sample had at least two spectra taken ) . two sorts of corrections were applied to the dispersions and linestrengths : ( i ) an aperture correction , based on that of jrgensen ( 1995 ) , to account for different effective apertures sampling different parts of the galaxy profile , and ( ii ) a run correction to remove systematic errors between different observing setups . after applying these corrections , individual measurements for each galaxy were combined using a weighting scheme based on the estimated errors and the overall quality of the spectrum . = = the median estimated errors in the final combined values are @xmath21 ( @xmath22 dex ) , @xmath23 mag and @xmath24 mag . the distribution of estimated errors for each quantity is shown in the upper panel of figure [ fig : errsum ] . the lower panel of the figure shows how the error estimates were calibrated against the repeat observations : the distribution of the ratio of rms error to estimated error for objects with repeat measurements is compared to the predicted distribution assuming the estimated errors are the true errors . the initial error estimates from the simulations have been re - scaled to give the best match ( under a k - s test ) to the rms errors from the repeat measurements . a re - scaling by factors of 0.85 and 1.15 respectively gives good agreement for the errors in @xmath11 and ; adding 0.005 mag likewise gives good agreement for the errors in . a comparison with the literature ( paper 2 , figure 13 ) shows that our dispersions are consistent with previous measurements by davies ( 1987 ) , guzmn ( 1993 ) , jrgensen ( 1997 ) , lucey ( 1997 ) and whitmore ( 1985 ) . for the subset of galaxies in common , we compared our linestrengths with the definitive lick system measurements of trager ( 1998 ) in order to derive the small zeropoint corrections required to calibrate our measurements to the lick system ( paper 2 , figures 14 & 15 ) ; the overlap of our measurements with those of lucey ( 1997 ) also shows consistency ( figure 16 , paper 2 ) . in this section we investigate the global relation found amongst the entire sample of efar galaxies with early - type morphological classifications ( cd , e or e / s0 ; see definitions in paper 3 ) for which we obtained linestrength measurements . the relation is shown in figure [ fig : mgsig]a and the relation in figure [ fig : mgsig]b . in order to fit linear relations with intrinsic scatter in the presence of significant measurement errors in both variables , arbitrary censoring of the dataset and a broad sample selection function , we have developed a comprehensive maximum likelihood ( ml ) fitting procedure ( saglia , in preparation ) . excluding galaxies with dispersions less than 100or selection probabilities less than 10% , and also outliers with low likelihoods , the ml fits to the relation ( 490 galaxies ) and the relation ( 423 galaxies ) are : @xmath25 these fits are shown in figure [ fig : mgsig ] as solid lines . the ratio of the slopes of these relations is consistent with the -relation we obtained in paper 2 : @[email protected] . monte carlo simulations of the dataset and fitting process , the results of which are displayed in figure [ fig : mgsigsim ] , show that there is no bias in the ml estimates of the slopes and zeropoints , and provide reliable estimates of the uncertainties in the fit . = + = the ml fits can be compared to simple regressions of and on @xmath26 . these regressions are shown in the figure as dashed lines , and are : @xmath27 as expected , the simple regressions yield slopes which are biased low due to the presence in the data of errors in the abscissa as well as the ordinate , and also the intrinsic scatter in the relation . slightly less - biased results are obtained by least squares regression minimising the orthogonal residuals ( jrgensen 1996 ) : @xmath28 these least squares fits and their uncertainties are obtained using the slopes regression program written by e.d.feigelson and described in isobe ( 1990 ) and feigelson & babu ( 1992 ) . the uncertainties are under - estimated because these regressions do not properly account for the measurement errors or the selection functions . we conclude that previous determinations of the slope of the relation are likely to be biased low whenever the dataset being fitted had significant errors in the velocity dispersions ( as is generally the case ) . hereafter we adopt the ml fits to the relation . the distributions of the residuals in and about the ml fits are shown in the insets to figures [ fig : mgsig]a and [ fig : mgsig]b . in order to minimise the effects of outliers , we robustly estimate the scatter about the relations as half the range spanned by the central 68% of the data points . we find an _ observed _ scatter of [email protected] mag about the relation and [email protected] mag about the relation . excluding outliers , the distributions of residuals are very well fitted by gaussians parametrised by the median residual and the robustly estimated scatter . there is no evidence for a tail of negative residuals such as noted by burstein ( 1988 ) and jrgensen ( 1996 ) . as the latter authors point out , the presence of such a tail is sensitive to the adopted slope of the relation . some giant ellipticals do , however , have intrinsically weak mg linestrengths for their velocity dispersions ( schweizer 1990 ) . the estimates of the _ intrinsic _ scatter about the relations that are provided by the ml fit may be exaggerated by outliers or by deviations of the underlying distribution of galaxies in the plane from a bivariate gaussian . we therefore drop the assumption of an intrinsic bivariate gaussian distribution in the plane and use monte carlo simulations based on the observed distribution of dispersions and linestrengths and their estimated errors ( accounting for both measurement errors and run correction errors ) . these simulations assume only that there is a global linear relation about which there is gaussian intrinsic scatter . we vary this intrinsic scatter and compute the robust estimate of the observed scatter about the fit ( the half - width of the central 68% of the residuals ) for the simulated distributions . the results of these simulations are presented in figure [ fig : mgscat ] , which shows the normalised likelihood distributions for the intrinsic scatter in and given the observed scatter . we find that to account for the observed scatter in the relations we require an intrinsic scatter of [email protected] mag for and [email protected] mag for . the ratio of the intrinsic scatter in to the intrinsic scatter in is slightly lower than expected from the observed relation , @[email protected] ( see paper 2 ) . = .comparison of relation fits [ cols="<,^,^,^,^ " , ] = = table [ tab : mgsig ] compares our fits to the relations obtained by other authors , and gives the observed scatter @xmath30mg@xmath31 and the intrinsic scatter @xmath30mg@xmath32 in the relations obtained in each case . for both and the slopes we obtain are about 25% steeper than those obtained by most previous authors . this is not due to a difference in our data , but stems from our use of the ml method rather than regressions . in this situation regressions are biased towards flatter slopes than the true relation because they ignore the intrinsic scatter , the presence of errors in both variables and the selection function of the dataset . the standard or orthogonal regression fits to our data , which our simulations show under - estimate the slope of the relations , give results very similar to those obtained by other authors . if we divide the sample by morphological type , we find that the cds have a zeropoint which is 0.009 mag higher than that of the other early - type galaxies in , and 0.014 mag higher in . these differences in the zeropoints are readily apparent in the distributions of residuals about the global relations ( see the insets to figures [ fig : mgsig]a&b ) , and are significant at the 3@xmath11-level . despite these zeropoint offsets , including or excluding the cds changes the scatter about the ml fit by less than its uncertainty , as they make up only 10% of the whole sample . we find no significant differences , however , if we compare the relations for the two volumes of space , the hercules - corona - borealis and perseus - pisces - cetus regions , from which our sample is drawn . the two regions have relations with slopes and zeropoints which are consistent both with each other and with the overall relations , providing a check that there are no gross systematic environmental differences between these two regions . we do not have enough galaxies per cluster to fit both the slope and the zeropoint of the relations on a cluster - by - cluster basis , even in our best - sampled clusters . we therefore limit ourselves to investigating the variation in the zeropoint . to this end we measured the median offset in and from the global fits given above for the clusters with three or more linestrength measurements ( 75 clusters for and 72 for ) . note that we only used galaxies that are cluster members based on their redshifts ( see paper 2 ) . the results are not changed significantly if we use all clusters , or only clusters with five or more measurements . = = the top panels of figure [ fig : mgclus ] show these zeropoint offsets as a function of cluster i d number ( cid ) , while the middle panels show the distributions of the offset values . the robustly - estimated scatter in the zeropoint offsets is [email protected] mag in and [email protected] mag in , showing that the relations are remarkably uniform among the aggregates of galaxies in the efar sample . the bottom panels in the figure plot the same offsets as a function of redshift , showing that there is no dependence of the relations on relative distance within the sample . this scatter in the zeropoint offsets could purely be a consequence of a galaxy - to - galaxy scatter in a global relation , or it could also require a variation in the zeropoint of the relation from cluster to cluster . these possibilities were examined by extending the simulations described in the previous section , adding a further source of scatter to the relation in the form of an intrinsic variation between clusters in the zeropoint of the relation . for simplicity we assume that this variation also has a gaussian distribution . we find that if we make the extreme assumption that there is cluster - to - cluster scatter but no intrinsic scatter between galaxies within a cluster , then zeropoint variations between clusters with an rms of 0.009 mag in and 0.015 mag in are required to recover the observed cluster - to - cluster scatter . however this model under - predicts the observed scatter about the global relation , giving [email protected] mag for and [email protected] mag for compared to the actual values of [email protected] mag and [email protected] mag . on the other hand , if we assume that there is no zeropoint variation between clusters , then the intrinsic scatter between galaxies required to recover the observed scatter in the global relation ( 0.016 mag in and 0.023 mag in ; see previous section ) predicts a scatter in the cluster zeropoints of [email protected] mag in and [email protected] mag in , which is consistent with the observed values of [email protected] mag and [email protected] mag within the joint errors . we conclude that there is no evidence for significant intrinsic zeropoint variations between clusters , since sampling a galaxy population drawn from a single global relation with intrinsic scatter consistent with the observations can account for the zeropoint differences between our clusters . as there is very little change in the zeropoint of the relation from cluster to cluster , it follows that there can be at most only a weak dependence of the zeropoint on the properties of the clusters . here we investigate the effect of cluster properties on the stellar populations as reflected in the zeropoints , considering cluster velocity dispersions , x - ray luminosities and x - ray temperatures ( all indicators of cluster mass ) . the cluster dispersions come from table 7 of paper 2 , using redshifts both from efar and from the zcat catalogue ( huchra 1992 ; version of 1997 may 29 ) . x - ray luminosities and temperatures are available for 26 of our 84 clusters in the homogeneous and flux - limited catalogue of x - ray properties of abell clusters by ebeling ( 1996 ) based on rosat all - sky survey data . the x - ray luminosities are determined to a typical precision of about 20% . in order to have comparable precision in the cluster velocity dispersions , we only use clusters with dispersions computed from at least 20 galaxy redshifts ; this also leaves 26 clusters , 17 of which are in common with the x - ray subsample . figure [ fig : mgrich ] shows the offsets in the relations as functions of @xmath33 , @xmath34 and @xmath35 . applying the spearman rank correlation statistic , we find that there is no significant correlation between the offsets and any of these quantities , and thus no evidence for a trend in the zeropoint of the relation with cluster mass . weighted regressions give best - fit relations and their uncertainties : @xmath36 if we take a complementary approach , splitting the clusters into two subsamples about the median value of @xmath37 and fitting the relations to the galaxies of the high-@xmath37 and low-@xmath37 clusters separately , we again find no significant differences in the slopes or the zeropoints of the fits , which are compatible with the global fits obtained above . there are at least four main questions which can be addressed using the above results . \(i ) what are the theoretical implications of the lack of correlation between the mass of a cluster and the zeropoint of the relation for cluster galaxies ? \(ii ) what effect do the stellar population differences implied by the observed variations in the relation have on fundamental plane estimates of distances and peculiar velocities ? \(iii ) what constraint does the intrinsic scatter about the relation place on the spread in age , metallicity and mass - to - light ratio amongst early - type galaxies in clusters ? \(iv ) what further constraints on these quantities result from combining the scatter about the relation with the intrinsic scatter about the fundamental plane ? the small scatter in the zeropoint of the relation from cluster to cluster , and in particular the lack of correlation between the zeropoint and the cluster mass , seems to imply that the mass over - density on mpc scales in which an early - type galaxy is found has little connection with its stellar population and star - formation history . the variation of the relation with cluster properties has previously been studied in a sample of 11 nearby clusters by jrgensen ( 1996 ) and jrgensen ( 1997 ) . following guzmn ( 1992 ) , these authors look for a trend in offsets with the ` local density ' _ within _ clusters . the estimator of local density used is @xmath38 , where @xmath39 is the projected distance of the galaxy from the cluster centre . since @xmath39 is only a lower limit on the galaxy s true distance from the cluster centre , this is a rather poor estimator of the true local density . jrgensen find that the residuals in show a weak trend0.009 ( jrgensen , priv.comm . ) ] with local density , @xmath40 . since the residuals do _ not _ correlate with radius within the cluster ( see figure 3 of jrgensen ( 1997 ) ) , but _ do _ show a significant correlation with cluster velocity dispersion , @xmath41 ( least - squares fit to the data in figure 5 of jrgensen ( 1997 ) ) , we would argue that a more straightforward interpretation of their results is a correlation of zeropoint with total cluster mass rather than local density . a correlation of this amplitude is formally consistent at the 2@xmath11 level with the distribution of offsets versus @xmath33 for the efar data ( see equation [ eqn : clusfitmg2 ] ) ; transforming jrgensen s result via the relation gives a correlation which is consistent at the 1.4@xmath11 level with equation [ eqn : clusfitmgb ] . we conclude that any correlation between the relation zeropoint and the cluster mass is sufficiently weak ( of order @xmath42 or less ) that it is not reliably established by the existing data , which are consistent with no correlation at all . semi - analytic models for the formation of elliptical galaxies , which previously neglected metallicity effects ( see kauffmann 1996 , baugh 1996 ) , are only now beginning to incorporate chemical enrichment and successfully reproduce the general form of the observed colour magnitude and relations ( kauffmann & charlot 1998 ) . in consequence , there are as yet no reliable predictions for the variation of the relation zeropoint with cluster mass . the limits given above , together with limits on the difference in zeropoints for field and cluster ellipticals ( burstein 1990 , de carvalho & djorgovski 1992 , jrgensen 1997 ) , should provide valuable additional constraints and encourage further development of chemical enrichment models within a hierarchical framework for galaxy and cluster formation . we now consider the effects on fp distance estimates of systematic differences in the stellar populations of early - type galaxies from cluster to cluster . in [ ssec : clusvars ] we found that the observed cluster - to - cluster variations in the zeropoint were consistent with sampling a single global relation with intrinsic scatter between galaxies , and did not _ require _ intrinsic scatter between clusters . here we turn the question around and ask how much intrinsic cluster - to - cluster scatter is _ allowed _ by the observations . from simulations using the model described in [ ssec : clusvars ] , incorporating intrinsic scatter both between galaxies and between clusters , we find that the maximum cluster - to - cluster scatter allowed within the 1@xmath11 uncertainties in the scatter in the global relation and the cluster zeropoints is approximately 0.005 mag in and 0.010 mag in . for our best - fit ml relations and a fp given by @xmath39@xmath43@xmath44 with @[email protected] , this level of cluster - to - cluster scatter would lead to rms errors in fp distance estimates of up to 10% . these systematic errors , resulting from differences in the mean stellar populations between clusters , would apply even to clusters in which the fp distance errors due to stellar population differences between galaxies had been made negligible by observing many galaxies in the cluster . we emphasise that our results here do not _ require _ any cluster - to - cluster scatter , but are _ consistent _ with cluster - to - cluster scatter corresponding to systematic distance errors between clusters with an rms of up to 10% . we therefore can not determine from the relation _ alone _ whether systematic differences in the mean stellar populations between clusters contribute significantly ( or at all ) to the errors in fp estimates of distances and peculiar velocities . a more effective way of testing for such systematic differences is by directly comparing each cluster s zeropoint offset from the global relation to the ratio of its fp and hubble distance estimates ; this approach will be investigated in a future paper . to answer the questions concerning the typical age , metallicity and mass - to - light ratio of early - type galaxies which were raised at the start of this discussion , we need to employ stellar population models . we use the predictions from the single stellar population models of worthey ( 1994 ) and vazdekis ( 1996 ) , noting the many caveats given by these authors regarding their models . to simplify our analysis , we fit , and @xmath46 as linear functions of logarithmic age ( @xmath47 , with @xmath48 in gyr ) and metallicity ( @xmath49 ) , for galaxies with ages greater than 4 gyr and metallicities in the range @xmath80.5 to @xmath500.5 . for the model of worthey ( 1994 ; salpeter imf ) we obtain @xmath51 figure [ fig : mgbmlrw ] compares this fit to worthey s model in the case of the predicted dependence of and @xmath46 on age and metallicity . the figure shows that for ages of 5 gyr or greater the fit and the model are in satisfactory agreement for all metallicities . = for the model of vazdekis ( 1996 ; bimodal imf , @xmath52=1.35 ) we have @xmath53 in agreement with the fit obtained by jrgensen ( 1997 ) . there is good agreement between the predictions of the two models for the dependence of and on age and metallicity , and moderately good agreement for the dependence of @xmath54 . note that the same change in the mg indices is produced by changes in age , @xmath55 , and metallicity , @xmath56 , if @xmath57 . this is the ` 3/2 rule ' of worthey ( 1994 ) , which applies to many of the lick line indices , leaving them degenerate with respect to variations in age and metallicity . however age and metallicity produce the same change in @xmath46 only if @xmath58 or 1/4 , so that measurements of mass - to - light ratios can in principle be combined with mg linestrengths to break the age / metallicity degeneracy . in the following analysis we infer the dispersion in the ages and metallicities of early - type galaxies by comparing the scatter in the relation with the predictions of the single stellar population models described in the previous section . this analysis uses the stellar population models to predict differential changes in the quantities of interest , and not absolute values . it is also important to remember that by the dispersion in age or metallicity we mean the dispersion in these quantities at fixed @xmath26 or , equivalently , the dispersion after the overall trend with @xmath26 is accounted for . thus the dispersion in age or metallicity we infer is the dispersion at fixed galaxy mass , not the distribution of ages and metallicities as a function of galaxy mass ( which is related to the slope of the relation and the distribution of galaxies along it ) . single stellar populations models specified by ( amongst other parameters ) a unique age and a unique metallicity can only provide an approximation to real galaxies , whose stellar contents must necessarily span a range ( though perhaps a narrow one ) of ages and metallicities . since the global mg indices can be quite sensitive to the detailed metallicity distribution ( greggio 1997 ) , some of the scatter we observe may be due to galaxy - to - galaxy differences in the shape of the metallicity distribution rather than a dispersion in the mean metallicity or age . a further complication is presented by the over - abundance of mg with respect to fe ( compared to the solar ratio ) in the cores of early - type galaxies ( peletier 1989 , gorgas 1990 , worthey 1992 ) . as a comparison of figures [ fig : mgsig ] & [ fig : mgbmlrw ] shows , the models discussed in the previous section are unable to account for the highest observed mg linestrengths . tantalo ( 1998 ) have produced single stellar population models including the effects of [ mg / fe ] variations and find that @xmath59 + 0.089 \delta\log t + 0.166 \delta\log z / z_\odot \label{eqn : tantalo}\ ] ] comparing this equation with those above , we see that the differential dependence on age and metallicity is similar to that predicted by worthey ( 1994 ) and vazdekis ( 1996 ) . however , any intrinsic scatter in the [ mg / fe]@xmath11 relation will contribute additionally to the intrinsic scatter in the relation and reduce the dispersion in age and metallicity required to account for the observations . for these reasons , and also because of other potential sources of intrinsic scatter such as dark matter , rotation , anisotropy , projection effects and broken homology , the estimates of the dispersion in age and metallicity derived here must be considered as upper limits . with these caveats in mind , we proceed to use the model fits given in the previous section to infer the dispersion in age or metallicity based on the observed intrinsic scatter of 0.016 mag in and 0.023 mag in . for ease of interpretation we quote the dispersions in age and metallicity as the fractional dispersions @xmath60 and @xmath61 . in applying the models in what follows , we adopt the mean of the coefficients for the two models and give the dispersions in age and metallicity corresponding to the intrinsic scatter about the relation . using the intrinsic scatter obtained from the relation would give results that are @xmath1030% smaller , since the observed ratio of the intrinsic scatters is @[email protected] , rather than about 2 as would be expected either from the observed relation or from the models . we use the scatter in rather than because our goal is to establish upper limits on the dispersions in age and metallicity . the estimated errors in the intrinsic scatter lead to uncertainties in the dispersions of 510% . if age variations in single stellar populations are the only source of scatter then the dispersion in age is @xmath3=67% , whereas if metallicity variations are the sole source then the dispersion in metallicity is @xmath4=43% . similarly , the observed difference in the relation zeropoint for the cd galaxies implies that these objects are either older or more metal - rich than normal e or e / s0 galaxies . if the zeropoint differences are interpreted as age differences , cds are on average 40% older than typical e or e / s0 galaxies ( as old as the oldest early - type galaxies ) ; if the zeropoint differences are interpreted as metallicity differences , cds have metallicities on average 25% higher than typical e or e / s0 galaxies ( as high as the most metal - rich early - type galaxies ) . we can also use the model fits to estimate the approximate change in @xmath54 corresponding to a change in the mg line indices . if these changes are caused by age variations alone , then we find that @xmath63 and @xmath64 ; if , however , they are due only to variations in metallicity we have @xmath65 and @xmath66 . thus the change in @xmath67 is about 5 times larger if the observed change in the mg indices is due to age differences rather than metallicity differences . the intrinsic scatter in the relation implies a dispersion in mass - to - light ratio of 50% if due to age variations , but only 10% if due to metallicity variations . this predicted scatter in @xmath2 is in fact a scatter in luminosity or surface brightness ( since that is all the models deal with ) . we can therefore readily establish the effect of this scatter on distances estimated using the fundamental plane ( fp ) if the scatter in @xmath2 is uncorrelated with the galaxies sizes and dispersions , as indeed is the case for the efar sample ( at least for galaxies with @xmath11@xmath68100 ) . for a fp given by @xmath39@xmath43@xmath69 , with @xmath39 the effective radius and @xmath70 the mean surface brightness within this radius , if the scatter in @xmath2 is simply a scatter in @xmath70 we have @xmath71 . most determinations of the fp , including our own , yield @xmath72 ( dressler 1987 , jrgensen 1996 , saglia 1998 ) . combining this relation with the dependence of @xmath2 on the mg line indices obtained above , we find that the scatter in the relation corresponds to an intrinsic scatter in relative distances estimated from the fp of 40% if due to age variations , or 8% if due to metallicity variations . as the intrinsic scatter in the fp is found to be in the range 1020% ( djorgovski & davis 1987 , jrgensen 1993 , jrgensen 1996 ) , one can not explain both the scatter in the relation and the scatter in the fp as the result of age variations alone or metallicity variations alone ( unless the single stellar population models are incorrect or there are significant galaxy - to - galaxy differences in the metallicity distributions ) . suitable combinations of age variations and metallicity variations _ can _ , however , account for the measured intrinsic scatter in both the and fp relations . as a simple model , we assume that the scatter in the fp and the relations ( at fixed @xmath26 ) is entirely due to variations in age and metallicity ( at fixed galaxy mass ) . these variations are further assumed to have gaussian distributions in @xmath47 and @xmath49 with dispersions @xmath73 and @xmath74 and correlation coefficient @xmath7 ( @xmath81@xmath75@xmath7@xmath751 ) . while a gaussian distribution of metallicities at fixed galaxy mass is a reasonable initial hypothesis for describing variations in the chemical enrichment process , the single - peaked shape of the assumed lognormal distribution for the mean ages may not realistically represent the star - formation history ( even for galaxies of the same mass ) . the dispersion in age inferred under this model should therefore be considered only as a general indication of the time - span over which early - type galaxies of fixed mass formed the bulk of their stellar population . writing the scatter in mg linestrengths and fp residuals as @xmath76 and @xmath77 and the dispersion in @xmath47 and @xmath78 as @xmath79 and @xmath80 , this simple model relates the scatter in the observed quantities to the dispersion in age and metallicity by : @xmath81 here @xmath82 and @xmath83 are the coefficients of @xmath47 and @xmath78 for mg , and @xmath84 and @xmath85 the coefficients for @xmath86 , derived from the mean of the linear fits to the two stellar population models given in [ ssec : models ] . = = figure [ fig : scatter ] shows the constraints on the variations in age and metallicity ( assumed for now to be uncorrelated ) which are imposed by the measured intrinsic scatter in the relations and the intrinsic dispersion in @xmath46 inferred from the intrinsic scatter in the fp . the intrinsic scatter we find about the and relations is then consistent with dispersions in age and metallicity on an elliptical locus defined by equation [ eqn : modelmg ] ( with @xmath7=0 ) in the @xmath3@xmath4 plane . the different loci for and ( the solid lines in figure [ fig : scatter ] ) result from the difference between the observed ratio of the scatter in to that in and the predicted ratio from the model , and give some indication of uncertainties both in the intrinsic scatter about the relations and in the model predictions . a second constraint is similarly obtained from the intrinsic scatter in distance ( in @xmath87 ) about the fp using equation [ eqn : modelfp ] ( again with @xmath7=0 ) . the dashed lines in figure [ fig : scatter ] correspond to intrinsic scatter about the fp of 10% , 15% and 20% . the important point to note about the figure is that , as mentioned in [ ssec : models ] , the dependences of the mg linestrengths and mass - to - light ratio on age and metallicity are quite different , so that ( if variations in age and metallicity are uncorrelated ) the two sets of constraints are nearly orthogonal . thus the region of the @xmath5@xmath4 plane that is consistent with the scatter in both the relation and the fp is quite limited . if we use the intrinsic scatter in the relation and assume a 20% intrinsic scatter in @xmath87 about the fp ( at the upper end of the quoted range see , , djorgovski & davis ( 1987 ) or jrgensen ( 1996 ) ) , we obtain approximate upper limits on the dispersions in age and metallicity of @xmath3=32% and @xmath4=38% . if , however , we use the intrinsic scatter in the relation and adopt an intrinsic fp scatter of 10% ( as obtained for coma by jrgensen 1993 ) , then we obtain approximate lower limits of @xmath3=15% and @xmath88=27% . similar arguments allow us to evaluate the relative contributions of the dispersions in age and metallicity to the errors in distance estimates derived from the fp . for the fiducial case ( @xmath30fp=20% , @xmath30=0.016 mag and @xmath7=0 ) , where @xmath3=32% and @xmath4=38% , the mean stellar population model implies that the dispersion in age gives an intrinsic fp scatter of 19% while the dispersion in metallicity gives 7% . in fact for most of the plausible range of dispersions in age and metallicity shown in figure [ fig : scatter ] , it is the dispersion in age which dominates the intrinsic scatter about the fp . only for the lowest plausible age dispersion and the highest plausible metallicity dispersion ( @xmath5=11% and @xmath4=43% , corresponding to @xmath30fp=10% and @xmath30=0.016 mag ) does the contribution to the fp scatter from the dispersion in metallicity achieve equality with the contribution from the dispersion in age . = the constraints on the dispersions change if there is a significant correlation ( or anti - correlation ) between the variations in age and metallicity . figure [ fig : scatter_rho ] shows how the constraints corresponding to the upper limits @xmath30fp=20% and @xmath30=0.016 mag ( corresponding to the thick lines in figure [ fig : scatter ] ) are modified as the correlation coefficient @xmath7 varies over its full range from @xmath81 to 1 . note that for @xmath89 we have @xmath90 . the main point to extract from this figure is that if the variations in age and metallicity have a correlation coefficient in the range @xmath80.5@xmath91@xmath7@xmath911 , then the dispersions in age and metallicity vary by only @xmath296% and @xmath2912% respectively about the values inferred in the uncorrelated case . only if the age and metallicity variations are strongly anti - correlated ( @xmath7@xmath6@xmath81 ; younger galaxies are more metal - rich ) do we obtain significantly different solutions , with a broader allowed range in both age and metallicity ( @xmath3 as large as 57% and @xmath4 as large as 80% ) . this conclusion is complementary to that reached by ferreras ( 1998 ) , who find that the apparently passive evolution of the colour magnitude relation observed in high - redshift clusters does not necessarily imply a common epoch of major star - formation if younger galaxies are on average more metal - rich . we can test the degree of correlation between the variations in age and metallicity by examining the joint distribution of residuals about the and fp relations . this distribution is shown for the efar data set in figure [ fig : dmgdfp]a . there is no evidence for a correlation between the residuals in this figure ; the spearman rank correlation coefficient between the residuals is 0.084 , and is not significant at the 2@xmath11 level . in order to investigate the expected distribution of residuals in the presence of the estimated measurement errors , we have performed monte carlo simulations of the efar data using the models for the dispersion in age and metallicity discussed above . figure [ fig : dmgdfp]b shows a simulation with @xmath3=32% and @xmath4=38% ; these are the values derived from the intrinsic scatter in the relation and a fp scatter of 20% when there is no correlation between age and metallicity . the simulated distribution resembles the observed distribution , although there is a weak but significant anti - correlation between the residuals ( due to the dominance of the age variations in the fp residuals ) which is not apparent in the efar data . over 100 such simulations , a two - dimensional k - s test ( press 1992 ) gives a median probability of 0.3% that this distribution and the observed distribution are the same . figures [ fig : dmgdfp]c&d show simulated distributions for the cases where the intrinsic scatter in is due to age alone or metallicity alone . neither case is consistent with the observed distribution , supporting the claim that neither age nor metallicity can be solely responsible for the scatter in both the relation and the fp . figures [ fig : dmgdfp]e h show simulated distributions for four cases where the variations in age and metallicity are correlated ( with @xmath7=+1 , + 0.5 , @xmath80.5 and @xmath81 respectively ) . the perfectly correlated and perfectly anti - correlated cases are not consistent with the observed distribution . however figure [ fig : dmgdfp]g shows that a distribution with no significant correlation between the and fp relation residuals is produced when @[email protected] . a two - dimensional k - s test gives a median probability over 100 such simulations of 1.7% that this distribution and the observed distribution are the same . this relatively low probability may reflect a problem with the model , although it may simply be due to sampling uncertainty ( the probabilities under this test vary between simulations with an rms of a factor of 6 ) or non - gaussian outliers in the efar residuals . the point to be emphasised is that a model with a moderate degree of anti - correlation between age and metallicity appears to give significantly better agreement with the observed distribution than a model in which age and metallicity are uncorrelated . we have examined the relation for early - type galaxies in the efar sample . we fit global and relations ( equations [ eqn : mlmgbp ] and [ eqn : mlmgtwo ] ) that have slopes about 25% steeper than those obtained by most previous authors . this difference results not from the data itself but from an improved fitting procedure : we apply a comprehensive maximum likelihood approach which correctly accounts for the biases introduced by both the sample selection function and the significant errors in both mg and @xmath11 . the _ observed _ scatter about the relations is 0.022 mag in and 0.031 mag in ; the _ intrinsic _ scatter in the global relations , estimated from monte carlo simulations , is 0.016 mag in and 0.023 mag in . with too few galaxies per cluster to reliably determine the full relation for each cluster separately , we fix the slopes of the relations at their global values in order to investigate the variation in the zeropoint from cluster to cluster . we find that the zeropoint has an observed scatter between clusters of 0.012 mag in and 0.019 mag in , and that this observed scatter is consistent with the small number of galaxies sampled in each cluster being drawn from a single global relation with intrinsic scatter between galaxies as given above the observations do not _ require _ any scatter in the zeropoint between clusters . the _ allowed _ range for the intrinsic scatter between clusters corresponds to cluster - to - cluster systematic errors in fundamental plane distances and peculiar velocities with an rms anywhere in the range 010% . we therefore can not determine from the relation _ alone _ whether systematic differences in the mean stellar populations between clusters contribute significantly ( or at all ) to the errors in distances and peculiar velocities obtained using the fundamental plane . we have also examined the variation in the relation with cluster properties . our cluster sample ranges from poor clusters to clusters as rich as coma , having velocity dispersions from 300 to 1000 and x - ray luminosities spanning 0.38@xmath9210@xmath93ergs@xmath16 . we do not detect a significant correlation of zeropoint with cluster velocity dispersion , x - ray luminosity or x - ray temperature , nor is there any significant difference in the relations obtained by fitting the galaxies in the high-@xmath37 clusters and low-@xmath37 clusters separately . the predominant factor in the production of mg in these early - type galaxies ( and presumably other @xmath45-elements and perhaps their metallicity and star - formation history in general ) is thus _ galaxy _ mass and not _ cluster _ these observations place constraints on semi - analytic models for the formation of elliptical galaxies , which are now beginning to incorporate chemical enrichment and should soon be able to make reliable predictions for the variation of the relation with cluster mass . we apply the single stellar population models of worthey ( 1994 ) and vazdekis ( 1996 ) to place upper limits on the global dispersion in the ages , metallicities and @xmath2 ratios of early - type galaxies of given mass using the intrinsic scatter in the global relation . we infer an upper limit on the dispersion in @xmath54 of 50% if the scatter in is due to age differences alone , or 10% if it is due to metallicity differences alone . these correspond to upper limits on the dispersion in relative galaxy distances estimated from the fundamental plane ( fp ) of 40% ( age alone ) or 8% ( metallicity alone ) . since the intrinsic scatter in the fp is found to be 1020% , one can not ( within the context of the single stellar population models ) explain both the scatter in the relation and the scatter in the fp as the result of age variations alone or metallicity variations alone . we therefore determine the joint range of dispersions in age and metallicity which are consistent with the measured intrinsic scatter in both the and fp relations . for a simple model in which the galaxies have independent gaussian distributions in @xmath47 and @xmath78 , we find upper limits of @xmath3=32% and @xmath88=38% at fixed galaxy mass . if the variations in age and metallicity are not independent , but have correlation coefficient @xmath7 , we find that so long as @xmath7 is in the range @xmath80.5 to 1 these limits on the dispersions in age and metallicity change by only @xmath296% and @xmath2912% respectively . only if the age and metallicity variations are strongly anti - correlated ( @xmath7@xmath6@xmath81 ) do we obtain significantly higher upper limits , with @xmath3 as large as 57% and @xmath4 as large as 80% . the distribution of the residuals from the and fp relations is only marginally consistent with a model having no correlation between age and metallicity , and is better - matched by a model in which age and metallicity variations are moderately anti - correlated ( @xmath3@xmath640% , @xmath4@xmath650% and @xmath7@[email protected] ) , with younger galaxies being more metal - rich . stronger bounds on the dispersion in age and metallicity amongst early - type galaxies of given mass will require more precise measurements of the deviations from the relation and the fundamental plane and also improved models for the dependence of the line indices and mass - to - light ratio on age and metallicity . further powerful constraints can also be obtained by measuring the intrinsic scatter in the and fp relations at higher redshifts , since the linestrengths and mass - to - light ratio have different dependences on age . mmc acknowledges the support of a dist collaborative research grant . db was partially supported by nsf grant ast90 - 16930 . rld thanks the lorenz centre and prof . de zeeuw . rkm received partial support from nsf grant ast90 - 20864 . rps acknowledges the financial support by the deutsche forschungsgemeinschaft under sfb 375 . gw is grateful to the serc and wadham college for a year s stay in oxford , to the alexander von humboldt - stiftung for making possible a visit to the ruhr - universitt in bochum and to nsf grants ast90 - 17048 and ast93 - 47714 for partial support . the entire collaboration benefitted from nato collaborative research grant 900159 and from the hospitality and monetary support of dartmouth college , oxford university , the university of durham and arizona state university . support was also received from pparc visitors grants to oxford and durham universities and pparc rolling grant ` extragalactic astronomy and cosmology in durham 1994 - 98 ' . we thank the referee , prof . alvio renzini , for a critique which resulted in substantial improvements to the paper . baugh c.m . , cole s. , frenk c.s . , 1996 , mnras , 283 , 1361 bender r. , burstein d. , faber s.m . , 1993 , apj , 411 , 153 bender r. , saglia r.p . , ziegler b. , belloni p. , greggio l. , hopp u. , 1998 , apj , 493 , 529 bender r. , ziegler b. , bruzual g. , 1996 , apj , 463 , l51 bower r.g . , lucey j.r . , ellis r.s . , 1992 , mnras , 254 , 601 bower r.g . , kodama t. , terlevich a. , 1998 , mnras , in press burstein d. , faber s.m . , dressler a. , 1990 , apj , 354 , 18 burstein d. , davies r.l . , dressler a. , faber s.m . , lynden - bell d. , terlevich r.j . , wegner g. , 1988 , in kron r.g . , renzini a. , eds , towards understanding galaxies at large redshifts , kluwer , dordrecht , p17 davies r.l , burstein d. , dressler a. , faber s.m . , lynden - bell d. , terlevich r.j . , wegner g. , 1987 , apjs , 64 , 581 de carvalho r.r . , djorgovski s. , 1992 , apj , 389 , l49 djorgovski s. , davis m. , 1987 , apj , 313 , 59 dressler a. , lynden - bell d. , burstein d. , davies r.l . , faber s.m . , terlevich r.j . , wegner g. , 1987 , apj , 313 , 42 ebeling h. , voges w. , bhringer h. , edge a.c . , huchra j.p . , briel u.g . , 1996 , mnras , 281 , 799 ellis r.s . , smail i. , dressler a. , couch w.c . , oemler a. , butcher h. , sharples r.m . , 1997 , apj , 483 , 582 feigelson e.d . , babu g.j . , 1992 , apj , 397 , 55 ferreras i. , charlot s. , silk j. , 1998 , apj , in press gonzlez j.j . , 1993 , phd thesis , university of california , santa cruz gorgas j. , efstathiou g. , aragn - salamanca a. , 1990 , mnras , 245 , 217 greggio l. , 1997 , mnras , 285 , 151 guzmn r. , 1993 , phd thesis , university of durham guzmn r. , lucey j.r . , carter d. , terlevich r.j . , 1992 , mnras , 257 , 187 huchra j. , geller m. , clemens c. , tokarz s. , michel a. , 1992 , bull.c.d.s . , 41 , 31 isobe t. , feigelson e.d . , akritas m.g . , babu g.j . , 1990 , apj , 364 , 104 jrgensen i. , 1997 , mnras , 288 , 161 jrgensen i. , franx m. , kjrgaard p. , 1995 , mnras , 276 , 1341 jrgensen i. , franx m. , kjrgaard p. , 1996 , mnras , 280 , 167 kauffmann g. , 1996 , mnras , 281 , 487 kauffmann g. , charlot s. , 1998 , mnras , 294 , 705 kelson d.d . , van dokkum p.g . , franx m. , illingworth g.d . , fabricant d. , 1997 , apj , 478 , l13 kodama t. , arimoto n. , 1997 , a&a , 320 , 41 kodama t. , arimoto n. , barger a. , aragn - salamanca a. , 1998 , a&a , in press lucey j.r . , guzmn r. , steel j. , carter d. , 1997 , mnras , 287 , 899 peletier r.f . , 1989 , phd thesis , university of groningen press w.h . , teukolsky s.a . , vetterling w.t . , flannery b.p . , 1992 , numerical recipes in c , 2nd edn , cambridge university press , p645 renzini a. , ciotti l. , 1993 , apj , 416 , l49 saglia r.p . , burstein d. , baggley g. , bertschinger e. , colless m.m . , davies r.l . , mcmahan r.k . , wegner g. , 1997 , mnras , 292 , 499 ( paper 3 ) saglia r.p . , colless m.m . , burstein d. , davies r.l . , mcmahan r.k . , watkins r. , wegner g. , 1998 , in renzini a. , white s.d.m . , eds , evolution of large scale structure , twin press , in press schweizer f. , seitzer p. , faber s.m , burstein d. , dalle ore c.m . , gonzlez j.j . , 1990 , apj , 364 , l33 shioya y. , bekki k. , 1998 , apj , in press stanford s.a . , eisenhardt p.r . , dickinson m. , 1998 , apj , 492 , 461 tantalo r. , chiosi c. , bressan a. , 1998 , a&a , 333 , 419 trager s.c . , 1997 , phd thesis , university of california , santa cruz trager s.c . , worthey g. , faber s.m . , burstein d. , gonzlez j.j . , 1998 , apjs , 116 , 1 van dokkum p.g . , franx m. , 1996 , mnras , 281 , 985 van dokkum p.g . , franx m. , kelson d.d . , illingworth g.d . , 1998 , apjl , in press vazdekis a. , casuso e. , peletier r.f . , beckman j.e . , 1996 , apjs , 106 , 307 wegner g. , colless m.m . , baggley g. , davies r.l . , bertschinger e. , burstein d. , mcmahan r.k . , saglia r.p . , 1996 , apjs , 106 , 1 ( paper 1 ) wegner g. , colless m.m . , saglia r.p . , mcmahan r.k . , davies r.l . , burstein d. , baggley g. , 1998 , mnras , submitted ( paper 2 ) whitmore b.c . , mcelroy d.b . , tonry j.l . , 1985 , apjs , 59 , 1 worthey g. , 1994 , apjs , 95 , 107 worthey g. , faber s.m . , gonzlez j.j . , 1992 , apj , 398 , 69 ziegler b.l . , bender r. , 1997 , mnras , 291 , 527

we have examined the relation for early - type galaxies in the efar sample and its dependence on cluster properties . a comprehensive maximum likelihood treatment of the sample selection and measurement errors gives fits to the global relation of @xmath0 and @xmath1 . the slope of these relations is 25% steeper than that obtained by most other authors due to the reduced bias of our fitting method . the intrinsic scatter in the global relation is estimated to be 0.016 mag in and 0.023 mag in . the relation for cd galaxies has a higher zeropoint than for e and s0 galaxies , implying that cds are older and/or more metal - rich than other early - type galaxies with the same velocity dispersion . we investigate the variation in the zeropoint of the relation between clusters . we find it is consistent with the number of galaxies observed per cluster and the intrinsic scatter between galaxies in the global relation . we find no significant correlation between the zeropoint and the cluster velocity dispersion , x - ray luminosity or x - ray temperature over a wide range in cluster mass . these results provide constraints for models of the formation of elliptical galaxies . however the relation on its own does not place strong limits on systematic errors in fundamental plane distance estimates due to stellar population differences between clusters . we compare the intrinsic scatter in the and fundamental plane ( fp ) relations with stellar population models in order to constrain the dispersion in ages , metallicities and @xmath2 ratios for early - type galaxies at fixed velocity dispersion . we find that variations in age alone or metallicity alone can not explain the measured intrinsic scatter in both and the fp . we derive the joint constraints on the dispersion in age and metallicity implied by the scatter in the and fp relations for a simple gaussian model . we find upper limits on the dispersions in age and metallicity at fixed velocity dispersion of 32% in @xmath3 and 38% in @xmath4 if the variations in age and metallicity are uncorrelated ; only strongly anti - correlated variations lead to significantly higher upper limits . the joint distribution of residuals from the and fp relations is only marginally consistent with a model having no correlation between age and metallicity , and is better - matched by a model in which age and metallicity variations are moderately anti - correlated ( @xmath5@xmath640% , @xmath4@xmath650% and @xmath7@[email protected] ) , with younger galaxies being more metal - rich . galaxies : distances and redshifts galaxies : elliptical and lenticular , cd galaxies : stellar content galaxies : formation galaxies : evolution

astro-ph9811089

nuclear multifragmentation is presently intensely studied both theoretically and experimentally . due to the similitude existent between the nucleon - nucleon interaction and the van der waals forces , signs of a liquid - gas phase transition in nuclear matter are searched . while the theoretical calculations concerning this problem started at the beginning of 1980 @xcite , the first experimental evaluation of the nuclear caloric curve was reported in 1995 by the aladin group @xcite . a wide plateau situated at around 5 mev temperature lasting from 3 to 10 mev / nucleon excitation energy was identified . the fact was obviously associated with the possible existence of a liquid - gas phase transition in nuclear matter and generated new motivations for further theoretical and experimental work . similar experiments of eos @xcite and indra @xcite followed shortly . using different reactions they obtained slightly different caloric curves , the plateau - like region being absent in the majority of cases . factors contributing to these discrepancies are both the precision of the experimental measurements and the finite - size effects of the caloric curve manifested through the dependency of the equilibrated sources [ @xmath5 sequence on the reaction type . concerning the first point of view , recent reevaluations of the aladin group concerning the kinetic energies of the emitted neutrons brought corrections of about 10 @xmath6 ( in the case of the reaction @xmath1au+@xmath1au , 600 mev / nucleon ) . more importantly however it was proven that the energies of the spectator parts are growing with approximately 30 @xmath6 in the bombarding energy interval 600 to 1000 mev / nucleon . on the other side , the universality of the quantity @xmath7 subject to the bombarding energy variation ( which was theoretically proven @xcite to be a signature of statistical equilibrium ) suggests that for the above - mentioned reactions the equilibrated sources sequence [ @xmath5 should be the same . consequently , we deal with an important nonequilibrium part included in the measured source excitation energies which may belong to both pre - equilibrium or pre - break - up stages @xcite . the smm calculations suggest a significant quantity of nonequilibrium energy even in the case of the 600 mev / nucleon bombarding energy reaction @xcite . thus , the necessity of accurate theoretical descriptions of the break - up stage and of the sequential secondary particle emission appears to be imperative in order to distinguish between the equilibrium and nonequilibrium parts of the measured excitation energies . these approaches should strictly obey the constrains of the physical system which , in the case of nuclear multifragmentation , are purely microcanonic . as we previously underlined @xcite , in spite of their success in reproducing some experimental data , the two widely used statistical multifragmentation models ( smm @xcite and mmmc @xcite ) are not strictly satisfying the microcanonical rules . the present paper describes some refinements and improvements brought to the sharp microcanonical multifragmentation model proposed in @xcite and also the employment of the model in its new version in the interpretation of the recent experimental data of the aladin group @xcite . the improvements brought to the model @xcite are presented in section ii . section iii presents the new evaluations of temperature curves and the first evaluations ( performed with this model ) of heat capacities at constant volume ( @xmath8 ) represented as a function of system excitation energy and temperature and also the comparison between the model predictions and the recent experimental heli isotopic temperature curve [ @xmath9 @xcite . conclusions are drawn in section iv . the improvements brought to the microcanonical multifragmentation model concerns both the _ break - up _ stage and the _ secondary particle emission _ stage . + ( i ) _ primary break - up refinements _ + comparing to the version of ref.@xcite the present model has the following new features : + ( a ) the experimental discrete energy levels are replacing the level density for fragments with @xmath10 ( in the previous version of the model a thomas fermi type level density formula was used for all particle excited states ) . in this respect , in the statistical weight of a configuration and the correction factor formulas @xcite the level density functions are replaced by the degeneracies of the discrete levels , @xmath11 ( here @xmath12 denotes the spin of the @xmath13th excited level ) . as a criterion for level selection ( i.e. the level life - time must be greater than the typical time of a fragmentation event ) we used @xmath14 1 mev , where @xmath15 is the width of the energy level . + ( b ) in the case of the fragments with @xmath16 the level density formula is modified as to take into account the strong decrease of the fragments excited states life - time ( reported to the standard duration of a fragmentation event ) with the increase of their excitation energy . to this aim the thomas fermi type formula @xcite is completed with the factor @xmath17 ( see ref.@xcite ) : @xmath18 where @xmath19 , @xmath20 and @xmath21 . + ( ii ) _ inclusion of the secondary decay stage _ + for the @xmath22 nuclei it was observed that the fragments excitation energies are sufficiently small such as the sequential evaporation scheme is perfectly applicable . according to weisskopf theory @xcite ( extended as to account for particles larger than @xmath23 ) , the probability of emitting a particle @xmath24 from an excited nucleus is proportional to the quantity : @xmath25 where @xmath26 are the stable excited states of the fragment @xmath24 subject to particle emission ( their upper limit is generally around 7 - 8 mev ) , @xmath27 is the kinetic energy of the formed pair in the center of mass ( c.m . ) frame , @xmath28 is the degeneracy of the level @xmath13 , @xmath29 and @xmath30 are respectively the reduced mass of the pair and the separation energy of the particle @xmath24 and finally @xmath31 is the inverse reaction cross - section . due to the specificity of the multifragmentation calculations we considered the range of the emitted fragments @xmath24 up to the @xmath32 limit . for the inverse reaction cross - section we have used the optical model based parametrization from ref . the sequential evaporation process is simulated by means of standard monte carlo ( see for example @xcite ) . for nuclei with @xmath33 ( the only excited states of @xmath34 nuclei taken into consideration are few states higher than 20 mev belonging to the @xmath23 particle ) depending on their amount of excitation we consider _ secondary break - up _ for @xmath35 and weisskopf evaporation otherwise ( here @xmath36 is the excitation energy of the fragment @xmath37 and @xmath38 is its binding energy ) . the microcanonical weight formulas have the usual form @xcite excepting the level density functions which are here replaced by the discrete levels degeneracies . due to the reduced dimensions of the @xmath39 systems , the break - up channels are countable ( and a classical monte carlo simulation is appropriate ) when a mean field approach is used for the coulomb interaction energy . in this respect , the wigner - seitz approach @xcite is employed for the coulomb interaction : @xmath40 where @xmath41 and @xmath42 denotes the mass and the charge of the source nucleus , the resulting fragments have the index @xmath13 , @xmath43^{1/3}$ ] and @xmath44 . here @xmath45 denotes the break - up volume and @xmath46 the volume of the nucleus at normal density . it should be added that @xmath47 is the radius of the source nucleus at break - up and @xmath48 is the radius of fragment @xmath13 at normal density . for each event of the primary break - up simulation , the entire chain of evaporation and secondary break - up events is monte carlo simulated . using the improved version of the microcanonical multifragmentation model , the caloric curves corresponding to two freeze - out radii ( r=2.25 a@xmath49 and r=2.50 a@xmath49 fm ) are reevaluated for the case of the source nucleus ( 70 , 32 ) ( the microcanonical caloric curves evaluated with the initial version of the model are given in ref . these are presented in fig . 1 ( a ) . one can observe that the main features of the caloric curve from refs . @xcite are reobtained . thus , one can recognize the liquid - like region at the beginning of the caloric curve , then a large plateau - like region and finally the linearly increasing gas - like region . one may also notice that the caloric curve behavior at the freeze - out radius variation is maintained : the decrease of the freeze - out radius leads to a global lifting of the caloric curve . as it is well known , the curves of the constant volume heat capacity ( @xmath8 ) as a function of system excitation energy ( @xmath50 ) and as a function of temperature ( @xmath51 ) may provide important information concerning the transition region and the transition order . for this reason the curves @xmath52 and @xmath53 have been evaluated ( see fig . 1 ( a ) and fig . 1 ( b ) ) . we remind that the constant volume heat capacity ( @xmath8 ) is calculable in the present model using the formula @xcite : @xmath54 ^ 2\right>+t^2\left<\left(\frac32 n_c- \frac52\right ) \frac1{k^2}\right>.\ ] ] it can be observed that the @xmath52 curve has a sharp maximum around 4.5 mev / nucleon excitation energy for both considered freeze - out radii . this suggests that a phase transition exists in that region . the transition temperatures can be very well distinguished by analyzing the @xmath53 . one can observe two sharp - peaked maxima pointing the transition temperatures corresponding to the two considered freeze - out radii . in order to make a direct comparison between the calculated heli isotopic temperature and the recent experimental results @xcite one has to deduce the sequence of excitation energy as a function of the system dimension [ @xmath5 . this is done as in refs . @xcite using as matching criterion the simultaneously reproduction of the @xmath55 and @xmath56 curves . this couple of curves can fairly well identify the dimension and the excitation of the equilibrated nuclear source @xcite . here @xmath2 stands for the multiplicity of intermediate mass fragments and is defined as the number of fragments with @xmath57 from a fragmentation event while @xmath3 denotes the charge asymmetry of the two largest fragments and , for one fragmentation event is defined as @xmath58 with @xmath59 where @xmath60 is the maximum charge of a fragment and @xmath61 is the second largest charge of a fragment in the respective event . @xmath4 represents the _ bound charge _ in one fragmentation event and is defined as the sum of the charges of all fragments with @xmath62 . the simultaneous fit of the calculated curves @xmath55 and @xmath56 on the corresponding experimental data ( @xmath1au+@xmath1au at 1000 mev / nucleon ) is given in fig . the agreement is very good . the equilibrated source sequence [ @xmath5 we used for this purpose is given in fig . 3 together with the experimental evaluations of the excitation energies as a function of source dimension for the reaction @xmath1au+@xmath1au at 600 , 800 and 1000 mev / nucleon . the theoretically obtained sequence is relatively close to the experimental line corresponding to 600 mev / nucleon bombarding energy . the deviations between the calculated equilibrated source sequence and the three experimental lines suggest that the experimental evaluations contain a quantity of non - equilibrium energy which grows with increasing the bombarding energy . as suggested in ref . @xcite , its origin may be situated in both the pre - equilibrium and pre - break - up stage . these deviations are exclusively due to the neutron kinetic energies which , reevaluated @xcite from the 1995 data @xcite , are much larger . it should also be pointed that apart from the smm predictions @xcite , the quantity of non - equilibrium energy predicted by the present model is smaller and thus the model predicted equilibrated source sequence is closer to the experimental line of the 600 mev / nucleon bombarding energy reaction . after evaluating the sequence of the equilibrated sources a direct comparison the heli calculated isotopic temperature curve with the ones recently evaluated by the aladin group @xcite is performed . to this purpose the uncorrected albergo temperature is used : @xmath63 $ ] , the experimental predictions being divided by @xmath64 ( which is the factor used in the aladin evaluation of the heli caloric curve chosen as to average the qsm , gemini and mmmc models predictions ) . the result is represented in fig . 4 as a function of @xmath4 . it can be observed that the agreement between the calculated @xmath65 and the experimental data corresponding to the @xmath1au+@xmath1au reaction at 600 and 1000 mev / nucleon bombarding energy is excellent on the entire range of @xmath4 . in comparison , the smm model predicts in the region @xmath66 a curve steeper than the experimental data . sumarizing , the microcanonical multifragmentation model from ref . @xcite is improved by refining the primary break - up part and by including the secondary particle emission . the caloric curve rededuced with the new version of the model preserves its general aspect @xcite manifesting an important plateau - like region . the transition regions are clearly indicated by the sharp maxima of the @xmath52 and @xmath53 curves . the model proves the ability of simultaneously fitting the `` definitory '' characteristics of the nuclear multifragmentation phenomenon @xmath55 and @xmath56 . evaluating the equilibrated source sequence @xmath67 [ by using the criterion of reproducing both @xmath68 and @xmath69 versus @xmath70 , a nonequilibrium part of the experimentally evaluated excitation energy growing with the increase of the bombarding energy is identified . the direct comparison of the calculated heli caloric curve shows an excellent agreement with the experimental heli curves recently evaluated by the aladin group . g. fai and j. randrup , nucl . a * 404 * , 551 ( 1983 ) ; comp . phys . comm . * 42 * , 385 ( 1986 ) ; l. p. csernai , g. fai and j. randrup , phys . b * 140 * , 149 ( 1984 ) ; g. fai , l. p. csernai , j. randrup and h. stocker , phys . b * 164 * , 265 ( 1985 ) . j. pochodzalla _ et al . _ , lett . * 75 * , 1040 ( 1995 ) . j. a. hauger _ et al . _ , lett . * 77 * , 235 ( 1996 ) . y. g. ma _ et al . b * 390 * , 41 ( 1997 ) . a. schuttauf _ _ , nucl . phys . * a607 * , 457 ( 1996 ) . j. pochodzalla , progr . . phys . * 39 * , 443 ( 1997 ) . w. trautmann _ et al . _ , gsi - preprint 98 - 20 , march 1998 . hongfei xi _ et al . _ , a * 359 * , 397 ( 1997 ) . w. f. j. mueller , proceedings of the workshop on multifragmentation , hirschegg ( austria ) ( 1999 ) , gsi . al . h. raduta and ad . r. raduta , phys . c * 59 * , 323 ( 1999 ) . al . h. raduta and ad . r. raduta , nucl . phys . * a647 * , 12 ( 1999 ) . j. p. bondorf , a. s. botvina , a. s. iljinov , i. n. mishustin and k. sneppen , phys . rep . * 257 * , 133 ( 1995 ) . d. h. e. gross , rep . . phys . * 53 * , 605 ( 1990 ) . al . h. raduta and ad . r. raduta , phys . c * 55 * , 1344 ( 1997 ) . h. raduta and ad . r. raduta , phys . c * 56 * , 2059 ( 1997 ) . j. randrup and s. koonin , nucl . a471 * , 355c ( 1987 ) . v. weisskopf , phys . rev . * 52 * , 295 ( 1937 ) . a. s. botvina , a. s. iljinov , i. n. mishustin , j. p. bondorf , r. donangelo and k. sneppen , nucl . phys . a475 * , 663 ( 1987 ) . i. dostrovsky , z. fraenkel and g. friedlander , phys . rev . * 116 * , 683 ( 1959 ) . al . h. raduta and ad . r. raduta , phys . c * 59 * , r1855 ( 1999 ) .

the microcanonical multifragmentation model from [ al . h. raduta and ad . r. raduta , phys . rev . c * 55 * , 1344 ( 1997 ) ; * 56 * , 2059 ( 1997 ) ; * 59 * , 323 ( 1999 ) ] is refined and improved by taking into account the experimental discrete levels for fragments with @xmath0 and by including the stage of sequential decay of the primary excited fragments . the caloric curve is reevaluated and the heat capacity at constant volume curve is represented as a function of excitation energy and temperature . the sequence of equilibrated sources formed in the reactions studied by the aladin group ( @xmath1au+@xmath1au at 600 , 800 and 1000 mev / nucleon bombarding energy ) is deduced by fitting simultaneously the model predicted mean multiplicity of intermediate mass fragments ( @xmath2 ) and charge asymmetry of the two largest fragments ( @xmath3 ) versus bound charge ( @xmath4 ) on the corresponding experimental data . calculated heli isotopic temperature curves as a function of the bound charge are compared with the experimentally deduced ones .

nucl-th9908088

the pair - correlation density @xmath1 is one of the key concepts in describing the correlation effects , arising from pauli exclusion principle and coulomb interaction , in the homogeneous electron liquid ( or gas).@xcite it also plays a significant role in the constructions of the exchange - correlation energy density functionals in density - functional theory ( dft),@xcite since in such constructions the homogeneous electron system is conventionally taken as a reference system . a great deal of theoretical progress has recently been made in giving an accurate evaluation of @xmath1 , or the more specific spin - resolved pair - correlation densities @xmath3 , with @xmath4$].@xcite in particular , @xmath0 , the on - top pair - correlation density , which arises totally from @xmath5 since @xmath6 , has been well known to play a special role in dft.@xcite the important implication of @xmath0 was also realized in many - body theory long ago because the random phase approximation ( rpa),@xcite due to its lack of accurate description of the short - range electron correlations , yields erroneous negative values for @xmath0 when the electron densities are not sufficiently high.@xcite it is well known that , in many - body theory , the long - range correlations can be rather successfully taken into account in the rpa , while the short - range correlations can be properly described by the ladder theory ( lt).@xcite in this paper , we attempt to investigate the short - range correlations in terms of @xmath7 in the lt , in both three dimensions ( 3d ) and two dimensions ( 2d ) . in fact , investigations in this direction date back long ago , and a great deal of achievement has been made . @xcite it is necessary here to give some introduction to the lt . the effective interaction @xmath8 in the lt between two scattering electrons with respective momenta @xmath9 and @xmath10 satisfies the following bethe - goldstone equation:@xcite @xmath11 where @xmath12 is the fourier transform of the coulomb potential , @xmath13 is the momentum distribution in the noninteracting ground state and @xmath14 is the fermi momentum , and @xmath15 . as mentioned above , the rpa gives poor description of the short - range correlations of the electrons , especially for @xmath1 as @xmath16 . in fact , the results for @xmath17 in the rpa violate the following cusp condition : @xcite @xmath18 where @xmath19 is the number of spatial dimensions , and @xmath20 is the bohr radius . it was shown recently @xcite that the pair - correlation density obtained from the first order perturbation calculation does not satisfy the cusp condition either . in this paper , we prove that @xmath21 calculated from @xmath22 of eq . ( [ bg1 ] ) satisfies the cusp condition . this indicates the reliablity of the lt in the calculations of the pair - correlation density at short range . the short - range structure of the pair - correlation density is determined by the behavior of the effective potential @xmath8 at large momentum transfer @xmath23 . in the limiting case , one therefore can approximately replace the momenta of the scattering electrons by zero in eq . ( [ bg1 ] ) , @xmath24 a frequently used approach to solving eq . ( [ bg2 ] ) in the literature is making the following approximation in the coulomb kernel in the momentum summation:@xcite @xmath25 with the preceding approximation , an analytical solution for @xmath26 was obtained which yields the following well - known result for @xmath5 in 3d , @xcite @xmath27 ^ 2 , \end{aligned}\ ] ] where @xmath28 with @xmath29 and @xmath30 . a similar result was obtained in 2d,@xcite @xmath31^{-2},\end{aligned}\ ] ] where @xmath32 with @xmath33 in 2d . in eqs . ( [ yasuhara ] ) and ( [ nagano ] ) , @xmath34 is the @xmath35th order modified bessel function . in this paper we have managed to solve exactly eq . ( [ bg2 ] ) , i.e. , without making the approximation of eq . ( [ kernel ] ) . our results for @xmath5 are @xmath36 ^ 2 , \end{aligned}\ ] ] in 3d , and @xmath37 ^ 2 , \end{aligned}\ ] ] in 2d . equations ( [ qian3d ] ) and ( [ qian2d ] ) are the main results of this paper . the paper is organized as follows : in sect . ii , we solve eq . ( [ bg2 ] ) exactly both in 3d and 2d . in sect . iii , we derive analytically the expressions of eqs . ( [ qian3d ] ) and ( [ qian2d ] ) for @xmath38 . we then compare our results with previous ones in the literature in sect . iv . sect . v is devoted to conclusions . some technical points on the solutions for the coefficients of the large momentum expansions of the effective potentials are given in appendix a. in appendix b , we prove the cusp condition in the lt . in this section , we present our solution to eq . ( [ bg2 ] ) at large momentum transfer @xmath23 in the effective potential in both 3d and 2d . to this end , we denote @xmath26 as @xmath39 , and reduce the momenta with unit @xmath14 , and potentials with @xmath40 , respectively . we present our solution for the 3d case in subsection a , and the 2d case in subsection b , separately . [ [ d ] ] 3d ~~ after carrying out the angular integrations in the summation of the momentum @xmath41 , eq . ( [ bg2 ] ) becomes @xmath42 we expand @xmath39 in the powers of @xmath43 , @xmath44 it can be easily confirmed by iteration that no odd power terms in the expansion of @xmath39 exist in the solution to eq . ( [ bgsolution ] ) . the erroneous odd power terms introduced into @xmath39 in refs . @xcite are purely due to the approximation made in the coulomb kernel in eq . ( [ kernel ] ) . we substitute eq . ( [ veffsolution ] ) into eq . ( [ bgsolution ] ) , and obtain @xmath45 where @xmath46 by carrying through partial integration on the right hand side of eq . ( [ m2n+1 m ] ) , one has , @xmath47,\end{aligned}\ ] ] where @xmath48 @xmath49 defined in the preceding equation can be evaluated to be @xmath50 substituting eq . ( [ phi ] ) into eq . ( [ m2n+1 mm ] ) yields , @xmath51 , ~ n \ge 0 .\end{aligned}\ ] ] finally , substituting eq . ( [ m2n+3 ] ) into eq . ( [ m2n+1 ] ) , and comparing the same power orders of @xmath43 , one obtains the following equations for @xmath52 : @xmath53 and @xmath54 equations ( [ a0equ ] ) and ( [ anequ ] ) for @xmath52 can be solved exactly in principle . in fact , by making the truncation of @xmath55 for @xmath56 , a nearly exact solution can be obtained as @xmath57 @xmath58 and @xmath59 where @xmath60 in appendix a , we show that the preceding solution for @xmath61 , which is directly related to @xmath5 , as shown in the next section , is very close to the exact numerical solution to eqs . ( [ a0equ ] ) and ( [ anequ ] ) . in fact , the large momentum behavior of @xmath39 is dominated by the leading terms in the large @xmath62 expansion of @xmath39 in eq . ( [ veffsolution ] ) , and hence a truncation solution like the preceding one is almost exact . [ [ d-1 ] ] 2d ~~ in 2d , we make use of the following expression , @xmath63 and rewrite eq . ( [ bg2 ] ) as follows : @xmath64 carrying out the angular integrations of @xmath41 and @xmath65 , we have @xmath66 where @xmath67 is the @xmath35th order bessel function . we expand @xmath39 in the powers of @xmath43 as follows : @xmath68 no even power terms exist in the solution to eq . ( [ bgsolution2 ] ) . again , the erroneous even power terms @xcite appear in @xmath39 due to the approximation made in eq . ( [ kernel ] ) . we substitute eq . ( [ veffsolution2 ] ) into eq . ( [ bgsolution2 ] ) , and obtain @xmath69 where @xmath70 carrying out the integration over @xmath71 in eq . ( [ n2nn ] ) , one obtains , @xmath72 the integral on the right hand side of eq . ( [ j0 ] ) can be expressed in terms of the hypergeometric function as follows,@xcite @xmath73 where @xmath74 is the gamma function . therefore , one has , @xmath75 substituting eq . ( [ n2n ] ) into eq . ( [ n2n+2 ] ) , and comparing the same power orders of @xmath43 , one finally gets @xmath76 and @xmath77 for @xmath78 . similarly to the 3d case , eqs . ( [ c0equ ] ) and ( [ cnequ ] ) can be solved exactly in principle . in fact , a nearly exact solution can be obtained as follows by the truncation of @xmath79 for @xmath56 : @xmath80 @xmath81 and @xmath82 where @xmath83 the spin - antiparallel pair - correlation density in the lt can be shown to be @xcite @xmath85 where the prime on the summations over @xmath9 , @xmath10 means the restrictions @xmath86 , and @xmath87 is defined as , @xmath88 below we present the results for the 3d and 2d cases in subsection a and b , respectively . we will reduce @xmath84 with unit @xmath89 . [ [ d-2 ] ] 3d ~~ using the approximate solution @xmath90 for @xmath8 , one obtains @xcite @xmath91 ^ 2 .\end{aligned}\ ] ] trivially , @xmath92 ^ 2 .\end{aligned}\ ] ] with the expression of eq . ( [ veffsolution ] ) , one has @xmath93 equation ( [ a0equ ] ) has been made use of in obtaining the preceding result . the expression for @xmath61 is given in eq . ( [ a0 ] ) , with which we obtain the final result of eq . ( [ qian3d ] ) . furthermore , it is straightforward to show , from eq . ( [ gud(r ) ] ) , that at small @xmath84 , @xmath94 [ [ d-3 ] ] 2d ~~ in 2d , one has , @xmath95 ^ 2 .\end{aligned}\ ] ] similar derivation to that in the 3d case leads to @xmath96 or , by the use of eq . ( [ c0 ] ) , the final result of eq . ( [ qian2d ] ) . furthermore , from eq . ( [ gud2(r ) ] ) , one can obtain @xmath97 evidently , the cusp condition of eq . ( [ cusp ] ) is satified both in eq . ( [ gud(r)1 ] ) and eq . ( [ gud(r)2 ] ) . in fact , in appendix b we shall show that , eq . ( [ cusp ] ) is satisfied , in general , in the full ladder theory . ( 5.0,6.0 ) ( -5.0,-4.0 ) ( 7.0,8.0 ) first of all , at limiting high density , we have , from eq . ( [ qian3d ] ) , @xmath98 in 3d . equation ( [ highdensity ] ) is the same as the corresponding yasuhara s result.@xcite we note that , the first order perturbation calculation , @xcite which is believed to approach to the exact result at high density limit , yields a result of @xmath99 . we plot @xmath100 calculated from eq . ( [ qian3d ] ) in fig . 1 , in comparison with that calculated from eq . ( [ yasuhara ] ) @xcite . notice that the discrepancy between eq . ( [ qian3d ] ) and eq . ( [ yasuhara ] ) , which appears not minor , arises purely from the approximation of eq . ( [ kernel ] ) made in obtaining eq . ( [ yasuhara ] ) in yasuhara s theory . in effect , lowy and brown @xcite had thrown doubt on the validity of the approximation of eq . ( [ kernel ] ) . we hence justify their doubt , at least for the limiting short range correlations . the result based on overhauser s proposal ( eq . ( 26 ) in ref . @xcite ) is also shown in fig . the comparison hence indicates that the coincidence between overhauser s result ( and the corresponding numerical result of gori - giorgi and perdew @xcite ) and yasuhara s is accidental . ( 5.0,6.0 ) ( -5.0,-4.0 ) ( 7.0,8.0 ) in fig . 2 , we plot @xmath100 in 2d calculated from eq . ( [ qian2d ] ) , together with that from eq . ( [ nagano]).@xcite once again , we emphasize that the discrepancy is totally due to the approximation of eq . ( [ kernel ] ) made in obtaining eq . ( [ nagano ] ) . however , at limiting high density , both equations yield the following same result : @xmath101 for a comparison , we have also shown in fig . 2 the result of eq . ( 17 ) in ref . @xcite , which was proposed by polini _ based on an interpolation between the first - order ( second - order in terms of the correlation energy ) calculation for the weak - coupling limit and overhauser type calculation @xcite for the strong - coupling limit . the proper approach to the short - range electron correlations in many - body theory is the ladder theory , in which the effective potential between two scattering particles satisfies the bethe - goldstone equation of eq . ( [ bg1 ] ) . in this paper , we have proved that , the ladder theory satisfies the cusp condition for the pair - correlation density in the homogeneous electron liquid . this enhances our belief in the capability of the ladder theory in describing the short - range correlations , especially in calculating the pair - correlation density . the main results obtained in this paper are , in effect , eq . ( [ qian3d ] ) and eq . ( [ qian2d ] ) given in the introduction , in three dimensions and two dimensions respectively , for the on - top pair - correlation density in the homogeneous electron liquid . these results have been derived by solving eq . ( [ bg2 ] ) , in which the two scattering particles in the bethe - goldstone equation are approximately taken to be static . this approximation should be reasonable since the limiting short range structure of the pair - correlation is determined by the large transfer momentum behavior of the effective potential . the major theoretical progress made in this paper is that we have removed the approximation of eq . ( [ kernel ] ) frequently made in the literature for the coulomb kernel in solving eq . ( [ bg2 ] ) . our solution to eq . ( [ bg2 ] ) is thus exact . this work was supported by the chinese national science foundation under grant no . ( 5.0,6.0 ) ( -5.0,-4.0 ) ( 7.0,8.0 ) ( -1.4,0.0 ) ( -4.4,0.77 ) ( 7.0,8.0 ) [ figure3 ] a nearly exact solution for @xmath61 to eqs . ( [ a0equ ] ) and ( [ anequ ] ) has been given in eq . ( [ a0 ] ) in sect . iii by the truncation of @xmath102 for @xmath56 . below we give the solution for @xmath52 by the truncation of @xmath103 for @xmath104 . @xmath105 @xmath106 @xmath107 and @xmath108 where @xmath109 in fig . 3 , we plot the results for @xmath61 calculated from eqs . ( [ a0 ] ) and ( [ a00 ] ) , together with the corresponding exact numerical solution to eqs ( [ a0equ ] ) , and ( [ anequ ] ) . there is basically no difference among them . we present the expressions of eqs . ( [ a00 ] ) , ( [ a10 ] ) , ( [ a20 ] ) , ( [ a30 ] ) above for possible future reference . similar expressions for the 2d case are given below . @xmath110 @xmath111 @xmath112 and @xmath113 where @xmath114 the corresponding illustration is given in fig . ( 5.0,6.0 ) ( -5.0,-4.0 ) ( 7.0,8.0 ) ( -1.4,0.0 ) ( -4.5,0.75 ) ( 7.0,8.0 ) [ figure4 ] due to the singularity of the coulomb potential between electrons , the many - body schrdinger wavefunction has a cusp when any two electrons coalesce.@xcite this fact leads to the cusp condition of eq . ( [ cusp ] ) @xcite for the pair - correlation density ( also known as kimball relation in the literature of many - electron theory ) . recently it was claimed that eq . ( [ cusp ] ) is not satisfied in the lt.@xcite in this appendix , we give a rigorous proof for eq . ( [ cusp ] ) in the lt . the proof will be formulated in 3d . we start with the definition of the spin - parallel static structure factor as follows : @xmath115 where @xmath116 is the spin - resolved density operator and @xmath117 is the particle number . it has been shown that the spin - antiparallel static structure factor in the lt can be expressed in terms of the effective potential of eq . ( [ bg1 ] ) as @xcite @xmath118.\end{aligned}\ ] ] next we examine the large momentum structure of @xmath119 . for @xmath120 and @xmath121 , one has , from eqs . ( [ bg1 ] ) and ( [ d ] ) , @xmath122,\end{aligned}\ ] ] which evidently goes to zero in the order of @xmath123 . therefore @xmath124 \nonumber \\ & & \times \sum_{{\bf k } ' } d({\bf p } , { \bf p } ' ; { \bf k } ' ) v_{eff } ( { \bf p } , { \bf p } ' ; { \bf k}').\end{aligned}\ ] ] in obtaining eq . ( [ dvdv ] ) , we have used the following relation , @xmath125 if @xmath126 . it seems that a mistake occurs in ref . @xcite due to a possible miss of the factor @xmath127 on the right hand side of the preceding equation , as it was employed to derive eq . ( 12 ) from eq . ( 11 ) in ref . @xcite . substituting eqs . ( [ dv ] ) and ( [ dvdv ] ) into eq . ( [ sud1 ] ) , one has @xmath128 ^ 2.\end{aligned}\ ] ] on the other hand , from eq . ( [ gud1 ] ) , we have @xmath129 ^ 2.\end{aligned}\ ] ] comparing eq . ( [ sud ] ) and eq . ( [ gud0 ] ) yields @xmath130 combining the preceding result with the following well - known relation @xcite , @xmath131 one proves the cusp condition of eq . ( [ cusp ] ) in the lt . the above proof can be straightforwardly extended to the 2d case . in fact , in 2d , it can be similarly shown @xmath132 in the lt . combining the above result with the following relation @xcite @xmath133 leads to eq . ( [ cusp ] ) for the 2d case . for reviews , see k. s. singwi and m. p. tosi , in _ solid state physics _ , edited by h. ehrenreich , f. seitz , and d. turnbull ( academic , new york , 1981 ) , vol . * 36 * , p. 177 ; s. ichimaru , rev . mod . phys . * 54 * , 5017 ( 1982 ) . p. hohenberg and w. kohn , phys . rev . * 136 * , b864 ( 1964 ) ; w. kohn and l. j. sham , phys . rev . * 140 * , a1133 ( 1965 ) ; o. gunnarsson and b. i. lundqvist , phys . rev . b * 13 * , 4724 ( 1976 ) ; s. lundqvist and n. h. march , _ theory of inhomogeneous electron gas _ ( plenum , new york , 1983 ) . a. glick and r. a. ferrell , ann . phys . * 11 * , 359 ( 1959 ) ; s. ueda , prog . * 26 * , 45 ( 1961 ) . for later theoretical developments , singwi , m. p. tosi , r. h. land , and a. sjlander , phys . rev . * 176 * , 589 ( 1968 ) .

the ladder theory , in which the bethe - goldstone equation for the effective potential between two scattering particles plays a central role , is well known for its satisfactory description of the short - range correlations in the homogeneous electron liquid . by solving exactly the bethe - goldstone equation in the limit of large transfer momentum between two scattering particles , we obtain accurate results for the on - top pair - correlation density @xmath0 , in both three dimensions and two dimensions . furthermore , we prove , in general , the ladder theory satisfies the cusp condition for the pair - correlation density @xmath1 at zero distance @xmath2 .

cond-mat0509177

waves propagating in a curved spacetime develop `` tails '' . in particular , it is well established that the _ dominant _ late - time behaviour of massless fields propagating in black - hole spacetimes is a power - law tail . price @xcite was the first to analyze the mechanism by which the spacetime outside a ( nearly spherical ) collapsing star divests itself of all radiative multipole moments , and leaves behind a schwarzschild black hole ; it was demonstrated that all radiative perturbations decay asymptotically as an inverse power of time . physically , these inverse power - law tails are associated with the backscattering of waves off the effective curvature potential at asymptotically far regions @xcite . the analysis of price was extended by other authors . bik @xcite generalized the analysis and studied the dynamics of a scalar field in a _ charged _ reissner - nordstrm spacetime . he also found an asymptotic inverse power - law decay of the field , with the _ same _ power indices as in the schwarzschild spacetime ( with the exception of the _ extremal _ reissner - nordstrm black hole @xcite ) . in a brilliant work , leaver @xcite demonstrated that the late - time tail can be associated with the existence of a branch cut in the green s function for the wave propagation problem . gundlach , price , and pullin @xcite showed that these inverse power - law tails also characterize the late - time evolution of radiative fields at future null infinity , and at the black - hole outer horizon . furthermore , they showed that power - law tails are a genuine feature of gravitational collapse the existence of these tails was demonstrated in full non - linear numerical simulations of the spherically symmetric collapse of a self - gravitating scalar field @xcite ( this was later reproduced in @xcite ) . our current understanding of the late - time tail is , however , somewhat unsatisfactory . the ( _ leading order _ ) power - law tails in black - hole spacetimes are well established @xcite , but the resultant formulae are only truly useful at very _ late _ times . in a typical evolution scenario there is a considerable time window in which the signal is no longer dominated by the quasinormal modes @xcite , but the leading order power - law tail has not yet taken over @xcite . the purpose of this paper is to derive analytic expressions for the _ higher - order corrections _ which `` contaminate '' the well - known power - law tail in a spherically symmetric gravitational collapse . the determination of these higher - order terms is important from several points of view : the analyses of bik @xcite and gundlach et . @xcite established the fact that the leading - order power - law tail is _ universal _ in the sense that it is _ independent _ of the black - hole electric charge ( i.e. , the power index in a _ charged _ reissner - nordstrm spacetime was shown to be identical with the one found by price @xcite for the neutral schwarzschild black hole ) . this observation begs the question : what fingerprints ( if any ) does the black - hole electric charge leave on the field s decay ? moreover , the calculation of higher - order corrections to the leading order power - law tail is also of practical importance ; this is especially crucial for the determination of the power index from numerical simulations . the dominant inverse power - law tail is _ `` contaminated '' _ by higher - order terms , whose effect become larger as the aveliable time of integration decreases . the precise power index is expected only at infinitely - late time . thus , in practice , the _ limited _ time of integration introduces an inherent error in the determination of the power index . the only systematic approach to _ quantify _ the errors which are introduced by the finite integration time is to study _ higher - order corrections_. if one computes the contaminated part of the late - time tail , then the ratio of the corrections to the leading order term is a systematic , quantitative , indication of the error caused by the _ finite_-time numerical calculation . these questions and several others are addressed in the present paper . the plan of the paper is as follows . in sec . [ sec2 ] we give a short description of the physical system and formulate the evolution equation considered . in sec . [ sec3 ] we give an analytical description of the late - time evolution of scalar fields in black - hole spacetimes . in sec . [ sec4 ] we confirm our analytical results by numerical simulations . we conclude in sec . [ sec5 ] with a brief summary of our results and their implications . we consider the evolution of a spherically symmetric massless scalar field in a spherically symmetric charged background ( a collapsing star or a fixed black hole ) . the external gravitational field of a spherically symmetric charged object of mass @xmath4 and charge @xmath5 is given by the reissner - nordstrm metric @xmath6 using the tortoise radial coordinate @xmath7 , which is defined by @xmath8 , the line element becomes @xmath9 where @xmath10 . the wave equation @xmath11 for the scalar field in the black - hole background is @xmath12 where @xmath13 in terms of the tortoise coordinate @xmath7 and for @xmath14 the curvature potential eq . ( [ eq4 ] ) reads @xmath15 the general solution to the wave - equation ( [ eq3 ] ) can be written as a series depending on two arbitrary functions @xmath16 and @xmath17 @xcite @xmath18}\ .\end{aligned}\ ] ] here @xmath19 is a retarded time coordinate and @xmath20 is an advanced time coordinate . for any function @xmath21 , @xmath22 is the @xmath23th derivative of @xmath24 ; negative - order derivatives are to be interpreted as integrals . the first two terms in eq . ( [ eq6 ] ) represent the zeroth - order solution ( with @xmath25 ) . the star begins to collapse at a retarded time @xmath26 . the world line of the stellar surface is asymptotic to an ingoing null line @xmath27 , while the variation of the field on the stellar surface is asymptotically infinitely redshifted @xcite . this effect is caused by the time dilation between static frames and infalling frames . a static external observer sees all processes on the stellar surface become `` frozen '' as the star approaches the horizon . thus , he sees all physical quantities approach a constant . we therefore make the explicit assumption that after some retarded time @xmath28 on @xmath27 . this assumption has been proven to be very successful @xcite . we begin with the first stage of the evolution , i.e. , the scattering of the field in the region @xmath29 . the first two terms in eq . ( [ eq6 ] ) represent the primary waves in the wave front , while the sum represents backscattered waves . the interpretation of these integral terms as backscatter comes from the fact that they depend on data spread out over a _ section _ of the past light cone , while outgoing waves depend only on data at a fixed @xmath30 @xcite . after the passage of the primary waves there is no outgoing radiation for @xmath31 , aside from backscattered waves . this means that @xmath32 . hence , for a large @xmath7 at @xmath33 , the dominant term in eq . ( [ eq6 ] ) is @xmath34 . the functions @xmath35 satisfy the recursion relation @xmath36 for @xmath37 , where @xmath38 , and @xmath39 . thus , one finds @xmath40g^{(-1)}(u_1 ) \big[1+o(m / y ) \big]\ .\ ] ] this is the dominant backscatter of the primary waves . with this specification of characteristic data on @xmath41 , we shall next consider the asymptotic evolution of the field . we confine our attention to the region @xmath42 , @xmath43 . in this region the spacetime is approximated as flat @xcite . ( the validity of this approximation is ultimately justified by numerical simulations ) . thus , the solution for @xmath44 can be written as @xmath45 comparing eq . ( [ eq8 ] ) with the initial data on @xmath41 eq . ( [ eq7 ] ) , one finds @xmath46 where @xmath47 for late times @xmath48 we can expand @xmath49 and similarly for @xmath50 . using these expansions we can rewrite eq . ( [ eq8 ] ) as @xmath51}\ , \ ] ] where the coefficients @xmath52 are those given in @xcite . using the boundary conditions for small @xmath53 ( regularity as @xmath54 , at the horizon of a black hole , or at @xmath55 for a stellar model ) , one finds that at late times the terms @xmath56 and @xmath57 must be of the same order ( see @xcite for additional details ) . thus , we conclude that @xmath58 and @xmath59 we therefore find that the late - time behaviour of the field for @xmath60 is @xmath61 \big[1+o(m / t)\big ] \ . \end{aligned}\ ] ] this is the late - time behaviour of the field at a fixed radius . it is straightforward to integrate eq . ( [ eq3 ] ) using the method described in @xcite . we have used , however , a modified version of the numerical code used in @xcite , which is essential to achieve the extremely high accuracy needed for the computation ( see @xcite for additional details ) . the late - time evolution of the scalar field is independent of the form of the initial data used . the results presented here are for a gaussian pulse on @xmath62 @xmath63^{2 } \right \}\ , \ ] ] with a center at @xmath64 and a width @xmath65 . the black - hole mass is set equal to @xmath66 ; this corresponds to the freedom to rescale the coordinates by an overall length scale . the temporal evolution of the field at a fixed radius @xmath67 is shown in the top panel of fig . the dominant _ power - law _ fall off is manifest at asymptotic late times . in order to study the contamination effect of higher - order terms ( [ eq14 ] ) ] , we use the notion of a _ local power index _ @xmath68 , defined by @xmath69 @xcite . taking cognizance of eq . ( [ eq14 ] ) we find @xmath70 the approach of the local power index to its well - known asymptotic value @xmath71 is depicts in the bottom panel of fig . the plot shows that @xmath72 from above , with a qualitative agreement with eq . ( [ eq16 ] ) . in order to establish _ quantitatively _ the physical picture presented in sec . [ sec3 ] , we define the quantity @xmath73 . figure [ fig2 ] . depicts @xmath74 as a function of @xmath75 at three surfaces of constant radius @xmath76 , and @xmath77 ( from bottom to top ) . the numerical result @xmath78 ( independently of the value of @xmath7 ) is in excellent agreement with the _ analytically _ predicted behaviour @xmath79 [ see eq . ( [ eq16 ] ) ] . thus , fig . [ fig2 ] . establishes the existence of the contamination term of order @xmath0 . it should be noted that the value of @xmath80 used in this figure is @xmath81 rather than the theoretical value @xmath82 . this slight deviation from the theoretical value is expected due to the corrections of order @xmath83 in the expression for @xmath68 eq . ( [ eq16 ] ) . the purpose of this paper was to derive analytic expressions for the _ higher - order corrections _ which `` contaminate '' the well - known power - law tail in a spherically symmetric gravitational collapse . we have shown , both analytically and numerically , that the dominant correction dies off at late times as @xmath0 . this late - time decay of the contamination is much _ slower _ than has been considered so far @xcite ( see the discussion in appendix b ) . aside from being _ theoretically _ important , the result eq . ( [ eq14 ] ) is also of _ practical _ importance . it follows that an ` exact ' ( numerical ) determination of the power index demands extremely long integration times . the most accurate method for determining the power index experimentally applies to the concept of the _ local _ power index . in this way one discard the relatively large contamination which characterizes the early stages of the evolution . still , it follows from eq . ( [ eq16 ] ) that a determination of the power index to within @xmath1 requires an integration time of order @xmath84 . the dominant power - law tail is known to be universal in the sense that it is _ independent _ of the black - hole parameters @xcite . we have shown , however , that this universality is removed once we consider higher - order corrections terms the leading order fingerprint of the black - hole electric _ charge _ behaves as @xmath3 . * acknowledgments * i thank tsvi piran for discussions . this research was supported by a grant from the israel science foundation . it follows from eqs . ( [ eq8 ] ) , ( [ eq9 ] ) , and ( [ eq12 ] ) that the asymptotic behaviour of the field at future null infinity @xmath85 ( i.e. , at @xmath86 ) is @xmath87 we finally consider the behaviour of the field at the black - hole outer horizon @xmath88 . as @xmath89 the curvature potential eq . ( [ eq4 ] ) is exponentially small , and the general solution to eq . ( [ eq3 ] ) can be written as @xmath90 . on @xmath91 we take @xmath92 ( for @xmath93 ) . thus , @xmath94 must be a constant , and with no loss of generality we can choose it to be zero . we next expand @xmath95 for @xmath96 as @xmath97 in order to match the @xmath98 solution eq . ( [ eqa2 ] ) with the @xmath99 solution eq . ( [ eq14 ] ) , we make the ansatz @xmath100 $ ] for the solution in the region @xmath98 and @xmath96 . in other words , we assume that the solution in the @xmath98 region has the same late time @xmath75 dependence as the @xmath14 solution . this assumption has been proven to be very successful for the leading order behaviour of both neutral @xcite and charged @xcite fields . using this assumption , one finds @xmath101 $ ] , where @xmath102 is a constant . thus , the asymptotic behaviour of the field at the black - hole horizon is @xmath103\ .\ ] ] the first attempt to calculate higher - order corrections to the dominant power - law tail was made by andersson @xcite ( in the context of the schwarzschild spacetime ) . this analysis was based on an _ approximated _ curvature potential in the region far away from the black hole [ see eq . ( 26 ) of @xcite ] . while this approximated potential simplifies the analysis , it actually _ misses _ the genuine leading order corrections terms . the leading order correction in @xcite was found to be of order @xmath104 . this ( artificial ) result is caused by the fact that terms of order @xmath105 ( and smaller ) were totally neglected by the approximated curvature potential used in @xcite . moreover , the approximated approach presented in @xcite , _ if _ extended along the same lines to a reissner - nordstrm spacetime would imply that the influence of the black - hole electric charge on the late - time tail vanishes _ identically _ ( i.e. , it would vanish to _ any _ order in @xmath106 ) . this result is again a direct consequence of the fact that the approximated approach of @xcite does not take into account curvature terms of order @xmath107 ( and @xmath108 ) which appear in the ( exact ) curvature potential . temporal evolution of the scalar field , evaluated at @xmath67 in a schwarzschild spacetime with @xmath66 . the initial data is a gaussian distribution with @xmath109 and @xmath110 . asymptotic _ power - law fall off is manifest at late times ( top panel ) . the bottom panel depicts the evolution of the local power index @xmath111 . the power index approaches the well - known asymptotic value @xmath112 at late times.,width=642 ] the time evolution of the quantity @xmath73 , evaluated at @xmath113 , and @xmath77 ( from bottom to top ) . the asymptotic value @xmath78 is in excellent agreement with the _ analytically _ predicted behaviour @xmath79 at late times . this result establishes the existence of the contamination term of order @xmath0.,width=642 ]

it is well known that the late - time behaviour of gravitational collapse is _ dominated _ by an inverse power - law decaying tail . we calculate _ higher - order corrections _ to this power - law behaviour in a spherically symmetric gravitational collapse . the dominant `` contamination '' is shown to die off at late times as @xmath0 . this decay rate is much _ slower _ than has been considered so far . it implies , for instance , that an ` exact ' ( numerical ) determination of the power index to within @xmath1 requires extremely long integration times of order @xmath2 . we show that the leading order fingerprint of the black - hole electric _ charge _ is of order @xmath3 .

gr-qc9907044

recently unconventional electronic properties of mono - crystalline graphenes attracts much attention motivated by experimental fabrication,@xcite although they were already the subject of theoretically study prior to the fabrication.@xcite multilayer films which contain more than two layers can also be synthesized , and various phenomena depending on the layer number have been reported.@xcite in this paper we present a theoretical study on the orbital magnetism in multilayer graphenes . the electronic structure of the monolayer graphene is quite different from conventional metals , because the conduction and valence bands touch at @xmath1 and @xmath2 points in the brillouin zone , around which the dispersion becomes linear like a relativistic particle . in multi - layer graphenes , the interlayer coupling makes a complex structure around the band touching . the electronic properties of graphene bilayer were theoretically studied for the band structure @xcite and the transport properties.@xcite for few - layered graphenes of more than two stacks , the electronic structure is investigated theoretically in a @xmath3 approximation,@xcite a density functional calculation,@xcite and a tight - binding model.@xcite on the experimental side , the band structures of graphenes from one to four layers were recently measured using angle - resolved photoemission spectroscopy.@xcite the orbital magnetism in graphene - based systems was first studied for a monolayer as a simple model to explain the large diamagnetism of graphite.@xcite it was found that the susceptibility becomes highly diamagnetic at @xmath4 ( band touching point ) even though the density of states vanishes there . the calculation was extended to graphite @xcite and to few - layered graphenes as a model of graphite intercalation compounds.@xcite the fermi surface of the graphite is known to be trigonally warped around the band touching point @xcite and the effect of the warping on magnetization was discussed within the perturbational approach.@xcite recently , the disorder effects on the magnetic oscillation @xcite and on the susceptibility @xcite were studied for the monolayer graphene . here we present a systematic study on the orbital magnetism for multilayer graphenes with arbitrary layer numbers in the effective mass approximation . we show that the hamiltonian of a multilayer graphene can be decomposed into those equivalent to monolayer or bilayer , which allows us to study the dependence of the susceptibility on layer numbers . we take the trigonal warping effect into the calculation and show that the fine structure around zero energy gives rise to singular magnetic properties . we introduce the model hamiltonian and its decomposition into subsystems in sec . [ sec_form ] , and present the calculation of the magnetization in sec . [ sec_mag ] . the discussion and summary are given in sec . [ sec_disc ] . we consider a multilayer graphene composed of @xmath5 layers of a carbon hexagonal network , which are arranged in the ab ( bernal ) stacking , as shown in fig . [ fig_schem ] . a unit cell contains @xmath6 and @xmath7 atoms on the layer @xmath8 . for convenience we divide carbon atoms into two groups as @xmath9 the atoms of group i are arranged along vertical columns normal to the layer plane , while those in group ii are above or below the center of hexagons in the neighboring layers . the lattice constant within a layer is given by @xmath10 nm and the distance between adjacent layers @xmath11 nm . the system can be described by a * k*@xmath12*p * hamiltonian closely related to a three - dimensional ( 3d ) graphite model.@xcite the low energy spectrum is given by the states in the vicinity of @xmath1 and @xmath2 points in the brillouin zone . let @xmath13 and @xmath14 be the bloch functions at the @xmath1 point , corresponding to the @xmath15 and @xmath16 sublattices , respectively , of layer @xmath17 . for monolayer graphene , the hamiltonian around @xmath1 point for the basis @xmath18 , @xmath19 is written as @xcite @xmath20 where @xmath21 and @xmath22 with @xmath23 being the nearest - neighbor coupling in a single layer . we cite the experimental estimation @xmath24 ev.@xcite for the inter - layer coupling , we include parameters @xmath25 , and @xmath26 , following the hamiltonian previously derived for a bilayer graphene.@xcite here @xmath25 represents the coupling between vertically neighboring atoms in group i ( @xmath27 ) , and @xmath26 between group ii atoms on neighboring layers ( @xmath28 ) , which are estimated to @xmath29 ev,@xcite and @xmath30 ev.@xcite if we look at the interaction between layers 1 and 2 , the matrix element @xmath31 , corresponding to the vertical bond , becomes @xmath25 not accompanying in - plane bloch number . the matrix element @xmath32 is written as @xmath33 with @xmath34 similar to the intra - layer term @xmath35 , as the in - plane vector components from @xmath36 to @xmath37 are identical to those from @xmath38 to @xmath36 . accordingly , if the basis is taken as @xmath39 ; @xmath40 ; @xmath41 ; @xmath42 , the hamiltonian for the multilayer graphene around the @xmath1 point becomes @xmath43 with @xmath44 the effective hamiltonian for @xmath2 is obtained by exchanging @xmath45 and @xmath46 and replacing @xmath25 with @xmath47 . the derivation of the effective hamiltonian based on a tight - binding model is presented in appendix [ sec_app_a ] . we show in the following that the hamiltonian matrix ( [ eq_h ] ) can be block - diagonalized into smaller matrices by choosing an appropriate basis independent of @xmath48 . first , we arrange the basis in the order of group i and then group ii , i.e. , @xmath19 , @xmath49 , @xmath50 , @xmath41 ; @xmath18 , @xmath51 , @xmath52 , @xmath41 . then , eq . ( [ eq_h ] ) becomes @xmath53 with @xmath54 being @xmath55 matrices defined as @xmath56 where the upper and lower signs correspond to odd and even @xmath5 , respectively . if we set @xmath57 ( the @xmath1 point ) , @xmath58 and @xmath59 vanish . remaining @xmath60 is equivalent to the hamiltonian of a one - dimensional tight - binding chain with the nearest - neighbor coupling @xmath25 , giving a set of eigenenergies @xmath61 with @xmath62 here , @xmath63 is an odd integer when the layer number @xmath5 is even , while @xmath63 is even when @xmath5 is odd , and therefore @xmath64 is allowed only for odd @xmath5 . the corresponding wave function is explicitly written as @xmath65 , \label{eq_psi}\ ] ] where @xmath66 represents the amplitudes at @xmath19 , @xmath49 , @xmath50 , @xmath41 and satisfies @xmath67 we have a relation between the wave functions @xmath68 and @xmath69 as @xmath70 now we construct the basis by assigning @xmath71 to the atoms of group i and ii as @xmath72 and attempt to rewrite the hamiltonian ( [ eq_h ] ) . the matrix elements within group i come from @xmath60 and become diagonal as is obvious from the definition , @xmath73 off - diagonal elements between @xmath74 and @xmath75 are written from @xmath58 as @xmath76 in the second equality we used relation ( [ eq_bipartite ] ) and orthogonality ( [ eq_orthogonality ] ) . lastly , the matrix elements within group ii are obtained from @xmath59 as @xmath77 \nonumber \\ + \!\ ! & \!\!\!\ ! & \gamma ' i k_y \sum_{j=1}^{n-1 } ( -1)^j \left [ \psi_{m'}^*(j+1 ) \psi_{m}(j ) - \psi_{m'}^*(j ) \psi_{m}(j+1 ) \right ] \nonumber\\ = & \!\!\!\ ! & \gamma ' \lambda_{n , m } ( k_x \delta_{m , m ' } + i k_y \delta _ { m ,- m ' } ) . \ ] ] the hamiltonian is thus closed in the subspace @xmath78 for each @xmath79 . particularly , @xmath64 is special in that the subspace is spanned with only two bases @xmath80 , while this is absent in even - layer graphenes . the sub - matrix is written as @xmath81 which is independent of @xmath25 and @xmath26 , and equivalent to the hamiltonian of the monolayer graphene . for @xmath82 , we rearrange the basis as @xmath83 where we take @xmath84 without loss of generality . we then obtain @xmath85 with @xmath86 . this is equivalent to the hamiltonian of a bilayer graphene except that @xmath25 and @xmath87 @xmath88 are multiplied by @xmath89 . thus the hamiltonian of odd - layered graphene is composed of one monolayer - type and @xmath90 bilayer - type subbands while that of even - layered graphene is composed of @xmath91 bilayers but no monolayer . the similar idea was previously proposed for trilayer graphene without @xmath26 , where it was shown that the energy spectrum becomes a superposition of that for a monolayer and for a bilayer.@xcite here we have extended this argument to decomposition of the hamiltonian matrix , and to systems with arbitrary number of layers including the trigonal warping . we also note that @xmath48-independence of the basis becomes important in the following sections , since this enables us to write the magnetization as a sum over contributions from sub - hamiltonians , which are independently calculated . many other parameters were introduced for the description of the band structure of bulk graphite.@xcite the parameter @xmath92 couples group i and ii atoms sitting on the neighboring layers , such as @xmath93 or @xmath94 . this parameter does not change the qualitative feature of the low - energy spectrum and therefore is not important.@xcite parameters @xmath95 and @xmath96 represent vertical hoppings between the second - nearest neighboring layers for group ii and i atoms , respectively . further , @xmath97 is an energy difference between the group i and ii atoms due to difference in the chemical environment . inclusion of these parameters @xmath95 , @xmath96 , and @xmath97 causes opening up of small energy gaps between the conduction and the valence bands . however , these gaps do not play important roles in the magnetization as will be discussed in the following . in 3d limit , @xmath98 , the eigenstate becomes a superposition of opposite traveling waves with @xmath99 along the stacking direction . the relation between the index @xmath63 and @xmath100 is obtained by comparing the eigenenergy of @xmath60 , eq . ( [ eq_1d ] ) , to that of the 3d limit , @xmath101 , as @xmath102 the band structure of the hamiltonian ( [ eq_hm ] ) can be obtained by replacing @xmath25 by @xmath103 and @xmath26 by @xmath104 in that of the bilayer.@xcite we plot in fig . [ fig_band0 ] the dispersion for @xmath105 , which has the maximum trigonal warping . the middle two subbands stick together at @xmath106 while the remaining two bands appear only in the energy range @xmath107 . if we neglect @xmath26 , the effective hamiltonian for @xmath108 becomes @xmath109 which works for the reduced basis @xmath110 giving a rotationally symmetric dispersion with the effective mass @xmath111 ) with @xmath105 and @xmath112 . @xmath113 is shown as horizontal dotted lines . right panel shows zoom out of the left . ( bottom ) 3d plot of the lower second band around the band touching point . four fermi points at @xmath4 indicated by dots . , width=302 ] the term proportional to @xmath26 is responsible for the trigonal warping effect , which is most remarkable around the band sticking point @xmath114 . let us define @xmath115 in the energy range @xmath116 , the fermi line splits into four separated pockets , one center part and three leg parts located trigonally , which shrink into four fermi points linearly with @xmath117 . we note that @xmath118 is proportional to @xmath119 and thus very sensitive to @xmath89 , while the energy of the higher - band bottom , @xmath103 , behaves linear to @xmath89 . the maximum of @xmath89 approaches 2 as the layer number increases , so that @xmath118 becomes as large as @xmath120 mev . figure [ fig_band ] shows the band structures around the @xmath1 point along the @xmath121 axis in the multilayer graphenes with @xmath122 , 3 , 4 , and 5 and @xmath123 . the lists of @xmath124 are given as @xmath125 while we have included @xmath23 , @xmath25 , and @xmath26 in our graphene model , the extra parameter neglected here may make some changes in the electronic structure . the energy band of a few - layered graphene has been calculated in the density functional calculation @xcite and the tight - binding model.@xcite those results differ from ours mainly in that the band centers relatively shift depending on @xmath63 , and that a narrow gap opens where the conduction and valence bands ( within a single @xmath63 ) touch , and where different bands ( with different @xmath63 s ) cross . gaps are attributed to effects of couplings such as @xmath95 , @xmath96 , and @xmath97 , which are mentioned above . in terms of the effective mass hamiltonian ( [ eq_h ] ) , those parameters appear as matrix elements without being multiplied by the wave number @xmath121 and @xmath126 , since they are associated with a hopping along the @xmath127 axis or a diagonal element . thus , they do not vanish at @xmath128 ( @xmath1 or @xmath2 ) and lift the degeneracy to open a gap . apart from the gap opening , the main feature of the trigonal warping is well described in the present model . it should also be mentioned that an energy gap is induced by an electric field perpendicular to the layer stacking direction,@xcite where the electrostatic potential appears as matrix elements independent of @xmath121 and @xmath126 as well . with @xmath123 , around the @xmath1 point ( taken as origin ) along the @xmath121 axis . right panel shows a zoom - out of the left . numbers assigned to curves indicate @xmath63 . , width=302 ] for the magnetic susceptibility , we use the general expression based on the linear response theory,@xcite @xmath129 with @xmath130 where @xmath131 is the valley degeneracy , @xmath132 is the spin degeneracy , and @xmath133 is the system size . we defined here @xmath134 , @xmath135 , @xmath136 , and @xmath137^{-1}$ ] with the chemical potential @xmath138 and the temperature @xmath139 . the formula valid also for the hamiltonian ( [ eq : bilayer_low_energy ] ) is discussed in appendix [ sec_app_b ] . by integration by parts in eq . ( [ eq_chi ] ) , we have @xmath140 showing that the susceptibility at non - zero temperature is written in terms of that at zero temperature . the integration of @xmath141 over @xmath138 is independent of @xmath139 . we include the impurity scattering effects by introducing a self - energy @xmath142 in the green s function , i.e. , @xmath143 in ( [ eq_chi ] ) is replaced by @xmath144 . here we simply assume the scattering rate @xmath145 to be independent of energy . using the decomposition of the hamiltonian , the magnetization of the @xmath5-layered graphene can be written as a summation over each sub - hamiltonian . the contribution from @xmath64 is exactly equivalent to the susceptibility of a monolayer graphene,@xcite which becomes at zero temperature and in the clean limit , @xmath146 thus , the odd - layer graphene always has a large diamagnetic peak at zero energy . the delta - function dependence of @xmath147 agrees with the general property of the susceptibility in systems described by the @xmath148-linear hamiltonian , as discussed in sec . [ sec_disc ] . in the presence of disorder , the delta - function is broadened into a lorentzian with width @xmath149 and the same area,@xcite i.e. @xmath150 within the present model assuming a constant @xmath149 . the shape of the peak itself depends on the model disorder and we may have some different manner of broadening in a more realistic treatment . in fact , in the monolayer graphene it was shown in a self - consistent born approximation@xcite that @xmath141 has a much sharper peak at @xmath151 than the lorentzian and also a large tail proportional to @xmath152 for @xmath153.@xcite in multi - layer cases effects of disorder are more complicated because of the presence of other bands . this problem is out of the scope of this work . the susceptibility of a bilayer graphene described by the hamiltonian ( [ eq_hm ] ) was analytically calculated for the case of @xmath154.@xcite the expression for @xmath155 and @xmath156 is given by @xmath157 with @xmath86 , where @xmath158 is a step function defined by @xmath159 the susceptibility diverges logarithmically toward @xmath0 , becomes slightly positive for @xmath160 , and vanishes for @xmath161 where the higher subband enters . in the presence of disorder , the logarithmic peak is broadened approximately as @xmath162 . the integration of @xmath141 in eq . ( [ eq_chi_bi ] ) over the fermi energy becomes @xmath163 independent of @xmath25 , which is exactly twice as large as that of the monolayer graphene ( [ eq_chi_mono ] ) . this arises due to the fact that the integral of @xmath141 over the fermi energy is determined only by terms of the hamiltonian matrix , proportional to @xmath121 or @xmath126 , and is independent of terms independent of @xmath121 and @xmath126 . a proof of this important property is presented in sec . [ sec_disc ] . if we include the extra band parameter @xmath26 , the low - energy structure of the susceptibility ( [ eq_chi_bi ] ) drastically changes due to the fine structure around the band touching point . to demonstrate this , we numerically calculate @xmath141 for the hamiltonian ( [ eq_hm ] ) in the case of the maximum trigonal warping , @xmath164 . figure [ fig_chi ] shows the susceptibility as a function of @xmath165 with several values of @xmath149 . we take @xmath166 , where the fermi line splitting occurs in lower than @xmath167 . for reference we also plot the result without the trigonal warping , @xmath168 , as a dashed curve . ) in the multilayer graphene with @xmath169 , @xmath123 and several disorder strengths @xmath149 . energy is scaled in units of @xmath167 . dashed curves show plots for @xmath170 . inset at the top is a zoom out of the top panel ( @xmath171 ) with units of energy @xmath25 . , width=302 ] when we go down from high energy in the top panel ( the smallest @xmath149 ) , the susceptibility gradually deviates downward from the logarithmic dependence of @xmath168 , and takes a sharp dip at @xmath172 . remarkably we have a strong peak centered on @xmath4 , which is regarded as the effect of the linear dispersions around zero energy . the integral of @xmath141 over the fermi energy is almost constant @xmath173 as discussed in sec . [ sec_disc ] , showing that the reduction in higher energies compensates the zero energy peak . as @xmath149 becomes larger , the peak begins to cancel with the reduction in high energy and the effect of @xmath26 eventually disappears when @xmath174 . in contrast , the peak associated with the monolayer band @xmath64 becomes broad in @xmath149 but never vanishes , as shown in eq.([eq : monolayer_with_broadening ] ) . figure [ fig_chi_multi ] shows @xmath175 of graphenes with layer number from @xmath176 to 5 with several disorder strengths @xmath149 . for @xmath177 , insets show the contributions from each of bilayer - type bands . the result of odd @xmath5 always contains a monolayer - like component , which is exactly the same as @xmath176 and thus omitted in the inset . we can see that odd - layered graphenes exhibit a particularly large peak , which mainly comes from the monolayer - type band . a bilayer - like component contains a central peak due to the trigonal warping and a logarithmic tail in high energies , in accordance with fig . [ fig_chi ] . the layer - number dependence of the susceptibility in multilayer graphene has been studied for the graphite intercalation compounds.@xcite this system can be viewed as independent multilayer graphenes bound by the intercalant layers , but the intercalants give a strong electrostatic potential along the stacking direction , leading to the charge redistribution among different layers.@xcite as a result the band structure and the magnetization are considerably different from our system with a uniform electrostatic potential in the vertical direction . in isolated multilayer graphenes realized in recent experiments , we may have some potential difference among layers depending on the experimental environment , and this can also be tuned by the external electric field as mentioned . in sec . [ sec_disc ] , we will show that , as long as the potential is not too strong to alter the entire band structure , this does not change the qualitative feature of the magnetization . the zero energy peak in the bilayer - type subband originates in dirac - like dispersions appearing around four fermi points . using the known results in a bilayer,@xcite we can show that the sequence of the landau levels in the center pocket approximately becomes @xmath178 with @xmath179 , and those in the three leg parts @xmath180 , where @xmath181 is the magnetic length . since the susceptibility is determined solely by landau level energies , we compare this to the monolayer s sequence @xmath182 and obtain @xmath141 from each pocket by substituting @xmath183 in eq . ( [ eq_chi_mono ] ) . we end up with @xmath184 except for a constant coming from the integral over the lower energy states . the zero - energy peak in fig . [ fig_chi ] fits well to the lorentzian with width @xmath149 and the area of the delta - function ( [ eq_chi_fine ] ) , as long as @xmath185 . the factor attached to @xmath147 becomes as large as 0.4 when @xmath164 and @xmath123 , and therefore the singularity is not too small compared with that of the monolayer . for @xmath5-layered graphene , a simple relation @xmath186 leads to the summation of ( [ eq_chi_fine ] ) over all the bilayer - type subbands , @xmath187 in fig . [ fig_chi_multi ] the peak height becomes a little larger than this estimation due to mixing with the logarithmic tail . the delta - function dependence of @xmath141 in monolayer graphene is a characteristic property common to general @xmath148-linear hamiltonian . this can be shown using the scaling argument . we consider a hamiltonian @xmath188 which contains only terms linear in @xmath121 and @xmath126 . we change the energy and wave number scales by an arbitrary factor @xmath189 as @xmath190 then the hamiltonian becomes formally identical under this transformation , since the coefficients of @xmath148-linear terms in the hamiltonian remain unchanged . going back to the definition of @xmath141 in ( [ eq_chi ] ) and ( [ eq_f ] ) , @xmath191 is scaled as @xmath192 the function @xmath193 should depend only on the coefficients of @xmath148-linear terms and natural constants , and thus is invariant under the scale transformation , namely we have @xmath194 . with ( [ eq_f_scale ] ) , we come up with a equation , @xmath195 which is satisfied solely by @xmath196 a constant @xmath15 is related to the integral of the susceptibility @xmath175 over the fermi energy @xmath165 . from ( [ eq_chi ] ) , we generally have @xmath197 where the integral path @xmath198 is a circle with an infinite radius with anti - clockwise direction . in the present system , ( [ eq_f_explicit ] ) immediately gives the integral as @xmath199 . this is an integral of the real function @xmath200 and thus is real . substituting eq . ( [ eq_f_explicit ] ) with real @xmath15 in eq . ( [ eq_chi ] ) , we finally obtain the explicit form of the zero - temperature susceptibility as @xmath201 as discussed in sec . [ sec_form ] , the band structure in more realistic models has an energy gap around zero energy due to extra band parameters neglected in the present model . it was also mentioned that the external electric field along stacking direction opens an energy gap . one might think that the gap would strongly reduce the large diamagnetism at the band touching point . however , we can show within the effective mass approximation that the integral of susceptibility over @xmath165 is independent of any kind of matrix elements without @xmath121 and @xmath126 , which are responsible for gap opening . this is obvious from the general expression ( [ eq_chi_oint ] ) ; even if the hamiltonian contains @xmath148-independent terms in addition to @xmath148-linear terms , they can be safely neglected in the integral as they are infinitesimal compared to @xmath202 on the path @xmath198 . in the effective - mass model of the multilayer graphene we immediately conclude that the integral is independent of @xmath25 , @xmath95 , @xmath96 , @xmath97 , and any other parameters independent of the wave vector . thus we expect that the large diamagnetic peak is still visible even when a gap opens , while it may get broadened in energy by the gap width . the diamagnetism in narrow gap systems is known in bismuth@xcite and recently studied for the gapped dirac fermion.@xcite any further discussion requires a direct computation of the magnetization including extra parameters , but we leave this for the future study . the integral of the susceptibility can be calculated by the hamiltonian with the @xmath148-independent terms dropped , and thus depends only on the band parameters associated with @xmath148-linear terms . in our model ( [ eq_h ] ) , the value is mainly determined by the dominant parameter @xmath23 , while @xmath26 gives a correction at most of the order of @xmath203 . the correction must be the second order in @xmath26 because we can change @xmath26 to @xmath204 in the hamiltonian with a unitary transformation multiplying the base on layer @xmath17 by @xmath205 . as a result , the integral of @xmath175 for the bilayer - type hamiltonian becomes almost twice as large as the monolayer s , and the summation over all the subsystems in @xmath5-layered graphene becomes approximately @xmath5 times as large as the monolayer s . it is instructive to derive the susceptibility starting from the landau - level energies . in the monolayer graphene , the thermodynamic function @xmath206 is given by @xmath207 \big\ } , \end{aligned}\ ] ] where @xmath208 , @xmath209 with @xmath210 , and @xmath211 a cutoff function which gradually decays to zero for @xmath212 with cutoff energy @xmath213 . we can rewrite this as @xmath214 where @xmath215 , \ ] ] with @xmath216 . expanding the integral @xmath217 with respect to @xmath218 , we immediately have @xmath219 \nonumber \\ = & \!\!\!\!\!\ ! & \int_0^\infty h(x ) d x - { 1\over 12 } h^2 \big [ h'(0 ) \!+\ ! { 1\over 2 } h'(\infty ) \big ] , \\end{aligned}\ ] ] up to the second order in @xmath218 or in @xmath16 . then , we have @xmath220 where @xmath221 is the thermodynamic function in the absence of a magnetic field and @xmath222 ^ 2 } \nonumber\\ & = & { g_{\rm v } g_{\rm s } \gamma^2 \over 12 \pi l^4 } \int_{-\infty}^{\infty } \big ( \!-\ ! { \partial f(\varepsilon ) \over \partial \varepsilon } \big ) \delta(\varepsilon ) d \varepsilon . \label{eq : thermodynamic_potential:_monolayer_graphene}\end{aligned}\ ] ] applying the relation @xmath223 , we obtain @xmath147 given by ( [ eq_chi_mono ] ) at zero temperature . we should note that the thermodynamic function in the absence of a magnetic field is given by @xmath224 for contributions of states with @xmath225 , we can expand the above with respect to @xmath226 and have to the lowest order in the field strength @xmath16 @xmath227 \nonumber\\ & \ ! \approx \ ! & { g_{\rm v } g_{\rm s } \over 2\pi l^2 } { ( \hbar\omega_b)^2 \over 48 } \!\ ! \lim_{\delta\rightarrow+0 } \big ( \int_\delta^\infty [ \varepsilon^{-2 } f(\varepsilon ) \!-\ ! \varepsilon^{-1 } f'(\varepsilon ) ] d \varepsilon \nonumber \\ & & \qquad\qquad\qquad\quad - \int_{-\infty}^{-\delta } [ \varepsilon^{-2 } f(\varepsilon ) \!-\ ! \varepsilon^{-1 } f'(\varepsilon ) ] d \varepsilon \big ) \nonumber \\ & \ ! = \ ! & - { g_{\rm v } g_{\rm s } \over 2\pi l^2 } { 1\over 24 } ( \hbar\omega_b)^2 \int_{-\infty}^\infty \big ( \!-\ ! { \partial f(\varepsilon ) \over \partial \varepsilon } \big ) \delta(\varepsilon ) d \varepsilon .\end{aligned}\ ] ] this gives a `` paramagnetic '' susceptibility . for @xmath228 , on the other hand , the change in the thermodynamic potential is calculated as @xmath229 the sum of these two contribution is the same as eq . ( [ eq : thermodynamic_potential:_monolayer_graphene ] ) , as is expected . in the bilayer graphene , the landau level with @xmath154 in the region @xmath230 can be calculated from the hamiltonian ( [ eq : bilayer_low_energy ] ) as @xcite @xmath231 where @xmath232 with @xmath233 defined in eq . ( [ eq_mstar ] ) , @xmath234 , and @xmath235 . we have doubly degenerate levels at zero energy ( @xmath236 , @xmath234 ) , while the spacing gradually becomes constant as @xmath237 goes higher . in a similar but more complicated manner , the susceptibility is calculated as @xmath238 which correctly describes the logarithmic divergence around zero energy in the rigorous expression ( [ eq_chi_bi ] ) , as expected . a constant term independent of energy is missing in eq . ( [ eq_chi_bi_alt ] ) since this depends on all the low - energy bands which are neglected in this calculation . we can understand the logarithmic dependence intuitively by looking into the landau - level sequence . the landau level energy can be expanded for large @xmath237 as @xmath239,\ ] ] where the first term gives the constant interval , and the second gives a shift toward zero energy , which is rewritten as @xmath240 . for @xmath241 , for example , the change in the total energy due to the energy shift is calculated as @xmath242 giving the @xmath243 dependence of the susceptibility . for the hamiltonian ( [ eq : bilayer_low_energy ] ) containing terms proportional to @xmath244 , the susceptibility formula ( [ eq_chi ] ) with ( [ eq_f ] ) is no longer valid , since this was originally derived for systems in which @xmath245 commutes with @xmath246 and @xmath247 . the modified formula should be @xmath248 this is derived in appendix [ sec_app_b ] . a scaling argument similar to the case of the monolayer graphene then gives @xmath249 . this again leads to the logarithmic dependence of @xmath141 on the fermi energy , which coincides with ( [ eq_chi_bi_alt ] ) apart from a constant . the experimental measurements of the magnetization of two - dimensional electron systems were performed on the semiconductor heterostructures , by using the superconducting quantum interference device ( squid)@xcite or using the torque magnetometer.@xcite we expect that the detection of the graphene magnetism is also feasible with those techniques . we have studied the orbital magnetism of multilayer graphene with the bernal stacking in the effective mass approximation . we have demonstrated that the hamiltonian and thus the susceptibility can be decomposed into those equivalent to the monolayer or bilayer bands . the monolayer - like band exists only in odd - layered graphenes and gives a strong diamagnetic peak at @xmath250 . the bilayer - like bands always exist and present a strong diamagnetism in the vicinity of zero energy , unless the fine band structure caused by @xmath26 is destroyed by the disorder . this work has been supported in part by the 21st century coe program at tokyo tech nanometer - scale quantum physics and by grants - in - aid for scientific research from the ministry of education , culture , sports , science and technology , japan . we derive in the following the effective mass equation eq . ( [ eq_h ] ) describing states in the vicinity of @xmath1 point in a multilayer graphene , by starting from the one - orbital tight - binding model . the following is nothing but a straightforward extension of the monolayer case.@xcite in a tight - binding model , the wave function is written as @xmath251,\end{aligned}\ ] ] where @xmath252 is the layer index , @xmath253 is the wave function of the @xmath254 orbital of a carbon atom located at the origin , as a function of three - dimensional position @xmath255 . @xmath256 is the three - dimensional position of the site @xmath257 , and @xmath258 is a two - dimensional component of @xmath259 , parallel to the layer . in the model including hopping parameters @xmath23 , @xmath25 , and @xmath26 defined in sec . [ sec_form ] , schrdinger s equation can be written as follows : for odd @xmath17 , @xmath260 , \\ \vare \psi_{b_{j}}({\mbox{\boldmath$\rho$}}_{b_{j } } ) & = & -\gamma_0 \sum_{l=1}^3 \psi_{a_{j}}({\mbox{\boldmath$\rho$}}_{b_{j } } + { \mbox{\boldmath$\tau$}}_l ) \nonumber\\ & & \hspace{0 mm } + \gamma_1 \left [ \psi_{a_{j+1}}({\mbox{\boldmath$\rho$}}_{b_{j } } ) + \psi_{a_{j-1}}({\mbox{\boldmath$\rho$}}_{b_{j } } ) \right ] . \quad \label{eq_sch1}\end{aligned}\ ] ] for even @xmath17 , @xmath261 , \quad \\ \vare \psi_{b_{j}}({\mbox{\boldmath$\rho$}}_{b_{j } } ) & = & -\gamma_0 \sum_{l=1}^3 \psi_{a_{j}}({\mbox{\boldmath$\rho$}}_{b_{j } } + { \mbox{\boldmath$\tau$}}_l ) \nonumber\\ & & \hspace{-20 mm } + \gamma_3 \sum_{l=1}^3 \left [ \psi_{a_{j+1}}({\mbox{\boldmath$\rho$}}_{b_{j } } - { \mbox{\boldmath$\tau$}}_l ) + \psi_{a_{j-1}}({\mbox{\boldmath$\rho$}}_{b_{j } } - { \mbox{\boldmath$\tau$}}_l ) \right ] . \label{eq_sch2}\end{aligned}\ ] ] here we introduced the vectors from b site to the nearest neighboring a sites as @xmath262 , @xmath263 , and @xmath264 , and we set @xmath265 . the states around @xmath1 point can be expressed in terms of the slowly - varying envelope functions @xmath266 as @xmath267 where @xmath268 , and @xmath269 , @xmath270 are phase factors defined by @xmath271 with @xmath272 . when @xmath273 is much smaller than the length scale of the envelope functions , we have @xmath274 with @xmath275 or @xmath7 . by substituting eq . ( [ eq_env ] ) with ( [ eq_approx ] ) , into schrdinger s equations ( [ eq_sch1 ] ) and ( [ eq_sch2 ] ) , we have for odd @xmath17 , @xmath276 \nonumber\\ \vare f_{b_j}({\mbox{\boldmath$\rho$ } } ) & = & \gamma k_+ f_{a_j}({\mbox{\boldmath$\rho$ } } ) + \gamma_1 \left [ f_{a_{j-1}}({\mbox{\boldmath$\rho$ } } ) + f_{a_{j+1}}({\mbox{\boldmath$\rho$ } } ) \right ] , \nonumber\\\end{aligned}\ ] ] and for even @xmath17 , @xmath277 , \nonumber\\ \vare f_{b_j}({\mbox{\boldmath$\rho$ } } ) & = & \gamma k_+ f_{a_j}({\mbox{\boldmath$\rho$ } } ) + \gamma ' k_- \left [ f_{a_{j-1}}({\mbox{\boldmath$\rho$ } } ) + f_{a_{j+1}}({\mbox{\boldmath$\rho$ } } ) \right],\nonumber\\\end{aligned}\ ] ] where @xmath21 with @xmath278 , @xmath279 . we also used an identity @xmath280 if we rewrite this set of equations into the matrix form for a vector @xmath281 , we finally obtain the hamiltonian matrix ( [ eq_h ] ) . the effective hamiltonian for another valley @xmath282 can be derived in a parallel way , while @xmath283 and @xmath284 are exchanged in eq . ( [ eq_c ] ) . the susceptibility formula ( [ eq_chi ] ) with ( [ eq_f ] ) has been derived for the luttinger - kohn representation of the bloch function@xcite and therefore in systems described by the hamiltonian consisting of the free electron kinetic energy @xmath285 and terms linear in @xmath286.@xcite we derive here the susceptibility formula which is valid in the general hamiltonian @xmath287 which includes @xmath148-square terms in off - diagonal matrix elements as well as @xmath148-linear terms . we shall confine ourselves to the case without electron - electron interaction for simplicity . consider the system described by the schrdinger equation @xmath288 with @xmath289 and @xmath290 being the vector potential . the thermodynamic function @xmath206 is given by @xmath291 \big\ } \nonumber\\ & = & - k_{\rm b } t \ , g_{\rm s } \frac{1}{v } \int d\varepsilon \ , \big(\!-\!{1\over\pi}\big ) { \rm i m } \ , { \rm tr } \ , \frac{1}{\vare -{\cal h } + i0 } \nonumber\\ & & \qquad\qquad \times\ln \big\ { 1 \!+\ ! \exp[\beta(\mu \!-\ ! \varepsilon ) ] \big\ } , \label{eq_omega}\end{aligned}\ ] ] where @xmath292 is the spin degeneracy and @xmath293 is the system volume . we consider an isotropic system and assume the vector potential @xmath294 , with @xmath295 where we are going to take the long wavelength limit @xmath296 , for which the field causes the response the same as that due to a spatially uniform magnetic field . in the presence of this vector potential , the hamiltonian changes from @xmath297 to @xmath298 , with @xmath299 , where @xmath300 . the hamiltonian can be expanded as @xmath301 where @xmath302 , @xmath303 , and @xmath304 . note that in general @xmath148-square hamiltonian @xmath305 does not commute with @xmath246 but does with @xmath306 . expanding the hamiltonian up to the second order in the strength of the magnetic field @xmath16 , we have @xmath307 where @xmath308 is the magnetic length , @xmath309 , @xmath310 , and we assumed that the system is translational invariant ( after the configuration average in the presence of impurities ) . we then expand @xmath311 up to the second order in @xmath312 , to have @xmath313 with @xmath314 . this immediately gives the change of the thermodynamic potential @xmath315 with eq . ( [ eq_omega ] ) . the susceptibility @xmath141 is obtained by a relation @xmath316 as @xmath317 this gives eq . ( [ eq_chi ] ) with ( [ eq_f_gen ] ) for the multilayer graphene . d. p. divincenzo and e. j. mele , phys . b * 29 * , 1685 ( 1984 ) . j. gonzalez , f. guinea , and m. a. h. vozmediano , nucl . b * 424 * , 595 ( 1994 ) . j. gonzalez , f. guinea , and m. a. h. vozmediano , phys . lett . * 77 * , 3589 ( 1996 ) . n. h. shon and t. ando , j. phys . * 67 * , 2421 ( 1998 ) . j. gonzalez , f. guinea , and m. a. h. vozmediano , phys . b * 63 * , 134421 ( 2001 ) . y. zheng and t. ando , phys . b * 65 * , 245420 ( 2002 ) . t. ando , y. zheng , and h. suzuura , j. phys . jpn . * 71 * , 1318 ( 2002 ) . h. suzuura and t. ando , phys . lett . * 89 * , 266603 ( 2002 ) . t. ando , j. phys . * 74 * , 777 ( 2005 ) . k. s. novoselov , e. mccann , s. v. morozov , v. i. falko , m. i. katsnelson , u. zeitler , d. jiang , f. schedin , and a. k. geim , nat . phys . * 2 * , 177 ( 2006 ) . t. ohta , a. bostwick , t. seyller , k. horn , and e. rotenberg , science * 313 * , 951 ( 2006 ) ; t. ohta , a. bostwick , j. l. mcchesney , t. seyller , k. horn , and e. rotenberg , phys . lett . * 98 * , 206802 ( 2007 ) . e. mccann and v. i. falko , phys . lett . * 96 * , 086805 ( 2006 ) .

we present a theoretical study on the orbital magnetism in multilayer graphenes within the effective mass approximation . the hamiltonian and thus susceptibility can be decomposed into contributions from sub - systems equivalent to monolayer or bilayer graphene . the monolayer - type subband exists only in odd layers and exhibits a delta - function susceptibility at @xmath0 . the bilayer - type subband appearing in every layer number gives a singular structure in the vicinity of @xmath0 due to the trigonal warping as well as a logarithmic tail away from @xmath0 . the integral of the susceptibility over energy is approximately given only by the layer number .

0712.2783

recently a new method for analyzing multifractal functions @xmath1 was introduced @xcite . it exploits the fact that the fractional derivative of order @xmath2 ( denoted here by @xmath3 ) of @xmath1 has , for a suitable range of @xmath2 , a power - law tail in its cumulative probability @xmath4 the exponent @xmath5 is the unique solution of the equation @xmath6 where @xmath7 is the scaling exponent associated to the behavior at small separations @xmath8 of the structure function of order @xmath9 , i.e. @xmath10 . it was also shown that the actual observability of the power - law tail when multifractality is restricted to a finite range of scales is controlled by how much @xmath7 departs from linear dependence on @xmath9 . the larger this departure the easier it is to observe multifractality . so far the theory of such power - law tails has been developed only for synthetic random functions , in particular the random multiplicative process @xcite for which kesten - type maps @xcite and large deviations theory can be used . it is our purpose here to test the fractional derivative method for invariant measures of dissipative dynamical systems , in particular for the feigenbaum invariant measure which appears at the accumulation point of the period doubling cascade where the orbit has period @xmath11 @xcite . its multifractality was proven rigorously in ref . @xcite using a thermodynamic formalism . for the feigenbaum measure all scaling exponents can be determined with arbitrary accuracy . there is an important difference in the way one processes functions and invariant measures to determine their multifractal properties and in particular the spectrum of singularities , usually denoted @xmath12 for functions @xcite and @xmath0 for measures @xcite . for a function @xmath1 one uses the moments or the pdfs of the increments @xmath13 to determine the scaling exponents , whereas for an invariant measure @xmath14 one works with integrals over intervals or boxes of different sizes . in the one - dimensional case the two approaches become equivalent by introducing the cumulative distribution function @xmath15 hence we shall apply the fractional derivative method to the integral of the invariant measure . the organization of the paper is the following . section [ s : thermo ] is devoted to the thermodynamic formalism for the feigenbaum attractor . in section [ ss : formalism ] , we recall the method used in ref . @xcite . in section [ ss : connection ] we show how this formalism , based on the study of the geometrical properties of the attractor , is actually connected to the standard multifractal formalism which focusses on the statistical properties of the invariant measure @xcite . to the best of our knowledge the exact relation between the two formalisms is discussed here for the first time . then , in section [ ss : numericalfreeenergy ] we calculate numerically the free energy and accordingly the scaling exponents @xmath7 for the integral of the invariant measure ; this is done by a very accurate transfer - matrix - based method . fractional derivatives are discussed in section [ s : fraclap ] . in section [ ss : fraclap_pheno ] we briefly recall the phenomenology of power - law tails in the distribution of fractional derivatives and the limits on observability . the fractional derivative analysis of the feigenbaum measure is presented in section [ ss : fraclap_numerics ] . concluding remarks are made in section [ s : concl ] . in this section we give a brief description of the thermodynamic formalism for the invariant measure of the feigenbaum map ( see ref . @xcite for the mathematical details ) and show how one can use it in order to study the multifractal properties of the hlder exponents . by feigenbaum attractor we understand the attractor of the one - dimensional mapping @xmath16 \to [ 0,1]$ ] , where @xmath17 is the solution of the feigenbaum cvitanovi doubling equation : @xmath18 equation ( [ g ] ) is known to have the unique solution in the class of smooth unimodal maps ( that is , maps having one critical point ) with a non - degenerate maximum . this solution is called the feigenbaum map . it is holomorphic in some complex neighborhood of @xmath19 $ ] and the first few terms in the power series expansion are @xcite @xmath20 the value of the universal constant @xmath21 which is the inverse of the feigenbaum scaling constant @xmath22 is approximately equal to @xmath23 . an attractor @xmath24 for the map @xmath25 can be constructed in the following way . for each @xmath26 define a collection of intervals of @xmath27th level : @xmath28 , \nonumber \\ & & \delta^{(n)}_i = g^{(i)}(\delta^{(n)}_0 ) \equiv \underbrace{g \circ g \circ \cdots \circ g}_{i } ( \delta_0^{(n ) } ) \quad ( 1 \leq i\leq 2^n-1 ) . \label{delta}\end{aligned}\ ] ] the following properties of the intervals @xmath29 are easy consequences of the doubling equation ( [ g ] ) : ( a ) intervals @xmath30 are pairwise disjoint . ( b ) @xmath31 . ( c ) each interval of @xmath27th level @xmath29 contains exactly two intervals of @xmath32th level , @xmath33 and @xmath34 . ( d ) @xmath35 , where @xmath36 denotes the length of the interval . the first three levels of the intervals are shown in fig . [ f : dynamicalpartition ] . the feigenbaum cvitanovi map @xmath17 and the first three levels of the partitions @xmath37 . for @xmath17 we used the expansion ( [ g1 ] ) , introduced in ref . @xcite up to @xmath38 . ] 65 10 dynamical partitions the properties above imply that it is natural to use a dyadic representation for the intervals @xmath29 . let @xmath39 , where @xmath40 . then we can use a sequence @xmath41 as a symbolic coding for intervals @xmath29 : @xmath42 . now we can define the feigenbaum attractor @xmath43 the set @xmath24 is isomorphic to the set of all infinite dyadic sequences @xmath44 . such sequences can be considered as a symbolic coordinate system on @xmath24 . in this new coordinate system the map @xmath25 acts as the dyadic addition of the sequence @xmath45 . notice that topologically @xmath24 is a cantor set . it is easy to see that @xmath24 is indeed an attractor for all but countably many initial points @xmath46 $ ] : @xmath47 as @xmath48 . the exceptional set of initial points consists of all unstable periodic orbits and their preimages . as we have seen above , all intervals @xmath29 have exponentially small lengths but the exponent varies from @xmath49 to @xmath50 . notice that exponents @xmath51 give all possible scalings of the fractal set @xmath24 . the basic ingredient which is needed for the multifractal analysis is the control over the spectrum of possible scalings corresponding to exponents , i.e. @xmath51 . such control can be achieved with the help of the thermodynamic formalism . the thermodynamic formalism which was constructed in ref . @xcite is based on the gibbsian description for the lengths of the intervals @xmath29 . it is shown in ref . @xcite that there exists a function @xmath52 ( thermodynamic potential ) which is defined on all infinite dyadic sequences such that : i. there exists a constant @xmath53 for which @xmath54 \ii . for any two dyadic sequences @xmath55 which coincide on the first @xmath56 positions @xmath57 with @xmath58 @xmath59\leq \frac{|\delta^{(n)}_{\epsilon_0 , \dots , \epsilon_{n-1}}|}{|\delta^{(n-1)}_{\epsilon_0 , \dots , \epsilon_{n-2}}|}\exp(-u(\epsilon_{n-1 } , \dots , \epsilon_{1},1,0,\dots , 0,\dots ) ) \leq \exp[c(2\gamma)^{\frac{n}{3 } } ] \ , .\ ] ] it immediately follows from ( [ exp1 ] ) that for @xmath60 $ ] @xmath61}\leq c_1 \ , .\ ] ] the condition @xmath62 corresponding to intervals @xmath63 with odd @xmath64 s plays only a technical role and it is not essential for our further analysis since the odd intervals contain information about the lengths of the even ones . indeed , it is very easy to see that for every odd @xmath64 the intervals @xmath63 and @xmath65 have lengths of the same order . we next introduce a parameter @xmath66 ( inverse temperature ) and define the partition function @xmath67 \label{part}\ ] ] and the free energy @xmath68 it immediately follows from ( [ part ] ) and ( [ free ] ) that @xmath69.\ ] ] in the thermodynamic limit @xmath70 the probability distributions @xmath71 \label{gibbs}\ ] ] tend to a limiting distribution @xmath72 which can be considered as a gibbs measure with the potential @xmath73 , inverse temperature @xmath66 and the boundary condition @xmath74 . this gibbs distribution generates the probability measure on @xmath75 which is the part of the whole attractor @xmath24 corresponding to intervals @xmath29 with odd numbers @xmath64 . we shall denote this gibbs measure on @xmath76 by @xmath77 . notice that @xmath78 corresponds to a unique invariant measure and @xmath79 gives a conditional distribution corresponding to lebesgue measure on @xmath19 $ ] . the free energy @xmath80 contains all information about the multifractal properties of the feigenbaum attractor . notice that the thermodynamic formalism leads to one - dimensional statistical mechanics with exponential decay of interactions and hence without phase transitions . this implies that @xmath80 is a smooth function . in fact it is holomorphic in some complex neighborhood of the real axis . denote @xmath81 using relations @xmath82 \nonumber \\ & = & \left<\frac{\ln |\delta^{(n)}_{1,\epsilon_1 , \dots , \epsilon_{n-1}}|}{n}\right>_{\nu^{(n)}_\beta } + o\left(\frac{1}{n}\right)\end{aligned}\ ] ] and @xmath83 \nonumber \\ & & -\left(\frac{1}{z_n(\beta)}\sum_{\epsilon_{n-1 } , \dots , \epsilon_0 = 0,1}\frac{1}{n}h(\epsilon_{n-1 } , \dots , \epsilon_1 ) \exp\left[\beta h(\epsilon_{n-1 } , \dots , \epsilon_1)\right]\right]^2 \nonumber \\ & & = \left<\left(\frac{\ln |\delta^{(n)}_{1,\epsilon_1 , \dots , \epsilon_{n-1}}|}{n}\right)^2\right>_{\nu^{(n)}_\beta } - \left(\left<\frac{\ln |\delta^{(n)}_{1,\epsilon_1 , \dots , \epsilon_{n-1}}|}{n}\right>_{\nu^{(n)}_\beta}\right)^2 + o\left ( \frac{1}{n}\right),\end{aligned}\ ] ] we conclude that @xmath80 is a monotone decreasing convex function . we shall also use the spectral representation for the free energy . consider the transfer - matrix operator @xmath84 : @xmath85h(\epsilon^{(0)},\epsilon^{(1 ) } , \epsilon^{(2 ) } , \dots , \epsilon^{(n ) } , \dots ) \ , .\ ] ] since @xmath84 is a positive linear operator , its largest eigenvalue @xmath86 is strictly positive and simple . it is easy to see that @xmath87 for an arbitrary point @xmath88 , denote by @xmath89 the interval of the @xmath56th level which contains @xmath90 . it follows from ( [ monotonicity ] ) that for points @xmath90 which are typical with respect to @xmath77 ( that is corresponding to a set of full @xmath77-measure . ) latexmath:[\[\label{typ1 } precisely , for @xmath77-almost all @xmath92 @xmath93 we next find the total number @xmath94 of the intervals of @xmath27th level whose length is of the order @xmath95 $ ] . we have @xmath96\ , \sim \ , z_n(\beta ) \ , \sim \ , \exp[f(\beta)n]\ ] ] which gives @xmath97 \ , .\ ] ] using ( [ n ] ) we can find the hausdorff dimension @xmath98 of the set of points @xmath88 which are typical with respect to the measure @xmath99 . since @xmath100\ ] ] we conclude that @xmath101 which immediately implies @xmath102 the hausdorff dimension @xmath103 of the whole attractor @xmath24 is equal to the maximum of @xmath98 over all @xmath104 . let @xmath105 be the unique solution of the equation @xmath106 . it is easy to see that @xmath107 . the integral @xmath1 of the feigenbaum invariant measure calculated with @xmath108 bins of uniform length @xmath109 in @xmath19 $ ] . inset : the invariant measure smoothed over the distance @xmath110 calculated as a frequency histogram . ] 65 10 feigenbaum measure , u(x ) we next discuss multifractal properties associated with the hlder exponents . consider the integral @xmath1 of the invariant measure @xmath14 , defined by ( [ distr ] ) , which is plotted in fig . [ f : feigendistribution ] . the attractor being topologically a cantor set , @xmath1 is a variant of the devil s staircase ( see ref . @xcite , section 8.2 ) . to find its spectrum of hlder exponents , notice that for every interval @xmath29 the increase of @xmath1 along the interval is equal to @xmath111 . hence @xmath29 corresponds to a hlder exponent @xmath112 where @xmath113 . this implies @xmath114 using ( [ typ1 ] ) , ( [ hold ] ) we conclude that the hlder exponent @xmath22 corresponds to an inverse temperature @xmath115 such that @xmath116 this gives @xmath117 where @xmath118 is the inverse function to @xmath119 . we can now find the hausdorff dimension @xmath120 of the set of points @xmath92 for which the hlder exponent of @xmath1 is equal to @xmath22 : @xmath121 notice that the analysis presented above can be made completely rigorous ( see , for example , refs . @xcite ) . it is quite interesting to compare the multifractal analysis which we presented above with the one introduced in ref . @xcite . although we restrict ourselves here to the case of the feigenbaum attractor , the result presented below holds in a much more general setting . basically our analysis is valid whenever the system under consideration can be described with the help of the thermodynamic formalism . the basic object for our analysis is the feigenbaum attractor itself and the method is based on the construction of the thermodynamic formalism for the lengths of the elements of dynamic partitions @xmath29 . the thermodynamic formalism uses considerable dynamical information about the map @xmath25 . in contrast , the analysis in ref . @xcite is carried out for fractal measures and does not directly use the dynamical information about the system . in the period - doubling setting the fractal measure is @xmath14 . it is the unique invariant measure for @xmath25 acting on @xmath24 ( see ref . it also can be considered as a physical or sinai ruelle bowen ( srb ) measure on @xmath19 $ ] . this means that under dynamics given by the map @xmath25 any initial absolutely continuous distribution @xmath122 on @xmath19 $ ] converges to @xmath14 : @xmath123 as @xmath70 . the multifractal analysis in ref . @xcite is based on a function @xmath124 which can be defined in the following way . consider a partition of the interval @xmath19 $ ] into subintervals @xmath125 of length @xmath8 . then @xmath126 it follows from ( [ hjkps1 ] ) that @xmath127 another characteristic of a multifractal measure is given by its spectrum of dimensions @xmath0 which is just the legendre transform of @xmath128 : @xmath129 \ , .\ ] ] the dual legendre relation allows one to find @xmath128 from @xmath0 : @xmath130 \ , .\ ] ] we next find a correspondence between the pair @xmath131 and the pair of thermodynamic functions @xmath132 . we shall show that @xmath133 where @xmath134 is an inverse function to the free energy @xmath80 . to derive the first relation we consider the dynamical partition @xmath135 and assume that @xmath136 but @xmath137 . for each @xmath63 define @xmath138 notice that the asymptotic behavior of @xmath139 depends only on asymptotic scalings of smaller elements of the dynamical partitions @xmath140 inside @xmath63 . the thermodynamic formalism constructed above implies that asymptotically those scalings are completely determined by the potential @xmath73 and hence they do not depend on @xmath64 . rescaling the invariant measure inside @xmath63 by a factor @xmath141 we conclude that @xmath142 where @xmath143 is a total number of the intervals @xmath144 inside @xmath63 . taking the sum over @xmath64 and using ( [ free1 ] ) we have @xmath145\ , .\end{aligned}\ ] ] this together with ( [ hjkps11 ] ) immediately gives @xmath146=2^p\ ] ] which implies the first relation in ( [ hjkps3 ] ) . we next show that the second relation holds . using ( [ hjkps2 ] ) we have @xmath147=\inf_p \ [ \alpha p - ( -f^{-1}(p\ln 2)]=\inf_z \ \left[\frac{\alpha}{\ln 2}z + f^{-1}(z)\right ] \nonumber \\ & = & \inf_\beta \ \left[\frac{\alpha}{\ln 2}f(\beta ) + \beta \right ] = \frac{\alpha}{\ln 2}\inf_\beta\ \left[\frac{\ln 2}{\alpha}\beta + f(\beta)\right ] \ , .\end{aligned}\ ] ] it is easy to see that the extremum in ( [ hjkps8 ] ) corresponds to @xmath148 which implies @xmath149 finally we express the scaling exponents @xmath7 for the structure functions through the thermodynamic characteristics . the exponent @xmath7 is defined by the scaling relation @xmath150 in terms of the integral @xmath1 of the invariant measure . let @xmath151 be a partition of @xmath19 $ ] into intervals of length @xmath8 . then @xmath152 which together with ( [ hjkps3 ] ) gives @xmath153 using ( [ hjkps22 ] ) one can also write @xmath7 in the following form : @xmath154 = 1 + \inf_{\alpha } \ [ \alpha p - d_h(\alpha ) ] \ , .\ ] ] at @xmath155 the infimum ( [ structure3 ] ) is attained , we can then write the relation between @xmath9 and @xmath156 as @xmath157 the scaling exponents for the structure functions are hence obtained as @xmath158 we now turn to concrete calculations for the scheme presented above . it is easy to see that all the thermodynamic functions can be effectively approximated numerically . the first step is to find approximations for the thermodynamic potential @xmath52 . we shall use markov approximations @xmath159 which are defined by the following formula : @xmath160 it was shown in @xcite that the limit in ( [ kapprox ] ) exists and @xmath161 we can construct a markov approximation for the transfer - matrix operator @xmath84 . namely , we define a finite dimensional linear operator @xmath162 : @xmath163h(\epsilon^{(0)},\epsilon^{(1 ) } , \epsilon^{(2 ) } , \dots , \epsilon^{(k-1 ) } ) \ , .\ ] ] in the matrix representation the operator @xmath162 corresponds to a certain @xmath164 matrix . denote by @xmath165 its largest eigenvalue . then @xmath166 is a natural approximation for the free energy @xmath80 . it follows from ( [ kapprox1 ] ) that @xmath167 converges to @xmath80 exponentially fast in @xmath168 topology . using @xmath167 we can effectively approximate all the multifractal functions which we discussed above . the corresponding numerical results are presented in the next section . here we show how to construct the transfer - matrix operator @xmath169 starting from @xmath170 to general @xmath171 . for @xmath170 , the matrix operator @xmath172 is just a scalar . denoting @xmath173 by @xmath174 for clarity , let us consider @xmath175 = \lim_{n \to \infty } \frac{|{\delta^{({n})}({1 , \overbrace{0 , \ldots , 0}^{n - 2 } , \epsilon^{(0)}})}| } { |{\delta^{({n - 1})}({1 , \underbrace{0 , \ldots , 0}_{n - 2}})}| } , \label{e : eu1 } \end{aligned}\ ] ] whose analytical expression is easy to calculate . the 0-th order approximation of the free energy is given by @xmath176.\end{aligned}\ ] ] for @xmath177 , since @xmath178 $ ] and @xmath179 $ ] , we have @xmath180 ) of @xmath17 . thus @xmath181 . therefore , the corresponding component of @xmath182 is @xmath183 before considering the second term in the argument of the logarithm in ( [ twoterms ] ) , we recall the relation @xmath184 which played a primary role in the proof of theorem 4.1 . of ref . since @xmath185 , in view of ( [ e : nsim ] ) , we have @xmath186 ^ 2 - 1\ } + o(\alpha^{-4(n - 1)})|.\end{aligned}\ ] ] thus for @xmath187 , ( [ e : eu1 ] ) can be rewritten as @xmath188 ^ 2 - 1\ } + o(\alpha^{-4(n - 1)})| } { |c \alpha^{-2(n - 1 ) } + o(\alpha^{-4(n - 1)})| } = 1 - [ g(\alpha^{-1})]^2.\end{aligned}\ ] ] we now arrive at the expression for the @xmath189-th order approximation of the free energy : @xmath190 ^ 2\}^{\beta } ] . \label{e : f0beta } \end{aligned}\ ] ] next , consider the @xmath191 approximation to the transfer matrix @xmath192 . from ( [ lk ] ) , it can be written in standard matrix notation as @xmath193 it then follows that the free energy is given by @xmath194 \nonumber \\ & & - \ln 2.\end{aligned}\ ] ] the four components of the transfer matrix require the evaluation of suitable exponential terms , expressible by ( [ kapprox ] ) , from @xmath195 = \lim_{n \to \infty } \frac{|{\delta^{({n})}({1 , \overbrace{0 , \ldots , 0}^{n - 3 } , \epsilon^{(1 ) } , \epsilon^{(0)}})}| } { |{\delta^{({n - 1})}({1 , \underbrace{0 , \ldots , 0}_{n - 3 } , \epsilon^{(1)}})}|}.\end{aligned}\ ] ] each of these terms is calculated in the same manner as for the @xmath170 case : @xmath196 ^ 2 - [ g(\alpha^{-1})]^2 \nonumber\\ & = & 1 - [ g(\alpha^{-1})]^2,\\ e^{u_2(0 , 1 ) } & = & \lim_{n \to \infty } \frac{|{\delta^{({n})}({1 , 0 , \dots , 0 , 1 , 0})}| } { |{\delta^{({n - 1})}({1 , 0 , \dots , 0 , 1 } ) } | } \nonumber \\ & = & \frac{|[g(0)]^2 - [ g(\alpha^{-2})]^2| } { |[g(0)]^2 - [ g(\alpha^{-1})]^2| } = \frac{1 - [ g(\alpha^{-2})]^2 } { 1 - [ g(\alpha^{-1})]^2},\\ e^{u_2(1 , 1 ) } & = & \lim_{n \to \infty } \frac{|{\delta^{({n})}({1 , 0 , \dots , 0 , 1 , 1})}| } { |{\delta^{({n - 1})}({1 , 0 , \dots , 0 , 1 } ) } | } \nonumber \\ & = & \frac{|[g^{(3)}(0)]^2 - [ g^{(3)}(\alpha^{-2})]^2| } { |[g(0)]^2 - [ g(\alpha^{-1})]^2|}. \end{aligned}\ ] ] for @xmath197 , we just write down the transfer matrix @xmath198 : @xmath199 now we are in a position to calculate the transfer matrix @xmath162 for general @xmath171 . let us consider the component for @xmath200 : @xmath201 & = & \lim_{n \to \infty } \frac{|{\delta^{({n})}({1 , \overbrace{0 , \ldots , 0}^{n - k - 2 } , \epsilon^{(k ) } , \ldots , \epsilon^{(1 ) } , \epsilon^{(0)}})}| } { |{\delta^{({n - 1})}({1 , \underbrace{0 , \ldots , 0}_{n - k - 2 } , \epsilon^{(k ) } , \ldots , \epsilon^{(1)}})}| } \nonumber \\ & = & \lim_{n \to \infty } \frac{|g({\delta^{({n})}({\overbrace{0 , \ldots , 0}^{n - k - 2 } , \epsilon^{(k ) } , \ldots , \epsilon^{(1 ) } , \epsilon^{(0)}})})| } { |g({\delta^{({n - 1})}({\underbrace{0 , \ldots , 0}_{n - k - 2 } , \epsilon^{(k ) } , \ldots , \epsilon^{(1)}})})|}. \label{e : compo } \end{aligned}\ ] ] hence it is enough to calculate @xmath202 . for this we use the following relation : @xmath203 by ( [ e : similarity ] ) the numerator in ( [ e : compo ] ) is given by @xmath204 ^ 2\ } - \{1 - c\ , \alpha^{-2(n - k - 1)}[g^{(j)}(\alpha^{-(k + 1)})]^2\ } + o(\alpha^{-4(n - k)})|\ } \nonumber \\ & = & c \ , \alpha^{-2(n - k - 1)}|[g^{(j)}(0)]^2 - [ g^{(j)}(\alpha^{-(k + 1)})]^2| + o(\alpha^{-4(n - k)}).\end{aligned}\ ] ] therefore the component ( [ e : compo ] ) is expressed as @xmath201 = \left| \frac{[g^{(j)}(0)]^2 - [ g^{(j)}(\alpha^{-(k + 1)})]^2 } { [ g^{(j')}(0)]^2 - [ g^{(j')}(\alpha^{-k})]^2 } \right| , \label{e : concretecompo } \end{aligned}\ ] ] where @xmath205 and @xmath206 . in other words , if the coordinates of the end points of the intervals are given by @xmath207 $ ] and @xmath208 $ ] , the component ( [ e : compo ] ) can be rewritten as @xmath209 = \left| \frac{\left(a_j^{(k + 1)}\right)^2 - \left(b_j^{(k + 1)}\right)^2 } { \left(a_{j'}^{(k ) } \right)^2 - \left(b_{j'}^{(k ) } \right)^2 } \right| . \label{e : ab}\end{aligned}\ ] ] if the transfer matrix is written as an ordinary @xmath210 matrix , it is easily found that the row and columns indices are given by @xmath211 , where @xmath212 for numerical calculation of the functions @xmath167 and @xmath7 associated to @xmath1 , we use the expansion ( [ g1 ] ) for @xmath17 , as given in ref . we include terms up to @xmath38 . numerical calculation is done with standard double precision ( 15 significant digits ) . for obtaining the largest eigenvalue , we use the power method for matrices @xcite . the multiplication of the matrix is stopped when the relative error of the most dominant eigenvalue becomes less than @xmath213 , thus giving @xmath213 absolute error on the @xmath80 function . an alternative approach , also based on the thermodynamic formalism and yielding 10-digit accuracy , may be found in ref . approximate free energies @xmath167 , with @xmath171 up to @xmath214 , calculated by the transfer - matrix method are shown in fig . [ f : free - energy ] . the @xmath171-th order approximation to the free energy @xmath167 . inset : enlargement of the region @xmath215 ; when increasing @xmath171 , alternate convergence is observed . ] 65 10 free energy we note that the @xmath170 approximation ( [ e : f0beta ] ) already gives a reasonable estimate . the discrepancy of the free energy between various orders of approximations is visible at large @xmath66 . however the @xmath216 region is irrelevant as far as @xmath7 for positive @xmath9 is concerned ( see eq . ( [ 1-beta ] ) ) . the corresponding @xmath217 are calculated from the @xmath167 by ( [ structure2 ] ) for different values of @xmath171 ; the results , which hardly depend on @xmath171 , are shown in fig . [ f : feigen - zetap](a ) . ( a ) scaling exponents @xmath7 of structure functions obtained by two methods . open circles : data obtained by a least - square fit of the slopes of the directly measured structure functions shown in ( b ) . lines : data calculated from the free energy @xmath167 using ( [ structure2 ] ) ( curves for different @xmath171 are essentially indistinguishable ) . , title="fig : " ] ( a ) scaling exponents @xmath7 of structure functions obtained by two methods . open circles : data obtained by a least - square fit of the slopes of the directly measured structure functions shown in ( b ) . lines : data calculated from the free energy @xmath167 using ( [ structure2 ] ) ( curves for different @xmath171 are essentially indistinguishable ) . , title="fig : " ] 65 10 feigen : structure func , zetap we also determined the structure functions of @xmath1 with @xmath218 uniform bins in @xmath219 $ ] ; they are plotted in fig . [ f : feigen - zetap](b ) . the exponents @xmath7 are then obtained by a least square fit of the structure functions over the range @xmath220 . with this number of bins , the quality of the fit begins to somewhat deteriorate beyond @xmath221 , but otherwise there is rather good agreement between the two methods of determining @xmath7 . note that the `` @xmath222-intercept '' of the graph of @xmath7 , namely @xmath223 , which is the codimension of the support of the invariant measure @xmath224 , is positive and its numerical value is slightly under one half @xcite . this will be important in the sequel . in this section we briefly recall the phenomenological approach to multifractality via fractional derivatives @xcite and adapt it to a multifractal measure . we therefore work , not with the measure @xmath224 itself , but with its integral @xmath1 . singularity exponents @xmath22 may be viewed as local hlder exponents of @xmath1 , i.e. , @xmath225 for @xmath226 . we turn to fractional derivatives of order @xmath2 defined , as in ref . @xcite , as the multiplication in the fourier space by @xmath227 by @xmath228 ( see ref . @xcite for precise definition ) . an isolated non - oscillatory singularity with exponent @xmath22 at a point @xmath90 implies @xmath229 if @xmath230 , as we shall assume hereafter , the exponent is negative , the fractional derivative can become arbitrarily large and thus contributes to the tail - behavior of the probability . a key assumption in the phenomenology is that this argument can be carried over to non - isolated multifractal singularities , provided we take all types of singularities into account . for the feigenbaum invariant measure , we know the hausdorff dimension @xmath120 of the set of points having a singularity with exponent @xmath22 . assuming that we can also use @xmath120 as a covering dimension , we can express the probability to have a singularity of exponent @xmath22 contributing a fractional derivative of order @xmath2 which exceeds ( in absolute value ) a given large value @xmath231 , that is we require @xmath232 in terms of the codimension of the set @xmath233 , the probability to satisfy ( [ e : y - x ] ) is written as @xmath234 here @xmath235 is the spatial dimension ( @xmath236 ) . taking now into account the singularities with all possible exponents @xmath22 , the tail of the cumulative probability of the fractional derivative of order @xmath2 is given , to the leading order , by the following power law @xmath237 an easy calculation shows that @xmath156 corresponding to the infimum in ( [ e : inf ] ) satisfies @xmath238 which immediately gives @xmath239 . on the other hand , we know that @xmath240 here the infimum is given by an @xmath22 satisfying the very same relation @xmath239 . hence , @xmath241 using ( [ alpha_star ] ) , we get @xmath242 where the second relation follows from @xmath239 . the geometrical interpretation of this equation is that the ( negative ) exponent of the power - law tail for the fractional derivative of order @xmath2 is the @xmath9-value of the intersection of the graph of @xmath7 and of a straight line of slope @xmath2 through the origin . as shown in ref . @xcite , in the presence of the finite range of scaling , the power - law tail ( [ e : cprob ] ) emerges only if the multifractality is sufficiently strong . this strength is given by the multifractality parameter @xmath243 , a measure of how strongly the data depart from being self similar ( which would imply @xmath244 ) : @xmath245 where @xmath246 . it was shown that observability of the power - law requires a sufficiently large value for the product @xmath247 , where @xmath56 is the number of octaves over which the data present multifractal scaling . in practice it was found in ref . @xcite that @xmath248 for example , fully - developed turbulence velocity data @xcite have typical @xmath249 values of the order of @xmath250 , thereby requiring a monstrous inertial range of about 300 octaves for observability of power - law tails . as we shall see , the situation is much more favorable for the feigenbaum invariant measure . before turning to numerical questions , we comment on an issue raised by an anonymous referee who worried about the nonlocal character of the fractional derivative and wrote in essence that our approach makes sense , strictly speaking , only for ( statistically ) translationally invariant in space systems : otherwise , if the system consists of components whose `` fractal properties '' are rather different the results will be smeared out . our feeling about such matters is summarized as follows . first one can observe that , of course , the attractor for the feigenbaum map is not homogeneous ( translation - invariant ) but after zooming in it becomes increasingly so ; the fractional derivative is not a local operator but the tail of its pdf is likely to be dominated by strongly localized events . second , a more technical observation . the idea of the multi - fractal analysis is based on the fact that the dynamics of a system determines a variety of scales . it is important that these scales do not depend on a particular place in the phase space . on the contrary , they are present and `` interact '' with each other everywhere . in the case of the feigenbaum attractor the scales depend on a symbolic location in a system of partitions . in physical systems , like homogeneous turbulence , such partitions are difficult to define rigorously . however , the invariance with respect to the space coordinate is still present and forms a basis for applicability of the multifractal calculus . the phenomenological arguments presented in the previous section suggest that we should find power - law tails in the cumulative probability for fractional derivatives of @xmath1 for suitable orders @xmath2 . inspection of fig . [ f : feigen - zetap ] indicates that @xmath2 should be between the minimum slope of the graph and unity . the value @xmath251 is of course not a fractional order but , as we shall see , it is associated with a power - law tail of exponent minus one .. ( putting @xmath252 into the partition function @xmath253 , we have @xmath254 . ) ] the minimum slope can be easily found . indeed , @xmath80 takes large values when @xmath66 is large negative . in this case the main contribution to @xmath80comes from the shortest interval of the partition with the length of the order of @xmath50 . hence , @xmath255 in the limit @xmath256 . this gives the following lower bound of the differentiation order : @xmath257 we have already observed that , because the @xmath7 graph does not pass through the origin , substantial values can be expected for the multifractality parameter @xmath243 . the actual values of @xmath249 , associated to values of @xmath9 ranging from @xmath258 to 3 by increments of @xmath259 are shown in table [ t : fzetap ] , together with the number @xmath56 of scaling octaves needed determined by @xmath260 ( cf . ( [ e : criterion ] ) ) . .[t : fzetap ] for the feigenbaum invariant measure we show the scaling exponents @xmath7 , the corresponding inverse temperature @xmath66 s , the multifractality parameter @xmath249 and the number of scaling octaves needed . [ cols="^,^,^,^,^,^,^",options="header " , ] in practice , on a 32 bit machine , we are limited to about 25 octaves of dynamical range in resolution over the interval @xmath261 $ ] . this should be enough to observe power - law tails . cumulative probabilities of absolute values of fractional derivatives of various orders @xmath262 for the feigenbaum invariant measure . each function displays a power - law tail with an exponent fairly close to the predicted value @xmath263 . insets : corresponding @xmath7 graphs . ( a ) @xmath264 . ( b ) @xmath265 . ( c ) @xmath266 . ( d ) @xmath267 . ( e ) @xmath268 . ] cumulative probabilities of absolute values of fractional derivatives of various orders @xmath262 for the feigenbaum invariant measure . each function displays a power - law tail with an exponent fairly close to the predicted value @xmath263 . insets : corresponding @xmath7 graphs . ( a ) @xmath264 . ( b ) @xmath265 . ( c ) @xmath266 . ( d ) @xmath267 . ( e ) @xmath268 . , title="fig : " ] cumulative probabilities of absolute values of fractional derivatives of various orders @xmath262 for the feigenbaum invariant measure . each function displays a power - law tail with an exponent fairly close to the predicted value @xmath263 . insets : corresponding @xmath7 graphs . ( a ) @xmath264 . ( b ) @xmath265 . ( c ) @xmath266 . ( d ) @xmath267 . ( e ) @xmath268 . , title="fig : " ] cumulative probabilities of absolute values of fractional derivatives of various orders @xmath262 for the feigenbaum invariant measure . each function displays a power - law tail with an exponent fairly close to the predicted value @xmath263 . insets : corresponding @xmath7 graphs . ( a ) @xmath264 . ( b ) @xmath265 . ( c ) @xmath266 . ( d ) @xmath267 . ( e ) @xmath268 . , title="fig : " ] cumulative probabilities of absolute values of fractional derivatives of various orders @xmath262 for the feigenbaum invariant measure . each function displays a power - law tail with an exponent fairly close to the predicted value @xmath263 . insets : corresponding @xmath7 graphs . ( a ) @xmath264 . ( b ) @xmath265 . ( c ) @xmath266 . ( d ) @xmath267 . ( e ) @xmath268 . , title="fig : " ] 65 10 feigen : cumulative probability indeed , fig . [ f : feigen - cump ] shows five instances of cumulative probabilities of fractional derivatives with power - law tails , corresponding to the values of the exponent @xmath9 listed in table [ t : fzetap ] . the corresponding order of differentiation @xmath2 ranges between @xmath269 and @xmath258 . is too small , e.g. for @xmath270 , no power - law tail is observed . ] since the function @xmath1 which we are analyzing is not periodic , we resort to the hann windowing technique employed previously in ref . @xcite ( section 13.4 ) . also , we use rank ordering to avoid binning . the power - law behavior observed is consistent with the phenomenological theory presented in section [ ss : fraclap_pheno ] , the residual discrepancies being due to the resolution of @xmath271 bins . we have found solid numerical evidence for the presence of power - law tails in the cumulative distribution of fractional derivatives for the integral @xmath1 of the invariant measure of the feigenbaum map . furthermore the exponents measured are consistent with those predicted by phenomenological arguments from the spectrum of singularities . since we have a fairly deep understanding of the structure of the attractor , thanks in particular to the thermodynamic formalism , a reasonable goal may be to actually prove the results . the main difficulty is that the operation of fractional derivative is non - local . however , we believe that a rigorous analysis here is still possible due to the quite simple spectral structure of the dynamical system corresponding to the feigenbaum attractor . we are grateful to rahul pandit for useful remarks . computational resources were provided by the yukawa institute ( kyoto ) . this research was supported by the european union under contract hprn - ct-2000 - 00162 and by the indo - french centre for the promotion of advanced research ( ifcpar 2404 - 2 ) . vul , ya.g . sinai and k.m . khanin , feigenbaum universality and the thermodynamic formalism , _ russian math . surveys _ * 39*:140 ( 1984 ) . g. parisi and u. frisch , on the singularity structure of fully developed turbulence , in _ turbulence and predictability in geophysical fluid dynamics _ , proceedings of international school of physics enrico fermi , jun . 1424 1983 , varenna , italy , m. ghil , r. benzi and g. parisi , eds . , pp . 8487 , north holland ( 1985 ) .

it is shown that fractional derivatives of the ( integrated ) invariant measure of the feigenbaum map at the onset of chaos have power - law tails in their cumulative distributions , whose exponents can be related to the spectrum of singularities @xmath0 . this is a new way of characterizing multifractality in dynamical systems , so far applied only to multifractal random functions ( frisch and matsumoto ( _ j . stat . phys . _ * 108*:1181 , 2002 ) ) . the relation between the thermodynamic approach ( vul , sinai and khanin ( _ russian math . surveys _ * 39*:1 , 1984 ) ) and that based on singularities of the invariant measures is also examined . the theory for fractional derivatives is developed from a heuristic point view and tested by very accurate simulations . _ j. stat . phys . in press _ * keywords : * chaotic dynamics , multifractals , thermodynamic formalism .

nlin0309068

for more than 60 years it has been well known that the quiet solar corona is heated to a temperature of about 12 million kelvins while the visible surface of the sun is roughly 250 times cooler ( grotrian 1939 ; edlen 1942 ; phillips , 1995 ) . it has been also recognized that magnetic fields or waves play a key rle in the heating of the solar corona so that somehow convective energy in the photosphere is converted to thermal energy in the corona via magnetic fields or wave energy . the primary energy source for this heating must lie in the convection zone below the solar photosphere ( e.g. bray et al . , 1991 ; golub & pasachoff 1998 ; aschwanden 2004 ) where there is 100 times as much energy available than that required to heat the corona ( @xmath0300w / m@xmath1 : withbroe & noyes 1977 , aschwanden 2001 ) . currently , the debate centres on whether the energy to heat the corona derives from dissipation of magneto - hydrodynamic ( mhd ) waves ( e.g. hollweg , 1981 ) or from numerous small - scale magnetic reconnection events giving rise to nanoflares ( parker , 1988 , aschwanden 2004 ) . it has been found theoretically that the interaction of the magnetic field with convective flows in or below the photosphere can produce two types of magnetic disturbances in coronal structures . firstly , the buffeting of magnetic flux concentrations in the photosphere by granulation generates mhd waves which can propagate into magnetic flux tubes and dissipate their energy in the chromosphere or corona ( e.g. ofman et al . 1998 ) . secondly , in coronal loops the random motions of magnetic loop foot - points can produce twisting and braiding of coronal field lines , which generates field - aligned electric currents that can be dissipated resistively ( e.g. parker 1972 , 1983 ; heyvaerts & priest 1983 ; van ballegooijen 1990 ) . the main difference between these processes is that plasma inertia plays a key rle in wave propagation , but is unimportant for the dynamics of field - aligned currents along coronal loops . thus these types of magnetic heating mechanisms can be crudely classified as either wave - heating or current - heating mechanisms . there are theoretical arguments for both mechanisms , but the observational evidence for nano - flare heating is perhaps looking less convincing than before . extrapolation of the number spectra of small flares down to microflares has been made to nano - flares but the total energy , while tantalisingly close , is most probably less than the required amount ( parnell et al . several theoretical studies showed that only high - frequency mhd waves ( @xmath2hz ) are capable of significant heating ( e.g. porter et al . 1994 , aschwanden 2004 ) . these waves have been sought using the fe xiv green " line at 530.3 nm ( emitted at @xmath3 k ) and the fe x red " line at 637.5 nm ( @xmath4 k ) . observations of high - frequency intensity oscillations of the coronal structures have been made by pasachoff and colleagues ( green line : pasachoff et al . , 1995 , 2000 , 2002 ) , by ruin and minarovjech ( green and red lines : ruin & minarovjech , 1991 , 1994 ) , by rudawy , phillips and colleagues ( green line , total eclipses in 1999 and 2001 ; phillips et al . , 2000 , williams et al . , 2001 ; rudawy et al . , 2001 ; williams et al . , 2002 ; rudawy et al . , 2004 ) , and by singh et al . ( green and red lines : singh et al . , 2009 ) . phillips and rudawy and their colleagues with their secis ( solar eclipse coronal imaging system ) ccd camera instrument have obtained the highest time resolution up to now . the results of these investigations are somewhat contradictory , with both positive and negative observations of oscillations ( e.g. pasachoff & landman 1984 ; koutchmy et al . 1994 ; cowsik et al . 1999 ; williams et al . 2001 , 2002 ; rudawy et al . space missions capable of comparable time resolution measurements have not been available up to the present time , so using ground - based equipment is still the only way of making such observations ( aschwanden , 2004 , klimchuk , 2006 ) . in this paper , we describe a set - up that will be used to search for high - frequency coronal oscillations . we shall be making observations using the mid - sized coronagraph ( belonging to the astronomical institute of slovak academy of sciences in tatranska lomnica : lexa , 1963 ) at lomnicky peak observatory . observations will be made in the fe xiv green line using the secis instrument ( phillips et al . 2000 ) , now owned by the astronomical institute at the university of wrocaw , poland . the observational system on the lomnicky peak observatory consists of 3 main instrumental parts : the 20-cm lyot - type coronagraph , secis instrument ( two fast - frame - rate ccd cameras , auxiliary electronics systems and dedicated computer ) and a special opto - mechanical interface between the coronagraph and secis cameras . the lomnicky peak coronagraph ( lexa , 1963 ) , made by carl zeiss jena , is located at the summit of lomnicky peak ( 2634 m altitude ) , allowing observations in the light of prominent coronal visible - light emission lines out to a significant distance beyond the solar limb . the front part of its optical system consists of a single objective lens ( bk7 glass , @xmath5=1.71 m , @xmath6=17.0 m , @xmath7200 mm , @xmath83 m ) and a primary diaphragm which obscures the lens to a final clear aperture of 195 mm . the focal lengths of the objective for the wavelength of the green line are 2975 and 2980 mm for the axial and outer light beams respectively . the central part of the optical system has an artificial moon ( the occulting disk ) which is a fat mirror inclined with respect to the optical axis and reflects the solar disk light out of the coronagraph tube . the artificial moon is fixed in front of a field lens in a hole in the center of the lens , and can be changed to similar ones with various diameters . behind the field lens there is a re - imaging triplet lens in order to correct , at least partially , geometric aberrations of the primary lens and to focus a difraction image of the primary diaphragm on lyot s stop . lyot s stop lies between the second and the third lenses of the triplet and blocks the scattered light coming from the primary diaphragm . a particular feature of the optical system is a four - lens imaging objective of 9 cm aperture . the combined action of both objectives creates the final image of the corona with diameter @xmath9 mm . the coronagraph is equipped with a fast optical automatic guider . it detects offsets of the actual pointing using two photodiode pairs in an anti - parallel connection providing closed - loop correction signals to the drives . it precisely stabilizes the relative position of the occulting disk against the solar image , ensuring stable position of the field of view . the two secis cameras are connected to the lomnicky peak coronagraph using a new , special opto - mechanical interface . the optical set - up is shown schematically in fig.1 , with fig.2 showing the components in the rigid light - tight box . the coronal image formed by the coronagraph is to the left in fig.1 , at the focus of the entrance lens ( marked l300/82 in fig.1 : diameter and focal length are d=82 mm , f=300 mm ) . this lens forms a parallel light - beam which then passes to a beam splitter ( bspl , shorter dimension = 50 mm ) . the reflected beam from the beam splitter then passes through a broad - band and neutral density filter combination ( marked nd / wb filter " in the figure ) , is reflected again from a flat mirror ( m3 ) , and finally is brought to a focus on to the ccd camera ( marked ccd wl " ) by an achromatic lens ( l120/50 ) . the parallel beam transmitted by the beam splitter is reflected by a flat mirror ( m1 ) , passes a narrow - band fexiv 530.3 nm interference filter ( fe5303 " ) , is reflected by a second flat mirror ( m2 ) , then brought to a focus on to the ccd camera ( marked ccd fe5303 " ) by the acromatic lens ( l120/50 ) . fig.1 shows the ray diagram for the configuration . in this figure , an on - axis pair of rays is shown by thick lines . the thin lines represent rays from one extreme of the coronagraph image . the angle between them ( greatly exaggerated in the figure ) is no more than about 0.5 degrees . with such a small angle , a negligible wavelength shift is produced by the interference filter ( fe5303 ) . the optical system was designed taking into account the actual optical parameters of the lomnicky peak coronagraph and the desired spatial scale of the images on the ccd chips . it was optimized to avoid any vignetting ( to keep entire fields of view as bright as possible ) and to limit geometric or chromatic aberrations of the system . in order to minimize the total cost of the interface , all optical and mechanical elements as well as adjustable optical mounts were general - purpose stock elements , selected from the melles griot company catalogue . the beam splitter ( bspl ) reflects only 10% of the light to the broad - band channel , transmitting the remaining light to the narrow - band ( green - line ) channel . the broad - band filter was selected to be centred on the green line wavelength but having a larger range of transmission ( central wavelength 530.0 nm , bandpass fwhm 10 nm ) . a neutral - density filter ( nd ) in the white - light optical channel was selected to ensure equal exposure times in the narrow - band and broad - band channels . final images in both channels are formed by lenses with the same focal lengths , so giving the same spatial scale . the entire optical system is mounted in a rigid box attached to a rear plate of the coronagraph . a light - tight black aluminium box acting as a rigid optical frame for the optical components was manufactured by the workshop of the astronomical institute at the university of wrocaw ( fig.2 ) . the optical system was assembled and pre - aligned in the box ; apart from focusing of the system , no other special alignments of the optical components are needed at the telescope ( fig.3 ) . three narrow - band filters with passbands around the fexiv 530.3 nm green line are available ( two made by barr associates , inc . , and one by andover corporation ) . all filters have fwhm passbands of @xmath10 nm and diameters of 50 mm . the filter chosen for use in the optical system is fitted with a thermostatic device to maintain the required temperature under variable ambient temperatures . the secis instrument was built , tested , and calibrated between 1997 and 1999 in a british polish collaboration to search for short - period coronal light oscillations . it was used with great success during the total eclipse seen from bulgaria in 1999 , from zambia in 2001 ( see phillips et al . , 2000 ; williams et al . , 2001 ; williams et al . , 2002 ; rudawy et al . , 2004 ) , and most recently from libya in 2006 . the ccd cameras ( manufactured by eev , chelmsford , u.k . ) are high - performance cameras designed specifically for scientific and machine vision applications . the image sensor is a monochrome 512@xmath11512 pixel frame transfer ccd . this device has square ( 15@xmath12 m @xmath13 m ) pixels , and can be driven at a non - interlaced frame rate up to 70 frames per second . the cameras digitise the signal from the ccd to nominally 12 bits and provide a real dynamic range of over 1000:1 . the camera electronics operate the cameras in an asynchronous " mode , where a trigger pulse from the control electronics commands one of the cameras to capture an image at a precise moment and with a precise exposure period . this feature allows the two cameras to capture accurately synchronised images . the data are captured and stored on a personal computer . the computer system captures the synchronised digital video streams from the two ccd cameras and reconstitutes the video images , storing them for more detailed analysis . the computer has dual pentium processors , 128 megabytes of memory , and four 9 gb disk drives . it is able to run a set of observations consisting of up to about 10000 images for each camera . the image processing software allows the replay of the video , and the cropping to sub - sequences and regions of interest . these selections can then be exported to files in fits format . more detailed information about secis and its first scientific application are described by phillips et al . ( 2000 ) . since 2003 secis has also been used for making high time resolution spectral observations of solar flares over the profile of the h-@xmath14 ( 656.3 nm ) line using the multi - channel subtractive double pass ( msdp ) imaging - spectrograph ( mein , 1977 and 1991 ) and large coronagraph ( with 530 mm main objective ) or horizontal telescope ( with 150 mm main objective ) installed at biakow observatory ( astronomical institute at the university of wrocaw ) ( radziszewski et al . , 2006 , 2007a , 2007b , 2008 ) . preliminary tests of the entire system were performed in april 2009 , though the lack of solar activity at that time meant that no prominent coronal structures were detectable . the filters manufactured by barr associates , inc . , have 50 mm clear diameter and have quarter - wave flatness specification . they are coated with ion - assist refractory oxide coatings that greatly reduce the wavelength shift with ambient temperature and filter longevity and produce higher transmittance . to investigate possible degradation since their last use during the 2006 eclipse , the filter passband widths and central wavelengths were tested for thermal stability using disk centre solar light and a small spectrograph with a dispersion of 0.9nm / mm connected directly to the coronagraph ( minarovjech , 2009 ) . examples of the mean reference disk centre spectra taken with and without the narrow - band filter are shown in fig.4 . the tests showed that the filters have to be heated to a fairly high temperature ( 4550 degrees celsius ) in order to tune the filter passband to the green - line wavelength . the dependence of the main passband parameters on the filter temperature is displayed in fig.5 . first , the filter passband transmission , the width ( fwhm ) of its passband , and its central wavelength averaged over the length ( 7.2 mm ) of the spectrograph slit were examined as a function of filter temperature ; these are indicated by the thick lines in fig.5 . these measurements show that a typical maximum transmission of the filters is @xmath010 % , and that the fwhm of the passband is about 0.31 nm for an ambient temperature resulting in the filter central wavelength to equal the green line wavelength . by taking short lengths of 0.8 mm at nine positions along the spectrograph slit ( which is aligned along the radial direction of the filter ) , we also examined the variation of the same quantities as a function of radial distance over the filter ; these are the thin lines plotted in fig.5 . there is a similar dependence on temperature in these individual measurements to the averaged results . the filter passband transmission varies by up to about 10% from the mean value , the passband wavelength position by up to 0.02 nm , and the passband width by up to only 4% . the background is very stable apart from one outlying measurement . test observations were taken in both the broad - band and narrow - band channels . with solar activity at an extremely low level , no coronal structures were visible in the green line at that time ( 2009 april 7 at 06:40 ut ) but the tests were nevertheless useful in that the optical and photometric quality of the data could be examined . examples of snapshots selected from the data in both channels are shown in figs.6 and 7 . inspection of the images shows that the image quality was very good . moreover , the different character of the radial gradients illustrates the very good spectral blocking of the narrow - band filter . the internal instrumental scattered light in the coronagraph and all the optical parts of secis was found to be sufficiently low to allow the required data acquisition . this confirms that the instrument itself is ready to measure prominent active region coronal loops above the solar limb when they appear under coronal " skies , i.e. with low degree of light scatter by the earth atmosphere . more detailed inspection of the data ( figs.8 and 9 ) shows that the noise level was low . the photon count level in the narrow - band channel within the portion of the image occupied by the artificial moon was measured to be at a very low level , averaging 8 dn s@xmath15 , only slightly more than the dark current level of 23 dn s@xmath15 . a bright coronal active region is expected to have a high signal - to - noise ratio , though experience from the 1999 and 2001 eclipses with the secis cameras suggests that the cameras are very unlikely to reach saturation levels . the secis instrument installed at the lomnicky peak observatory lyot coronagraph will allow data to be acquired that may result in an improved knowledge of where in the corona mhd waves are generated and/or dissipated . in particular , the signatures of high - frequency mhd waves involved in coronal heating may be observed . a considerable improvement in our knowledge of a long - standing problem of solar physics could be made by such observations , with implications for the physics of active regions , flares , the solar wind , and solar activity , as well as mechanisms of solar - terrestrial relationships . we acknowledge the anonymous referee for comments which helped to improve the paper . the work of j. a. and j. r. was supported partly by the slovak research and development agency under the contract no . apvv-0066 - 06 which also covered all expenses related to the secis instrument at the lomnicky peak observatory ( slovakia ) . authors are obliged for the support of the astronomical institute , slovak academy of sciences staff , namely assistants k. mank , r. maura , p. havrila , p. bendk , and the workshop assistant j. klein . p.r . was supported by the polish ministry of science and higher education , grant number n203 022 31/2991 . this research has made use of nasa s astrophysics data system .

heating mechanisms of the solar corona will be investigated at the high - altitude solar observatory lomnicky peak of the astronomical institute of sas ( slovakia ) using its mid - size lyot coronagraph and post - focal instrument secis provided by astronomical institute of the university of wrocaw ( poland ) . the data will be studied with respect to the energy transport and release responsible for heating the solar corona to temperatures of mega - kelvins . in particular investigations will be focused on detection of possible high - frequency mhd waves in the solar corona . the scientific background of the project , technical details of the secis system modified specially for the lomnicky peak coronagraph , and inspection of the test data are described in the paper .

1004.3454

start from the innocuous looking recurrence , @xmath0 which determines a sequence of integers with initial values @xmath1 these numbers will be called _ pseudo - factorials _ , since the omission of the sign alternation @xmath2 in equation determines the sequence of factorials , @xmath3 at the suggestion of one of us ( bacher , october 2004 ) , they have been included as sequence * a098777 * in the _ on - line encyclopedia of integer sequences _ @xcite , which is brilliantly maintained by sloane and a gang of dedicated volunteers . the purpose of this note is to show that these numbers , though not classical , have a host of interesting properties . the _ exponential generating function _ , @xmath4 is fundamental to our treatment . ( the fact that the absolute values @xmath5 are dominated by factorials implies that @xmath6 is analytic at least in @xmath7 . ) we first elucidate the relation between @xmath6 and elliptic functions of a kind introduced by alfred cardew dixon in 1890 ( _ vide _ @xcite ) , then show the reduction to the more common weierstra framework of @xmath8functions : this forms the subject of sections [ dixon - sec ] and [ weier - sec ] . a simple consequence of the elliptic connections , worked out in section [ lat - sec ] , is an expression of pseudo - factorials as sums over a hexagonal lattice . next , we establish a continued fraction of a new type , relative to the _ ordinary generating function _ of pseudo - factorials , @xmath9 to be taken in the sense of formal power series since its radius of convergence is 0 . to wit : @xmath10 where the denominators are successively @xmath11 , and the numerators are @xmath12 . such a repetitive pattern of order 2 in what is known as a _ jacobi fraction _ is somewhat unusual . the fraction was first discovered experimentally proving it in sections [ sradd - sec][proofadd - sec ] is the central theme of the present study . finally , the convergents of the continued fraction can be made explicit , via generating functions : this is conducive to what seems to be a new class of `` elliptic polynomials '' in the sense of lomont brillhart @xcite ; see section [ ortho - sec ] . in section [ cong - sec ] , we then draw several consequences of the previous developments , in the form of hankel determinant evaluations and elementary congruence properties of pseudo - factorials . as kindly suggested by a referee and by the editor of _ the ramanujan journal _ , we offer a few comments relating to the motivations behind studies , such as the present one . one primary motivation is to gain a better understanding of a still mysterious class of continued fractions , the ones that have coefficients that are polynomial ( or rational ) functions in the depth @xmath13 . this highly general but somewhat impenetrable class of special functions seems to have been first noticed by pollaczek in @xcite . it is piquant to note that apery s continued fraction @xcite , which first led to an irrationality proof of riemann s @xmath14 , has cubic denominators and sextic denominators , a feature also shared by the conrad continued fractions @xcite relative to the dixonian functions @xmath15 , themselves closely related to pseudo - factorials ( section [ dixon - sec ] ) . our main continued fraction ( theorem [ main - thm ] ) is pollaczek in a mildly extended sense : it has polynomial coefficients , but modulated by an unsual periodicity of order 2 . on another register , the relation of elliptic functions to continued fractions is an old subject , going back at least to eisenstein , stieljes , rogers , and ramanujan . whereas the @xmath16series and @xmath17function aspects are not immediately relevant to our discussion , we may observe that the continued fractions relative to the jacobian functions @xmath18 have arithmetic content : together with the hankel determinant product evaluations that they imply , they have been used by milne in his penetrating study @xcite of representations of integers as sums of @xmath19 or @xmath20 square or triangular numbers . in this spirit , we offer some new hankel determinant evaluations in section [ cong - sec ] , but will leave to others the task of determining whether they are of some arithmetic interest . the class of orthogonal polynomials associated to any continued fraction , whose coefficients are rational - in - the - depth , is also of interest . many instances of low degree have been categorized by chihara in @xcite . `` elliptic polynomials '' , that is , orthogonal polynomials associated with series expansions of elliptic functions have a tradition that goes back at least to carlitz @xcite and they form the topic of the entire book by lomont and brillhart @xcite . the discovery of a new class ( section [ ortho - sec ] ) is of interest in this context . finally , the pseudo - factorials are tightly coupled with @xmath21 , as seen from our discussion of lattice sums in section [ lat - sec ] . in this context , it is well worth noting a spectacular arithmetic continued fraction for @xmath22 , recently obtained by tanguy rivoal @xcite , which is of the sextic duodecimic type ( ! ) . this and previous observations reflect the fact that our understanding of an orbit of questions , surrounding pollaczek continued fractions , elliptic functions , elliptic polynomials , and diophantine approximation properties , is still fragmentary ; but they also suggest that `` hidden '' structures are yet to be discovered in this area ( see also our brief conclusion in section [ concl - sec ] ) . this section serves to establish the first connection between pseudo - factorials and elliptic functions . the starting point is the exponential generating function defined by ; it satisfies a functional equation , @xmath23 which directly translates the defining recurrence . to make @xmath6 explicit , take the functional equation and differentiate once , so that @xmath24 since , by again , one has @xmath25 and @xmath26 . in order to solve the nonlinear differential equation , `` cleverly '' multiply by @xmath27 to get @xmath28 which is integrated to give @xmath29 with @xmath30 a yet unspecified constant . equivalently , one has @xmath31 which upon one more integration gives @xmath32 where use has been made of the initial condition @xmath33 . the constant @xmath30 is finally identified by means of a second order expansion ( with @xmath34 , @xmath35 ) , to the effect that one must have @xmath36 . ( the computations parallel those of @xcite . ) in view of our subsequent treatment , it is convenient to standardize . a linear change of variables yields @xmath37 where we have taken into account that @xmath36 . throughout this study , a fundamental constant is @xmath38 ( a period of the function @xmath39 defined below in ) , @xmath40 where the evaluation results from the classical eulerian integral @xcite : @xmath41 ( numerically , with @xmath42 a decimal fraction , indicates that @xmath42 is an approximation of @xmath43 to the last digit stated . ] , we find @xmath44 . ) a simple computation shows that @xmath45 so that we can write @xmath46 the function @xmath6 can now be expressed in terms of specific elliptic functions introduced by a. c. dixon in his original memoir @xcite . define the function @xmath47 by the equation @xmath48 then , a comparison of and permits us to identify @xmath6 . [ dix - thm ] the exponential generating function of pseudo - factorials satisfies @xmath49 where the dixonian elliptic function @xmath47 is as in and @xmath38 is the constant . we can offer a few comments regarding dixonian functions . there is actually a _ pair _ of `` higher - order trigonometric '' functions , @xmath39 and @xmath50 , where @xmath39 and @xmath50 are evocative of a sine and a cosine function , respectively . their properties can be developed from first principles , as done by dixon followed by conrad flajolet @xcite , starting with the differential system , @xmath51 and initial conditions @xmath52 , @xmath53 . ( see also the works of lundberg @xcite and the recent developments by lindquist and peetre @xcite for a yet more general approach . ) for the record , we note that @xmath54 whose coefficients ( @xmath55 and @xmath56 ) are respectively * a104133 * and * a104134 * of sloane s _ encyclopedia_. the works of conrad and flajolet @xcite provide continued fraction expansions for the ordinary generating function of these coefficients , but these are then relative to the expansions of @xmath15 at @xmath57 , and _ not _ at @xmath58 , as in . finally , we observe that , by the calculation and by , we have the identity @xmath59 so that the pair @xmath60 parametrizes the _ fermat cubic _ @xmath61 defined by the equation @xmath62 , which is of genus 1 . it is _ a priori _ possible to reduce the exponential generating function @xmath6 of pseudo - factorials to any of the several canonical forms of elliptic functions . here , we show , by elementary calculations similar to the ones of the previous section , how to arrive directly at an expression involving the weierstra function @xmath8 . we recall that this function @xmath63 is classically defined by the nonlinear differential equation @xcite @xmath64 together with the initial condition @xmath65 as @xmath66 . by design , the pair @xmath67 parametrizes the elliptic curve @xmath68 , with invariants @xmath69 . the starting point is the fundamental relation , namely , @xmath70 . we first claim the identity @xmath71 the proof is obtained by verifying that the derivative of the left - hand side is 0 , @xmath72 ( the final reduction uses ) , combined with the initial condition @xmath33 . next , we have @xmath73 again by way of . now set @xmath74 the basic elliptic connection is provided by the differential relation @xmath75 which is clearly of the weierstra type . to see it , it suffices to square the two sides of the identities and , then compare the outcomes . equation ( 20 ) then shows that @xmath76 is closely related to the elliptic curve @xmath77 defined by @xmath78 the curve @xmath77 contains six integral points , since it is known that the mordell curve @xmath79 has six rational points ; see , for instance , the simath tables that are accessible at tnt.math.metro - u.ac.jp / simath / mordell / mordell+ . ] ( including the point at infinity corresponding to the identity element of the underlying group ) forming a cyclic group of order six . the non - trivial points of this group are : @xmath80 ( of order @xmath81 ) , @xmath82 ( of order @xmath83 ) and @xmath84 ( of order @xmath85 ) . since @xmath86 , the series @xmath76 represents the expansion of the weierstra function @xmath8 around the unique real @xmath81-torsion point @xmath80 of @xmath77 . the following result recovers @xmath6 from @xmath87 and constitutes the main result of this section . [ weier - thm ] let @xmath88 be the weierstra function with invariants @xmath89 and @xmath90 and smallest positive real period are _ not _ the shortest non - zero elements of the period lattice for @xmath8 . they generate a sublattice of index @xmath83 in the period lattice of @xmath8 whose shortest elements are given by the six numbers @xmath91 ; see also section [ lat - sec ] and figure [ lat - fig ] . ] @xmath92 with @xmath38 as in . the exponential generating function of pseudo - factorials satisfies @xmath93 we first establish the expression of the real period of @xmath94 , only making use of the most basic properties of elliptic functions ( * ? ? ? let us denote temporarily the _ real half - period _ by @xmath95 . by general properties of elliptic functions since @xmath96 is odd ; hence @xmath97 must be a root of the third - degree polynomial associated with @xmath8 ; here , @xmath98 . ] , we have @xmath99 , while @xmath97 is a real root of @xmath100 ; that is , @xmath98 . thus , since @xmath8 is the inverse of an elliptic integral , we must have @xmath101 & = & \ds \frac{1}{6}\b\left(\frac13,\frac12\right)+\frac{1}{6}\b\left(\frac13,\frac16\right)= { 2^{-7/3}3^{1/2}}{\pi}^{-1}\gamma\left(\frac13\right)^3 , \end{array } \ ] ] as shown by the changes of variables @xmath102 and @xmath103 ( respectively ) in the last two integrals of the first line , followed by eulerian beta function evaluations . we henceforth denote the quantity @xmath95 by @xmath104 . the function @xmath105 is determined by @xmath33 and by the functional equation deduced from : @xmath106 we proceed to verify . since @xmath107 and @xmath108 , we first have @xmath109 in view of , the equality then reduces to proving the identity @xmath110 since @xmath94 is an even function , which is @xmath111periodic , the left - hand side of can be put under the rational form @xmath112 , where the numerator @xmath113 involves @xmath114 at @xmath115 . writing @xmath114 for @xmath116 , it remains to verify that @xmath117 vanishes identically . derivation of the differential equation @xmath118 for @xmath8 yields @xmath119 and we indeed obtain @xmath120 which concludes the proof of the statement . from the dixonian as well as the weierstra connections discussed in the previous section , expressions of the @xmath121 as _ lattice sums _ can be developed . [ lat - thm ] the pseudo - factorials are expressible as lattice sums involving a twelfth root of unity : with @xmath122 and @xmath123 as in eq . , one has , for any @xmath124 : @xmath125^{n+1 } } \ , , \qquad \zeta:=e^{i\pi/6}.\ ] ] the formula implies explicit asymptotics for @xmath126 , namely , @xmath127 once only the relevant dominant poles are retained . this explains in particular the regular sign pattern `` + --+ '' observed in . let @xmath128 be an elliptic ( i.e. , meromorphic , doubly periodic ) function that has only simple poles . let @xmath129 be its lattice of periods and let @xmath130 be the set of poles contained in a fundamental domain of @xmath131 . then , if @xmath132 , one has @xmath133,\ ] ] where @xmath134 represents the residue of @xmath128 at @xmath135 . theorem [ weier - thm ] and the formula show that it suffices , up to an affine transformation , to work out the singular structure of @xmath136 in order to deduce the partial fraction expansion of @xmath6 , from which the lattice sum expression will result . the function @xmath137 has lattice of periods @xmath138 and @xmath139 has simple poles at @xmath140 and @xmath141 . given the series expansion @xmath142 of @xmath8 around 0 , we find the residue of @xmath139 at @xmath57 to be @xmath143 by the discussion following , we also have @xmath144 . moreover , we have @xmath145 and @xmath146 , hence @xmath147 and @xmath148 . the expansion of @xmath94 around @xmath149 is thus @xmath150 from which a simple computation provides the residue of @xmath139 at @xmath149 , and , similarly , the residue at @xmath151 : @xmath152 ( 6,6 ) ( 0,0)_left : _ the `` primary '' lattice of periods [ thick lines ] of @xmath6 , where a fundamental domain is obtained by the union of two adjacent equilateral triangles ( one grey , one white ) ; the poles ( three per fundamental domain ) are represented by small discs and form a `` secondary '' hexagonal lattice [ thin lines ] . _ right _ : a diagram showing the three poles ( circled ) of a fundamental domain of @xmath6 around the origin . , title="fig : " ] ( -0.4,3.0)(1,0)7.1 ( 6.15,3.2)@xmath153 ( 3.75,0.0)(0,1)6.0 ( 3.77,0.0)(0,1)6.0 ( 2.9,5.8)@xmath154 now , by , , and , the singular structure of @xmath139 is entirely known . since @xmath155 , an affine transformation ( composed of a translation , a rotation , and a dilation ) provides the singular structure of @xmath6 itself we abbreviate the discussion , which is routine . as represented in figure [ lat - fig ] , the lattice of periods @xmath131 of @xmath6 is a hexagonal lattice with generators @xmath156 ; we may call it the `` primary '' hexagonal lattice . there are three poles of @xmath6 in the fundamental domain , at @xmath157 and @xmath158 , which , upon translation by @xmath131 , generate a `` secondary '' hexagonal lattice . the residue of @xmath6 ( deduced from and ) , is then found to be of the form @xmath159 at a point of the form @xmath160 $ ] , which corresponds to a 3colouring of the secondary hexagonal lattice ( since @xmath161 is a third root of unity ) . the corresponding partial fraction decomposition results for @xmath6 . finally , the factf$ ] represent the coefficient of @xmath162 in the formal power series or analytic function @xmath163 . ] that , for @xmath124 , @xmath164 \left(\frac{1}{z - a}+\frac{1}{a}+\frac{z}{a^2}\right ) = -\frac{1}{a^{n+1}}.\ ] ] yields the stated lattice - sum formula for @xmath121 . the sum establishes the pseudo - factorials as a two - dimensional analogue of bernoulli and euler numbers , one that is relative to the hexagonal lattice . it might be of interest to investigate systematically continued fractions relative to other hexagonal lattice sums . it is worthy of note that arithmetic properties of analogous `` lemniscatic '' sums , relative to the square lattice , have been studied by hurwitz @xcite . this section serves to introduce the basic technology needed to develop an explicit continued fraction expansion from an addition theorem of a suitable form . an experimental approach specialized to pseudo - factorials follows , in section [ experadd - sec ] . we can then reap the crop in section [ proofadd - sec ] and finally prove our main continued fraction result ( theorem [ main - thm ] ) . stieltjes and rogers independently discovered that the continued fraction expansion of an ordinary generating function , @xmath165 , is closely related to _ addition formulae _ satisfied by the corresponding exponential generating function , @xmath166 . first , a definition . let @xmath167 be a formal power series . it is said to satisfy an _ addition formula of the stieltjes rogers type _ if there exist nonzero constants @xmath168 and formal power series @xmath169 , such that @xmath170 where @xmath171 has valuation @xmath123 and is normalized by @xmath172 . in , the valuation condition on @xmath173 is essential , the normalization @xmath174 being a mere convenience for what follows . the addition formula gives rise to an algorithm for computing the @xmath175 , knowing @xmath176 , either symbolically or via some series expansion . first , setting @xmath177 in the addition formula shows that @xmath176 must be equal to @xmath178 ( this makes use of the normalization @xmath179 ) . next , assume that the functions @xmath180 and the coefficients @xmath181 have already been determined . then , by differentiating @xmath182 times the addition formula with respect to @xmath183 , then setting @xmath177 , one finds @xmath184_{y=0 } = \omega_\ell\varphi_\ell(x).\ ] ] upon comparing the coefficient of @xmath185 in the taylor expansions of both sides , we see that _ at most _ one nonzero coefficient @xmath186 can satisfy the equation , given the normalization @xmath187 . once this choice has been fixed , then @xmath188 is uniquely determined as a linear combination of the previous @xmath189 , together with derivatives of @xmath190 , and the process can continue . this construction also determines the broad class of functions in which the @xmath175 live : _ each @xmath175 belongs to the vector space generated over @xmath191 by the first derivatives @xmath192 of the function @xmath176_. this algorithm , albeit suboptimal from a computational point of view , permits us to experiment with addition formulae relative to the generating function of any given number sequence @xmath193 , a fact that will prove especially valuable in section [ experadd - sec ] . addition formulae of the stieltjes rogers type are logically equivalent to continued fraction expansions as expressed by the following central theorem originally due to stieltjes @xcite and rogers @xcite ; see also perron @xcite and wall @xcite . [ sradd - thm ] let the _ exponential _ generating function @xmath167 satisfy an addition formula of the form . then , the corresponding _ ordinary _ generating function @xmath194 admits a jacobi - type continued fraction expansion fraction , or an associated continued fraction . ] , @xmath195 where the coefficients are determined by @xmath196 there , @xmath197\varphi_j(z)$ ] , @xmath198 , and @xmath199 . as an illustration , following stieltjes and rogers , we examine the case of @xmath200 each @xmath201 ( provided it exists ) must then _ a priori _ be of the form @xmath202 , where @xmath203 is a polynomial satisfying @xmath204 . in a simple case like this , classical trigonometric identities yield @xmath205 which , in normalized form , becomes @xmath206 with @xmath207\sec(z)$ ] an euler number , theorem [ sradd - thm ] then provides the continued fraction expansion : @xmath208 where the coefficients @xmath209 , are obtained here as quotients of consecutive squared factorials . the absence of linear terms reflects the parity of @xmath210 . section [ sradd - sec ] has shown that , in order to approach the construction of a continued fraction expansion for pseudo - factorials , we need to develop a suitably constrained addition formula for their exponential generating function @xmath6 , which is elliptic . we proceed here in an experimental manner in order to infer the likely shape of such an addition formula . once this had been done , the proof of our main continued fraction reduces to purely mechanical verifications to be carried out in the next section . a first idea that comes to mind is to look for an addition formula of a kind similar to the secant case , namely , @xmath211 for some function ( or power series ) @xmath212 . however , all such formulae can only arise from a class of special functions that comprises _ five _ parametrized subclasses , of which prototypes are @xmath213 ( this is a rephrasing of a classification of orthogonal polynomial systems due to meixner @xcite . ) obviously , elliptic functions are not amongst this group . another source of inspiration is a continued fraction relative to elliptic functions , which is also due to stieltjes and rogers . with @xmath18 the jacobian elliptic functions , as classically defined , where we leave the modulus @xmath214 implicit , we have @xmath215 see for instance @xcite . this can be put into an equivalent stieltjes rogers form ( up to normalization ) , namely , @xmath216 corresponding to the continued fraction expansion @xmath217 where @xmath218\cn ( z)$ ] . we now turn to the continued fraction expansion relative to pseudo - factorials which , by theorem [ sradd - thm ] , involves determining the right addition formula for @xmath6 . based on experiments under the maple system as well as on induction from the secant and jacobian cases , we started searching for an addition formula of the form @xmath219 for some power series @xmath220 , with ( at least ) @xmath221 , and @xmath222 all being @xmath223 . since some binary pattern is present in the continued fraction , it is natural further to suppose that @xmath224 , @xmath225 , which then corresponds to an `` odd - even '' addition formula , @xmath226 where @xmath227\psi(w)$ ] and @xmath228\xi(w)$ ] . in the notations of , we thus hope for an addition formula of the form @xmath229 where @xmath230 means that the ratio @xmath231 is a constant . the pleasant feature of the conjectured expansions and is that their plausibility can be effectively tested . indeed , from the previous section , we have available an algorithm that can determine the ( unique ) @xmath232 corresponding to @xmath233 , this to any desired precision . it then suffices to check that @xmath234 verification of these relations for about a dozen of the @xmath235 and till orders in the range 50100 convinces us that we are on the right tracks . in fact , we found experimentally that @xmath236 up to @xmath237 , the series starting as @xmath238 also , the function @xmath239 that appears in must be proportional to @xmath240 of the addition formula , whose expansion starts as @xmath241 . that function @xmath240 must itself , on general grounds , be a linear combination of @xmath6 and @xmath242 without constant term , so that @xmath243 finally , assuming the observed law of the coefficients in the continued fraction to hold forever , we can deduce the only possible shape of the @xmath244 and @xmath245 functions . the function @xmath246 is then inferred on the basis of the fact that @xmath247 must be a linear combination of @xmath248 . all in all , every ingredient of an addition formula of type is in place , and we are eventually led to conjecturing an addition formula for @xmath6 @xmath249 which we shall establish in the next section . at this stage , we know that establishing the continued fraction relative to the ordinary generating function @xmath250 of pseudo - factorials reduces to deciding the validity of the conjectured addition formula for the exponential generating function @xmath6 . the proof we propose is a computer - assisted verification . as we shall explain , it only involves routine algebraic manipulations ; namely , rational function operations , normalizations , substitutions , as well as multivariate polynomial divisions . the calculations were performed using the maple computer algebra engine ( version 11 ) . without any attempt at optimization ( we purposely wanted our program to rely solely on the most basic algebraic operations ) , the mechanical verification reduces to the mere execution of a few billion machine instructions currently , just a few seconds of elapsed time . [ add - prop ] the function @xmath6 satisfies the following addition formula : @xmath251 we can _ a priori _ appeal to either the dixon or the weierstra framework , and we have opted for the latter . the weierstra @xmath8function , @xmath252 satisfies the two algebraic relations @xmath253 the first one ( def ) is the basic differential equation , which serves as _ definition _ of @xmath8 ; the second one ( add ) is the familiar _ addition _ theorem of elliptic function theory @xcite . both are `` known '' to maple ; both can be viewed as deterministic rewrite rules permitting one to expand and simplify expressions involving @xmath8 . let @xmath254 be the fundamental constant ( real period ) of section [ weier - sec ] . we know that @xmath145 . the addition rule ( add ) combined with the expression of @xmath6 stated in theorem [ weier - thm ] , equation , then mechanically expresses @xmath6 as a rational fraction in @xmath255 and @xmath256 , where @xmath257 . similar expressions are obtained for @xmath242 ( by standard derivation rules combined with partial reductions by ( def ) ) and @xmath258 ( since @xmath8 is an even function , while @xmath96 is odd ) . in this way , one automatically obtains rational forms in @xmath259 for @xmath260 where the last one necessitates a substitution @xmath261 , followed by an application of the ( add ) rule . let now @xmath262 be the difference between the left - hand side and the right - hand side of the relation to be proved , equation . by the process described above , @xmath262 becomes a rational function , with coefficients in @xmath263 , in the _ four _ quantities @xmath264 , where @xmath265 , @xmath266 , and similarly for @xmath267 , with @xmath183 replacing @xmath43 . the ( rather large ) rational fraction normalizes to the form @xmath268 , with numerator @xmath113 and denominator @xmath269 involving , respectively , 2388 and 1256 monomials . one can then operate with the rule ( def ) , instantiated as @xmath270 the corresponding reductions being simply effected by multivariate polynomial divisions . when this is done , we find that @xmath113 reduces to 0 , while @xmath269 is reduced to a _ nonzero _ polynomial , which is of degree 1 in @xmath271 and of degree 8 in @xmath272 . the verification of the addition formula for @xmath6 is thereby completed . a direct application of theorem [ sradd - thm ] to the addition formula expressed by proposition [ add - prop ] gives rise to our main continued fraction . [ main - thm ] the ordinary generating function of the pseudo - factorials satisfies @xmath273 where the coefficients are , with the notations of the jacobi form : @xmath274 we make use of the addition formula , which , taken under the form , yields @xmath275 with the @xmath276 and @xmath277 determined by @xmath278 we have , with the notations of , @xmath279 , as well as @xmath280 and @xmath281 . the required normalization of a stieltjes rogers addition formula , @xmath282 , combined with the low - order expansions of @xmath283 , gives us @xmath284 as well as @xmath285 we thus have @xmath286 and , for @xmath287 : @xmath288 by theorem [ sradd - thm ] , the last two formulae conclude the proof of . our goal in this section consists in finding an explicit form for the polynomials that appear in the convergents of the main continued fraction of theorem [ main - thm ] , this by way of their exponential generating function . we focus our attention on the denominator polynomials , precisely , on their reciprocals , which form a family of formally _ orthogonal polynomials _ that appears to be new . we start by specializing to the continued fraction under consideration some well - known algebraic properties found in @xcite that hold for an arbitrary jacobi fraction . the convergents of are obtained by truncating the infinite fraction before a numerator . in this way , a collection of rational fractions @xmath289 of increasing degrees is obtained , @xmath290 so that @xmath291 , @xmath292 , and so on . the denominator polynomials @xmath293 satisfy a `` three - term recurrence '' relation , @xmath294 ( the @xmath203 satisfy the same recurrence , but with initial conditions @xmath295 , @xmath296 . ) the reciprocal polynomials defined by @xmath297 then satisfy the recurrence @xmath298 with the @xmath299 as in . on general grounds , they are _ formally orthogonal _ with respect to a ( formal ) measure whose moments coincide with the pseudo - factorials , @xmath126 . in other words , they are orthogonal with respect to the bilinear form @xmath300 observe finally that , once the @xmath293 are known , the @xmath203 can somehow be regarded as known . indeed , relative to , one has , in the sense of formal power series , @xmath301 so that the coefficients of the @xmath203 are expressible as a convolution of the two sequences @xmath302q_k(z)$ ] and @xmath303f(z)$ ] . we have the following characterization . [ qegf ] let @xmath304 be the exponential generating function of the reciprocal polynomials @xmath305 of , with coefficients : @xmath306 consider the algebraic curve @xmath307 which is of genus @xmath57 and is parametrized by @xmath308 and let @xmath309 be the branch that satisfies @xmath310 : @xmath311 define @xmath312 and introduce the fundamental elliptic integral @xmath313 then , the generating function @xmath314 satisfies @xmath315 equation was first arrived at by a combination of induction and of partly heuristic calculations , based on `` guessing '' intermediate differential equations as well as on maple s symbolic integration capabilities . rather than offering a heavy proof by successive transformations of the defining recurrence , we have opted to present a computer - assisted verification of . in this way , we feel we save symbols , hence pages , hence trees . the price to be paid was only a few hours of interaction with the symbolic engine and ( eventually ! ) a few seconds of computer processing time . as in the previous section , only well - specified totally - algorithmic steps are eventually used . once more , there is no difficulty in checking intermediate steps against series expansions up to order 100 and beyond . our proof of the identity eventually boils down to exhibiting a fourth - order differential operator in @xmath316 , with coefficients in @xmath317 , that is satisfied by the difference between the two sides of . it is then sufficient to check that both sides satisfy the same initial conditions given by the coefficients of @xmath318 up to @xmath319 . the entire process relies on the _ holonomic framework _ pioneered by zeilberger @xcite , with supporting theorems to be found in works of stanley , such as @xcite and ( * ? ? ? let @xmath320 be a ground field , which we take here to be @xmath317 , the field of rational fractions in @xmath321 . ( throughout , we treat the quantity @xmath321 as a parameter . ) a formal power series of @xmath322\!]$ ] , simply called `` function '' , is _ holonomic _ ( alternative names are differentiably finite , @xmath323finite , @xmath324finite ) if it satisfies a linear differential equation with coefficients in the rational field @xmath325 . equivalently , @xmath326 is holonomic if the vector space over @xmath325 spanned by all the derivatives @xmath327 is finite - dimensional . holonomic functions are known to be closed under sum , product , differentiation , integration , and algebraic substitutions ( i.e. , substitutions of algebraic functions in place of variables ) . finally , if @xmath326 is holonomic , its sequence of coefficients @xmath328h)$ ] satisfies a linear recurrence relation with coefficients in @xmath329 . clearly , a holonomic function is determined by a finite amount of information ; namely , a defining differential equation supplemented by sufficiently many initial conditions . given two holonomic functions @xmath330 , one can then verify their conjectured identity as follows . * compute a differential equation , of order @xmath331 , say , that is satisfied by the difference @xmath332 . * check the coincidence of the expansions of @xmath113 and @xmath269 up to terms of order @xmath333 . by the finiteness of the underlying vector spaces , the process constitutes a valid _ proof _ of @xmath334 , in `` non - singular '' cases at leasth(t)$ ] , of a solution @xmath335 and having coefficients in @xmath336 $ ] ) has no root in @xmath337 . in the case of a non - singular operator of order @xmath331 , the number of needed initial conditions equals @xmath331 . ( in the `` singular '' case , a higher , but still effectively computable , number may be needed . ) ] . in our context , it could be carried out comparatively easily , thanks to the powerful gfun library developed by salvy and zimmermann @xcite . in what follows , we use @xmath338 to represent the differential operator @xmath339 ; we denote by @xmath340 the shift operator on infinite sequences @xmath341 such that @xmath342 . we let @xmath343 generically represent a polynomial of degree @xmath123 , either in @xmath316 ( for differential operators ) or in @xmath13 ( for difference operators ) , with coefficients in @xmath320 . as indicated before , the quantity @xmath321 is treated as a parameter . our purpose is to prove @xmath334 , where @xmath113 is the left - hand side of and @xmath269 is the right - hand side : @xmath344 and @xmath345 as defined in the statement . see figure [ ab - fig ] for a summary of the main steps of our proof . _ the left - hand side @xmath346 . _ the parity inherent in the coefficients suggests to introduce the subsequences @xmath347 and @xmath348 . the basic recurrence then relates @xmath349 to @xmath350 and @xmath351 to @xmath352 ; hence , by substitution , the fact that the vector @xmath353 depends linearly on @xmath354 via a matrix , whose coefficients are polynomial in @xmath13 ( and the parameter @xmath321 ) . by instantiating this last relation at @xmath355 and @xmath356 , and using back substitution , there results that @xmath349 and @xmath351 satisfy explicit linear recurrences of order 2 with coefficients that are polynomial in @xmath13 . the difference operators annihilating @xmath357 and @xmath358 are found in this way to be of the form @xmath359 equivalently , the exponential generating functions @xmath360 are found to satisfy @xmath361=0 $ ] and @xmath362=0 $ ] , where @xmath363 and @xmath364 are each of the form @xmath365 in fact , second - order operators @xmath366 and @xmath367 that appear to cancel @xmath368 and @xmath369 , respectively , can be guessed ( roughly , by the method of indeterminate coefficients cleverly implemented in maple s gfun ) . the guesses can then be turned into full - fledged proofs by checking ( with maple s ore_algebra , see @xcite ) the operator divisibility relations : @xmath370 and @xmath371 . once this is done , a differential operator that annihilates @xmath314 can be obtained by making use of properties of holonomic functions ( effective closure under sum ) and a further round of simplification based on guessing . it is found in this way that the second - order operator=0 $ ] , which involved terms of the rough form @xmath372 and eventually led us to infer . ] @xmath373 with @xmath374 , annihilates @xmath304 . @xmath375 a\equiv \upsilon(z , t ) \ , : & \pi_6 \partial^2y+\pi_5\partial y+\pi_4 y & = & 0;&\hbox{see~$\frak{a}$ in~\eqref{aop}}\\[2 mm ] \eta(t ) , \chi(t ) \ , : & \pi_6 \partial^2y+\pi_5\partial y+\pi_4 y & = & 0;&\hbox{$\eta,\chi$ algebraic~\eqref{etadef } , \eqref{chidef}}\\[2 mm ] \left\{\!\begin{array}{c } \ds \exp\left(\pm { z}j(t)\right ) \\[1 mm ] \ds \cosh , \sinh\left({z}j(t)\right ) \end{array}\!\right\}\ ! : \quad & \pi_4 \partial^2 y + \pi_3\partial y -z^2 y & = & 0 ; & \hbox{see~\eqref{2ndo}}\\[4 mm ] \left\{\begin{array}{c } \ds b^+\equiv \eta(t)\cosh\left({z}j(t)\right ) \\[1 mm ] \ds b^-\equiv \chi(t)\sinh\left({z}j(t)\right ) \end{array}\right\}\ ! : & \pi_{12}\partial^4 y + \cdots + \pi_{8 } y & = & 0 ; & \hbox{closure algorithm ( $ \times$)}\\[4 mm ] \delta:=a-(b^++b^- ) & \pi_{12}\partial^4 y + \cdots + \pi_{8 } y & = & 0 ; & \hbox{closure algorithm ( $ + $ ) } \\[2 mm ] \hline \end{array}\ ] ] _ the right - hand side @xmath376 . _ we can build up differential equations starting with the explicit expression of @xmath269 in : see again figure [ ab - fig ] for a summary . given a quantity @xmath377 that depends on @xmath321 , we set @xmath378 and @xmath379 , defining its `` odd '' and `` even '' parts ( in @xmath321 ) , respectively . with @xmath380 , we then consider @xmath381 and @xmath382 , and proceed to construct the corresponding annihilators , @xmath383 and @xmath384 . first , we observe that if @xmath385 is an arbitrary polynomial , then @xmath386 this equation is invariant by @xmath387 , so that it is also satisfied when the exponential in is replaced by @xmath388 . next , the function @xmath309 , given by a cubic algebraic equation , is found to satisfy a second - order differential equation with coefficients that are of degree at most 6 . the application of closure rules for products of holonomic functions then provides for @xmath389 a differential operator @xmath390 that is of order 4 , with coefficients of degree at most 12 . we can then proceed to construct the annihilator @xmath384 of the odd part @xmath391 . it turns out that the algebraic function @xmath392 defined in satisfies _ the same differential equation _ as @xmath309 , but with different initial conditions ( @xmath393 , @xmath394 ) . there now results from this fact and the comments accompanying that we can take @xmath395 as annihilator of the right - hand side ( @xmath269 ) of . _ the comparison . _ finally , it remains to verify that @xmath334 . the operator @xmath396 is defined to annihilate @xmath397 , where @xmath398 and @xmath399 are the vector spaces of solutions of @xmath400=0 $ ] and @xmath401=0 $ ] , respectively . by construction , the difference @xmath332 is such that @xmath402=0 $ ] . the operator , obtained by holonomic closure under sums , is of type @xmath403 the associated recurrence operator is found to be of the form @xmath404 with leading coefficient @xmath405 so that @xmath396 is non - singular at @xmath57 . it thus suffices to verify that the expansions of @xmath406 and of the right - hand size @xmath269 in coincide till terms of order @xmath407 , @xmath408 so as to complete the proof that @xmath334 . equation is now established . orthogonal polynomials attached to continued fraction expansions relative to elliptic functions , have been first studied by carlitz and al - salam ( see @xcite for some more recent developments ) , and , as already mentioned , they form the subject of the monograph _ elliptic polynomials _ by lomont and brillhart @xcite . the family made explicit by theorem [ qegf ] does not appear to be captured by their classification and hence seems to be new . remarkably , in connection with birth - and - death processes having cubic weights , gilewicz _ et al . _ @xcite have recently discovered another new family of orthogonal polynomials , related to the expansion of the dixonian function sm taken at 0 ( as in @xcite ) , rather than at the point @xmath58 that is needed here ( cf theorem [ dix - thm ] ) . the continued fraction of theorem [ main - thm ] has several interesting by - products that we now examine . these include an explicit evaluation of hankel determinants , as well as elementary congruence properties of pseudo - factorials . [ [ hankel - determinants . ] ] * _ hankel determinants . _ * + + + + + + + + + + + + + + + + + + + + + + + + it is well known that , generally , coefficients of a jacobi fraction can be expressed as determinants . this fact is classically derived from stieltjes s matrix version of the addition theorem @xcite ; it is equivalent to the @xmath409 decomposition of the gram matrix @xmath410 , with entries @xmath411 ( for @xmath412 ) , which is also known as the hankel matrix of the sequence @xmath413 ; see for instance @xcite . conversely , any known continued fraction yields an explicit hankel determinant evaluation . given this , an immediate consequence of theorem [ main - thm ] is the following . let @xmath414 be a positive integer . the hankel determinant of pseudo - factorials @xmath415 admits the closed form @xmath416 where the @xmath417 are the continued fraction numerators of . [ [ congruences . ] ] * _ congruences . _ * + + + + + + + + + + + + + + + + [ cols=">,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>,>",options="header " , ] a cursory examination of the @xmath121 suggests clear divisibility patterns ; for instance , from , we immediately expect the @xmath121 to be divisible by 10 , for @xmath13 large enough . figure [ cong - fig ] tabulates arithmetic congruence properties of the @xmath121 for small values of the modulus @xmath418 and of the index @xmath13 . the table obviously has much structure : the sequence @xmath126 appears to be eventually @xmath57 modulo the numbers @xmath419 ; there are obvious periodically reproducing patterns , such as @xmath420 ( mod 3 ) , @xmath421 ( mod 6 ) , or the more recondite , and curiously repetitive , @xmath422 of length 36 corresponding to modulus 7 . we state here a simple consequence of our main continued fraction in theorem [ main - thm ] . [ cong - thm ] @xmath423 the sequence @xmath126 of pseudo - factorials is eventually periodic modulo any integer @xmath424 . @xmath425 for each @xmath426 , the sequence @xmath126 satisfies modulo @xmath427 a linear recurrence with constant coefficients that is of order at most @xmath414 . the statement is an instance of the general fact that @xmath428fraction expansions with integer coefficients automatically imply congruence properties @xcite . @xmath423 the main continued fraction representing the ordinary generating function @xmath250 of pseudo - factorials has a factor of @xmath429 at its numerator of rank @xmath430 . in particular , the contributions induced by the stages @xmath430 , @xmath431 , and so on , of this continued fraction are zero modulo @xmath429 . in other words , @xmath250 is congruent modulo @xmath429 to the @xmath418th convergent of the continued fraction . thus , it satisfies , modulo @xmath429 , a linear recurrence of order at most @xmath418 . hence it is eventually periodic modulo @xmath418 . @xmath425 the estimate above of the order of the recurrence satisfied by modular reductions of the pseudo - factorials can be vastly improved @xcite . by the classical @xmath432determinant identity of orthogonal polynomials and convergent denominators , the difference of two successive convergents of satisfies the identity @xmath433 this specializes to the @xmath428fraction relative to @xmath250 , when the @xmath434 are taken to be as in . by expressing that @xmath250 is the sum of the differences of its successive convergents , we then obtain , for any @xmath435 , @xmath436 in particular , since the @xmath434 are all integers and the @xmath437 are integral with @xmath438 , we have , @xmath439 in the sense that coefficients of both series are equal , after reduction modulo @xmath418 . thus , modulo @xmath418 , the @xmath121 satisfy a linear recurrence whose characteristic polynomial is exactly the denominator polynomial @xmath440 ( reduced mod @xmath418 ) . as an illustration , corresponding to @xmath441 , we find the congruences @xmath442 which already justify the data of figure [ cong - fig ] for moduli @xmath443 ; for instance , from the convergent @xmath444 , we find @xmath445 . for @xmath446 , the form @xmath447 explains the observed patterns of @xmath126 modulo 7 and the period equal to 36 . by contrast , for @xmath448 , we find that @xmath449 ( @xmath450 reduces to 1 modulo 11 ) , thereby establishing that the @xmath121 with @xmath451 are all divisible by 11 . as the previous discussion suggests , congruence properties of pseudo - factorials are tightly linked to arithmetic properties of the @xmath452 polynomials whose exponential generating function has been determined in theorem [ qegf ] . let again @xmath343 denote an unspecified polynomial of degree @xmath123 . without attempting a general discussion , we only remark here the existence of striking regularities , as summarized by the following data . first , for @xmath414 a prime of the form @xmath453 : @xmath454 \mod~7 & \mod~ 13 & \mod~ 19 & \mod~31\\ \hline \ds \frac{p_7}{q_7}\equiv \frac{\pi_6}{1 + 4z^6 } & \ds \frac{p_{13}}{q_{13}}\equiv \frac{\pi_{12}}{1 + 11z^{12 } } & \ds \frac{p_{19}}{q_{19}}\equiv \frac{\pi_{18}}{1 + 11z^{18 } } & \ds \frac{p_{31}}{q_{31}}\equiv \frac{\pi_{30}}{1 + 4z^{30}}.\\[2.5 mm ] \hline\hline\end{array } $ } \ ] ] finally , for @xmath414 a prime of the form @xmath455 : @xmath454 \mod~5 & \mod~ 11 & \mod~ 17 & \mod~23\\ \hline \ds \frac{p_5}{q_5}\equiv { \pi_4 } & \ds \frac{p_{11}}{q_{11}}\equiv { \pi_{10 } } & \ds \frac{p_{17}}{q_{17}}\equiv { \pi_{16 } } & \ds \frac{p_{23}}{q_{23}}\equiv { \pi_{22}}.\\[2.5 mm ] \hline\hline\end{array } $ } \ ] ] the relation between elliptic functions and continued fractions is an old subject , one that is especially rich . connections are manifest with the theta function framework in the form of various types of @xmath16series expansions , starting with eisenstein and including the celebrated rogers ramanujan identities ( * ? ? ? closer to our perspective are early contributions due to stieltjes and rogers regarding the jacobian sn , cn framework . conrad , first in his dissertation @xcite then in collaboration with flajolet @xcite , has elicited new connections with the dixonian framework of the sm , cm functions . as the present work supplemented by further investigations of ours indicate , there are new continued fractions to be explored , attached to the weierstra and dixonian frameworks . in this vein , we have recently discovered new elliptic continued fractions , relative to `` equiharmonic '' and `` lemniscatic '' numbers , which are lattice analogues of the pseudo - factorials we plan to report on these in a future publication . * acknowledgements . * the work of p. flajolet was supported in part by the sada and lareda projects of the french national research agency ( anr ) . the authors are grateful to frdric chyzak , manuel kauers , and bruno salvy for insightful discussions relative to computer verification of identities . they are also indebted to bruno salvy for making available his upgraded version of the maple gfun package on which the developments of section [ ortho - sec ] could be most effectively built . thanks finally to robin chapman , an anonymous referee , and the editor for encouraging remarks and constructive suggestions . mourad e. h. ismail and d. masson . some continued fractions related to elliptic functions . in b. berndt and f. gesztesy , editors , _ continued fractions : from analytic number theory to constructive approximation _ , volume 236 of _ contemporary mathematics _ , pages 149166 , providence , ri , 1999 . ams . erik lundberg . om hypergoniometriska funktioner af komplexa variabla . manuscript , 1879 . english translation , `` on hypergoniometric functions of complex variables '' available from jaak peetre s web page .

this study presents miscellaneous properties of pseudo - factorials , which are numbers whose recurrence relation is a twisted form of that of usual factorials . these numbers are associated with special elliptic functions , most notably , a dixonian and a weierstra function , which parametrize the fermat cubic curve and are relative to a hexagonal lattice . a continued fraction expansion of the ordinary generating function of pseudo - factorials , first discovered empirically , is established here . this article also provides a characterization of the associated orthogonal polynomials , which appear to form a new family of `` elliptic polynomials '' , as well as various other properties of pseudo - factorials , including a hexagonal lattice sum expression and elementary congruences .

0901.1379

particle and transverse energy production in the central rapidity region of heavy ion collisions can be treated as a combination of hard / semihard parton production and soft particle production . with increasing energies , the semihard qcd - processes are expected to become increasingly important . this is due to two reasons : firstly , already in @xmath2 collisions the rapid rise of the total and inelastic cross sections can be explained by copious production of semihard partons , _ minijets _ , with transverse momenta @xmath3 gev @xcite . this is also expected to happen in @xmath4 collisions at very high energies . secondly , the semihard particle production scales as @xmath5 , so that for large nuclei the importance of semihard partons is increased further @xcite . the soft , non - perturbative , particle production in ultrarelativistic heavy ion collisions can be modelled _ e.g. _ through strings @xcite or through a decaying strong background colour field @xcite . the time scale for producing partons and transverse energy into the central rapidity region by semihard collisions is short , typically @xmath6 @xmath1 , where @xmath7 gev is the smallest transverse momentum included in the computation . the soft processes are completed at later stages of the collision , at @xmath8 @xmath1 . if the density of partons produced in the hard and semihard stages of the heavy ion collision becomes high enough - as will be the case - a saturation in the initial parton production can occur @xcite , and softer particle production will be screened . the fortunate consequence of this is that a larger part of parton production in the central rapidities can be _ computed _ from perturbative qcd ( pqcd ) at higher energies and the relative contribution from soft collisions with @xmath9 gev becomes smaller . typically , the expectation is that at the sps ( pb+pb at @xmath10 @xmath11gev ) , the soft component dominates , and at the lhc ( pb+pb at @xmath12 @xmath11tev ) the semihard component is the dominant one . at the rhic ( au+au at @xmath13 @xmath11gev ) one will be in the intermediate region , and both components should be taken into account . a lot of effort has also been devoted for building event generators @xcite addressing the dominance of semihard processes in nuclear collisions at high energies . these have generated also new insight and very useful discussion during the recent years . also recently , a promising novel approach to minijet production has been developed @xcite . i have divided this talk basically into two halves . in the first one , i will recapitulate the basic features of semihard parton production and review our latest results @xcite . the main goal of these studies is to find out the initial conditions for early qgp - formation at @xmath14 @xmath1 , including the transverse energy deposited into the mid - rapidity region , chemical composition of the parton plasma , and , to study the possibility of a very rapid thermalization and estimate the initial net baryon - to - entropy ratio . it is vitally important to study the early formation of strongly interacting partonic matter , since the later evolution of the qgp , the final state global observables , and the suggested signals of the plasma will strongly depend on the initial conditions . the second half i will devote for discussion of an additional mechanism for parton and transverse energy production : minijets from a bfkl - ladder @xcite . especially , i will estimate the maximum amount of transverse energy one should expect from the bfkl - minijets in heavy ion collisions . hadronic jets originating from high @xmath15 quarks and gluons are clearly observed experimentally but when the partons have @xmath16 gev the jets become very difficult to distinguish @xcite from the underlying event . in heavy ion collisions , where we expect hundreds ( rhic ) or thousands ( lhc ) of minijets with @xmath17 gev be produced , detection of individual minijets will be impossible . however , the semihard partons are expected to contribute dramatically to the early formation of qgp . the idea of multiple production of semihard gluons and quarks in @xmath18 and @xmath4 collisions is based on a picture of independent binary parton - parton collisions . the key quantity is the integrated jet cross section , @xmath19 where @xmath20 are the fractional momenta of the incoming partons @xmath21 and @xmath22 , and @xmath23 are the parton distributions in @xmath24 ( @xmath25 ) . the factor 2 comes from the fact that , in the lowest order ( lo ) pqcd , there are two partons produced in each semihard subcollision . in the eikonal models for @xmath18 collisions @xcite the ratio @xmath26 can be interpreted as the average number of semihard events in one inelastic collision . the results i will be quoting in the following @xcite are obtained with the mrsh @xcite and mrsd- @xcite parton distributions with a scale choice @xmath27 . more detailed formulation can be found in refs . @xcite , and numerical evaluation of eq . ( [ sigmajet ] ) in ref . @xcite . the formula above is defined in the lowest order ( lo ) , @xmath28 . often a constant factor @xmath29 is used to simulate the effects of nlo terms . studies of the nlo jet cross section @xmath30 @xcite show that ( with a scale choice @xmath27 and with a jet size @xmath31 ) this is a reasonable approximation @xcite . strictly speaking , however , a theoretical @xmath32-factor can only be defined for quantities where a well - defined , infrared - safe measurement function can be applied @xcite . for @xmath33-production in nuclear collisions , an acceptance window in the whole central rapidity unit defines such a function but for this acceptance criteria and for @xmath17 gev the exact nlo contribution has not been computed yet . the first estimate of the average number of produced semihard partons with @xmath34 in an @xmath4 collision at a fixed impact parameter @xmath35 can be obtained as @xcite @xmath36 and the average transverse energy carried by these partons as @xcite @xmath37 where @xmath38 is the nuclear overlap function @xcite which scales @xmath39 , describing thus the typical scaling of hard processes in nuclear collisions . the normalization is @xmath40 and , for large nuclei with woods - saxon nuclear densities , @xmath41 . the acceptance criteria imposed for the quantities @xmath42 and for @xmath43 will be @xmath44 , and the corresponding cuts will be made in @xmath45 and @xmath46 . in eqs . ( [ n ] ) and ( [ et ] ) above , @xmath47 is the average number of semihard collisions and @xmath48 is the average transverse energy carried by the partons produced in each of these collisions . we fix @xmath49 gev , _ i.e. _ we describe the initial conditions at @xmath50 @xmath1 . the predictions for the central rapidity unit in pb - pb collisions at the rhic and lhc energies are summarized in tables 1 . also , contributions from gluon , quark and antiquark production are shown separately @xcite . at @xmath51 can be found in ref . ] in the results given , we have neglected nuclear effects in parton distributions : @xmath52 . in reality , however , in the typical @xmath53-region @xmath54 there are quite strong shadowing corrections @xcite , especially for the lhc nuclear collisions . also , the scale evolution of nuclear gluon shadowing was shown to be potentially important in the analysis in ref . @xcite . however , a re - analysis with the input from hera at small-@xmath53 @xcite has to be performed before getting a solid quantitative prediction of the shadowing effects on minijet production . the rapid rise of the structure function @xmath55 at small values of @xmath53 observed at hera @xcite does not affect the bulk of the 2 gev minijets at rhic energies very much but obviously has quite dramatic consequences at the lhc energies . as demonstrated in @xcite , there is a clear enhancement of minijet production due to the new parton distributions . however , the more rapidly the gluon distributions rise , the more there should be nuclear shadowing due to the glrmq - fusions @xcite . again , a more quantitative prediction depends on the scale evolution of nuclear gluon shadowing as well . the average numbers of semihard partons at @xmath0 @xmath1 with @xmath44 and @xmath56 gev in central pb - pb collisions , as given by eq . ( [ n ] ) . shadowing is not included and @xmath57 . the upper values are obtained with mrsh and the lower ones with mrsd- parton distributions . * ( b ) * the average transverse energy carried by these partons , as predicted by eq . ( [ et ] ) . [ cols="^,^,^,^,^,^",options="header " , ] [ table1 ] let us now have a closer look at the results in table 1 . there are four important observations . firstly , the gluons clearly dominate both the initial parton and transverse energy production : the initial parton system is about 80 % glue . secondly , the effective transverse area of the produced semihard partons is @xmath58 . comparing this with the effective nuclear transverse area , @xmath59 , we notice that @xmath60 _ i.e. _ the parton system at the lhc at 0.1 @xmath1 is already dense enough so that a saturation of parton production can take place @xcite . in this way , the scale @xmath61 acquires also _ dynamical _ significance . at rhic , since @xmath62 , saturation occurs at smaller values of @xmath15 ( at @xmath63 @xmath1 ) , possibly in the region where pqcd can not be trusted . this qualitative argumentation is supported by a more quantitative , although still phenomenological , analysis of ref . @xcite , where we suggested that at sufficiently large energies ( lhc ) and large nuclei ( @xmath64 ) , a dynamical screening mass is generated , causing a saturation in the minijet cross sections @xcite at a perturbative scale like @xmath65 . the consequence is that the softer parton production is screened and its relative importance becomes smaller . the third interesting observation is that the gluonic subsystem in the central rapidity unit @xmath66 may thermalize very fast , at least in the lhc nuclear collisions . in the perturbatively produced system the ( transverse ) energy per gluon is @xmath67 gev and the energy density of the system at @xmath68 fm/@xmath69 is @xmath70 . the temperature @xmath71 of an ideal ( massless boson ) gas in a complete thermal (= both kinetic and chemical ) equilibrium with this energy density can be computed from @xmath72 , and we get @xmath73 0.988 ( 1.14 ) gev with the mrsh ( mrsd- ) densities . especially , we find @xcite @xmath74 so that at the lhc the average energy of gluons is already as in an ideal gas in thermal equilibrium . only isotropization is needed , and a rapid thermalization is indeed possible . an instant thermalization would in turn have profound consequences on _ e.g. _ thermal dileptons , for which a high initial temperature plays a crucial role @xcite . note that our conclusion of the possibility of an almost instant thermalization is due to the small-@xmath53 enhancement in the hera gluon densities . from the energy / gluon viewpoint it also seems that thermalization for rhic is going to happen somewhat later . note however , that above i did not consider isotropization of the system at all . in the simplified picture presented here , the transit time of the colliding nuclei , @xmath75 , and the initial parton spread , @xmath76 for the partons which will be produced in the mid - rapidity , are neglected . then a bjorken - like boost - invariant picture is possible , and in the central rapidities a proper time @xmath77 is a relevant variable . for a more thorough discussion of isotropization , a more detailed microscopic space - time picture has to be specified , as done in refs . @xcite ( see also the discussion in @xcite ) . the fourth observation @xcite is that initially , at @xmath14 fm/@xmath69 , the net baryon number density in the central rapidity unit is very small as compared to the gluon density but _ larger _ than the nuclear matter density ( 0.17 @xmath78 ) , even though the colliding nuclei are practically already far apart , especially at the lhc where @xmath79 . more precisely , we estimate @xmath80 with the mrsh ( mrsd- ) parton distributions . computing the net baryon - to - entropy ratio by using @xmath81 for a thermal boson gas gives _ initially _ , at @xmath0 fm/@xmath69 : @xmath82 for lhc , and @xmath83 for rhic . we conclude that at the future colliders we are still relatively far away from the extreme conditions of the early universe , where the inverse of the specific entropy is @xmath84 @xcite . for the lhc , assuming an instant thermalization of the gluon system at @xmath0 @xmath1 , and an adiabatic evolution thereafter , the final entropy can be approximated by the initial entropy of gluons @xcite . the non - perturbative mechanism for particle production will not increase the entropy much but does increase the net baryon number . if the non - perturbative contribution to the net baryon number production is assumed to be of the same order of magnitude as in the current pb+pb collisions at sps , the _ final _ baryon - to - entropy ratio for the lhc will be @xmath85 . for the rhic nuclear collisions , thermalization is most likely not as instantaneous , but following nevertheless the same line of arguments , and taking into account that the non - perturbative component becomes important also for entropy production , we estimate @xmath86 for the final net baryon - to - entropy ratio . minijet production i have considered above is based on collinear factorization , where the perturbative partonic cross sections are factorized at a momentum scale @xmath87 from the parton distributions with nonperturbative input . next , i will discuss an additional mechanism for minijet and transverse energy production , where factorization is not used . the small-@xmath53 rise in the structure function @xmath88 observed at hera @xmath89 gev@xmath90 @xcite can be explained by the leading @xmath91 dglap - evolution @xcite and also by the leading @xmath92 evolution @xcite . also a power - like behaviour , @xmath93 , expected in the leading @xmath94 bfkl - approach @xcite , does not contradict the data . in the following , let us assume that the small-@xmath53 increase is entirely due to the bfkl - physics . then , with this assumption , we will study what is the _ maximum _ transverse energy deposit in the central rapidity unit due to the minijets emitted from a bfkl - ladder in the lhc nuclear collisions . at rhic the bfkl - minijets are not expected to contribute in any significant manner because the bfkl - enhancement takes place only at @xmath95 . therefore , this latter part of my talk , which is based on ref . @xcite , will be relevant only for the lhc . it is instructive to start from a case of fully inclusive minijets with two tagging jets separated by a wide rapidity gap , as studied by mueller and navelet @xcite . the ( summed ) subprocess is also shown fig . 1a , where the incoming partons have momentum fractions @xmath96 and @xmath97 , the tagging jets rapidites @xmath98 and @xmath99 ( @xmath100 ) and transverse momenta @xmath101 and @xmath102 , respectively . between the tagging jets there are @xmath103 gluons emitted , labeled by 1 ... @xmath103 . thus each final state is described by a feynman graph with 2 incoming and @xmath104 outgoing on - shell gluons . the colour singlet hard bfkl - pomeron ladder arises when these feynman graphs are squared and summed . in the kinematic region we will be interested in , the rapidities are strongly ordered , @xmath105 , but the transverse momenta are not , @xmath106 . then only the transverse degrees of freedom of the momenta of the virtual legs become important . the tagging jets of fig . 1a have transverse momenta at a perturbative scale , so that one may use collinear factorization to write the cross section down as : @xmath107 where only gluons are considered . the strong rapidity ordering simplifies the momentum fractions to @xmath108 and @xmath109 , and the parton densities factor out of the sum . for the process @xmath110 the leading contribution in the large @xmath111 limit comes from the @xmath112-channel amplitude . in a physical gauge , this amplitude is also gauge invariant up to the subleading terms . the matrix element @xmath113 consists then of the following building blocks : in the leading @xmath111 approximation , in a physical gauge and with the strong rapidity ordering , each gluon can be regarded as emitted from an effective non - local lipatov - vertex , where bremsstrahlung from initial and final legs and emission from the exchanged gluon are summed . these are described by the black blobs in fig 1 . also the propagators are effective ones since they are exponentiated ( reggeized ) after computing the leading virtual corrections to the @xmath112-channel gluon exchange . the effective propagators are drawn by thicker ( vertical ) lines in fig . 1 . original references , detailed discussion and derivation of these concepts can be found in the useful lecture notes by del duca @xcite . next , we square each matrix element @xmath113 , and due to the strong ordering in rapidities , colour singlet ladders with @xmath104 rungs are formed . the colour algebra can be performed by summing ( averaging ) over the final ( initial ) state colours , and the polarization sums can be done . with help of _ e.g. _ laplace - transformation ( in @xmath114 ) , the rapidity integrals can be disentangled . finally , by summing over @xmath103 , one obtains an iterative integral equation , the inhomogeneous bfkl - equation @xcite ( see also @xcite ) , which describes an addition of one rung into the colour - singlet hard pomeron ladder . the bfkl - ladder is denoted by @xmath115 in fig 1a . the cross section ( [ fact ] ) then becomes : @xmath116 where @xmath117 and @xmath118 . if all the virtual corrections and the real emissions are neglected , the ladder reduces into @xmath119 , and the born limit for the two jets separated by a large rapidity interval is recovered @xcite . let us then study the case with tagging jets further by fixing one step of the ladder , as shown in fig . it is straightforward to sum the graphs with gluon emissions between the tagging jet @xmath120 and the fixed minijet @xmath69 , and , between the minijet @xmath69 and the tagging jet @xmath121 . this creates a ladder on each side of the fixed rung . especially , we learn that a generic factor @xmath122 , which includes the phase - space factor and contraction of the two lipatov vertices associated with the step @xmath69 , arises from fixing the the rung @xmath69 . the cross section becomes @xcite @xmath123 our goal is to study the leading bfkl minijet production mechanism which is @xmath124 . as illustrated in fig . 2 , we therefore relax the requirement of having tagging jets . then coupling of the pomeron ladder to the hadron becomes essentially non - perturbative and a form - factor , or , rather , a parton distribution , will be needed . also , now that we do not require any tagging jets , we have to give up collinear factorization . we do not have any perturbative born limit to compare with , either . therefore , the best we can do is to adopt the procedure for deep inelastic scattering ( dis ) in @xcite , where an addition of each rung into the pomeron ladder between the two hadrons or nuclei is expected to be described by the _ homogeneous _ bfkl equation for the unintegrated gluon density @xmath125 , @xmath126 . \label{hbfkl}\ ] ] normalization for this scale - invariant equation is given by the gluon distributions @xmath127 determined from experimental input @xcite . by using the knowledge of the factor arising from fixing one rung of the bfkl - pomeron ladder , the inclusive minijet cross section from the bfkl - ladder can now be written down as @xcite @xmath128 where @xmath15 and @xmath129 are the transverse momentum and the rapidity ( in the hadron cms ) of the minijet . from momentum conservation and multi - regge kinematics the momentum fractions become @xmath130 . due to the fact that in this case we do not have an `` external '' hard probe like the virtual photon with an associated quark box as in dis , nor an on - shell born cross section to relax into , we can not determine the overall dimensionless normalization constant @xmath131 exactly . however , we are able to fix the _ slope _ of the minijet @xmath15-distribution , which will be sufficient for estimating the upper limit of transverse energy production from the bfkl - ladder . the minijet cross section of eq . ( [ bfklmini ] ) is shown in fig . 3 @xcite . in the bfkl - computation we have used the unintegrated gluon densities compatible with the small-@xmath53 rise in the set mrsd- @xcite . for comparison , the more traditional ( cfltlo ) minijet cross sections , discussed in the first half of the talk , are also shown with the mrsd- parton distributions . the nlo jet analysis @xcite indicates that lo+nlo calculation with collinear factorization reproduces the measured jet cross sections well . therefore , at @xmath132 gev , the bfkl - minijet contribution should be less than the collinearly factorized . we can thus argue that @xmath133 . the transverse energy production due to the minijets from the bfkl - ladders at @xmath134 in @xmath4 collisions can now be estimated @xcite from @xmath135 the coherence of the bfkl ladder is broken when we fix a rung , and the cross section diverges at @xmath136 . a cut - off is , unfortunately , needed also in the bfkl case . the saturation of the cfltlo - minijet cross section in the lhc pb - pb collisions ( as considered in the first half of the talk ) , implies that the bfkl - cross section should not grow much larger than the curve 2 in fig . [ dsdpt ] . therefore , we do not trust the bfkl - computation with @xmath137 below @xmath138 gev . with these values , we find for central pb - pb collisions at the lhc , @xmath139 gev with fixed @xmath140 , and , 4940 gev with ( _ ad hoc _ ) running @xmath141 . comparing these numbers to the value 15330 gev in table 1b for gluons , we see that the bfkl - contribution is at most a few 10% correction to the results in @xcite . on the other hand , one should perhaps compare the results at the same level of approximations , ( only lo gluons , @xmath142 fixed ) _ i.e. _ curves 3 and a in fig . [ dsdpt]a . then the two contributions become of similar magnitude . in this case , however , the @xmath61 in the cfltlo - computation should be lower than 2 gev , and the bfkl contribution would again be the smaller one . we worked under the assumption that the bfkl - evolution is responsible for _ all _ the small-@xmath53 rise at hera , _ i.e. _ we studied the _ maximum _ contribution from the kinematical region relevant for the hard bfkl - pomeron . since the hera results can be explained by the leading @xmath143 and/or the leading @xmath144 approximations , the leading @xmath94-contribution is obviously not the dominant mechanism at the present values of @xmath53 . thus , my conclusion is that the bfkl - minijets certainly bridge the way towards softer physics at @xmath145 gev , but the initial conditions relevant for the early qgp - formation in the lhc nuclear collisions are dominantly given by the minijets computed in collinear factorization . * acknowledgements . * the results discussed in this talk are based on refs . i would like to thank k. kajantie , a. leonidov , b. mller , v. ruuskanen and x .- wang for fruitful collaboration . i also owe special thanks to a. leonidov and v. ruuskanen for getting our bfkl - study started and finally finished . blaizot and a. mueller , nucl . phys . * b289 * ( 1987 ) 847 . k. kajantie , p. v. landshoff and j. lindfors , phys . * 59 * ( 1987 ) 2517 ; + k.j . eskola , k. kajantie and j. lindfors , nucl . phys . * b323 * ( 1989 ) 37 . eskola , k. kajantie and v. ruuskanen , phys . b332 * ( 1994 ) 191 . b. andersson , g. gustafson , b. nilsson - almqvist , nucl . b281 * ( 1986 ) 289 . k. werner , phys . ( 1993 ) 87 . wang and m. gyulassy , phys . * d44 * ( 1991 ) 3501 ; _ ibid . _ * d45 * , 844 ( 1992 ) 844 ; + phys . * 68 * ( 1992 ) 1480 ; comput . phys . commun . * 83 * ( 1994 ) 307 . k. kajantie and t. matsui , phys . lett . * b164*(1985 ) 373 ; g. gatoff , a. k. kerman and t. matsui , phys . rev . * d36 * ( 1987 ) 114 ; k.j . eskola and m. gyulassy , phys . rev . * c47 * ( 1993 ) 2329 . l. v. gribov , e. m. levin and m. g. ryskin , phys . rep . * 100 * ( 1983 ) 1 . eskola , b. mller and x .- wang , phys . b374 ( 1996 ) 20 ; b. mller , in this workshop . eskola and k. kajantie , preprint cern - th/96 - 259 , september 1996 . k. geiger and b. mller , nucl . b369 * ( 1992 ) 600 ; k. geiger , phys . rev . * d47 * ( 1993 ) 133 . a. kovner , l. mc lerran and h. weigert , phys . rev . * d52 * ( 1995 ) 3809 , 6231 ; + r. venugopalan , preprint doe / er/ 40561 - 251-int96 - 00 - 120 , march 1996 . eskola , a. leonidov and p.v . ruuskanen , preprint cern - th/96 - 124 . ua1 collaboration , c. albajar _ et al . _ , nucl . phys . * b309 * ( 1988 ) 405 . martin , w.j . stirling and r.g . roberts , ral preprint 93 - 077 ( 1993 ) . martin , w.j . stirling and r.g . roberts , phys . b306 * ( 1993 ) 145 . eskola and x .- n . wang , int . j. mod . phys . * a10 * ( 1995 ) 2881 . ellis , z. kunszt and d.e . soper , phys . lett . * 69 * ( 1992 ) 1496 ; + z. kunszt and d. e. soper , phys . rev . * d46 * ( 1992 ) 192 . nm collaboration , p. amaudruz _ et al . _ , z. phys . * c51 * ( 1991 ) 387 . eskola , nucl . b400 * ( 1993 ) 240 . h1 collaboration , i. abt _ b407 * ( 1993 ) 515 ; t. ahmed _ et al . _ , * b439 * ( 1995 ) 471 ; zeus collaboration , m. derrick _ et al . _ , b316 * ( 1993 ) 412 ; z. phys . * c65 * ( 1995 ) 379 . a. mueller and j. qiu , nucl . phys . * b268 * ( 1986 ) 427 . levin , in this workshop . ruuskanen , proc . _ quark matter 91 _ , nucl . phys . * a544 * ( 1992 ) 169c . eskola and x .- wang , phys . rev . * d49 * ( 1994 ) 1284 ; + t.s . bir , e. van doorn , b. mller , m.h . thoma and x .- wang , phys . rev . * c48 * ( 1993 ) 1275 . u. heinz , `` heavy ion physics at the lhc '' , in proc . of _ towards the lhc experimental programme _ , ecfa meeting , 5 - 8 march 1992 , evian - les - bains , france , p. 95 dokshitzer , sov . jetp * 46 * ( 1977 ) 1649 ; v. n. gribov and l. n. lipatov , sov . nucl . phys . * 15 * ( 1972 ) 438 , 675 ; g. altarelli , g. parisi , nucl . phys . * b126 * ( 1977 ) 298 . r. d. ball and s. forte , phys . b351 * ( 1995 ) 313 ; _ ibid _ * b359 * ( 1995 ) 362 ; _ ibid _ * b336 * ( 1994 ) 77 . e. a. kuraev , l. n. lipatov and v. s. fadin , sov . jetp * 45 * ( 1977 ) 199 ; + ya . balitskij and l. n. lipatov , sov . j. nucl * 28 * ( 1978 ) 822 . a. h. mueller and h. navelet , nucl . phys * b282 * ( 1987 ) 727 . v. del duca , `` an introduction to the perturbative qcd pomeron and to jet physics at large rapidities '' , desy 95 - 023 , february 1995 . l. v. gribov , e. m. levin and m. g. ryskin , phys lett . * b100 * ( 1981 ) 173 ; phys lett . * b121 * ( 1983 ) 65 ; e. m. levin and m. g. ryskin , phys . rep . * 189 * ( 1990 ) 267 . v. del duca , m. e. peskin and w .- k . tang , phys . lett . * b306 * ( 1993 ) 151 . a. j. askew , j. kwieciski , a. d. martin and p. j. sutton , phys . rev . * d49 * ( 1994 ) 4402 .

recent results on minijet production in nuclear collisions at the rhic and lhc energies are reviewed . initial conditions of the qgp at @xmath0 @xmath1 , especially parton chemistry , thermalization and net baryon number - to - entropy ratio are discussed . also , contribution of minijets from a hard bfkl - pomeron ladder will be estimated . = -1 cm = -1 cm = 16.5 cm = 22.8 cm * minijets in ultrarelativistic heavy ion collisions at rhic and lhc * k.j.eskola _ cern / th , ch-1211 geneve 23 , switzerland _

hep-ph9610365

the primary goal of tomography is to determine the internal structure of an object without cutting it , namely using data obtained by methods that leave the object under investigation undamaged . these data can be obtained by exploiting the interaction between the object and various kinds of probes including x - rays , electrons , and many others . after its interaction with the object under investigation , the probe is detected to produce what we call a projected distribution or tomogram , see fig . [ fig : profiles ] . tomography is a rapidly evolving field for its broad impact on issues of fundamental nature and for its important applications such as the development of diagnostic tools relevant to disparate fields , such as engineering , biomedical and archaeometry . moreover , tomography can be a powerful tool for many reconstruction problems coming from many areas of research , such as imaging , quantum information and computation , cryptography , lithography , metrology and many others , see fig . [ fig : tomography ] . from the mathematical point of view the reconstruction problem can be formulated as follows : one wants to recover an unknown function through the knowledge of an appropriate family of integral transforms . it was proved by j. radon @xcite that a smooth function @xmath0 on @xmath1 can be determined explicitly by means of its integrals over the lines in @xmath1 . let @xmath2 denote the integral of @xmath3 along the line @xmath4 ( tomogram ) . then @xmath5 where @xmath6 is the laplacian on @xmath1 , and its square root is defined by fourier transform ( see theorem [ thm : inversioneformula ] ) . we now observe that the formula above has built in a remarkable duality : first one integrates over the set of points in a line , then one integrates over the set of lines passing through a given point . this formula can be extended to the @xmath7-dimensional case by computing the integrals of the function @xmath3 on all possible hyperplanes . this suggests to consider the transform @xmath8 defined as follows . if @xmath3 is a function on @xmath9 then @xmath10 is the function defined on the space of all possible @xmath11-dimensional planes in @xmath9 such that , given a hyperplane @xmath12 , the value of @xmath13 is given by the integral of @xmath3 along @xmath12 . the function @xmath10 is called _ radon transform _ of @xmath3 . there exist several important generalizations of the radon transform by john @xcite , gelfand @xcite , helgason @xcite and strichartz @xcite . more recent analysis has been boosted by margarita and volodya manko and has focused on symplectic transforms @xcite , on the deep relationship with classical systems and classical dynamics @xcite , on the formalism of star product quantization @xcite , and on the study of marginals along curves that are not straight lines @xcite . in quantum mechanics the radon transform of the wigner function @xcite was considered in the tomographic approach to the study of quantum states @xcite and experimentally realized with different particles and in diverse situations . for a review on the modern mathematical aspects of classical and quantum tomography see @xcite . good reviews on recent tomographic applications can be found in @xcite and in @xcite , where particular emphasis is given on maximum likelihood methods , that enable one to extract the maximum reliable information from the available data can be found . as explained above , from the mathematical point of view , the internal structure of the object is described by an unknown function @xmath3 ( density ) , that is connected via an operator to some measured quantity @xmath14 ( tomograms ) . the tomographic reconstruction problem can be stated as follows : for given data @xmath14 , the task is to find @xmath3 from the operator equation @xmath15 . there are many problems related to the implementation of effective tomographic techniques due to the instability of the reconstruction process . there are two principal reasons of this instability . the first one is the ill - posedness of the reconstruction problem : in order to obtain a satisfactory estimate of the unknown function it is necessary an extremely precise knowledge of its tomograms , which is in general physically unattainable @xcite . the second reason is the discrete and possibly imperfect nature of data that allows to obtain only an approximation of the unknown function . the first question is whether a partial information still determines the function uniquely . a negative answer is given by a theorem of smith , solomon and wagner @xcite , that states : `` a function @xmath3 with compact support in the plane is uniquely determined by any infinite set , but by no finite set of its tomograms '' . therefore , it is clear that one has to abandon the request of uniqueness in the applications of tomography . thus , due to the ill - posedness of reconstruction problem and to the loss of uniqueness in the inversion process , a regularization method has to be introduced to stabilize the inversion . a powerful approach is the introduction of a mumford - shah ( ms ) functional , first introduced in a different context for image denoising and segmentation @xcite . the main motivation is that , in many practical applications , one is not only interested in the reconstruction of the density distribution @xmath3 , but also in the extraction of some specific features or patterns of the image . an example is the problem of the determination of the boundaries of inner organs . by minimizing the ms functional , one can find not only ( an approximation of ) the function but also its sharp contours . very recently a ms functional for applications to tomography has been introduced in the literature @xcite . some preliminary results in this context are already available but there are also many interesting open problems and promising results in this direction , as we will try to explain in the second part of this article . the article is organized as follows . section [ sec : radon ] contains a short introduction to the radon transform , its dual map and the inversion formula . section [ sec : ill ] is devoted to a brief discussion on the ill - posedness of the tomographic reconstruction and to the introduction of regularization methods . in section [ sec : ms ] a ms functional is applied to tomography as a regularization method . in particular , in subsection [ subsec : ms ] the piecewise constant model and known results are discussed together with a short list of some interesting open problems . finally , in section [ sec:3dinterpretation ] we present an electrostatic interpretation of the regularization method based on the ms functional , which motivates us to introduce an improved regularization method , based on the blake - zisserman functional @xcite , as a relaxed version of the previous one . consider a body in the plane @xmath1 , and consider a beam of particles ( neutrons , electrons , x - rays , etc . ) emitted by a source . assume that the initial intensity of the beam is @xmath16 . when the particles pass through the body they are absorbed or scattered and the intensity of the beam traversing a length @xmath17 decreases by an amount proportional to the density of the body @xmath18 , namely @xmath19 so that @xmath20 a detector placed at the exit of the body measures the final intensity @xmath21 and then from @xmath22 one can record the value of the density @xmath18 integrated on a line . if another ray with a different direction is considered , with the same procedure one obtains the value of the integral of the density on that line . the mathematical model of the above setup is the following : given a smooth function @xmath23 on the plane , @xmath24 , and a line @xmath12 , consider its tomogram , given by @xmath25 where @xmath26 is the euclidean measure on the line @xmath12 . in this way , we have defined an operator @xmath27 that maps a smooth function @xmath3 on the plane @xmath1 into a function @xmath10 on @xmath28 , the manifold of the lines in @xmath1 . we ask the following question : if we know the family of tomograms @xmath29 , can we reconstruct the density function @xmath3 ? the answer is affirmative and in the following we will see how to obtain this result . let us generalize the above definitions to the case of an @xmath7-dimensional space . let @xmath3 be a function defined on @xmath9 , integrable on each hyperplane in @xmath9 and let @xmath30 be the manifold of all hyperplanes in @xmath9 . the radon transform of @xmath3 is defined by eq . ( [ sharp lambda ] ) , where @xmath26 is the euclidean measure on the hyperplane @xmath12 . thus we have an operator @xmath27 , the _ radon transform _ , that maps a function @xmath3 on @xmath9 into a function @xmath10 on @xmath31 , namely @xmath32 . its dual transform , also called _ back projection operator _ , @xmath33 associates to a function @xmath14 on @xmath31 the function @xmath34 on @xmath9 given by @xmath35 where @xmath36 is the unique probability measure on the compact set @xmath37 which is invariant under the group of rotations around @xmath38 . using its signed distance @xmath39 from the origin and a unit vector @xmath40 perpendicular to @xmath12.,scaledwidth=45.0% ] let us consider the following covering of @xmath30 @xmath41 where @xmath42 is the unit sphere in @xmath9 . thus , the equation of the hyperplane @xmath12 is @xmath43 with @xmath44 denoting the euclidean inner product of @xmath45 . [ fig : radon nd ] . observe that the pairs @xmath46 are mapped into the same hyperplane @xmath47 . therefore ( [ double covering ] ) is a double covering of @xmath30 . thus @xmath30 has a canonical manifold structure with respect to which this covering mapping is differentiable . we identify continuous ( differentiable ) functions @xmath14 on @xmath30 with continuous ( differentiable ) functions @xmath14 on @xmath48 satisfying @xmath49 . we will momentarily work in the schwartz space @xmath50 of complex - valued rapidly decreasing functions on @xmath9 . in analogy with @xmath50 we define @xmath51 as the space of @xmath52 functions @xmath14 on @xmath48 which for any integers @xmath53 , any multiindex @xmath54 , and any differential operator @xmath55 on @xmath42 satisfy @xmath56 the space @xmath57 is then defined as the set of @xmath58 satisfying @xmath59 . now we want to obtain an inversion formula , namely we want to prove that one can recover a function @xmath3 on @xmath9 from the knowledge of its radon transform . in order to get this result we need a preliminary lemma , whose proof can be found in @xcite , which suggests an interesting physical interpretation . [ sharp flat ] let @xmath60 and @xmath61 , @xmath62 , @xmath63 . then @xmath64 where @xmath65 depends only on the dimension @xmath7 , and @xmath66 denotes the convolution product , @xmath67 is the potential at @xmath38 generated by the charge distribution @xmath3.,scaledwidth=45.0% ] a physical interpretation of lemma [ sharp flat ] is the following : if @xmath3 is a charge distribution , then the potential at the point @xmath38 generated by that charge is exactly @xmath68 , see fig . [ fig : potential ] . notice , however , that the potential of a point charge scales always as the inverse distance _ independently _ of the dimension @xmath7 , and thus it is coulomb only for @xmath69 . the only dependence on @xmath7 is in the strength of the elementary charge @xmath65 . this fact is crucial : indeed , the associated poisson equation involves an @xmath7-dependent ( fractional ) power of the laplacian , which appears in the inversion formula for the radon transform . [ thm : inversioneformula ] let @xmath60 . then @xmath70 where @xmath71 , with @xmath72 , is a pseudodifferential operator whose action is @xmath73 where @xmath74 is the fourier transform of @xmath3 , @xmath75 the proof of theorem [ thm : inversioneformula ] can be found in @xcite . equation ( [ inversion formula0 ] ) says that , modulo the final action of @xmath76 , the function @xmath3 can be recovered from its radon transform @xmath10 by the application of the dual mapping @xmath77 : first one integrates over the set of points in a hyperplane and then one integrates over the set of hyperplanes passing through a given point . explicitly we get @xmath78 which has the following remarkable interpretation . note that if one fixes a direction @xmath79 , then the function @xmath80 is constant on each plane perpendicular to @xmath40 , i.e. it is a ( generalized ) plane wave . therefore , eq . ( [ inversion formula ] ) gives a representation of @xmath3 in terms of a continuous superposition of plane waves . a well - known analogous decomposition is given by fourier transform . when @xmath81 , one recovers the inversion formula ( [ eq : radoninversion ] ) originally found by radon @xcite . we have defined the radon transform of any function @xmath82 as @xmath10 . the following theorem @xcite contains the characterization of the range of the radon linear operator @xmath27 and the extension of @xmath27 to the space of square integrable functions @xmath83 . [ th : radonbijection ] the radon transform @xmath27 is a linear one - to - one mapping of @xmath50 onto @xmath84 , where the space @xmath84 is defined as follows : @xmath85 if and only if @xmath86 and for any integer @xmath87 the integral @xmath88 is a homogeneous polynomial of degree @xmath89 in @xmath90 . moreover , the radon operator @xmath27 can be extended to a continuous operator from @xmath83 and @xmath91 . in medical imaging , computerized tomography is a widely used technique for the determination of the density @xmath3 of a sample from measurements of the attenuation of x - ray beams sent through the material along different angles and offsets . the measured data @xmath14 are connected to the density @xmath3 via the radon transform @xmath92 . to compute the density distribution @xmath3 the equation @xmath93 has to be inverted . unfortunately it is a well known fact that @xmath27 is not continuously invertible on @xmath91 @xcite , and this imply that the problem of inversion is ill - posed . for this reason , regularization methods have to be introduced to stabilize the inversion in the presence of data noise . we discuss ill - posed problems only in the framework of linear problems in hilbert spaces @xcite . let @xmath94 be hilbert spaces and let @xmath95 be a linear bounded operator from @xmath96 into @xmath97 . the problem @xmath98 is called well - posed by hadamard ( 1932 ) if it is uniquely solvable for each @xmath99 and if the solution depends continuously on @xmath14 . otherwise , ( [ prob : inverseproblem ] ) is called ill - posed . this means that for an ill - posed problem the operator @xmath100 either does not exist , or is not defined on all of @xmath97 , or is not continuous . the practical difficulty with an ill - posed problem is that even if it is solvable , the solution of @xmath101 need not be close to the solution of @xmath102 if @xmath103 is close to @xmath14 . in general @xmath100 is not a continuous operator . to restore continuity we introduce the notion of a regularization of @xmath100 . this is a family @xmath104 of linear continuous operators @xmath105 which are defined on all @xmath97 and for which @xmath106 on the domain of @xmath100 . obviously @xmath107 as @xmath108 if @xmath100 is not bounded . with the help of a regularization we can solve ( [ prob : inverseproblem ] ) approximately in the following sense . let @xmath109 be an approximation to @xmath14 such that @xmath110 . let @xmath111 be such that , as @xmath112 , @xmath113 then , as @xmath112 , @xmath114 hence , @xmath115 is close to @xmath116 if @xmath103 is close to @xmath14 . the number @xmath117 is called a _ regularization parameter_. determining a good regularization parameter is one of the crucial points in the application of regularization methods . there are several methods for constructing a regularization as the truncated singular value decomposition , the method of tikhonov - phillips or some iterative methods @xcite . in the following section we present a regularization method based on the minimization of a mumford - shah type functional . in many practical applications one is not only interested in the reconstruction of the density distribution @xmath3 but also in the extraction of some specific features within the image which represents the density distribution of the sample . for example , the planning of surgery might require the determination of the boundaries of inner organs like liver or lung or the separation of cancerous and healthy tissue . segmenting a digital image means finding its _ homogeneous regions _ and its _ edges _ , or _ boundaries_. of course , the homogeneous regions are supposed to correspond to meaningful parts of objects in the real world , and the edges to their apparent contours . the mumford - shah variational model is one of the principal models of image segmentation . it defines the segmentation problem as a joint smoothing / edge detection problem : given an image @xmath118 , one seeks simultaneously a `` piecewise smoothed image '' @xmath119 with a set @xmath120 of abrupt discontinuities , the `` edges '' of @xmath14 . the original mumford - shah functional @xcite , is the following : @xmath121 where * @xmath122 is an open set ( _ screen _ ) ; * @xmath123 is a closed set ( _ set of edges _ ) ; * @xmath124 ( _ cartoon _ ) ; * @xmath125 denotes the distributional gradient of @xmath126 ; * @xmath127 is the datum ( _ digital image _ ) ; * @xmath128 are parameters ( _ tuning parameters _ ) ; * @xmath129 denotes the @xmath11-dimensional hausdorff measure . the squared @xmath130 distance in ( [ defn : jms ] ) plays the role of a fidelity term : it imposes that the cartoon @xmath126 approximate the image @xmath14 . the second term in the functional imposes that the cartoon @xmath126 be piecewise smooth outside the edge set @xmath120 . in other word this term favors sharp contours rather than zones where a thin layer of gray is used to pass smoothly from white to black or viceversa . finally the third term in the functional imposes that the contour @xmath120 be `` small '' and as smooth as possible . what is expected from the minimization of this functional is a sketchy , cartoon - like version of the given image together with its contours . see fig . [ fig : cartooneye ] . the minimization of the @xmath131 functional represents a compromise between accuracy and segmentation . the compromise depends on the tuning parameters @xmath132 and @xmath133 which have different roles . the parameter @xmath132 determines how much the cartoon @xmath126 can vary , if @xmath132 is small some variations of @xmath126 are allowed , while as @xmath132 increases @xmath126 tends to be a piecewise constant function . the parameter @xmath133 represents a scale parameter of the functional and measure the amount of contours : if @xmath133 is small , a lot of edges are allowed and we get a fine segmentation . as @xmath133 increases , the segmentation gets coarser . for more details on the model see the original paper @xcite , and the book @xcite . ) . center : contours of the image in the mumford - shah model ( edges @xmath120 ) . right : piecewise smooth function approximating the image ( cartoon @xmath126 ) @xcite.,scaledwidth=45.0% ] the minimization of the @xmath134 functional in ( [ defn : jms ] ) is performed among the admissible pairs @xmath135 such that @xmath120 is closed and @xmath136 . it is worth noticing that in this model there are two unknowns : a scalar function @xmath126 and the set @xmath120 of its discontinuities . for this reason this category of problems is often called `` free discontinuities problem '' . existence of minimizers of the @xmath134 functional in ( [ defn : jms ] ) was proven by de giorgi , carriero , leaci in @xcite in the framework of bounded variation functions without cantor part ( space sbv ) introduced by ambrosio and de giorgi in @xcite . further regularity properties for optimal segmentation in the mumford - shah model were shown in @xcite . here we present a variation of the ms functional , adapted to the inversion problem of the radon transform . more precisely , we consider a regularization method that quantifies the edge sets together with images , i.e. a procedure that gives simultaneously a reconstruction and a segmentation of @xmath3 ( assumed to be supported in @xmath137 ) directly from the measured tomograms @xmath14 , based on the minimization of the mumford - shah type functional @xmath138 the only difference between the functionals @xmath134 and @xmath139 is the first term , i.e. the fidelity term , that ensures that the reconstruction for @xmath3 is close enough to a solution of the equation @xmath140 , whereas the other terms play exactly the same role explained for the functional @xmath134 . as explained above , in addition to the reconstruction of the density @xmath3 , we are interested in the reconstruction of its singularity set @xmath120 , i.e. the set of points where the solution @xmath3 is discontinuous . the main difference with respect to the standard mumford - shah functional ( [ defn : jms ] ) is that we have to translate the information about the set of sharp discontinuities of @xmath14 ( and hence on the space of the radon transform ) into information about the strong discontinuities of @xmath3 . here we will review the results obtained by ramlau and ring @xcite concerning the minimization of ( [ def : msfunctional ] ) restricted to piecewise constant functions @xmath3 , and then consider some interesting open problems . for medical applications , it is often a good approximation to restrict the reconstruction to densities @xmath3 that are constant with respect to a partition of the body , as the tissues of inner organs , bones , or muscles have approximately constant density . we introduce the space @xmath141 as the space of piecewise constant functions that attain at most @xmath142 different function values , where @xmath55 is an open and bounded subset of @xmath9 . in other words , each @xmath143 is a linear combination of @xmath142 characteristic functions @xmath144 of sets @xmath145 which satisfy @xmath146 we assume that the @xmath147 s are open relatively to @xmath55 and we set @xmath148 for the boundary of @xmath147 with respect to the topology relative to the open domain @xmath55 . in this situation the edge set will be given by the union of the boundaries of @xmath147 s . for technical reasons it is necessary to assume a _ nondegeneracy _ condition on the admissible partitions of @xmath55 : @xmath149 for some @xmath150 , for all @xmath151 , where @xmath152 denotes the lebesgue measure on @xmath9 . it turns out to be convenient to split the information encoded in a typical function , @xmath153 into a `` geometrical '' part described by the @xmath142-tuple of pairwise disjoint sets @xmath154 which cover @xmath55 up to a set of measure zero and a `` functional '' part given by the @xmath142-tuple of values @xmath155 . we also use the notation @xmath156 , for the boundaries @xmath157 of @xmath147 . as usual when dealing with inverse problems , we have to assume that the data @xmath14 are not exactly known , but that we are only given noisy measured tomograms @xmath103 of a ( hypothetical ) exact data set @xmath14 with @xmath158 . if we restrict the functional ( [ def : msfunctional ] ) to functions in @xmath141 we obtain that the second term ( involving the derivatives of @xmath3 ) disappears , therefore it remains to minimize the functional @xmath159 over @xmath141 , with respect to the functional variable @xmath160 ( a vector of @xmath142 components ) and the geometric variable @xmath161 ( a partition of the domain @xmath55 with at most @xmath142 distinct regions satisfying the non degeneracy condition ( [ nondegeneracy - cond ] ) ) . so the problem is to find @xmath162 such that @xmath163 where @xmath164 it is clear that @xmath165 will depend on the regularization parameter @xmath133 and on the error level @xmath166 . now we can state the results concerning the functional @xmath167 in ( [ functional : jbeta ] ) . there are several technical details necessary for the precise statement and proof of the theorems , for which we refer to the original paper @xcite . here we will give a simplified version of the theorems with the purpose of explain the main goal , without too many technical details . the first result is about the existence of minimizers of the functional @xmath167 in ( [ functional : jbeta ] ) . for all @xmath168 there exists a minimizer @xmath169 of the functional @xmath167 in ( [ functional : jbeta ] ) , with @xmath170 . the second result regards the stable dependence of the minimizers of the functional @xmath167 in ( [ functional : jbeta ] ) on the error level @xmath166 . let @xmath171 be a sequence of functions in @xmath172 and let @xmath168 . for all @xmath173 , let @xmath174 denote the minimizers of the functional @xmath167 with initial data @xmath175 . if @xmath176 in @xmath172 , as @xmath177 , then there exists a subsequence of @xmath174 such that @xmath178 as @xmath179 , and @xmath180 is a minimizer of @xmath167 with initial data @xmath103 . moreover , the limit of each convergent subsequence of @xmath174 is a minimizer of @xmath167 with initial data @xmath103 . finally the last theorem is a regularization result . let @xmath181 be given , @xmath182 and let @xmath183 . assume we have noisy data @xmath168 with @xmath184 . let us choose the parameter @xmath185 satisfying the conditions @xmath186 and @xmath187 as @xmath112 . for any sequence @xmath188 , let @xmath189 denote the minimizers of the functional @xmath190 with initial data @xmath175 and regularization parameter @xmath191 . then there exists a convergent subsequence of @xmath189 . moreover , for every convergent subsequence with limit @xmath192 the function @xmath193 is a solution of the equation @xmath194 with a minimal perimeter . moreover if @xmath195 is the unique solution of this equation then the whole sequence converges @xmath196 when @xmath197 . finally , let us list some open problems in this context : * is the nondegeneracy condition ( [ nondegeneracy - cond ] ) necessary ? * can one find an a priori optimal value for the number @xmath142 of different values ? * is it possible to give an a priori estimate on the @xmath198-norm of the solution ( maximum principle ) ? * and finally , it would be very important for applications to prove the existence of minimizers of the functional @xmath139 not restricted to piecewise constant functions @xmath3 . we observe that all these problems are quite natural , and have been completely solved in the case of the standard mumford - shah functional @xmath134 in ( [ defn : jms ] ) , see e.g. @xcite . in this section we restrict our attention to the @xmath199-dimensional case . we propose an electrostatic interpretation of the regularization method based on the functional @xmath139 discussed in the previous section . the intent is to give a physical explanation of the fidelity term @xmath200 in the functional ( [ def : msfunctional ] ) , that provide the intuition for an improved regularization method . for @xmath69 , the inversion formula ( [ inversion formula0 ] ) and the electrostatic identity ( [ eq : electid ] ) particularize , respectively , as follows : for all @xmath201 one gets @xmath202 and @xmath203 where @xmath61 and @xmath204 is a constant . we present two preliminary lemmas . [ lemma : normrf ] for all real valued @xmath201 one has @xmath205 we know that @xmath206 , therefore @xmath207 ( x)\ , { \mathcal{i}}({\mathcal{r}}{f } ) ( x ) \ ; \textrm{d}x \nonumber \\ & & = \frac{1}{2(2\pi)^2 } \int_{{\mathbb{r}}^3}| \nabla { \mathcal{i}}({\mathcal{r}}{f } ) ( x)|^2\ ; \textrm{d}x . \nonumber\end{aligned}\ ] ] [ lemma : electricfield ] for all real valued @xmath201 define @xmath208 and @xmath209 then @xmath210 @xmath211 where we used the inversion formula ( [ inversionformula_again ] ) . now we consider a measured tomogram @xmath212 and let us assume that @xmath213 for some @xmath214 . by lemma [ lemma : normrf]-[lemma : electricfield ] it follows immediately that the fidelity term @xmath215 can be rewritten as follows : @xmath216 where @xmath217 are the corresponding electric fields , while @xmath218 are the corresponding potentials . with respect to the standard mumford - shah functional @xmath134 in ( [ defn : jms ] ) , the new fidelity term in the functional @xmath139 in ( [ def : msfunctional ] ) controls the distance between the radon transform of @xmath3 and the tomographic data @xmath14 . the relevant difference with respect to the original functional is that the function @xmath3 and its radon transform @xmath10 are defined in different spaces . let us try to interpret the fidelity term @xmath219 from a physical point of view . a key ingredient for this goal is the electrostatics formulation of the radon transform . this formulation can be summarized as follows : if we consider , in dimension @xmath199 , a function @xmath3 , we can think at it as a charge distribution density ; if we apply to @xmath3 first the radon operator @xmath27 and then its adjoint @xmath77 we obtain , up to a constant , the electrostatic potential generated by the charge distribution @xmath3 . this formulation can be stated in any dimension @xmath7 : the difference with general potential theory in dimension @xmath7 is that , in tomography , the potential produced by a point charge always scales like @xmath220 , which is the case of electrostatic potential only in dimension @xmath199 . from the electrostatic formulation of the radon transform we can prove that the fidelity term in the functional @xmath139 actually imposes that the electric field produced by the charge distribution @xmath3 must be close to the `` measured electric field '' . therefore we conclude that the term @xmath221 is a fidelity term in this weaker sense . using this property based on the electrostatic interpretation of the tomographic reconstruction , we can try to minimize some appropriate functionals in the new variables @xmath222 ( electric field ) or @xmath223 ( electric potential ) and then compute the corresponding @xmath3 ( charge density ) . we manipulate the functional @xmath139 as follows : @xmath224 where @xmath225 is a new functional depending on a vector function @xmath222 and on a set @xmath120 , and we used the fact that @xmath226 , since @xmath222 is conservative . we observe that the functional @xmath227 is a second order functional for a vector field @xmath222 in which appears the measure of the set @xmath120 that is the set of discontinuities of @xmath3 and thus is the set of discontinuities of @xmath228 . in the functional @xmath225 we recognize some similarities with a famous second - order free - discontinuity problem : the blake - zisserman model . this model is based on the minimization of the blake - zisserman functional @xmath229 among admissible triplets @xmath230 , where * @xmath122 is an open set ; * @xmath231 are closed sets ; * @xmath232 is the set of discontinuities of @xmath233 ( jump set ) , and @xmath234 the the set of discontinuities of @xmath235 ( crease set ) ; * @xmath236 , @xmath237 is a scalar function ; * @xmath238 denotes the distributional laplacian of @xmath233 ; * @xmath239 is the datum ( grey intensity levels of the given image ) ; * @xmath240 are parameters ; * @xmath129 denotes the @xmath11-dimensional hausdorff measure . the blake - zisserman functional allows a more precise segmentation than the mumford - shah functional in the sense that also the curvature of the edges of the original picture is approximated . on the other hand , minimizers may not always exist , depending on the values of the parameters @xmath241 and on the summability assumption on @xmath242 . we refer to @xcite for motivation and analysis of variational approach to image segmentation and digital image processing . in particular see @xcite for existence of minimizer results and @xcite for a counterexample to existence and @xcite for results concerning the regularity of minimizers . equation ( [ eqnaeeay_manipulation ] ) implies that the functional @xmath139 can be rewritten in terms of the vector field @xmath222 and of the discontinuities set of @xmath228 , i.e. the set of creases of @xmath222 , using the terminology of the blake - zisserman model . the fact that in the functional @xmath225 the discontinuities set of @xmath222 is not present depends on the fact that we are assuming that the charge density @xmath3 in the functional @xmath139 do not concentrate on surfaces or on lines . if we admit concentrated charge layers we can consider the blake - zisserman model for the vector function @xmath222 as a relaxed version of the mumford - shah model for the charge @xmath3 . in other words we propose to investigate the connections between minimizers of @xmath139 and minimizers of the higher order functional @xmath243 : @xmath244 with the additional constraint @xmath245 . the main advantage of this approach is that the functional @xmath243 is a purely differential functional , while the functional @xmath139 is an integro - differential one . we expect that some results about the blake - zisserman model that could be rephrased into tomographic terms would provide immediately new results in tomography . conversely all the peculiar tomographic features as the intrinsic vector nature of the variable @xmath222 , the fact that its support can not be bounded and the extra - constraint @xmath246 , motivate new research directions in the study of free - discontinuities problems . for example , an interesting result in this context would be the determination of a good hypothesis on the datum @xmath247 that ensure that the charge density @xmath3 do not concentrate . we conclude this section with some comments : * we proved that the measured data @xmath14 are actually the measured electric field produced by the unknown charge density , so the term @xmath200 in the functional is a fidelity term in a weak sense . * the problem of the reconstruction of the charge can be rephrased into a reconstruction problem for the electric field . the electric field is an irrotational vector field , so the new minimization problem is actually a constrained minimization . in order to avoid this constraint one could reformulate the reconstruction problem in terms of the electric potential @xmath223 ( @xmath248 ) obtaining a third - order functional in which the fidelity term is @xmath249 where the potentials are given by ( [ eq : potentials ] ) . * all this considerations hold true in dimension @xmath199 . in a generic dimension @xmath250 the situation is quite different because the inversion formula for the radon transform involves a ( possibly fractional ) power of the laplacian . in this case the electrostatic description of tomography given in this section fails . in order to restore it , it is necessary to consider another radon - type transform which involves integrals of @xmath3 over linear manifolds with codimension @xmath251 such that @xmath252 , i.e. @xmath253 , see e.g. @xcite . we thank g. devillanova , g. florio and f. maddalena for for helpful discussions . this work was supported by `` fondazione cassa di risparmio di puglia '' and by the italian national group of mathematical physics ( gnfm - indam ) .

in this article we present a review of the radon transform and the instability of the tomographic reconstruction process . we show some new mathematical results in tomography obtained by a variational formulation of the reconstruction problem based on the minimization of a mumford - shah type functional . finally , we exhibit a physical interpretation of this new technique and discuss some possible generalizations . i _ keywords _ : radon transform ; integral geometry ; image segmentation ; calculus of variations .

1501.07385

reciprocity , which was first found by lorentz at the end of 19th century , has a long history@xcite and has been derived in several formalisms . there are two typical reciprocal configurations in optical responses as shown in fig . the configurations in figs . [ fig1](a ) and [ fig1](b ) are transmission reciprocal and those in figs . [ fig1](a ) and [ fig1](c ) are reflection reciprocal . as shown in fig . [ fig1 ] , we denote transmittance by @xmath0 and reflectance by @xmath1 ; the suffice k and @xmath2 stand for incident wavenumber vector and angle , respectively . the reciprocal configurations are obtained by symmetry operations on the incident light of the wavenumber vector : ( @xmath3 ) or ( @xmath4 ) . reciprocity on transmission means that @xmath5 , and that on reflection is expressed as @xmath6 , which is not intuitively obvious and is frequently surprising to students . the most general proof was published by petit in 1980,@xcite where reciprocal reflection as shown in fig . [ fig1 ] is derived for asymmetric gratings such as an echelette grating . on the basis of the reciprocal relation for the solutions of the helmholtz equation , the proof showed that reciprocal reflection holds for periodic objects irrespective of absorption . it seems difficult to apply the proof to transmission because it would be necessary to construct solutions of maxwell equations that satisfy the boundary conditions at the interfaces of the incident , grating , and transmitted layers . the history of the literature on reciprocal optical responses has been reviewed in ref . since the 1950s , scattering problems regarding light , elementary particles , and so on have been addressed by using scattering matrix ( s - matrix ) . in the studies employing the s - matrix , it is assumed that there is no absorption by the object . the assumption leads to the unitarity of the s - matrix and makes it possible to prove reciprocity . the reciprocal reflection of lossless objects was verified in this formalism.@xcite in this paper we present a simple , direct , and general derivation of the reciprocal optical responses for transmission and reflection relying only on classical electrodynamics . we start from the reciprocal theorem described in sec . [ thm ] and derive the equation for zeroth order transmission and reflection coefficients in sec . [ proof ] . the equation is essential to the reciprocity . a numerical and experimental example of reciprocity is presented in sec . [ example ] . the limitation and break down of reciprocal optical responses are also discussed . the reciprocal theorem has been proved in various fields , such as statistical mechanics , quantum mechanics , and electromagnetism.@xcite here we introduce the theorem for electromagnetism . when two currents exist as in fig . [ fig2 ] and the induced electromagnetic ( em ) waves travel in linear and locally responding media in which @xmath7 and @xmath8 , then @xmath9 equation is the reciprocal theorem in electromagnetism . the proof shown in ref . exploits plane waves and is straightforward . equation ( [ reci ] ) is valid even for media with losses . the integrands take non - zero values at the position @xmath10 where currents exist , that is , @xmath11 . the theorem indicates the reciprocity between the two current sources @xmath12 ( @xmath13 ) and the induced em waves @xmath14 which are observed at the position of the other source @xmath15 ( @xmath16 ) . in this section , we apply the reciprocal theorem to optical responses in both transmission and reflection configurations . first , we define the notation used in the calculations of the integrals in eq . ( [ reci ] ) . an electric dipole oscillating at the frequency @xmath17 emits dipole radiation , which is detected in the far field . when a small dipole @xmath18 along the @xmath19 axis is located at the origin , it is written as @xmath20 and @xmath21 , where @xmath22 denotes the unit vector along the @xmath19 axis and @xmath23 the magnitude of the dipole . the dipole in vacuum emits radiation , which in the far field is @xmath24 where polar coordinates ( @xmath25 , @xmath2 , @xmath26 ) are used , a unit vector is given by @xmath27 , and @xmath28 . because the dipole @xmath18 is defined by @xmath29 and conservation of charge density is given by @xmath30 , we obtain the current @xmath31 associated with the dipole @xmath18 : @xmath32 consider two arrays of @xmath33 dipoles ( long but finite ) in the @xmath34 plane as shown in fig . the two arrays have the same length , and the directions are specified by normalized vectors @xmath35 ( @xmath36 ) and @xmath37 . in this case , the current is @xmath38 . if the dipoles coherently oscillate with the same phase , then the emitted electric fields are superimposed and form a wave front at a position far from the array in the @xmath34 plane as drawn in fig . [ fig3 ] . the electric field vector of the wave front , @xmath39 , satisfies @xmath40 and travels with wavenumber vector @xmath41 . thus , if we place the dipole arrays far enough from the object , the induced em waves become slowly decaying incident plane waves in the @xmath34 plane to a good approximation . the arrays of dipoles have to be long enough to form the plane wave . for the transmission configuration , we calculate @xmath42 ( @xmath43 and @xmath44 ) . figure [ fig3 ] shows a typical transmission configuration , which includes an arbitrary periodic object asymmetric along the @xmath19 axis . the relation between the current @xmath12 , the direction @xmath35 of the dipole , and the wavenumber vector @xmath41 of the wave front is summarized as @xmath45 and @xmath46 . it is convenient to expand the electric field into a fourier series for the calculation of periodic sources : @xmath47 where @xmath48 is the fourier coefficient of @xmath49 , @xmath50 ( @xmath51 ) , and @xmath52 is the periodicity of the object along the @xmath53 axis ( see fig . [ fig3 ] ) . the @xmath19 component is expressed in homogeneous media in vacuum as @xmath54 , where the signs correspond to the directions along the @xmath19 axis . when the dipole array is composed of sufficiently small and numerous dipoles , the integration can be calculated to good accuracy as @xmath55 where @xmath56 . to ensure that the integration is proportional to @xmath57 , the array of dipoles has to be longer than @xmath58 : @xmath59 where @xmath60 is the least common multiple of the diffraction channels which are open at the frequency @xmath17 . this condition would usually be satisfied when @xmath39 forms a plane wave . by permutating 1 and 2 in eq . ( [ j1e2 ] ) , we obtain @xmath61 . equation ( [ j1e2 ] ) and the reciprocal theorem in eq . ( [ reci ] ) lead to the equation @xmath62 each electric vector @xmath63 ( @xmath36 ) is observed at the position @xmath10 where there is another current @xmath64 ( @xmath65 ) . the integral in eq . ( [ reci ] ) is reduced to eq . ( [ j1e2 ] ) which is expressed only by the zeroth components of the transmitted electric field . the reciprocity is thus independent of higher order harmonics , which are responsible for the modulated em fields in structured objects . when there is no periodic object in fig . [ fig3 ] , a similar relation holds : @xmath66 the transmittance @xmath67 is given by @xmath68 from eqs . ( [ e_reci])([t_reci ] ) , we finally reach the reciprocal relation @xmath69 . the feature of the proof that @xmath69 is independent of the detailed evaluation of @xmath63 and therefore makes the proof simple and general . the proof can be extended to two - dimensional periodic structure by replacing the one - dimensional periodic structure in fig . [ fig3 ] by two - dimensional one . although we have considered periodic objects , the proof can also be extended to non - periodic objects . to do this extension , eq . ( [ e_expand ] ) has to be expressed in the general form @xmath70 , and a more detailed calculation for @xmath71 is required . reciprocity for transmission thus holds irrespective of absorption , diffraction , and scattering by objects . in fig . [ fig3 ] the induced electric fields @xmath14 are polarized in the @xmath34 plane . the polarization is called tm polarization in the terminology of waveguide theory and is also often called @xmath72 polarization . for te polarization ( which is often called @xmath73 polarization ) for which @xmath14 has a polarization parallel to the @xmath74 axis , the proof is similar to what we have described except that the dipoles are aligned along the @xmath74 axis . reciprocal reflection is also shown in a similar way . the configuration is depicted in fig . the two sources have to be located to satisfy the mirror symmetry about the @xmath19 axis . the calculation of @xmath71 leads to the reciprocal relation for reflectance @xmath75 . note that @xmath76 in eq . ( [ e0_reci ] ) has to be evaluated by replacing the periodic object by a perfect mirror . an example of reciprocal optical response is shown here . figure [ fig5](a ) displays the structure of the sample and reciprocal transmission configuration . the sample consists of periodic grooves etched in metallic films of au and cr on a quartz substrate . the periodicity is 1200 nm , as indicated by the dotted lines in fig . [ fig5](a ) . the unit cell has the structure of au : air : au : air = 3:1:4:5 . the thickness of au , cr , and quartz is 40 nm , 5 nm , and 1 mm , respectively . the structure is obviously asymmetric about the @xmath19 axis . the profile was modeled from an afm image of the fabricated sample . figure [ fig5](b ) shows our numerical results . the incident light has @xmath77 and tm polarization ( the electric vector is in the @xmath34 plane ) . the numerical calculation was done with an improved s - matrix method@xcite the permittivities of gold and chromium were taken from refs . and ; the permittivity of quartz is well known to be 2.13 . in the numerical calculation , the incident light is taken to be a plane wave , and harmonics up to @xmath78 in eq . ( [ e_expand ] ) were used , which is enough to obtain accurate optical responses . the result indicates that transmission spectra ( lower solid line ) are numerically the same in the reciprocal configurations , while reflection ( upper solid line ) and absorption ( dotted line ) spectra show a definite difference . the absorption is plotted along the left axis . the difference implies that surface excitations are different on each side and absorb different numbers of photons . nonetheless , the transmission spectra are the same for incident wavenumber vectors @xmath79 and @xmath80 . experimental transmission spectra are shown in fig . [ fig5](c ) and are consistent within experimental error . reciprocity is thus confirmed both numerically and experimentally . there have a few experiments on reciprocal transmission ( see references in ref . ) . in comparison with these results , fig . [ fig5](c ) shows the excellent agreement of reciprocal transmission and is the best available experimental evidence supporting reciprocity . we note that transmission spectra in figs . [ fig5](b ) and [ fig5](c ) agree quantitatively above 700 nm . on the other hand , they show a qualitative discrepancy below 700 nm . the result could come from the difference between the modeled profile in fig . [ fig5](a ) and the actual profile of the sample . the dip at 660 nm stems from a surface plasmon at the metal - air interface , so that the measured transmission spectra would be affected significantly by the surface roughness and the deviation from the modeled structure . as described in sec . [ thm ] , the reciprocal theorem assumes that all media are linear and show local response . logically , it can happen that the reciprocal optical responses do not hold for nonlinear or nonlocally responding media . reference discusses an explicit difference of the transmittance for a reciprocal configuration in a nonlinear optical crystal of knbo@xmath81:mn . the values of the transmittance deviate by a few tens of percent in the reciprocal configuration . the crystal has a second - order response such that @xmath82 . the break down of reciprocity comes from the nonlinearity . does reciprocity also break down in nonlocal media ? in nonlocal media the induction * d * is given by @xmath83 . although a general proof for this case has not been reported to our knowledge , it has been shown that reciprocity holds in a particular stratified structure composed of nonlocal media.@xcite in summary , we have presented an elementary and heuristic proof of the reciprocal optical responses for transmittance and reflectance . when the reciprocal theorem in eq . ( [ reci ] ) holds , the reciprocal relations come from geometrical configurations of light sources and observation points , and are independent of the details of the objects . transmission reciprocity has been confirmed both numerically and experimentally . we thank s. g. tikhodeev for discussions . one of us ( m. i. ) acknowledges the research foundation for opto - science and technology for financial support , and the information synergy center , tohoku university for their support of the numerical calculations . 20 r. j. potton,``reciprocity in optics , '' rep . . phys . * 67 * , 717754 ( 2004 ) . r. petit , `` a tutorial introduction , '' in _ electromagnetic theory of gratings _ , edited by r. petit ( springer , berlin , 1980 ) , p. 1 . n. a. gippius , s. g. tikhodeev , and t. ishihara , `` optical properties of photonic crystal slabs with an asymmetric unit cell , '' phys b * 72 * , 045138 - 17 ( 2005 ) . l. d. landau , e. m. lifshitz , and l. p. pitaevskii , _ electrodynamics of continuous media _ ( pergamon press , ny , 1984 ) , 2nd ed . j. d. jackson , _ classical electrodynamics _ ( john wiley & sons , nj , 1999 ) , 3rd ed . s. g. tikhodeev , a. l. yablinskii , e. a. muljarov , n. a. gippius , and t. ishihara , `` quasiguided modes and optical properties of photonic crystal slabs , '' phys . rev . b * 66 * , 045102 - 117 ( 2002 ) . l. li , `` use of fourier series in the analysis of discontinuous periodic structures , '' j. opt . a , * 13 * , 18701876 ( 1996 ) . p. b. johnson and r. w. christy , `` optical constants of the noble metals , '' phys . b * 6 * , 43704379 ( 1972 ) . p. b. johnson and r. w. christy , `` optical constants of transition metals : ti , v , cr , mn , fe , co , ni , and pd , '' phys . b * 9 * , 50565070 ( 1974 ) . m. z. zha and p. gnter , `` nonreciprocal optical transmission through photorefractive knbo@xmath81:mn , '' opt * 10 * , 184186 ( 1985 ) . h. ishihara , `` appearance of novel nonlinear optical response by control of excitonically resonant internal field , '' in _ proceedings of 5th symposium of japanese association for condensed matter photophysics _ ( 1994 ) , pp . 287281 ( in japanese ) . reciprocal configurations . ( a ) and ( b ) show reciprocal configurations for transmission . @xmath84 in ( a ) denotes transmittance for incident wavenumber vector @xmath85 . @xmath86 in ( b ) is defined similarly . the reciprocal relation is @xmath5 . ( a ) and ( c ) are reciprocal for reflection . @xmath87 in ( a ) is reflectance for incident wavenumber vector @xmath88 and @xmath89 in ( c ) for @xmath90 . the reciprocal relation is @xmath6.,width=283 ] schematic drawing of reciprocal configuration for transmission . the object has an arbitrary periodic structure , which is asymmetric along the @xmath19 axis . currents @xmath12 induce electric fields @xmath39 ( @xmath36).,width=245 ] schematic configuration for reciprocal reflection . the object has an arbitrary periodic structure , which consists of asymmetric unit cells . the currents @xmath12 yield electric fields @xmath39 ( @xmath36).,width=264 ] ( a ) schematic drawing of metallic grating profile modeled from afm images . the periodicity is 1200 nm . the dotted lines show the unit cells in which the ratio is au : air : au : air = 3:1:4:5 . the thickness of au , cr , and the quartz substrate is 40 nm , 5 nm , and 1 mm , respectively . ( b ) numerically calculated spectra for 10@xmath91 incidence of @xmath79 ( upper panel ) and @xmath80 ( lower panel ) of tm polarization . in both panels the reflectance ( upper solid line ) and absorption ( dotted line ) are plotted using the left axis , while the transmittance ( lower solid line ) uses the right axis . ( c ) measured transmittance spectra , corresponding to the transmittance spectra in ( b).,width=264 ]

we present an elementary proof concerning reciprocal transmittances and reflectances . the proof is direct , simple , and valid for the diverse objects that can be absorptive and induce diffraction and scattering , as long as the objects respond linearly and locally to electromagnetic waves . the proof enables students who understand the basics of classical electromagnetics to grasp the physical basis of reciprocal optical responses . in addition , we show an example to demonstrate reciprocal response numerically and experimentally .

physics0610143

turbulence is a key element of the dynamics of astrophysical fluids , including those of interstellar medium ( ism ) , clusters of galaxies and circumstellar regions . the realization of the importance of turbulence induces sweeping changes , for instance , in the paradigm of ism . it became clear , for instance , that turbulence affects substantially star formation , mixing of gas , transfer of heat . observationally it is known that the ism is turbulent on scales ranging from aus to kpc ( see armstrong et al 1995 , elmegreen & scalo 2004 ) , with an embedded magnetic field that influences almost all of its properties . the issue of quantitative descriptors that can characterize turbulence is not a trivial one ( see discussion in lazarian 1999 and ref . therein ) . one of the most widely used measures is the turbulence spectrum , which describes the distribution of turbulent fluctuations over scales . for instance , the famous kolmogorov model of incompressible turbulence predicts that the difference in velocities at different points in turbulent fluid increases on average with the separation between points as a cubic root of the separation , i.e. @xmath0 . in terms of direction - averaged energy spectrum this gives the famous kolmogorov scaling @xmath1 , where @xmath2 is a _ 3d _ energy spectrum defined as the fourier transform of the correlation function of velocity fluctuations @xmath3 . note that in this paper we use @xmath4 to denote averaging procedure . quantitative measures of turbulence , in particular , turbulence spectrum , became important recently also due to advances in the theory of mhd turbulence . as we know , astrophysical fluids are magnetized , which makes one believe that the correspondence should exist between astrophysical turbulence and mhd models of the phenomenon ( see vazquez - semadeni et al . 2000 , mac low & klessen 2004 , bellesteros - paredes et al . 2007 , mckee & ostriker 2007 and ref . therein ) . in fact , without observational testing , the application of theory of mhd turbulence to astrophysics could always be suspect . indeed , from the point of view of fluid mechanics astrophysical turbulence is characterized by huge reynolds numbers , @xmath5 , which is the inverse ratio of the eddy turnover time of a parcel of gas to the time required for viscous forces to slow it appreciably . for @xmath6 we expect gas to be turbulent and this is exactly what we observe in hi ( for hi @xmath7 ) . in fact , very high astrophysical @xmath5 and its magnetic counterpart magnetic reynolds number @xmath8 ( that can be as high as @xmath9 ) present a big problem for numerical simulations that can not possibly get even close to the astrophysically - motivated numbers . the currently available 3d simulations can have @xmath5 and @xmath8 up to @xmath10 . both scale as the size of the box to the first power , while the computational effort increases as the fourth power ( 3 coordinates + time ) , so the brute force approach can not begin to resolve the controversies related , for example , to ism turbulence . we expect that observational studies of turbulence velocity spectra will provide important insights into ism physics . even in the case of much more simple oceanic ( essentially incompressible ) turbulence , studies of spectra allowed to identify meaningful energy injection scales . in interstellar , intra - cluster medium , in addition to that , we expect to see variations of the spectral index arising from the variations of the degree of compressibility , magnetization , interaction of different interstellar phases etc . how to get the turbulence spectra from observations is a problem of a long standing . while density fluctuations are readily available through both interstellar scincillations and studies of column density maps , the more coveted velocity spectra have been difficult to obtain reliably until very recently . turbulence is associated with fluctuating velocities that cause fluctuations in the doppler shifts of emission and absorption lines . observations provide integrals of either emissivities or opacities , both proportional to the local densities , at each velocity along the line of sight . it is far from trivial to determine the properties of the underlying turbulence from the observed spectral line . centroids of velocity ( munch 1958 ) have been an accepted way of studying turbulence , although it was not clear to when and to what extend the measure really represents the velocity . recent studies ( lazarian & esquivel 2003 , henceforth le03 , esquivel & lazarian 2005 , ossenkopf et al 2006 , esquivel et al . 2007 ) have showed that the centroids are not a good measure for supersonic turbulence , which means that while the results obtained for hii regions ( odell & castaneda 1987 ) are probably ok , those for molecular clouds are unreliable . an important progress in analytical description of the relation between the spectra of turbulent velocities and the observable spectra of fluctuations of spectral intensity was obtained in lazarian & pogosyan ( 2000 , henceforth lp00 ) . this description paved way to two new techniques , which were later termed velocity channel analysis ( vca ) and velocity coordinate spectrum ( vcs ) . the techniques provide different ways of treating observational data in position - position - velocity ( ppv ) data cubes . while vca is based on the analysis of channel maps , which are the velocity slices of ppv cubes , the vcs analyses fluctuations along the velocity direction . if the slices have been used earlier for turbulence studies , although the relation between the spectrum of intensity fluctuations in the channel maps and the underlying turbulence spectrum was unknown , the analysis of the fluctuations along the velocity coordinate was initiated by the advent of the vcs theory . with the vca and the vcs one can relate both observations and simulations to _ turbulence theory_. for instance , the aforementioned turbulence indexes are very informative , e.g. velocity indexes steeper than the kolmogorov value of @xmath11 are likely to reflect formation of shocks , while shallower indexes may reflect scale - dependent suppression of cascading ( see beresnyak & lazarian 2006 and ref . therein ) . by associating the variations of the index with different regions of ism , e.g. with high or low star formation , one can get an important insight in the fundamental properties of ism turbulence , its origin , evolution and dissipation . the absorption of the emitted radiation was a concern of the observational studies of turbulence from the very start of the work in the field ( see discussion in munch 1999 ) . a quantitative study of the effects of the absorption was performed for the vca in lazarian & pogosyan ( 2004 , henceforth lp04 ) and for the vcs in lazarian & pogosyan ( 2006 , henceforth lp06 ) . in lp06 it was stressed that absorption lines themselves can be used to study turbulence . indeed , the vcs is a unique technique that does not require a spatial coverage to study fluctuations . therefore individual point sources sampling turbulent absorbing medium can be used to get the underlying turbulent spectra . however , lp06 discusses only the linear regime of absorption , i.e. when the absorption lines are not saturated . this substantially limits the applicability of the technique . for instance , for many optical and uv absorption lines , e.g. mg ii , sii , siii the measured spectra show saturation . this means that a part of the wealth of the unique data obtained e.g. by hst and other instruments can not be handled with the lp06 technique . the goal of this paper is to improve this situation . in particular , in what follows , we develop a theoretical description that allows to relate the fluctuations of the absorption line profiles and the underlying velocity spectra in the saturated regime . below , in 2 we describe the setting of the problem we address , while our main derivations are in 3 . the discussion of the new technique of turbulence study is provided in 4 , while the summary is in 5 . while in our earlier publications ( lp00 , lp04 , lp06 ) concentrated on emission line , in particular radio emission lines , e.g. hi and co , absorption lines present the researchers with well defined advantages . for instance , they allow to test turbulence with a pencil beam , suffer less from uncertainties in path length . in fact , studies of absorption features in the spectra of stars have proven useful in outlining the gross features of gas kinematics in milky way . recent advances in sensitivity and spectral resolution of spectrographs allow studies of turbulent motions . among the available techniques , vcs is the leading candidate to be used with absorption lines . indeed , it is only with extended sources that the either centroid or vca studies are possible . at the same time , vcs makes use not of the spatial , but frequency resolution . thus , potentially , turbulence studies are possible if absorption along a single line is available . in reality , information along a few lines of sight , as it shown in fig 1 is required to improve the statistical accuracy of the measured spectrum . using the simulated data sets chepurnov & lazarian ( 2006ab ) experimentally established that the acceptable number of lines ranges from 5 to 10 . for weak absorption , the absorption and emission lines can be analyzed in the same way , namely , the way suggested in lp06 . for this case , the statistics to analyse is the squared fourier transform of the doppler - shifted spectral line , irrespectively of the fact whether this is an emission or an absorption spectral line . such a `` spectrum of spectrum '' is not applicable for saturated spectral lines , which width is still determined by the doppler broadening . it is known ( see spitzer 1978 ) that this regime corresponds to the optical depth @xmath12 ranging from 10 to @xmath13 . the present paper will concentrate on this regime larger than @xmath14 the line width is determined by atomic constants and therefore it does not carry information about turbulence . ] . consider the problem in a more formal way . intensity of the absorption line at frequency @xmath15 is given as @xmath16 where @xmath17 is the optical depth . in the limit of vanishing intrinsic width of the line @xmath18 , the frequency spread of @xmath19 is determined solely by the doppler shift of the absorption frequency from moving atoms . the number density of atoms along the line of sight moving at required velocity @xmath20 is @xmath21 where @xmath22 is the thermal distribution centered at every point @xmath23 at the local mean velocity that is determined by the sum of turbulent and regular flow at that point . this is the density in ppv coordinate that we introduced in lp00 , so @xmath24 . the intrinsic line width is accounted for by the convolution @xmath25 or , in more detail , @xmath26 with intrinsic profile given by the lorenz form @xmath27 , the inner integral gives the shifted voigt profile @xmath28 so we have another representation @xmath29 we clearly see from eq . ( [ eq : tau_h ] ) that the line is affected both by doppler shifts and atomic constants . the optical depth as a function of frequency contains fluctuating component arising from turbulent motions and associated density inhomogeneities of the absorbers . statistics of optical depth fluctuations along the line of sight therefore carries information about turbulence in ism . the optical depth is determined by the density of the absorbers in the ppv space , @xmath30 . in our previous work we have studied statistical properties of @xmath30 in the context of emission lines , using both structure function and power spectrum formalisms . absorption lines demonstrate several important differences that warrant separate study . firstly , our ability to recover the optical depth from the observed intensity @xmath31 depends on the magnitude of the absorption as well as sensitivity of the instrument and the level of measurement noise @xmath32 . for lines with low optical depth @xmath33 we can in principle measure the optical depth throughout the whole line . at higher optical depths , the central part of the line is saturated below the noise level and the useful information is restricted to the wings of the line . this is the new regime that is the subject of this paper . in this regime the data is available over a window of frequencies limited to velocities high enough so that @xmath34 but not as high as to have lorentz tail define the line . higher the overall optical depth , narrower are the wings ( following spitzer , at @xmath35 the wings are totally dominated by lorentz factor ) . we shall denote this window by @xmath36 where @xmath37 is the velocity that the window is centered upon ( describing frequency position of the wing ) and @xmath38 is the wing width . it acts as a mask on the `` underlying '' data @xmath39 secondly , fluctuations in the wings of a line are superimposed on the frequency dependent wing profile . in other words , the statistical properties of the optical depth are inhomogeneous in this frequency range , with frequency dependent statistical mean value . while _ fluctuations _ of the optical depth @xmath40 that have origin in the turbulence can still be assumed to be statistically homogeneous , the mean profile of a wing must be accounted for . what statistical descriptors one should chose in case of line of sight velocity data given over limited window ? primary descriptors of a random field , here @xmath41 , are the ensemble average product of the values of the field at separated points the two point correlation function @xmath42 and , reciprocally , the average square of the amplitudes of its ( fourier ) harmonics decomposition the power spectrum @xmath43 in practice these quantities are measurable if one can replace ensemble average by averaging over different positions which relies on some homogeneity properties of stochastic process . we assume that underlying turbulence is homogeneous and isotropic . this does not make the optical depth to be statistically homogeneous in the wings of the line , but allows to introduce the fluctuations of @xmath12 on the background of the mean profile @xmath44 , @xmath45 , which are ( lp04 ) @xmath46 homogeneous correlation function depends only on a point separation and amplitudes of distinct fourier harmonics are independent . the obvious relations are @xmath47 although mathematically the power spectrum is just a fourier transform of the correlation function @xmath48 which of them is best estimated from data depends on the properties of the signal and the data . the power spectrum carries information which is localized to a particular scale and as such is insensitive to processes that contribute outside the range of scales of interest , in particular to long - range smooth variations . on the other hand , determination of fourier harmonics is non - local in configuration space and is sensitive to specifics of data sampling the finite window , discretization , that all lead to aliasing of power from one scales to another . the issue is severe if the aliased power is large . conversely , the correlation function is localized in configuration space and can be measured for non - uniformly sampled data . however , at each separation it contains contribution from all scales and may mix together the physical effects from different scales . in particular , @xmath49 is not even defined for power law spectra @xmath50 with index @xmath51 ( for one dimensional data ) .. ] this limitation is relieved if one uses the structure function @xmath52 instead , which is well defined for @xmath53 . the structure function can be thought of as regularized version of the correlation function @xmath54 that is related to the power spectrum in the same way as the correlation function , if one excludes the @xmath55 mode . velocity coordinate spectrum studies of lp06 demonstrated that the expected one dimensional spectrum of ppv density fluctuations along velocity coordinate that arise from turbulent motions is @xmath56 where @xmath57 is the index of line - of - sight component of the velocity structure function . for kolmogorov turbulence @xmath58 and for turbulent motions dominated by shocks @xmath59 . these spectra are steep @xmath60 which makes the direct measurement of the structure functions impractical ( although for @xmath61 the structure function can be defined ) . at the same time , in our present studies we deal with a limited range of data in the wings of the absorption lines , which complicates the direct measurements of the power spectrum . below we first describe the properties of the power spectrum @xmath62 in this case , and next develop the formalism of higher order structure functions . let us derive the power spectrum of the optical depth fluctuations , @xmath63 here @xmath64 is wave number reciprocal to the velocity ( frequency ) separation between two points on the line - of - sight and angular brackets denote an ensemble averaging .. here we restrict ourselves to diagonal terms only . ] fourier transform of the eq . ( [ eq : tau_h ] ) with respect to velocity is @xmath65 and the power spectrum @xmath66 } w(k_v - k_v^\prime ) w^*(k_v - k_v^{\prime\prime } ) \right\rangle \nonumber\end{aligned}\ ] ] which is useful to express using average velocity @xmath67 and velocity difference @xmath68 , as well as correspondent variables for the wave numbers @xmath69 and @xmath70 , as @xmath71 the fluctuating , random quantities , over which the averaging is performed are the density @xmath72 and the line - of - sight component of the velocity of the absorbers @xmath73 , varying along the line of sight . in our earlier papers ( see lp00 , lp04 ) we argued that in many important cases they can be considered as uncorrelated between themselves , so that @xmath74 } ~ ~ , \label{eq : maxprof_average}\ ] ] where @xmath75 is the correlation function of the density of the absorbers and @xmath76 is the structure function of their line - of - sight velocity due to turbulent motions . @xmath77 is expected to saturate at the value @xmath78 for separations of the size of the absorbing cloud @xmath79 . the dependence of @xmath80 and @xmath81 only on spatial separation between a pair of absorbers reflects the assumed statistical homogeneity of the turbulence model . introducing @xmath82 and performing integration over @xmath83 one obtains @xmath84 where @xmath85 , @xmath86 and symmetrized window @xmath87 is defined in the appendix . if one has the whole line available for analysis , the masking window will be flat with @xmath88-function like fourier transform . the combination of the windows in the power spectrum will translate to @xmath89 and @xmath90 masking the data has the effect of aliasing modes of the large scales that exceed the available data range , to shorter wavelength . this is represented by the convolution with fourier image of the mask . secondary effect is the contribution of the modes with different wave numbers @xmath91 to the diagonal part of the power spectrum . this again reflects the situation that different fourier components are correlated in the presence of the mask . to illustrate the effects of the mask , let us assume that we select the line wing with the help of a gaussian mask centered in the middle of the wing at @xmath92 @xmath93 that gives @xmath94 all integrals can then be carried out to obtain @xmath95 } { \sqrt{(d^+ + 2 \delta^2)(d^- + 2 \delta^2 ) } } \exp\left[-\frac{\delta^2 d^-}{d^- + 2 \delta^2 } k_v^2\right ] \\ & & \times \exp\left[\frac{2 a^2}{d^-+2 \delta^2}\right ] \left\ { \exp\left[\frac{-4 a \delta^2 k_v}{d^- + 2\delta^2}\right ] \mathrm{erfc } \left[\frac{\sqrt{2}(a-\delta^2 k_v)}{\sqrt{d^-+2\delta^2 } } \right ] + ( k_v \to -k_v ) \right\ } \nonumber\end{aligned}\ ] ] the following limits ( taking @xmath96 ) are notable : @xmath97 \nonumber \\ a \to 0 & : & \\ p(k_v ) & \propto & \alpha(\nu_0)^2 s \int_0^s dz ( s - z ) \xi(z ) \frac{\delta^2 \exp\left[-\frac{1}{4}\frac{v_1 ^ 2}{d^+ + \delta^2}\right ] } { \sqrt{(d^+ + 2 \delta^2)(d^- + 2 \delta^2 ) } } \exp\left[-\frac{\delta^2 d^-}{d^- + 2 \delta^2 } k_v^2\right ] \nonumber\end{aligned}\ ] ] the last expression particularly clearly demonstrates the effect of the window , which width in case of the line wing is necessarily necessarily limited by @xmath98 . the power spectrum is corrupted at scales @xmath99 , but still maintains information about turbulence statistics for @xmath100 . indeed , in our integral representation the power spectrum at @xmath64 is determined by the linear scales such that @xmath101 which translates into @xmath102 . thus , if @xmath103 over all scales defining power at @xmath64 one has @xmath104 and there is no significant power aliasing . varies little , @xmath105 in the interval @xmath106 . ] for intermediate scales there is a power aliasing as numerical results demonstrate in figure [ fig : spectrum ] . , @xmath38 , @xmath107 , @xmath92 are dimensionless , in the units of @xmath108 , the variance of the turbulent velocity at the scale of the cloud . intrinsic line broadening is neglected . only the effect of the turbulent motions and not spatial inhomogeneity of the absorbers is taken into account . the underlying scaling of the turbulent velocities is kolmogorov , @xmath58 . the left panel illustrates the power aliasing due to finite width of the window . the power spectrum is plotted , from top to bottom , for @xmath109 , i.e the widths of the wing ranges from the complete line to one - tenth of the line width . the straight line shows the power law @xmath110 expected under ideal observational circumstances . one finds that for the ideal gaussian mask the underlying spectrum is recovered for @xmath111 . the right panel shows the modification of the spectrum due to thermal broadening , which is taken at the level @xmath112 . thermal effects must be accounted for for @xmath113.,title="fig : " ] , @xmath38 , @xmath107 , @xmath92 are dimensionless , in the units of @xmath108 , the variance of the turbulent velocity at the scale of the cloud . intrinsic line broadening is neglected . only the effect of the turbulent motions and not spatial inhomogeneity of the absorbers is taken into account . the underlying scaling of the turbulent velocities is kolmogorov , @xmath58 . the left panel illustrates the power aliasing due to finite width of the window . the power spectrum is plotted , from top to bottom , for @xmath109 , i.e the widths of the wing ranges from the complete line to one - tenth of the line width . the straight line shows the power law @xmath110 expected under ideal observational circumstances . one finds that for the ideal gaussian mask the underlying spectrum is recovered for @xmath111 . the right panel shows the modification of the spectrum due to thermal broadening , which is taken at the level @xmath112 . thermal effects must be accounted for for @xmath113.,title="fig : " ] doppler broadening described by @xmath114 incorporates both turbulent and thermal effects . thermal effects are especially important in case of narrow line wings , since the range of the wavenumbers relatively unaffected by both thermal motions and the mask is limited @xmath115 and exists only for relatively wide wings @xmath116 . for narrower wings the combined turbulent and thermal profile must be fitted to the data , possibly determining the temperature of the absorbers at the same time . this recipe is limited by the assumption that the temperature of the gas is relatively constant for the absorbers of a given type . we should note that the gaussian window provides one of the ideal cases , limiting the extend of power aliasing since the window fourier image falls off quickly . one of the worst scenarios is represented by sharp top hat mask , which fourier image falls off only as @xmath117 spreading the power from large scales further into short scales . for steep spectra that we have in vcs studies all scales may experience some aliasing . this argues for extra care while treating the line wings through power spectrum or for use of alternative approaches . second order structure function provides an alternative to power spectrum measurement in case of steep spectra with the data limited to the section of the lines . the second order structure function of the fluctuations of the optical depth can be defined as @xmath118 it represents additional regularization of the correlation function beyond the ordinary structure function @xmath119 @xmath120 is proportional to three dimensional velocity space density structure function at zero angular separations @xmath121 , discussed in lp06 . using the results of lp06 for @xmath121 we obtain @xmath122 \nonumber \\ & \propto & \frac{{\bar \rho}^2 s^2 } { d_z(s ) } \frac{1}{m } \left(\frac{r_0}{s}\right)^\gamma \left [ \hat v^{2p } \gamma(-p ) \left(2^{p-1}-2^{1-p}\right ) + \frac{2^{4 p-6}}{p-2 } \hat v^4 + o ( \hat v^6 ) \right ] \label{eq : d_vv}\end{aligned}\ ] ] where @xmath123 and @xmath124 is the correlation index that describes spatial inhomogeneities of the absorbers . to shorten intermediate formulas , the dimensionless quantities @xmath125 , @xmath126 , @xmath127 are introduced . the first term in the expansion contains information about the underlying field , while the power law series represent the effect of boundary conditions at the cloud scale . in contrast to ordinary correlation function they are not dominant until @xmath128 , i.e for @xmath129 the second order structure function is well defined . when turbulent motions provide the dominant contribution to optical depth fluctuations , @xmath130 , we see that measuring the @xmath131 one can recover the turbulence scaling index if @xmath132 , which includes both interesting cases of kolmogorov turbulence and shock - dominated motions . this condition is replaced by @xmath133 if the density fluctuations , described by correlation index @xmath124 , are dominant . at sufficiently small scales the second order structure function has the same scaling as the first order one @xmath134 a practical issue of measuring the structure functions directly in the wing of the line is to take into account the line profile . the directly accessible @xmath135 is related to the structure function of the fluctuations as @xmath136 where the mean profile of the optical depth @xmath137 is related to the mean profile of ppv density @xmath138 given in the appendix b of lp06 . at small separations @xmath139 , the correction to the structure function due to mean profile behaves as @xmath140 and is subdominant . the price one pays when utilizing higher - level structure function is their higher sensitivity to the noise in the data . while correlation function itself is not biased by the noise except at at zero separations ( assuming noise is uncorrelated @xmath141[\tau(v_1+v)+n(v_1+v)]\right\rangle = \xi_\tau(v ) + \langle n^2 \rangle \delta(v)\ ] ] already the structure function is biased by the noise which contributes to all separations @xmath142 ^ 2\right\rangle = d_\tau(v ) + 2 \langle n^2 \rangle\ ] ] this effect is further amplified for the second order structure function @xmath143 ^ 2 \right\rangle = dd_\tau(v ) + 3 \langle n^2 \rangle\ ] ] the error in the determination of the structures function of higher order due to noise also increases . structure functions and power spectra are used interchangeably in the theory of turbulence ( see monin & yaglom 1975 ) . however , complications arise when spectra are `` extremely steep '' , i.e. the corresponding structure function of fluctuation grows as @xmath144 , @xmath145 . for such random fields , one can not use ordinary structure functions , while the one dimensional fourier transforms that is employed in vcs corresponds to the power spectrum of @xmath146 is well defined . as a rule , one does not have to deal with so steep spectra in theory of turbulence ( see , however , cho et al . 2002 and cho & lazarian 2004 ) . within the vcs , such `` extremely steep '' spectra emerge naturally , even when the turbulence is close to being kolmogorov . this was noted in lp06 , where the spectral approach was presented as the correct one to studying turbulence using fluctuations of intensity along v - coordinate . the disadvantage of the spectral approach is when the data is being limited by a non - gaussian window function . then the contributions from the scales determined by the window function may interfere in the obtained spectrum at large @xmath64 . an introduction of an additional more narrow gaussian window function may mitigate the effect , but limits the range of @xmath64 for which turbulence can be studied . thus , higher order structure functions ( see the subsection above ) , is advantageous for the practical data handling . in terms of the vca theory , we used mostly spectral description in lp00 , while in lp04 , dealing with absorption , we found advantageous to deal with real rather than fourier space . in doing so , however , we faced the steepness of the spectrum along the v - coordinate and provided a transition to the fourier description to avoid the problems with the `` extremely steep '' spectrum . naturally , our approach of higher order structure functions is applicable to dealing with the absorption within the vca technique . in the paper above we have discussed the application of vcs to strong absorption lines . the following assumption were used . first of all , considering the radiative transfer we neglected the effects of stimulated emission . this assumption is well satisfied for optical or uv absorption lines ( see spitzer 1979 ) . then , we assumed that the radiation is coming from a point source , which is an excellent approximation for the absorption of light of a star or a quasar . moreover , we disregarded the variations of temperature in the medium . within our approach the last assumption may be most questionable . indeed , it is known that the variations of temperature do affect absorption lines . nevertheless , our present study , as well as our earlier studies , prove that the effects of the variations of density are limited . it is easy to see that the temperature variations can be combined together with the density ones to get effective renormalized `` density '' which effects we have already quantified . our formalism can also be generalized to include a more sophisticated radiative transfer and the spatial extend of the radiation source . in the latter case we shall have to consider both the case of a narrow and a broad telescope beam , the way it has been done in lp06 . naturally , the expressions in lp06 for a broad beam observations can be straightforwardly applied to the absorption lines , substituting the optical depth variations instead of intensities . the advantage of the extended source is that not only vcs , but also vca can be used ( see deshpande et al . 2000 ) . as a disadvantage of an extended source is the steepening of the observed v - coordinate spectrum for studies of unresolved turbulence . this , for instance , may require employing even higher order structure functions , if one has to deal with windows arising from saturation of the absorption line . in lp06 we have studied the vcs technique in the presence of absorption and formulated the criterion for the fluctuations of intensity to reliably reflect the fluctuations in turbulent velocities . in this paper , however , we used the logarithms of intensities and showed that this allows turbulence studies beyond the regime , at which fluctuations of intensity would be useful . the difficulty of such an approach is the uncertainty of the base level of the signal . taking logarithm is a non - linear operation that may distort the result , if the base level of signal is not accounted for properly . however , the advantage of the approach that potentially it allows studies of velocity turbulence , when the traditional vca and vcs fail . further research should clarify the utility of this approach . the study of turbulence using the modified vcs technique above should be reliable for optical depth @xmath12 up to @xmath13 . for this range of optical depth , the line width is determined by doppler shifts rather than the atomic constants . while formally the entire line profile provides information about the turbulence , in reality , the flat saturated part of the profile will contain only noise and will not be useful for any statistical study . thus , the wings of the lines will contain signal . as several absorption lines can be available along the same line of sight , this allows to extend the reliability of measurements combining them together . we believe that piecewise analyses of the wings belonging to different absorption lines is advantageous . the actual data analysis may employ fitting the data with models , that , apart from the spectral index , specify the turbulence injection scale and velocity dispersion , as this is done in chepurnov et al . ( 2006 ) . note , that measurements of turbulence in the same volume using different absorption lines can provide complementary information . formally , if lines with weak absorption , i.e. @xmath148 are available , there is no need for other measurements . however , in the presence of inevitable noise , the situation may be far from trivial . naturally , noise of a constant level , e.g. instrumental noise , will affect more weak absorption lines . the strong absorption lines , in terms of vcs sample turbulence only for sufficiently large @xmath64 . this limits the range of turbulent scales that can be sampled with the technique . however , the contrast that is obtained with the strong absorption lines is higher , which provides an opportunity of increasing signal to noise ratio for the range of @xmath64 that is sampled by the absorption lines . if , however , a single strong absorption line is used , an analogy with a two dish radio interferometer is appropriate . every dish of the radio interferometer samples spatial frequencies in the range approximately @xmath149 $ ] , where @xmath150 is the operational wavelength , @xmath151 is the diameter of the dish . in addition , the radio interferometer samples the spatial frequency @xmath152 , where @xmath153 is the distance between the dishes . similarly , a strong absorption line provides with the information on turbulent velocity at the largest spatial scale of the emitting objects , as well as the fluctuation corresponding to the scales @xmath154 . in lp06 we concentrated on obtaining asymptotic regimes for studying turbulence . at the same time in chepurnov et al . ( 2006 ) fitting models of turbulence to the data was attempted . in the latter approach non - power law observed spectra can be used , which is advantageous for actual data , for which the range of scales in @xmath64 is rather limited . indeed , for hi with the injection velocities of 10 km / s and the thermal velocities of 1 km / s provides an order of magnitude of effective `` inertial range '' . correcting for thermal velocities one can increase this range by a factor , which depends on the signal to noise ratio of the data . using heavier species rather than hydrogen one can increase the range by a factor @xmath155 . this may or may not be enough for observing good asymptotics . we have seen in [ ] that for absorption lines the introduction of windows determined by the width of the line wings introduces additional distortions of the power spectrum . however , this is not a problem if , instead of asymptotics , fitting of the model is used . compared to the models used in chepurnov et al . ( 2006 ) the models for absorption lines should also have to model the window induced by the absorption . the advantage is , however , that absorption lines provide a pure pencil beam observations . formally , there exists an extensive list of different tools to study turbulence that predated our studies ( see lazarian 1999 and ref . therein ) . however , a closer examination shows that this list is not as impressive as it looks . moreover , our research showed that some techniques may provide confusing , if not erroneous , output , unless theoretical understanding of what they measure is achieved . for instance , we mentioned in the introduction an example of the erroneous application of velocity centroids to supersonic molecular cloud data . note , that clumps and shell finding algorithms would find a hierarchy of clumps / shells for synthetic observations obtained with _ incompressible _ simulations . this calls for a more cautious approach to the interpretation of the results of some of the accepted techniques . for instance , the use of different wavelets for the analysis of data is frequently treated in the literature as different statistical techniques of turbulence studies ( gill & henriksen 1990 , stutzki et al . 1998 , cambresy 1999 , khalil et al . 2006 ) , which creates an illusion of an excessive wealth of tools and approaches . in reality , while fourier transforms use harmonics of @xmath156 , wavelets use more sophisticated basis functions , which may be more appropriate for problems at hand . in our studies we also use wavelets both to analyze the results of computations ( see kowal & lazarian 2006a ) and synthetic maps ( ossenkopf et al . 2006 , esquivel et al . 2007 ) , along with or instead of fourier transforms or correlation functions . wavelets may reduce the noise arising from inhomogeneity of data , but we found in the situations when correlation functions of centroids that we studied were failing as the mach number was increasing , a popular wavelet ( @xmath38-variance ) was also failing ( cp . esquivel & lazarian 2005 , ossenkopf et al . 2006 , esquivel et al . 2007 ) . while in wavelets the basis functions are fixed , a more sophisticated technique , principal component analysis ( pca ) , chooses basis functions that are , in some sense , the most descriptive . nevertheless , the empirical relations obtained with pca for extracting velocity statistics provide , according to padoan et al . ( 2006 ) , an uncertainty of the velocity spectral index of the order @xmath157 ( see also brunt et al . 2003 ) , which is too large for testing most of the turbulence theories . in addition , while our research in lp00 shows that for density spectra @xmath158 , for @xmath159 both velocity and density fluctuations influence the statistics of ppv cubes , no dependencies of ppv statistics on density have been reported so far in pca studies . this also may reflect the problem of finding the underlying relations empirically with data cubes of limited resolution . the latter provides a special kind of shot noise , which is discussed in a number of papers ( lazarian et al . 2001 , esquivel et al . 2003 , chepurnov & lazarian 2006a ) . _ spectral correlation function ( scf ) _ ( see rosolowsky et al . 1999 for its original form ) is another way to study turbulence . further development of the scf technique in padoan et al . ( 2001 ) removed the adjustable parameters from the original expression for the scf and made the technique rather similar to vca in terms of the observational data analysis . indeed , both scf and vca measure correlations of intensity in ppv `` slices '' ( channel maps with a given velocity window @xmath160 ) , but if scf treats the outcome empirically , the analytical relations in lazarian & pogosyan ( 2000 ) relate the vca measures to the underlying velocity and density statistics . mathematically , scf contains additional square roots and normalizations compared to the vca expressions . those make the analytical treatment , which is possible for simpler vca expressions , prohibitive . one might speculate that , similar to the case of conventional centroids and not normalized centroids introduced in lazarian & esquivel ( 2003 ) , the actual difference between the statistics measured by the vca and scf is not significant . in fact , we predicted several physically - motivated regimes for vca studies . for instance , slices are `` thick '' for eddies with velocity ranges less than @xmath160 and `` thin '' otherwise . vca relates the spectral index of intensity fluctuations within channel maps to the thickness of the velocity channel and to the underlying velocity and density in the emitting turbulent volume . in the vca these variations of indexes with the thickness of ppv `` slice '' are used to disentangle velocity and density contributions . we suspect that similar thick " and thin " slice regimes should be present in the scf analysis of data , but they have not been reported yet . while the vca can be used for all the purposes the scf is used ( e.g. for an empirical comparisons of simulations and observations ) , the opposite is not true . in fact , padoan et al . ( 2004 ) stressed that vca eliminates errors inevitable for empirical attempts to calibrate ppv fluctuations in terms of the underlying 3d velocity spectrum . _ vcs _ is a statistical tool that uses the information of fluctuations along the velocity axis of the ppv . among all the tools that use spectral data , including the vca , it is unique , as it _ does not _ require spatial resolution . this is why , dealing with the absorption lines , where good spatial coverage is problematic , we employed the vcs . potentially , having many sources sampling the object one can create ppv cubes and also apply the vca technique . however , this requires very extended data sets , while for the vcs sampling with 5 or 10 sources can be sufficient for obtaining good statistics ( chepurnov & lazarian 2006a ) . we feel that dealing with the ism turbulence , it is synergetic to combine different approaches . for the wavelets used their relation with the underlying fourier spectrum is usually well defined . therefore the formulation of the theory ( presented in this work , as well as , in our earlier papers in terms of the fourier transforms ) in terms of wavelets is straightforward . at the same time , the analysis of data with the wavelets may be advantageous , especially , in the situations when one has to deal with window functions . in the paper above we have shown that * studies of turbulence with absorption lines are possible with the vcs technique if , instead of intensity @xmath161 , one uses the logarithm of the absorbed intensity @xmath162 , which is equivalent to the optical depth @xmath163 . * in the weak absorption regime , i.e. when the optical depth at the middle of the absorption line is less than unity , the analysis of the @xmath17 coincides with the analysis of intensities of emission for ideal resolution that we discussed in lp06 . * in the intermediate absorption retime , i.e. when the optical depth at the middle of the absorption line is larger than unity , but less than @xmath13 , the wings of the absorption line can be used for the analysis . the saturated part of the line is expected to be noise dominated . * the higher the absorption , the less the portion of the spectrum corresponds to the wings available for the analysis . in terms of the mathematical setting this introduces and additional window in the expressions for the vcs analysis . however , the contrast of the small scale fluctuations increases with the decrease of the window . * for strong absorption regime , the broadening is determined by lorentzian wings of the line and therefore no information on turbulence is available . following eqns . ( [ eq : ptau_gen],[eq : maxprof_average ] ) the power spectrum of the optical depth is @xmath165 where @xmath85 and @xmath86 while @xmath69 and @xmath166 . since the mask is real , @xmath167 . to deal with absolute values in the lorentz transform , we split integration regions in quadrants i @xmath168 , ii @xmath169 , iii @xmath170 and iv @xmath171 . integration over quadrants iii and iv can be folded into integration over regions i and ii respectively by substitution @xmath172 . writing out only integration over @xmath64 @xmath173 \nonumber \\ ii+iv & : & \int_0^{\infty } d k_v^\prime \int_{-\infty}^0 \!\ ! d k_v^{\prime\prime } e^{-\frac{1}{2}{k_v^+}^2 d^- } e^{-\frac{1}{4}{k_v^-}^2 d^+ } e^ { -a k_v^- } \\ & & \times \left [ w\left(k_v - k_v^+-\frac{k_v^-}{2}\right ) w\left(k_v^+ -k_v -\frac{k_v^-}{2}\right ) + w\left(k_v^+ + k_v + \frac{k_v^-}{2}\right ) w\left(-k_v - k_v^+ + \frac{k_v^-}{2}\right ) \right ] \nonumber\end{aligned}\ ] ] changing variables of integration to @xmath174 and @xmath175 @xmath176 \nonumber \\ ii+iv & : & \int_{-\infty}^\infty d k_v^+ e^{-\frac{1}{2}{k_v^+}^2 d^- } \int_{|2 k_v^+|}^\infty \!\ ! d k_v^- e^{-\frac{1}{4}{k_v^-}^2 d^+ } e^ { -a k_v^- } \\ & & \times \left [ w\left(k_v - k_v^+-\frac{k_v^-}{2}\right ) w\left(k_v^+ -k_v -\frac{k_v^-}{2}\right ) + w\left(k_v + k_v^+ + \frac{k_v^-}{2}\right ) w\left(-k_v - k_v^+ + \frac{k_v^-}{2}\right ) \right ] \nonumber\end{aligned}\ ] ] at the end , the integrals can be combined into the main contribution and the correction that manifests itself only when lorentz broadening is significant . @xmath177 where symmetrized window is @xmath178 the final expression for @xmath179 is then @xmath180 armstrong , j. w. , rickett , b. j. , & spangler , s. r. 1995 , apj , 443 , 209 ballesteros - paredes , j. , klessen , r. , mac low , m. & vasquez - semadeni , e. 2006 , in `` protostars and planets v '' , reipurth , d. jewitt , and k. keil ( eds . ) , university of arizona press , tucson , 951 pp . , 2007 . , p.63 - 80 cho , j. , & lazarian , a. 2003 , , 345 , 325 cho , j. , & lazarian , a. 2004 , , 615 , l41 cho , j. , & lazarian , a. 2005 , theoretical and computational fluid dynamics , 19 , 127 cho , j. , lazarian , a. , honein , a. , knaepen , b. , kassinos , s. , & moin , p. 2003 , apj , 589 , l77 esquivel , a. , lazarian , a. , pogosyan , d. , & cho , j. 2003 , mnras , 342 , 325 falgarone , e. 1999 , in _ interstellar turbulence _ , ed . by j. franco , a. carraminana , cup , ( henceforth _ interstellar turbulence _ ) p.132 lazarian , a. , pogosyan , d. , & esquivel , a. 2002 , in asp conf . 276 , seeing through the dust , ed . r. taylor , t. l. landecker , & a. g. willis ( san francisco : asp),182 lazarian , a. , pogosyan , d. , vzquez - semadeni , e. , & pichardo , b. 2001 , , 555 , 130 lazarian , a. , vishanic , e. , cho , j. 2004 , , 603 , 180 lazarian , a. & yan , h. 2004 , in `` astrophysical dust '' eds . a. witt & b. draine , aps , v. 309 , p.479 maron , j. & goldreich , p. 2001 , apj , 554 , 1175 monin , a.s . & yaglom , a. m. 1975 , statistical fluid mechanics : mechanics of turbulence , vol . 2 ( cambridge : mit press ) munch , g. 1999 , in `` interstellar turbulence '' , eds . j. franco and a. carraminana , cup , p. 1 munch , g. 1958 , rev . , 30 , 1035 narayan , r. , & goodman , j. 1989 , mnras , 238 , 963 pudritz , r. e. 2001 , from darkness to light : origin and evolution of young stellar clusters , asp , vol . 243 . eds t. montmerle and p. andre . san francisco , p.3 spangler , s.r . , & gwinn , c.r . 1990 , apj , 353 , l29 stanimirovi , s. , & lazarian , a. , 2001 , , 551 , 53 stutzki , j. 2001 , astrophysics and space science supplement , 277 , 39 sunyaev , r.a . , norman , m.l . , & bryan , g.l . 2003 , astronomy letters , 29 , 783 von hoerner , s. 1951 , zeitschrift fr astrophysics , 30 , 17 wilson , o.c . , munch , g. , flather , e.m . , & coffeen , m.f . 1959 , apjs , 4 , 199

we continue our work on developing techniques for studying turbulence with spectroscopic data . we show that doppler - broadened absorption spectral lines , in particularly , saturated absorption lines , can be used within the framework of the earlier - introduced technique termed the velocity coordinate spectrum ( vcs ) . the vcs relates the statistics of fluctuations along the velocity coordinate to the statistics of turbulence , thus it does not require spatial coverage by sampling directions in the plane of the sky . we consider lines with different degree of absorption and show that for lines of optical depth less than one , our earlier treatment of the vcs developed for spectral emission lines is applicable , if the optical depth is used instead of intensity . this amounts to correlating the logarithms of absorbed intensities . for larger optical depths and saturated absorption lines , we show , that the amount of information that one can use is , inevitably , limited by noise . in practical terms , this means that only wings of the line are available for the analysis . in terms of the vcs formalism , this results in introducing an additional window , which size decreases with the increase of the optical depth . as a result , strongly saturated absorption lines carry the information only about the small scale turbulence . nevertheless , the contrast of the fluctuations corresponding to the small scale turbulence increases with the increase of the optical depth , which provides advantages for studying turbulence combining lines with different optical depths . we show that , eventually , at very large optical depths the lorentzian profile of the line gets important and extracting information on velocity turbulence , gets impossible . combining different absorption lines one can tomography turbulence in the interstellar gas in all its complexity

0801.1151

the effect of temperature and angular momentum on pairing properties is an interesting subject in the study of nuclear structure . because of its simplicity , the bcs theory is often used , which offers a good description of pairing correlation in the macroscopic systems such as metallic superconductors . it predicts a collapse of the pairing gap at @xmath0 , which signals the sharp superfluid - normal ( sn ) phase transition at finite temperature . the bcs theory , however , ignores quantal and thermal fluctuations , which are significant in finite small systems . therefore , it needs to be corrected for the application to finite nuclei . various theoretical approaches have been proposed to study the effects of fluctuations on nuclear pairing @xcite . their results show that , at zero angular momentum , thermal fluctuations smear out the sharp sn phase transition , resulting in a pairing gap , which does not collapse at finite temperature . in rotating nuclei , a phenomenon of temperature induced pair correlations , which reflects the strong fluctuations of the order parameter in small systems , has also been predicted @xcite . the recent microscopic approach , called the modified bcs ( mbcs ) theory @xcite has shown , for the fist time , that the microscopic source causing the non - collapsing pairing gap is the quasiparticle - number fluctuation ( qnf ) . recently , we proposed the self - consistent quasiparticle random - phase approximation ( scqrpa ) @xcite , which includes the qnf as well as the quantal fluctuations due to dynamic coupling to pair vibrations . the purpose of present work is to extend this approach to finite temperature and finite angular momentum . the pairing hamiltonian is considered , which describes a system of @xmath1 particles interacting via a pairing force with the parameter @xmath2 and rotating with angular velocity @xmath3 and a fixed angular momentum projection @xmath4 on the laboratory ( or body ) fixed @xmath5 axis : @xmath6 where @xmath7 ( @xmath8 ) is the operator that creates ( annihilates ) a particle with angular momentum @xmath9 , spin projection @xmath10 or @xmath11 , and energy @xmath12 . for simplicity , the subscripts @xmath9 label the single - particle states @xmath13 with @xmath14 0 , whereas @xmath15 denote the time - reversal states @xmath16 . the particle number operator @xmath17 is defined as @xmath18 , whereas @xmath19 is the @xmath20-projection of total angular momentum . the variational procedure is applied to minimize the expectation value of this hamiltonian in the grand canonical ensemble . the result yields the final equations for the pairing gap , particle number and total angular momentum , which include the effect of qnf in the form @xmath21 @xmath22~ , \hspace{5 mm } m = \sum_k m_k(n_k^{+ } - n_k^{-})~ , \label{nm}\ ] ] where the quasiparticle energy @xmath23 and renormalized single - particle energy @xmath24 are given as @xmath25 @xmath26 with @xmath27 , and @xmath28 . the expectation values @xmath29 and @xmath30 are evaluated by solving a set of coupled equations , which contain the scqrpa @xmath31 and @xmath32 amplitudes . the qnf is given as @xmath33 , where the quasiparticle occupation numbers @xmath34 are found from the integral equations @xmath35^{2}+[\gamma_{k}^{\pm}(\omega)]^2}d\omega~ , \label{nkcoupling}\ ] ] with the mass operators @xmath36 obtained by solving the set of equations for double - time quasiparticle green s functions and those of a quasiparticle coupled with scqrpa pair vibrations . the quasiparticle dampings are given as @xmath37 $ ] . the proposed approach is called the ftbcs1+scqrpa theory . neglecting the coupling to scqrpa , i.e. the factors @xmath29 and @xmath30 , it becomes the ftbcs1 theory , which is different from the conventional ftbcs theory by the presence of the qnf . the violation of particle number at zero angular momentum is approximately removed by applying the lipkin - nogami ( ln ) method . the corresponding approaches are called the ftln1+scqrpa and ftln1 . the numerical calculations are carried out within the @xmath38 doubly degenerate equidistant model with the number @xmath38 of levels equal to that of particles , @xmath1 , as well as for @xmath39o , @xmath40ca , @xmath41fe , and @xmath42sn . the results obtained show that , at zero angular momentum , under the effect of qnf within the ftbcs1 ( ftln1 ) , the sharp sn phase transition predicted by the ftbcs theory is smoothed out . as the result , the pairing gap does not collapse at @xmath43 , but has a tail , which extends to high @xmath44 . the dynamic coupling to the scqrpa vibrations significantly improves the agreement with the exact results for the total energies and heat capacities obtained for @xmath45 as well as those obtained @xmath41fe within the finite - temperature quantum monte carlo method @xcite [ figs . [ fig ] ( a ) [ fig ] ( c ) ] . however , for heavy nuclei such as @xmath42sn , the scqrpa corrections are found to be negligible in comparison with the ftbcs1 ( ftln1 ) results . for @xmath39o and @xmath40ca , the ftbcs1 pairing gaps , obtained at different @xmath4 , decreases as @xmath44 increases and do not collapses at high @xmath44 . at @xmath4 higher than the critical value @xmath46 , where the ftbcs gap for @xmath47 disappears , there appear thermally assisted pairing correlations , in which the ftbcs1 gap reappears at a given @xmath48 , and remains finite at @xmath49 [ fig . [ fig ] ( d ) ] . this phenomenon is caused by the qnf within the ftbcs1 theory . at @xmath47 , the qnf is zero , so the ftbcs and ftbcs1 gaps are the same as functions of @xmath4 ( or @xmath3 ) , and both collapse at @xmath50 . however , with increasing @xmath44 , the ftbcs1 gaps , which are obtained at different @xmath44 , collapse at @xmath51 , and remain finite even at very high @xmath44 , whereas those given by the conventional ftbcs theory vanish at @xmath52 and @xmath53 [ figs . ( e ) and [ fig ] ( f ) ] . 0 l. g. moretto , phys . b * 35 * , 397 ( 1971 ) ; nucl . phys . a * 185 * , 145 ( 1971 ) . l. g. moretto , phys . b * 40 * , 1 ( 1972 ) . a. l. goodman , nucl . a * 352 * , 30 ( 1981 ) ; phys . c * 29 * , 1887 ( 1984 ) . r. rossignoli , p. ring , and n.d . dang , phys . b * 297 * , 9 ( 1992 ) . s. frauendorf , _ et . b * 68 * , 024518 ( 2003 ) . n.d . dang and v. zelevinsky , phys . c * 64 * , 064319 ( 2001 ) ; n.d . dang and a. arima , phys . rev . c * 67 * , 014304 ( 2003 ) ; n. d. dang , nucl . a * 784 * , 147 ( 2007 ) . n.q . hung and n.d . dang , phys . c * 76 * , 054302 ( 2007 ) , ibid . * 77 * , 029905(e ) ( 2008 ) . s. rombouts , k. heyde , and n. jachowicz , phys . c * 58 * , 3295 ( 1998 ) .

an approach is proposed to nuclear pairing at finite temperature and angular momentum , which includes the effects of the quasiparticle - number fluctuation and dynamic coupling to pair vibrations within the self - consistent quasiparticle random - phase approximation . the numerical calculations of pairing gaps , total energies , and heat capacities are carried out within a doubly folded multilevel model as well as several realistic nuclei . the results obtained show that , in the region of moderate and strong couplings , the sharp transition between the superconducting and normal phases is smoothed out , causing a thermal pairing gap , which does not collapse at a critical temperature predicted by the conventional bardeen - cooper - schrieffer s ( bcs ) theory , but has a tail extended to high temperatures . the theory also predicts the appearance of a thermally assisted pairing in hot rotating nuclei .

0806.1795

water vapor is an important molecule for the chemistry of interstellar and circumstellar clouds . the 6@xmath8 - 5@xmath9 masing transition of h@xmath3o at 22 ghz , which arises from levels around 700k , has been used since its detection by cheung et al . ( 1969 ) to trace high excitation gas around star forming regions and evolved stars . the size of the emitting regions at that frequency is typically of the order of a few milliarcseconds ( a few 10@xmath10 cm ) . hence , no information has been obtained from this line on the role of h@xmath3o at large spatial scales . other h@xmath3o lines have been detected from ground or airborne based telescopes like the 3@xmath11 - 2@xmath12 transition at 183 ghz ( waters et al . , 1980 ; cernicharo et al . 1990 , 1994 , 1996 ; gonzlez - alfonso et al . 1994 , 1998 ) , the 4@xmath13 - 3@xmath14 transition at 380 ghz ( phillips , kwan and huggins 1980 ) , the 10@xmath15 - 9@xmath16 transition at 321 ghz ( menten , melnick and phillips 1990a ) and the 5@xmath17 - 4@xmath18 transition at 325 ghz ( menten et al . also , the 1@xmath19 - 2@xmath20 transition of h@xmath21o at 547 ghz has been observed by zmuidzinas et al . ( 1994 ) . among these lines only the 3@xmath11 - 2@xmath12 transition at 183 ghz has been used to map the emission of h@xmath3o at very large spatial scale ( cernicharo et al . 1994 , hereafter referred to as cer94 ) . the map of the orion molecular cloud shown in cer94 is 6 orders of magnitude larger than the size of the spots detected at 22 ghz and for the first time an h@xmath3o abundance estimate was derived for the different large scale components of the orion molecular cloud . the iso satellite has provided the opportunity to observe thermal lines of water in the middle and far - infrared ( see the reviews by van dishoeck 1997 ; cernicharo 1997 ; and cernicharo et al 1997a,1998 ) . mapping of the sgrb2 molecular cloud by cernicharo et al . ( 1997b ) has definitely shown that water vapor is an ubiquitous molecule in molecular clouds with an abundance of 10@xmath7 . maps of the emission of several h@xmath3o lines in orion irc2 have been obtained by cernicharo et al . ( 1997a , 1998 , 1999 ) . observations of the central position have been also obtained by van dishoeck et al . ( 1998 ) , gonzlez - alfonso et al . ( 1998 ) and harwit et al . however , the iso observations of h@xmath3o have drawbacks . in addition to the limited spectral resolution and the high opacity of the thermal lines of h@xmath3o , the poor angular resolution provided by iso in the far - infrared prevents any detailed study of the spatial structure and physical conditions of the h@xmath3o emitting regions . an important aid in deriving h@xmath3o abundances could come from the observation of another masing transition of h@xmath3o with similar properties to those of the 183 ghz line . the 325 ghz line of h@xmath3o was observed by menten et al ( 1990b ) in the direction of orion irc2 and other molecular clouds . however , no maps were obtained . here we report the detection of extended water emission at 325 ghz and show that the h@xmath3o abundance is @xmath4 10@xmath5 in the * plateau*. the present data show the importance of ground - based observations of h@xmath3o in deriving the abundance of h@xmath3o in molecular clouds and in providing useful contraints on the physical conditions of the emitting regions . our ground - based observations provide much finer spatial resolution than iso or swas , and an estimate of x(h@xmath3o ) as accurate as that obtained from the extremely optically thick h@xmath3o lines observed in the submillimeter and far infrared domains . the observations were performed with the 10.4 m telescope of the caltech submillimeter observatory at the summit of mauna kea ( hawaii ) on april 1@xmath22 1998 . the receiver , a helium - cooled sis mixer operating in double - sideband mode ( dsb ) , was tuned at the frequency of the 5@xmath04@xmath1 line of h@xmath23o ( 325.152919 ghz ) . the h@xmath3o line was placed in the upper sideband ( usb ) to minimize atmospheric noise from the image sideband ( which was at 322.35 ghz ) . lines in the signal sideband are severely attenuated relative to those in the image sideband , due to the atmospheric h@xmath24 line . therefore it was necessary to check the sideband origin of a given line . the tuning frequency was shifted by 100 mhz to confirm that the central feature in the spectrum of orion - irc2 was the water vapor line . the backend consisted of a 1024 channel acousto - optic spectrometer covering a bandwidth of 500 mhz ( @xmath25v=1.1 kms@xmath26 ) . figure 1 shows the observed spectrum which matches very well the previous observation by menten et al . ( 1990b ) except for the line intensity ( see below ) . the pointing was determined by observing the same line towards the o - rich evolved star vy cma and was found to be accurate to 5 . the weather conditions were very stable during the observations with an atmospheric pressure and temperature of 620 mb and -1.4@xmath27c respectively . the relative humidity was measured to be 4 - 5% . the measured opacity from tipping scans at 225 ghz was @xmath40.025 . during the same night we performed broadband fourier transform spectroscopy ( fts ) measurements of the atmospheric absorption with the fts described in serabyn and weisstein ( 1995 ) . model calculations using the multi - layer atmospheric radiative transfer model atm ( cernicharo 1985,1988 ; pardo 1996 ) yielded an estimated precipitable water vapor column above the telescope of @xmath28200 @xmath29 m , which corresponds to a zenith transmission at 325.15 ghz of @xmath28 60% ( the corresponding value for the image side bande was @xmath28 87% , hence the line intensities for the image sideband features are overstimated by a factor @xmath28 2 ) . the heterodyne h@xmath3o data were calibrated using an absorber at ambient temperature . the calculated system noise temperature , for the signal sideband , was @xmath42100 k. the orion spectrum shown in figure 1 shows that the lines from the image sideband are weaker than the 325 ghz h@xmath3o line , i.e. , just the opposite of that occurring in the spectrum of menten et al . ( 1990b ) . rather than a variation of the maser emission ( the h@xmath3o line profile in figure 1 is identical to that shown in menten et al . 1990b ) we think that this difference is due to much better atmospheric transmission during our observations . we estimate that our intensity scale is correct to within 20 - 30% . the orion - irc2 map was carried out in position switching mode by using the on - the - fly procedure with an off position 5 away in azimuth . the spatial distribution of the 325 ghz emission is shown in figure 2 together with that of ch@xmath2oh ( from the image side band ) and the 183 ghz emission from cer94 . integrated intensity maps for selected velocity intervals are also shown in figure 2 . in order to compare the line profiles of the h@xmath3o lines at 183 and 325 ghz we reobserved a few positions at 183 ghz with the 30-m iram telescope in january 1999 . the weather conditions were also excellent with a zenith opacity at this frequency of @xmath4 1 . the spectra , together with those obtained in 1994 , are shown in figure 3 . the observed 5@xmath17 - 4@xmath18 line profile towards the center position looks similar to that of the 3@xmath11 - 2@xmath12 line observed by cer94 ( see spectra in figure 3 ) . however , the antena temperature of the line is 20 times weaker and the line profile , although covering the same velocity range , is shifted towards the red . taking into account the different beam sizes of the iram-30 m telescope at 183 ghz and the cso at 325 ghz , and the extension of the emission in the latter line , we estimate that the main beam brightness temperature ratio , @xmath30=t@xmath31(183)/t@xmath31(325 ) is 10 - 20 if both lines were observed with a telescope of @xmath32 beam size . on the other hand , t@xmath31(183)@xmath33 k. both @xmath30 and t@xmath31(183 ) are well determined and can not be related to a calibration problem as the atmospheric conditions were extremely good during our observations at both frequencies . similar values for * r * , i.e. , @xmath341020 , are also found at other positions in the cloud ( see figure 2 ) and represent a real difference in the brightness temperatures of the two lines . only at position @xmath35=-12 , @xmath36=48 the peak temperature of the 5@xmath17 - 4@xmath18 transition approaches that of the 3@xmath11 - 2@xmath12 ( the 5@xmath17 - 4@xmath18 line is , however , narrower ) . the 325 ghz emission at this position presents a local maximum clearly visible in the velocity maps ( see figure 2 ) . like the 3@xmath11 - 2@xmath12 line , the 5@xmath17 - 4@xmath18 transition is masing in nature . there are some narrow features at 325 ghz but with intensities of only a few k , i.e. , much weaker than those reported at 183 ghz by cer94 . these features agree in velocity with those found at 183 ghz . however , @xmath30 changes drastically from feature to feature , a fact that reveals the maser nature of the emission . outside the central region the lines are very narrow ( @xmath373 - 5 kms@xmath26 ) . some of the 3@xmath11 - 2@xmath12 narrow velocity components have antenna temperatures above 2000 k and are probably a few arcseconds in size ( cer94 , gonzlez - alfonso 1995 ) . the observations at this frequency performed in january 1999 clearly indicate a variation in the intensity of some of these features with respect to those of cer94 . however , at positions where the line is dominated by the plateau emission ( @xmath35=28 , @xmath36=-16 and @xmath35=12 , @xmath36=-16 for example ; see figure 3 ) and outside the central region ( @xmath35=-12 , @xmath36=48 ; figure 3 ) the line shape and intensity do not show any significant change between both epochs . a possible explanation for the 183 ghz and 325 ghz emission being spatially extended could be that it arises from many masing point like sources strongly diluted in the beam . this is ruled out by the results of cer94 where even the strong features at 183 ghz ( t@xmath382000 - 4000 k ) show indication of some spatial extent ( see above ) . the densities needed to reproduce the observed brightness temperatures of the maser spots at 22 ghz would result on a thermal or suprathermal 183 ghz line . consequently , if the 183 ghz emission was arising from the same region than that of the 22 ghz line very large column densities will be needed to reproduce the observed 183 ghz and 325 ghz intensities . in addition , the weak and extended emission observed at 183 ghz by cer94 clearly indicates the presence of water vapour coexistent with the molecular gas in the orion molecular ridge . in order to understand the behavior of the two masing lines we have modeled the radiative transfer of the rotational levels of p - h@xmath3o for the physical conditions of the orion molecular cloud . the radiative transfer method is described in gonzlez - alfonso & cernicharo ( 1997 ) , and the model consists of a molecular shell with diameter of @xmath39 cm ( size of 15@xmath40 at 450 pc ) which expands at a constant velocity of 25 km s@xmath26 . collisional rates between water vapor and helium were taken from _ green _ , maluendes & mclean ( 1993 ) . the helium abundance was assumed to be 0.1 , and the rates were corrected to take into account the collisions between h@xmath23o and h@xmath23 . we calculated the statistical equilibrium populations of the lowest 45 rotational levels of para - h@xmath3o for different temperatures ( t@xmath41=100 , 150 , 200 and 300 k ) , column densities n(p - h@xmath3o ) , and volume densities ( n(h@xmath3)=@xmath42 , @xmath43 and @xmath44 @xmath45 ) . figure 4 shows the main beam brightness temperatures ( t@xmath31 ) ( as observed by a telescope of 15@xmath40 beam size ) for the 183 and 325 ghz para - water lines ( thin and broad lines , respectively ) , together with @xmath30 ( dashed lines ) . these t@xmath31 were computed from the integrated intensity by assuming that the spectral emission is gaussian - shaped . inspection of fig . 4 indicates that the line intensity ratio @xmath30 increases with n(p - h@xmath3o ) for low column densities ( which depend on t@xmath41 and n(h@xmath3 ) ) . both masers are unsaturated in these conditions , but the higher opacity of the 183 ghz transition makes this line more sensitive to variations of n(p - h@xmath3o ) . for higher values of n(p - h@xmath3o ) , the 183 ghz line becomes saturated and the exponential amplification of the 325 maser line yields a decrease of @xmath30 . finally , when both the 183 and 325 ghz lines are saturated , @xmath30 approaches a nearly constant value or even decreases below 1 for high n(h@xmath3 ) and low t@xmath41 . the maser at 183 ghz is quenched for these later conditions , although t@xmath31 can still remain above t@xmath41 due to the suprathermal excitation of the line ( see cer94 ) . even for relatively low column densities ( n(p - h@xmath3o)@xmath46 @xmath47 ) , low temperatures ( 100 k ) and volume density ( @xmath42 @xmath45 ) , the 183 ghz line has an intensity larger than 10 k ( see cer94 ) . however , the possibility of appreciable amplification for the 5@xmath17 - 4@xmath18 line is much more restricted than for the 183 ghz line , due to the high energy of the levels involved in the 325 ghz line ( @xmath48 k ) , and to the higher frequency and einstein coefficient of this transition . this fact explains the difference in spatial extent between both transitions , so that the the 325 ghz line is spatially restricted to the plateau while the 183 ghz line is in addition detected in the ridge ( cer94 ) . the water vapor column density that fits the observed log * r*@xmath49 depends strongly on the assumed values of n(h@xmath3 ) and t@xmath41 . the higher n(h@xmath3 ) and t@xmath41 , the lower n(p - h@xmath3o ) that is needed to obtain an appreciable amplification of the 325 ghz line . 4 shows that * r*@xmath50 is obtained in different panels for n(p - h@xmath3o ) ranging from @xmath51 @xmath47 ( n(h@xmath3)=@xmath44 @xmath45 , t@xmath41=300 k ) to @xmath52 @xmath47 ( n(h@xmath3)=@xmath42 @xmath45 , t@xmath41=100 k ) . however , some of these models are not compatible with the observed intensities . for n(h@xmath3)@xmath53@xmath43 @xmath45 and t@xmath41@xmath54150 k , we find that * r*@xmath55 yields t@xmath31(325 ) in excess of @xmath56 k and t@xmath31(183)@xmath54@xmath57 k. both the observed intensities and * r * are only compatible with more moderate values of n(h@xmath3 ) and/or t@xmath41 . the physical reason is that , for high values of n(h@xmath3 ) and/or t@xmath41 , the collisional pumping of the 183 ghz line becomes so efficient that the emission in this line reaches high intensities for column densities that provide t@xmath31(325 ) of 50100 k. of course , the plateau may have regions with very high densities and temperatures ( which will give rise , for example , to the emission at 22 ghz and to the narrow spectral features observed at 183 and 325 ghz ) , but these will be much smaller than the observed size of the cloud . the widespread emission from the plateau we observe at 183 ghz and 325 ghz is only compatible with moderate values of n(h@xmath3 ) and t@xmath41 . in our models , the 22 ghz line will have intensities similar to those already observed in orion ( genzel et al , 1981 ) only for the highest kinetic temperatures and volume densities in figure 4 . the brightest spots at 22 ghz could be correlated with the narrow features at 183 ghz , and with the relatively weak lines at 325 ghz . brightness temperatures above 10@xmath58 k can be obtained for large column densities , t@xmath59@xmath54150 k and n(h@xmath3)@xmath5410@xmath58 @xmath45 . lower limits for n(h@xmath3 ) and t@xmath41 can be obtained from the observations of other molecular lines ( e.g. , blake et al . 1987 ) , so that we adopt n(h@xmath3)@xmath53@xmath43 @xmath45 and t@xmath41=100150 k. for these conditions we obtain n(p - h@xmath3o ) in the range @xmath60@xmath61 @xmath47 , and hence n(h@xmath3o ) in the range @xmath62@xmath52 @xmath47 . the water vapor abundance can be derived from the co data taken with similar angular resolution ( see cer94 ) . for the intermediate velocity wind cer94 derived a co column density of 10@xmath63 @xmath47 . hence , the x(h@xmath3o)/x(co ) abundance ratio in the plateau is around 1 , i.e. , x(h@xmath3o)@xmath28@xmath64 . in the ridge molecular cloud the 325 ghz is very weak . our models and the 183 ghz data provide an estimate for x(h@xmath3o ) of a few 10@xmath6 - 10@xmath7 ( see cer94 ) which is in good agreement with our iso results ( see cernicharo et al . 1998 , 1999 ) . the comparison of several masing transitions arising in relatively low energy levels of h@xmath3o allows us to constrain the physical conditions of the different emitting regions . so far , ground - based observations of these transitions with large radio telescopes are the only means to obtain the spatial distribution of h@xmath3o in interstellar clouds . j. cernicharo and e. gonzlez - alfonso acknowledge spanish dges for this research under grants pb96 - 0883 and esp98 - 1351e . pardo gratefully acknowledges the financial support of the _ observatoire de paris - meudon _ , _ cnes _ and _ mto - france_. the cso is supported by nsf contract # ast-9615025 .

we present observations of the 5@xmath04@xmath1 transition of water vapor _ at 325.15 ghz _ taken with the cso telescope towards orion irc2 . the emission is more extended than that of other molecular species such as ch@xmath2oh . however , it is much less extended than the emission of water vapor at 183.31 ghz reported by cernicharo et al ( 1994 ) . a comparison of the line intensities at 325.15 ghz and 183.31 ghz puts useful constraints on the density and temperature of the emitting regions and allows an estimate of h@xmath3o abundance , x(h@xmath3o ) , of @xmath410@xmath5 in the plateau and @xmath410@xmath6 - 10@xmath7 in the ridge . 21.75 cm -0.5 cm = 20pt ism : molecules ism : individual ( orion irc2 ) line : profiles masers radio lines : ism submillimeter

astro-ph9906037

threshold nets are obtained by assigning a weight @xmath1 , from a distribution @xmath2 , to each of @xmath3 nodes and connecting any two nodes @xmath4 and @xmath5 whose combined weights exceed a certain threshold , @xmath6 : @xmath7 @xcite . threshold nets can be produced of ( almost ) arbitrary degree distributions , including scale - free , by judiciously choosing the weight distribution @xmath2 and the threshold @xmath6 , and they encompass an astonishingly wide variety of important architectures : from the star graph ( a simple cartoon " model of scale - free graphs consisting of a single hub ) with its low density of links , @xmath8 , to the complete graph . studied extensively in the graph - theoretical literature @xcite , they have recently come to the attention of statistical and non - linear physicists due to the beautiful work of hagberg , swart , and schult @xcite . , and ( b ) its box representation , highlighting modularity . nodes are added one at a time from bottom to top , @xmath9 s on the left and @xmath10 s on the right.,scaledwidth=35.0% ] hagberg _ et al_. , exploit the fact that threshold graphs may be more elegantly encoded by a two - letter sequence , corresponding to two types of nodes , @xmath9 and @xmath10 @xcite . as new nodes are introduced , according to a prescribed sequence , nodes of type @xmath9 connect to none of the existing nodes , while nodes of type @xmath10 connect to all of the nodes , of either type : @xmath11 and @xmath12 . in fig . [ graph_box](a ) we show an example of the threshold graph obtained from the sequence @xmath13 . note the _ modular _ structure of threshold graphs : a subsequence of @xmath14 consecutive @xmath10 s gives rise to a @xmath15-clique , while nodes in a subsequence of @xmath9 s connect to @xmath10 nodes thereafter , but not among one another . we highlight this modularity with a diagram of boxes ( similar to @xcite ) : oval boxes enclose nodes of type @xmath9 , that are not connected among themselves , while rectangular boxes enclose @xmath16-cliques of @xmath10-nodes @xcite . a link between two boxes means that all of the nodes in one box are connected to all of the nodes in the other , fig . [ graph_box](b ) . given the sequence of a threshold net , there exist fast algorithms to compute important structural benchmarks , besides its modularity , such as degree distribution , triangles , betweenness centrality , and the spectrum and eigenvectors of the graph laplacian @xcite . the latter are a crucial determinant of dynamics and synchronization and have applications to graph partitioning and mesh processing @xcite . perhaps more importantly , it becomes thus possible to _ design _ threshold nets with a particular degree distribution , spectrum of eigenvalues , etc . , @xcite . despite their malleability , threshold nets are limited in some obvious ways , for example their diameter is 1 or 2 , regardless of the number of nodes @xmath3 . our idea consists of studying the broader class of nets that can be constructed from a sequence ( formed from two or more letters ) by deterministic rules of connectivity on their own right . it is truly this property that gives the nets all their desired attributes : modularity ( as in everyday life complex nets ) , easily computable structural measures including the possibility of design and a high degree of compressibility . roughly speaking , each additional letter to the alphabet allows for an increase of one link in the nets diameter , so that the three - letter nets possess diameter 3 or 4 ( some of the new types of two - letter nets have diameter 3 ) . this modest increase is very significant , however , in view of the fact that the diameter of many everyday life complex nets is not much larger than that @xcite . sequence nets gain us much latitude in the types of nets that can be described in this elegant fashion , while retaining much of the analytical appeal of threshold nets . another unusual property of sequence nets is that any ensemble of sequence nets admits a natural ordering ; simply list them alphabetically according to their sequences . one may use this ordering for exploring eigenvalues and other structural properties of sequence nets . in this paper , we make a first stab at the general class of _ sequence nets_. in section [ two - letter ] we explore systematically all of the possible rules for creating connected sequence nets from a two - letter alphabet . applying symmetry arguments , we find that threshold nets are only one of three equivalence classes , characterized by the highest level of symmetry . we then discuss the remaining two classes , showing that also then there is a high degree of modularity and that various structural properties can be computed easily . curiously , the new classes of two - letter sequence nets can be related to a generalized form of threshold nets , where the difference @xmath17 , rather than the sum of the weights , is the one compared to the threshold @xmath6 . in section [ three - letter ] we derive all possible forms of connected three - sequence nets . symmetry arguments lead us to the discovery of 30 distinct equivalence classes . among these classes , we identify a natural extension of threshold nets to three - letter sequence nets . despite the enlarged alphabet , 3-letter sequence nets do retain many of the desirable properties of threshold and 2-letter sequence nets . we also show that at least some of the 3-letter sequence nets can be mapped into threshold nets with _ two _ thresholds , instead of one . we conclude with a summary and discussion of open problems in section [ conclude ] . consider graphs that can be constructed from sequences @xmath18 of the two letters @xmath9 and @xmath10 . we can represent any possible rule by a @xmath19 matrix * r * whose elements indicate whether nodes of type @xmath4 connect to nodes of type @xmath5 : @xmath20 if the nodes connect , and 0 otherwise ( @xmath21 stands for @xmath22 , respectively ) . [ graph_box ] gives an example of the graph obtained from the sequence @xmath13 , applying the _ threshold _ rule @xmath23 . since each element can be @xmath24 or @xmath25 independently of the others , there are @xmath26 possible rules . we shall disregard , however , the four rules that fail to connect between @xmath9 and @xmath10 , @xmath27 for they yield simple _ disjoint _ graphs of the two types of nodes : @xmath28 yields isolated nodes only , @xmath29 yields one complete graph of type @xmath9 and one of type @xmath10 , @xmath30 yields a complete graph of type @xmath9 and isolated nodes of type @xmath10 , etc . applied to the sequence @xmath13 ( a ) , and from @xmath31 applied to the reverse - inverted sequence @xmath32 ( b ) , are identical.,scaledwidth=35.0% ] the list of remaining rules can be shortened further by considering two kinds of symmetries : ( a ) permutation , and ( b ) time reversal . _ permutation _ is the symmetry obtained by permuting between the two types of nodes , @xmath33 . thus , a permuted rule ( @xmath34 and @xmath35 ) acting on a permuted sequence ( @xmath36 ) yields back the original graph @xcite . _ time reversal _ is the symmetry obtained by reversing the arrows ( time " ) in the connectivity rules , or taking the transpose of @xmath37 . the transposed rule acting on the reversed sequence @xmath38 yields back the original graph . the two symmetry operations are their own inverse and they form a symmetry group . in particular , one may combine the two symmetries : a rule with @xmath34 applied on a reversed sequence with inverted types @xmath39 yields back the original graph , see fig . [ time_reversal ] . all of the four rules @xmath40 are equivalent and generate threshold graphs . @xmath41 is the rule for threshold graphs exploited by hagberg et al . , @xcite , and @xmath42 is equivalent to it by permutation . @xmath31 is obtained from @xmath41 by time reversal and permutation ( fig . [ time_reversal ] ) , and @xmath43 is obtained from @xmath41 by time reversal . the two rules @xmath44 are equivalent , by either permutation or time reversal , and generate non - trivial bipartite graphs that are different from threshold nets ( fig . [ abgraphs ] ) . the rule @xmath45 generates complete bipartite graphs . however , the complete bipartite graph @xmath46 can also be produced by applying @xmath47 to the sequence @xmath48 of @xmath49 @xmath9 s followed by @xmath50 @xmath10 s , so the rule @xmath51 is a `` degenerate '' form of @xmath47 . one could see that this is the case at the outset , because of the symmetrical relations @xmath52 , @xmath11 : these render the ordering of the @xmath9 s and @xmath10 s in the graph s sequence irrelevant . by the same principle , @xmath53 and @xmath54 are degenerate forms of @xmath41 and @xmath42 , respectively . they yield threshold graphs with segregated sequences of @xmath9 s and @xmath10 s . the two rules @xmath55 are equivalent , by either permutation or time reversal , and generate non - trivial graphs different from threshold graphs and graphs produced by @xmath47 ( fig . [ abgraphs ] ) . finally , the rule @xmath56 is a degenerate form of @xmath57 ( or @xmath58 ) and yields only complete graphs ( which are threshold graphs , so @xmath59 is subsumed also in @xmath60 ) . , applying rules @xmath47 ( a ) , @xmath41 ( b ) , and @xmath57 ( c ) . note the figure - background symmetry of ( a ) and ( c ) : the graphs are the inverse , or complement of one another ( see text ) . the inverse of the threshold graph ( b ) is also a ( two - component ) threshold graph , obtained from the same sequence and applying the rule @xmath42 ( @xmath41 s complement).,scaledwidth=47.0% ] to summarize , @xmath41 , @xmath47 , and @xmath57 are the only two - letter rules that generate different classes of non - trivial connected graphs . there is yet another amusing type of symmetry : applying @xmath47 and @xmath57 to the same sequence yields _ complement _ , or _ inverse _ graphs nodes are adjacent in the inverse graph if and only if they are _ not _ connected in the original graph . the figure - background symmetry manifest in the rules @xmath47 and @xmath57 ( @xmath61 ) is also manifest in the graphs they produce ( fig . [ abgraphs]a , c ) . on the other hand , the inverse of threshold graphs are also threshold graphs . also , the complement of a threshold rule applied to the complement ( inverted ) sequence yields back the original graph . in this sense , threshold graphs have maximal symmetry . @xmath47-graphs are typically less dense , and @xmath57-graphs are typically denser than threshold graphs . possible connections between nodes of type @xmath9 and @xmath10 . ( b ) three equivalent representations of the threshold rule @xmath41 . the second and third diagram are obtained by label permutation and time - reversal , respectively . ( c ) diagrams for @xmath47 and @xmath57 . note how they complement one another to the full set of connections in part ( a).,scaledwidth=25.0% ] the connectivity rules have an additional useful interpretation as directed graphs , where the nodes represent the letters of the sequence alphabet , a directed link , e , g . , from @xmath9 to @xmath10 indicates the rule @xmath52 , and a connection of a type to itself is denoted by a self - loop ( fig . [ graph_notation ] ) . because the rules are the same under permutation of types , there is no need to actually label the nodes : all graph isomorphs represent the same rule . likewise , time - reversal symmetry means that graphs with inverted arrows are equivalent as well . note that the direction of self - loops is irrelevant in this respect , so we simply take them as undirected . we shall make use of this notation , extensively , for the analysis of 3-letter sequence nets in section [ three - letter ] . a very special property of sequence nets is the fact that any arbitrary ensemble of such nets possesses a natural ordering , simply listing the nets alphabetically according to their sequences . in contrast , think for example of the ensemble of erds - rnyi random graphs of @xmath3 nodes , where links are present with probability @xmath49 : there is no natural way to order the @xmath62 graphs in the ensemble @xcite . plotting a structural property against the alphabetical ordering of the ensemble reveals some inner structure of the ensemble itself , yielding new insights into the nature of the nets . as an example , in fig . [ eigs_2threshold ] we show @xmath63 , the second smallest eigenvalue , for the ensemble of connected threshold nets containing @xmath64 nodes ( there are @xmath65 graphs in the ensemble , since their sequences must all start with the letter @xmath9 ) . notice the beautiful pattern followed by the eigenvalues plotted in this way , which resembles a fractal , or a cayley tree : the values within the first half of the graphs in the @xmath1-axis repeat in the second half , and the pattern iterates as we zoom further into the picture . nodes , plotted against their alphabetical ordering.,scaledwidth=45.0% ] structural properties of the new classes of two - letter sequence nets , @xmath47 and @xmath57 , are as easily derived as for threshold nets . here we focus on @xmath47 alone , which forms a subset of bipartite graphs . the analysis for @xmath57 is very similar and often can be trivially obtained from the complementary symmetry of the two classes . all connected sequence nets in the @xmath47 class must begin with the letter @xmath9 and end with the letter @xmath10 . a sequence of this sort may be represented more compactly @xcite by the numbers of @xmath9 s and @xmath10 s in the alternating layers , @xmath66 . we assume that there are @xmath3 nodes and @xmath14 layers ( @xmath14 is even ) . we also use the notation @xmath67 and @xmath68 for the total number of @xmath9 s and @xmath10 s , as well as @xmath69 and likewise for @xmath70 . finally , since all the nodes in a layer have identical properties we denote any @xmath9 in the @xmath4-th layer by @xmath71 and any @xmath10 in the @xmath5-th layer by @xmath72 . with this notation in mind we proceed to discuss several structural properties . : since @xmath9 s connect only to subsequent @xmath10 s ( and @xmath10 s only to preceding @xmath9 s ) the degree @xmath73 of the nodes is given by @xmath74 : there are no triangles in @xmath47 nets so the clustering of all nodes is zero . : every @xmath9 is connected to the last @xmath10 , so the distance between any two @xmath9 s is 2 . every @xmath10 is connected to the first @xmath9 in the sequence , so the distance between any two @xmath10 s is also 2 . the distance between @xmath75 and @xmath76 is 1 if @xmath77 ( they connect directly ) , and 3 if @xmath78 ( @xmath75 links to @xmath79 , that links to @xmath80 , that links to @xmath76 ) . : because of the time - reversal symmetry between @xmath9 and @xmath10 , it suffices to analyze @xmath10 nodes only . the result for @xmath9 can then be obtained by simply reversing the creation sequence and permuting the letters . the vertex betweenness @xmath81 of a node @xmath82 is defined as : @xmath83 where @xmath84 is the number of shortest paths from node @xmath85 to @xmath86 ( @xmath87 ) , excluding the cases that @xmath88 or @xmath89 . @xmath90 is the number of shortest paths from @xmath85 to @xmath86 that goes through @xmath82 . the factor @xmath91 appears for undirected graphs since each pair is counted twice in the summation . the betweenness of @xmath10 s can be calculated from lower layers to higher layers recursively . in the first b - layer @xmath92 and @xmath93 for @xmath94 . the second term on the rhs accounts for the shortest paths from layer @xmath95 to itself and all previous layers of @xmath9 , and the third term corresponds to paths from @xmath95 to @xmath72 to @xmath71 ( @xmath96 ) to @xmath97 . although this recursion can be solved explicitly it is best left in this form , as it thus highlights the fact that the betweenness centrality increases from one layer to the next . in other words , the networks are _ modular _ , where each additional @xmath10-layer dominates all the layers below . : unlike threshold nets , for @xmath47 nets the eigenvalues are _ not _ integer , and there seems to be no easy way to compute them . instead , we focus on the second smallest and largest eigenvalues , @xmath63 and @xmath98 , alone , for their important dynamical role : the smaller the ratio @xmath99 the more susceptible the network is to synchronization @xcite . consider first @xmath63 . for @xmath47 it is easy to show that both the _ vertex _ and _ edge connectivity _ are equal to @xmath100 . then , following an inequality in @xcite , @xmath101 the upper bound seems stricter and is a reasonable approximation to @xmath63 ( see fig . [ l2bounds ] ) . nets with @xmath64 against their alphabetical ordering ( solid curve ) , and their upper and lower bounds ( broken lines).,scaledwidth=40.0% ] for @xmath98 , using theorem 2.2 of @xcite one can derive the bounds @xmath102 but they do not seems very useful , numerically . playing with various structural properties of the nets , plotted against their alphabetical ordering , we have stumbled upon the approximation @xmath103 where @xmath104 is the average degree of the graph , see fig . [ l2approx ] . the approximation is exact for bipartite _ complete _ graphs ( @xmath105 ) and the relative error increases slowly with @xmath3 ; it is roughly at 10% for @xmath106 . nets with @xmath64 against their alphabetical ordering ( solid curve ) , and its approximated value ( broken line).,scaledwidth=40.0% ] in @xcite it was shown that threshold graphs have a mapping to a sequence net , with a unique sequence ( under the threshold rule " @xmath41 ) ; and conversely , for any @xmath41-sequence net there exists a set of weights @xmath107 of the nodes ( not necessarily unique ) , such that connecting any two nodes that satisfy @xmath7 reproduces the sequence net . here we establish a similar relation between @xmath47- ( or @xmath57- ) sequence nets and a different kind of threshold net , where connectivity is decided by the difference @xmath17 rather than the sum of the weights . we begin with the mapping of a weighted set of nodes to a @xmath47-sequence net . let a set of @xmath3 nodes have weights @xmath107 ( @xmath108 ) , taken from some probability density , and we assume @xmath109 , without loss of generality . denote nodes with @xmath110 as type @xmath9 and nodes with @xmath111 as type @xmath10 . finally , connect any two nodes @xmath4 and @xmath5 that satisfy @xmath112 . the resulting graph can be constructed by a unique sequence under the rule @xmath47 , obtained as follows . for convenience , rewrite the set of weights as @xmath113 where the first @xmath114 weights correspond to @xmath9-nodes and the rest to @xmath10-nodes . denote the creation sequence by @xmath18 and determine the @xmath115 by the algorithm ( in pseudo - code ) : set @xmath116 , @xmath117 for @xmath118 , do : 0.4 cm if @xmath119 0.8 cm set @xmath120 and @xmath121 0.4 cm else 0.8 cm set @xmath122 and @xmath123 end . it is understood that if the @xmath124 are exhausted before the end of the loop , the remainder @xmath10-nodes are automatically affixed to the end of the sequence ( and similarly for the @xmath125 ) . for example , using this algorithm we find that the difference - threshold " graph resulting from the set of weights @xmath1261,2,3,5,7,16,17,20@xmath127 and @xmath128 , can be reproduced from the sequence @xmath13 , with the rule @xmath47 . consider now the converse problem : given a graph created from the sequence @xmath18 with the rule @xmath47 , we derive a ( non - unique ) set of weights @xmath129 such that connecting any two nodes with @xmath112 results in the same graph . rewrite first the creation sequence into its compact form @xmath130 , and assign weights @xmath131 for nodes @xmath9 in layer @xmath131 , weights @xmath132 for nodes @xmath10 in layer @xmath0 , and set the threshold at @xmath133 . for example , the sequence @xmath13 has a compact representation @xmath134 , with @xmath135 layers , so the three @xmath9 s in layer @xmath25 have weights @xmath25 , the two @xmath10 s in layer @xmath136 have weights @xmath137 , the two @xmath9 s in layer @xmath138 have weights @xmath138 , and the single @xmath10 in layer @xmath139 has weight @xmath140 . the weights @xmath141 , with connection threshold @xmath142 , reproduce the original graph . sequence graphs obtained from the rule @xmath57 can be also mapped to difference threshold graphs in exactly the same way , only that the criterion for connecting two nodes is then @xmath143 , instead of @xmath112 , as for @xmath47 . the mapping of sequence nets to generalized threshold graphs may be helpful in the analysis of some of their properties , for example , for finding the _ isoperimetric number _ of a sequence graph @xcite . with a three - letter alphabet , @xmath144 , there are at the outset @xmath145 possible rules . again , these can be reduced considerably , due to symmetry . because the rule matrix has 9 entries ( an odd number ) no rule can be identical to its complement . thus , we can limit ourselves to rules with no more than 4 non - zero entries and apply symmetry arguments to reduce their space at the very end we can then add the complements of the remaining rules . in fig . [ 3nets ] we list all possible three - letter rules with two , three , and four interactions . rules that lead to disconnected graphs , and symmetric rules ( by label permutation or time - reversal ) have been omitted from the figure . and @xmath146 ) , and rules 3 , 12 , 13 , and 14 are degenerate cases of rules 2 , 6 , 7 , and 6 , respectively . this leaves us with fifteen distinct three - letter rules ( underlined ) , and their fifteen complements , for a total of 30 different classes of three - letter sequence nets.,scaledwidth=40.0% ] rule 2 @xcite is in fact not new : identifying nodes of type @xmath9 and @xmath146 ( as marked in rule 1 of the figure ) we can easily see that the rule is identical to the two - letter rule 8 . in the same fashion , rule 7 is the same as the two - letter threshold rule 4 . rule 3 is a degenerate form of 2 : because of the double connection @xmath147 and @xmath148 , the order at which @xmath10 and @xmath146 appear in the sequence relative to one another is inconsequential . ( on the other hand , the order of the @xmath10 s relative to @xmath9 s _ is _ important , since @xmath9 s connect only to those @xmath10 s that appear earlier in the sequence . ) then , given a sequence one can rearrange it by moving all the @xmath146 s to the end of the list . if we now apply 2 , @xmath52 and @xmath148 , then we get the same graph as from the original sequence under the rule 3 . the same consideration applies to rules , and , that are degenerate forms of 6 , 7 and 8 ( or 6 ) , respectively . we are thus left with only 15 distinct rules with fewer than 5 connections . to these one should add their complements , for a total of 30 distinct three - letter rules . note the resemblance of , , and to two - letter threshold nets . seems like a particularly symmetrical generalization and we will focus on it in much of our discussion below . while one can easily establish wether a graph is connected or not , _ a posteriori _ , with a burning algorithm that requires @xmath149 steps , it is useful to have shortcut rules that tell us how to avoid bad sequences at the outset : knowing that two - letter threshold graphs are connected if and only if their sequence ends with @xmath10 , deals with the question most effectively . analogous criteria exist for three - letter sequence graphs but they are a bit more complicated . for example , three - letter sequences interpreted with lead to connected graphs if and only if they satisfy : _ ( 1 ) the first a and the first c in the sequence appear before the last b. ( 2 ) the sequence does not start with b_. ( we assume that the sequence contains all three letters . ) for 1the requirements are : _ ( 1 ) the first a in the sequence must appear after the first b. ( 2 ) the last c in the sequence must appear before the last b. ( 3 ) the last a in the sequence must appear after the first c , and there ought to be at least one b between the two . _ similar criteria exist for all other three - letter rules and can be found by inspection . structural properties of three - letter sequence nets are analyzed as easily as those of two - letter nets , here we list , as an example , a few basic attributes of sequence nets . we use a notation similar to that of section [ new_classes ] . : @xmath9 and @xmath146 nodes form complete subgraphs , while @xmath10 nodes connect to all preceding @xmath9 s and @xmath146 s . thus the degree of the nodes are : @xmath150 : since the @xmath9 nodes make a subset complete graph @xmath151 , and likewise for @xmath146 , @xmath152 . the @xmath10 s do not connect among themselves , but they all connect to the nodes in the first layer ( which does not consist of @xmath10 s ) , so @xmath153 . for the distance of @xmath9 nodes from @xmath10 , we have @xmath154 where @xmath155 is the index of the first @xmath9-layer and @xmath156 is the index of the last @xmath10-layer . the first line follows since @xmath10 s are directly connected to preceding @xmath9 s and @xmath146 s . the second , and third and fourth lines are illustrated in fig . [ distance]a and b , respectively . the distance @xmath157 follows the very same pattern . finally , inspecting all different cases one finds @xmath158 in nets . ( a ) if @xmath159 and the first @xmath9 is below @xmath72 the distance is 2 . ( b ) if the first @xmath9 is above @xmath72 , then the first @xmath146 must be below ( @xmath10 ca nt start the sequence ) ; in that case if @xmath71 is below the last @xmath10 the distance is 3 , and otherwise the distance is 4 . only the relevant parts of the complete net are shown.,scaledwidth=40.0% ] : we have found no obvious way to compute the eigenvalues , despite the similarities between nets and two - letter threshold nets . however , plots of the eigenvalues against the alphabetical ordering of the nets once again reveals intriguing fractal patterns , and one can hope that these might be exploited at the very least to produce good bounds and approximations . in fig . [ r_r18 ] we plot the ratio @xmath160 for nets with @xmath161 against their alphabetical ordering . the @xmath1-axis includes sequences of nets that are not connected : in this case @xmath162 and synchronization is not possible . these cases show as gaps in the plot , for example , the big gap in the center corresponds to disconnected sequences that start with the letter @xmath10 ( see section [ connect ] ) . for nets consisting of @xmath161 nodes , against their alphabetical ordering . note the gap near the center , which corresponds to sequences of disconnected graphs . note also the mirror symmetry this is due to the mirror symmetry of the rule itself.,scaledwidth=40.0% ] some of the three - letter sequence nets can be mapped to generalized forms of threshold nets . for example , the following scheme yields a _ two_-threshold net , equivalent to three - letter sequence nets generated by the rule . let the nodes be assigned weights @xmath163 , from a random distribution , and connect any two nodes @xmath4 and @xmath5 that satisfy @xmath164 or @xmath165 . identifying nodes with weight @xmath166 with @xmath9 , nodes with @xmath167 with @xmath10 , and nodes with @xmath168 with @xmath146 , we see that all @xmath9 s connect to one another and all @xmath146 s connect to one another but the @xmath10 s do not , and @xmath9 s and @xmath146 s do not connect ; nodes of type @xmath9 and @xmath10 may or may not connect , and likewise for nodes of type @xmath146 and @xmath10 . to reflect the actual connections , the nodes of type @xmath9 and @xmath10 may be arranged in a sequence according to the algorithm in @xcite , for the threshold rule . also the nodes of type @xmath146 and @xmath10 may be arranged in a sequence , to reflect the actual connections , with the very same algorithm . because there are no connections between @xmath9 and @xmath146 the two results may be trivially merged . note , however , that once the @xmath9-@xmath10 sequence is established the order of the @xmath10 s is set , so the direction of connections between @xmath146 and @xmath10 ( @xmath169 or @xmath170 ) is _ not _ arbitrary . in our example , the mapping is possible to but not to . we have introduced a new class of nets , sequence nets , obtained from a sequence of letters and fixed rules of connectivity . two - letter sequence nets contain threshold nets , and in addition two newly discovered classes . the class can be mapped to a difference - threshold " net , where nodes @xmath4 and @xmath5 are connected if their weights difference satisfies @xmath143 . this type of net may be a particularly good model for social nets , where the weights might measure political leaning , economical status , number of offspring , etc . , and agents tend to associate when they are closer in these measures . we have shown that the structural properties of the new classes of two - letter sequence nets can be analyzed with ease , and we have introduced an ordering in ensembles of sequence nets that is useful in visualizing and studying their various attributes . we have fully classified 3-letter sequence nets , and looked at a few examples , showing that they too can be analyzed simply . the diameter of sequence nets grows linearly with the number of letters in the alphabet and for a 3-letter alphabet it is already 3 or 4 , comparable to many everyday life complex nets . realistic diameters might be achieved with a modest expansion of the alphabet . there remain numerous open questions : applying symmetry arguments we have managed to reduce the class of 3-leter nets to just 30 types , but we have not ruled out the possibility that some overlooked symmetry might reduce the list further ; the question of which sequences lead to connected nets can be studied by inspection for small alphabets , but we have no comprehensive approach to solve the problem in general ; we have shown how to map sequence nets to generalized types of threshold nets , in some cases is such a mapping always possible ? is there a systematic way to find such mappings for any sequence rule ? ; what kinds of nets would result if the connectivity rules applied only to the @xmath50 preceding letters , instead of to _ all _ preceding letters ? etc . we hope to tackle some of these questions in future work .

we study a new class of networks , generated by sequences of letters taken from a finite alphabet consisting of @xmath0 letters ( corresponding to @xmath0 types of nodes ) and a fixed set of connectivity rules . recently , it was shown how a binary alphabet might generate threshold nets in a similar fashion [ hagberg et al . , phys . rev . e 74 , 056116 ( 2006 ) ] . just like threshold nets , sequence nets in general possess a modular structure reminiscent of everyday life nets , and are easy to handle analytically ( i.e. , calculate degree distribution , shortest paths , betweenness centrality , etc . ) . exploiting symmetry , we make a full classification of two- and three - letter sequence nets , discovering two new classes of two - letter sequence nets . the new sequence nets retain many of the desirable analytical properties of threshold nets while yielding richer possibilities for the modeling of everyday life complex networks more faithfully .

0804.3776

as the number of agn surveyed by _ xmm - newton _ and _ chandra _ increases , it is becoming apparent that the vast majority show evidence for a narrow ( unresolved by _ xmm _ ) line at @xmath1 6.4 kev , due to emission from neutral iron ; recent papers describing such lines include gondoin ( 2003 ; ngc 3227 ) , page ( 2003 ; mrk 896 ) , pounds ( 2003 ; mrk 766 ) , petrucci ( 2002 ; mrk 841 ) , turner ( 2002 ; ngc 3516 ) , obrien ( 2001 ; mrk 359 ) , reeves ( 2001 ; mrk 205 ) and many others . in a number of cases ( e.g. , ngc 3783 kaspi 2002 ; ngc 5548 yaqoob 2001 ) the lines have actually been resolved by _ , with fwhm velocities typically @xmath5 5000 km s@xmath6 . iron k@xmath0 emission was first identified as a common feature by _ ginga _ ( pounds 1990 ; nandra & pounds 1994 ) ; observations by _ asca _ tended to find relatively broad profiles , although re - analysis of some of the data indicates that the lines _ may _ be narrower than originally measured ( lubinski & zdziarski 2001 ) . there is some disagreement over these results , however , with yaqoob ( 2002 ) stating that the _ asca _ calibration changes have a negligible effect on the line profiles . very few broad lines have been found in _ xmm _ data to date , examples being mcg @xmath76@xmath730@xmath715 ( fabian 2002 ) , mrk 205 ( reeves 2001 ) and mrk 509 ( pounds 2001 ) . however , not all these line profiles are the same , with mcg @xmath76@xmath730@xmath715 showing a strongly asymmetric line , presumably due to the strong gravitational forces and high velocity in the inner accretion disc . mrk 205 , however , is only well - fitted by a relativistic disc - line model if the disc is strongly ionised , since the broad emission peaks at @xmath1 6.7 kev . this is not the only conceivable explanation , as reeves ( 2001 ) discuss : the emission could come from a spherical distribution of clouds , rather than the planar accretion disc ; alternatively , the broad line may actually consist of a blend of different ionisation narrow lines . the broad line in mrk 509 is also apparently ionised . the narrow emission lines observed by _ xmm - newton _ and _ chandra _ are interpreted as fe k fluorescence from cold ( neutral ) matter far from the inner accretion disc . suggestions for the origin of the narrow line include the putative molecular torus , the broad line region ( blr ) or the very outer - most reaches of the accretion disc . the baldwin effect is well known for optical / uv emission lines , with baldwin ( 1977 ) first reporting that the ew of the c iv ( @xmath8 = 1549 ) line decreased with increasing uv luminosity . since then , significant anti - correlations have been found between the luminosity and various other ions , such as n v , he ii , c iii ] , mg ii and ly@xmath0 ( e.g. , tytler & fan 1992 ; zamorani 1992 ; green , forster & kuraszkiewicz 2001 ; dietrich 2002 ) , although the strengths of these correlations are still unclear . it should be noted that , although the baldwin effect is generally accepted to be an anti - correlation between equivalent width and luminosity , green ( 2001 ) claim that ew is actually more strongly correlated with redshift than luminosity for their data . however , croom ( 2002 ) find that , for 12 of the 14 lines tested , the stronger correlation is with absolute magnitude rather than redshift . iwasawa & taniguchi ( 1993 ) reported an x - ray baldwin effect in the fe k lines found in _ ginga _ observations of agn . they find a strong relationship for their seyfert sample , but were unable to conclude that it holds for qsos , due to poor constraints ; there is also a baldwin effect for the broad iron lines found in _ asca _ data ( nandra 1997 ) . such broad lines are thought to be produced through fluorescence of the accretion disc itself . nandra suggest , based on an earlier paper ( nandra 1995 ) , that this baldwin effect is due to the presence of an ionised skin on the accretion disc , with the degree of ionisation increasing with luminosity ; see also nayakshin ( 2000a , b ) . nandra ( 1997 ) also find that the narrow line core drops in intensity as the luminosity increases , but conclude that the entire baldwin effect in their data can be attributed to the broader line alone . in this paper we show , and attempt to explain , a baldwin effect for the narrow fe k@xmath0 lines measured in _ xmm _ data , for a sample including both seyfert galaxies and qsos . this sample consists of 53 type i agn ( table [ objects ] ) , these being mainly a combination of our own proprietary targets and public observations obtained from the online _ xmm _ science archive . a literature search also revealed four more objects for which the relevant data had been published ( ngc 5506 matt 2001 ; ngc 3516 turner 2002 ; 1h 0707@xmath7495 boller 2002 ; ton s180 vaughan 2002 and vaughan 2003 , private communication ) . .the type i agn included in this sample , ordered by redshift within the radio - quiet and radio - loud groups . the luminosities were calculated for the 210 kev rest frame and 90 per cent errors / upper limits are given for the rest - frame equivalent widths . @xmath9 narrow line seyfert 1 galaxies ; @xmath10 broad line seyfert 1 galaxies ; @xmath11 qsos [ cols= " < , < , < , < , < " , ] figure [ ew - lum ] plots the rest - frame ew against the de - reddened 210 kev luminosity . it can clearly be seen that there is a decrease in the ew as the luminosity increases the ` x - ray baldwin effect ' . the asurv ( astronomy survival analysis ; feigelson & nelson 1985 ) package can be used in the presence of censored ( upper limit ) data . this allows the spearman rank ( sr ) statistic to be applied to the complete dataset , and gives an anti - correlation between the ew and luminosity of @xmath4 99.98 per cent ( @xmath1 98.5 per cent if the upper limits are dropped ) . due to selection effects , luminosity and redshift are very strongly correlated , as shown in figure [ lum - z ] . hence , it is often difficult to determine whether the correlation in question is with luminosity or redshift . using the simple spearman rank , a weaker correlation ( 99 per cent ; 79.5 per cent if the upper limits are excluded ) was found between the line strength and the redshift . an alternative method for checking involves the partial spearman rank , which gives an indication as to which of the two relationships is stronger . in this case , agreeing with the simple spearman rank , the ew - luminosity correlation appears to be the dominant relationship . weighted linear regression can be used to find the slope of the best - fit line to the log - log plot of ew against luminosity that is , the power - law fit . this was performed using two different methods . firstly the upper limit measurements were completely removed ; this gave a value of @xmath70.17 @xmath12 0.08 . the second method used the linear regression option within asurv ; although upper limit values are accounted for , this method does not include the errors on the actual measurements . using this method , the slope is very similar , with a value of @xmath70.18 @xmath12 0.04 . as a further check to determine whether the baldwin effect could be due to evolution , the objects at higher redshifts were progressively removed from the sample . it was found , however , that even considering only those agn at z @xmath5 0.1 , there still remained an inverse correlation between ew and luminosity which was consistent in magnitude ( slope of 0.15 @xmath12 0.08 using asurv ) with that found for the complete sample . this implies that the decrease in line strength is predominantly a luminosity - dependent effect . there is a possible complication in that the highest luminosity agn tend to be radio - loud , since the average x - ray emission from radio - quiet agn is about three times lower than that from radio - loud objects ( zamorani 1981 ; worrall 1987 ) . the underlying worry , therefore , is that the ew is very low at the highest luminosities because of dilution through beaming effects . however , as fig . [ ew - lum ] shows , there are lower - luminosity radio - loud agn in this sample which have correspondingly higher ews . conversely , pb 5062 and pg 1634 + 706 radio - quiet agn at luminosities of @xmath1 10@xmath13 erg s@xmath6 have low upper limits of ew @xmath5 63 ev for their lines . hence , the anti - correlation observed is not simply due to beaming . to show the overall effect more clearly , figure [ bins ] was plotted . this shows the mean ew for a number of luminosity bins considering all the objects ( top plot ) and just the radio - quiet ones ( bottom ) . in both plots , the decrease in ew is obvious . to produce figure [ bins ] , upper limits were taken to be half of the value , together with an equally sized error bar ; a similar decrease in ew is also observed if the upper limits are dropped , although the luminosity range covered is lower . figure [ ew - gamma ] plots the ew against the power - law slope measured over the 210 kev rest - frame energy band . since flatter slopes are often taken to be indicative of reflection , it might be expected that they would correspond to stronger emission lines ( since the fluorescence line may come about through reflection processes ) , although the 210 kev band is not very sensitive to reflection components . however , this is clearly not the case ; if anything , there is a slight trend in the opposite direction , with some of the steeper slopes showing the highest ews . suggestions for the origin of the neutral fe k@xmath0 line include the putative molecular torus and the blr ; these theories will be investigated in the following sections , considering both the feasibility of the line production and possible explanations for the x - ray baldwin effect . yaqoob ( 2001 ) give the following equation for the ew of the narrow line in agn , for low optical depths : @xmath14 @xmath15 where e@xmath16 is the energy of fe k edge ( 7.11 kev for neutral iron ) and e@xmath17 the central energy of the line , both in units of kev . f@xmath18 is the fraction of the sky covered by blr clouds ; a@xmath19 , the abundance of iron relative to hydrogen , where 4.68 @xmath20 10@xmath21 is the solar value ( anders & grevesse 1989 ) ; n@xmath22 , the column density of the clouds in @xmath23 and @xmath24 , the fluorescence yield (= 0.34 for neutral fe ) . based on a particular blr model , yaqoob find that the strength of the fe k@xmath0 line in ngc 5548 is likely to be too large to come simply from the blr ; the predicted ew is 36 ev , whereas the measured value is almost four times larger , at 133 ev . they suggest that these differences could be explained by a decline in the continuum level shortly before the observation ; in this event , the line would not have had time to respond to the smaller continuum level and , so , would appear unusually strong . whatever the explantion for ngc 5548 , if the blr origin is correct , the reason why most of the objects in table [ objects ] have relatively high ew values compared to those predicted by the above equation must be explained . it must also be determined whether a variation in the parameters could explain the observed ew - luminosity correlation . the iron k line is produced by a different physical process ( fluoresence ) from the optical / uv emission lines ( photoionisation ) . nevertheless , changes in covering factor or abundance , for example , might be reflected in the blr optical / uv lines , so it is briefly examined whether such correlations exist . one possibility is that the covering fraction of the blr clouds is systematically underestimated in the above formula and decreases with increasing luminosity . there is no independent evidence for very high blr covering factors , however , and the observed ew of optical / uv lines can be reproduced using modest values ( few 10s of percent ) . for the agn sample used here , no correlation is found between the ew of the narrow iron line and those of civ @xmath81549 and ly@xmath0 ( values taken from wang , lu & zhou 1998 and constantin & shields 2003 ) . likewise , no correlation is found with the h@xmath25/civ ratio , which can be used as a ionisation diagnostic , or with the h@xmath25 fwhm ( obtained from dewangan 2002 ; kaspi 2000 ) or ew ( marziani 2003 ; v ' eron - cetty , v ' eron & gonalves 2001 ) ; this is shown in figure [ hbetafwhm ] . h@xmath25 line fluxes for the h@xmath25/civ ratio were taken from marziani ( 2003 ) , cruz - gonz ' alez ( 1994 ) and mulchaey ( 1994 ) . comparing the slope of the baldwin relation is problematic , as different optical / uv lines show different slopes . for example , using the same cosmology as in this paper , croom ( 2002 ) find a correlation for the civ line , produced in the blr , of ew @xmath2 l@xmath26 . they also find ew @xmath2 l@xmath27 for the ciii ] blend ( ciii ] , aliii and siiii ] ) . dietrich ( 2002 ) give a list of baldwin effect slopes for different emission lines , varying between @xmath1 @xmath70.24 and + 0.02 , for h@xmath28 = 65 km s@xmath6 mpc@xmath6 and @xmath29 = 0 . they mention that including a cosmological constant of 0.7 changes the luminosities by @xmath5 20 per cent . green et al . ( 2001 ) find a similar range , using h@xmath28 = 50 km s@xmath6 mpc@xmath6 and @xmath29 = 0 . the overall impression in the optical / uv is of a correlation between the slope of the baldwin relation and the ionisation potential , suggesting a change in the average continuum shape with luminosity or redshift ( depending which dominates the correlation found in a particular study ) . it is possible that the x - ray baldwin effect found here is produced by a corresponding variation in the shape of the continuum illuminating the fluorescing material , although one might expect then to see some correlations with the observed optical / uv lines . the range of luminosities of the agn in this sample over - laps with those investigated by dietrich ( 2002 ) . upon comparison of all the objects with those with log@xmath8l@xmath30(1450 ) @xmath4 44 , they find little change in the slope of the baldwin relation for the different ranges , so the fact that qsos tend to have larger l@xmath31/l@xmath32 ratios than seyfert galaxies is not likely to be an important factor . alternatively it could be argued that the iron abundance has been underestimated and varies with luminosity . predicting the trend of metallicity in agn is complicated since their evolutionary history is not known sufficiently well . blr metallicities appear to be near - solar , but comparison of blr observations with detailed photoionisation models suggest that blr metallicity does increase with luminosity ( e.g. , hamann & ferland 1993 ; shemmer & netzer 2002 ) . this is oppposite to the trend expected if the x - ray baldwin effect is to be explained by an abundance effect . deriving the iron abundance from optical line ratios is made difficult due to the thermostatic effect of fe ii ( which is a major coolant ) . for the present sample , there is no correlation between the strength of the x - ray fe k@xmath0 line and the optical fe ii feature at @xmath84590 ( figure [ feii ] ; fe ii ews from marziani 2003 ) . the predicted ew is linearly dependent on column density in the above equation , but yaqoob et al . ( 2001 ) note that the approximations are only valid for n@xmath22 @xmath5 5.6 @xmath20 10@xmath33 @xmath23 ( i.e. assuming an fe k absorption optical depth much less than unity ) . although some blr models suggest higher column densities ( e.g. , radovich & rafanelli 1994 ; recondo - gonzalez 1997 ) , values @xmath34 @xmath23 are thought ` normal ' . typical blr velocities are of the order 5000 km s@xmath6 , which is resolvable by _ chandra_. to date , the velocity width of the narrow iron line has only been measured a few times , and the results are not entirely consistent . kaspi ( 2002 ) resolved the line profile of ngc 3783 with the _ chandra _ high energy transmission grating spectrometer ( hetgs ) , finding a fwhm velocity of @xmath1 1700 km s@xmath6 , which is low for a blr but consistent with originating in the inner part of the torus . likewise , ogle ( 2000 ) obtain a fwhm value of 1800 @xmath12 200 km s@xmath6 for the narrow fe k@xmath0 line detected in ngc 4151 . the width of the iron lines in mcg@xmath76@xmath730@xmath715 have also been measured with the hetgs . the situation is complicated by the presence of the various different components of the line in this object , both resolved and unresolved , narrow and broad . however , lee ( 2002 ) obtain a width of 11 000@xmath35 km s@xmath6 for the resolved narrow component when considering the full observation ; this drops to 3600@xmath36 km s@xmath6 for the ` high ' flux state . within the errors , these measurements are consistent with a fwhm of @xmath1 6000 km s@xmath6 , which would indicate that the line is not formed in the torus . yaqoob ( 2001 ) measure a fwhm of @xmath1 4500 km s@xmath6 for ngc 5548 , which also supports an origin in the blr . it should be noted that , although the measurements are all broadly consistent , and indicate an origin in either the outer blr or torus , there are relatively few grating measurements of the line , so it is unclear how narrow the line core truly is . an alternative possibility is that the ` narrow ' lines may actually be formed in the accretion disc itself . nayakshin ( 2000a , b ) discusses how a baldwin effect could be produced by the ionisation of the skin of the disc , as the luminosity increases . a recent paper by yaqoob et al . ( 2003 ) suggests that such narrow lines may come from a disc which is viewed face - on and has a flat emissivity profile . similar suggestions have been made for optical / uv blr lines . however , as mentioned above , no correlation is found here between the narrow iron line ew and the h@xmath25 parameters . there is a difference between the blrs found in narrow line and broad line seyfert galaxies the nls1s show much narrower emission lines , hence their name . it has been previously noted that nls1s tend to show weaker h@xmath25 emission ( e.g. , gaskell 1985 ) and this was confirmed upon comparing the h@xmath25 ew of samples of nls1s and bls1s ( taken from marziani 2003 ) student s t - test gave a very low probability of the two groups originating in the same population , with the nls1s having a much lower mean ew . considering the few objects in the present sample , this difference in line strengths is also found . when comparing the fe k@xmath0 ews , however , there is no appreciable difference between the narrow- and broad - line samples , shown both by the t - test and asurv ( to account for the upper limits ) . possible support for the accretion disc origin lies in the variability of the narrow line in mrk 841 ( petrucci 2002 ) . they find that the line varied between two observations separated by 15 hours , although , within the errors , the change in line flux is not large . such a rapid variation is hard to explain if the line is formed in distant reprocessing matter , indicating its origin might be closer to the central engine ; the line width is only constrained to be @xmath5 170 ev . to date , mrk 841 is the only object in which this variation has been observed . overall , while the blr remains a possible source for some of the narrow iron emission , we are unable to explain how the blr alone could produce the strongest observed narrow lines nor why the baldwin relation should exist . in particular , it is unclear why the most luminous objects should have fairly normal blrs compared to the lower - luminosity objects , as judged from their optical / uv blr lines , yet have no detectable narrow 6.4 kev iron line . krolik , madau & . zycki ( 1994 ) discuss the production of the iron k@xmath0 line in the torus . they compute that for a thomson optical depth , @xmath37 , of 0.51 , the ew of the line should be of the order 100 ev , assuming an unobscured view ; for @xmath37 = 2 , this value approximately halves , to 55 ev . they also find a small decrease in ew for increasing opening angle of the torus . the value of @xmath1 100 ev , for a low optical depth , is , as they point out , in broad agreement with the ew measured in many seyfert 1 galaxies , suggesting that the fe k@xmath0 emission line could be due solely to reflection / fluorescence from the torus and not linked to the accretion disc or blr at all . ghisellini , haardt & matt ( 1994 ) also mention that the torus could produce an emission line of ew @xmath1 90 ev , if the column density is @xmath38 10@xmath39 @xmath23 . it should be noted that both of these papers used a value of 3.31 @xmath20 10@xmath21 for the solar abundance of iron , rather than the 4.68 @xmath20 10@xmath21 assumed by yaqoob ( 2000 ) . assuming the ew scales linearly with the abundance , the values from krolik ( 1994 ) and ghisellini ( 1994 ) should be scaled to @xmath1 141 or @xmath1 127 ev , respectively , to compare to yaqoob s result . as mentioned earlier , the presence of a neutral reflection component would indicate the existence of cool , compton thick material ( guilbert & rees 1988 ; lightman & white 1988 ; george & fabian 1991 ) , such as the torus . some of the spectra here show evidence for such reflection , although the values are generally poorly constrained . this is not entirely surprising though , since most of the objects are at fairly low redshift , meaning that the rest frame bands covered by _ xmm _ do not extend much above @xmath1 10 kev , while the compton reflection hump is expected to peak around 3050 kev ; reflection components may not be detected in the present data , but that is not to say that they do not exist . _ if _ , however , reflection components are not found in the spectra , this does not necessarily rule out the torus as the origin of the narrow line . instead , the material of the torus could be compton thin ; if this were the case , no strong reflection component would be expected . matt , guainazzi & maiolino ( 2003 ) show that , for a compton thin torus with n@xmath22 = 2 @xmath20 10@xmath33 @xmath23 , a fluorescent iron line with ew @xmath1 80 ev can be formed . this is , therefore , a further possibility for production of the narrow line . knigl & kartje ( 1994 ) discuss a dusty disc - driven hydromagnetic wind model for the torus , finding that , if l@xmath40 @xmath38 1.5 @xmath20 10@xmath41(m/10@xmath42 ) erg s@xmath6 , where l@xmath40 is the 2100 @xmath43 m infrared luminosity , then the radiation pressure force could be expected to flatten the dust distribution ; this causes the opening angle of the torus to increase , leading to a reduction in the covering factor . this relates to krolik ( 1994 ) , who found that the ew of the fe k@xmath0 line decreases slightly with increasing opening angle . in this way , an increase in luminosity and subsequent decrease in covering factor could lead to a smaller ew . this result is similar to work done by mushotzky & ferland ( 1984 ) , who proposed a luminosity - dependent ionisation parameter and , hence , covering factor to explain the original baldwin effect . an alternative explanation , suggested by ohsuga & umemura ( 2001 ) , is that relatively low - luminosity agn are more likely to contain dusty walls of gas , supported by radiation pressure from a circumnuclear starburst ; the stronger radiation pressure from the more luminous agn may prevent these from forming . if the x - ray baldwin effect is , indeed , caused by a decrease in the covering factor of the torus , then this has implications for the number of type-2 qsos . many seyfert 2 galaxies are known , but very few obscured qsos have yet been discovered ( e.g. , derry 2003 and references therein ) . a luminosity - dependent drop in the covering factor provides goes some way towards explaining the lack of high luminosity obscured sources , although obscuration should still arise due to mergers and the birth of the qsos . there is an apparent x - ray baldwin effect for the narrow , neutral iron line observed in many agn : as the 210 kev rest frame luminosity increases , the equivalent width of the line drops . the reason for this correlation is uncertain , but one possibility is a decline in the covering factor of the putative molecular torus as the luminosity increases . this decrease in covering fraction could be due to increased radiation pressure flattening the torus , leading to an increased opening angle and , hence , smaller covering factor . although the blr remains a possibility for the origin of some of the narrow line , it is difficult to explain how the strongest lines could be formed , or what it is that leads to the baldwin effect . the negative correlation between ew of the line and the luminosity of the object is not solely due to a dilution effect by beaming in the radio - loud sources , since the effect is also observed in the radio - quiet objects alone . given the resolution of epic and the _ chandra _ hetgs , it is not currently possible to rule out the origin of the line being either the blr clouds or reflection off distant matter ( i.e. , the torus ) . future observations with calorimeter - based x - ray detectors ( such as the x - ray spectrometer on - board _ astro - e2 _ , or _ xeus _ ) will provide sufficient spectral resolution , of the order a few hundred km s@xmath6 , to be able to resolve this issue . the work in this paper is based on observations with _ xmm - newton _ , an esa science mission , with instruments and contributions directly funded by esa and nasa . the authors would like to thank the epic consortium for all their work during the calibration phase , and the soc and ssc teams for making the observation and analysis possible . this research has made use of the nasa / ipac extragalactic database ( ned ) , which is operated by the jet propulsion laboratory , california institute of technology , under contract with the national aeronautics and space administation . support from a pparc studentship is acknowledged by klp . anders e. , grevesse n. , 1989 , geochim . acta , 53 , 197 baldwin j.a . , 1977 , apj , 214 , 679 blustin a.j . , branduardi - raymont g. , behar e. , kaastra j.s . , , page m. j. , sako m. , steenbrugge k.c . , 2002 , a&a , 392 , 453 boller th . , 2002 , mnras , 329 , l1 constantin a. , shields j.c . , 2003 , pasp , 115 , 592 croom s.m . , 2002 , mnras , 337 , 275 cruz - gonz ' alez i. , carrasco l. , serrano a. , guichard j. , dultzin - hacyan d. , bisiacchi g.f . , 1994 , apjs , 94 , 47 dewangan g.c . , 2002 , apj , 581 , l71 derry p.m. , obrien p.t . , reeves j.n . , , imanishi m. , uemo s. , 2003 , mnras , in press ( astro - ph/0304550 ) dietrich m. , hamann f. , shields j.c . , constantin a. , vestergaard m. , chaffee f. , foltz c.b . , junkkarinen v.t . , 2002 , apj , 581 , 912 fabian a.c . , 2002 , mnras , 335 , l1 feigelson e.d . , nelson p.i . , 1985 , apj , 293 , 192 gaskell c.m . , 1985 , apj , 291 , 112 gaskell c.m . wampler e.j . , shields g.a . , 1981 , apj , 249 , 443 george i.m . , fabian a.c . , 1991 , mnras , 249 , 352 ghisellini g. , haardt f. , matt g. , 1994 , mnras , 267 , 743 gondoin p. , orr a. , lumb d. , siddiqui h. , 2003 , a&a , 397 , 883 green p.j . , forster k. , kuraszkiewicz j. , 2001 , apj , 556 , 727 guilbert p.w . , rees m.j . , 1988 , mnras , 233 , 475 hamann f. , ferland g. , 1993 , apj , 418 , 11 iwasawa k. , taniguchi y. , 1993 , apj , 413 , l15 kaspi s. , smith p.s . , netzer h. , maoz d. , jannuzi b.t . , giveon u. , 2000 , apj , 533 , 631 kaspi s. , 2002 , apj , 574 , 643 knigl a. , kartje j.f . , 1994 , apj , 434 , 446 krolik j.h . , madau p. , . zycki p.t . , 1994 , apj , 420 , 57 kuraszkiewicz j.k . , green p.j . , forster k. , aldcroft t.l . , evans i.n . , koratkar a. , 2002 , apjs , 143 , 257 lee j.c . , iwasawa k. , houck j.c . , fabian a.c . , marshall h.l . , canizares c.r . , 2002 , apj , 570 , l47 lightman a.p . , white t.r . , 1988 , apj , 335 , 57 lubinski p. , zdziarski a.a . , 2001 , mnras , 323 , l37 magdziarz p. , zdziarski a.a . , 1995 , mnras , 273 , 837 marziani p. , sulentic j.w . , zamanov r. , calvani m. , dultzin - hacyan d. , bachev r. , zwitter t. , 2003 , apjs , 145 , 199 matt g. , guainazzi m. , maiolino r. , mnras , in press ( astro - ph/0302328 ) matt g. , guainazzi m. , perola g.c . , fiore f. , nicastro f. , cappi m. , piro l. , 2001 , a&a , 377 , l31 mulchaey j.s . , koratkar a. , ward m.j . , wilson a.s . , whittle m. , antonucci r.r.j . , kinney a.l . , hurt t. , 1994 , apj , 436 , 586 mushotzky r. , ferland g.j . , 1984 , apj , 278 , 558 nandra k. , pounds k.a . , 1994 , mnras , 268 , 405 nandra k. , fabian a.c . , brandt w.n . , kunieda h. , matsuoka m. , mihara t. , ogasaka y. , terashima y. , 1995 , mnras , 276 , 1 nandra k. , george i.m . , mushotzky r.f . , turner t.j . , yaqoob t. , 1997 , apj , 488 , l91 nayakshin s. , 2000a , apj , 534 , 718 nayakshin s. , 2000b , apj , 540 , l37 obrien p.t . , page k. , reeves j.n . , pounds k. , turner m.j.l . , puchnarewicz e.m . , 2001 , mnras , 327 , l37 ogle p.m. , marshall h.l . , , canizares c.r . , 2000 , apjl , 545 , l81 ohsuga k. , umemura m. , 2001 , a&a , 371 , 890 page k.l . , obrien p.t . , reeves j.n . breeveld a.a . , 2003 , mnras , 340 , 1052 page m.j . , 2001 , a&a , 365 , l152 petrucci p.o . , 2002 , a&a , 388 , l5 pounds k.a . , reeves j.n . , page k.l . , , obrien p.t . , 2003 , mnras , in press ( astro - ph/0302151 ) pounds k. , reeves j. , obrien p. , page k. , turner m. , nayakshin s. , 2001 , apj , 559 , 181 pounds k.a . , 1990 , nature , 344 , 132 radovich m. , rafanelli p. , 1994 , an , 315 , 265 recondo - gonzalez m.c . , wamsteker w. , clavel j. , rodriguez - pascual p.m. , vio r. , ting - gui w. , santos - lleo m. , makino f. , 1997 , a&as , 121 , 461 reeves j. n. , turner m. j. l. , pounds k. a. , obrien p. t. , boller th . , ferrando p. , kendziorra , e. ; vercellone , s. , 2001 , a&a , 365 , l134 reeves j.n . , turner m.j.l . , ohashi t. , kii t. , 1997 , mnras , 292 , 468 shemmer o. , netzer h. , 2002 , apj , 567 , l19 turner t.j . , 2002 , apj , 574 , l123 tytler d. , fan x .- m . , 1992 , apjs , 79 , 1 vaughan s. , boller th . , fabian a.c . , ballantyne d.r . , brandt w.n . , trmper j. , 2002 , mnras , 337 , 247 v ' eron - cetty m .- p . , v ' eron p. , gonalves a.c . , 2001 , a&a , 372 , 730 wang t .- g . , lu y .- j . , zhou y .- y . , 1998 , apj , 493 , 1 wilms j. , reynolds c.s . , begelman m.c . , reeves j. , molendi s. , staubert r. , kendziorra e. , 2001 , mnras , 328 , l27 worrall d.m . , tananbaum h. , giommi p. , zamorani g. , 1987 , apj , 313 , 596 yaqoob t. , george i.m . , kallman t.r . , padmanabhan u. , weaver k.a . , turner t.j . , 2003 , apj , in press ( astro - ph/0306428 ) yaqoob t. , padmanabhan u. , dotani t. , nandra k. , 2002 , apj , 569 , 487 yaqoob t. , george i.m . , nandra k. , turner t.j . , serlemitsos p.j . , mushotzky r.f . , 2001 , apj , 546 , 759 zamorani g. , 1981 , apj , 245 , 357 zamorani g. , marano b. , mignoli m. , zitelli v. , boyler b.j . , 1992 , mnras , 256 , 238

the majority of active galactic nuclei ( agn ) observed by _ xmm - newton _ reveal narrow fe k@xmath0 lines at @xmath1 6.4 kev , due to emission from cold ( neutral ) material . there is an x - ray baldwin effect in type i agn , in that the equivalent width of the line decreases with increasing luminosity , with weighted linear regression giving ew @xmath2 l@xmath3 ( spearman rank probability of @xmath4 99.9 per cent ) . with current instrumental capabilities it is not possible to determine the precise origin for the narrow line , with both the broad line region and putative molecular torus being possibilities . a possible explanation for the x - ray baldwin effect is a decrease in covering factor of the material forming the fluorescence line . [ firstpage ] galaxies : active x - rays : galaxies quasars : emission lines

astro-ph0309394

the use of lasers in atomic physics often demands long term stability of the central frequency of the light emission . for metrological applications the stabilization technique @xcite should be very carefully chosen and applied , frequently controlling the laser linewidth and avoiding to introduce any artificial shift in the laser emission frequency . moreover , locking to the center of an atomic or molecular transition usually requires modulation techniques and lock - in detection . on the other hand , for many scientific and technical applications one only needs to avoid frequency drifts and , sometimes , the desired laser frequency does not lay at the maximum of an atomic lineshape , but rather at a displaced frequency , as for instance , when operating an optical cooler @xcite . for such applications a few simple and reliable techniques were developed @xcite and allow on e to deal with lasers in various long run experiments . the main idea behind many of these techniques is to generate a dispersive lineshape that will produce an error signal . in particular , for the dichroic atomic vapor laser lock ( davll ) @xcite and its variants @xcite , the stabilization frequency may easily be chosen around the center of the doppler - broadened line . however , a relatively uniform external magnetic field is needed to generate the zeeman split of the probed hyperfine transition and a double detection with well balanced photodetectors is also necessary . + in this work we report on a simple method to generate a dispersive signal in a very direct way , and therefore of easy implementation . our technique explores the dispersive signal obtained when a gaussian - profile light beam is sent through an atomic vapor cell and is detected after spatial filtering by an aperture ( fig . [ setup ] ) . we call this method angells , an acronym for atomic non - linearly generated laser locking signal . + the third order susceptibility term of an induced atomic vapor polarization by a laser beam results on a non - linear refractive index term , proportional to the laser intensity . the total refractive index of the vapor can thus be written as @xmath1 and the radial intensity gradient of a gaussian - profile beam will induce a radial refractive index gradient in the medium . this index gradient will in turn act as a lens for the gaussian beam , which will therefore suffer ( self)-focusing or ( self)-defocusing , depending on the sign of the nonlinear refractive index @xcite . the nonlinear index changes sign across a sharp resonance of the nonlinear medium . if on one side of the resonance frequency the index increment is positive ( maximum on the beam axis ) , the medium behaves as a converging lens and the power of a initially collimated beam transmitted through an aperture will increase ( peak of the dispersive lineshape ) . on the other side of the resonance frequency the laser - induced increment is negative ( minimum on the axis ) , the medium behaves as a divergent lens and the transmission through an aperture yields a correspondingly diminished signal ( valley of the dispersive lineshape ) . in other words , the nonlinear medium acts as a lens which focal length depends on the laser frequency . for a hot atomic vapor , for instance , the non - linear refractive index can be written as @xcite : @xmath2 where @xmath3 is the doppler width , @xmath4 is the vapor temperature , @xmath5 is the boltzmann constant , @xmath6 is the light wavenumber , @xmath7 is the atomic mass and @xmath8 is the laser frequency detuning relative to the atomic transition . for red detuning frequencies ( @xmath9 ) , @xmath10 is negative while for blue detuning ( @xmath11 ) , @xmath10 is positive . the power transmitted through the aperture is thus modulated when the frequency is scanned around _ an atomic transition _ , resulting in a dispersive - like lineshape with _ doppler width_. + in our technique , the nonlinear medium is a resonant atomic vapor , placed in the laser beam path past a converging lens to enhance nonlinear effects with higher light intensity radial gradients . our experimental setup is sketched in fig . [ setup ] . a 852 nm cw tunable semiconductor laser beam is splitted by a 90/10 beam splitter . the lower - intensity beam ( @xmath12 ) , of approximately gaussian spatial profile ( no spatial filter needed ) and of slightly saturating intensity , is focused by a 150 mm - focal - length lens . a warm ( 40 to @xmath13 60@xmath14c , corresponding to densities of @xmath15 - @xmath16 ) atomic cesium vapor @xcite contained in a 1-mm - long optical cell @xcite is placed close ( @xmath13 20 mm ) to the focus of the laser beam . we detect the transmission of the laser beam through an aperture adjusted so as to capture @xmath17 of the beam power ( typically 2-mm aperture for a beam of diameter 6 mm ) . when the frequency is scanned around the cs @xmath18 doppler transition , the nonlinear refraction turns from self - focusing to self - defocusing . this gives rise to a dispersive - like lineshape superimposed to a non - zero offset corresponding to the out - of - resonance aperture transmission ( no vapor - induced modifications ) . very small structures on these spectra are attributed to non - linear effects due to the beam reflection on the cell windows . the comparable dimensions of the beam diameter and the cell thickness makes the prevention of this high - order interaction difficult . however , this does not pose any additional problem to lock the laser at any position in the broad range inside the doppler width . + the error signal is the subtraction of a reference voltage ( corresponding to a frequency within the doppler line ) from the photodetector amplifier voltage . such an error signal is amplified and sent to the control of the laser frequency to correct for laser frequency drifts . in semiconductor lasers , the frequency is changed mostly through the injection current , the junction temperature or , in the case of extended cavity configuration , the external diffraction grating angle . we worked with a dfb laser diode resonant with the cs d@xmath19 line and a fabry - prot semiconductor laser with extended cavity , emitting around the rb d@xmath19 line . the electronic correction signal is fed back in the junction current in the dfb or in the piezoelectric actuator in the extended - cavity laser @xcite . for the sake of simplicity , we have operated both systems with a home - made electronic circuit having only proportional and integral gains . + a scheme of the locking circuit is shown in figure [ circuit ] . a voltage ramp generator allows one to scan the laser frequency ( through current or cavity pzt modulation ) around the atomic resonance . we choose a locking frequency with the help of a reference saturated absorption ( sa ) spectrum carried out in an extra vapor cell and exhibiting characteristic sub - doppler features ( see fig . [ locked]a ) . we use the sa signal obtained in this additional cell as frequency reference ( fig . [ locked]a ) as well as to monitor the locking performance ( fig . [ locked]c ) . the locking procedure follows some basic steps : the ramp is turned off and the offset finely tuned until the laser frequency is at the desired locking point * ( @xmath20 ) * , marked by dots in figs . [ locked]a and [ locked]b . the error signal is then brought to zero by adjusting the reference voltage ( fig . [ locked]b ) and a switch closes the loop , ultimately locking the laser at the desired frequency ( fig . [ locked]c ) . modifying the reference voltage allows one to lock the laser at any point within the doppler width and thus to explore the different hyperfine transitions shown in the sa spectrum . notice that the reference voltage brings the error signal around zero and so compensates for the non - resonant background signal . + ) , ( b ) dispersion curve for the error signal as a function of frequency detuning relative to the atomic resonance , and ( c ) the angells - frequency - locked as well as an unlocked sa signal . the chosen frequency of stabilization ( @xmath20 ) is marked by a red dot in spectra ( a ) and ( b),width=302 ] figure [ locked]c exhibits the sa signal with the laser locked at the selected frequency , over a period of a few minutes , as well as the sa signal for an unlocked laser . the system remains locked for hours even after we have strongly and repeatedly hit our home - made optical table . the short - term rms frequency width is the same for the locked and the unlocked laser , i.e. of the order of 2 mhz or less , as measured using the saturated absorption line flank as a frequency discriminator . the long - term frequency fluctuations of the locked laser remains limited to less than 2 mhz rms , while the frequency of the laser unlocked for a few minutes fluctuates in excess of 20 mhz . + , dashed line ) and after ( @xmath21 , solid line ) the beam minimum waist.,width=302 ] we checked the stabilization sensitivity to vapor density and cell alignment along the beam . although the vapor temperature has been varied between 45 and 63 @xmath14c , the lineshape of the generated signal is stable against temperature changes as shown in fig . [ dispersion]a . the doppler - profile center position changes very little ( @xmath22mhz/@xmath14c ) over this 20 @xmath14c temperature variation . each lineshape in fig . [ dispersion]a has been recorded at a given temperature ( values indicated in the figure frame ) , fluctuating less than 1 @xmath14c from its reference value . thus , the laser frequency locking , particularly at the doppler center , is not affected by small temperatures fluctuations of the vapor which lead to frequency drifts of the order of magnitude or less than the rms frequency width , as evidenced in fig.[locked]c . even to lock the laser at frequencies other than at the line center , we have only monitored the cell temperatur e , without active control . more noticeable modifications appear on the profile wings , that do not play a role in the stabilization process inside the doppler width . similarly , the error signal remains approximately unchanged over @xmath13 2-mm displacements along the beam , around the optimal position of the cell ( @xmath13 20 mm on either side from focal point ) . another characteristic of using the angells technique is the possibility of choosing the sign of the error signal slope by purely optical means , as shown in figure [ dispersion]b . the dispersive curve gets inverted when the vapor is displaced _ across the focal position _ ( see fig.[setup ] ) . we also emphasize here the fact that semiconductor lasers are well known to exhibit very stable amplitude @xcite , allowing us to disregard amplitude noise in the detected signal . for lasers with higher intensity fluctuations a second photodetector may be used to normalize the frequency error signal . in summary , we have presented an opto - electronic stabilization method based on the direct generation of an optical dispersive - like signal . the angells technique has proved to be an easy and robust locking method against diode laser frequency drifts . for similar performance , the set - up is simpler than traditional locking techniques , not requiring magnetic fields or beam modulation . the set - up is flexible : different combinations of laser power , beam diameter , focalization , cell length and vapor density have been used , the values given in the article corresponding to those used for the presented results . acknowledgment : this work was partially funded by conselho nacional de desenvolvimento cientfico e tecnolgico ( cnpq , contract 472353/2009 - 8 ) and coordenao de aperfeioamento de pessoal de nvel superior ( capes ) . , i.v . and m.c . acknowledge grants by cnpq . 99 see , for instance , j.l . hall , l .- s . ma , m. taubman , b. tiemann , f .- hong , o. pfister , j. ye , ieee trans . on instr . and meas . * 48 * , 583 ( 1999 ) c. salomon , d. hils , j. hall , j. opt . b * 5 * , 1576 ( 1988 ) the dispersive force in an optical molasses is maximized at a frequency red shifted half a linewidth from the cyclic cooling transition frequency . see , for instance , p.d . lett , w.d . phillips , s.l . rolston , c.e . tanner , r.n . watts , c.i . westbrook , j. opt . b * 6 * , 2084 ( 1989 ) b. chron , h. gilles , j. havel , o. moreau , h. sorel , j. phys . iii * 4 * , 401 ( 1994 ) k.l . corwin , z .- t . hand , r.j . epstein , c.e . wieman , appl . opt . * 37 * , 3295 ( 1998 ) c.p . pearman , c.s . adams , s.g . cox , p.f . griffin , d.a . smith , i.g . hughes , j. phys . b : at . mol . . phys . * 35 * , 5141 ( 2002 ) j. e. debs , n.p . robins , a. lance , m.b . kruger , j.d . close , appl . opt . * 47 * , 5163 ( 2008 ) n.p . robins , b.j.j . slagmolen , d.a . shaddock , j.d . close , m. b. gray , opt . lett . * 27 * , 1905 ( 2002 ) e.a . gazazyan , a.v . papoyan , d. sarkisyan , a. weis , laser phys . lett . * 4 * , 801 ( 2007 ) w. s. martins , m. grilo , m. brasileiro , o. di lorenzo , m. ori , m. chevrollier , appl * 49 * , 871 ( 2010 ) see , e. g. , t. ackemann , t. scholz , ch . vorgerd , j. nalik , l.m . hoffer , g.l . lippi , opt . commun . * 147 * , 411 ( 1998 ) c.f . mccormick , d.r . solli , r.y . chiao , j.m . hickmann , phys . a * 69 * , 023804 ( 2004 ) we have performed similar experiments of laser stabilization using an atomic rb vapor to lock a 780 nm diode laser to different transitions of the two rubidium isotopes . the cell length was chosen to have a thickness small compared to the lens focal length . stabilization was also observed with longer cells ( 10 mm and 20 mm ) , however thin cells help enhance dispersive effects over absorptive ones . our home - made external cavity is tuned thanks to a low - voltage piezoelectric actuator ( displacement of about 6@xmath23m / volt ) . k. peterman , _ laser diode modulation and noise _ ( kluwer ac . , dordrecht,1991 )

we report on a simple and robust technique to generate a dispersive signal which serves as an error signal to electronically stabilize a monomode cw laser emitting around an atomic resonance . we explore nonlinear effects in the laser beam propagation through a resonant vapor by way of spatial filtering . the performance of this technique is validated by locking semiconductor lasers to the cesium and rubidium @xmath0 line and observing long - term reduction of the emission frequency drifts , making the laser well adapted for many atomic physics applications .

1204.0506

the study of quasinormal modes ( qnms ) of a black hole ( bh ) has long history @xcite . the reason behind this interest is that the qnms offer a direct way of studying the key features of the physics of compact massive objects , without the complications of the full 3d general relativistic simulations . for example , by comparing the theoretically obtained gravitational qnms with the frequencies of the gravitational waves , one can confirm or refute the nature of the central engines of many astrophysical objects , since those modes differ for the different types of objects black holes , superspinars ( naked singularities ) , neutron stars , black hole mimickers etc . @xcite . to find the qnms , one needs to solve the second - order linear differential equations describing the linearized perturbations of the metric : the regge - wheeler equation ( rwe ) and the zerilli equation for the schwarzschild metric or the teukolsky radial equation ( tre ) for the kerr metric and to impose the appropriate boundary conditions the so - called black hole boundary conditions ( waves going simultaneously into the horizon and into infinity)@xcite . additionally , one requires a regularity condition for the angular part of the solutions . and then , one needs to solve a connected problem with two complex spectral parameters the frequency @xmath2 and the separation constant @xmath3 ( @xmath4 real for a nonrotating bh , with @xmath5 the angular momentum of the perturbation ) . this system was first solved by chandrasekhar & detweiler@xcite and teukolsky & press @xcite and later developed through the method of continued fractions by leaver @xcite . for more recent results , see also @xcite . because of the complexity of the differential equations , until now , those equations were solved either approximately or numerically meeting an essential difficulty @xcite . the indirect approaches like the continued fractions method have some limitations and are not directly related with the physics of the problem . the rwe , the zerilli equation and tre , however , can be solved analytically in terms of confluent heun functions , as done for the first time in @xcite . imposing the boundary conditions on those solutions _ directly _ ( see @xcite ) one obtains a system of spectral equations and featuring the confluent heun functions which can be solved numerically . in this article , for the first time we present finding @xmath5 and @xmath2 _ directly _ in the case for gravitational perturbation @xmath6 in a schwarzschild metric , i.e. we solve the rwe and tre analytically in terms of confluent heun functions and we use a newly developed method ( the two - dimensional generalization of the mller method described in the internal technical report @xcite ) to solve the system of two transcendental equations with two complex variables . then we use the epsilon method to study the stability of the solutions with respect to small variations in the phase condition . the results are compared with already - published ones and are found to coincide with at least 8 digits for the rwe and 6 digits for the tre . for the first time , the so - called algebraically special mode @xmath0 is evaluated with precision of more than 6 digits , and it is shown to have a nonzero real part . this firmly refutes the hypothetical relation of this mode with the algebraically special once . also demonstrated is the nontrivial dependence on @xmath1 of the first 11 modes in both cases . the angular equation for both cases is the solution of the teukolsky angular equation when there is no rotation ( @xmath7 ) : @xmath8 where @xmath9 $ ] is the angle . the results for the qnms should be independent of the choice of @xmath10 in the spectral conditions . in our numerical experiments , we use @xmath11 . the general form of the radial equations is obtained from the solutions of the rwe and tre written in terms of the confluent heun functions according to @xcite , on which the black hole boundary conditions have been imposed . the choice of the local solution in terms of the heun function takes into account the boundary condition on the horizon . then , it remains to impose the following boundary condition on the space infinity ( for details see @xcite ) : @xmath12 where @xmath13 is the confluent heun function as defined in maple and the parameters @xmath14 differ for the two equations . the values of the parameters when the bh mass is @xmath15 and , if we choose @xmath16 which turns out to be large enough to simulate numerically the actual infinity , are ( @xcite ) : 1 . for the solutions of the regge - wheeler equation : @xmath17 2 . for the solutions of the teukolsky radial equation : @xmath18 where @xmath19 is the separation constant . the parameters were obtained by solving the teukolsky radial equation and substituting @xmath20 and they are clearly different from those in the regge - wheeler case . hence , it is important to check whether both methods give the same results for qnm and with what precision . for values of the parameters @xmath21 of general type , the confluent heun function @xmath22 has branching points in the complex z - plane at the singular points @xmath23 and @xmath24 . in the maple package , as a branch cut is chosen the semi - infinite interval @xmath25 on the real axis . the presence of the branch cut may lead to the disappearance of some modes or their translation , since by changing the phase of the complex variable @xmath26 , we may make a transition to another sheet of the multivalued function . to avoid this , we use the epsilon method with which one can find the correct sheet and remain on it . this is done by introducing a small variation ( @xmath27 ) in the phase condition @xmath28 ( defined by the direction of steepest descent , see @xcite ) , with which one can move the branch cuts farther from the roots and thus avoid the jump discontinuity in the function . for more information on the epsilon method and the numerical procedures , see @xcite . from the angular equation , it is clear that it can be solved explicitly without solving the system and and the values of @xmath5 are known : @xmath29 . in this paper , only the first value , @xmath30 , is used to find the qnms with both radial equations . one can then either solve only the radial equations or solve the systems and with the appropriate values of the parameters . if one solves the problem as a two - dimensional system , making calculations with 15 digits of precision ( and 32 software floating - point digits ) , one obtains as expected , @xmath31 with the first digit different from digit 9 being the 17th . the numerical results for the frequencies are summed in table [ table1 ] . .a list of the frequencies obtained for the qnms of schwarzschild black hole using the regge - wheeler equation and the teukolsky equation . the modes with @xmath32 are found for @xmath33 , modes from @xmath34 with @xmath35 . the first 5 frequencies ( @xmath36 ) were obtained also by fiziev in @xcite using exact solutions of rwe in the heun functions [ cols="<,<,<,<",options="header " , ] from the table , one can see that the frequencies from the two types of equations coincide with at least 6 digits . a comparison between the rwe frequencies and the ones published by andersson @xcite , published in @xcite shows that the difference between the two results is smaller than @xmath37 in most cases and is due to the numerical reasons . there are two important results from this study . first , as seen from table [ table1 ] for both the rwe and the tre , the mode number 8 has a small but nonzero real part . according to leaver s evaluations this mode should be equal to @xmath38 @xcite , with an exactly zero real part , if it is to correspond to the so - called _ algebraically special mode_. algebraically special ( as ) modes have a special place in the qnm studies @xcite . the andersson method is not applicable for them and these are excluded from his consideration . berti , cardoso and starinets ( @xcite ) make a review on the results so far concerning these modes . theoretically the 9th mode ( @xmath0 ) should be purely imaginary with value @xmath39 , if it indeed corresponds to the as case . in our results , even though purely imaginary modes do not pose a problem for the method , the real part of the 9th mode is distinctly not zero , and it has at least 7 stable digits when changing @xmath1 in the interval discussed below for both rwe and tre . this clearly shows that this mode does not agree with the hypothesis for the as mode , which is to be expected since the as mode should correspond to different boundary conditions those of the so - called totally - transmission modes ( @xcite . the second important result is the dependence of the frequencies @xmath40 on @xmath1 . the direction of steepest descent is supposed to be the optimal direction in which the solutions satisfy the black hole boundary conditions on infinity in the first term approximation for asymptotic series for the heun functions @xcite . the validity of steepest descent method in its simplest form for the radial equations in both cases under variations in this condition , however , is still an open problem studied here for the first time . using the @xmath1 method , one can explore the intervals for @xmath1 in which each mode can be found . the results for both rwe and tre , as expected , coincide . generally , the intervals into which each mode can be found narrow down when increasing @xmath41 . while for the first 5 modes it is possible to find @xmath42 for positive and negative values of @xmath1 in a certain interval , as follows : for @xmath43 , @xmath44 $ ] , for @xmath45 $ ] , for @xmath46 $ ] , for @xmath47 $ ] , for @xmath48 $ ] , where the first sign corresponds to frequencies with a positive real part and the second sign to those with negative real parts . the imaginary parts for each mode @xmath41 coincide . ] for @xmath49 , ( but @xmath50 ) the modes with a positive real part can be found only for negative values of @xmath1 , and the dependency becomes @xmath51 . for @xmath0 , the mode has different behavior with respect to @xmath1 for @xmath52 $ ] , one finds a mode with _ negative _ real part and vice versa : ( @xmath53 ) . the so - found relation @xmath54 needs to be examined further . for the case @xmath55 , similar ( to some extent ) behavior was mentioned also in @xcite ( and discussed in @xcite ) . it was suggested that there are two as modes which are symmetrical to the imaginary axis and perhaps may be related with the branch cut in the asymptotic of the rwe potential when @xmath2 is purely imaginary . using the @xmath1 method applied on the asymptotics of the confluent heun functions , one can directly obtain the place of the branch cut on the real axis as a function of @xmath1 and they can be easily visualized plotting the solution @xmath56 . therefore , the use of the confluent heun functions and the @xmath1 method offers a direct way to examine the solutions and their properties in relation to the branch cut in the complex r - plane , something that can not be readily done in the continued fractions method generally used to obtain the qnms . further exploration of the dependence @xmath57 ( or @xmath58 ) in the intervals mentioned above shows that , for both the rwe and the tre , it is approximately a periodic function with amplitude @xmath59 and period @xmath60 which change with @xmath41 in a nontrivial way ( fig.[fig1 ] and fig . [ fig2 ] ) . for @xmath61 , from the rwe and the tre one obtains @xmath62 , @xmath63 and those values remain approximately constant with respect to @xmath41 ( @xmath61 ) . for @xmath64 , the dependence of @xmath59 and @xmath60 on @xmath41 becomes more pronounced : the amplitudes and the periods of the rwe increase with @xmath41 until they reach @xmath65 , @xmath66 for @xmath67 . for the tre , the amplitude and the period decrease to @xmath68 , @xmath69 . for @xmath0 , the two periodic behaviors have approximately equal amplitudes @xmath70 . those results hint that , although the so - obtained frequencies are stable with at least 6 digits with respect to @xmath1 , there is also some finer dependence , the origin of which should be carefully investigated . @xmath71{\includegraphics[height=120px , width=110px]{fig1a.eps } } & \subfigure[]{\includegraphics[height=120px , width=110px]{fig1b.eps } } \\ \subfigure[]{\includegraphics[height=120px , width=110px]{fig1c.eps } } & \subfigure[]{\includegraphics[height=120px , width=110px]{fig1d.eps}}\\ \end{array } $ ] in this paper , were presented the qnms for a schwarzschild bh obtained from the rwe and the tre , by solving the differential equations analytically in terms of confluent heun functions . the qnms from the tre for the case @xmath6 were calculated for the first time and were found to coincide with the well - known qnms from the rwe with precision of 6 digits . we demonstrated a new method for studying the stability of the qnm calculations . the results show nontrivial dependence on small variation in the phase condition ( the @xmath1 method ) which requires additional investigation . for the first time , the mode @xmath0 was obtained directly from the spectral condition on the exact analytical solutions of rwe and tre and was found to have a nonzero real part , which proves that this mode is not the algebraically special mode . the mode in question is stable with 6 digits of significance with respect to changes in @xmath1 , which proves that its real part is indeed not zero . those results presented here show the strength of using confluent heun functions to find qnms of nonrotating bhs and are encouraging in continuing this work in finding qnms of rotating bhs . this article was supported by the foundation `` theoretical and computational physics and astrophysics '' , by the bulgarian national scientific fund under contracts do-1 - 872 , do-1 - 895 , do-02 - 136 , and sofia university scientific fund , contract 185/26.04.2010 . p.f . chose the evaluation of the qnms of non - rotating bhs as a test of the two - dimensional mller algorithm , proposed the epsilon method , as a generalization of the previous work and he supervised the project . d.s . is responsible for the calculation of qnms , based on the implementation of confluent heun functions and for the exploration and analysis of the @xmath1-method in the sector of the complex r - plane where qnms can be found . 00 , _ the quasi - normal modes of the schwarzschild black hole _ , proc . london a*344 * : 441 - 452 ( 1975 ) , _ black holes and gravitational waves . iii - the resonant frequencies of rotating holes _ , apj:*239 * , 292 - 295 , ( 1980 ) , _ the mathematical theory of black holes _ , clarendon press / oxford university press ( international series of monographs on physics . volume 69 ) , ( 1983 ) , _ quasinormal modes of black holes and black branes _ , class . quantum grav . * 26 * 163001 ( 108pp ) ( 2009 ) , _ black hole quasinormal modes : hints of quantum gravity ? _ , in international workshop on dynamics and thermodynamics of black holes and naked singularities , milan , italy , 2004 ( to be published ) , arxiv : gr - qc/0411025v1 ( 2004 ) , _ quasi - normal modes and gravitational wave astronomy _ , gen.rel.grav.*40 * : 945 - 970 ( 2008 ) , arxiv:0709.0657v2 [ gr - qc ] ( 2008 ) , _ quasinormal modes of black holes : from astrophysics to string theory _ , reviews of modern physics , * 83 * : 793 - 836 , issue 3 , ( 2011 ) , arxiv:1102.4014v1 [ gr - qc ] ( 2011 ) , _ on gravitational - wave spectroscopy of massive black holes with the space interferometer lisa _ , phys.rev.d * 73*:064030 , ( 2006 ) , arxiv:0512160v2 [ gr - qc ] , _ will einstein have the last word on gravity ? _ , astro2010 : the astronomy and astrophysics decadal survey , arxiv : 0903.0100v1 [ gr - qc ] , _ how to tell gravastar from black hole _ , class . quant . grav . * 24 * : , 4191 - 4206 , ( 2007 ) , arxiv:0706.1513v2 [ gr - qc ] , _ ergoregion instability in rotating gravastars _ , phys.rev.d * 78*:084011 , ( 2008 ) , arxiv:0808.4080v1 [ gr - qc ] , _ gravitational wave signatures of the absence of an event horizon : nonradial oscillations of a thin - shell gravastar _ , phys.rev.d * 80*:124047,(2009 ) , arxiv:0909.0287v2 [ gr - qc ] , _ the spectrum of electromagnetic jets from kerr black holes and naked singularities in the teukolsky perturbation theory_,astrophysics and space science , * 332 * , pp.385 - 401 , arxiv:1002.0480v2 [ astro-ph.he ] , ( 2010 ) , _ perturbations of a rotating black hole . iii - interaction of the hole with gravitational and electromagnetic radiation _ , apj * 193 * : 443 ( 1974 ) , _ an analytic representation for the quasi - normal modes of kerr black holes _ , proc . london a*402 * : 285 - 298 ( 1985 ) , _ classes of exact solutions to the teukolsky master equation _ class . quantum grav . * 27 * 135001 ( 2010 ) , arxiv:0908.4234v4 [ gr - qc ] , _ exact solutions of regge - wheeler equation and quasi - normal modes of compact objects _ , class . * 23 * 2447 - 2468 ( 2006 ) , arxiv:0509123v5 [ gr - qc ] , _ novel relations and new properties of confluent heun s functions and their derivatives of arbitrary order _ , j. phys . a : math * 43 * ( 2010 ) 035203 , arxiv:0904.0245 [ math - ph ] , _ teukolsky - starobinsky identities : a novel derivation and generalizations _ , * , 124001 ( 2009 ) , arxiv:0906.5108 [ gr - qc ] , _ two - dimensional generalization of the muller root - finding algorithm and its applications _ ( 2011 ) , arxiv:1005.5375v2 [ cs.na ] , _ a numerically accurate investigation of black - hole normal modes _ , proc . london a**439 * * no.1905 : 47 - 58 ( 1992 ) . _ analytic treatment of black - hole gravitational waves at the algebraically special frequency _ , phys d * 62 * 064009 ( 2000 ) . _ unconventional gravitational excitation of a schwarzschild black hole _ , class.quant.grav . * 20 * l217 ( 2003 ) , arxiv : gr - qc/0301018v4

although finding numerically the quasinormal modes of a nonrotating black hole is a well - studied question , the physics of the problem is often hidden behind complicated numerical procedures aimed at avoiding the direct solution of the spectral system in this case . in this article , we use the exact analytical solutions of the regge - wheeler equation and the teukolsky radial equation , written in terms of confluent heun functions . in both cases , we obtain the quasinormal modes numerically from spectral condition written in terms of the heun functions . the frequencies are compared with ones already published by andersson and other authors . a new method of studying the branch cuts in the solutions is presented the epsilon - method . in particular , we prove that the mode @xmath0 is not algebraically special and find its value with more than 6 firm figures of precision for the first time . the stability of that mode is explored using the @xmath1 method , and the results show that this new method provides a natural way of studying the behavior of the modes around the branch cut points .

1109.1532

variable selection methods based on penalty theory have received great attention in high - dimensional data analysis . a principled approach is due to the lasso of @xcite , which uses the @xmath0-norm penalty . @xcite also pointed out that the lasso estimate can be viewed as the mode of the posterior distribution . indeed , the @xmath1 penalty can be transformed into the laplace prior . moreover , this prior can be expressed as a gaussian scale mixture . this has thus led to bayesian developments of the lasso and its variants @xcite . there has also been work on nonconvex penalization under a parametric bayesian framework . @xcite derived their local linear approximation ( lla ) algorithm by combining the expectation maximization ( em ) algorithm with an inverse laplace transform . in particular , they showed that the @xmath2 penalty with @xmath3 can be obtained by mixing the laplace distribution with a stable density . other authors have shown that the prior induced from a penalty , called the nonconvex log penalty and defined in equation ( [ eqn : logp ] ) below , has an interpretation as a scale mixture of laplace distributions with an inverse gamma mixing distribution @xcite . recently , @xcite extended this class of laplace variance mixtures by using a generalized inverse gaussian mixing distribution . related methods include the bayesian hyper - lasso @xcite , the horseshoe model @xcite and the dirichlet laplace prior @xcite . in parallel , nonparametric bayesian approaches have been applied to variable selection @xcite . for example , in the infinite gamma poisson model @xcite negative binomial processes are used to describe non - negative integer valued matrices , yielding a nonparametric bayesian feature selection approach under an unsupervised learning setting . the beta - bernoulli process provides a nonparametric bayesian tool in sparsity modeling @xcite . additionally , @xcite proposed a nonparametric approach for normal variance mixtures and showed that the approach is closely related to lvy processes . later on , @xcite constructed sparse priors using increments of subordinators , which embeds finite dimensional normal variance mixtures in infinite ones . thus , this provides a new framework for the construction of sparsity - inducing priors . specifically , @xcite discussed the use of @xmath4-stable subordinators and inverted - beta subordinators for modeling joint priors of regression coefficients . @xcite established the connection of two nonconvex penalty functions , which are referred to as log and exp and defined in equations ( [ eqn : logp ] ) and ( [ eqn : exp ] ) below , with the laplace transforms of the gamma and poisson subordinators . a subordinator is a one - dimensional lvy process that is almost surely non - decreasing @xcite . in this paper we further study the application of subordinators in bayesian nonconvex penalization problems under supervised learning scenarios . differing from the previous treatments , we model latent shrinkage parameters using subordinators which are defined as stochastic processes of regularization parameters . in particular , we consider two families of compound poisson subordinators : continuous compound poisson subordinators based on a gamma random variable @xcite and discrete compound poisson subordinators based on a logarithmic random variable @xcite . the corresponding lvy measures are generalized gamma @xcite and poisson measures , respectively . we show that both the gamma and poisson subordinators are limiting cases of these two families of the compound poisson subordinators . since the laplace exponent of a subordinator is a bernstein function , we have two families of nonconvex penalty functions , whose limiting cases are the nonconvex log and exp . additionally , these two families of nonconvex penalty functions can be defined via composition of log and exp , while the continuous and discrete compound poisson subordinators are mixtures of gamma and poisson processes . recall that the latent shrinkage parameter is a stochastic process of the regularization parameter . we formulate a hierarchical model with multiple regularization parameters , giving rise to a bayesian approach for nonconvex penalization . to reduce computational expenses , we devise an ecme ( for expectation / conditional maximization either " ) algorithm @xcite which can adaptively adjust the local regularization parameters in finding the sparse solution simultaneously . the remainder of the paper is organized as follows . section [ sec : levy ] reviews the use of lvy processes in bayesian sparse learning problems . in section [ sec : gps ] we study two families of compound poisson processes . in section [ sec : blrm ] we apply the lvy processes to bayesian linear regression and devise an ecme algorithm for finding the sparse solution . we conduct empirical evaluations using simulated data in section [ sec : experiment ] , and conclude our work in section [ sec : conclusion ] . our work is based on the notion of bernstein and completely monotone functions as well as subordinators . let @xmath5 with @xmath6 . the function @xmath7 is said to be completely monotone if @xmath8 for all @xmath9 and bernstein if @xmath10 for all @xmath9 . roughly speaking , a _ subordinator _ is a one - dimensional lvy process that is non - decreasing almost surely . our work is mainly motivated by the property of subordinators given in lemma [ lem : subord ] @xcite . [ lem : subord ] if @xmath11 is a subordinator , then the laplace transform of its density takes the form @xmath12 where @xmath13 is the density of @xmath14 and @xmath15 , defined on @xmath16 , is referred to as the _ laplace exponent _ of the subordinator and has the following representation @xmath17 \nu ( d u).\ ] ] here @xmath18 and @xmath19 is the lvy measure such that @xmath20 . conversely , if @xmath15 is an arbitrary mapping from @xmath21 given by expression ( [ eqn : psi ] ) , then @xmath22 is the laplace transform of the density of a subordinator . it is well known that the laplace exponent @xmath15 is bernstein and the corresponding laplace transform @xmath23 is completely monotone for any @xmath24 @xcite . moreover , any function @xmath25 , with @xmath26 , is a bernstein function if and only if it has the representation as in expression ( [ eqn : psi ] ) . clearly , @xmath15 as defined in expression ( [ eqn : psi ] ) satisfies @xmath27 . as a result , @xmath15 is nonnegative , nondecreasing and concave on @xmath16 . we are given a set of training data @xmath28 , where the @xmath29 are the input vectors and the @xmath30 are the corresponding outputs . we now discuss the following linear regression model : @xmath31 where @xmath32 , @xmath33^t$ ] , and @xmath34 is a gaussian error vector @xmath35 . we aim at finding a sparse estimate of the vector of regression coefficients @xmath36 by using a bayesian nonconvex approach . in particular , we consider the following hierarchical model for the regression coefficients @xmath37 s : @xmath38 & \stackrel{iid}{\sim } p(\eta_j ) , \\ \sigma & \sim\iga(\alpha_{\sigma}/2 , \beta_{\sigma}/2),\end{aligned}\ ] ] where the @xmath39 s are referred to as latent shrinkage parameters , and the inverse gamma prior has the following parametrization : @xmath40 furthermore , we regard @xmath39 as @xmath41 , that is , @xmath42 . here @xmath43 is defined as a subordinator . let @xmath44 , defined on @xmath16 , be the laplace exponent of the subordinator . taking @xmath45 , it can be shown that @xmath46 defines a nonconvex penalty function of @xmath47 on @xmath48 . moreover , @xmath46 is nondifferentiable at the origin because @xmath49 and @xmath50 . thus , it is able to induce sparsity . in this regard , @xmath51 forms a prior for @xmath47 . from lemma [ lem : subord ] it follows that the prior can be defined via the laplace transform . in summary , we have the following theorem . [ thm : lapexp00 ] let @xmath15 be a nonzero bernstein function on @xmath16 . if @xmath52 , then @xmath46 is a nondifferentiable and nonconvex function of @xmath47 on @xmath53 . furthermore , @xmath54 where @xmath43 is some subordinator . recall that @xmath14 is defined as the latent shrinkage parameter @xmath55 and in section [ sec : blrm ] we will see that @xmath56 plays the same role as the regularization parameter ( or tuning hyperparameter ) . thus , there is an important connection between the latent shrinkage parameter and the corresponding regularization parameter ; that is , @xmath57 . because @xmath58 , each latent shrinkage parameter @xmath39 corresponds to a local regularization parameter @xmath59 . therefore we have a nonparametric bayesian formulation for the latent shrinkage parameters @xmath39 s . it is also worth pointing out that @xmath60 where @xmath61 denotes a laplace distribution with density given by @xmath62 , then @xmath63 defines the proper density of some random variable ( denoted @xmath64 ) . subsequently , we obtain a proper prior @xmath65 for @xmath47 . moreover , this prior can be regarded as a laplace scale mixture , i.e. , the mixture of @xmath66 with mixing distribution @xmath67 . if @xmath68 , then @xmath69 is not a proper density . thus , @xmath70 is also improper as a prior of @xmath47 . however , we still treat @xmath70 as the mixture of @xmath66 with mixing distribution @xmath67 . in this case , we employ the terminology of pseudo - priors for the density , which is also used by @xcite . obviously , @xmath71 is bernstein . it is an extreme case , because we have that @xmath72 , @xmath73 and that @xmath74 , where @xmath75 denotes the dirac delta measure at @xmath56 , which corresponds to the deterministic process @xmath76 . we can exclude this case by assuming @xmath77 in expression ( [ eqn : psi ] ) to obtain a strictly concave bernstein function . in fact , we can impose the condition @xmath78 . this in turn leads to @xmath77 due to @xmath79 . in this paper we exploit laplace exponents in nonconvex penalization problems . for this purpose , we will only consider a subordinator without drift , i.e. , @xmath77 . equivalently , we always assume that @xmath80 . we here take the nonconvex log and exp penalties as two concrete examples ( also see * ? ? ? the log penalty is defined by @xmath81 while the exp penalty is given by @xmath82 clearly , these two functions are bernstein on @xmath16 . moreover , they satisfy @xmath27 and @xmath83 . it is also directly verified that @xmath84 \nu(du ) } , \ ] ] where the lvy measure @xmath19 is given by @xmath85 the corresponding subordinator @xmath86 is a gamma subordinator , because each @xmath14 follows a gamma distribution with parameters @xmath87 , with density given by @xmath88 we also note that the corresponding pseudo - prior is given by @xmath89 furthermore , if @xmath90 , the pseudo - prior is a proper distribution , which is the mixture of @xmath91 with mixing distribution @xmath92 . as for the exp penalty , the lvy measure is @xmath93 . since @xmath94 d b } = \infty,\ ] ] then @xmath95 $ ] is an improper prior of @xmath47 . additionally , @xmath96 is a poisson subordinator . specifically , @xmath14 is a poisson distribution with intensity @xmath97 taking values on the set @xmath98 . that is , @xmath99 which we denote by @xmath100 . in this section we explore the application of compound poisson subordinators in constructing nonconvex penalty functions . let @xmath101 be a sequence of independent and identically distributed ( i.i.d . ) real valued random variables with common law @xmath102 , and let @xmath103 be a poisson process with intensity @xmath104 that is independent of all the @xmath105 . then @xmath106 , for @xmath24 , follows a compound poisson distribution with density @xmath107 ( denoted @xmath108 ) , and hence @xmath43 is called a compound poisson process . a compound poisson process is a subordinator if and only if the @xmath105 are nonnegative random variables @xcite . it is worth pointing out that if @xmath109 is the poisson subordinator given in expression ( [ eqn : possion ] ) , it is equivalent to saying that @xmath14 follows @xmath110 . we particularly study two families of nonnegative random variables @xmath111 : nonnegative continuous random variables and nonnegative discrete random variables . accordingly , we have continuous and discrete compound poisson subordinators @xmath109 . we will show that both the gamma and poisson subordinators are limiting cases of the compound poisson subordinators . in the first family @xmath111 is a gamma random variable . in particular , let @xmath112 and the @xmath111 be i.i.d . from the @xmath113 distribution , where @xmath114 , @xmath115 and @xmath116 . the compound poisson subordinator can be written as follows @xmath117 the density of the subordinator is then given by @xmath118 we denote it by @xmath119 . the mean and variance are @xmath120 respectively . the laplace transform is given by @xmath121 where @xmath122 is a bernstein function of the form @xmath123.\ ] ] the corresponding lvy measure is given by @xmath124 notice that @xmath125 is a gamma measure for the random variable @xmath126 . thus , the lvy measure @xmath127 is referred to as a generalized gamma measure @xcite . the bernstein function @xmath128 was studied by @xcite for survival analysis . however , we consider its application in sparsity modeling . it is clear that @xmath128 for @xmath114 and @xmath116 satisfies the conditions @xmath129 and @xmath130 . also , @xmath131 is a nonnegative and nonconvex function of @xmath47 on @xmath48 , and it is an increasing function of @xmath132 on @xmath133 . moreover , @xmath131 is continuous w.r.t . @xmath47 but nondifferentiable at the origin . this implies that @xmath131 can be treated as a sparsity - inducing penalty . we are interested in the limiting cases that @xmath134 and @xmath135 . [ pro : first ] let @xmath136 , @xmath128 and @xmath137 be defined by expressions ( [ eqn : first_tt ] ) , ( [ eqn : first ] ) and ( [ eqn : first_nu ] ) , respectively . then 1 . @xmath138 and @xmath139 ; 2 . @xmath140 and @xmath141 ; 3 . @xmath142 and @xmath143 . this proposition can be obtained by using direct algebraic computations . proposition [ pro : first ] tells us that the limiting cases yield the nonconvex log and exp functions . moreover , we see that @xmath14 converges in distribution to a gamma random variable with shape @xmath144 and scale @xmath145 , as @xmath146 , and to a poisson random variable with mean @xmath144 , as @xmath147 . it is well known that @xmath148 degenerates to the log function @xcite . here we have shown that @xmath122 approaches to exp as @xmath147 . we list another special example in table [ tab : exam ] when @xmath149 . we refer to the corresponding penalty as a _ linear - fractional _ ( lfr ) function . for notational simplicity , we respectively replace @xmath150 and @xmath151 by @xmath145 and @xmath152 in the lfr function . the density of the subordinator for the lfr function is given by @xmath153 we thus say each @xmath14 follows a squared bessel process without drift @xcite , which is a mixture of a dirac delta measure and a randomized gamma distribution @xcite . we denote the density of @xmath14 by @xmath154 . lllll & bernstein functions & lvy measures @xmath137 & subordinators @xmath14 & priors + log & @xmath155 & @xmath156 & @xmath157 & proper@xmath158 + exp & @xmath159 $ ] & @xmath160 & @xmath161 & improper + lfr & @xmath162 & @xmath163 & @xmath164 & improper + cel & @xmath165 $ ] & @xmath166 & @xmath167 & improper + + + in the second case , we consider a family of discrete compound poisson subordinators . particularly , @xmath111 is discrete and takes values on @xmath168 . and it is defined as logarithmic distribution @xmath169 , where @xmath170 and @xmath171 , with probability mass function given by @xmath172 moreover , we let @xmath173 have a poisson distribution with intensity @xmath174 , where @xmath114 . then @xmath14 is distributed according to a negative binomial ( nb ) distribution @xcite . the probability mass function of @xmath14 is given by @xmath175 which is denoted as @xmath176 . we thus say that @xmath14 follows an nb subordinator . let @xmath177 and @xmath178 . it can be verified that @xmath179 has the same mean and variance as the @xmath119 distribution . the corresponding laplace transform then gives rise to a new family of bernstein functions , which is given by @xmath180.\ ] ] we refer to this family of bernstein functions as _ compound exp - log _ ( cel ) functions . the first - order derivative of @xmath181 w.r.t . @xmath182 is given by @xmath183 the lvy measure for @xmath181 is given by @xmath184 the proof is given in appendix 1 . we call this lvy measure a _ generalized poisson measure _ relative to the generalized gamma measure . like @xmath128 , @xmath181 can define a family of sparsity - inducing nonconvex penalties . also , @xmath181 for @xmath114 , @xmath185 and @xmath116 satisfies the conditions @xmath186 , @xmath187 and @xmath188 . we present a special cel function @xmath189 as well as the corresponding @xmath14 and @xmath137 in table [ tab : exam ] , where we replace @xmath190 and @xmath150 by @xmath152 and @xmath145 for notational simplicity . we now consider the limiting cases . [ pro:8 ] assume @xmath137 is defined by expression ( [ eqn : second_nu ] ) for fixed @xmath185 and @xmath116 . then we have that 1 . @xmath191 and @xmath192 . @xmath193 and @xmath194 . 3 . @xmath195 and @xmath142 . 4 . @xmath196 and @xmath197 notice that @xmath198 this shows that @xmath127 converges to @xmath199 , as @xmath200 . analogously , we obtain the second part of proposition [ pro:8]-(d ) , which implies that as @xmath200 , @xmath14 converges in distribution to a gamma random variable with shape parameter @xmath144 and scale parameter @xmath145 . an alternative proof is given in appendix 2 . proposition [ pro:8 ] shows that @xmath181 degenerates to exp as @xmath147 , while to log as @xmath200 . this shows an interesting connection between @xmath128 in expression ( [ eqn : first ] ) and @xmath181 in expression ( [ eqn : second ] ) ; that is , they have the same limiting behaviors . we note that for @xmath201 , @xmath202\ ] ] which is a composition of the log and exp functions , and that @xmath203\ ] ] which is a composition of the exp and log functions . in fact , the composition of any two bernstein functions is still bernstein . thus , the composition is also the laplace exponent of some subordinator , which is then a mixture of the subordinators corresponding to the original two bernstein functions @xcite . this leads us to an alternative derivation for the subordinators corresponding to @xmath122 and @xmath204 . that is , we have the following theorem whose proof is given in appendix 3 . [ thm : poigam ] the subordinator @xmath14 associated with @xmath128 is distributed according to the mixture of @xmath205 distributions with @xmath206 mixing , while @xmath14 associated with @xmath181 is distributed according to the mixture of @xmath207 distributions with @xmath208 mixing . additionally , the following theorem illustrates a limiting property of the subordinators as @xmath145 approaches 0 . [ thm : limit ] let @xmath209 be a fixed constant on @xmath210 $ ] . 1 . if @xmath211 where @xmath212 $ ] or @xmath213 , then @xmath14 converges in probability to @xmath56 , as @xmath214 . 2 . if @xmath215 where @xmath216\ ] ] or @xmath213 , then @xmath14 converges in probability to @xmath56 , as @xmath214 . the proof is given in appendix 4 . since @xmath14 converges in probability to @xmath56 " implies @xmath14 converges in distribution to @xmath56 , " we have that @xmath217 finally , consider the four nonconvex penalty function given in table [ tab : exam ] . we present the following property . that is , when @xmath213 and for any fixed @xmath116 , we have @xmath218 \leq\frac{s}{\gamma s { + } 1 } \leq\frac{1}{\gamma } [ 1 { - } \exp ( { - } \gamma s ) ] \leq\frac { 1}{\gamma } \log\big({\gamma } s { + } 1 \big ) \leq s,\ ] ] with equality only when @xmath219 . the proof is given in appendix 5 . this property is also illustrated in figure [ fig : penalty ] . in table [ tab : exam ] with @xmath220 and @xmath71 . ] we apply the compound poisson subordinators to the bayesian sparse learning problem given in section [ sec : levy ] . defining @xmath221 , we rewrite the hierarchical representation for the joint prior of the @xmath37 under the regression framework . that is , we assume that @xmath222 & \stackrel{ind}{\sim } & l(b_j|0 , \sigma ( 2\eta_j)^{-1 } ) , \\ f_{t^{*}(t_j)}(\eta_j ) & { \propto } & \eta_j^{-1 } f_{t(t_j)}(\eta_j),\end{aligned}\ ] ] which implies that @xmath223 the joint marginal pseudo - prior of the @xmath37 s is given by @xmath224 we will see in theorem [ thm : poster ] that the full conditional distribution @xmath225 is proper . thus , the maximum _ a posteriori _ ( map ) estimate of @xmath226 is based on the following optimization problem : @xmath227 clearly , the @xmath59 s are local regularization parameters and the @xmath228 s are latent shrinkage parameters . moreover , it is interesting that @xmath43 ( or @xmath55 ) is defined as a subordinator w.r.t . @xmath56 . the full conditional distribution @xmath229 is conjugate w.r.t . the prior , which is @xmath230 . specifically , it is an inverse gamma distribution of the form @xmath231.\ ] ] in the following experiment , we use an improper prior of the form @xmath232 ( i.e. , @xmath233 ) . clearly , @xmath229 is still an inverse gamma distribution in this setting . additionally , based on @xmath234 \prod_{j=1}^p \exp(-\frac { \eta _ j}{\sigma } |b_j|)\vadjust{\eject}\ ] ] and the proof of theorem [ thm : poster ] ( see appendix 6 ) , we have that the conditional distribution @xmath235 is proper . however , the absolute terms @xmath236 make the form of @xmath237 unfamiliar . thus , a gibbs sampling algorithm is not readily available and we resort to an em algorithm to estimate the model . notice that if @xmath238 is proper , the corresponding normalizing constant is given by @xmath239 d |b_j|= 2 \int_{0}^{\infty } \exp\big [ -t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big ] d ( |b_j|/\sigma),\ ] ] which is independent of @xmath240 . also , the conditional distribution @xmath241 is independent of the normalizing term . specifically , we always have that @xmath242 which is proper . as shown in table [ tab : exam ] , except for log with @xmath243 which can be transformed into a proper prior , the remaining bernstein functions can not be transformed into proper priors . in any case , our posterior computation is directly based on the marginal pseudo - prior @xmath244 . we ignore the involved normalizing term , because it is infinite if @xmath244 is improper and it is independent of @xmath240 if @xmath244 is proper . given the @xmath245th estimates @xmath246 of @xmath247 in the e - step of the em algorithm , we compute @xmath248 p(\eta_j|b_j^{(k ) } , \sigma ^{(k ) } , t_j ) } d \eta_j + \log p(\sigma ) \\ & \propto-\frac{n+\alpha_{\sigma}}{2 } \log\sigma{- } \frac{\|{\bf y } { -}{\bf x}{\bf b}\|_2 ^ 2 + \beta_{\sigma}}{2 \sigma } - ( p+1 ) \log \sigma \\ & \quad- \frac{1 } { \sigma } \sum_{j=1}^p here we omit some terms that are independent of parameters @xmath240 and @xmath226 . in fact , we only need to compute @xmath249 in the e - step . considering that @xmath250 and taking the derivative w.r.t . @xmath236 on both sides of the above equation , we have that @xmath251 the m - step maximizes @xmath252 w.r.t.@xmath253 . in particular , it is obtained that : @xmath254 the above em algorithm is related to the linear local approximation ( lla ) procedure @xcite . moreover , it shares the same convergence property given in @xcite and @xcite . subordinators help us to establish a direct connection between the local regularization parameters @xmath59 s and the latent shrinkage parameters @xmath39 s ( or @xmath41 ) . however , when we implement the map estimation , it is challenging how to select these local regularization parameters . we employ an ecme ( for expectation / conditional maximization either " ) algorithm @xcite for learning about the @xmath37 s and @xmath59 s simultaneously . for this purpose , we suggest assigning @xmath59 gamma prior @xmath255 , namely , @xmath256 because the full conditional distribution is also gamma and given by @xmath257 \sim\ga\big(\alpha_{t } , 1/[\psi(|b_j|/\sigma ) + \beta_{t}]\big).\ ] ] recall that we here compute the full conditional distribution directly using the marginal pseudo - prior @xmath238 , because our used bernstein functions in table [ tab : exam ] can not induce proper priors . however , if @xmath238 is proper , the corresponding normalizing term would rely on @xmath59 . as a result , the full conditional distribution of @xmath59 is possibly no longer gamma or even not analytically available . figure [ fig : graphal0]-(a ) depicts the hierarchical model for the bayesian penalized linear regression , and table [ tab : alg ] gives the ecme procedure where the e - step and cm - step are respectively identical to the e - step and the m - step of the em algorithm , with @xmath258 . the cme - step updates the @xmath59 s with @xmath259 in order to make sure that @xmath260 , it is necessary to assume that @xmath261 . in the following experiments , we set @xmath262 . we conduct experiments with the prior @xmath263 for comparison . this prior is induced from the @xmath264-norm penalty , so it is a proper specification . moreover , the full conditional distribution of @xmath59 w.r.t . its gamma prior @xmath265 is still gamma ; that is , @xmath257 \sim\ga\big({\alpha_t}{+}2 , \ ; 1/({\beta_t } { + } \sqrt{|b_j|/\sigma})\big).\ ] ] thus , the cme - step for updating the @xmath59 s is given by @xmath266 the convergence analysis of the ecme algorithm was presented by @xcite , who proved that the ecme algorithm retains the monotonicity property from the standard em . moreover , the ecme algorithm based on pseudo - priors was also used by @xcite . . the basic procedure of the ecme algorithm [ cols= " < , < " , ] our analysis is based on a set of simulated data , which are generated according to @xcite . in particular , we consider the following three data models small , " medium " and large . " data s : : : @xmath267 , @xmath268 , @xmath269 , and @xmath270 is a @xmath271 matrix with @xmath272 on the diagonal and @xmath273 on the off - diagonal . data m : : : @xmath274 , @xmath275 , @xmath276 has @xmath277 non - zeros such that @xmath278 and @xmath279 , and @xmath280 . data l : : : @xmath281 , @xmath282 , @xmath283 , and @xmath284 ( five blocks ) . for each data model , we generate @xmath285 data matrices @xmath286 such that each row of @xmath286 is generated from a multivariate gaussian distribution with mean @xmath287 and covariance matrix @xmath270 , @xmath288 , or @xmath289 . we assume a linear model @xmath290 with multivariate gaussian predictors @xmath286 and gaussian errors . we choose @xmath240 such that the signal - to - noise ratio ( snr ) is a specified value . following the setting in @xcite , we use @xmath291 in all the experiments . we employ a standardized prediction error ( spe ) to evaluate the model prediction ability . the minimal achievable value for spe is @xmath272 . variable selection accuracy is measured by the correctly predicted zeros and incorrectly predicted zeros in @xmath292 . the snr and spe are defined as @xmath293 for each data model , we generate training data of size @xmath294 , very large validation data and test data , each of size @xmath295 . for each algorithm , the optimal global tuning parameters are chosen by cross validation based on minimizing the average prediction errors . with the model @xmath292 computed on the training data , we compute spe on the test data . this procedure is repeated @xmath296 times , and we report the average and standard deviation of spe and the average of zero - nonzero error . we use `` '' to denote the proportion of correctly predicted zero entries in @xmath226 , that is , @xmath297 ; if all the nonzero entries are correctly predicted , this score should be @xmath298 . we report the results in table [ tab : toy2 ] . it is seen that our setting in figure [ fig : graphal0]-(a ) is better than the other two settings in figures [ fig : graphal0]-(b ) and ( c ) in both model prediction accuracy and variable selection ability . especially , when the size of the dataset takes large values , the prediction performance of the second setting becomes worse . the several nonconvex penalties are competitive , but they outperform the lasso . moreover , we see that log , exp , lfr and cel slightly outperform @xmath264 . the @xmath264 penalty indeed suffers from the problem of numerical instability during the em computations . as we know , the priors induced from lfr , cel and exp as well as log with @xmath299 are improper , but the prior induced from @xmath264 is proper . the experimental results show that these improper priors work well , even better than the proper case . vs. @xmath300 on data s " and data m " where @xmath301 is the permutation of @xmath302 such that @xmath303 . ] recall that in our approach each regression variable @xmath37 corresponds to a distinct local tuning parameter @xmath59 . thus , it is interesting to empirically investigate the inherent relationship between @xmath37 and @xmath59 . let @xmath304 be the estimate of @xmath59 obtained from our ecme algorithm ( alg 1 " ) , and @xmath305 be the permutation of @xmath306 such that @xmath307 . figure [ fig : tb1 ] depicts the change of @xmath308 vs.@xmath300 with log , exp , lfr and cel on data s " and data m. " we see that @xmath308 is decreasing w.r.t . moreover , @xmath308 becomes 0 when @xmath300 takes some large value . a similar phenomenon is also observed for data l. " this thus shows that the subordinator is a powerful bayesian approach for variable selection . in this paper we have introduced subordinators into the definition of nonconvex penalty functions . this leads us to a bayesian approach for constructing sparsity - inducing pseudo - priors . in particular , we have illustrated the use of two compound poisson subordinators : the compound poisson gamma subordinator and the negative binomial subordinator . in addition , we have established the relationship between the two families of compound poisson subordinators . that is , we have proved that the two families of compound poisson subordinators share the same limiting behaviors . moreover , their densities at each time have the same mean and variance . we have developed the ecme algorithms for solving sparse learning problems based on the nonconvex log , exp , lfr and cel penalties . we have conducted the experimental comparison with the state - of - the - art approach . the results have shown that our nonconvex penalization approach is potentially useful in high - dimensional bayesian modeling . our approach can be cast into a point estimation framework . it is also interesting to fit a fully bayesian framework based on the mcmc estimation . we would like to address this issue in future work . consider that @xmath310 & = \log\big[1-\frac{1}{1{+}\rho } \exp(-\frac{\rho}{1{+}\rho } \gamma s)\big ] - \log\big[1-\frac{1}{1{+}\rho}\big ] \\ & = \sum_{k=1}^{\infty } \frac{1}{k ( 1{+}\rho)^k } \big[1- \exp\big ( { -}\frac{\rho}{1{+}\rho } k \gamma s\big)\big ] \\ & = \sum_{k=1}^{\infty } \frac{1}{k ( 1{+}\rho)^k } \int_{0}^{\infty } ( 1- \exp(- u s ) ) \delta_{\frac{\rho k \gamma}{1{+}\rho}}(u ) d u.\end{aligned}\ ] ] we thus have that @xmath311 . we here give an alternative proof of proposition [ pro:8]-(d ) , which is immediately obtained from the following lemma . let @xmath312 take discrete value on @xmath313 and follow negative binomial distribution @xmath314 . if @xmath315 converges to a positive constant as @xmath316 , @xmath317 converges in distribution to a gamma random variable with shape @xmath315 and scale @xmath272 . since @xmath318 we have that @xmath319 notice that @xmath320 and @xmath321 this leads us to @xmath322 similarly , we have that @xmath323 consider a mixture of @xmath324 with @xmath325 mixing . that is , @xmath326 letting @xmath327 , @xmath328 and @xmath329 , we have that @xmath330 we now consider a mixture of @xmath331 with @xmath332 which is @xmath333 . let @xmath334 , @xmath335 , @xmath336 and @xmath337 . thus , @xmath338 since @xmath339=1 $ ] , we only need to consider the case that @xmath213 . recall that @xmath119 , whose mean and variance are @xmath340 whenever @xmath213 . by chebyshev s inequality , we have that @xmath341 hence , we have that @xmath342 similarly , we have part ( b ) . we first note that @xmath343 which implies that @xmath344 for @xmath345 . subsequently , we have that @xmath346 \leq0 $ ] . as a result , @xmath347 for @xmath345 . as for @xmath348 , it is directly obtained from that @xmath349 since @xmath350 = \frac{\gamma}{\exp(\gamma s ) } - \frac{\gamma}{1+\gamma s}<0 $ ] for @xmath345 , we have that @xmath351 for @xmath345 . first consider that @xmath352 \prod_{j=1}^p \sigma^{-1 } \exp\big(-t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big).\ ] ] to prove that @xmath353 is proper , it suffices to obtain that @xmath354 \prod_{j=1}^p \sigma^{-1 } \exp \big(-t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big ) d { \bf b } < \infty}.\ ] ] it is directly computed that @xmath355 \nonumber \\ & = \exp\big [ { - } \frac{1}{2 \sigma } ( { \bf b}{- } { \bf z})^t { \bf x}^t { \bf x } ( { \bf b}- { \bf z } ) \big ] \times\exp\big[- \frac{1}{2 \sigma } { \bf y}^t ( { \bf i}_n - { \bf x } ( { \bf x}^t { \bf x})^{+ } { \bf x}^t ) { \bf y}\big],\end{aligned}\ ] ] where @xmath356 and @xmath357 is the moore - penrose pseudo inverse of matrix @xmath358 @xcite . here we use the well - established properties that @xmath359 and @xmath360 . notice that if @xmath358 is nonsingular , then @xmath361 . in this case , we consider a conventional multivariate normal distribution @xmath362 . otherwise , we consider a singular multivariate normal distribution @xmath363 @xcite , the density of which is given by @xmath364.\ ] ] here @xmath365 , and @xmath366 , @xmath367 , are the positive eigenvalues of @xmath358 . in any case , we always write @xmath368 . thus , @xmath369 d{\bf b } < \infty}$ ] . it then follows the propriety of @xmath370 because @xmath371 \prod _ { j=1}^p \exp\big ( { - } t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big)\leq\exp\big [ { - } \frac{1}{2 \sigma } \|{\bf y}- { \bf x}{\bf b}\|_2 ^ 2 \big].\ ] ] we now consider that @xmath372 \prod_{j=1}^p \exp\big(-t_j \psi \big(\frac{|b_j| } { \sigma } \big ) \big).\ ] ] let @xmath373 { \bf y}$ ] . since the matrix @xmath374 is positive semidefinite , we obtain @xmath375 . based on expression ( [ eqn : pf01 ] ) , we can write @xmath376 \varpropto n({\bf b}|{\bf z } , \sigma({\bf x}^t { \bf x})^{+ } ) { \iga}(\sigma|\frac{\alpha_{\sigma } { + } n{+}2p{-}q}{2 } , \nu{+ } \beta_{\sigma}).\ ] ] subsequently , we have that @xmath377 d { \bf b } d \sigma } < \infty,\ ] ] and hence , @xmath377 \prod_{j=1}^p \exp\big(-t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big ) d { \bf b}d \sigma } < \infty.\ ] ] therefore @xmath378 is proper . thirdly , we take @xmath379 } { \sigma ^{\frac{n+\alpha_{\sigma}+2p}{2 } + 1 } } \prod_{j=1}^p \big\{\exp \big({-}t_j \psi\big(\frac{|b_j| } { \sigma } \big ) \big ) \frac { t_j^{{\alpha_t}{- } 1 } \exp({- } { \beta_t } t_j)}{\gamma({\alpha_t } ) } \big\ } \\ & \triangleq f({\bf b } , \sigma , { \bf t}).\end{aligned}\ ] ] in this case , we compute @xmath380 } { \sigma^{\frac{n+\alpha_{\sigma}+2p}{2 } + 1 } } \prod _ { j=1}^p \frac{1 } { \big({\beta_t } { + } \psi\big(\frac{|b_j| } { \sigma } \big ) \big)^{{\alpha_t } } } d { \bf b}d \sigma}.\ ] ] similar to the previous proof , we also have that @xmath381 because @xmath382 . as a result , @xmath383 is proper . finally , consider the setting that @xmath384 . that is , @xmath385 and @xmath386 . in this case , if @xmath387 , we obtain @xmath388 and @xmath389 . as a result , we use the inverse gamma distribution @xmath390 . thus , the results still hold . polson , n. g. and scott , j. g. ( 2010 ) . `` shrink globally , act locally : sparse bayesian regularization and prediction . '' in bernardo , j. m. , bayarri , m. j. , berger , j. o. , dawid , a. p. , heckerman , d. , smith , a. f. m. , and west , m. ( eds . ) , _ bayesian statistics 9_. oxford university press . the authors would like to thank the editors and two anonymous referees for their constructive comments and suggestions on the original version of this paper . the authors would especially like to thank the associate editor for giving extremely detailed comments on earlier drafts . this work has been supported in part by the natural science foundation of china ( no . 61070239 ) .

in this paper we discuss bayesian nonconvex penalization for sparse learning problems . we explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one - dimensional lvy processes . we particularly study a family of continuous compound poisson subordinators and a family of discrete compound poisson subordinators . we exemplify four specific subordinators : gamma , poisson , negative binomial and squared bessel subordinators . the laplace exponents of the subordinators are bernstein functions , so they can be used as sparsity - inducing nonconvex penalty functions . we exploit these subordinators in regression problems , yielding a hierarchical model with multiple regularization parameters . we devise ecme ( expectation / conditional maximization either ) algorithms to simultaneously estimate regression coefficients and regularization parameters . the empirical evaluation of simulated data shows that our approach is feasible and effective in high - dimensional data analysis .

1308.6069

* the anomalous x - ray pulsars ( axps ) * are a group of x - ray pulsars whose spin periods fall in a narrow range ( @xmath0 s ) , whose x - ray spectra are very soft , and which show no evidence that they accrete from a binary companion ( see mereghetti 1999 for a recent review ) . these objects may be isolated neutron stars with extremely strong ( @xmath1 g ) surface magnetic fields , or they may be accreting from a `` fallback '' accretion disk . optical measurements could potentially help discriminate between these models . an optical counterpart to one axp , 4u 0142 + 61 , has recently been identified and shown to have peculiar optical colors ( hulleman et al . * the radio - quiet neutron stars ( rqnss ) * are a group of compact x - ray sources found near the center of young supernova remnants . their x - ray spectra are roughly consistent with young , cooling neutron stars , but they show no evidence for the non - thermal emission associated with `` classical '' young pulsars like the crab ( see brazier & johnston 1999 for a review ) . the x - ray spectral properties of the rqnss and the axps are similar ( see , e.g. , chakrabarty et al . below in table 1 , the general properties of the three rqnss as our targets in the southern sky are listed . clcccc & & @xmath2 & age & @xmath3 & + source & snr & ( kpc ) & ( @xmath4 yr ) & ( kev ) & refs + 1e 08204247 & pup a & 2.0 & 3.7 & 0.28 & 1 - 3 + 1e 16145055 & rcw 103 & 3.3 & 1 - 3 & 0.56 & 4 - 6 + 1e 12075209 & pks 120952 & 1.5 & 7 & 0.25 & 7 - 9 + + + + our observations were made using the magellan instant camera ( magic ) on the magellan-1/walter baade 6.5-meter telescope at las campanas observatory , chile . magic is a ccd filter photometer built by mit and cfa for the @xmath5 focus of the baade telescope . the current detector is a 2048@xmath62048 site ccd with a 69 mas / pixel scale and a 142@xmath6142 arcsec field of view . we used the sloan filter set , which have the following central wavelengths ( fukugita et al . 1996 ) : @xmath7=3540 ; @xmath8=4770 ; @xmath9=6230 ; @xmath10=7620 ; and @xmath11=9130 . brazier , k.t.s . , & johnston , s. 1999 , mnras , 303 , l1 bignami , g.f . , caraveo , g.a . , & mereghetti , s. 1992 , apj , 389 , l67 chakrabarty , d. et al . 2001 , apj , 548 , 800 fukugita , m. et al . 1996 , aj , 111 , 1748 garmire , g.p . , pavlov , g.g . , & garmire , a.b . 2000 , iauc , 7350 , 2 gotthelf , e.v . , petre , r. , & hwang , u. 1997 , 487 , l175 helfand , d.j . , & becker , r.h . 1984 , nature , 307 , 215 hulleman , f. , kerkwijk , m.h . , & kulkarni , s.r . 2000 , nature , 408 , 689 mereghetti , s. 1999 , in the neutron star black hole connection , ed . c. kouveliotou et al . , ( dordrecht : kluwer ) mereghetti , s. , caraveo , p. , & bignami , g.f . 1992 , a & a , 263 , 172 mereghetti , s. , bignami , g.f . , & caraveo , p.a . 1996 , apj , 464 , 842 pavlov , g. g. , zavlin , v.e . , & trmper , j. 1999 , apj , 511 , l45 petre , r. , becker , c.m . , & winkler , p.f . 1996 , apj , 465 , l43 petre , et al . 1982 , apj , 258 , 22 seward , f.d . 1990 , apjss , 73 , 781 tuohy , i. , & garmire , g. 1980 , apj , 239 , 107

we report on our search for the optical counterparts of the southern hemisphere anomalous x - ray pulsar 1e1048.1 - 5937 and the radio - quiet neutron stars in supernova remnants puppis a , rcw 103 , and pks 1209 - 52 . the observations were carried out with the new mit / cfa magic camera on the magellan - i 6.5 m telescope in chile . we present deep multiband optical images of the x - ray error circles for each of these targets and discuss the resulting candidates and limits . # 1_#1 _ # 1_#1 _ = # 1 1.25 in .125 in .25 in

astro-ph0112125

synchronization in chaotic systems is a surprising phenomenon , which recently received a lot of attention , see e.g. @xcite . even though the heuristic theory and the classification of the synchronization phenomena are well studied and reasonably well understood , a mathematically rigorous theory is still lacking . generally speaking , a standard difficulty lies in the fact that the phenomenon involves the dynamics of non - uniformly chaotic systems , typically consisting of different sub - systems , whose long - time behavior depends crucially on the sign of the `` central '' lyapunov exponents , i.e. of those exponents that are zero in the case of zero coupling , and become possibly non - trivial in the presence of interactions among the sub - systems . the mathematical control of such exponents is typically very hard . progress in their computation is a fundamental preliminary step for the construction of the srb measure of chains or lattices of chaotic flows , which may serve as toy models for extensive chaotic systems out - of - equilibrium ( i.e. they may serve as standard models for non - equilibrium steady states in non - equilibrium statistical mechanics ) . in a previous paper @xcite , we introduced a simple model for phase synchronization in a three - dimensional system consisting of the suspension flow of arnold s cat map coupled with a clock . the coupling in @xcite was unidirectional , in the sense that it did not modify the suspension flow , but only the clock motion . notwithstanding its simplicity , the model has a non - trivial behavior : in particular , it exhibits phase locking and in @xcite we constructed the corresponding attractive invariant manifold via a convergent expansion . however , because of unidirectionality , the lyapunov spectrum in @xcite was very simple : the `` longitudinal '' exponents ( i.e. , those corresponding to the motion on the invariant manifold ) coincided with the unperturbed ones , and the central exponent was expressed in the form of a simple integral of the perturbation over the manifold . in this paper , we extend the analysis of @xcite to a simple bidirectional model , for which the lyapunov spectrum is non - trivial , and we show how to compute it in terms of a modified expansion , which takes the form of a decorated tree expansion discussed in detail in the following . the model is defined as follows . take arnold s cat map @xmath0 and denote by @xmath1 and @xmath2 the eigenvalues and eigenvectors , respectively , of @xmath3 : @xmath4 with @xmath5 , so that @xmath6 are normalized . we let the suspension flow of arnold s cat be defined as @xmath7 , with @xmath8 , if @xmath9 @xmath10 . formally , @xmath11 is the solution to the following differential equation instead of , but throughout the paper we only used the fact that at all times @xmath12 the variable @xmath13 jumped abruptly from @xmath14 to @xmath15 , and besides these discontinuities the flow was smooth . therefore , all the results and statements of @xcite are correct , modulo this re - interpretation of the flow equation ( * ? ? ? * ( 2.1 ) ) , where @xmath16 should be replaced by @xmath17 . ] on @xmath18 : x=(t)(s ) x , [ 1.susf]where @xmath19 is the @xmath20-periodic delta function such that @xmath21 for all @xmath22 . the model of interest is obtained by coupling the suspension flow of arnold s cat map with a clock by a regular perturbation , so that on @xmath23 the evolution equation is @xmath24x+\varepsilon f(x , w , t ) , & \\ \dot{w}=1+\varepsilon g(x , w , t ) , \end{cases}\ ] ] where @xmath25 and @xmath26 , @xmath27 are @xmath20-periodic in their arguments . for @xmath28 the motions of @xmath13 and @xmath29 are independent . therefore , the relative phase @xmath29 mod @xmath30 among the two flows is arbitrary . if @xmath31 and if the interaction is dissipative ( in a suitable sense , to be clarified in a moment ) , then the phases of the two sub - systems can lock , so that the limiting motion in the far future takes place on an attractor of dimension smaller than 3 , for all initial data in an open neighborood of the attractor . in @xcite , we explicitly constructed such an attractor in terms of a convergent power series expansion in @xmath32 , for @xmath33 and a special class of dissipative functions @xmath27 . in this paper , we generalize the analysis of @xcite to @xmath34 . our first result concerns the construction of the attractive invariant manifold for @xmath34 . [ prop:1 ] let @xmath35 be the flow on @xmath36 associated with the dynamics , with @xmath26 and @xmath27 analytic in their arguments . set @xmath37 and assume there exists @xmath38 such that @xmath39 and @xmath40 , independently of @xmath41 . then there are constants @xmath42 such that for @xmath43 there exist a homemorphism @xmath44 and a continuous function @xmath45 , both hlder - continuous of exponent @xmath46 , such that the surface @xmath47 is invariant under the poincar map @xmath48 and the dynamics of @xmath48 on @xmath49 is conjugated to that of @xmath50 on @xmath51 , i.e. @xmath52 the proof of this theorem is constructive : it provides an explicit algorithm for computing the generic term of the perturbation series of @xmath53 with respect to @xmath32 , it shows how to estimate it and how to prove convergence of the series . as a by - product , we show that the invariant manifold is holomorphic in @xmath32 in a suitable domain of the complex plane , whose boundary contains the origin . the construction also implies that @xmath54 is an attractor . we denote by @xmath55 its basin of attraction and by @xmath56 an arbitrary open neighborood of @xmath54 contained in @xmath55 such that @xmath57 , with @xmath58 the lesbegue measure on @xmath59 . in addition to the construction of the invariant surface , in this paper we show how to compute the invariant measure on the attractor and the lyapunov spectrum , in terms of convergent expansions . more precisely , let @xmath60 be the lesbegue measure restricted to @xmath56 , i.e. , denoting by @xmath61 the characteristic function of @xmath56 , @xmath62 , for all measurable @xmath63 . the `` natural '' invariant measure on the attractor , @xmath64 , is defined by @xmath65 for all continuous functions @xmath66 and @xmath60-a.e . @xmath67 , where @xmath68 . the limiting measure @xmath64 is supported on @xmath54 and such that @xmath69 . on the attractor , @xmath64-a.e . point defines a dynamical base , i.e. a decomposition of the tangent plane as @xmath70 , such that @xmath71 the constants of motion @xmath72 are the lyapunov exponents , and we suppose them ordered as @xmath73 ; in the following we shall call @xmath74 the _ central _ lyapunov exponent . our second main result is the following . [ prop:2 ] there exists @xmath75 such that the following is true . let @xmath66 be an hlder continuous function on @xmath59 . then @xmath76 is hlder continuous in @xmath32 , for @xmath77 . if @xmath66 is analytic , then @xmath76 is analytic in @xmath32 , for @xmath78 and a suitable @xmath66-dependent constant @xmath79 . moreover , the lyapunov exponents @xmath72 , @xmath80 , are analytic in @xmath32 for @xmath77 . in particular , the central lyapunov exponent is negative : @xmath81 , while @xmath82 . the paper is organized as follows . theorem [ prop:1 ] is proved in section [ sec:2 ] below . the proof follows the same strategy of @xcite : ( 1 ) we first write the equations for the invariant surface and solve them recursively at all orders in @xmath32 ; ( 2 ) then we express the result of the recursion ( which is not simply a power series in @xmath32 ) in terms of tree diagrams ( planar graphs without loops ) ; trees with @xmath83 nodes are proportional to @xmath84 times a _ tree value _ , which is also a function of @xmath32 ; ( 3 ) finally , using the tree representation , we derive an upper bound on the tree values . the fact that the dissipation is small , of order @xmath85 , produces bad factors @xmath86 in the bounds of the tree values , for some @xmath87 depending on the tree therefore , we need to show that for any tree @xmath87 is smaller than a fraction of @xmath83 , if @xmath83 is the number of nodes in the tree . this is proved by exhibiting suitable cancellations , arising from the condition @xmath39 . theorem [ prop:2 ] is proved in section [ sec:3 ] . the proof adapts the tree expansion to the computation of the local lyapunov exponents @xmath88 on the invariant surface , in the spirit of ( * ? ? ? * chapter 10 ) and @xcite . the positive local lyapunov exponent @xmath89 plays the role of the gibbs potential for the invariant measure @xmath90 . therefore , given a convergent expansion for @xmath89 , @xmath90 can be constructed by standard cluster expansion methods , as in ( * ? ? ? * chapter 10 ) . finally , @xmath72 can be expressed as averages of the local exponents over the stationary distribution . in section [ sec:4 ] , we present some numerical evidences for a fractal to non - fractal transition of the invariant manifold , and formulate some conjectures . in this section , we define the equations for the invariant manifold , by introducing a conjugation that maps the dynamics restricted to the attractor onto the unperturbed one . the conjugation is denoted by @xmath91 , with @xmath92 where @xmath93 is the identity in @xmath94 . let @xmath95 and @xmath96 be the initial conditions at time @xmath97 . we will look for a solution to of the form @xmath98 for @xmath99 , with boundary conditions @xmath100 the evolution equation for @xmath29 will be written by `` expanding the vector field at first order in @xmath101 and at zeroth order in @xmath102 '' , i.e. as @xmath103 where @xmath104 and g(,t):=g(s h()+a(,t),w_0+t+u(,t),t)-_0(,t)- _ 1(,t)u(,t).[2.g ] the logic in the rewriting is that the ( linear ) approximate dynamics obtained by neglecting @xmath105 is dissipative , with contraction rate proportional to @xmath32 , thanks to the second condition in : this will allow us to control the full dynamics as a perturbation of the approximate one . the approximation obtained by neglecting @xmath105 is the simplest one displaying dissipation . in principle we could have expanded the dynamics at first order both in @xmath101 and in @xmath106 , but the result would be qualitatively the same . we now set @xmath107 and fix @xmath108 such that @xmath109 ; then we obtain @xmath110 the equation for @xmath13 , if expressed in terms of @xmath111 , gives , after integration , @xmath112 for @xmath113 , these give [ eq:2.7 ] @xmath114 it is useful to introduce an auxiliary parameter @xmath115 , to be eventually set equal to @xmath32 , and rewrite and as [ eq:2.8 ] @xmath116 the idea is to first consider @xmath115 as a parameter independent of @xmath117 , then write the solution in the form of a power series in @xmath32 , with coefficients depending on @xmath60 , and finally show that the ( @xmath115-dependent ) radius of convergence of the series in @xmath32 behaves like @xmath118 , @xmath119 , at small @xmath60 : this implies that we will be able to take @xmath120 without spoiling the summability of the series . summarizing , we will look for a solution of in the form @xmath121 with [ eq:2.11 ] @xmath122 and : @xmath123 , with @xmath11 and @xmath124 as in ; @xmath105 is defined in ; @xmath60 must be eventually set equal to @xmath32 . the solution to - is looked for in the form of a power series expansion in @xmath117 ( at fixed @xmath60 , in the sense explained after ) . therefore , we write [ eq:2.10 ] @xmath125 and insert these expansions into . by the analyticity assumption on @xmath26 and @xmath27 , we may expand ( defining @xmath126 and @xmath127 ) @xmath128 where @xmath129 in the first sum denotes the constraint @xmath130 , and @xmath131 and similarly for @xmath132 ( recall that @xmath133 and @xmath6 are the eigenvalues and eigenvectors of @xmath134 ) ; here and henceforth we are denoting by @xmath135 the standard scalar product in @xmath136 . moreover @xmath137 where @xmath138 and @xmath139 . for future reference , we note since now that the analyticity of @xmath26 and @xmath27 yields , by the cauchy inequality , @xmath140 for some constant @xmath141 , uniformly in @xmath142 ( here @xmath143 ) . define @xmath144 , @xmath145 , and @xmath146 , with @xmath147 ) . setting @xmath148 and plugging into , we find for @xmath149 [ eq:2.12 ] @xmath150 introduce the notation @xmath151 where @xmath152 . here and henceforth , if @xmath153 , the product @xmath154 should be interpreted as 1 , and similarly for the other products in the case that @xmath155 and/or @xmath156 . then , defining @xmath157 , we find , for @xmath158 , [ eq:2.13 ] @xmath159 where @xmath160 . we now want to bound the generic term in the series originating from the recursive equations ; the goal is to show that the @xmath83-th order is bounded proportionally to @xmath161 , with @xmath162 and @xmath163 . we find convenient to represent graphically the coefficients in in terms of rooted trees ( or simply trees , in the following ) as in @xcite . we refer to ( * ? ? ? * section v ) for the definition of trees and notations . with respect to the trees in @xcite in the present case there are four _ types _ of nodes . we use the symbols , , and , calling them nodes of type @xmath164 , @xmath165 , @xmath166 and @xmath167 , respectively : they correspond to contributions to @xmath168 , respectively . the constraint @xmath129 forbids a node of type 0 or 1 to be immediately preceded by exactly one node of these two types . recall that a _ tree _ is a partially ordered set of nodes and lines ; the partial ordering relation is denoted by @xmath169 and each line will be drawn as an arrow pointing from the node it exits to the node it enters . we call @xmath170 the set of nodes and @xmath171 the set of lines of the tree @xmath172 . as in @xcite we denote by @xmath173 the node such that @xmath174 for any node @xmath175 : @xmath173 will be called the _ special node _ and the line exiting @xmath173 will be called the _ root line_. the root line can be imagined to enter a further point , called the root , which , however , is not counted as a node . with each node we associate a label @xmath176 to denote its type and a time variable @xmath177 $ ] . with the nodes of types 2 and 3 we also associate a label @xmath178 . denoting by @xmath179 the path of lines connecting @xmath29 to @xmath180 , with both @xmath181 and @xmath29 included , set @xmath182 where @xmath183 if @xmath184 , @xmath185 if @xmath186 , and @xmath187 otherwise . given a node @xmath181 , we denote by @xmath188 the unique node immediately following it ; moreover , we let @xmath189 be the number of nodes of type @xmath190 immediately preceding it ; if @xmath191 , we also define @xmath192 , @xmath193 , to be the number of nodes @xmath29 of type @xmath190 with @xmath194 immediately preceding it ( i.e. such that @xmath195 ) . finally , we let @xmath196 , with @xmath197 , and @xmath198 . a node @xmath181 is called an _ end - node _ if @xmath199 , while it is called an _ internal node _ if it is not an end - node . the _ node factor _ @xmath200 is defined as @xmath201 where @xmath202 is interpreted as equal to @xmath203 , while the _ node integral _ is @xmath204 with the definitions above , we denote by @xmath205 the set of labelled trees with @xmath83 nodes , @xmath206 , and the constraint that nodes of type @xmath164 or @xmath165 can not be immediately preceded by exactly one node of type @xmath164 or @xmath165 ; if @xmath191 , we also denote by @xmath207 the subset of by @xmath205 with @xmath208 . then , one can prove by induction that [ eq:2.14 ] @xmath209 where @xmath210 with the integrals to be performed by following the tree ordering , i.e. by starting from the end - nodes and by moving towards the root . in figure [ fig : n=00003d1 ] the first order contributions are graphically represented , while the contributions of order @xmath211 are shown in figure [ fig : n=00003d2 ] . for each node @xmath181 , the label @xmath212 is drawn superimposed on the line exiting @xmath181 , for clarity purposes , while the label @xmath213 ( to be summed over ) is not explicitly shown . @c=2cm@r=0.5 cm @<-[r]^- _ ( 1.08)v_0 & * + [ o][f*:black ] & u^(1)()=@<-[r]^- _ ( 1.10)v_0 & * + [ o][f ] + @<-[r]^- _ ( 1.08)v_0 & * + [ f*:black ] & h^(1)_()=@<-[r]^- _ ( 1.12)v_0 & * + [ f ] @c=1.5cm@r=0.5 cm .1 cm ^(2)()=@<-[r]^- _ ( 1.12)v_0 & * + [ o][f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ f ] & + & @<-[r]^- _ ( 1.20)v_0 & * + [ o][f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ f*:black ] @c=1.5cm@r=0.5 cm u^(2)()=@<-[r]^- _ ( 1.12)v_0 & * + [ o][f]@<-[r]^- _ ( 1.20)v_1 & * + [ f ] & + & @<-[r]^- _ ( 1.20)v_0 & * + [ o][f]@<-[r]^- _ ( 1.20)v_1 & * + [ f*:black ] @c=1.5cm@r=0.5 cm a^(2)_()=@<-[r]^- _ ( 1.12)v_0 & * + [ f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ f ] & + & @<-[r]^- _ ( 1.20)v_0 & * + [ f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ f*:black ] + @<-[r]^- _ ( 1.14)v_0 & * + [ f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ o][f ] & + & @<-[r]^ - _ ( 1.20)v_0 & * + [ f*:black]@<-[r]^- _ ( 1.20)v_1 & * + [ o][f*:black ] @c=1.5cm@r=0.5 cm h^(2)_()=@<-[r]^- _ ( 1.16)v_0 & * + [ f]@<-[r]^- _ ( 1.20)v_1 & * + [ f ] & + & @<-[r]^- _ ( 1.20)v_0 & * + [ f]@<-[r]^- _ ( 1.20)v_1 & * + [ f*:black ] + @<-[r]^- _ ( 1.14)v_0 & * + [ f]@<-[r]^- _ ( 1.20)v_1 & * + [ o][f ] & + & @<-[r]^ - _ ( 1.20)v_0 & * + [ f]@<-[r]^- _ ( 1.20)v_1 & * + [ o][f*:black ] for @xmath214 a few contributions to @xmath215 are shown in figure [ fig : n=00003d3 ] . all the other contributions are obtained by replacing the nodes preceding the special node @xmath173 by nodes of a different type , with the constraint that if only one line enters @xmath173 then it exits a node of type 2 or 3 ; note that in @xcite the linear trees at the bottom of the figure were not possible . @r=0.5 cm & & * + [ o][f ] & & & & * + [ o][f*:black ] & & & & * + [ o][f*:black ] + @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[ru ] _ ( 1.14)v_1 @<-[rd]^- _ ( 1.16)v_2 & & + & @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[ru ] _ ( 1.14)v_1 @<-[rd ] _ ( 1.16)v_2 & & + & @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[ru ] _ ( 1.14)v_1 @<-[rd ] _ ( 1.16)v_2 & & + + & & * + [ f ] & & & & * + [ o][f ] & & & & * + [ o][f*:black ] & + & & * + [ f ] + @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[ru]^- _ ( 1.14)v_1 @<-[rd]^- _ ( 1.16)v_2 & & + & @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[r]^- _ ( 0.70)v_1 & * + [ f]@<-[r]^- _ ( 1.28)v_2 & * + [ o][f*:black ] & + & @<-[r ] _ ( 0.80)v_0 & * + [ o][f]@<-[r]^ _ ( 0.70)v_1 & * + [ f]@<-[r]^- _ ( 1.28)v_2 & * + [ f ] + & & * + [ f ] given @xmath216 , we let @xmath217 be the family of labelled trees differing from @xmath218 just by the choice of the labels @xmath219 . then , using and proceeding as in ( * ? ? ? * section vi ) , we obtain @xmath220 where @xmath221 is a suitable constant and @xmath222 is the number of internal nodes of type @xmath190 in @xmath218 ; see @xcite for details . we are then left with bounding @xmath223 . [ lem:2.1 ] for all @xmath224 one has @xmath225 . the proof is given in appendix [ app : a ] , where it is also shown that such an upper bound on @xmath226 is optimal , i.e. there are trees that saturate the inequality . combining lemma [ lem:2.1 ] with and recalling that the number of distinct families @xmath217 in @xmath227 is bounded by @xmath228 , for a suitable @xmath229 , we get @xmath230 is a positive constant and @xmath231 $ ] is the integer part . implies that the radius of convergence of the series is bounded by @xmath232 . therefore , we can take @xmath233 . hlder - continuity of @xmath234 and @xmath235 can be proved mutatis mutandis like in @xcite . this completes the proof of theorem [ prop:1 ] . in order to compute the lyapunov exponents , we need to understand how the vectors on the tangent space evolve under the interacting dynamics . to this purpose , we set @xmath236 , rewrite as @xmath237 , with @xmath238 , and write the dynamics on the tangent space as follows : @xmath239 + \e \boldsymbol{\partial f}({\boldsymbol{x}},t)\;,\ ] ] where @xmath240 , @xmath241 is the solution to found in section [ sec:2 ] , @xmath242 are the projections into the unperturbed eigendirections @xmath243 , @xmath133 are the corresponding unperturbed eigenvalues , and @xmath244 is the jacobian matrix of @xmath245 . integration of gives the tangent map @xmath246 . we denote by @xmath247 the solution to with initial condition @xmath248 at @xmath97 ( started at @xmath249 ) . for @xmath250 $ ] we obtain @xmath251 we look for a conjugation @xmath252 , @xmath253 where @xmath93 is the identity in @xmath254 and @xmath255 is a @xmath256 matrix , such that , by setting @xmath257 for a suitable @xmath256 matrix @xmath258 , to be determined , one has @xmath259 , that is @xmath260 of course only the part of involving the tangent dynamics has still to be solved , so we study the conjugation equation @xmath261 the matrix @xmath262 will be taken to be diagonal in the basis @xmath263 , where @xmath264 , and @xmath265 . then in the basis @xmath266 one has @xmath267 while the matrix @xmath268 takes diagonal form , with values @xmath269 along the main diagonal . from now on we shall use this basis , and implicitly assume that the indices @xmath270 , run over the values @xmath271 , unless stated otherwise . by setting @xmath272 , we obtain from @xmath273 note that , given a solution @xmath274 of , then also @xmath275 is a solution with @xmath276 replaced with @xmath277 , where @xmath278 are non - zero functions from @xmath51 to @xmath279 . therefore , with no loss of generality , we can require the diagonal elements of @xmath280 to vanish : hence @xmath280 will be looked for as an off - diagonal matrix . we look for the conjugation in the form - , with @xmath281 equations and give @xmath282 where @xmath283 can be computed iteratively via as & & -.8truecm ds_^(())= +_n1^n_0^d_1 ( s^_1_(()),_1 ) _ 0^_1d_2(s^_2_(()),_2 ) + & & 1.15truecm _ 0^_n-1d_n(s^_n_(()),_n):=(+m(,)).[eq:3.9bis]at first order in @xmath32 , gives ( with @xmath284 ) @xmath285 in particular , setting @xmath286 and recalling that @xmath287 is off - diagonal , we find @xmath288 while , for @xmath289 , we have to solve recursively for @xmath290 , the result being ( if @xmath193 ) : [ eq:3.11 ] @xmath291 in order to compute the higher orders , we insert in the left side of , thus getting @xmath292 where @xmath293 . note that , according to , @xmath294 is expressed as a series of iterated integrals of @xmath295 , where @xmath244 is analytic in its argument . therefore , the power series expansion in @xmath32 of @xmath294 can be obtained ( and its @xmath83-th order coefficient can be bounded ) by using the corresponding expansions for the components of @xmath296 ; here the functions @xmath297 , @xmath298 , @xmath299 and @xmath300 are as in with the coefficients given by and bounded as in . we write @xmath301 by using the very definition of @xmath294 and the bounds , it is straightforward to prove that ^(n ) ( ) : = _ i , j \{+,-,3 } | _ i , j ( ) | c_4^n^-[(2n-1)/3][e3.14]for a suitable @xmath302 . now , if @xmath303 , the diagonal part of gives @xmath304 while the off - diagonal part can be solved in a way similar to , i.e. , if @xmath193 , [ eq:3.16 ] @xmath305 , \label{eq:3.16a } \\ k_{\alpha,3}^{(n)}(\varphi ) & = -\a\l_\a^{-1}\sum_{m\in\mathbb z_\a}\l_\a^{-m } \big [ \mathfrak{m}_{\alpha,3}^{(n)}(s^m\f ) + \mathcal q_{\a,3}^{(n)}(s^m\f ) \bigr ] , \label{eq:3.16b } \\ k_{3,-\alpha}^{(n)}(\varphi ) & = - \a\sum_{m\in\mathbb z_{\a}}\l_{-\a}^{m } \big[\mathfrak{m}_{3,-\alpha}^{(n)}(s^m\f)\l_{-\a } + \mathcal q_{3,-\a}^{(n)}(s^m\f ) \bigr ] , \label{eq:3.16c}\end{aligned}\ ] ] where we have set @xmath306 in the simple case that @xmath307 , @xmath308 , while @xmath309 thus recovering the formula for @xmath310 given in ( * ? ? ? * section vii ) . in figure [ fig : rappresentazione - grafica - di gamma^n ] and [ fig : rappresentazione - grafica - di k^n ] we give a graphical representation of and , respectively . the representation of and is the same as in figure [ fig : rappresentazione - grafica - di k^n ] , simply with the labels @xmath311 replaced by @xmath312 and @xmath313 , respectively . @r=0.5cm@c=1.0 cm & & & & & & & & * + [ o][f**:black ] + _ i^(n)()=@<-[r ] ^-ii ^(1.24)(n ) & * + [ f**:black ] & = @<-[r]^-ii _ ( 0.75)v_0 & * + [ f**:black]@ < [ r]^<<^(0.6)ii ^(1.4)(n ) & * + [ o][f**:black ] & + & @<-[r]^-ii _ ( 0.80)v_0 & * + [ f**:black ] @ < [ ru]^-ij_(1.2)(n_1)@<-[rd]^(0.2)^-ji^(1.2)(n_2 ) & + & & & & & & & & * + [ f**:white ] @r=0.5cm@c=1.0 cm & & & & & & & & * + [ o][f**:black ] + k_i , j^(n)()=@<-[r ] ^-i j ^(1.24)(n ) & * + [ f**:white ] & = @<-[r]^-i j _ ( 0.75)v_0 & * + [ f**:white]@ < [ r]^<<^(0.7)i j ^(1.4)(n ) & * + [ o][f**:black ] & + & @<-[r]^-i j _ ( 0.80)v_0 & * + [ f**:white ] @ < [ ru]^-i j_(1.2)(n_1 ) @<-[rd]^(0.2)^(0.6)j j^(1.2)(n_2 ) & + & & & & & & & & * + [ f**:white ] + & & * + [ f**:white ] + + @<-[r]^-i j _ ( 0.80)v_0 & * + [ f**:white ] @<-[ru]^-i j_(1.2)(n_1 ) @<-[rd]^(0.25)^(0.6)j j^(1.2)(n_2 ) & + & & * + [ f**:black ] to iterate the graphical construction and provide a tree representation for both @xmath262 and @xmath280 , we need a few more definitions . we identify three types of _ principal nodes _ , that we call of type @xmath314 , @xmath315 and @xmath316 , and represent graphically , respectively , by , and . with any such node @xmath181 , we associate a label @xmath317 , to denote its type , and two labels @xmath318 , which will be drawn superimposed to the line exiting @xmath181 ; if @xmath181 is of type @xmath315 , then @xmath319 , while if @xmath181 is of type @xmath314 , then @xmath320 . a node is of type @xmath316 if and only if it is an end - node . furthermore , with each node @xmath181 with @xmath321 , we associate a label @xmath322 , while we set @xmath323 for all nodes @xmath181 with @xmath324 ; with each node @xmath181 with @xmath325 , we associate a label @xmath326 such that either @xmath327 or @xmath328 ( recall that if @xmath181 is a nodes of type @xmath314 then @xmath320 , so that either @xmath329 or @xmath330 are @xmath331 ) , and a label @xmath332 ; if @xmath181 is of type @xmath315 or @xmath316 we define @xmath187 . if @xmath333 denotes the number of lines entering @xmath181 and @xmath334 the number of lines of type @xmath190 entering @xmath181 , we have the constraints @xmath335 and @xmath336 . moreover : @xmath337 ; @xmath338 . if @xmath339 and @xmath340 is the node immediately preceding @xmath181 on has @xmath341 and @xmath342 . if @xmath343 , let @xmath344 be the two nodes immediately preceding @xmath181 ; if @xmath345 , with no loss of generality we assume that @xmath340 is of type @xmath346 ( so that @xmath347 is of type @xmath314 ) ; if @xmath348 , with no loss of generality we assume that @xmath340 is of type @xmath314 ( so that @xmath347 is of type @xmath315 ) ; in both cases we impose the constraints that @xmath349 , @xmath350 and @xmath351 . denoting by @xmath188 the node immediately following @xmath181 , we set @xmath352 , and @xmath353 we are finally ready to define the _ node factors _ associated with the nodes : @xmath354 then , by iterating the graphical representation in figures [ fig : rappresentazione - grafica - di gamma^n ] and [ fig : rappresentazione - grafica - di k^n ] , we end up with trees like that in figure [ fig : esempi - di - alberi 1 e 2 ] for @xmath289 ; note that the end - nodes are all of type @xmath316 . if @xmath286 the only difference is that the special node @xmath355 is of type @xmath315 . with the definitions above , we denote by @xmath356 the set of labelled trees such that @xmath357 , @xmath206 , and the constraints and properties described above . then it is straightforward to prove by induction that @xmath358 where @xmath359 given @xmath360 , we let @xmath217 be the family of labelled trees differing from @xmath218 just by the choice of the labels @xmath219 . then , using , it is easy to see that @xmath361},\ ] ] which immediately implies that @xmath362},\ ] ] for a suitable constant @xmath363 . therefore , the radius of convergence in @xmath32 of the series for @xmath255 and @xmath364 is proportional to @xmath365 , which allows us to fix eventually @xmath366 . the lyapunov exponents @xmath367 are the time average of the quantities @xmath368 . however , if @xmath369 denotes the restriction of @xmath370 on the attractor @xmath49 , the dynamical system @xmath371 is conjugated to an asonov system and hence it is ergodic : therefore time - averaged observables are @xmath41-independent . furthermore , there exists a unique srb measure @xmath372 such that @xmath373 the measure @xmath374 can be computed by reasoning as in ( * ? ? ? * chapter 10 ) . let @xmath375 be a markov partition for @xmath134 on @xmath376 and set @xmath377 . call @xmath378 the symbolic code induced by the markov partition @xmath379 and denote by @xmath380 the symbolic representation of a point @xmath381 , i.e. @xmath382 . then the expansion rate of @xmath369 along the unstable manifold of @xmath383 is @xmath384 , where @xmath385 with @xmath386 denoting the shift map and @xmath387 . if @xmath374 denotes the gibbs distribution for the energy function @xmath388 ( see ( * ? ? ? * chapter 5 ) ) , then the srb distribution @xmath389 for the system @xmath371 is @xmath374 and can be computed accordingly ( see ( * ? ? ? * chapter 6 ) ) . moreover , by construction , @xmath388 is analytic in @xmath32 and hlder - continuous in @xmath380 . therefore , for any hlder - continuous function @xmath390 , the expectation value @xmath391 is hlder - continuous in @xmath32 for @xmath77 . if @xmath66 is analytic in @xmath32 , then there exists a positive constant @xmath392 , depending on @xmath66 , such that @xmath393 is analytic for @xmath394 . in particular the lyapunov exponents are analytic in @xmath32 and , from , one finds @xmath395 this completes the proof of theorem [ prop:2 ] . in this section , we discuss informally some of the consequences of our main theorem , and formulate a conjecture about the transition from fractal to smooth(er ) behavior , which is suggested by our result . from theorem [ prop:1 ] we know that the surface @xmath54 of the attractor is h@xmath396lder continuous , but we do not have any control on its possible differentiability . this means that our attractor may be fractal , and we actually expect this to be the case for @xmath32 positive and small enough . an analytic estimate of the fractal dimension of the attractor in terms of the lyapunov exponents is provided by the _ lyapunov dimension _ @xmath397 , which is defined as follows . consider an ergodic dynamical system admitting an srb measure on its attractor , and let @xmath398 be its lyapunov exponents , counted with their multiplicities . then , @xmath399 where @xmath400 is the largest integer such that @xmath401 . the _ kaplan - yorke conjecture _ @xcite states that @xmath402 coincides with the hausdorff dimension of the attractor ( also known as the _ information dimension _ , see , e.g. , ( * ? ? ? * chapt.5.5.3 ) for a precise definition ) . in this section , we take @xmath397 as a heuristic estimate of the fractal dimension of the attractor , without worrying about the possible validity of the conjecture ( which has been rigorously proven only some special cases , see e.g. ) . specializing the expression of @xmath397 to our context , we find that , for @xmath32 sufficiently small , d_l=2+=3++r_2(),[eq : fract]where @xmath403 is the taylor remainder of order 2 in @xmath32 , which is computable explicitly in terms of the convergent expansion derived in the previous sections . note that @xmath404 , so that @xmath405 is smaller than 3 ( as desired ) and is decreasing in @xmath32 , for @xmath32 small . therefore , combined with the kalpan - yorke conjecture , ( [ eq : fract ] ) suggests that the attractor is fractal for @xmath32 small , and its fractal dimension decreases ( as expected ) by increasing the strength @xmath32 of the dissipative interaction . it is now tempting to extrapolate to larger values of @xmath32 ( possibly beyond the range of validity of theorem [ prop:2 ] ) , up to the point where , possibly , the relative ordering of @xmath74 and @xmath406 changes . in the simple case that @xmath33 ( which is the case considered in @xcite ) , the lyapunov exponents @xmath407 are independent of @xmath32 : @xmath408 . therefore , on the basis of , we conjecture that by increasing @xmath32 the hausdorff dimension of the attractor decreases from @xmath167 to @xmath166 until @xmath32 reaches the critical value @xmath409 , where @xmath410 . formally , this critical point is @xmath411(higher orders ) , the higher orders being computable via the expansion described in the previous sections . for @xmath412 , we expect the attractor to be a smooth manifold of dimension two . the transition is illustrated in fig.[fig1 ] and [ fig2 ] for the simple case that @xmath33 and @xmath413 , in which case the expected critical point is @xmath414 . ( 100,30 ) ( 0,0 ) @xmath33 and @xmath413 . a ) : @xmath415 . b ) : @xmath416.,title="fig:",width=264 ] ( 50,0 ) @xmath33 and @xmath413 . a ) : @xmath415 . b ) : @xmath416.,title="fig:",width=264 ] ( 100,30 ) ( 0,0 ) @xmath33 and @xmath27 as in fig . a ) : @xmath417 . b ) : @xmath418.,title="fig:",width=264 ] ( 50,0 ) @xmath33 and @xmath27 as in fig . a ) : @xmath417 . b ) : @xmath418.,title="fig:",width=264 ] if @xmath419 , on the basis of numerical simulations , the attractor does not seem to display a transition from a fractal set to a smooth manifold . still , for suitable choices of @xmath419 , we expect the attractor to display a `` first order phase transition '' , located at the value of @xmath32 where @xmath420 , to be called again @xmath409 . at @xmath421 , the derivative of the hausdorff dimension of the attractor with respect to @xmath32 is expected to have a jump . a possible scenario is that the attractor is fractal both for @xmath422 and for @xmath423 , but it is `` smoother '' at larger values of @xmath32 , in the sense that its closure may be a regular , smooth , manifold of dimension two . an illustration of this smoothing " mechanism is in fig.[fig3 ] . ( 100,30 ) ( 0,0 ) . a ) : @xmath416 . b ) : @xmath424.,title="fig:",width=264 ] ( 50,0 ) . a ) : @xmath416 . b ) : @xmath424.,title="fig:",width=264 ] it would be interesting to investigate the nature of this transition in a more quantitative way , by comparing a numerical construction of the attractor with the theory proposed here , obtained by extrapolating the convergent expansion described in this paper to intermediate values of @xmath32 . such a comparison goes beyond the purpose of this paper , and we postpone the discussion of this issue to future research . one has @xmath426 for @xmath149 and @xmath427 for @xmath211 . given a tree of order @xmath428 one proceed by induction . assume that @xmath429 for all the trees @xmath430 of order @xmath431 and consider a tree @xmath172 of order @xmath83 . let @xmath355 be the special node of @xmath172 , and call @xmath432 the subtrees entering @xmath173 , with @xmath433 . if @xmath355 is a node of type 1 , then @xmath434 , so that the bound follows for @xmath435 . if @xmath436 , then the node preceding @xmath355 can not be of type 1 . call @xmath437 the subtrees entering @xmath438 , with @xmath439 . then one has @xmath440 for all @xmath441 . finally if @xmath355 is not a node of type 1 , then it has not to be counted and the argument follows by using the inductive bounds for the subtrees entering @xmath355 . moreover the bound in lemma [ lem:2.1 ] is optimal . indeed there are trees @xmath172 of order @xmath83 such that @xmath442 . define recursively the level @xmath443 of a node @xmath181 by setting @xmath444 if @xmath181 is an end - node and @xmath445 if at least one line entering @xmath181 exits a node @xmath29 with level @xmath446 . then consider a tree in which all nodes except the end - nodes are circles ( thais is of type 1 or 2 ) and have two entering lines except those with level @xmath165 which have only one entering line ; see figure [ fig:3.8 ] for an example with @xmath447 ( means we can have any kind of square node ) . for such trees one has @xmath448 , where @xmath449 , @xmath450 is the number of internal nodes and @xmath451 is the number of end - nodes . hence @xmath452 . @r=0.5 cm & & & * + [ o][f]@<-[r ] & * + < 3.0pt>[f**:black]+[f**:white ] + & & * + [ o][f]@<-[ru]@<-[rd ] & & + & & & * + [ o][f]@<-[r ] & * + < 3.0pt>[f**:black]+[f**:white ] + @<-[r ] & * + [ o][f]@<-[ruu]@<-[rdd ] & & & + & & & * + [ o][f]@<-[r ] & * + < 3.0pt>[f**:black]+[f**:white ] + & & * + [ o][f]@<-[ru]@<-[rd ] & & + & & & * + [ o][f]@<-[r ] & * + < 3.0pt>[f**:black]+[f**:white ] 10 adkmz a. arenas , a. daz - guilera , j. kurths , y. moreno , ch . zhou , _ synchronization in complex networks _ phys . rep . * 469 * ( 2008 ) , no . 3 , 93 - 153 . blech i.i . blekhman , _ synchronization in science and technology _ , asme press , new york , 1988 . bkovz s. boccaletti , j. kurths , g. osipov , d.l . valladares , c.s . zhou , _ the synchronization of chaotic systems _ , phys . * 366 * ( 2002 ) , no . 1 - 2 , 1 - 101 . bfg04 f. bonetto , p. falco , a. giuliani , _ analyticity of the srb measure of a lattice of coupled anosov diffeomorphisms of the torus _ , j. math * 45 * ( 2004 ) , no . 8 , 3282 - 3309 . eck - ruelle j .- eckmann , d. ruelle , _ ergodic theory of chaos and strange attractors _ , rev . modern phys . * 57 * ( 1985 ) , no . 3 , 617 - 656 . farmer - ott - yorke j.d . farmer , e. ott , j.a . yorke , _ the dimension of chaotic attractors _ , phys . d * 7 * ( 1983 ) , no . 1 - 3 , 153 - 180 . g. gallavotti , _ foundations of fluid dynamics _ , springer - verlag , berlin heidelberg , 2002 . gbg g. gallavotti , f. bonetto , g. gentile , _ aspects of the ergodic , qualitative and statistical theory of motion _ , springer , berlin , 2004 . ggg g. gallavotti , g. gentile , a. giuliani , _ resonances within chaos _ , chaos * 22 * , 026108 ( 2012 ) , 6 pages . gonzalez j.m . gonzlez - miranda , _ synchronization and control of chaos . an introduction for scientists and engineers _ , imperial college press , london , 2004 . kaplan - yorke l. kaplan , j.a . yorke , _ chaotic behavior of multidimensional difference equations _ , functional differential equations and approximation of fixed points , lecture notes in mathematics 730 , 204 - 227 , eds . peitgen , h .- o . walther , springer , berlin , 1979 . pcjmh l. pecora , th.l . carroll , g.a . johnson , d.j . mar , j.f . heagy , _ fundamentals of synchronization in chaotic systems , concepts , and applications _ , chaos * 7 * ( 1997 ) , no . 4 , 520 - 543 . a. pikovsky , m. rosenblum , j. kurths , _ synchronization . a universal concept in nonlinear sciences _ , cambridge university press , cambridge , 2001 .

we consider a three - dimensional chaotic system consisting of the suspension of arnold s cat map coupled with a clock via a weak dissipative interaction . we show that the coupled system displays a synchronization phenomenon , in the sense that the relative phase between the suspension flow and the clock locks to a special value , thus making the motion fall onto a lower dimensional attractor . more specifically , we construct the attractive invariant manifold , of dimension smaller than three , using a convergent perturbative expansion . moreover , we compute via convergent series the lyapunov exponents , including notably the central one . the result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model . the main novelty of the current construction relies in the computation of the lyapunov spectrum , which consists of non - trivial analytic exponents . some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed . * _ keywords : _ * partially hyperbolic systems ; anosov systems ; synchronization ; phase - locking ; lyapunov exponents ; fractal attractor ; srb measure ; tree expansion ; perturbation theory .

1506.07573

in high energy physics ( hep ) , unfolding ( also called unsmearing ) is a general term describing methods that attempt to take out the effect of smearing resolution in order to obtain a measurement of the true underlying distribution of a quantity . typically the acquired data ( distorted by detector response , inefficiency , etc . ) are binned in a histogram . the result of some unfolding procedure is then a new histogram with estimates of the true mean bin contents prior to smearing and inefficiency , along with some associated uncertainties . it is commonly assumed that such unfolded distributions are useful scientifically for comparing data to one or more theoretical predictions , or even as quantitative measurements to be propagated into further calculations . since an important aspect of the scientific enterprise is to test hypotheses , we can ask : `` should unfolded histograms be used to test hypotheses ? '' if the answer is yes , then one can further ask if there are limitations to the utility of testing hypotheses using unfolded histograms . if the answer is no , then the rationale for unfolding would seem to be limited . in this note we illustrate an approach to answering the title question with a few variations on a toy example that captures some of the features of real - life unfolding problems in hep . the goal of the note is to stimulate more interest in exploring what one of us ( rc ) has called a _ bottom - line test _ for an unfolding method : _ if the unfolded spectrum and supplied uncertainties are to be useful for evaluating which of two models is favored by the data ( and by how much ) , then the answer should be materially the same as that which is obtained by smearing the two models and comparing directly to data without unfolding _ this is a different emphasis for evaluating unfolding methods than that taken in studies that focus on intermediate quantities such as bias and variance of the estimates of the true mean contents , and on frequentist coverage of the associated confidence intervals . while the focus here is on comparing two models for definiteness , the basic idea of course applies to comparing one model to data ( i.e. , goodness of fit ) , and to more general hypothesis tests . recently zech @xcite has extended the notion of the bottom - line test to parameter estimation from fits to unfolded data , and revealed failures in the cases studied , notably in fits to the width of a peak . we adopt the notation of the monograph _ statistical data analysis _ by glen cowan @xcite ( suppressing for simplicity the background contribution that he calls @xmath0 ) : @xmath1 is a continuous variable representing the _ true _ value of some quantity of physical interest ( for example momentum ) . it is distributed according to the pdf @xmath2 . @xmath3 is a continuous variable representing the _ observed _ value of the same quantity of physical interest , after detector smearing effects and loss of events ( if any ) due to inefficiencies . @xmath4 is the resolution function of the detector : the conditional pdf for observing @xmath3 , given that the true value is @xmath1 ( and given that it was observed somewhere ) . @xmath5 contains the expectation values of the bin contents of the _ true _ ( unsmeared ) histogram of @xmath1 ; @xmath6 contains the bin contents of the _ observed _ histogram ( referred to as the _ smeared histogram _ , or occasionally as the _ folded _ histogram ) of @xmath3 in a single experiment ; @xmath7 contains the expectation values of the bin contents of the _ observed _ ( smeared ) histogram of @xmath3 , including the effect of inefficiencies : @xmath8 $ ] ; @xmath9 is the response matrix that gives the probability of an event in true bin @xmath10 being observed in bin @xmath11 after smearing : @xmath12 ; @xmath13 contains the point estimates of @xmath14 that are the output of an unfolding algorithm . @xmath15 is the covariance matrix of the estimates @xmath16 : @xmath17 $ ] . the estimate of @xmath15 provided by an unfolding algorithm is @xmath18 . thus we have @xmath19 as discussed by cowan and noted above , @xmath9 includes the effect of the efficiency @xmath20 , i.e. , the effect of events in the true histograms not being observed in the smeared histogram . the only efficiency effect that we consider here is that due to events being smeared outside the boundaries of the histogram . ( that is , we do not consider an underflow bin or an overflow bin . ) the response matrix @xmath9 depends on the resolution function and on ( unknown ) true bin contents ( and in particular on their true densities @xmath2 _ within _ each bin ) , and hence @xmath9 is either known only approximately or as a function of assumptions about the true bin contents . the numbers of bins @xmath21 and @xmath22 need not be the same . ( @xmath23 is often suggested , while @xmath24 leaves the system of equations under - determined . ) for the toy studies discussed here , we set @xmath25 , so that @xmath9 is a square matrix that typically has an inverse . in the smeared space , we take the observed counts @xmath26 to be independent observations from the underlying poisson distributions : @xmath27 the unfolding problem is then to use @xmath9 and @xmath26 as inputs to obtain estimates @xmath16 of @xmath14 , and to obtain the covariance matrix @xmath15 of these estimates ( or rather an estimate of @xmath15 , @xmath18 ) , ideally taking in account uncertainty in @xmath9 . when reporting unfolded results , authors report @xmath16 , ideally along with @xmath18 . ( if only a histogram of @xmath16 with `` error bars '' is displayed , then only the diagonal elements of @xmath18 are communicated , further degrading the information . ) the `` bottom line test '' of an application of unfolding is then whether hypothesis tests about underlying models that predict @xmath14 can obtain meaningful results if they take as input @xmath16 and @xmath18 . for the null hypothesis @xmath28 , we consider the continuous variable @xmath1 to be distributed according the true pdf @xmath29 where @xmath30 is known , and @xmath31 is a normalization constant . for the alternative hypothesis @xmath32 , we consider @xmath1 to be distributed according the true pdf @xmath33 where @xmath30 is the same as in the null hypothesis , and where @xmath34 is a pdf that encodes a departure from the null hypothesis . in this note , we assume that both @xmath34 and @xmath35 are known , and lead to potentially significant departures from the null hypothesis at large @xmath1 . the constant @xmath35 controls the level of such departures . figure [ truepdfs ] displays the baseline pdfs that form the basis of the current study , for which we take @xmath36 to be a normalized gamma distribution , @xmath37 and @xmath38 . is represented by @xmath39 , shown in red . the alternative hypothesis @xmath32 has an additional component shown in dashed blue , with the sum @xmath40 in solid blue . , title="fig:",scaledwidth=49.0% ] is represented by @xmath39 , shown in red . the alternative hypothesis @xmath32 has an additional component shown in dashed blue , with the sum @xmath40 in solid blue . , title="fig:",scaledwidth=49.0% ] for each hypothesis , the true bin contents @xmath14 are then each proportional to the integral of the relevant @xmath2 over each bin . for both hypotheses , we take the smearing of @xmath3 to be the gaussian resolution function , @xmath41 where @xmath42 is known . for baseline plots , we use the values shown in table [ baseline ] , and the study the effect of varying one parameter at a time . for both @xmath3 and @xmath1 , we consider histograms with 10 bins of width 1 spanning the interval [ 0,10 ] . the default @xmath42 is half this bin width . the quantities @xmath14 , @xmath9 , and @xmath43 are then readily computed as in ref . figure [ histos ] displays @xmath14 and @xmath43 ( in solid histograms ) , while fig . [ responsepurity ] displays the response matrix as well as the source bin of events that are observed in each bin . in each simulated experiment , the total number of events is sampled from a poisson distribution with mean given in table [ baseline ] . .values of parameters used in the baseline unfolding examples [ cols="<,<,<",options="header " , ] and smeared @xmath43 , for ( left ) the null hypothesis @xmath28 and ( right ) the alternative hypothesis @xmath32 . data points : in mc simulation a set @xmath44 of true points is chosen randomly and then smeared to be the set @xmath45 . the three points plotted in each bin are then the bin contents when @xmath1 and @xmath3 are binned , followed by the unfolded estimate for bin contents . , title="fig:",scaledwidth=49.0% ] and smeared @xmath43 , for ( left ) the null hypothesis @xmath28 and ( right ) the alternative hypothesis @xmath32 . data points : in mc simulation a set @xmath44 of true points is chosen randomly and then smeared to be the set @xmath45 . the three points plotted in each bin are then the bin contents when @xmath1 and @xmath3 are binned , followed by the unfolded estimate for bin contents . , title="fig:",scaledwidth=49.0% ] for default parameter values in table [ baseline ] . ( right ) for each bin in the measured @xmath1 value , the fraction of events that come from that bin ( dominant color ) and from nearby bins . , title="fig:",scaledwidth=49.0% ] for default parameter values in table [ baseline ] . ( right ) for each bin in the measured @xmath1 value , the fraction of events that come from that bin ( dominant color ) and from nearby bins . , title="fig:",scaledwidth=49.0% ] boundary effects at the ends of the histogram are an important part of a real problem . in our simplified toy problems , we use the same smearing for events near boundaries as for all events ( hence not modeling correctly some physical situations where observed values can not be less than zero ) ; events that are smeared to values outside the histogram are considered lost and contribute to the inefficiencies included in @xmath9 . these toy models capture some important aspects of real problems in hep . for example , one might be comparing event generators for top - quark production in the standard model . the variable @xmath1 might be the transverse momentum of the top quark , and the two hypotheses might be two calculations , one to higher order . another real problem might be where @xmath1 represents transverse momentum of jets , the null hypothesis is the standard model , and the alternative hypothesis is some non - standard - model physics that turns on at high transverse momentum . ( in this case , it is typically not the case that amplitude @xmath35 of additional physics is known . ) in a typical search for non - standard - model physics , the hypothesis test of @xmath28 vs. @xmath32 is formulated in the smeared space , i.e. , by comparing the histogram contents @xmath26 to the mean bin contents @xmath43 predicted by the true densities @xmath2 under each hypothesis combined with the resolution function and any efficiency losses . the likelihood @xmath46 for the null hypothesis is the product over bins of the poisson probability of obtaining the observed bins counts : @xmath47 where the @xmath48 are taken from the null hypothesis prediction . likelihoods for other hypotheses , such as @xmath49 , are constructed similarly . for testing goodness of fit , it can be useful @xcite to use the observed data to construct a third hypothesis , @xmath50 , corresponding the _ saturated model _ @xcite , which sets the predicted mean bin contents to be exactly those observed . thus @xmath51 is the upper bound on @xmath52 for any hypothesis , given the observed data . the negative log - likelihood ratio @xmath53 is a goodness - of - fit test statistic that is asymptotically distributed as a chisquare distribution if @xmath28 is true . similarly one has @xmath54 for testing @xmath32 . an alternative ( in fact older ) goodness - of - fit test statistic is pearson s chisquare @xcite , @xmath55 yet another alternative , generally less favored , is known as neyman s chisquare @xcite , @xmath56 ref . @xcite argues that eqn . [ baker ] is the most appropriate gof statistic for poisson - distributed histograms , and we use it as our reference point in the smeared space . figure [ nullgofsmeared ] shows the distributions of @xmath57 and @xmath58 , and their difference , for histograms generated under @xmath28 . both distributions follow the expected @xmath59 distribution with 10 degrees of freedom ( dof ) . in contrast , the histogram of @xmath60 ( figure [ nullgofsmeared ] ( bottom left ) ) has noticeable differences from the theoretical curve . , in the smeared space with default value of gaussian @xmath42 , histograms of the gof test statistics : ( top left ) @xmath57 , ( top right ) @xmath58 , and ( bottom left ) @xmath60 . the solid curves are the chisquare distribution with 10 dof . ( bottom right ) histogram of the event - by - event difference in the two gof test statistics @xmath58 and @xmath57 . , title="fig:",scaledwidth=49.0% ] , in the smeared space with default value of gaussian @xmath42 , histograms of the gof test statistics : ( top left ) @xmath57 , ( top right ) @xmath58 , and ( bottom left ) @xmath60 . the solid curves are the chisquare distribution with 10 dof . ( bottom right ) histogram of the event - by - event difference in the two gof test statistics @xmath58 and @xmath57 . , title="fig:",scaledwidth=49.0% ] , in the smeared space with default value of gaussian @xmath42 , histograms of the gof test statistics : ( top left ) @xmath57 , ( top right ) @xmath58 , and ( bottom left ) @xmath60 . the solid curves are the chisquare distribution with 10 dof . ( bottom right ) histogram of the event - by - event difference in the two gof test statistics @xmath58 and @xmath57 . , title="fig:",scaledwidth=49.0% ] , in the smeared space with default value of gaussian @xmath42 , histograms of the gof test statistics : ( top left ) @xmath57 , ( top right ) @xmath58 , and ( bottom left ) @xmath60 . the solid curves are the chisquare distribution with 10 dof . ( bottom right ) histogram of the event - by - event difference in the two gof test statistics @xmath58 and @xmath57 . , title="fig:",scaledwidth=49.0% ] for testing @xmath28 vs. @xmath32 , a suitable test statistic is the likelihood ratio @xmath61 formed from the probabilities of obtaining bin contents @xmath26 under each hypothesis : @xmath62 where the second equality follows from eqn . [ baker ] . figure [ lambdah0h1 ] shows the distribution of @xmath63 for events generated under @xmath28 and for events generated under @xmath32 , using the default parameter values in table [ baseline ] . for events generated under @xmath28 ( in blue ) and @xmath32 ( in red ) . , scaledwidth=49.0% ] we would assert that these results obtained in the smeared space are the `` right answers '' for chisquare - like gof tests of @xmath28 and @xmath32 ( if desired ) , and in particular for the likelihood - ratio test of @xmath28 vs @xmath32 in fig . [ lambdah0h1 ] . given a particular observed data set , such histograms can be used to calculate @xmath64-values for each hypothesis , simply by integrating the appropriate tail of the histogram beyond the observed value of the relevant likelihood ratio @xcite . in frequentist statistics , such @xmath64-values are typically the basis for inference , especially for the simple - vs - simple hypothesis tests considered here . ( of course there is a vast literature questioning the foundations of using @xmath64-values , but in this note we assume that they can be useful , and are interested in comparing ways to compute them . ) we compare @xmath65 , @xmath58 , @xmath60 , and the generalization of eqn . [ chisq ] including correlations in various contexts below . for poisson - distributed data , arguments in favor of @xmath65 when it is available are in ref . @xcite . in the usual @xmath59 gof test with ( uncorrelated ) estimates @xmath66 having _ gaussian _ densities with standard deviations @xmath67 , one would commonly have @xmath68 although not usually mentioned , this is equivalent to a likelihood ratio test with respect to the saturated model , just as in the poisson case . the likelihood is @xmath69 where for @xmath46 one has @xmath48 predicted by @xmath28 , and for the saturated model , one has @xmath70 . thus @xmath71 and hence @xmath72 ( it is sometimes said loosely and incorrectly that for the gaussian model , @xmath73 , but clearly the ratio is necessary to cancel the normalization factor . ) there is also a well - known connection between the usual gaussian @xmath59 of eqn . [ chisq ] and pearson s chisquare in eqn . [ pearson ] : since the variance of a poisson distribution is equal to its mean , a naive derivation of eqn . [ pearson ] follows immediately from eqn . [ chisq ] . if one further approximates @xmath48 by the estimate @xmath74 , then one obtains neyman s chisquare in eqn . [ neyman ] . if one unfolds histograms and then compares the unfolded histograms @xmath16 to ( never smeared ) model predictions @xmath75 , even informally , then one is implicitly assuming that the comparison is scientifically meaningful . for this to be the case , we would assert that the results of comparisons should not differ materially from the `` right answers '' obtained above in the smeared space . here we explore a few test cases . given the observed histogram contents @xmath26 , the likelihood function for the unknown @xmath76 follows from eqn . [ poisprob ] and leads to the maximum likelihood ( ml ) estimates @xmath77 , i.e. , @xmath78 one might then expect that the ml estimates of the unknown means @xmath14 can be obtained by substituting @xmath66 for @xmath26 in eqn . [ nurmu ] . if @xmath9 is a square matrix , as assumed here , then this yields @xmath79 these are indeed the ml estimates of @xmath14 as long as @xmath9 is invertible and the estimates @xmath80 are positive @xcite , which is generally the case in the toy problem studied here . the covariance matrix of the estimates @xmath16 in terms of @xmath9 and @xmath76 is derived in ref . @xcite : @xmath81 where @xmath82 . since the true values @xmath76 are presumed unknown , it is natural to substitute the estimates from eqn . [ nun ] , thus obtaining an estimate @xmath18 . consequences of this approximation are discussed below . in all cases ( even when matrix inversion fails ) , the ml estimates for @xmath14 can be found to desired precision by the iterative method variously known as @xcite expectation maximization ( em ) , lucy - richardson , or ( in hep ) the iterative method of dagostini @xcite . because the title of ref . @xcite mentions bayes theorem , in hep the em method is unfortunately ( and wrongly ) referred to as `` bayesian '' , even though it is a fully frequentist algorithm @xcite . as discussed by cowan @xcite , the ml estimates are unbiased , but the unbiasedness can come at a price of large variance that renders the unfolded histogram unintelligible to humans . therefore there is a vast literature on `` regularization methods '' that reduce the variance at the price of increased bias , such that the mean - squared - error ( the sum of the bias squared and the variance ) is ( one hopes ) reduced . the method of regularization popularized in hep by dagostini @xcite ( and studied for example by bohm and zech @xcite ) is simply to stop the iterative em method before it converges to the ml solution . the estimates @xmath83 then retain some memory of the starting point of the solution ( typically leading to a bias ) and have lower variance . the uncertainties ( covariance matrix ) also depend on when the iteration stops . our studies in this note focus on the ml and truncated iterative em solutions , and use the em implementation ( unfortunately called roounfoldbayes ) in the roounfold @xcite suite of unfolding tools . this means that for the present studies , we are constrained by the policy in roounfold to use the `` truth '' of the training sample to be the starting point for the iterative em method ; thus we have not studied convergence starting from , for example , a uniform distribution . useful studies of the bias of estimates are thus not performed . other popular methods in hep include variants of tikhonov regularization , such as `` svd '' method advocated by hocker and kartvelishvili @xcite , and the implementation included in tunfold @xcite . the relationship of these methods to those in the professional statistics literature is discussed by kuusela @xcite . figure [ histos ] shows ( in addition to the solid histograms mentioned above ) three points with error bars plotted in each bin , calculated from a particular set of simulated data corresponding to one experiment . the three points are the bin contents when the sampled values of @xmath1 and @xmath3 are binned , followed by that bin s components of the set of unfolded estimates @xmath16 . figure [ matricesinvert](left ) shows the covariance matrix @xmath18 for the estimates @xmath16 obtained for the same particular simulated data set , unfolded by matrix inversion ( eqn . [ nurmuinv ] ) to obtain the ml estimates . figure [ matricesinvert ] ( right ) shows the corresponding correlation matrix with elements @xmath84 . figure [ matricesiterative ] shows the corresponding matrices obtained when unfolding by the iterative em method with default number of iterations . for the ml solution , adjacent bins are negatively correlated , while for the em solution with default ( 4 ) iterations , adjacent bins are positively correlated due to the implicit regularization . for unfolded estimates , as provided by the ml estimates ( matrix inversion ) . ( right ) the correlation matrix corresponding to @xmath18 , with elements @xmath84 . , title="fig:",scaledwidth=49.0% ] for unfolded estimates , as provided by the ml estimates ( matrix inversion ) . ( right ) the correlation matrix corresponding to @xmath18 , with elements @xmath84 . , title="fig:",scaledwidth=49.0% ] for unfolded estimates , as provided by the default iterative em method . ( right ) the correlation matrix corresponding to @xmath18 , with elements @xmath84 . , title="fig:",scaledwidth=49.0% ] for unfolded estimates , as provided by the default iterative em method . ( right ) the correlation matrix corresponding to @xmath18 , with elements @xmath84 . , title="fig:",scaledwidth=49.0% ] figure [ converge ] shows an example of the convergence of iterative em unfolding to the ml solution for one simulated data set . on the left is the fractional difference between the em and ml solutions , for each of the ten histogram bins , as a function of the number of iterations , reaching the numerical precision of the calculation . on the right is the covariance matrix @xmath18 after a large number of iterations , showing convergence to that obtained by matrix inversion in fig . [ matricesinvert](left ) . , title="fig:",scaledwidth=49.0% ] ( left ) . , title="fig:",scaledwidth=49.0% ] although the ml solution for @xmath16 may be difficult for a human to examine visually , if the covariance matrix @xmath15 is well enough behaved , then a computer can readily calculate a chisquare gof test statistic in the unfolded space by using the generalization of eqn . [ chisq ] , namely the usual formula for gof of gaussian measurements with correlations @xcite , @xmath85 if unfolding is performed by matrix inversion ( when equal to the ml solution ) , then substituting @xmath86 from eqn . [ nurmuinv ] , @xmath87 from eqn . [ nurmu ] , and @xmath88 from eqn . [ covmu ] , yields @xmath89 so for @xmath82 as assumed by cowan , this @xmath90 calculated in the unfolded space is equal to pearson s chisquare ( eqn . [ pearson ] ) in the smeared space . if however one substitutes @xmath91 for @xmath43 as in eqn . [ nun ] , then @xmath90 in the unfolded space is equal to neyman s chisquare in the smeared space ! this is the case in the implementation of roounfold that we are using , as noted below in the figures . for events unfolded with the ml estimates , figure [ nulgofunfoldedinvert ] ( top left ) shows the results of such a @xmath90 gof test with respect to the null hypothesis using same events used in fig . [ nullgofsmeared ] . as foreseen , the histogram is identical ( apart from numerical artifacts ) with the histogram of @xmath60 in fig . [ nullgofsmeared ] ( bottom left ) . figure [ nulgofunfoldedinvert ] ( top right ) show the event - by - event difference of @xmath90 and pearson s @xmath59 in the smeared space , and figure [ nulgofunfoldedinvert ] ( bottom ) is the difference with respect to @xmath57 in the smeared space . figure [ nulgofunfolded ] shows the same quantities calculated after unfolding using the iterative em method with default iterations . for these tests using ml unfolding , the noticeable difference between the gof test in the smeared space with that in the unfolded space is directly traced to the fact that the test in the unfolded space is equivalent to @xmath60 in the smeared space , which is an inferior gof test compared to the likelihood ratio test statistic @xmath92 . it seems remarkable that , even though unfolding by matrix inversion would appear not to lose information , in practice the way the information is used ( linearizing the problem via expressing the result via a covariance matrix ) already results in some failures of the bottom - line test of gof . this is without any regularization or approximate em inversion . that tests for compatibility with @xmath28 in the unfolded space , for the same events generated under @xmath28 as those used in the smeared - space test of fig . [ nullgofsmeared ] . ( top right ) for these events , histogram of the difference between @xmath90 in the unfolded space and @xmath58 in the smeared space . ( bottom ) for these events , histogram of the difference between @xmath90 in the unfolded space and the gof test statistic @xmath92 in the smeared space . , title="fig:",scaledwidth=49.0% ] that tests for compatibility with @xmath28 in the unfolded space , for the same events generated under @xmath28 as those used in the smeared - space test of fig . [ nullgofsmeared ] . ( top right ) for these events , histogram of the difference between @xmath90 in the unfolded space and @xmath58 in the smeared space . ( bottom ) for these events , histogram of the difference between @xmath90 in the unfolded space and the gof test statistic @xmath92 in the smeared space . , title="fig:",scaledwidth=49.0% ] that tests for compatibility with @xmath28 in the unfolded space , for the same events generated under @xmath28 as those used in the smeared - space test of fig . [ nullgofsmeared ] . ( top right ) for these events , histogram of the difference between @xmath90 in the unfolded space and @xmath58 in the smeared space . ( bottom ) for these events , histogram of the difference between @xmath90 in the unfolded space and the gof test statistic @xmath92 in the smeared space . , title="fig:",scaledwidth=49.0% ] , here calculated after unfolding using the iterative em method with default ( four ) iterations . , title="fig:",scaledwidth=49.0% ] , here calculated after unfolding using the iterative em method with default ( four ) iterations . , title="fig:",scaledwidth=49.0% ] , here calculated after unfolding using the iterative em method with default ( four ) iterations . , title="fig:",scaledwidth=49.0% ] for the histogram of each simulated experiment , the gof statistic @xmath90 is calculated with respect to the prediction of @xmath28 and also with respect to the prediction of @xmath32 . the difference of these two values , @xmath93 , is then a test statistic for testing @xmath28 vs. @xmath32 , analogous to the test statistic @xmath63 . figure [ delchi ] shows , for the same events as those used in fig . [ lambdah0h1 ] , histograms of the test statistic @xmath93 in the unfolded space for events generated under @xmath28 and under @xmath32 , with @xmath9 calculated using @xmath28 and using @xmath32 . for the default problem studied here , the dependence on @xmath9 is not large . thus unless otherwise specified , all other plots use @xmath9 calculated under @xmath28 . , histogram of the test statistic @xmath93 in the unfolded space , for events generated under @xmath28 ( in blue ) and @xmath32 ( in red ) , with @xmath9 calculated using @xmath28 . ( right ) for the same events , histograms of the test statistic @xmath93 in the unfolded space , with @xmath9 calculated using @xmath32 . , title="fig:",scaledwidth=49.0% ] , histogram of the test statistic @xmath93 in the unfolded space , for events generated under @xmath28 ( in blue ) and @xmath32 ( in red ) , with @xmath9 calculated using @xmath28 . ( right ) for the same events , histograms of the test statistic @xmath93 in the unfolded space , with @xmath9 calculated using @xmath32 . , title="fig:",scaledwidth=49.0% ] figure [ deldel ] shows , for the events in figs . [ lambdah0h1 ] and in [ delchi ] , histograms of the event - by - event difference of @xmath63 and @xmath93 . the red curves correspond to events generated under @xmath28 , while the blue curves are for events generated under @xmath32 . the unfolding method is ml on the left and iterative em on the right . this is an example of a _ bottom - line test _ : does one obtain the same answers in the smeared and unfolded spaces ? there are differences apparent with both unfolding techniques . since the events generated under both @xmath28 and @xmath32 are shifted in the same direction , the full implications are not immediately clear . thus we turn to roc curves or equivalent curves from neyman - pearson hypothesis testing . and in [ delchi](left ) , histogram of the event - by - event difference of @xmath63 and @xmath93 . in the left histogram , ml unfolding is used , while in the right histogram , iterative em unfolding is used . , title="fig:",scaledwidth=49.0% ] and in [ delchi](left ) , histogram of the event - by - event difference of @xmath63 and @xmath93 . in the left histogram , ml unfolding is used , while in the right histogram , iterative em unfolding is used . , title="fig:",scaledwidth=49.0% ] we can investigate the effect of the differences apparent in fig . [ deldel ] by using the language of neyman - pearson hypothesis testing , in which one rejects @xmath28 if the value of the test statistic ( @xmath63 in the smeared space , or @xmath93 in the unfolded space ) is above some critical value @xcite . the type i error probability @xmath94 is the probability of rejecting @xmath28 when it is true , also known as the `` false positive rate '' . the type ii error probability @xmath95 is the probability of accepting ( not rejecting ) @xmath28 when it is false . the quantity @xmath96 is the _ power _ of the test , also known as the `` true positive rate '' . the quantities @xmath94 and @xmath95 thus follow from the cumulative distribution functions ( cdfs ) of histograms of the test statistics . in classification problems outside hep is it common to make the roc curve of true positive rate vs. the false positive rate , as shown in fig . figure [ alphabeta ] shows the same information in a plot of @xmath95 vs. @xmath94 , i.e. , with the vertical coordinate inverted compared to the roc curve . figure [ alphabetaloglog ] is the same plot as fig . [ alphabeta ] , with both axes having logarithmic scale . the result of this `` bottom line test '' does not appear to be dramatic in this first example , and appear to be dominated by the difference between the poisson - based @xmath63 and @xmath93 already present in the ml unfolding solution , rather than by the additional differences caused by truncating the em solution . unfortunately no general conclusion can be drawn from this observation , since as mentioned above the em unfolding used here starts from the true distribution as the first estimate . it is of course necessary to study other initial estimates . and [ delchi](left ) , roc curves for classification performed in the smeared space ( blue curve ) and in the unsmeared space ( red curve ) . ( left ) unfolding by ml , and ( right ) unfolding by iterative em . , title="fig:",scaledwidth=49.0% ] and [ delchi](left ) , roc curves for classification performed in the smeared space ( blue curve ) and in the unsmeared space ( red curve ) . ( left ) unfolding by ml , and ( right ) unfolding by iterative em . , title="fig:",scaledwidth=49.0% ] and [ delchi](left ) , plots of @xmath95 vs. @xmath94 , for classification performed in the smeared space ( blue curve ) and in the unsmeared space ( red curve ) . ( left ) unfolding by ml , and ( right ) unfolding by iterative em . , title="fig:",scaledwidth=49.0% ] and [ delchi](left ) , plots of @xmath95 vs. @xmath94 , for classification performed in the smeared space ( blue curve ) and in the unsmeared space ( red curve ) . ( left ) unfolding by ml , and ( right ) unfolding by iterative em . , title="fig:",scaledwidth=49.0% ] vs. @xmath94 as in fig . [ alphabeta ] , here with logarithmic scale on both axes . , title="fig:",scaledwidth=49.0% ] vs. @xmath94 as in fig . [ alphabeta ] , here with logarithmic scale on both axes . , title="fig:",scaledwidth=49.0% ] with the above plots forming a baseline , we can ask how some of the above plots vary as we change the parameters in table [ baseline ] . figure [ sigmaparam ] shows , as a function of the gaussian smearing parameter @xmath42 , the variation of the gof results shown for @xmath97 in 1d histograms in figs . [ nulgofunfoldedinvert ] ( top left ) and [ nulgofunfoldedinvert ] ( bottom ) . the events are generated under @xmath28 . used in smearing ( vertical axis ) . the horizontal axes are the same as those in the 1d histograms in figs . [ nulgofunfoldedinvert ] ( top left ) and [ nulgofunfoldedinvert ] ( bottom ) , namely @xmath90 in the unfolded space ; and the difference with respect to @xmath57 in the smeared space ; for gof tests with respect to @xmath28 using events generated under @xmath28 . , title="fig:",scaledwidth=49.0% ] used in smearing ( vertical axis ) . the horizontal axes are the same as those in the 1d histograms in figs . [ nulgofunfoldedinvert ] ( top left ) and [ nulgofunfoldedinvert ] ( bottom ) , namely @xmath90 in the unfolded space ; and the difference with respect to @xmath57 in the smeared space ; for gof tests with respect to @xmath28 using events generated under @xmath28 . , title="fig:",scaledwidth=49.0% ] figure [ deldelsigma ] shows the variation of the 1d histogram in fig [ deldel ] with the gaussian @xmath42 used in smearing , for both ml and em unfolding . used in smearing ( vertical axis ) of the 1d histogram in fig [ deldel ] of the event - by - event difference of @xmath63 and @xmath93 . ( right ) the same quantity for iterative em unfolding . , title="fig:",scaledwidth=49.0% ] used in smearing ( vertical axis ) of the 1d histogram in fig [ deldel ] of the event - by - event difference of @xmath63 and @xmath93 . ( right ) the same quantity for iterative em unfolding . , title="fig:",scaledwidth=49.0% ] figures [ deldelb ] and [ deldelbiter ] show , for ml and em unfolding respectively , the result of the bottom - line test of fig . [ deldel ] as a function of the amplitude @xmath35 of the extra term in @xmath98 in eqn . [ altp ] . as a function of the amplitude @xmath35 of the extra term in @xmath98 in eqn . [ altp ] , for ( left ) @xmath9 derived from @xmath28 and ( right ) @xmath9 derived from @xmath32 ; for ml unfolding . , title="fig:",scaledwidth=49.0% ] as a function of the amplitude @xmath35 of the extra term in @xmath98 in eqn . [ altp ] , for ( left ) @xmath9 derived from @xmath28 and ( right ) @xmath9 derived from @xmath32 ; for ml unfolding . , title="fig:",scaledwidth=49.0% ] , for iterative em unfolding . , title="fig:",scaledwidth=49.0% ] , for iterative em unfolding . , title="fig:",scaledwidth=49.0% ] figure [ deldelnummeas ] shows , for ml and em unfolding , the result of the bottom - line test of fig . [ deldel ] as a function of the mean number of events in the histogram of @xmath26 . as a function of the number of events on the histogram of @xmath26 , for ( left ) ml unfolding and ( right ) iterative em unfolding . , title="fig:",scaledwidth=49.0% ] as a function of the number of events on the histogram of @xmath26 , for ( left ) ml unfolding and ( right ) iterative em unfolding . , title="fig:",scaledwidth=49.0% ] figure [ deldelreg ] shows , for iterative em unfolding , the result of the bottom - line test of fig . [ deldel ] as a function of the number of iterations . as a function of number of iterations in ( left ) linear vertical scale and ( right ) logarithmic vertical scale . , title="fig:",scaledwidth=49.0% ] as a function of number of iterations in ( left ) linear vertical scale and ( right ) logarithmic vertical scale . , title="fig:",scaledwidth=49.0% ] this note illustrates in detail some of the differences that can arise with respect to the smeared space when testing hypotheses in the unfolded space . as the note focuses on a particularly simple hypotheses test , and looks only at the ml and em solutions , no general conclusions can be drawn , apart from claiming the potential usefulness of the `` bottom line tests '' . even within the limitations of the roounfold software used here ( in particular that the initial estimate for iterating is the presumed truth ) , we see indications of dangers of testing hypotheses after unfolding . perhaps the most interesting thing to note thus far is that unfolding by matrix inversion ( and hence no regularization ) yields , in the implementation studied here , a generalized @xmath93 test statistic that is identical to @xmath60 in the smeared space , which is intrinsically inferior to @xmath63 . the potentially more important issue of bias due to regularization affecting the bottom line test remains to be explored . such issues should be kept in mind , even in informal comparisons of unfolded data to predictions from theory . for quantitative comparison ( including the presumed use of unfolded results to evaluate predictions in the future from theory ) , we believe that extreme caution should be exercised , including performing the bottom - line - tests with various departures from expectations . this applies to both gof tests of a single hypothesis , and comparisons of multiple hypotheses . more work is needed in order to gain experience regarding what sort of unfolding problems and unfolding methods yield results that give reasonable performance under the bottom - line - test , and which cases lead to bad failures . as often suggested , reporting the response matrix @xmath9 along with the smeared data can facilitate comparisons with future theories in the folded space , in spite of the dependence of @xmath9 on the true pdfs . we are grateful to pengcheng pan , yan ru pei , ni zhang , and renyuan zhang for assistance in the early stages of this study . rc thanks the cms statistics committee and gnter zech for helpful discussions regarding the bottom - line test . this work was partially supported by the u.s . department of energy under award number de sc0009937 . louis lyons , `` unfolding : introduction , '' in proceedings of the phystat 2011 workshop on statistical issues related to discovery claims in search experiments and unfolding , edited by h.b . prosper and l. lyons , ( cern , geneva , switzerland , 17 - 20 january 2011 ) + https://cds.cern.ch/record/1306523 ( see end of section 5 . ) olive et al . ( particle data group ) , chin . c * 38 * 090001 ( 2014 ) and 2015 update . the likelihood - ratio gof test with saturated model is eqn . the @xmath59 test for gaussian data with correlations is eqn . pearson s @xmath59 is eqn . 38.48 . mikael kuusela , `` introduction to unfolding in high energy physics , '' lecture at advanced scientific computing workshop , eth zurich ( july 15 , 2014 ) + http://mkuusela.web.cern.ch/mkuusela/eth_workshop_july_2014/slides.pdf t. adye , `` unfolding algorithms and tests using roounfold , '' , in proceedings of the phystat 2011 workshop on statistical issues related to discovery claims in search experiments and unfolding , edited by h.b . prosper and l. lyons , ( cern , geneva , switzerland , 17 - 20 january 2011 ) https://cds.cern.ch/record/1306523 , p. 313 . we used version 1.1.1 from + http://hepunx.rl.ac.uk/~adye/software/unfold/roounfold.html , accessed dec .

in many analyses in high energy physics , attempts are made to remove the effects of detector smearing in data by techniques referred to as `` unfolding '' histograms , thus obtaining estimates of the true values of histogram bin contents . such unfolded histograms are then compared to theoretical predictions , either to judge the goodness of fit of a theory , or to compare the abilities of two or more theories to describe the data . when doing this , even informally , one is testing hypotheses . however , a more fundamentally sound way to test hypotheses is to smear the theoretical predictions by simulating detector response and then comparing to the data without unfolding ; this is also frequently done in high energy physics , particularly in searches for new physics . one can thus ask : to what extent does hypothesis testing after unfolding data materially reproduce the results obtained from testing by smearing theoretical predictions ? we argue that this `` bottom - line - test '' of unfolding methods should be studied more commonly , in addition to common practices of examining variance and bias of estimates of the true contents of histogram bins . we illustrate bottom - line - tests in a simple toy problem with two hypotheses .

1607.07038

active galactic nuclei ( agn ) involve the most powerful , steady sources of luminosity in the universe . it is believed that the center core of agn consist of super massive black hole ( smbh ) surrounded by an accretion disk . in some cases powerful collimated jets are found in agn , perpendicular to the plane of accretion disk . the origin of jets are still unclear . agns whose jets are viewed at a small angle to its axis are called blazars . the overall ( radio to @xmath4-ray ) spectral energy distribution ( sed ) of blazars shows two broad non - thermal continuum peaks . the low - energy peak is thought to arise from electron synchrotron emission . the leptonic model suggests that the second peak forms due to inverse compton emission . this can be due to upscattering , by the same non - thermal population of electrons responsible for the synchrotron radiation , and synchrotron photons ( synchrotron self compton : ssc ) @xcite . blazars often show violent flux variability , that may or may not appear correlated in the different energy bands . simultaneous observation are then crucial to understand the physics behind variability . in this section we discuss the code that we have used to obtain an estimation of the characteristic parameters of the ssc model . the ssc model assumes a spectrum for the accelerated electron density @xmath5 , which is a broken power law with exponents @xmath6 and @xmath7 . the minimum , maximum and break lorentz factors for the electrons are usually called @xmath8 , @xmath9 and @xmath10 respectively . the emitting region is considered to be a blob of radius @xmath11 moving with doppler factor @xmath3 with respect to the observer in a magnetic field of intensity @xmath1 . the model is thus characterized by nine free parameters . ' '' '' @xmath12 ' '' '' in the present work we have kept @xmath8 fixed and equal to unit , which is a satisfactory approximation already used in the literature . the determination of the remaining eight parameters has been performed by finding their best values and uncertainties from a @xmath13 minimization in which multi - frequency experimental points have been fitted to the ssc spectrum modelled as in @xcite . minimization has been performed using the levenberg - marquardt method @xcite , which is an efficient standard for non - linear least - squares minimization that smoothly interpolates between two different minimization approaches , namely the inverse hessian method and the steepest descent method . for completeness , we briefly present the pseudo - code for the algorithm in table i. a crucial point in our implementation is that from @xcite we can only obtain a numerical approximation to the ssc spectrum , in the form of a sampled sed . on the other hand , from table i , we understand that at each step the calculation of the @xmath14 requires the evaluation of the sed for all the observed frequencies . although an observed point will likely not be one of the sampled points coming from @xcite , it will fall between two sampled points , so that interpolation can be used to approximate the value of the sed . at the same time , the levenberg - marquardt method requires the calculation of the partial derivatives of @xmath14 with respect to the ssc parameters . these derivatives have also been obtained numerically by evaluating the incremental ratio of the @xmath14 with respect to a sufficiently small , dynamically adjusted increment of each parameter . this method could have introduced a potential inefficiency in the computation , due to the recurrent need to evaluate the sed at many , slightly different points in parameter space , this being the most demanding operation in terms of cpu time . for this reason we set up the algorithm to minimize the number of calls to @xcite across different iterations . the @xmath0 fit during different iterations are shown in fig.1 . .data sets used in this study . the observation period of each state can be found at fig.2 . [ cols="<,^,^,^",options="header " , ] [ l2ea4-t1 ] in order to study the behavior of parameters with source activity , we choose mrk421 ( table ii ) , considering the larger availability of mwl data sets and the lower redshift , hence less uncertainty after ebl correction of vhe data . the @xmath0 fitted seds are shown in fig.2 . in addition to the @xmath0 test , we also checked the goodness of the fit using the kolmogorov - smirnov ( ks ) test . considering the occurrence of different physical processes ( synchrotron and inverse compton , at substantially different energies ) , and the different quality of low- and high - energy data , we used a _ piecewise ks test _ , _ i.e. _ we applied the ks test separately to low- and high - energy data . then the ks test always confirms that the fit residuals are normal at 5% confidence level . our results suggest that in mkn421 , @xmath1 decreases with source activity whereas @xmath15 and @xmath3 increase ( fig.3 top ) . this can be interpreted in a frame where the synchrotron power and peak frequency remain constant with varying source activity by decreasing magnetic field and increasing the number of low energy electrons . this mechanism results in an increased electron - photon scattering efficiency and hence in an increased compton power . other emission parameters appear uncorrelated with source activity . in fig.3 ( bottom ) , the @xmath1-@xmath15 anti - correlation results from a roughly constant synchrotron peak frequency . the @xmath1-@xmath3 correlation suggests that the compton emission of mkn421 is always in the thomson limit . the @xmath3-@xmath15 correlation is an effect of the constant synchrotron and compton frequencies of the radiation emitted by a plasma in bulk relativistic motion towards the observer .

here we report our recent study on the spectral energy distribution ( sed ) of the high frequency bllac object mrk421 in different luminosity states . we used a full - fledged @xmath0-minimization procedure instead of more commonly used `` eyeball '' fit to model the observed flux of the source ( from optical to very high energy ) , with a synchrotron - self - compton ( ssc ) emission mechanism . our study shows that the synchrotron power and peak frequency remain constant with varying source activity , and the magnetic field ( @xmath1 ) decreases with the source activity while the break energy of electron spectrum ( @xmath2 ) and the doppler factor ( @xmath3 ) increase . since a lower magnetic field and higher density of electrons result in increased electron - photon scattering efficiency , the compton power increases , so does the total emission .

1111.0265

understanding and characterizing general features of the dynamics of open quantum systems is of great importance to physics , chemistry , and biology @xcite . the non - markovian character is one of the most central aspects of an open quantum process , and attracts increasing attentions @xcite . markovian dynamics of quantum systems is described by a quantum dynamical semigroup @xcite , and often taken as an approximation of realistic circumstances with some very strict assumptions . meanwhile , exact master equations , which describe the non - markovian dynamics , are complicated @xcite . based on the infinitesimal divisibility in terms of quantum dynamical semigroup , wolf _ et al . _ provided a model - independent way to study the non - markovian features @xcite . later , in the intuitive picture of the backward information flow leading to the increasing of distinguishability in intermediate dynamical maps , breuer , laine , and piilo ( blp ) proposed a measure on the degree of non - markovian behavior based on the monotonicity of the trace distance under quantum channels @xcite , as shown in fig . [ fig : sketch ] . the blp non - markovianity has been widely studied , and applied in various models @xcite . ( color online ) sketch of the information flow picture for non - markovianity @xcite . according to this scenario , the loss of distinguishability of the system s states indicates the information flow from the system to the reservoir . if the dynamics is markovian , the information flow is always outward , represented by the green thick arrow . non - markovian behaviors occurs when there is inward information flow , represented by the orange thin arrow , bringing some distinguishability back to the system.,width=226 ] unlike for classical stochastic processes , the non - markovian criteria for quantum processes is non - unique , and even controversial . first , the non - markovian criteria from the infinitesimal divisibility and the backward information flow are not equivalent @xcite . second , several other non - markovianity measures , based on different mechanism like the monotonicity of correlations under local quantum channels , have been introduced @xcite . third , even in the framework of backward information flow , trace distance is not the unique monotone distance for the distinguishability between quantum states . other monotone distances on the space of density operators can be found in ref . @xcite , and the statistical distance @xcite is another widely - used one . different distance should not be expected to give the same non - markovian criteria . the inconsistency among various non - markovianity reflects different dynamical properties . in this paper , we show that the blp non - markovianity can not reveal the infinitesimal non - divisibility of quantum processes caused by the non - unital part of the dynamics . besides non - markovianity , `` non - unitality '' is another important dynamical property , which is the necessity for the increasing of the purity @xmath0 under quantum channels @xcite and for the creating of quantum discord in two - qubit systems under local quantum channels @xcite . in the same spirit as blp non - markovianity , we define a measure on the non - unitality . as blp non - markovianity is the most widely used measure on non - markovianity , we also provide a measure on the non - unital non - markovianity , which can be conveniently used as a supplement to the blp measure , when the quantum process is non - unital . we also give an example to demonstrate an extreme case , where the blp non - markovianity vanishes while the quantum process is not infinitesimal divisible . this paper is organized as follows . in sec . [ review ] , we give a brief review on the representation of density operators and quantum channels with hermitian orthonormal operator basis , and various measures on non - markovianity . in sec . [ sec : non - unital - nm ] , we investigate the non - unitality and the non - unital non - markovianity and give the corresponding quantitative measures respectively . in sec . [ sec : example ] , we apply the non - unital non - markovianity measure on a family of quantum processes , which are constructed from the generalized amplitude damping channels . section [ sec : conclusion ] is the conclusion . the states of a quantum system can be described by the density operator @xmath1 , which is positive semidefinite and of trace one . quantum channels , or quantum operations , are completely positive and trace - preserving ( cpt ) maps from density operators to density operators , and can be represented by kraus operators , choi - jamiokowski matrices , or transfer matrices @xcite . in this work , we use the hermitian operator basis to express operators and represent quantum channels . let @xmath2 be a complete set of hermitian and orthonormal operators on complex space @xmath3 , i.e. , @xmath4 satisfies @xmath5 and @xmath6 . any operator @xmath7 on @xmath3 can be express by a column vector @xmath8 through @xmath9 with @xmath10 . every @xmath11 is real if @xmath7 is hermitian . in the meantime , any quantum channel @xmath12 can be represented by @xmath13 $ ] via @xmath14=t(\mathcal{e})r(\rho ) \label{eq:1},\ ] ] where @xmath15 is a @xmath16 real matrix with the elements @xmath17 furthermore , one can easily check that @xmath18 for the composition of quantum channels . here @xmath19 denotes the composite maps @xmath20 . taking into the normalization of the quantum states , i.e. , @xmath21 , @xmath22 can be fixed as @xmath23 for any density operator @xmath1 by choosing @xmath24 with @xmath25 the identity operator . in such a case , @xmath4 for @xmath26 are traceless and generate the algebra @xmath27 . this real parametrization @xmath28 for density operators is also called as coherent vector , or generalized bloch vector @xcite . in order to eliminate the degree of freedom for the fixed @xmath22 , we use the decomposition @xmath29 . therefore , any density operator @xmath1 can be expressed as @xmath30 with @xmath31 the generalized bloch vector and @xmath32 represents @xmath33 . under this frame , quantum channels can be represented by the affine map @xcite @xmath34 where @xmath35 is a real matrix with the dimension @xmath36 and the elements of the vector @xmath37 reads @xmath38_{\mu } = \left\langle\lambda_{\mu},\mathcal{e}(\openone)\right\rangle / d,\ ] ] for @xmath26 . comparing eq . ( [ eq:1 ] ) with eq . ( [ eq:2 ] ) , one could find that @xmath39_{\mu\nu},\ ] ] for @xmath40 . thus , @xmath15 can be decomposed into the following sub - blocks : @xmath41.\ ] ] reminding that a quantum channel @xmath42 is said to be unital if and only if @xmath43 @xcite , one could find that the necessary and sufficient condition for a unital map is that @xmath44 , namely , @xmath45 thus , @xmath37 describes the non - unital property of the quantum channel @xmath42 . the necessary and sufficient condition above could be easily proved by realizing that the bloch vector of @xmath46 is zero vector , i.e. , @xmath47 . based on the sub - block form of @xmath15 , @xmath44 is equivalent to that @xmath15 is block diagonal , i.e. , @xmath48 . whether a quantum channel @xmath42 is completely positive ( cp ) can be reflected by the choi - jamiokowski matrix @xcite @xmath49 where @xmath50 is the maximally entangled state . here @xmath51 is a basis in hilbert space . @xmath42 is cp if and only if the choi - jamiokowski matrix is positive . with the hermitian operator basis , @xmath52 is a @xmath16 matrix and can be written in the form @xcite @xmath53 substituting this formula into eq . ( [ eq : c ] ) and utilizing eq . ( [ eq : t ] ) , one could express the choi - jamiokowski matrix as @xmath54 if @xmath42 is unital , it can be reduced into @xmath55 without the presence of correlation between the open system and its environment in the initial states , the reduced dynamics for the open system from @xmath56 to any @xmath57 can be expressed as @xmath58,\ ] ] which is a quantum channel . this indicates that @xmath59 is cpt . the unitary operator @xmath60 describes the time evolution of the closed entirety , and @xmath61 is the initial state of the environment . a quantum process @xmath62 is said to be infinitesimal divisible , also called as time - inhomogeneous or time - dependent markovian , if it satisfies the following composition law @xcite @xmath63 for any @xmath64 , where @xmath65 is also completely positive and trace preserving . various measures on the degree of the non - markovian behavior of quantum processes have been proposed and investigated @xcite . almost all of the measures on the non - markovianity can be classified into three kinds , base on the degree of the violation of the following properties owned by the infinitesimal divisible quantum process : \(i ) monotonicity of distance @xmath66 under cpt maps . that is @xmath67 for any quantum channel @xmath42 , where @xmath68 is an appropriate monotone distance under cpt maps on the space of density operators @xcite , including trace distance , bures distance , statistical distance , relative entropy , and fidelity ( although fidelity itself is not a distance , it can be used to construct monotone distances ) and so on . some measures on non - markovianity by increasing of the monotone distance during the mediate dynamical maps @xmath65 have been given and discussed in refs . @xcite . the typical measure of this type , which would be used later in this paper , was first proposed by breuer , laine , and piilo in ref . @xcite , based on the monotonicity of trace distance @xcite @xmath69 where @xmath70 . interpreting the increase of the trace distance during the time evolution as the information flows from the environment back to the system , the definition of the blp non - markovianity is defined by @xmath71 where @xmath72 and @xmath73 for @xmath74 are two evolving states . \(ii ) positivity of the choi - jamiokowski matrix for cpt maps . the choi - jamiokowski matrix @xmath75 if and only if @xmath42 is a quantum channel , namely , @xmath42 is a cpt map . some measures on non - markovianity by the negativity of the choi - jamiokowski matrix for mediate dynamical maps @xmath65have been given and discussed in refs . @xcite . in this work we would use one of these measures , which was proposed by rivas , huelga and plenio ( rhp ) in ref . they utilize the negativity of the choi - jamiokowski matrix @xmath76 for the mediate dynamical maps with the definition @xmath77 where @xmath78 \(iii ) monotonicity of correlations @xmath79 under local quantum channels . that is @xmath80\leq e(\rho^{ab})$ ] for any local quantum channel @xmath42 , where @xmath79 is an appropriate measure for the correlations in the bipartite states @xmath81 , including entanglement entropy and the mutual information . the corresponding measures on non - markovianity are given and discussed in refs . the non - markovianity measure @xmath82 is available to capture the non - markovian behavior of the unital aspect of the dynamics . but for the non - unital aspect , it is not capable . to show this , we use the hermitian orthonormal operator basis to express states and quantum channels . utilizing eq . ( [ eq : rho ] ) , the trace distance between two states @xmath83 and @xmath84 is given by @xmath85\cdot\bm{\lambda}\big|.\end{aligned}\ ] ] therefore , for the two evolving states , we get @xmath86 \cdot\bm{\lambda}\big| , \label{eq : dis}\ ] ] where @xmath83 , @xmath84 are initial states of the system . from this equation one can see that the trace distance between any two evolved states is irrelevant to the non - unital part @xmath87 of the time evolution . then if there are two quantum channels , whose affine maps are @xmath88 and @xmath89 , respectively , the characteristic of trace distance between the evolving states from any two initial states can not distinguish these two channels . more importantly , @xmath87 may cause the non - divisibility of the quantum process @xmath90 , and this can not be revealed by @xmath82 . on the other hand , the non - unital part @xmath87 has its own physical meaning : @xmath91 is necessary for the increasing of the purity @xmath92 @xcite . in other words , @xmath93 besides the non - markovian feature , the non - unitality is another kind of general feature of quantum processes . in analogy to the definition of blp non - markovianity , we defined the following measure on the degree of the non - unitality of a quantum process : @xmath94 } { \mathrm{d}t } \right| \mathrm{d}t,\ ] ] where @xmath95 is the initial state . obviously , @xmath96 vanishes if @xmath97 . since the non - unital aspect of the dynamics , which is not revealed by the trace distance , has its own speciality , we aim to measure the effect of non - unitality on non - markovian behavior . however , a perfect separation of the non - unital aspect from the total non - markovianity may be infeasible . therefore we require a weak version @xmath98 for measuring non - unital non - markovianity to satisfy the following three conditions : ( i ) @xmath99 vanishes if @xmath90 is infinitesimal divisible , ( ii ) @xmath99 vanishes if @xmath90 is unital , ( iii ) @xmath100 should be relevant to @xmath87 . based on these conditions , we introduce the following measure @xmath101 where @xmath102 with @xmath103 is the set of the trajectory states which evolve from the maximally mixed state , and @xmath104,\ ] ] with @xmath68 an appropriate distance which will be discussed below . the first condition is guaranteed if we require that @xmath66 is monotone under any cpt maps , i.e. , @xmath105\leq d(\rho_{1},\rho_{2})$ ] for any quantum channel @xmath42 . for the unital time evolution , the set @xmath106 only contains the maximally mixed state , so the above defined @xmath100 vanishes , and the second condition is satisfied . the third condition excludes the trace distance . in this paper , we use the bures distance which is defined as @xmath107},\ ] ] where @xmath108 is the uhlmann fidelity @xcite between @xmath83 and @xmath84 . here bures distance is an appropriate distance for @xmath100 because it obeys the monotonicity under cpt maps @xcite and is relevant to @xmath87 . as here only the monotonicity of distance is relevant , for simplicity , we can also take the square of the bures distance or just the opposite value of unlmann fidelity as a simple version of monotone `` distance '' @xcite . quantum relative entropy @xcite @xmath110 $ ] , or its symmetric version @xmath111 , is another qualified candidate for the distance . noting that when the support of @xmath83 is not within the support of @xmath84 , namely , @xmath112 , @xmath113 will be infinite , so in such cases , quantum relative entropy will bring singularity to the measure of non - markovianity . also , hellinger distance @xcite is qualified . although all of these distances are monotone under cpt maps , they may have different characteristics in the same dynamics , see ref . @xcite . the difference between non - unital non - markovian measure defined by eq . ( [ eq : definition_nun ] ) and the blp - type measures , including those which use other alternative distances , is the restriction on the pairs of initial states . comparing with the blp - type measures relying on any pair of initial states , the non - unital non - markovianity measure only relies on the pairs consisting of the maximally mixed state and its trajectory states . on one hand , this restriction makes the non - unital non - markovianity measure vanish when the quantum processes are unital , no matter they are markovian or non - markovian ; on the other hand , this restriction reflects that non - unital non - markovianity measure reveals only a part of information concerning the non - markovian behaviors . to illustrate the non - unital non - markovian behavior , we give an example in this section . we use the generalized amplitude damping channel ( gadc ) as a prototype to construct a quantum process . the gadc can be described by @xmath114 with the kraus operators @xmath115 given by @xcite @xmath116 where @xmath117 and @xmath118 are real parameters . note that for any @xmath119 $ ] and any @xmath120 $ ] , the corresponding @xmath42 is a quantum channel . for a two - level system , the hermitian orthonormal operator basis can be chosen as @xmath121 , where @xmath122 is the vector of pauli matrices . with the decomposition in eq . ( [ eq : rho ] ) , the affine map for the bloch vector is given by @xmath123 @xcite , where @xmath124 the gadc is unital if and only if @xmath125 or @xmath126 . when @xmath126 , @xmath127 , the map is identity . a quantum process can be constructed by making the parameter @xmath117 and @xmath118 to be dependent on time @xmath128 . for simplicity , we take @xmath129 and @xmath130 , where @xmath131 is a constant real number . this is a legitimate quantum process , because @xmath90 is a quantum channel for every @xmath57 , and @xmath132 is the identity map . first , let us consider the @xmath82 for this quantum process . for any two initial states @xmath83 and @xmath84 , we have the trace distance @xmath133 & = \frac{1}{2}\mathrm{tr}\left|m(\mathcal{e}_{t})[\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})]\cdot\frac{\bm{\sigma}}{\sqrt{2}}\right|\nonumber \\ & = \frac{1}{\sqrt{2}}\left|m(\mathcal{e}_{t})[\mathbf{r}(\rho_{1})-\mathbf{r}(\rho_{2})]\right|,\end{aligned}\ ] ] where @xmath134 is the euclidean length of the vector @xmath31 , and we used the equality @xmath135 for pauli matrices . denoting @xmath136 by @xmath137 , we get @xmath138=\frac{e^{-t/2}}{\sqrt{2}}\sqrt{x^{2}+y^{2}+e^{-t}z^{2}},\label{eq : gadc_tr_distance}\ ] ] which implies @xmath139\leq0 $ ] for every time point @xmath57 and for any real numbers @xmath140 , @xmath141 , and @xmath142 . thus , the blp non - markovianity vanishes , i.e. , @xmath143 , although @xmath90 may be not infinitesimal divisible , which will become clear later . ( color online ) ( a ) evolution of trace distance and bures distance between two evolving states of a two - level system under the variant generalized amplitude damping channel , initially from the maximal mixed states @xmath144 and its trajectory state @xmath145 , respectively . ( b ) the evolution of @xmath146 defined by eq . ( [ eq : gt ] ) , whose integral with respect to time @xmath128 is rhp measure for non - markovianity . in these figures , the parameters are taken as @xmath147 and @xmath148.,width=188 ] in order to investigate whether @xmath90 is infinitesimal divisible or not , we shall apply @xmath100 in the above model . the trajectory of the maximally mixed state under @xmath90 reads @xmath149 where @xmath150 taking these trajectory states as the initial states , we get the corresponding evolving states : @xmath151 then the fidelity reads @xmath152=\frac{1}{2}(h_{+}+h_{-}),\ ] ] where @xmath153 to compare with the behavior of trace distance , we also get @xmath154=\left|\eta_{t}w_{\tau}\right|/2 $ ] . with the expressions @xmath130 and @xmath129 , it is @xmath155=\frac{e^{-t}}{2}\left|\cos2\omega\tau\right|(1-e^{-\tau}).\label{eq : gadc_nonunital}\ ] ] in fig . [ fig : db_dtr](a ) , we can see that while the trace distance between the evolving states @xmath156 and @xmath157 monotonously decreases with the time @xmath128 , the bures distance increases during some intermediate time intervals . from eq . ( [ eq : gadc_nonunital ] ) , one can see although @xmath158 $ ] depends on @xmath159 , it does not depend on @xmath160 . actually , from eq . ( [ eq : gadc_tr_distance ] ) one could find that for any two initial states , the trace distance between the evolving states is independent on @xmath160 . in this sense , the blp non - markovianity treats a family of quantum processes , which only differ with @xmath161 , as the same one . meanwhile , @xmath100 reveals the effects of @xmath161 on the infinitesimal non - divisibility and is capable of measuring it . in order to compare with bhp measure , we also calculate the @xmath146 defined by eq . ( [ eq : gt ] ) . we get @xmath162\ ] ] with @xmath163 the mediate dynamical maps @xmath164 with infinitesimal @xmath165 are not completely positive when @xmath166 . from fig . [ fig : db_dtr](b ) , we can see that the increasing of the bures distance occurs in the regimes where @xmath166 , which coincides with the monotonicity of bures distance under cpt maps . in conclusion , we have shown that the measure for non - markovianity based on trace distance can not reveal the infinitesimal non - divisibility caused by the non - unital part of the dynamics . in order to reflect effects of the non - unitality , we have constructed a measure on the non - unital non - markovianity , and also defined a measure on the non - unitality , in the same spirit as blp non - markovianity measure . like non - markovianity , the non - unitality is another interesting feature of the quantum dynamics . with the development of quantum technologies , we need novel theoretical approaches for open quantum systems . it is expected that some quantum information methods would help us to understand some generic features of quantum dynamics . we hope this work may draw attention to study more dynamical properties from the informational perspective . this work was supported by nfrpc through grant no . 2012cb921602 , the nsfc through grants no . 11025527 and no . 10935010 and national research foundation and ministry of education , singapore ( grant no . wbs : r-710 - 000 - 008 - 271 ) . m. m. wolf , quantum channels & operations guided tour , http://www-m5.ma.tum.de/ foswiki/ pub/ m5 /allgemeines/ michaelwolf/ qchannellecture.pdf[http://www-m5.ma.tum.de/ foswiki/ pub/ m5 /allgemeines/ michaelwolf/ qchannellecture.pdf ]

trace distance is available to capture the dynamical information of the unital aspect of a quantum process . however , it can not reflect the non - unital part . so , the non - divisibility originated from the non - unital aspect can not be revealed by the corresponding measure based on the trace distance . we provide a measure of non - unital non - markovianity of quantum processes , which is a supplement to breuer - laine - piilo ( blp ) non - markovianity measure . a measure on the degree of the non - unitality is also provided .

1301.5763

interplay between superconductivity and the onset of electronic spin and charge modulations in cuprate superconductors remains one of the intriguing and unresolved issues in the field of high - temperature superconductivity . manifestations of electronic modulations are reported in a broad doping range for several families of cuprates most noticeably around the doping level of 1/8@xcite . for 1/8-doped lanthanum cuprates , the modulated structure is widely believed to exhibit one - dimensional pattern often referred to as stripes " @xcite . yet the principal aspects of the same experimental evidence are also consistent with the possibility of two - dimensional modulations called checkerboards " @xcite . the experiment - based arguments discriminating between stripes and checkerboards in 1/8-doped lanthanum cuprates are at present rather indirect . at the same time , the issue can not be resolved on purely theoretical grounds , because it requires accuracy of the calculations of the ground state energy not achievable by first - principles theories . the checkerboard was , initially , less popular as an interpretation of experiments , in part , because of the perception that it inhibits the electronic transport in all lattice directions and hence prevents the onset of superconductivity as well . the latter point , however , was challenged by a model of superconductivity in the background of a checkerboard modulation @xcite . that model was based on a particular version of a checkerboard called grid " . later , the grid checkerboard was shown to be inconsistent with the results of spin - polarized neutron scattering experiment of ref.@xcite . this experiment , however , did not rule out another version of a checkerboard representing a two - dimensional arrangement of spin vortices@xcite shown in fig.[fig : checkers ] . somewhat similar noncollinear spin textures were also considered in refs . the general properties of spin - vortex checkerboard were analyzed in refs.@xcite . so far , however , the superconductivity model developed for grid@xcite has not been generalized to the spin - vortex checkerboard . in the present article , we introduce such a generalized model , find its mean - field solution and compare it with experiments . it should be remarked here that , experimentally , the onset of static spin modulations in 1/8-doped lanthanum cuprates spin vortices or stripes largely suppresses three - dimensional superconductivity but appears to coexist with two - dimensional superconductivity @xcite . for both spin vortices and stripes , the suppression of the three dimensional superconductivity can be explained by the displacement of the modulation patterns in the adjacent cuo@xmath1 planes . beyond 1/8-doped lanthanum cuprates , the generic situation possibly involves the fluctuating counterpart of the static spin - vortex texture used in the present work . these fluctuations are likely caused by system s proximity to the threshold of electronic phase separation @xcite . they are expected to couple spin , charge and lattice degrees of freedom see e.g.@xcite . in this respect , the spin - vortex checkerboard just suggests us the coupling connectivity between low - energy fermionic states . as it stands , the model illustrates the potential of the general two - component scenarios@xcite in the limit of initially localized components for describing the superconductivity in cuprates . [ fig : spinbackground ] with labels @xmath2 , @xmath3 and @xmath4 as introduced in the text . colors represent different quasiparticle states as follows : b - states with @xmath5 ( pink ) , @xmath6 ( yellow ) , @xmath7 ( green ) , and @xmath8 ( gray ) ; even a - state [ @xmath9 ( orange ) , odd a - state [ @xmath10 ( brown ) ; even c - state [ @xmath11 ( blue ) , odd c - state [ @xmath12 ( cyan).,scaledwidth=40.0% ] the model to be considered has two different kinds of fermionic states physically located in magnetic and non - magnetic parts of the underlying spin texture . the general reasoning for constructing the model is the same as in ref.@xcite . namely , the entire texture is divided into plaquets having different kind of spin background , and then , for each plaquet , only one - particle fermionic states closest to the chemical potential are retained for the model description . given that plaquets are rather small , it can be estimated@xcite that the spacing of one - particle energies within each plaquet is of the order of 40 mev , which implies that , for temperatures much smaller than 400k , it is appropriate to retain only the levels closest to the chemical potential . we expect that the lowest one - particle states in spin - polarized plaquets are non - spin - degenerate , and hence we include exactly one state per plaquet . as can be seen in fig . [ fig : checkers ] , the texture has four kinds of spin - polarized plaquets . we refer to two of the resulting states as a - states " and to the remaining two as c - states " . two different kinds a - states are distinguished by index @xmath13 , and c - states by index @xmath14 . two a - states or two c - states with different values of @xmath3 or @xmath4 respectively are expected to have orthogonal spin wave functions that can be obtained from each other by spin inversion . the lowest - energy states of spin - unpolarized plaquets around the cores of spin vortices are expected to be spin - degenerate . we , therefore , place two fermionic states on each such plaquet with spins up " or down " along any chosen direction . we call them b - states " . since the spin texture contains four nonequivalent kinds of spin - vortex cores , we distinguish the corresponding b - states by index @xmath15 and by spin index @xmath16 or @xmath17 see fig . [ fig : unitcell ] . we now construct the low - energy hamiltonian following the same reasoning as in ref.@xcite . namely , we assume that the direct one - fermion hopping between different plaquets is suppressed due to the differences of spin textures and due to the expected texture fluctuations . at the same time , we assume that fast fluctuations of the texture around the average pattern shown in fig . [ fig : checkers ] induce pair transitions that do not change the center - of - mass position of the two fermions involved . this leaves us only with terms representing on - site energies @xmath18 , @xmath19 and @xmath20 ( with @xmath21 ) and with the following two - kinds of effective interaction terms , namely : two a - states or two c - states adjacent to a given spin vortex core making transitions to the two b - states inside the core or vice versa . the resulting hamiltonian is : @xmath22 } } a_{k \text{o}_{[i,\alpha]}}+h.c\right)\right.\\ \left . + \left ( b^{\dagger}_{i \alpha \uparrow}b^{\dagger}_{i \alpha \downarrow } c_{m \text{e}_{[i,\alpha ] } } c_{n \text{o}_{[i,\alpha]}}+h.c\right ) \right ] , \end{gathered}\ ] ] where @xmath23 is the interaction constant , @xmath18 , @xmath19 and @xmath20 are on - site energies defined with respect to the chemical potential @xmath24 , which we set equal to zero , index @xmath25 labels unit cells depicted in fig [ fig : unitcell ] , and indices @xmath3 , @xmath2 and @xmath4 label the plaquets within the unit cell as illustrated in fig [ fig : unitcell ] . following ref.@xcite , whenever the specific value of subscripts @xmath3 or @xmath4 is fixed , as is the case in the interaction term of hamiltonian ( [ eq : originalhamlt ] ) , we use subscript e " for @xmath26 or @xmath27 referring to the corresponding plaquets as even " and subscript o " for @xmath28 or @xmath29 referring to the corresponding plaquets as odd " . double - subscripts notations such as @xmath30}}$ ] imply that a - states labeled as @xmath31 must be adjacent to the b - states labeled as @xmath32 . if all terms containing c - states are removed from hamiltonian ( [ eq : originalhamlt ] ) , the result would be exactly equivalent to the hamiltonian considered in ref @xcite . since c - states do not directly couple to a - states , and since c - states have the same connectivity with the b - states as a - states ( but shifted ) , the mean - field solutions of the two models are very similar with the only difference being that b - states now experience mean field from both a - states and c - states , which , in turn , makes that mean field two times larger , and , as a result , the value of the superconducting transition temperature becomes modified . since the entire derivation has nearly the same structure and logic as that of ref . @xcite , below we only include the formal structure of the derivation and the results , leaving the justification mostly to ref . @xcite . in the model considered , each of the fermionic states couples to relatively few other states , which makes a mean - field solution only very approximate . we nevertheless proceed with finding a mean - field solution , assuming that gives at least the right qualitative picture of model s behavior . the first step of this solution is to introduce the bogoliubov transformation for @xmath33-states within the same plaquet : @xmath34 where @xmath35 and @xmath36 are positive real numbers satisfying a constraint arising from canonical fermionic anticommutation relations @xmath37 and @xmath38 are the transformation phases , which are to be determined later by minimizing system s energy . substituting bogoliubov transformation for @xmath33-states ( [ eq : bgvtransb ] ) in ( [ eq : originalhamlt ] ) and keeping only the thermal averages of terms that do not change the occupations of b - states , we obtain partially averaged hamiltonian , @xmath39\\+ \epsilon_{a } \sum_{i,\eta } a^{\dagger}_{i\eta } a_{i\eta}+ \epsilon_{c } \sum_{i,\eta } c^{\dagger}_{i\zeta } c_{i\zeta}\\+gsw ( 1- 2n_{b } ) \mathlarger{\mathlarger{\sum}}_{\alpha } \bigg [ ( e^{-i \varphi_{\alpha } } \sum_{ij } a_{j \text{e}_{[i,\alpha ] } } a_{k \text{o}_{[i,\alpha ] } } + h.c ) \\ + ( e^{-i \varphi_{\alpha } } \sum_{ij } c_{m \text{e}_{[i,\alpha ] } } c_{n \text{o}_{[i,\alpha ] } } + h.c ) \bigg ] , \end{gathered}\ ] ] where @xmath40 is the energy of b - quasiparticles and @xmath41 is their occupation number . + as explained in ref.@xcite , in order to assure proper fermionic anticommutation relations for the bogoliubov counterparts of a- and c - states , the bogoliubov transformation for these states should be made in the quasimomentum space . therefore , we need to rewrite the hamiltonian in terms of the real - space fourier transforms for a- and c- operators . to do the fourier transforms we first change the notations from @xmath42 , @xmath43 to the notations @xmath44 , @xmath45 , where @xmath46 indicates the position of the center of the respective plaquet . for the following , we need to define the following vectors ( all in units of underlying crystal lattice ) : @xmath47 which connects an even a - state with an adjacent even c - state , and also vectors @xmath48 which connect an even a - state with four adjacent odd a - states . the subscript @xmath2 in @xmath49 is chosen such that @xmath2-th b - states are located between the pairs a - states connected by vector @xmath49 originated from an even a - state . now , we define the position of each unit cell by the position @xmath50 of an even a - state within this cell . therefore , even a - states are located at a set of positions @xmath51 , odd a - states at @xmath52 , even c - states at @xmath53 , and odd c - states at @xmath54 . finally we rewrite the hamiltonian ( [ eq : avbstate ] ) as follows , @xmath55 \\ + \sum_{\mathbf{r}_\text{e } } \left\ { \epsilon_{a } a^{\dagger}({\bf r}_\text{e } ) a({\bf r}_\text{e } ) + \epsilon_{a } a^{\dagger } ( { \bf r}_\text{e } + { \bf r}_{1 } ) a ( { \bf r}_\text{e } + { \bf r}_{1 } ) \right.\\ \left . + \epsilon_{c } c^{\dagger}({\bf r}_\text{e } + { \bf l } ) c({\bf r}_\text{e } + { \bf l } ) \right.\\ \left . + \epsilon_{c } c^{\dagger}({\bf r}_\text{e } + { \bf l } + { \bf r}_{1 } ) c({\bf r}_\text{e } + { \bf l } + { \bf r}_{1 } ) \right\ } \\ + gsw ( 1- 2n_{b})\sum_{\alpha } \bigg [ \left(e^{-i \varphi_{\alpha } } \sum_{{\bf r}_\text{e } } a({\bf r}_\text{e } ) a({\bf r}_\text{e } + { \bf r}_{\alpha } ) + h.c \right ) \\ + \left(e^{-i \varphi_{\alpha } } \sum_{{\bf r}_\text{e } } c({\bf r}_\text{e } + { \bf l } ) c({\bf r}_\text{e}+{\bf l } + { \bf r}_{5-\alpha } ) + h.c \right)\bigg ] , \end{gathered}\ ] ] we now explicitly write separate fourier transforms for even and odd a- and c- states as follows : @xmath56 since the superlattice periods for each of the above four kinds of states are the same , the sets of wave vectors @xmath57 are also the same for all four transformations , even though the corresponding states are shifted with respect to each other in real space . substituting these transformations to ( [ eq : evenodd ] ) we obtain @xmath58 + \epsilon_{a } \sum_{\mathbf{k } } a^{\dagger}_\text{e}(\mathbf{k } ) a_\text{e}(\mathbf{k } ) + \\ \epsilon_{a } \sum_{\mathbf{k } } a^{\dagger}_\text{o}(\mathbf{k } ) a_\text{o}(\mathbf{k})+\epsilon_{c } \sum_\mathbf{k } c^{\dagger}_\text{e}(\mathbf{k } ) c_\text{e}(\mathbf{k})+\epsilon_{c } \sum_{\mathbf{k } } c^{\dagger}_\text{o}(\mathbf{k } ) c_\text{o}(\mathbf{k})\\ + gsw ( 1- 2n_{b})\sum_{\mathbf{k } } \bigg [ \left ( a_\text{e}(\mathbf{k } ) a_\text{o}(\mathbf{-k})v(\mathbf{k } ) + h.c \right)\\+ ( c_\text{e}(\mathbf{k } ) c_\text{o}(\mathbf{-k})\tilde{v}(\mathbf{k } ) + h.c ) \bigg ] , \end{gathered}\ ] ] where , @xmath59 \cos \left[\mathbf{k}\mathbf{r}_{1 } + \frac{\varphi_{1 } - \varphi_{3}}{2}\right]\nonumber\\ & & + 2 \exp \left [ -i \frac{\varphi_{2 } + \varphi_{4}}{2 } \right ] \cos \left[\mathbf{k}\mathbf{r}_{2 } + \frac{\varphi_{2 } - \varphi_{4}}{2}\right],\nonumber \end{aligned}\ ] ] and , @xmath60 \cos \left[\mathbf{k}\mathbf{r}_{2 } + \frac{\varphi_{3 } - \varphi_{1}}{2}\right ] \nonumber\\ & & + 2 \exp \left [ -i \frac{\varphi_{2 } + \varphi_{4}}{2 } \right ] \cos \left[\mathbf{k}\mathbf{r}_{1 } + \frac{\varphi_{4 } - \varphi_{2}}{2}\right].\nonumber \end{aligned}\ ] ] bogoliubov transformations for a and c- states can now be defined as , @xmath61 where @xmath62 @xmath63 and @xmath64 @xmath65 are the real - valued coefficients for a and c - states respectively , subjected to a constraint arising from the fermionic canonical commutation relations for a- and c - operators : @xmath66 and @xmath67 and @xmath68 are complex phases all to be found by the energy minimization . we now complete the following steps : ( i ) we substitute the above canonical transformation for a and c - states ( [ eq : btacststes ] ) into hamiltonian ( [ eq : ftac ] ) , then ( ii ) obtain the energy of the system by summing over the thermal averages of the diagonal terms the result is given in appendix , and then ( iii ) minimize the resulting energy with respect to the choice of phases @xmath69 and @xmath70 , which , as explained in appendix a gives conditions : @xmath71 & = 1 , \\ \cos [ \phi_{\tilde{v}}(\mathbf{k})+ \phi_{c}(\mathbf{k } ) ] & = 1,\end{aligned}\ ] ] were @xmath72 and @xmath73 are the complex phases of @xmath74 and @xmath75 respectively , which , in turn , depend on phases @xmath76 . ( the actual values of phases @xmath77 and @xmath70 do not need to be obtained explicitly , because they will not enter any quantity further computed in this paper . ) with the above conditions , the expression for the energy of the system becomes : @xmath78 \\+ 2 \epsilon_{a } \sum_{\mathbf{k } } \left\lbrace u^{2}(\mathbf{k})n_{a}(\mathbf{k } ) + v^{2}(\mathbf{k})[1-n_{a}(\mathbf{k } ) ] \right\rbrace \\+2 \epsilon_{c } \sum_{\mathbf{k } } \left\lbrace p^{2}(\mathbf{k})n_{c}(\mathbf{k } ) + q^{2}(\mathbf{k})[1-n_{c}(\mathbf{k } ) ] \right\rbrace \\+ 2gsw \left ( 1- 2n_{b } \right ) \sum_{\mathbf{k } } \bigg [ u(\mathbf{k})v(\mathbf{k})(1 - 2n_{a}(\mathbf{k } ) ) |v(\mathbf{k})|\bigg . \\ \bigg.+ p(\mathbf{k})q(\mathbf{k})(1 - 2n_{c}(\mathbf{k}))]|\tilde{v}(\mathbf{k})| \bigg ] , \end{gathered}\ ] ] where , @xmath79 @xmath80 are the bogoliubov quasiparticle occupation number and @xmath81 , @xmath82 are their energies , to be given in the next section . as argued in ref @xcite , the chemical potential of the system is likely to coincide with either @xmath83 or @xmath84 ( same as @xmath85 ) , which , given our convention @xmath86 , means that either @xmath87 or @xmath88 . below , we treat these two cases separately , referring to them as case i " and case ii " respectively , and also refer to the case of @xmath89 as the critical case " . the coefficients of the bogoliubov transformations for both cases i and ii are obtained in appendices [ ap : casei ] and [ ap : caseii ] respectively by minimizing the total energy ( [ eq : energyfinal ] ) at fixed quasiparticle occupation numbers . we then substitute those coefficients back to eq ( [ eq : energyfinal ] ) and obtain the energy of a bogoliubov quasiparticle by taking derivative of the total energy ( [ eq : energyfinal ] ) with respect to the quasiparticle occupation numbers @xmath90 , @xmath91 or @xmath92 . in case i , the above procedure gives the following quasiparticle energies : @xmath94 @xmath95 @xmath96 . \end{gathered}\ ] ] the mean - field approach now requires finding a nontrivial solution for @xmath90 , @xmath91 , @xmath97 , @xmath81 , @xmath82 , and @xmath40 from eqs.([nb ] , [ eq : na ] , [ eq : nc ] , [ eq : eaenergyi ] , [ eq : ecenergyi ] , [ eq : ebenergyi ] ) . in general , it can only be done numerically , but one can also obtain a closed analytical equation for the critical temperature @xmath0 using the fact that , near the transition , the superconducting state is close to the normal states , which allows one to use the limits @xmath98 detailed calculations can be found in appendix b. this gives @xmath99 , \end{split } \label{tc1}\ ] ] from which , the mean- field @xmath100 can be obtained numerically . as explained in ref.@xcite , the density of a- and c- states has a van hove singularity located at the value corresponding to @xmath101 and @xmath102 . we identify this singularity with the superconducting gap , which we , therefore , give by formula @xmath103 as @xmath104 , it approaches not to zero but to @xmath105 , which we associate with the pseudogap . following the same procedure as for case i , we obtain : @xmath107 @xmath108 @xmath109 where @xmath110 detailed calculations can be found in appendix c. the same approach as in case i now gives the critical temperature @xmath111 and the superconducting gap parameter @xmath112 associated with van hove singularity for a- and b- states located at @xmath113 . in fig ( [ fig : figfamily ] ) , we present numerically computed temperature dependencies of superconducting gaps for cases i and ii given by eqs.([delta1 ] ) and ( [ delta2 ] ) respectively . . solid lines above the thick line represent case i and below the thick line case ii . the dashed line shows the standard result of the bardeen - cooper - schrieffer theory@xcite.,scaledwidth=48.0% ] the families of plots for cases i and ii are connected through the critical case @xmath114 , which is represented by the thick red line . this case corresponds to the ratio @xmath115 . plots above the critical - case line correspond to case i : at @xmath116 , they all end at nonzero values @xmath117 . plots below the critical - case line correspond to case ii : they all have @xmath118 , and , moreover approach closely the canonical bcs dependence for @xmath119 . thus , if the assumptions of the present model are valid , the critical - case ratio @xmath120 signifies the transition from the conventional behavior @xmath118 for @xmath121 to unconventional behavior @xmath122 for @xmath123 . the value of @xmath120 for the critical case makes important quantitative difference from the critical case result @xmath124 for the grid - based model of refs.@xcite , which involved only a- and b- states . such a difference was to be expected , because the coupling between b- and c - states in the present model leads to additional energy advantage for the superconducting state and hence higher superconducting transition temperature for the same value of the coupling constant @xmath23 . in fig . [ fig : exp ] , we show how the predictions of the present model for the temperature dependence of the superconducting gap compare with the available experimental results for break junctions ( bj)@xcite and the interlayer tunneling ( ilt ) @xcite in bismuth families of cuprates . this is the same set of experimental data as the one used in ref.@xcite for testing the grid - based model . the predictions of the grid - based model are also plotted in fig . [ fig : exp ] . the model predictions , when limited to cases i or ii only , require two input parameters @xmath125 and @xmath0 , which help us to determine @xmath23 and @xmath105 for case i , or @xmath126 for case ii . the choice between cases i and ii is made on the basis of the ratio @xmath127 being larger or smaller than @xmath128 . all plots in fig . [ fig : exp ] demonstrate either very good or satisfactory agreement between the predictions of the present model and the experiment . in comparison with the predictions of the grid - based model , the agreement with experiment has improved overall . specifically , it became better in frames ( a , b , c , d , e , f , g , o , q , r , s ) , remained about the same in frames ( k , l , m , p ) , and became worse in frames ( h , i , j ) . it should be remarked here that the experimental data themselves are subject to a number of uncertainties , including , in particular , the overheating effect for the ilt measurements @xcite . we further remark that , despite the significant experimental difficulty of measuring @xmath129 close to @xmath130 , the very notion of the existence fo the critical ratio @xmath127 which separates the dependencies ending with @xmath131 from those ending with @xmath122 appears to be reasonably supported by experiments , and , moreover , the value @xmath132 for such a critical ratio obtained in this work leads to more consistent predictions than the critical value 4 obtained in ref.@xcite for the grid - based model . in the present paper , we generalized the superconductivity model proposed in ref . @xcite for the grid background to the background formed by the checkerboard of spin vortices . the technical difference is that the former involves two kinds of fermionic states , while the later involves three , even though two of the three are similar . we have shown that the qualitative predictions of the grid - based model largely remain intact , which means that the detailed analysis of the grid - based model of ref.@xcite can also be extended to the spin - vortices - based model . therefore , we did not try to repeat it here . the most important difference between the spin - vortices - based model and the grid - based model turns out to be the critical ratio @xmath127 above which the temperature dependence of the superconducting gap ends at the value @xmath133 , which , in turn , is probably related to the pseudogap . for spin vortices , this critical value is @xmath132 , while , for grid , it is 4 . we have demonstrated that the predictions for the temperature evolution of the superconducting gap for the spin - vortices - based model exhibits good agreement with experiments , and moreover , this agreement is somewhat better than for the grid - based model . + + + + in the broader context of cuprate superconductivity , the model considered in this work is still rather oversimplified . however , one can use it to develop intuition about more realistic settings that must involve the fluctuations of the spin background , as well as other interactions between fermions . in this appendix , we elaborate on the derivation steps ( ii ) and ( iii ) mentioned after eq.([eq : uvpqcons ] ) . substituting the canonical transformation for a and c - states ( [ eq : btacststes ] ) in ( [ eq : ftac ] ) , we obtain @xmath134 \\ + 2 \epsilon_{a } \sum_{\mathbf{k } } \left\lbrace u^{2}(\mathbf{k})n_{a}(\mathbf{k } ) + v^{2}(\mathbf{k})[1-n_{a}(\mathbf{k } ) ] \right\rbrace \\+2 \epsilon_{c } \sum_{\mathbf{k } } \left\lbrace p^{2}(\mathbf{k})n_{c}(\mathbf{k } ) + q^{2}(\mathbf{k})[1-n_{c}(\mathbf{k } ) ] \right\rbrace \\ + 2gsw \left ( 1- 2n_{b } \right ) \sum_{\mathbf{k } } \\ \bigg [ u(\mathbf{k})v(\mathbf{k})(1 - 2n_{a}(\mathbf{k } ) ) |v(\mathbf{k})| \cos [ \phi_{v}(\mathbf{k})+ \phi_{a}(\mathbf{k } ) ] \bigg . + p(\mathbf{k})q(\mathbf{k})(1 - 2n_{c}(\mathbf{k}))|\tilde{v}(\mathbf{k})| \cos [ \phi_{\tilde{v}}(\mathbf{k})+ \phi_{c}(\mathbf{k})]\bigg ] , \end{gathered}\ ] ] where all variables are defined in section ( iii ) . two interaction terms in the above expression have phase - dependent factors @xmath135 $ ] and @xmath136 $ ] . eventually , the variational ground - state energy obtained by finding @xmath137 , @xmath63 , @xmath138 and @xmath65 will monotonically decrease with the increasing absolute value of these terms . this implies that the variational energy will be minimized for @xmath139|= 1 $ ] and @xmath140|=1 $ ] . choosing the sign of cosines in these relations is just a matter of sign convention for the bogoliubov transformation coefficients later converting into the sign of the products @xmath141 and @xmath142 . for @xmath93 , the bogoliubov transformation parameters @xmath35 and @xmath36 enter the energy ( [ eq : energyfinal ] ) only as a term proportional to @xmath143 . for such a case , given the constriant @xmath144 , the minimization of energy ( [ eq : energyfinal ] ) gives , @xmath145 @xmath146 . the relative negative sign of @xmath35 and @xmath36 implies later the positive relative sign for the pairs of transformation parameters @xmath147 and @xmath148 . the minimization of energy ( [ eq : energyfinal ] ) with respect to @xmath137 , @xmath149 , @xmath138 , and @xmath65 , finally , gives @xmath150 where , @xmath151 the total energy @xmath152 of the system in this case can be expressed as @xmath153 , \end{gathered}\ ] ] which is an implicit function of @xmath154 and @xmath155 both @xmath154 and @xmath156 are a function of four phases @xmath157 , @xmath158 , @xmath159 and @xmath160 entering eqs . ( [ eq : vk ] , [ eq : vkt ] ) . therefore the energy equation should be further minimized with respect to the values of these phases . such minimization imposes only one constraint @xmath161 in this case , @xmath162 in eq . and , as a result , the minimization of energy gives @xmath163 . the minimization with respect to @xmath35 subject to condition , now gives @xmath164 where we introduced the following parameters : the parameters @xmath166 and @xmath167 are defined by eqs.([eq : ca],[eq : cc ] ) . solving the bi - quadratic equation ( [ s4 ] ) , we obtain @xmath168 + we obtain @xmath40 by varying energy with respect to @xmath92 : @xmath169 + 2gsw ( -2 ) \sum_{\mathbf{k } } \left\ { u(\mathbf{k})v(\mathbf{k})(1 - 2n_{a}(\mathbf{k } ) ) |v(\mathbf{k})| \right . \\ \left . + p(\mathbf{k})q(\mathbf{k})(1 - 2n_{c}(\mathbf{k}))(\mathbf{k})]|\tilde{v}(\mathbf{k})| \right\}. \end{gathered}\ ] ] by substituting the parameters defined by eqs.([eq : sw],[eq : rcases ] ) , we obtain @xmath170 which , after some manipulations gives eq.([eq : ebenergyii ] ) . the quasiparticle excitation energies for a- and c- states are calculated in a similar way as @xmath171 and @xmath172 , which gives eqs.([eq : eaenergyii ] ) and ( [ eq : ecenergyii ] ) respectively . the total energy of the system in this case is @xmath173.\ ] ] in this case , phases @xmath38 also obey the constraint ( [ eq : phasecondition ] ) . j. m. tranquada , b. j. sternlieb , j. d. axe , y. nakamura , and s. uchida , nature ( london ) * 375 * , 561 ( 1995 ) . k. yamada , c. h. lee , k. kurahashi , j. wada , s. wakimoto , s. ueki , h. kimura , y. endoh , s. hosoya , g. shirane , r. j. birgeneau , m. greven , m. a. kastner , and y. j. kim , phys . rev . b * 57 * , 6165 ( 1998 ) . j. e. hoffman , e. w. hudson , k. m. lang , v. madhavan , h. eisaki , s. uchida , j. c. davis , science * 295 * , 5554 , 466 - 469 ( 2002 ) . k. mcelroy , r. w. simmonds , j. e. hoffman , d .- h . lee , j. orenstein , h. eisaki , s. uchida , and j. c. davis , nature * 422 * , 592 - 596 ( 2003 ) . m. vershinin , shashank misra , s. ono , y. abe , yoichi ando , ali yazdani , science * 303 * , 5666 , 1995 - 1998 ( 2004 ) . t. hanaguri , c. lupien , y. kohsaka , d .- h . lee , m. azuma , m. takano , h. takagi , and j. c. davis , nature * 430 * , 1001 - 1005 ( 2004 ) . p. abbamonte , a. rusydi , s. smadici , g. d. gu , g. a. sawatzky , and d. l. feng , nature physics * 1 * , 155 - 158 ( 2005 ) . seiki komiya , han - dong chen , shou - cheng zhang , and yoichi ando , phys . 94 * , 207004 ( 2005 ) . k. mcelroy , d .- h . lee , j. e. hoffman , k. m. lang , j. lee , e. w. hudson , h. eisaki , s. uchida , and j. c. davis , phys . lett . * 94 * , 197005 ( 2005 ) . w. d. wise , m. c. boyer , kamalesh chatterjee , takeshi kondo , t. takeuchi , h. ikuta , yayu wang , and e. w. hudson , nature physics * 4 * , 696 - 699 ( 2008 ) . edurado h. da silva neto , pegor aynajian , alex frano , riccardo comin , enrico schierle , eugen weschke , andrs gyenis , jinsheng wen , john schneeloch , zhijun xu , shimpei ono , gends gu , mathieu le tacon , ali yazdani , science * 343 * , 6169 , 393 - 396 ( 2014 ) . r. comin , r. sutarto , f. he , e. h. da silva neto , l. chauviere , a. fra , r. liang , w. n. hardy , d. a. bonn , y. yoshida , h. eisaki , a. j. achkar , d. g. hawthorn , b. keimer , g. a. sawatzky , and a. damascelli , nature materials * 14 * , 796 - 800 ( 2015 ) . j. m. tranquada , n. ichikawa , k. kakurai , and s. uchida , j. phys . solids * 60 * , 8 - 9 , 1019 - 1023 ( 1999 ) . k. mitsen and o. m. ivanenko , phys . usp . * 47 * , 493 ( 2004 ) . john a. robertson , steven a. kivelson , eduardo fradkin , alan c. fang , and aharon kapitulnik , phys . b * 74 * , 134507 ( 2006 ) . j. m. tranquada , neutron scattering studies of antiferromagnetic correlations in cuprates , in _ handbook of high - temperature superconductivity _ , editors : j. r. schrieffer and j. s. brooks , springer , ( 2006 ) . b. v. fine , phys . b. * 75 * , 060504(r ) ( 2007 ) . b. v. fine , j. supercond . * 24 * , 1207 ( 2011 ) . b. v. fine , j. supercond . * 26 * , 2621 - 2626 ( 2013 ) . g. seibold , phys . b * 58 * , 15520 ( 1998 ) . m. berciu and s. john , phys . b * 59 * , 15143 ( 1999 ) . c. timm and k. h. bennemann , phys . rev . lett . * 84 * , 4994 ( 2000 ) . h. koizumi , j. phys . 77 * , 034712 ( 2008 ) . j. a. wilson , j. phys . matter * 21 * , 245702 ( 2009 ) . m azzouz , b w ramakko , and g presenza - pitman , j. phys . matter * 22 * , 345605 ( 2010 ) . q. li , m. hcker , g. d. gu , a. m. tsvelik , and j. m. tranquada , phys . * 99 * , 067001 ( 2007 ) . e. berg , e. fradkin , e .- a . kim , s. a. kivelson , v. oganesyan , j. m. tranquada , and s. c. zhang , phys . . lett . * 99 * , 127003 ( 2007 ) . b. v. fine , phys . * 94 * , 157005 ( 2005 ) . j. m. tranquada , g. d. gu , m. hcker , q. jie , h .- j . kang , r. klingeler , q. li , n. tristan , j. s. wen , g. y. xu , z. j. xu , j. zhou , and m. v. zimmermann , phys . b * 78 * , 174529 ( 2008 ) . b. v. fine and t. egami , phys . rev . b * 77 * , 014519 ( 2008 ) . t. egami , b. v. fine , d. parshall , a. subedi and d. j. singh , adv . matter phys . * 2010 * , 164916 ( 2010 ) . j. ranninger and t. domanski , phys . b * 81 * , 014514 ( 2010 ) . g. pawlowski , r. micnas , and s. robaszkiewicz , phys . b * 81 * , 064514 ( 2010 ) . s. i. vedeneev , a. g. m. jansen , p. samuely , v. a. stepanov , a. a. tsvetkov , and p. wyder , phys . b * 49 * , 9823 ( 1994 ) . n. miyakawa , p. guptasarma , j. f. zasadzinski , d. g. hinks , and k. e. gray , phys . lett . * 80 * , 157 ( 1998 ) . akimenko , r. aoki , h. murakami , v.a . gudimenko , physica c ( amsterdam ) * 319 * , 59 - 72 ( 1999 ) . minoru suzuki , takao watanabe , and azusa matsuda , phys . 82 * , 5361 ( 1999 ) . minoru suzuki and takao watanabe , phys . * 85 * , 4787 ( 2000 ) . v. m. krasnov , a. yurgens , d. winkler , p. delsing , and t. claeson , phys . * 84 * , 5860 ( 2000 ) . a. yurgens , d. winkler , t. claeson , s. ono , and yoichi ando , phys . . lett . * 90 * , 147005 ( 2003 ) . yoshiharu yamada , kenkichi anagawa , takasada shibauchi , takenori fujii , takao watanabe , azusa matsuda , and minoru suzuki , phys . b * 68 * , 054533 ( 2003 ) . v. m. krasnov , physica c ( amsterdam ) * 372 - 376 * , 103 ( 2002 ) . v. m. krasnov , phys . b * 65 * , 140504 ( 2002 ) . v. m. krasnov , m. sandberg , and i. zogaj , phys . 94 * , 077003 ( 2005 ) . a. yurgens , d. winkler , t. claeson , s. ono , and yoichi ando , phys . . lett . * 92 * , 259702 ( 2004 ) . v. n. zavaritsky , phys . 92 * , 259701 ( 2004 ) . v. n. zavaritsky , physica c ( amsterdam ) * 404 * , 440 ( 2004 )

we introduce a microscopic model aimed at describing superconductivity that can possibly exist in the background of a magnetic texture called spin - vortex checkerboard " . this texture was proposed previously as a possible alternative to stripes to interpret the experimental phenomenology of spin and charge modulations in 1/8-doped lanthanum cuprates . the model involves two kinds of interacting fermionic excitations residing in spin - rich and spin - poor regions of the modulated structure . it is a generalization of another model developed earlier for the so - called grid checkerboard " . we present the mean - field solution of the model , from which we obtain model s predictions for the temperature evolution of the superconducting gap , compare these predictions with available experiments on high-@xmath0 cuprate superconductors and find a good overall agreement .

1703.09979

the purpose of this paper is to study some aspects of the relative motion of free test particles in the gravitational field of a rotating astronomical source . imagine , for the sake of concreteness , a free mass @xmath2 in orbit about the central source of mass @xmath3 and angular momentum @xmath4 , where @xmath2 , @xmath5 and @xmath4 are constants . let @xmath6 be the proper time along the orbit of mass @xmath2 and @xmath7 be a local orthonormal tetrad system that is parallel transported along this path . here @xmath8 is the four - velocity vector of @xmath2 and @xmath9 , @xmath10 , are unit spacelike gyro directions that determine the local spatial frame of @xmath2 . we employ units such that @xmath11 throughout ; moreover , the signature of the metric is @xmath12 in our convention . suppose that at @xmath13 a probe is launched from @xmath2 and for some time @xmath14 the motion of the probe is closely monitored by observers comoving with @xmath2 . to study relative motion in an invariant manner within the context of general relativity , the quasi - inertial fermi normal coordinate system @xcite is indispensable as it most closely corresponds to actual observations . let @xmath15 be the fermi coordinates of the probe at an event @xmath16 , then there exists a unique spacelike geodesic of length @xmath17 that connects @xmath16 orthogonally to the path of @xmath2 at an event @xmath18 with proper time @xmath6 such that @xmath19 and @xmath20 , where @xmath21 is the unit vector at @xmath18 that is tangent to the spacelike geodesic at @xmath18 and orthogonal to the worldline of @xmath2 . the metric in fermi coordinates is given by @xmath22 to second order in the distance away from @xmath2 that permanently lies at the spatial origin of the fermi coordinates by construction and has fermi coordinates @xmath23 . here @xmath24 is defined by @xmath25 which is the projection of the riemann curvature tensor on the tetrad frame @xmath26 along the reference worldline . the theoretical equation of motion of the probe relative to mass @xmath2 can be described by the reduced geodesic equation @xcite @xmath27 where @xmath28 is the fermi velocity of the probe relative to @xmath2 . the four - velocity of the probe is given by @xmath29 , where @xmath30 can be determined from @xmath31 since @xmath32 is a timelike unit vector . the fermi coordinates are geodesic coordinates based on an orthonormal tetrad frame @xmath33 that is parallel propagated along the orbit of @xmath2 . they are admissible within a cylindrical region of radius @xmath34 along the worldline of @xmath2 ; that is , @xmath35 , where @xmath36 is a certain radius of curvature of spacetime . the nature of @xmath36 has been discussed in detail in our recent paper on explicit fermi coordinates @xcite . for the sake of simplicity , we limit our considerations to @xmath37 . a general feature of the reduced geodesic is that it contains a critical speed given by @xmath38 , especially in the case of one - dimensional motion . the notion of critical speed in gravitational motion has been recently reviewed in ref . we are particularly interested in the role that the critical speed @xmath39 plays in the motion of the probe ; therefore the probe can be launched from @xmath2 at any initial speed @xmath40 , but for simplicity we concentrate on its motion only when it is relatively close to the reference particle @xmath2 . to first order in @xmath41 is the generalized jacobi equation @xcite @xmath42 which describes the timelike motion of the probe when @xmath43 for @xmath44 , we can drop all velocity - dependent terms in equations and ; then , the inequality in is satisfied and reduces to the jacobi equation @xmath45 suppose that the worldline of the probe can be determined in the fermi coordinate system on the basis of equations and ; then , it is in principle possible to find the representation of this worldline in the background coordinate system using the explicit coordinate transformation between the two systems of coordinates . in practice , however , it is not possible in general to find explicit expressions for fermi coordinates in terms of the background coordinates and therefore one has to resort to approximation schemes ; see ref . @xcite for a general discussion of this problem . nevertheless , it is important to recognize that once @xmath46 is determined from the generalized jacobi equation ( or the jacobi equation ) , the worldline of the probe is completely characterized . that is , if in the background coordinate system the worldline of @xmath2 is given by @xmath47 , then the worldline of the probe is @xmath48 , where @xmath49 and we have replaced the fermi temporal coordinate @xmath50 by @xmath6 in the argument of @xmath51 . this relation follows from the definition of fermi coordinates and the fact that @xmath52 for @xmath53 , since the ( generalized ) jacobi equation is linear in the distance away from the reference trajectory . alternatively , @xmath48 may be viewed as a perturbed geodesic orbit . depending upon whether @xmath51 is determined from the jacobi or the generalized jacobi equation , the results of this approach may be interpreted respectively in terms of the linear or nonlinear perturbations of geodesic orbits in the background gravitational field . regarding the source of the gravitational field and the reference trajectory , we assume that @xmath2 is on a stable equatorial circular geodesic orbit about a kerr source . this choice for the orbit to some extent complements our previous work on the motion of astrophysical jets @xcite , where the reference trajectory is a radial geodesic along the rotation axis of a kerr source . the geodesics of kerr spacetime have been discussed by a number of authors ; see , for instance , ref . @xcite and references therein . the characteristics of the source and the reference trajectory will be described in detail in the next section . various aspects of tidal dynamics in black - hole spacetimes have been discussed in a number of papers ; see , for instance @xcite and the references cited therein . furthermore , we treat free - particle orbits for the sake of simplicity ; a corresponding discussion of black - hole accretion disks @xcite is beyond the scope of this investigation . section [ s3 ] is devoted to the solution of the jacobi equation . we show in section [ s4 ] that a rotation introduced in section [ s3 ] leads to a complete transformation of the generalized jacobi equation to an autonomous form . this equation is then analyzed in detail . finally , section [ s5 ] contains a brief discussion of our results . the spacetime region of interest is the exterior kerr domain with the metric @xmath54 where @xmath55 is the specific angular momentum of the source . here @xmath56 are the standard boyer - lindquist coordinates and @xmath57 the reference trajectory is assumed to be a stable circular orbit of fixed radius @xmath58 in the equatorial plane @xmath59 ; in terms of @xmath58 , such orbits exist from infinity all the way down to the last stable circular orbits given by @xmath60 throughout this paper , the upper ( lower ) sign refers to orbits where @xmath2 rotates in the same ( opposite ) sense as the source . for @xmath58 less than the solution of , there are unstable circular orbits that end at the null circular orbits given by @xmath61 it is useful to define the keplerian frequency for the orbits under consideration here as @xmath62 let us note that in the orbital equations and , a prograde orbit becomes retrograde and vice versa when @xmath63 ; this circumstance explains why the combination @xmath64 usually appears in orbital equations . we assume that @xmath65 throughout ; therefore , the sign of @xmath66 indicates the sense of the orbit . it follows from the geodesic equation that along the reference worldline @xmath67 where @xmath68 is such that the null circular orbits are solutions of @xmath69 and we have assumed in that @xmath70 at @xmath71 . the constants of the motion for the reference worldline are the specific energy @xmath72 and orbital angular momentum @xmath73 , associated respectively with the timelike and azimuthal killing vectors @xmath74 and @xmath75 , and are given by @xmath76 to determine the orthonormal tetrad @xmath77 , let us first consider the tetrad @xmath78 carried by the fundamental static observers in the exterior kerr spacetime . in the equatorial plane and in terms of @xmath79 , the natural orthonormal tetrad of these observers along the coordinate directions is given by @xmath80 where @xmath81 next , we subject this tetrad to a lorentz boost with speed @xmath82 , @xmath83 , such that @xmath84 is the unit vector tangent to the worldline of the reference particle @xmath2 . the new orthonormal tetrad along the orbit of @xmath2 is @xmath85 , \quad \tilde{\lambda}^\mu _ { ( 1 ) } = \lambda^\mu _ { \hspace{\myspace } ( 1)},\\ \label{eq19}\tilde{\lambda}^\mu_{\hspace{\myspace}(2 ) } = \lambda^\mu _ { \hspace{\myspace}(2 ) } , \quad \tilde{\lambda}^\mu_{\hspace{\myspace } ( 3 ) } = \tilde{\gamma } [ \lambda^\mu_{\hspace{\myspace } ( 3 ) } + \tilde{\beta } \lambda^\mu_{\hspace{\myspace}(0 ) } ] , \end{aligned}\ ] ] where @xmath86 is the lorentz factor corresponding to @xmath82 . we find from @xmath87 that the lorentz pair @xmath88 is given by @xmath89 clearly , @xmath90 is positive ( negative ) for prograde ( retrograde ) orbits ; moreover , @xmath91 diverges at the null orbits ( @xmath69 ) . the spatial triad of @xmath92 is along the spherical polar coordinate directions , which are the radial , normal and tangential directions with respect to the orbit , and therefore needs to be rotated back such that the resulting tetrad would then be parallel propagated along the orbit as illustrated in figure [ fig:1 ] . thus @xmath93 it follows from the vanishing of the covariant derivative of @xmath94 along the orbit that @xmath95 where we have chosen the integration constant such that @xmath96 at @xmath13 . note that @xmath97 , which , to first order in @xmath98 and @xmath99 can be written as @xmath100 that is , the difference between these angles is due to a combination of geodetic and gravitomagnetic precessions . the final result for the tetrad frame in @xmath101 coordinates is then @xmath102 finally , we need to compute the projection of the riemann tensor along the orbit on this tetrad frame as in equation . the result may be expressed as a @xmath103 matrix @xmath104 , where @xmath105 and @xmath106 are indices that belong to the set @xmath107 . this matrix has the form @xmath108,\ ] ] where @xmath109 and @xmath110 are @xmath111 symmetric and traceless matrices containing respectively the electric and magnetic components of the riemann tensor . it turns out that in the case under consideration @xmath112,\qquad \mathcal{h } = \kappa \left [ \begin{array}{ccc } 0 & h & 0\\ h & 0 & h ' \\ 0 & h ' & 0\end{array}\right ] , \ ] ] where @xmath113 the proportionality @xmath114 is expected on the basis of the correspondence with newtonian tides . in , @xmath115 is constant and is given by @xmath116 , while @xmath117 here @xmath118 is a new lorentz pair , i.e. @xmath119 , given by @xmath120 which reduces to @xmath121 in the schwarzschild limit @xmath122 ; moreover , @xmath123 diverges at the null orbits ( @xmath124 . let us note that for the orbits under consideration and @xmath125 , @xmath126 $ ] ; that is , @xmath127 is positive ( negative ) for a prograde ( retrograde ) orbit and far from the source ( @xmath128 ) , @xmath129 , so that @xmath130 as @xmath131 , then @xmath132 monotonically increases from zero to @xmath133 at the last stable circular orbits . furthermore , for @xmath134 , @xmath127 is always negative for retrograde orbits , but it is not always positive for prograde orbits ; in fact , @xmath127 vanishes for a prograde orbit with radius @xmath135 . these curvature components can now be used in equations and to study the motion of the probe relative to @xmath2 . it should be mentioned that our results for the tidal tensor @xmath109 are consistent with the work of marck @xcite . let us now turn to the solution of the jacobi equation . for simplicity , we use instead of the fermi coordinates @xmath136 , the _ dimensionless _ fermi coordinates @xmath137 , @xmath138 where @xmath139 . the jacobi equation then takes the form @xmath140 = s \left [ \begin{array}{c } x\\ z\end{array}\right ] , \qquad s=\left [ \begin{array}{cc } 3\gamma^2 \cos ^2\eta-1 & 3\gamma^2 \sin \eta \cos \eta \\ 3\gamma^2 \sin \eta \cos \eta & 3\gamma^2 \sin ^2\eta -1 \end{array}\right],\\ \label{eq36 } \frac{d^2y}{d\eta^2 } + \sigma^2 y=0,\qquad \sigma = \sqrt{3\gamma^2 - 2}.\end{aligned}\ ] ] thus normal to the equatorial plane , we have a simple harmonic motion of frequency @xmath141 , where @xmath142 ranges from 1 to @xmath143 for @xmath125 ; that is , it increases monotonically from unity at @xmath144 to @xmath143 at the last stable circular orbits . the equations of relative motion in the equatorial plane can be put into an autonomous form by a rotation to the radial and tangential directions ( cf . ) . that is , let @xmath145 = r\left [ \begin{array}{c } x \\ z\end{array}\right ] , \qquad r=\left [ \begin{array}{cc } \cos \eta & \sin \eta \\ -\sin \eta & \cos \eta \end{array } \right].\ ] ] then , equation reduces to the autonomous system @xmath146 where an overdot denotes differentiation with respect to @xmath147 . assuming that at @xmath148 , @xmath149 and the initial velocity of the probe @xmath150 has components @xmath151 in the local @xmath152 system , we find @xmath153,\\ \label{eq41 } y=\frac{1}{\sigma } v_0 \cos \vartheta \sin \sigma \eta , \\ \label{eq42 } \zeta = - \frac{v_0\sin \vartheta}{\lambda^2 } [ ( 3\gamma^2\eta -\frac{4}{\lambda } \sin \lambda \eta ) \sin \varphi + 2 ( 1-\cos \lambda \eta ) \cos \varphi],\end{aligned}\ ] ] where @xmath154 . we note that @xmath155 , since @xmath156 so that for @xmath125 , @xmath157 ranges from zero to unity ; that is , it decreases monotonically from unity at @xmath144 to zero at the last stable circular orbits . if the reference trajectory is one of the last stable circular orbits , then @xmath158 , @xmath159 is given by with @xmath160 and @xmath161.\end{aligned}\ ] ] recalling the restrictions @xmath37 and @xmath44 , which in this case translate to @xmath162 and @xmath163 , we find that the validity of equations , and is limited in time due to the presence of secular terms . it is interesting to note that terms that appear in the solution of the jacobi equation in equations and with proper frequency @xmath164 have frequency @xmath165 with respect to the coordinate time @xmath166 ; to lowest order in @xmath167 and with @xmath168 , the deviation of this frequency from the keplerian frequency corresponds to the einstein precession frequency for an orbit of vanishing eccentricity , while the first - order correction in @xmath169 corresponds to the de sitter - lense - thirring precession frequency @xcite . finally , a remark is in order here regarding the boundary condition that @xmath170 at @xmath171 , which corresponds to our assumption that the probe is initially launched from @xmath2 . in fact , it is only necessary that the probe be near the reference particle @xmath2 , so that @xmath41 must be initially very small compared to @xmath34 ; for the sake of simplicity , we choose @xmath172 throughout this paper . this means that in terms of @xmath173 , we always assume that @xmath174 the reduction of the jacobi equation to the autonomous system is basically due to the fact that @xmath175 , where @xmath176 is a constant matrix . system then takes the form @xmath177 = d\left [ \begin{array}{c } \xi\\ \zeta \end{array}\right ] , \ ] ] which is equivalent to system . the transformation from @xmath178 to @xmath179 involves a rotation from a local inertial frame to rotating axes ; therefore , in equation we note the presence of coriolis and centripetal terms on the left - hand side of this system . moreover , this transformation is related to hill s contributions to the classical three - body problem @xcite ; indeed , system in the newtonian limit @xmath180 is equivalent to a limiting form of the hill system discussed in ref . @xcite , namely , system of ref . @xcite with @xmath181 . it can be shown , after much algebra , that under the same rotation as in equation , the nonlinear generalized jacobi equation can be rendered autonomous as well . this remarkable fact is due to the special symmetries of the kerr spacetime @xcite . introducing @xmath182 and @xmath183 , where @xmath184 = r\left [ \begin{array}{c } \dot{x}\\ \dot{z}\end{array}\right],\end{aligned}\ ] ] we find that @xmath185 , @xmath186 , and @xmath187 + 6\gamma^2\beta \xi \delta_\zeta\nonumber \\ - 2 \big ( \gamma^2\beta \delta^2_\xi -\frac{1}{3 } \sigma^2 \delta_\zeta \big ) ( \xi \delta_\zeta -\zeta \delta_\xi ) + 2 y\dot{y } \delta_\xi \big ( \gamma^2\beta \delta_\zeta + \sigma^2-\frac{1}{3}\big ) \nonumber\\ -2\dot{y}^2 \big ( \gamma^2\beta \zeta \delta_\xi -\frac{1}{3 } \xi \big ) , \label{eq48}\\ \fl \ddot{y } + \sigma^2 ( 1 - 2\dot{y}^2)y = \frac{2}{3 } y[\delta^2 _ \xi -(\sigma^2 + 1 ) \delta^2_\zeta -9\gamma^2 \beta \delta_\zeta ] -2\dot{y } \big [ \gamma^2\beta \delta_\xi ( \xi \delta_\zeta -\zeta \delta_\xi ) \nonumber\\ + \big ( 3\gamma^2-\frac{2}{3}\big ) \xi \delta_\xi -\big(\gamma^2+\frac{2}{3}\big ) \zeta \delta_\zeta \big ] + 2\gamma^2\beta \dot{y}^2(y\delta _ \zeta -\zeta \dot{y}),\label{eq49}\\ \fl\ddot{\zeta}+2\dot{\xi } = 2\delta_\zeta [ \zeta \delta_\zeta -(\sigma^2 + 1)\xi \delta_\xi ] -6\gamma^2\beta \xi \delta_\xi -2 \delta_\xi \big ( \gamma^2\beta \delta _ \zeta + \frac{1}{3 } \sigma^2\big ) ( \xi \delta_\zeta -\zeta \delta_\xi ) \nonumber\\ + \frac{2}{3 } ( \sigma^2 + 1)\dot y ( y\delta_\zeta -\zeta \dot{y})+2y\dot{y } [ \gamma^2\beta ( \delta^2_\zeta + 3 ) + \sigma^2\delta_\zeta ] -2\gamma^2\beta \dot{y}^2\zeta \delta_\zeta .\label{eq50}\end{aligned}\ ] ] starting with the exact solution of the jacobi equation as the unperturbed solution , it is straightforward to develop a solution of equations via the standard perturbation expansion in terms of @xmath188 . however , the results of such a perturbation scheme , based on the small parameter @xmath189 , should be used only in conjunction with the higher - order tidal terms that is , terms that have been neglected in the expansions of the gravitational potentials in equations and would result in terms in the equation of motion that go beyond the linear order in @xmath41 . it follows from equations , , and that the trajectory of the probe in boyer - lindquist coordinates , @xmath190 , is given by @xmath191 using @xmath192 , @xmath193 and @xmath194 from the ( generalized ) jacobi equation and replacing @xmath195 by @xmath196 , we can determine the path of the probe in the standard kerr coordinate system . this is due to the fact that the generalized jacobi equation is linear in the distance @xmath197 away from the reference geodesic ; therefore , @xmath198 may be viewed as a generalized jacobi field that is defined along the reference geodesic . it turns out that for small - amplitude perturbations @xmath199 ) , the radial ( @xmath200 ) and vertical ( @xmath159 ) motions in general contain the basic proper epicyclic frequencies @xmath201 and @xmath202 , respectively . the autonomous system of equations naturally splits into equations for the vertical and equatorial motions . that is , if the probe is launched vertically in the @xmath159 direction , then its motion will be confined to the @xmath159 direction according to the generalized jacobi equation . similarly , if the probe is launched in the orbital @xmath203 plane , then the probe will remain in this plane throughout its motion . if the probe is launched in the purely vertical direction relative to the reference orbit , then @xmath204 for all @xmath147 and hence the equations of motion reduce to @xmath205 the vertical motion is simply uniform for @xmath206 ; that is , equation exhibits the critical speed @xmath207 beyond which the character of the motion is opposite to the low - speed limit that agrees with newtonian expectations . if the probe is launched with @xmath208 , then it will begin to decelerate in agreement with newtonian gravity . however , for @xmath209 , the probe will accelerate , as treated in detail in @xcite . in fact , the vertical motion can be described exactly @xcite , since equation implies that @xmath210 whose solution can be determined by quadrature . this solution represents timelike geodesic motion if the following condition is satisfied @xmath211 for @xmath212 , condition is always satisfied and it follows from that the motion of @xmath159 is periodic and confined to the interval @xmath213 $ ] , where @xmath214 on the other hand , for @xmath215 the particle is accelerated to ( almost ) the local speed of light . in this way , the particle gains enormous tidal energy . according to condition , the range of @xmath159 over which this happens monotonically shrinks to zero as @xmath189 approaches unity . the question of the admissibility of fermi coordinates in such a case is a difficult one ; an unsuccessful attempt to answer such a question is contained in our recent work @xcite . it is essential for astrophysical applications to connect these local results to what distant observers would measure @xcite . indeed , the gravitational tidal energy of particles within the fermi coordinate system can be transferred to the outside world ( i.e. beyond the fermi system ) through collisions with other particles or emission of radiation . in our previous work involving tidal dynamics about a reference escape trajectory along the axis of rotation of the central source @xcite , we found that for relative speeds above the critical speed @xmath216 , tidal _ deceleration _ occurs in a cone of angle @xmath217 measured from the rotation axis , where @xmath217 is given by @xmath218 and corresponds to an angle of about @xmath219 , while tidal _ acceleration _ occurs outside this cone , i.e. within the complement of the critical - velocity cone involving latitudes from @xmath220 to @xmath221 , where @xmath222 and corresponds to an angle of about @xmath223 . based on these previous results , we expect that in the present situation the phenomenon of tidal acceleration for speeds above the critical speed may not be restricted to the vertical axis alone . to investigate this issue in the case under consideration , we have integrated the full system of equations with initial conditions . our results indicate that vertical acceleration either parallel or antiparallel to the kerr rotation axis occurs when the probe is launched with @xmath224 in a direction @xmath225 , where for a given @xmath226 , @xmath227 is restricted to a certain interval , namely , @xmath228 . to clarify the situation analytically , let us first note that if @xmath229 is a solution of equations , then so is @xmath230 . in particular , equation is clearly invariant under the parity transformation @xmath231 ; therefore , vertical motion is expected to occur symmetrically with respect to the equatorial plane . for this reason , we limit our considerations to @xmath232 , which corresponds to motion antiparallel to the kerr axis in our convention ( see figure [ fig:1 ] ) . furthermore , let us note that with initial conditions , vertical acceleration vanishes initially , i.e. @xmath233 at @xmath234 ; therefore , it proves useful to compute @xmath235 using equation . it turns out that @xmath236 where @xmath237 is a quadratic function of @xmath238 given by @xmath239 here @xmath240 , @xmath241 and @xmath242 are defined by @xmath243 for @xmath244 , @xmath235 is positive for @xmath224 ; in fact , this is the case for an extended domain about the kerr axis . the boundary of this domain is characterized by @xmath245 , i.e. @xmath246 given by @xmath247 and @xmath248 ( i.e. @xmath249 ) , where @xmath235 vanishes , as illustrated in figure [ fig:1.5 ] , where @xmath250 and @xmath251 . let us recall that for a prograde reference orbit of radius @xmath58 about a kerr black hole , @xmath127 increases monotonically from zero at @xmath252 to @xmath253 at the last stable circular orbit . for a fixed @xmath254 $ ] and @xmath209 , the tidal - acceleration domain expands as @xmath255 . on the other hand , for a fixed @xmath256 , the acceleration domain in figure [ fig:1.5 ] remains practically the same as @xmath257 . for @xmath130 , however , the shape of the domain changes significantly . in fact , for @xmath258 , it follows from equation that @xmath259 , where @xmath260 defined by @xmath261 is a positive constant such that @xmath262 . therefore , the acceleration domain in this case is within the intersection of the vertical strip defined by @xmath263 , where @xmath264 and @xmath265 , and the circle of unit radius centered at the origin of @xmath266 coordinates . though we have considered a simple circular equatorial orbit as the reference trajectory in this paper , it is not difficult to imagine that the tidal acceleration process described above within the context of the physics of an accretion disk about the source could possibly contribute to the formation of astrophysical jets . moreover , tidal acceleration occurs symmetrically with respect to the equatorial plane of the kerr source . this feature would be consistent with the observed occurrence of double jets , namely , a pair of relativistic outflows in opposite directions along the rotation axis of the central source . if @xmath267 for all @xmath147 , then the motion is confined to the @xmath268 plane and represents a timelike geodesic if @xmath269 a solution of the generalized jacobi equation ( gje ) in the @xmath270 plane is characterized by the parameters @xmath127 , @xmath189 and @xmath271 . for @xmath189 extremely small compared to unity , the solution of the gje is given essentially by the corresponding solution of the jacobi equation as in the previous section . in general , as @xmath189 slowly increases toward unity , the influence of the nonlinear terms in the gje can not be ignored . in fact , since @xmath272 in the jacobi case contains a secular term , namely , @xmath273 with @xmath274 , the solution of the gje in general grows rapidly and leaves the admissible region of the fermi coordinate system . this difficulty can be avoided , however , in the case of the purely radial variations with @xmath275 or @xmath276 ; in either case , the jacobi equation has periodic solutions corresponding to the stability of the circular reference orbit under small radial perturbations . the solution of the gje may then contain quasi - periodic oscillations exhibiting a complex beat phenomenon involving several frequencies as demonstrated in figure [ fig:2 ] . however , such oscillations appear to occur only for @xmath277 , in which case equations are generally not adequate physically , since higher - order tidal terms of comparable magnitude have been neglected in our analysis . even if a quasi - periodic oscillation of the type illustrated in figure [ fig:2 ] survives the inclusion of relevant higher - order tidal terms , the amplitude of the effect may not be large enough to be of any physical significance in connection with the observed quasi - periodic oscillations ( qpos ) @xcite . we have studied the generalized jacobi equation for a circular reference geodesic orbit in the equatorial plane of the exterior kerr spacetime . this equation has been reduced to an autonomous system ; the corresponding tidal dynamics can then be naturally divided into vertical and equatorial motions relative to the circular orbit . in connection with vertical motion , we have clarified the role of the critical speed @xmath207 in the character of the solutions of the generalized jacobi equation . the general approach developed in this work may be of interest in connection with relativistic physics inside a space station , the tracking of artificial earth satellites as well as satellite - to - satellite doppler tracking . moreover , in connection with the relativistic astrophysics of accretion disks around ( rotating ) astronomical sources , our results may be relevant for the gravitational aspects of the complex plasma physics that would be involved in the formation of high - energy jets @xcite . that is , the vertical acceleration phenomenon explored in this work may be related to how relativistic jets get started above and below an accretion disk around a kerr black hole . once the flows are magnetically confined to regions near the axis of rotation of the central source , the results of previous investigations @xcite involving tidal dynamics of ultrarelativistic particles would have to be taken into account for the parallel and antiparallel flows along the kerr rotation axis . xx synge j l 1960 _ relativity : the general theory _ ( amsterdam : north - holland ) chicone c and mashhoon b 2002 _ class . quantum grav . _ * 4231 chicone c and mashhoon b 2005 _ preprint _ gr - qc/0511129 mashhoon b 2005 _ int . j. mod . * 14 * 2025 chicone c and mashhoon b 2005 _ ann . ( leipzig ) _ * 14 * 290 + chicone c and mashhoon b 2004 _ class . quantum grav . _ * 21 * l139 + chicone c and mashhoon b 2005 _ class . quantum grav . _ * 22 * 195 + chicone c and mashhoon b 2005 _ astron . _ & _ astrophys . _ * 437 * l39 + chicone c and mashhoon b 2005 _ ann . ( leipzig ) _ * 14 * 751 chicone c , mashhoon b and punsly b ( 2004 ) _ int . j. mod . phys . d _ * 13 * 945 + chicone c , mashhoon b and punsly b 2005 _ phys . a _ * 343 * 1 chandrasekhar s 1992 _ the mathematical theory of black holes _ ( new york : oxford university press ) mashhoon b and theiss d s 1982 _ phys . lett . _ * 49 * 1542 marck j - a 1983 _ proc . london a _ * 385 * 431 bini d , geralico a and jantzen r t 2005 _ class . quantum grav . _ * 22 * 4729 ishii m , shibata m and mino y 2005 _ phys . d _ * 71 * 044017 kojima y and takami k 2006 _ class . quantum grav . _ * 23 * 609 kato s , fukue j and mineshige s 1998 _ black - hole accretion disks _ ( kyoto : kyoto university press ) aliev a n , galtsov d v and petukhov v i 1986 _ astrophys . _ & _ space sci . _ * 124 * 137 aliev a n and galtsov d v 1988 _ sov . * 14 * 48 chicone c , mashhoon b and retzloff d g 1996 _ ann . inst . h. poincar _ * 64 * 87 chicone c , mashhoon b and retzloff d g 1999 _ helv . phys . acta _ * 72 * 123 carter b 1973 _ black holes _ edited by c dewitt and b s dewitt ( new york : gordon and breach ) strohmayer t e 2001 _ astrophys . j. _ * 552 * l49 + strohmayer t e 2001 _ astrophys . j. _ * 554 * l169 + wagoner r v , silbergleit a s and ortega - rodriguez m 2001 _ astrophys . j. _ * 559 * l25 punsly b 2001 _ black hole gravitohydromagnetics _ ( new york : springer - verlag )

the motion of free nearby test particles relative to a stable equatorial circular geodesic orbit about a kerr source is investigated . it is shown that the nonlinear generalized jacobi equation can be transformed in this case to an autonomous form . tidal dynamics beyond the critical speed @xmath0 is studied . we show , in particular , that a free test particle vertically launched from the circular orbit parallel or antiparallel to the kerr rotation axis is tidally accelerated if its initial relative speed exceeds @xmath1 . possible applications of our results to high - energy astrophysics are briefly mentioned .

gr-qc0602071

the new scalar particle with mass @xmath10@xmath11 gev ( atlas ) @xcite , @xmath12 gev ( cms ) @xcite , recently found at the lhc , has properties consistent with those of the long - awaited higgs boson of the standard model ( sm ) @xcite . this discovery implies that the landau pole in the higgs self - interaction is well above the quantum gravity scale @xmath13 gev ( see , e.g. ref . moreover , within the sm , the vacuum is stable , or metastable with a lifetime exceeding that of the universe by many orders of magnitude @xcite . without the addition of any further new particles , the sm is therefore an entirely self - consistent , weakly - coupled , effective field theory all the way up to the planck scale ( see refs . @xcite for a recent discussion ) . nevertheless , it is clear that the sm is incomplete . besides a number of fine - tuning problems ( such as the hierarchy and strong cp problems ) , the sm is in conflict with the observations of non - zero neutrino masses , the excess of matter over antimatter in the universe , and the presence of non - baryonic dark matter . the most economical theory that can account simultaneously for neutrino masses and oscillations , baryogenesis , and dark matter , is the neutrino minimal standard model ( @xmath14msm ) @xcite . it predicts the existence of three heavy neutral leptons ( hnl ) and provides a guideline for the required experimental sensitivity @xcite . the search for these hnls is the focus of the present proposal . in addition to hnls , the experiment will be sensitive to many other types of physics models that produce weakly interacting exotic particles with a subsequent decay inside the detector volume , see e.g. refs . longer lifetimes and smaller couplings would be accessible compared to analogous searches performed previously by the charm experiment @xcite . in the remainder of this document the theoretical motivation for hnl searches is presented in section [ sec : theo ] and the limits from previous experimental searches are then detailed in section [ sec : exp_status ] . the proposed experimental set - up is presented in section [ sec : exp ] and in section [ sec : bg ] the background sources are discussed , before the expected sensitivity is calculated in section [ sec : sens ] . the conclusions are presented in section [ sec : conc ] . in type - i seesaw models ( for a review see ref . @xcite ) the extension of the sm fermion sector by three right - handed ( majorana ) leptons , @xmath15 , where @xmath16 , makes the leptonic sector similar to the quark sector ( see fig . [ ferm ] ) . irrespective of their masses , these neutral leptons can explain the flavour oscillations of the active neutrinos . four different domains of hnl mass , @xmath17 , are usually considered : msm all fermions have both left- and right - handed components and masses below the fermi scale.,title="fig:",scaledwidth=48.0% ] msm all fermions have both left- and right - handed components and masses below the fermi scale.,title="fig:",scaledwidth=48.0% ] * models with hnls with @xmath18gev @xcite are motivated by grand unified theories . in such theories the observed baryon asymmetry of the universe originates in cp - violating decays of the hnls , which produce a lepton asymmetry @xcite . this asymmetry is then converted into a baryon asymmetry by sphalerons @xcite . the large mass of the hnls results in a fine - tuning problem for the higgs mass . a natural solution is provided by low energy supersymmetry but at present this is not supported by experimental evidence . theories with very heavy neutral leptons are unable to account for dark matter and can not be directly probed by experiments ; * models with @xmath19 gev ( for a review see ref . @xcite ) are motivated by a possible solution to the hierarchy problem at the electroweak scale ( see e.g. ref . the baryon asymmetry of the universe can be produced via resonant leptogenesis and sphalerons @xcite . as above , there is no candidate for dark matter particles . a portion of the parameter space can be accessed by direct searches at the atlas and cms experiments @xcite ; * models with masses of the hnls below the fermi scale and roughly of the order of the masses of the known quarks and leptons , are able to account for neutrino masses and oscillations and can also give rise to the baryon asymmetry of the universe and can provide dark matter@xcite ( for a review see ref . the phenomenology of gev - scale hnls was previously studied in refs . @xcite . owing to its relatively large mass , the dark matter candidate the @xmath20kev hnl , does not contribute to the number of relativistic neutrino species measured recently by the planck satellite @xcite ; * models with @xmath21 ev @xcite are motivated by the @xmath22@xmath23 deviations observed in short - baseline neutrino - oscillation experiments @xcite , reactor neutrino experiments @xcite and gallium solar neutrino experiments @xcite . such neutral leptons are usually referred to as sterile neutrinos . theories involving these sterile neutrinos can explain neither the baryon asymmetry of the universe nor dark matter . the gev - scale hnls of category ( 3 ) are able to solve all major problems of the sm and the search for such particles is the focus of the present proposal . the most general renormalisable lagrangian of all sm particles and three singlet ( with respect to the sm gauge group ) fermions , @xmath15 , is @xmath24 where @xmath25 , @xmath26 are the sm lepton doublets , @xmath27 is the superscript denoting the charge conjugation , @xmath28 , where @xmath29 is the sm higgs doublet , and @xmath30 are the relevant yukawa couplings . the last term is the majorana mass term , which is allowed as the @xmath15 carry no gauge charges . when the sm higgs field gains a non - zero vacuum expectation value , @xmath31gev , the yukawa term in eqn . results in mixing between the hnls and the sm neutrinos in the mass basis , the massive active - neutrino states mix with each other , as is required to explain neutrino oscillations . the model given by eqn . contains eighteen new parameters compared to the sm ( six cp - violating phases , six mixing angles , three dirac masses , and three majorana masses ) . five of these parameters ( related to three mixing angles of the active neutrinos and two mass differences ) have been determined by low - energy neutrino experiments @xcite . the large number of cp - violating phases opens the possibility of significantly larger cp violation than that seen in the quark sector . in particular , the baryon asymmetry of the universe can be explained by a wide range of parameters @xcite . in fact , hnls with any mass difference and mixing angle allowed by experimental constraints can produce the necessary baryon asymmetry in the universe ( bau ) . the most interesting variant of this model is the @xmath14msm . in this model , the lightest singlet fermion , @xmath32 , has a very weak mixing with the other leptons , playing no role in active - neutrino mass generation . the lightest singlet @xmath32 is then sufficiently stable to be a dark matter candidate . this particle could be detected by searching for a narrow line in the x - ray spectrum coming from radiative decays @xmath33 ( for a review see ref . the reduced number of parameters that arises from requiring that @xmath32 be a dark matter candidate results in the further requirement that @xmath34 be nearly degenerate in mass . this enables cp violation to be enhanced to the level required to explain the baryon asymmetry in the universe . the order of magnitude of the different parameters in eqn . can be understood by considering the typical value of the dirac mass term , @xmath35 . the scale of the active neutrino masses is given by the seesaw formula , @xmath36 for the hnl mass @xmath37gev , and @xmath38ev , the value of @xmath39 is in the region of @xmath40kev , and the yukawa couplings are of the order of @xmath41 . denoting the mixing angles between @xmath15 and active neutrinos of flavour @xmath42 by @xmath43 , the integral mixing angle , @xmath44 measures the overall strength of interactions between the @xmath15 and the active neutrinos . at a given mass , @xmath45 , a larger @xmath46 yields stronger interactions ( mixing ) of the singlet fermions and the sm leptons . in the early universe , large @xmath46 would result in the @xmath34 particles coming into equilibrium above the electroweak temperature and would therefore erase any baryon asymmetry in the universe . a small mixing between the neutral leptons and the active neutrinos would enable hnls to generate the observed baryon asymmetry of the universe and would also explain why these particles have not yet been observed in experiments . , from big bang nucleosynthesis ( black line : `` bbn '' ) , from the baryon asymmetry of the universe ( `` bau '' ) and from the seesaw mechanism ( blue solid lines : `` bau n '' and `` seesaw n '' refer to a normal mass - hierarchy of active neutrinos and `` bau i '' and `` seesaw i '' refer to an inverted mass - hierarchy ) . the allowed region of the parameter space is shown in white for the normal hierarchy case . the limits from direct experimental searches are outlined in fig . [ exp ] . figure taken from ref . @xcite.,scaledwidth=75.0% ] in the @xmath14msm the hnl - neutrino mixing gives rise to hnl ( @xmath34 ) production in weak decays of heavy mesons . the same mixing gives rise to the decay of the hnls to sm particles . the allowed mixing angles are small and the @xmath34 particles are much longer - lived ( by a factor @xmath47 ) than weakly decaying sm particles of similar mass ( see fig . [ taun ] ) . for hnl masses below the charm threshold , the most relevant production mechanisms are shown in fig . [ production - and - decays ] ( left ) . the requirement for mixing into active neutrinos at both production and decay results in signal yields which depend on the fourth power of the hnl - neutrino mixing , @xmath48 . potential two- and three - body decay modes of @xmath34 are shown in fig . [ production - and - decays ] ( right ) . for @xmath49 mesons , the typical branching fractions expected for the upper and lower limits of the @xmath14msm parameter space are at the level of @xcite @xmath50 the three - body leptonic decay branching fractions depend on the flavour pattern of hnl - to - active neutrino mixing : @xmath51 ( @xmath52 ) . among the two - body decays the most promising for searches are @xmath53 , @xmath54 , @xmath55 , @xmath56 . branching fractions to @xmath57 and @xmath58 are always below 2% . the @xmath59 final state is the cleanest signature experimentally and is the focus of the studies below . the @xmath60 and @xmath61 final states provide additional experimental signatures that extend the sensitivity and could be used to constrain additional parameter space . assuming a branching fraction @xmath62 and a factor @xmath63 from the lifetime , an experiment to detect @xmath34 would require more than @xmath64 @xmath49 mesons in order to fully explore the parameter space with @xmath65gev . preliminary studies of an experimental design were described in ref . the region of the lifetime - mass @xmath66 plane consistent with the cosmological constraints is shown in fig . [ taun ] . figure [ exp ] shows the allowed region in the @xmath67 plane , given the constraints from particle physics experiments . for all points in fig . [ exp ] below the line marked `` seesaw '' , the mixing of the hnl with active neutrinos becomes too weak to produce the observed pattern of neutrino flavour oscillations . cosmological considerations result in additional limits . if the hnls are required to provide a mechanism for baryogenesis , their coupling with matter should be sufficiently weak such that they lie below the upper line marked `` bau '' . a hnl with the parameters to the left of the `` bbn '' line would live sufficiently long in the early universe to result in an overproduction of primordial helium-4 in big bang nucleosynthesis @xcite . the regions excluded by the charm @xcite , cern ps191 @xcite and nutev @xcite experiments are also shown . limits by bebc @xcite and ccfr @xcite are not shown in fig . [ exp ] . a detailed discussion of the experimental constraints ( including also those from peak search experiments @xcite ) is presented in refs . @xcite . in the ( @xmath68 , @xmath17 ) plane from cosmological considerations and neutrino mixing together with limits from previous experimental searches ( the solid and dashed lines indicate the dependence of these regions on the pattern of hnl mixing with the electron , muon and tau - neutrino ) . figures taken from ref . . a normal mass hierarchy of the neutrinos is shown on the left and an inverted hierarchy on the right.,title="fig:",scaledwidth=48.0%]0.04 in the ( @xmath68 , @xmath17 ) plane from cosmological considerations and neutrino mixing together with limits from previous experimental searches ( the solid and dashed lines indicate the dependence of these regions on the pattern of hnl mixing with the electron , muon and tau - neutrino ) . figures taken from ref . . a normal mass hierarchy of the neutrinos is shown on the left and an inverted hierarchy on the right.,title="fig:",scaledwidth=48.0% ] the combined experimental and theoretical constraints imply that the hnls must have masses larger than approximately @xmath69mev . the domain of masses accessible by the present experiment , i.e. masses below those of charm mesons , naturally appears under the assumption that the observed hierarchy in the masses of the different generations of quarks and charged leptons is preserved in the majorana sector of the theory . the proposed experiment will use a 400gev proton beam on a fixed target to produce a large number of charm mesons . the hnls from charm meson decays have a significant polar angle with respect to the beam direction , @xmath70mrad on average , as shown in fig . [ pthnl ] . in order to maximise the geometric acceptance for a given transverse size of the detector , the detection volume must therefore be placed as close as possible to the target . the production of the charm mesons is accompanied by copious direct production of pions , kaons and short - lived light resonances . the subsequent decays of these particles would result in a large flux of muons and neutrinos . to minimise these decays , a combination of a target and a hadron absorber of a few metres length , both made of as dense a material as possible , is required . to reduce the detector occupancy and backgrounds induced by the residual muon flux , a muon shield is required downstream of the hadron absorber . the experimental set - up must therefore balance the opposing requirements of locating the detector as close as possible to the target and of accommodating a sufficiently long muon shield upstream of the fiducial volume of the detector to reduce muon - induced backgrounds . the detector must be able to reconstruct the final state particles from @xmath71 decays , will be used to indicate @xmath34 . ] identify muons , and determine the @xmath72 invariant mass and parent particle flight direction with sufficient resolution to reject backgrounds . to be sensitive to the decays @xmath73 , @xmath74 and @xmath75 in addition to the @xmath76 decay , a magnetic spectrometer , an electromagnetic calorimeter and a muon detector are mandatory . the background neutrino flux and the residual muon flux in the detector constitute crucial parameters in the design of the experiment . interactions of neutrinos inside the fiducial volume can mimic signal events , in particular via charged - current interactions of muon - neutrinos . this motivates evacuating the fiducial volume to a level where such background events are negligible ( see section [ sec : bg ] ) . interactions of neutrinos and muons in the material near the fiducial volume , can produce long - lived @xmath77 mesons , such as neutral kaons , which can decay in the detector fiducial volume and mimic signal events . to suppress neutrino - induced @xmath77 background events from the downstream end of the muon shield , the neutrino flux from light meson decays must be minimised at the source . this is achieved by the use of target and hadron absorber materials with the smallest possible interaction length . in addition , the muon shield must be sufficiently long to reduce muon - induced @xmath77 backgrounds to a level that is comparable to or smaller than the background from neutrinos . shielding against cosmic rays is not required and the detector could therefore be located in an open space . a description of the proposed beam line and detector design is given below . in simulated hnl decays with @xmath78 gev.,title="fig:",scaledwidth=49.0% ] in simulated hnl decays with @xmath78 gev.,title="fig:",scaledwidth=49.0% ] the sps beam energy , beam intensity and the flexibility of the time structure offer a perfectly suitable production facility for the proposed experiment . the recently completed cern neutrinos to gran sasso ( cngs ) programme made use of @xmath79gev protons from the sps over a nominal year with 200 days of operation , 55% machine availability , and 60% of the sps supercycle . this produced a total of @xmath80 protons on target per year . the experiment described here assumes the same fraction of beam time as cngs and the current sps performance . in particular , the preliminary investigation of the operational mode assumes minimal modification to the sps , maximises the use of the existing transfer lines and is compatible with the operation of both the lhc and the north area fixed target programmes as they are currently implemented . increased experimental reach could come as a by - product of a future sps upgrade from a proton spill intensity of about @xmath81 protons per pulse to @xmath82 protons per pulse . the design of the proposed experiment aims at taking advantage of such an upgrade but the experiment does not rely on it . from the general considerations given above , the experimental requirements favour the choice of a relatively long extraction to reduce the detector occupancy per unit time , and to allow a simple target design based on dense materials . at the same time , the extraction type should not affect significantly the sps cycle time as compared to cngs , and should respect the constraints on the activation levels in the extraction region and the risk of damage to the extraction septum . several options exist , all of which may be acceptable from the point of view of the experiment , but the optimal choice requires further study : * slow extraction with a spill length of @xmath83s . this would mean a @xmath84s sps spill cycle , and consequently a reduction of the number of protons on target by 10% ; * fast non - coherent resonant extraction of o(1ms ) ; * fast extraction similar to that of cngs of o(@xmath85s ) . the first two options involve extraction with the sps rf system off and the beams debunched . this produces a quasi - continuous spill that is compatible with a continuous detector readout , as described in section [ ssec : synergy ] . the first two extraction methods result in a lower detector occupancy . with lifetimes of @xmath86s , the dominant fraction of @xmath49 mesons decay before interacting in the target , regardless of the target material . however , the flux of secondary pions and kaons , which give rise to a substantial muon flux from decays in flight , can be greatly reduced by the use of a dense target material with a short interaction length . for example , the use of tungsten , rather than graphite , reduces the number of decaying pions and kaons by an order of magnitude . a 50 cm long tungsten target would suffice . however , the beam energy density must then be diluted in order to avoid destructive thermal shock waves in the target and to allow effective heat extraction . it may be possible to extract the deposited beam energy using water cooling . since the neutrinos produced in charm decays , which can mix with hnls , have in any case a relatively high transverse momentum ( see fig . [ pthnl ] ) , there is no requirement to have a small beam spot . the beam line design can then be driven purely by the technical requirements and constraints . for the same reason , the experiment does not impose stringent constraints on the optical parameters of the extracted beam . the dilution of the beam energy deposition mentioned above may therefore be obtained by allowing the transverse size of the beam to increase in a dedicated section of the primary beam line , and by using a combination of orthogonally deflecting kicker magnets to produce a lissajous sweep across the target , similar to the lhc beam dump . preliminary investigation shows that it should be possible to achieve a beam spot size of about @xmath87 mm on the target and a sweep diameter of several centimetres , with the only constraint being a beam divergence less than the natural divergence of the produced hnls . beam losses in the straight section leading to the target must be minimised and monitored to reduce the background neutrino flux . the use of a slow extraction would simplify the target design . to prevent damage to the target caused by possible failures in the beam line , the beam extraction would need to be interlocked with the magnet currents in the dedicated transfer line . the proton target will be required to withstand a beam power of @xmath88kw , corresponding to @xmath89 protons per @xmath90s at @xmath79gev . detailed thermo - mechanical studies will be required in order to produce a technical design . to stop the remaining secondary pions and kaons before they decay , a hadron absorber will immediately follow the target . by surrounding the target , the absorber will also stop pions and kaons at large angles which may otherwise produce muons that enter the fiducial volume due to large angle scattering . the absorber will also provide the first level of lateral radiation shielding , as well as absorbing the residual non - interacting protons ( @xmath91 of the incident proton flux ) and the electromagnetic radiation generated in the target . the physical dimensions of the absorber are driven by the radiological requirements on the muon shield ( see section [ ssec : mushield ] below ) . as part of the absorber , a concrete shielding wall will close - off the target bunker volume from the downstream muon shield tunnel . the muon flux after the hadron absorber is estimated from a sample of @xmath92 @xmath93 events generated with pythia with a proton beam energy of 400gev and a fixed proton target . the prompt component of the muon flux originates from the electromagnetic decays of meson resonances , mainly @xmath94 , @xmath95 , @xmath96 , @xmath97 . the non - prompt component originates mainly from the decay in flight of charged pions and kaons . for the present estimates , only the decays of primary pions and kaons are considered as non - prompt sources . secondary , tertiary , or higher order pion or kaon decays should give a softer momentum spectrum that is more easily attenuated than the high - energy primary pion and kaon decays that drive the requirement on the muon shield length . the exact radial extent of the muon shield will be defined based on a detailed simulation to ensure that the muons from higher order interactions , which generally have larger polar angles , are also absorbed . the dense target and the hadron absorber described above will be designed to stop all hadrons . however , a fraction @xmath98 of the hadrons entering the absorber will decay into muons before interacting , where @xmath99 is the interaction length of hadrons in the hadron absorber ; @xmath100 is the decay length of the hadron and @xmath101 , @xmath102 and @xmath45 are the hadron lifetime , lab momentum and rest mass ; and @xmath103 is the branching fraction for the decay of the hadron to muons ( 100% for @xmath104 and 63% for @xmath105 , other muon - producing decay channels are neglected ) . a cross - check with the muon fluxes observed in the charm `` beam dump '' experiment @xcite indicates that the above method for estimating the `` first - generation '' muon flux gives the correct order of magnitude . the resulting muon flux is shown as a function of the muon momentum in fig . [ fig : muonfluxes ] . the muons from charm decays represent a negligible fraction of the total muon flux . @xmath93 events generated with pythia with a proton beam of 400 gev and a fixed proton target ( see text for details ) . the total flux of first - generation muons above a given momentum is shown . , scaledwidth=70.0% ] figure [ fig : muonfluxes ] indicates that if a shield were put in place to stop muons below 350gev , the muon flux would be reduced by about seven orders of magnitude to o(1 ) muon per @xmath92 protons on target . however , the muon - induced @xmath77 production rate from nuclei would still be larger than that of neutrino interactions that is computed in section [ sec : bg ] . therefore , the experiment will use a muon absorber that nominally stops muons with energies of up to 400gev , which requires a length of 52 m of uranium ( or 54 m of tungsten ) @xcite . m. , scaledwidth=100.0% ] the final configuration of the beam line and detector will be based on an optimisation of the proton spill duration , the target design , the hadron absorber and the muon shield lengths , and the number of protons on target . the general beam line layout as described above is shown in fig . [ fig : setup - sketch ] . the proposed beam line arrangement consists of a proton beam impinging on a @xmath106 m long target , followed by a hadron absorber of @xmath107 m length , a shielding wall to confine air and radiation , and a @xmath108 m long uranium or tungsten muon absorber enclosed in iron and concrete . the experiment requires a new switching section from an existing beam line , which can be very similar to the one already used for the cngs facility . a short section of transfer line , consisting almost entirely of drift length to allow the beam energy density to be reduced , will then bring the high energy protons to the target bunker that contains the target and the hadron absorber . the site of the experiment should be such that it allows use of existing infrastructure . in particular , the site will need to be provisioned with the equipment to handle air and water activation , to comply with radiation protection standards . the site should also be well within the cern boundaries . both construction and operation of the beam line and the detector can be such that the interference caused to running facilities is minimised . based on these requirements and constraints , the north area could provide a suitable location with a relatively short transfer line of a few hundred metres , branching - off from tt20 , with a new beam splitting near to the tdc2 splitter area . the beam deflection could then be arranged to allow the dedicated transfer line to point to a target bunker at sufficient depth and distance from the tcc2 target area to minimise the excavation in activated soil . soil tests will be needed to determine the exact location . a 60 m trenched tunnel after the target bunker would allow the installation of the muon shield , and an area at relatively low elevation just before the north area hall would provide a suitable site for the location of the surface detector building . the location of the proposed target bunker near to the tcc2 target area suggests that it may be possible to benefit from the general refurbishments that are already foreseen for the current target area in terms of activated water and air treatment . the detector consists of a long decay volume followed by a spectrometer . for a given detector length , the detector diameter should be maximised . in the discussion below the 5 m aperture of the lhcb spectrometer @xcite is taken as a realistic scale . figure [ length ] shows a scan of the length of the detector for both a single detector element and for two longitudinally arranged detector elements . for a given hnl lifetime and detector aperture , the number of hnls decaying in the apparatus with the decay products going through the spectrometer saturates as a function of the length of the detector . the use of two magnetic spectrometers increases the geometric acceptance by 70% compared to a single element . therefore , the proposed detector will have two almost identical detector elements as depicted in fig . [ detector ] . a diagram of a single detector element is also shown in fig . [ detector3d ] . to reduce to a negligible level the background caused by interactions of neutrinos with the remaining air inside the decay volume , a pressure of less than @xmath109 mbar will be required ( see section [ sec : bg ] ) . each detector element therefore consists of a @xmath11050 m long cylindrical vacuum vessel of 5 m diameter . the first @xmath111 m constitute the decay volume and the subsequent 10 m are used for the magnetic spectrometer . the combined calorimeter and muon detector have a length of 2 m . the magnetic spectrometer includes a 4 m long dipole magnet , two tracking layers upstream of the magnet , and two tracking layers downstream of the magnet ( see fig . [ detector ] ) . for the required level of vacuum , the tracking chamber thickness and resolution are matched to give a similar contribution to the overall spectrometer resolution ( see fig . [ magfield ] ) . using straw tubes with @xmath110120@xmath112 m resolution and with 0.5% @xmath113 , like those presently being produced for the na62 experiment @xcite , simulation studies indicate that 2.5 m is required between tracking chambers , giving @xmath114 m length for each magnetic spectrometer . an electromagnetic calorimeter is located behind each vacuum vessel for @xmath115 reconstruction and lepton identification . the calorimeter material is also part of the muon filter for the muon detector , which consists of an iron wall followed by a tracking station . an additional tracking station at the beginning of each decay vessel will be used to veto charged particles entering the fiducial volume . these stations will also reject upstream neutrino interactions . for a mass @xmath78gev , 75% of the @xmath72 decay products have both tracks with momentum @xmath116gev . the momentum and hence mass resolution scales with the integrated field of the magnets . a 0.5tm field integral results in a mass resolution of @xmath117mev for @xmath116gev tracks ( see fig . [ kl ] ) . for a 1gev hnl this provides ample separation between the signal peak and the high mass tail of partially reconstructed @xmath118 decays . further optimisation of the magnetic field will need to take into account the shape of the high mass tail from such decays which may enter the signal mass window . decay products with various momenta in the left and middle panels and the momentum spectrum of the decay products in the rightmost panel . the momentum window indicated is for the measured momenta and is required for the harder of the two decay products . a hnl with mass 1gev is assumed.,scaledwidth=95.0% ] a feasibility study of a dipole magnet , similar to the lhcb magnet @xcite , with a free aperture of almost 16m@xmath119 and a field integral of @xmath106tm , has been conducted . figure [ magnet - scketch ] shows a sketch of a magnet which fulfills the requirements of the proposed experiment @xcite . with a yoke with outer dimension of @xmath120m@xmath121 , and two al-99.7 coils , the proposed magnet provides a peak field of @xmath122 t , and a @xmath123tm over a length of @xmath124 m . for comparison , the lhcb magnet mentioned above contains @xmath125 more iron for its yoke , and dissipates three times more power . the proposed experiment will require two @xmath11050 m long , 5 m diameter vacuum tanks each of which will be similar to that used in the na62 experiment @xcite . the @xmath109 mbar vacuum required to suppress neutrino interactions ( see section [ sec : bg ] ) , is several orders of magnitude less demanding than the pressure used in the na62 vacuum tank ( @xmath126 mbar ) and should therefore not represent a technological challenge . the tracking stations of the magnetic spectrometer must provide good spatial resolution and minimise the contribution from multiple scattering . the na62 straw tracker tubes @xcite , which are manufactured from thin polyethylene terephthalate ( pet ) , are ideal to meet both of these goals . gas tightness of these tubes has been demonstrated in long term tests and the mass production procedure is also well established . the reconstruction of @xmath115 mesons and the identification of electrons would be required to reconstruct decays such as @xmath127 and @xmath128 . an electromagnetic calorimeter with a modest energy resolution will therefore substantially improve the discovery potential of the proposed experiment . the lhcb shashlik calorimeter has demonstrated an energy resolution of @xmath129 , which is comparable to the momentum resolution of the proposed magnetic spectrometer in the 10 to 20gev energy range . the shashlik technology also provides an economical solution with fine granularity , as well as time resolution better than a few ns , which will be needed to correlate the calorimeter and tracker information . the sps provides a quasi - continuous flux of protons on target over its extraction period . a dead - timeless readout system such as that envisaged for the lhcb upgrade @xcite would suffice for the proposed experiment . this readout system will record data continuously in 25ns time - slices . the data will be pushed out to read - out boards which in turn will push the data to a pc - farm to perform the event building . tracks with matching times will be combined to form two - prong vertices , and events with good vertices will be maintained for further analysis . this part of the selection will be executed on - line in the event building farm to reduce the data storage rate to a negligible level . it is estimated that storage of @xmath11010tb / year would be required . the muon shield described above is designed to stop muons with momenta of up to 400gev , thus reducing muon - induced backgrounds to a negligible level . the rate of charged - current neutrino interactions ( cc ) occurring at the downstream end of the muon shield is estimated by extrapolating from a measurement by charm @xcite , which used 400gev protons impinging on a cu target . to extrapolate this measurement to the proposed geometry , the angular and momentum distributions of the neutrinos are simulated using pythia @xcite . this results in an expected cc rate in the last interaction length of the muon shield of @xmath130 per @xmath131 protons on target if a cu target were used . as a cross - check geant @xcite is used to simulate the neutrino spectrum produced by a 400gev proton beam on a cu target . this yields a cc rate four times larger than the estimate based on the charm data . replacing the cu target by a w target lowers the cc rate by @xmath132% . the neutrinos from the geant simulation using a w target are passed to genie @xcite to simulate the cc and neutral - current ( nc ) neutrino interactions in the muon shield . this yields a cc(nc ) rate of @xmath133 per interaction length per @xmath131 protons on target . conservatively , this rate is used to evaluate the background . neutrino interactions in the decay volume could be a source of background . in a decay volume filled with air under atmospheric pressure , the above rate translates into @xmath134 neutrino interactions per @xmath131 protons on target . a pressure in the decay volume of 0.01 mbar reduces this rate to a negligible level . another source of background is the rate of the neutrino interactions that occur in the muon shield just upstream of the decay volume . a combination of geant and genie is used to predict that in @xmath114% of the neutrino interactions a @xmath135 or @xmath136 will be produced . this prediction is consistent with measurements by nomad @xcite . in the first 5 m of the decay volume two - prong vertices are mainly from @xmath135 and @xmath137 decays . for the remaining 35 m of the decay volume @xmath138% of two - prong vertices originate from @xmath139 decays . requiring one of the two decay tracks to be identified as a muon , yields @xmath140 two - prong vertices in @xmath131 protons on target . owing to their different kinematics , the geometrical acceptance of @xmath139 decay products is significantly smaller than that of hnl decay products . figure [ fig : kl - mass - ip ] shows the invariant mass of these candidates together with their distance of closest approach ( ip ) when extrapolated back to the w target . with a 1gev invariant mass for the signal . , title="fig:",scaledwidth=48.0% ] with a 1gev invariant mass for the signal . , title="fig:",scaledwidth=48.0% ] requiring ip@xmath141 m reduces this background to a handful of candidates for @xmath131 protons on target , while the ip of all signal candidates is below 1 m . the ip of candidates will also be used to reject possible background induced in neutrino interactions in the surrounding material , like the vacuum tank or the floor , and from cosmic rays . backgrounds that originate from neutrino interactions could be vetoed by detecting the associated activity produced in the associated shower , especially the lepton in cc interactions . this signature has not been exploited yet in the above yields and could be used to reduce the background to a negligible level . more detailed simulation studies will be undertaken to optimise the position of the veto stations at the entrance of the decay volume . if shown to be advantageous , the last interaction lengths of both the muon shield and the muon filter could be instrumented in order to detect the products of neutrino interactions . the sensitivity of the proposed experiment depends on the final states used to reconstruct the signal decays and the pattern of couplings between the neutrino generations and the hnls . the region in @xmath142 that can be probed by the proposed experiment is conservatively estimated by assuming only the decay @xmath143 is used and considering the production mechanism @xmath144 . the relationship between @xmath142 and @xmath68 is discussed in ref . @xcite . as detailed in section [ sec : exp_status ] , the present strongest limits on @xmath142 come from the charm experiment @xcite and are at the @xmath41 level for a mass of 1.5gev and a few times @xmath145 for a mass of 1gev . the theoretically allowed region for @xmath146 is between @xmath41 and @xmath147 . the expected number of signal events for the proposed experiment is given by @xmath148 where @xmath149 is the total number of protons on target , @xmath150 is the ratio of the @xmath151 production cross - section with respect to the total cross - section ( the factor 2 accounts for the fact that charm is produced in pairs ) , @xmath152 is the total efficiency and @xmath153 is the product of the production and decay branching fractions of the hnls . the quantity @xmath152 can be written as the product of the probability that the hnls decay in the fiducial volume and the trigger and reconstruction efficiencies . the probability of decay is estimated with simulated events for different values of @xmath142 and the efficiencies are assumed to be 100% . the dependence on the hnl lifetime introduces a @xmath142 factor in @xmath152 . since the quantities @xmath153 and @xmath152 each depend on a factor @xmath142 , this results in an overall @xmath154 dependence . assuming @xmath155 , a hnl mass of 1gev , @xmath156 , @xmath157s and @xmath158 , @xmath159 fully reconstructed @xmath160 events would be observed , four orders of magnitudes larger than the number that would be expected in the charm experiment . considering a point in the cosmologically favoured region with @xmath161 , @xmath162s and @xmath163gev , 120 fully reconstructed @xmath164 events would be expected in the proposed experiment . the electromagnetic calorimeter allows the reconstruction of decay modes with a neutral pion in the final state such as @xmath75 , where the @xmath165 , allowing the signal yields to be doubled . channels with electrons , such as @xmath166 , could also be studied , allowing a further increase in the yields and the parameter @xmath167 to be probed . other decay channels with a sm neutrino in the final state , e.g. @xmath168 , are more challenging to select , since the invariant mass will have a broad distribution but could be separated from the background by using particle identification information . both the background levels and the total neutrino flux will be measured by the experiment itself . the neutrino flux from charm - meson decays is more difficult to obtain experimentally . a study of the @xmath169 distribution of neutrino events in the calorimeter together with existing 400gev pcu data @xcite and a detailed simulation will provide a normalisation . in summary , for a hnl mass below 2gev the proposed experiment has discovery potential for the cosmologically favoured region with @xmath146 between @xmath41 and a few times @xmath170 . fixed target experiments of the type proposed could be performed using both the fermilab and kek proton beams . the beams considered are the 800gev and 120gev fnal beams with @xmath171 protons on target and @xmath172 protons on target , respectively ; and the kek 30gev beam with @xmath173 protons on target . at fnal , the 800gev beam would give a similar hnl flux to that of the proposed sps experiment , i.e. the lower proton intensity would be approximately compensated by the increase in the charm cross - section at higher energy @xcite . however , a significantly longer muon filter would be required due to the higher beam energy , which would be much more challenging , leading to a significant loss of acceptance . the fnal 120gev beam would have a factor ten lower event yield than in the proposed sps experiment , while the kek beam would have a factor 1.52 lower yield , the latter estimate has a large uncertainty due to the poor knowledge of the charm cross - section at low energy . the sensitivity of a colliding beam experiment at the cern lhc is estimated assuming a luminosity of 1000fb@xmath174 and an energy of 14tev , as is foreseen in three to four years of running for the high luminosity upgrade . the hnl decay volume is taken to be located 60 m away from the interaction region and 50mrad off - axis , in order to avoid the lhc beam line . the overall hnl event yield would be a factor approximately 200 smaller than in the proposed sps experiment . although masses of the hnls are expected to be around the gev - scale , it is possible that they are heavier than @xmath49 mesons . if the hnls are lighter than @xmath1105gev , they can be produced in beauty hadron decays . the most copious hnl production mechanism with @xmath175 mesons would be the semileptonic @xmath176 decays and the total available mass would therefore be restricted to less than @xmath177gev . the reduced cross - section for the production of beauty mesons with respect to charm mesons means that the limits that could be derived from a dedicated experiment at the lhc would be about four orders of magnitude weaker than those from charm decays . such limits would then be comparable to the upper limits from the theoretical consideration of the baryon asymmetry . the sps at cern is therefore the ideal facility to conduct the proposed experiment to search for hnls . the proposed experiment will search for new physics in the largely unexplored domain of new , very weakly interacting particles with masses below the fermi scale . this domain is inaccessible to the lhc experiments and to comparable experiments at other existing facilities . the proposed detector is based on existing technologies and therefore requires no substantial r&d phase . a moderately sized collaboration could construct the proposed detector in a few years . the design of the beam line is challenging , in particular , the beam extraction and beam target , as well as the radiological aspects require further study . the solutions proposed are being actively discussed with machine experts . the impact that a discovery of a hnl would have on particle physics is difficult to overestimate . in short , it could solve two of the most important shortcomings of the sm : the origin of the baryon asymmetry of the universe , the origin of neutrino mass . in addition , the results of this experiment , together with cosmological and astrophysical data , could be crucial to determine the nature of dark matter . we are grateful to g. arduini , m. calviani , d. grenier , e. gschwendtner and h. vincke for useful discussions and valuable input on the beam line and the target . we would like to thank f. rademakers for providing the three dimensional sketch of the experiment . w. flegel is warmly acknowledged for adapting the design of the lhcb magnet to our needs . we are grateful to s. gninenko and a. rozanov for stimulating discussions and to e. van herwijnen for setting up our web site . g. aad _ et al . _ , atlas collaboration , phys . * b 716 * ( 2012 ) 1 , arxiv : hep - ex/1207.7214 ; atlas collaboration , http://cds.cern.ch/record/1523727 . s. chatrchyan _ et al . _ , cms collaboration , phys . lett . * b 716 * , ( 2012 ) 30 , arxiv : hep - ex/1207.7235 ; cms collaboration , http://cds.cern.ch/record/1542387 . f. cerutti , talk at eps hep conference , stockholm , 2013 , https://cds.cern.ch/record/1598145 . j. ellis _ et al . _ , phys . * b 679 * ( 2009 ) 369 , arxiv : hep - ph/0906.0954 . d. buttazzo _ et al . _ , arxiv : hep - ph/1307.3536 . f. bezrukov , m.y . kalmykov , b.a . kniehl and m. shaposhnikov , jhep * 10 * ( 2012 ) 140 , arxiv : hep - ph/1205.2893 . t. asaka , s. blanchet and m. shaposhnikov , phys . lett . * b 631 * ( 2005 ) 151 , arxiv : hep - ph/0503065 . t. asaka and m. shaposhnikov , phys . * b 620 * ( 2005 ) 17 , arxiv : hep - ph/0505013 . d. gorbunov and m. shaposhnikov , jhep * 10 * ( 2007 ) 015 , arxiv : hep - ph/0705.1729 . b. batell , m. pospelov and a. ritz , phys . rev . * d83 * ( 2011 ) 054005 , + doi:10.1103/physrevd.83.054005 , arxiv : hep - ph/0911.4938 . s. n. gninenko , phys . rev . * 103 * ( 2009 ) 241802 , arxiv / hep - ph:0902.3802 ; s. n. gninenko and d. s. gorbunov , phys . rev . d * 81 * ( 2010 ) 075013 , arxiv : hep - ph/0907.4666 . aliev , a.s . cornell and n. gaur , jhep * 07 * ( 2007 ) 072 , doi:10.1088/1126 - 6708/2007/07/072 , arxiv : hep - ph/0705.4542 . f. bergsma _ et al . _ , charm collaboration , phys . * b 157 * ( 1985 ) 458 . mohapatra and p.b . pal , world sci . lect . notes phys . * 60 * ( 1998 ) 1 ; world sci . notes phys . * 72 * ( 2004 ) 1 . p. minkowski , phys . * b 67 * ( 1977 ) 421 . t. yanagida , progr . * 64 * ( 1980 ) 1103 ; m. gell - mann , p. ramond and r. slansky , in _ supergravity _ , north holland , amsterdam , 1980 ; r.n . mohapatra and g. senjanovic , phys . * 44 * ( 1980 ) 912 . m. fukugita and t. yanagida , phys . * b 174 * ( 1986 ) 45 . kuzmin , v.a . rubakov and m.e . shaposhnikov , phys . * b 155 * ( 1985 ) 36 . klinkhamer and n.s . manton , phys . * d 30 * ( 1984 ) 2212 . g. senjanovic , riv . nuovo cim . * 034 * ( 2011 ) 1 . giudice , in g. kane and a. pierce ( eds . ) , _ perspectives on lhc physics _ ( 2008 ) 155 , arxiv : hep - ph/0801.2562 . a. pilaftsis and t.e.j . underwood , nucl . phys . * b 692 * ( 2004 ) 303 , arxiv : hep - ph/0309342 . a. pilaftsis and t.e.j . underwood , phys . * d 72 * ( 2005 ) 113001 , arxiv : hep - ph/0506107 . akhmedov , v.a . rubakov and a.y . smirnov , phys . * 81 * ( 1998 ) 1359 , arxiv : hep - ph/9803255 . s. dodelson and l.m . widrow , phys . * 72 * ( 1994 ) 17 , arxiv : hep - ph/9303287 . shi and g.m . fuller , phys . * 82 * ( 1999 ) 2832 , arxiv : astro - ph/9810076 . a. boyarsky , o. ruchayskiy and m. shaposhnikov , ann . nucl . part . * 59 * ( 2009 ) 191 , arxiv : hep - ph/0901.0011 . shrock , phys . * d 24 * ( 1981 ) 1232 . shrock , phys . rev . * d 24 * ( 1981 ) 1275 . m. gronau , c.n . leung and j.l . rosner , phys . * d 29 * ( 1984 ) 2539 . johnson , d.w . mckay and t. bolton , phys . * d 56 * ( 1997 ) 2970 , arxiv : hep - ph/9703333 . ade _ et al . _ , planck collaboration , arxiv : astro - ph/1303.5076 . a. de gouvea , phys . * d 72 * ( 2005 ) 033005 , arxiv : hep - ph/0501039 . aguilar - arevalo _ et al . _ , lsnd collaboration , phys . * d 64 * ( 2001 ) 112007 , arxiv : hep - ex/0104049 . aguilar - arevalo _ et al . _ , miniboone collaboration , phys . * 110 * ( 2013 ) 161801 , arxiv : hep - ex/1207.4809 ; arxiv : hep - ex/1303.2588 . g. mention _ et al . _ , * d 83 * ( 2011 ) 073006 , arxiv : hep - ex/1101.2755 . abdurashitov _ et al . _ , phys . * c 73 * ( 2006 ) 045805 , arxiv : nucl - ex/0512041 . abdurashitov _ et al . _ , sage collaboration , phys . rev . * c 59 * ( 1999 ) 2246 , arxiv : hep - ph/9803418 . w. hampel _ et al . _ , gallex collaboration , phys . * b 420 * ( 1998 ) 114 . f. kaether _ et al . _ , phys . * b 685 * ( 2010 ) 47 , arxiv : hep - ex/1001.2731 . m. gonzalez - garcia , m. maltoni , j. salvado and t. schwetz , jhep * 12 * ( 2012 ) 123 , arxiv : hep - ph/1209.3023 . m. drewes and b. garbrecht , jhep * 03 * ( 2013 ) 096 , arxiv : hep - ph/1206.5537 . l. canetti , m. drewes , t. frossard and m. shaposhnikov , phys . * d 87 * ( 2013 ) 093006 , arxiv : hep - ph/1208.4607 ; l. canetti , m. drewes and m. shaposhnikov , phys . * 110 * ( 2013 ) 6 , 061801 , arxiv : hep - ph/1204.3902 . d. gorbunov and m. shaposhnikov , proposal submitted to european strategy group , 2012 ; s.n . gninenko , d.s . gorbunov and m.e . shaposhnikov , adv . high energy phys . * 2012 * ( 2012 ) 718259 , arxiv : hep - ph/1301.5516 . f. bergsma _ et al . _ , charm collaboration , phys . * b 166 * ( 1986 ) 473 . g. bernardi _ et al . _ , phys . lett . * b 166 * ( 1986 ) 479 ; g. bernardi _ et al . _ , phys . * b 203 * ( 1988 ) 332 ; f. vannucci , j. phys . ( 2008 ) , doi:10.1088/1742 - 6596/136/2/022030 . a. vaitaitis _ et al . _ , nutev collaboration , phys . * 83 * ( 1999 ) 4943 , arxiv : hep - ex/9908011 . a.m. cooper - sarkar _ et al . _ , wa66 collaboration , phys . * b 160 * ( 1985 ) 207 . mishra _ et al . _ , ccfr collaboration , phys . * 59 * ( 1987 ) 1397 . shrock , phys . lett . * b 96 * ( 1980 ) 159 ; r.e . shrock , phys . lett . * b 112 * ( 1982 ) 382 ; d.i . britton _ _ , phys . * d 68 * ( 1992 ) 3000 ; d.i . et al . _ , * d 46 * ( 1992 ) 3000 ; t. yamazaki _ et al . _ , conf . c840719 " ( 1984 ) 262 ; d.a . bryman and t. numao , phys . * d 53 * ( 1996 ) 558 ; r. abela _ et al . _ , phys . * b 105 * ( 1981 ) 263 ; m. daum _ et al . _ , phys . * d 36 * ( 1987 ) 2624 ; r.s . hayano _ et al . _ , phys . * 49 * ( 1982 ) 1305 ; m. aoki _ et al . _ , phys . * d 84 * ( 2011 ) 052002 .

a new fixed - target experiment at the cern sps accelerator is proposed that will use decays of charm mesons to search for heavy neutral leptons ( hnls ) , which are right - handed partners of the standard model neutrinos . the existence of such particles is strongly motivated by theory , as they can simultaneously explain the baryon asymmetry of the universe , account for the pattern of neutrino masses and oscillations and provide a dark matter candidate . cosmological constraints on the properties of hnls now indicate that the majority of the interesting parameter space for such particles was beyond the reach of the previous searches at the ps191 , bebc , charm , ccfr and nutev experiments . for hnls with mass below @xmath0gev , the proposed experiment will improve on the sensitivity of previous searches by four orders of magnitude and will cover a major fraction of the parameter space favoured by theoretical models . the experiment requires a 400gev proton beam from the sps with a total of @xmath1 protons on target , achievable within five years of data taking . the proposed detector will reconstruct exclusive hnl decays and measure the hnl mass . the apparatus is based on existing technologies and consists of a target , a hadron absorber , a muon shield , a decay volume and two magnetic spectrometers , each of which has a 0.5tm magnet , a calorimeter and a muon detector . the detector has a total length of about 100 m with a 5 m diameter . the complete experimental set - up could be accommodated in cern s north area . the discovery of a hnl would have a great impact on our understanding of nature and open a new area for future research . + cern - spsc-2013 - 024 / spsc - eoi-010 + + + proposal to search for heavy neutral leptons at the sps w. bonivento@xmath2 , a. boyarsky@xmath3 , h. dijkstra@xmath4 , u. egede@xmath5 , m. ferro - luzzi@xmath4 , b. goddard@xmath4 , a. golutvin@xmath5 , d. gorbunov@xmath6 , r. jacobsson@xmath4 , j. panman@xmath4 , m. patel@xmath5 , o. ruchayskiy@xmath7 , t. ruf@xmath4 , n. serra@xmath8 , m. shaposhnikov@xmath7 , d. treille@xmath9 +

1310.1762

in this section , we review the renormlization group ( rg ) equation for the generalized karder - parisi - zhang ( kpz ) equations . we first define scale - dependent parameters @xmath0 , @xmath69 , @xmath71 , and @xmath72 , and then introduce a perturbation theory leading to the equation for determining them . we start with the generating functional @xmath184 $ ] by which all statistical quantities of the kpz equations are determined . following the martin - siggia - rose - janssen - dedominicis ( msrjd ) formalism @xcite , @xmath184 $ ] is expressed as @xmath185 = \int \mathcal{d } [ h , { i\tilde{h } } ] \exp\biggl[-s[h,{i\tilde{h}};\lambda_0]+\int_{-\infty}^{\infty } d\omega \int_{-\lambda_0}^{\lambda_0 } dk \biggl({j}(k,\omega ) h(-k,-\omega)+ { \tilde{j}}(k,\omega ) { i\tilde{h}}(-k,-\omega ) \biggr ) \biggr ] , \label{eq : generatingfunctional}\ ] ] where @xmath186 is the auxiliary field , @xmath187 and @xmath188 are source fields , and @xmath189 $ ] is the msrjd action for the generalized kpz equation . throughout supplemental material , we use the notation @xmath190 for the fourier transform of @xmath191 for any field a. the action @xmath189 $ ] is explicitly written as @xmath192 = & \frac{1}{2}\int_{-\infty}^{\infty } \frac{d\omega}{2\pi } \int^{\lambda_0}_{-\lambda_0}\frac{d k}{2\pi } \begin{pmatrix } h(-k,-\omega ) & { i\tilde{h}}(-k,-\omega ) \end{pmatrix } g_0^{-1}(k,\omega ) \begin{pmatrix } h(k,\omega ) \\ { i\tilde{h}}(k,\omega ) \end{pmatrix } \nonumber \\ & + \frac{\lambda_0}{2 } \int_{-\infty}^{\infty } \frac{d\omega_1 d\omega_2}{(2\pi)^2 } \int^{\lambda_0}_{-\lambda_0}\frac{d k_1 d k_2}{(2\pi)^2 } k_1 k_2{i\tilde{h } } ( -k_1-k_2,-\omega_1-\omega_2 ) h(k_1 , \omega_1)h(k_2,\omega_2 ) , \label{eq : action}\end{aligned}\ ] ] where @xmath193 is the inverse matrix of the bare propagator @xmath194 here , we consider a coarse - grained description at a cutoff @xmath195 . let us define @xmath196 for any quantity @xmath190 , where @xmath197 is the heaviside step function . the statistical quantities of @xmath198 are described by the generating functional @xmath199 $ ] with replacement of @xmath200 by @xmath201 . we thus define the effective msrjd action @xmath202 $ ] by the relation @xmath203= \int \mathcal{d } [ h^ < , { i\tilde{h}}^ < ] \exp\biggl[-s[h^<,{i\tilde{h}}^<;\lambda]+\int_{-\infty}^{\infty } d\omega \int_{-\lambda}^{\lambda } dk \biggl({j}^<(k,\omega ) h^<(-k,-\omega)+ { \tilde{j}}^<(k,\omega ) { i\tilde{h}}^<(-k,-\omega ) \biggr ) \biggr].\end{aligned}\ ] ] we can then confirm that @xmath202 $ ] is determined as @xmath204 \biggr ] = \int \mathcal{d } [ h^ > , { i\tilde{h}}^ > ] \exp \biggl[-s[h^<+h^>,{i\tilde{h}}^<+{i\tilde{h}}^>;\lambda_0]\biggr ] . \end{aligned}\ ] ] then , the propagator and the three point vertex function for the effective msrjd action at @xmath23 are defined as @xmath205}{\delta({i\tilde{h}}(k_1 , \omega_1))\delta({h^<}(k_2 , \omega_2 ) ) } \right|_{{h^<}=0,{i\tilde{h}^<}=0 } , \label{eq : def of propagator}\\ ( g^{-1})_{\tilde{h}\tilde{h}}(k_1 , \omega_1 ; \lambda ) \delta(\omega_1+\omega_2)\delta(k_1+k_2 ) & \equiv \left . \frac{\delta^2 s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta({i\tilde{h}^<}(k_1 , \omega_1))\delta({i\tilde{h}^<}(k_2 , \omega_2))}\right|_{{h^<}=0,{i\tilde{h}^<}=0 } , \\ \gamma_{\tilde{h } h h}(k_1 , \omega_1;k_2,\omega_2 ; \lambda)\delta(\omega_1+\omega_2+\omega_3)\delta(k_1+k_2+k_3 ) & \equiv \left.\frac{\delta^3 s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta({i\tilde{h}^<}(k_1 , \omega_1))\delta({h^<}(k_2 , \omega_2))\delta({h^<}(k_3,\omega_3))}\right|_{{h^<}=0,{i\tilde{h}^<}=0}. \label{eq : def of vertex}\end{aligned}\ ] ] from these quantities , we define the parameters as @xmath206 in the next section , we will provide a non - perturbative proof for the claim that @xmath207 on the basis of symmetry properties @xcite . below , we derive a set of equations that determines @xmath0 , @xmath69 , @xmath71 , and @xmath72 , respectively . we can calculate @xmath208 by using the perturbation theory in @xmath209 . at the second - order level , the propagators are calculated as @xmath210 , \label{eq : propagator1}\\ ( g^{-1})_{\tilde{h}\tilde{h}}(\omega , k ; \lambda ) = & ( g_0^{-1})_{\tilde{h}\tilde{h}}(k,\omega ) \nonumber \\ & -2\lambda_0 ^ 2\int^{\infty}_{-\infty } \frac{d\omega}{2\pi } \int _ { \lambda \leq \vert q\vert \leq \lambda_0 } \frac{d q}{2\pi } q^2(k - q)^2 c_0(q,\omega ) c_0(k - q,\omega-\omega ) , \label{eq : propagator2}\end{aligned}\ ] ] where @xmath211 is the bare correlation function defined by @xmath212 in the calculation of eq . ( [ eq : propagator1 ] ) , one should carefully note the relation @xcite @xmath213 by setting @xmath214 for eqs . ( [ eq : definition of nu ] ) - ( [ eq : propagator2 ] ) , we obtain the rg equation @xmath215 , \label{eq : rgnu}\\ -\lambda\frac{d k(\lambda)}{d \lambda } & = k(\lambda)\biggl [ \frac{g}{2(1+f)^5}\biggl(26-f+2f^2+f^3+(2 - 21f+6f^2+f^3)\frac{h}{g}\biggr)\biggr ] , \label{eq : rgk}\\ -\lambda\frac{d d(\lambda)}{d \lambda } & = d(\lambda)\biggl [ \frac{g}{(1+f)^3}\biggl(1+\frac{h}{g}\biggr)^2\biggr ] , \label{eq : rgd}\\ -\lambda \frac{d d_d(\lambda)}{d \lambda } & = d_d(\lambda)\biggl [ \frac{g^2}{2h(1+f)^5}\biggl(16 + 3f+f^2 + 2(9 - 5f)\frac{h}{g } + ( 2 - 13f - f^2)\frac{h^2}{g^2}\biggr)\biggr ] \label{eq : rgdd},\end{aligned}\ ] ] where we have introduced the dimensionless parameters @xmath120 , @xmath216 and @xmath217 as @xmath218 we also obtain the following equations of @xmath120 , @xmath216 and @xmath217 from eqs . ( [ eq : rgnu ] ) - ( [ eq : rgdd ] ) : @xmath219 , \label{eq : rgf}\\ -\lambda \frac{d g}{d \lambda } & = 7g-\frac{g^2}{2(1+f)^5}\biggl [ 76 - 7f+4f^2 + 3f^3+(2 - 71f+14f^2 + 3f^3)\frac{h}{g}-2(1+f)^2\frac{h^2}{g^2}\biggr ] , \label{eq : rgg}\\ -\lambda \frac{d h}{d \lambda } & = 5h+\frac{g^2}{2(1+f)^5}\biggl [ 16 + 3f+f^2-(60 + 7f+6f^2 + 3f^3)\frac{h}{g}-(4 - 50f+19f^2 + 3f^3)\frac{h^2}{g^2}\biggr ] . \label{eq : rgh}\end{aligned}\ ] ] the stable fixed point of the equations eqs . ( [ eq : rgf ] ) - ( [ eq : rgh ] ) is found to be @xmath220 . by substituting the fixed point values to eqs . ( [ eq : rgnu])-([eq : rgdd ] ) and solving them , we obtain the scaling laws @xmath221 in the limit @xmath63 . here , @xmath56 , @xmath57 , @xmath58 and @xmath59 are constants which are determined by the initial parameter values . in this section , we prove @xmath222 for all generalized kpz equations , and @xmath223 for @xmath224 or @xmath225 , and @xmath226 for @xmath225 . these results are easily obtained from the following ward - takahashi identities @xcite : @xmath227 @xmath228 and @xmath229 these identities are relaed to invariance properties of the msrjd action for the shit transformation , tilt transformation , and the time - reversal transformation , respectively . in the next subsections , we will derive eqs . ( [ eq : wtid - shift1])-([eq : fdt1 ] ) following the arguments @xcite . here , we derive eqs . ( [ eq : lambda])-([eq : k / dd ] ) from eqs . ( [ eq : wtid - shift1])-([eq : fdt1 ] ) . first , by differentiating eq . ( [ eq : wtid - tilt ] ) with respect to @xmath230 and taking the limit @xmath231 , we have @xmath232 next , we substitute eq . ( [ eq : wtid - shift1 ] ) to eq . ( [ eq : wtid - tilt2 ] ) and take the limit @xmath233 . then , we obtain @xmath234 by recalling the definition eq . ( [ eq : definition of lambda ] ) , we find that this equality is eq . ( [ eq : lambda ] ) . second , we differentiate eq.([eq : fdt1 ] ) twice with respect to @xmath230 . then , we have @xmath235 by taking the limit @xmath236 and using eqs . ( [ eq : definition of nu ] ) and ( [ eq : definition of d ] ) , we obtain eq . ( [ eq : nu / d ] ) . finally , by differentiating eq . ( [ eq : fdt1 ] ) four times with respect to @xmath230 , we arrive at eq . ( [ eq : k / dd ] ) . we consider a shift transformation @xmath237 where @xmath238 is an infinitesimal parameter that depends on time . the variation of the msrjd action for the transformation is calculated as @xmath239 - s[h,{i\tilde{h}};\lambda_0]= \int dt dx { i\tilde{h}}(x , t ) \partial_t c(t ) . \label{eq : variation of shift}\end{aligned}\ ] ] it should be noted that this simple form comes from the invariance property of the msrjd action for the time - independent @xmath240 @xcite . then , the variation of the effective msrjd action is derived as @xmath241=&-\log \int \mathcal{d } [ h^ { > ' } , { i\tilde{h}}^{>'}]\exp\biggl[-s[h^{'},{i\tilde{h}}^ { ' } ; \lambda_0]\biggr ] , \nonumber \\ = & -\log \int \mathcal{d } [ h^ { > } , { i\tilde{h}}^{>}]\exp\biggl[-s[h,{i\tilde{h } } ; \lambda_0]-\int dtdx { i\tilde{h}}(x , t ) \partial_t c(t ) \biggr ] , \nonumber \\ = & \int dtdx { i\tilde{h}}^<(x , t ) \partial_t c(t ) -\log \int \mathcal{d } [ h^ { > } , { i\tilde{h}}^{>}]\exp\biggl[-s[h,{i\tilde{h } } ; \lambda_0]-\int dtdx { i\tilde{h}}^>(x , t ) \partial_t c(t ) \biggr ] , \nonumber \\ = & s[{h^<},{i\tilde{h}^ < } ; \lambda]+\int dtdx { i\tilde{h}}^<(x , t ) \partial_t c(t ) . \label{eq : variation}\end{aligned}\ ] ] when we obtain the fourth line in eq . ( [ eq : variation ] ) from the third line , we have used @xmath242 here , noting the trivial relation @xmath243=s[{h^<},{i\tilde{h}^ < } ; \lambda]+ \int dt dx \frac{\delta s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta { h^<}(x , t)}c(t),\ ] ] we rewrite eq . ( [ eq : variation ] ) as @xmath244}{\delta { h^<}(x , t)}c(t)-{i\tilde{h}^<}(x , t ) \partial_t c(t ) \biggr)=0,\end{aligned}\ ] ] which is further expressed as @xmath244}{\delta { h^<}(x , t)}+\partial_t{i\tilde{h}^<}(x , t ) \biggr)c(t)=0.\end{aligned}\ ] ] since this equality holds for any @xmath238 , we obtain @xmath245}{\delta { h^<}(x , t)}+\partial_t{i\tilde{h}^<}(x , t ) \biggr ) = 0 . \label{eq : wtid - shift2}\end{aligned}\ ] ] the differentiation of eq . ( [ eq : wtid - shift2 ] ) with respect to @xmath246 leads to @xmath247 by performing the fourier transformation , we arrive at eq . ( [ eq : wtid - shift1 ] ) . we consider a tilt transformation @xmath248 where @xmath249 is an infinitesimal parameter . the tilt transformation for their fourier transforms is expressed as @xmath250 we then find the symmetry property @xmath251=s[h^<+h^>,{i\tilde{h}}^<+{i\tilde{h}}^ > ; \lambda_0],\end{aligned}\ ] ] from which we obtain @xmath252&=-\log \int \mathcal{d } [ h^ > , { i\tilde{h}}^>]\exp[-s[h^{<}+h^{>},{i\tilde{h}}^{<}+{i\tilde{h}}^ { > } ; \lambda_0 ] ] , \nonumber \\ & = -\log \int \mathcal{jd } [ h^ { > ' } , { i\tilde{h}}^{>'}]\exp[-s[h^{<'}+h^{>'},{i\tilde{h}}^{<'}+{i\tilde{h}}^ { > ' } ; \lambda_0 ] ] , \nonumber \\ & = s[h^{<'},{i\tilde{h}}^ { < ' } ; \lambda]-\log\mathcal{j } , \label{eq : traction}\end{aligned}\ ] ] where @xmath253 is the jacobian for the tilt transformation , and @xmath254 is a field independent quantity . the expansion of eq . ( [ eq : traction ] ) in @xmath249 leads to the identity @xmath255}{\delta { h^<}(k , t ) } { h^<}(k , t ) + \frac{\delta s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta { i\tilde{h}^<}(k , t ) } { i\tilde{h}^<}(k , t)\biggr ) + i \delta(k)\partial_k \frac{\delta s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta { h^<}(k , t ) } -a \biggr]=0.\end{aligned}\ ] ] we differentiate this identity with respect to @xmath256 and @xmath257 . then , we have @xmath258}{\delta { h^<}(k , t)\delta { i\tilde{h}^<}(k_1,t_1 ) } \delta(k_2-k)\delta(t_2-t ) + \frac{\delta^3 s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta { h^<}(k , t)\delta { i\tilde{h}^<}(k_1,t_1 ) { h^<}(k_2,t_2 ) } { h^<}(k , t ) \nonumber \\ + & \frac{\delta^2 s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta { i\tilde{h}^<}(k , t ) { h^<}(k_2,t_2 ) } \delta(k_1-k)\delta(t_1-t ) + \frac{\delta^3 s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta { h^<}(k , t)\delta { i\tilde{h}^<}(k_1,t_1 ) { h^<}(k_2,t_2 ) } { i\tilde{h}^<}(k , t ) \biggr ) \nonumber \\ & + i \delta(k)\partial_k \frac{\delta s[{h^<},{i\tilde{h}^ < } ; \lambda]}{\delta { h^<}(k , t)\delta { i\tilde{h}^<}(k_1,t_1 ) { h^<}(k_2,t_2 ) } \biggr]=0.\end{aligned}\ ] ] by taking the limit @xmath259 and recalling the definitions given in eqs . ( [ eq : def of propagator ] ) - ( [ eq : def of vertex ] ) , we obtain @xmath260 the fourier transform of this equality is eq . ( [ eq : wtid - tilt ] ) . we consider a time - reversal transformation @xmath261 the variation of the action eq . ( [ eq : action ] ) under this transformation is calculated as @xmath262-s[h,{i\tilde{h}};\lambda_0],\nonumber \\ = & \biggl(\frac{d_0}{\nu_0}-\frac{d_{d0}}{k_0}\biggr)\frac{\nu_0 k_0}{d_0 } \int \frac{d \omega d k}{(2\pi)^2 } \biggl(\frac{\nu_0}{d_0}k^2h(-\omega ,- k ) h(\omega , k ) -2{i\tilde{h}}(-\omega , -k ) h(\omega , k)\biggr).\end{aligned}\ ] ] the generalized kpz equation is invariant when @xmath168 or @xmath224 . here , we focus on the case @xmath168 . then , we obtain @xmath263=s[h^<,{i\tilde{h}}^<;\lambda]-\log\mathcal{j},\end{aligned}\ ] ] where @xmath264 is the jacobian of the time - reversal transformation . by differentiating this equality with respect to @xmath265 and @xmath266 , we have @xmath267}{\delta({i\tilde{h}^<}(k_1 , \omega_1))\delta({h^<}(k_2 , \omega_2 ) ) } = & -\frac{\nu_0 k_1 ^ 2}{d_0}\frac{\delta^2 s[h^{<'},{i\tilde{h}}^ { < ' } ; \lambda]}{\delta ( { i\tilde{h}}^{<'}(k_1 , -\omega_1))\delta({i\tilde{h}}^{<'}(k_2 , -\omega_2 ) ) } \nonumber \\ & -\frac{\delta^2 s[h^ { < ' } , { i\tilde{h}}^ { < ' } ; \lambda]}{\delta ( h^{<'}(k_1 , -\omega_1))\delta({i\tilde{h}}^{<'}(k_2 , -\omega_2 ) ) } , \\\end{aligned}\ ] ] where we have used the relation @xmath268 by recalling the definition given in eqs . ( [ eq : def of propagator ] ) - ( [ eq : def of vertex ] ) , we obtain @xmath269 by rearranging eq . ( [ eq : fdt2 ] ) , we arrive at the identities eq . ( [ eq : fdt1 ] ) . we consider the dimensionless quantities given by @xmath270 substituting the scaling relations eqs . ( [ eq : scaling nu ] ) - ( [ eq : scaling dd ] ) to these equalities , we have @xmath271 on the other hand , @xmath272 takes the value @xmath273 in the limit @xmath63 . then , we arrive at @xmath274 this value is determined only by the fixed point . we consider the following transformation : @xmath275 where @xmath106 , @xmath276 and @xmath277 are constants . for the cases that @xmath278 and @xmath279 , the equation for @xmath280 is written as @xmath281 where we have introduced @xmath282 by imposing @xmath283 and @xmath284 , we obtain @xmath101 and @xmath102 . then , we have the relation @xmath285 which leads to @xmath286 that is , @xmath287 is invariant under the scale transformation . as shown in fig . [ fig : dd ] , the graphs of @xmath72 with @xmath177 at @xmath289 , @xmath290 and @xmath291 do not exhibit plateau . instead , the graphs at @xmath289 and @xmath290 quickly approach @xmath72 in the limit @xmath292 when @xmath23 is smaller than @xmath67 . therefore , we define the bare parameter as @xmath179 for such cases .

we study a noisy kuramoto - sivashinsky ( ks ) equation which describes unstable surface growth and chemical turbulence . it has been conjectured that the universal long - wavelength behavior of the equation , which is characterized by scale - dependent parameters , is described by a kardar - parisi - zhang ( kpz ) equation . we consider this conjecture by analyzing a renormalization - group equation for a class of generalized kpz equations . we then uniquely determine the parameter values of the kpz equation that most effectively describes the universal long - wavelength behavior of the noisy ks equation . _ introduction. _ eddy viscosity in turbulence , which can explain how a vortex pattern emerges in a non - uniform turbulent flow , depends on the observed length scales @xcite . as exemplified by the richardson law @xcite , there are cases in which a parameter of a macroscopic description is not given as a definite value , but is rather expressed as a function of length scale . another example of scale - dependent parameters has been observed in one- or two- dimensional fluid dynamics , where the viscosity is not uniquely defined in the hydrodynamic description @xcite . here , it seems reasonable to expect that such scale dependent parameters in a macroscopic description can be reproduced by an effective stochastic system @xcite . in this letter , we attempt to determine the effective stochastic system theoretically when scale - dependent parameters are observed . as the simplest example for scale - dependent parameters , we consider the one - dimensional kardar - parisi - zhang ( kpz ) equation @xcite . it is known that the effective surface tension @xmath0 at a scale @xmath1 for the equation is @xmath2 in the limit @xmath3 , which is similar to the richardson law for turbulence . recently , the kpz equation was rigorously derived from a stochastic many - particle model @xcite , and the so - called kpz class has been extensively discussed both theoretically and experimentally @xcite . however , in general , even if we find systems that may exhibit scale - dependent parameters similar to those for the kpz class , a method to determine the parameter values of the corresponding kpz equation has not yet been reported . specifically , let us consider a noisy kuramoto - sivashinsky ( ks ) equation , which exhibits spatially extended chaos in the noiseless limit @xcite . the model describes turbulent chemical waves and unstable interface motion , which are caused by negative surface tension . it has been conjectured that a kpz equation may be an effective model for describing the long - wavelength behavior of the noisy ks equation ; this conjecture is referred to as the _ yakhot conjecture _ @xcite . indeed , direct numerical simulations showed that statistical properties of the long wave length modes are similar to those of the kpz equations @xcite . here , one may recall the renormalization group ( rg ) , which is a standard method for studying scale - dependent parameters . for a given noisy ks equation , the rg equation was calculated using a perturbation theory @xcite . the infrared fixed point of the rg equation determines the scale - dependent behavior @xmath4 in the limit @xmath3 , which has the same power - law form as that for the kpz equations . nevertheless , as shown below , the analysis at the infrared fixed point of the rg equation can not determine the parameter values of the corresponding kpz equation . in this letter , we present a framework for studying the effective description . we study an rg equation for generalized kpz equations that include noisy ks equations and kpz equations . we then consider solution trajectories of the rg equation , in which each point flows to the infrared fixed point of the noisy ks equation we study . the solution trajectories also approach a subspace in the ultraviolet limit , which enables us to define a collection of bare parameters of the generalized kpz equations . by using the lowest perturbation theory for the rg equation , we uniquely determine the most effective model among such kpz equations that describes the infrared universal behavior of a noisy ks equation in the most efficient manner . _ setup. _ we study models for the stochastic growth of a surface . we assume that the time evolution of the height @xmath5 of the surface is described by a generalized kpz equation : @xmath6 where @xmath7 is the surface tension , @xmath8 is the surface diffusion constant , @xmath9 is the strength of the non - linearity , and @xmath10 is the noise satisfying @xmath11 . here , @xmath12 and @xmath13 are the strength of the noise . when @xmath14 , eq . ( [ eq : gkpz ] ) is the kpz equation , while when @xmath15 , eq . ( [ eq : gkpz ] ) with @xmath16 is the deterministic ks equation . we refer to eq . ( [ eq : gkpz ] ) with @xmath17 and @xmath18 , as the noisy ks equation . the five parameters in eq . ( [ eq : gkpz ] ) are collectively denoted by @xmath19 . more precisely , these parameters are defined for a field @xmath20 whose fourier transform @xmath21 is assumed to be zero for @xmath22 . @xmath23 is called a cut - off wavenumber . we explicitly express the cutoff dependence of the parameters as @xmath24 . here , for a given model with @xmath25 , we define a model with @xmath24 for @xmath26 by eliminating the contribution @xmath27 in the dynamics , which may be formally expressed as @xmath28 @xcite . this functional relation trivially satisfies @xmath29 from this , we obtain the rg equation @xmath30 which determines @xmath28 under the initial condition @xmath31 @xcite . an approximate form of @xmath32 with the perturbation theory @xcite was derived in @xcite . by introducing dimensionless parameters @xmath33 , @xmath34 , and @xmath35 , we express the result as @xmath36 , @xmath37 , @xmath38 , @xmath39 , and @xmath40 , where @xmath41 , @xmath42 , @xmath43 , @xmath44 , @xmath45 , @xmath46 , @xmath47 , @xmath48 , and @xmath49 . the result @xmath40 implies that the parameter @xmath9 is not renormalized , which is a consequence of the invariance of eq . ( [ eq : gkpz ] ) under the tilt transformation @xcite . here , from eq . ( [ eq : rgeq ] ) , we derive the autonomous equation for @xmath50 . the fixed point of the equation is calculated as @xmath51 @xcite . by using these values , we also obtain @xmath52 , @xmath53 , @xmath54 , and @xmath55 in the limit @xmath3 , where @xmath56 , @xmath57 , @xmath58 , and @xmath59 are constants that depend on the initial condition @xmath60 , while @xmath61 is independent of @xmath60 @xcite . the singular behavior @xmath62 implies that the effective surface tension depends on the observed scale @xmath23 . this is contrasted with cases in which each @xmath24 converges to a finite value in the limit @xmath63 . then , @xmath64 is interpreted as renormalized parameters measured in experiments . since the exponents characterizing the divergent behavior are common to all the models given by eq . ( [ eq : gkpz ] ) , we refer to the power - law region as the universal range . the smallest characteristic wavenumber scale is also denoted by @xmath65 , the value of which depends on @xmath60 . then , the universal range is defined as @xmath66 . as another common aspect of the rg equation ( [ eq : rgeq ] ) , we observe that @xmath24 shows a plateau region in the ultraviolet limit when @xmath67 is sufficiently large . this enables us to define a collection of bare parameters , which is denoted by @xmath68 . and @xmath69 for @xmath70 , respectively . the inset shows the graphs of @xmath71 ( solid red ) and @xmath72 ( dotted green ) . [ fig : nud - ks ] ] here , we focus on a specific model , a noisy ks equation with @xmath73 , defined at @xmath74 . in fig . [ fig : nud - ks ] , we display the numerical solution of eq . ( [ eq : rgeq ] ) for this initial condition @xmath70 . it can be seen that @xmath67 is in the plateau region . thus , the collection of the bare parameters @xmath75 is assumed to be identical to the initial condition @xmath70 without loss of accuracy @xcite . on the other hand , the numerical solution in the infrared limit obeys @xmath52 and @xmath53 in accordance with the analysis of the fixed point . now , for the noisy ks equation with @xmath75 , we consider the set @xmath76 of bare parameters @xmath68 , each of which has the same factors @xmath56 , @xmath57 , @xmath58 , and @xmath59 in the universal range and the same wavenumber scale @xmath65 as those for the noisy ks equation . the graph of @xmath24 for a given @xmath77 determines the wavenumber scale @xmath78 that represents the end of the ultraviolet plateau . note that the value of @xmath78 depends on @xmath77 . then , there is a special model with @xmath77 such that @xmath79 . for this model , as soon as the graph of @xmath24 exits from the ultraviolet plateau region , it enters the infrared universal range . in other words , this special model represents the universal behavior of the noisy ks equation in the most efficient manner . we refer to it as _ the most effective model for the universal range of the noisy ks equation _ with @xmath75 . below , we determine the most effective model . and @xmath80 for @xmath81 , respectively . @xmath82 and @xmath83 converge to the same value , @xmath84.[fig : fdt - ks ] ] _ representation of the parameter space. _ solution trajectories for the rg equation are expressed as curves in the five - dimensional parameter space consisting of @xmath85 . we attempt to simplify the representation of trajectories so as to determine the most effective model . first , recalling @xmath40 , we may restrict the parameter space into the subspace @xmath86 . next , as shown in fig . [ fig : fdt - ks ] , we find that @xmath82 and @xmath80 converge to the same value , 2.24 , in the universal range for the noisy ks equation . we can explain this phenomenon as follows . first , for the generalized kpz equations with @xmath68 satisfying @xmath87 , we can show the fluctuation - dissipation relation with the effective temperature @xmath88 fixed by using a time - reversal symmetry . this relation leads to the invariance property of @xmath82 and @xmath80 along solution trajectories of the rg equation ( [ eq : rgeq ] ) @xcite . for the other cases where @xmath89 including for noisy ks equations , @xmath82 and @xmath80 change in @xmath23 however , they satisfy @xmath90 in the universal range . therefore , it is reasonable to conjecture that the time - reversal symmetry emerges in the universal range . now , since the most effective model represents the universal behavior most efficiently , this special model should be in the subspace satisfying @xmath91 . on the basis of the results , we express the bare - parameter space by @xmath92 . for each value of @xmath93 , we have a model that exhibits the infrared universal behavior of @xmath75 . finally , for a generalized kpz equation with @xmath68 at @xmath67 in the ultraviolet plateau region , we consider the following scale transformation : @xmath94 , @xmath95 , and @xmath96 , which yields another generalized kpz equation with a different collection of bare parameters @xmath97 at @xmath98 in the ultraviolet plateau region . by imposing @xmath99 and @xmath100 , we obtain @xmath101 and @xmath102 . we then find that @xmath103 is invariant under the transformation @xcite . thus , we parameterize @xmath104 as @xmath105 . the next problem is to determine the values of @xmath106 and @xmath107 of the most effective model for the universal range of the noisy ks equation . _ most effective model. _ since @xmath107 is invariant under the scale transformation , the determination of @xmath107 can be separated from the determination of @xmath106 . here , we notice the condition @xmath108 for the most effective model . because this condition is invariant under the scale transformation , the value of @xmath107 is uniquely determined . furthermore , the condition @xmath109 fixes the value of @xmath106 . below , we explicitly calculate these values . as a function of @xmath110 for @xmath111 ( solid red line ) , @xmath112 ( green dash - dotted line ) , @xmath113 ( dotted blue line ) and @xmath114 ( dashed pink line ) . the inset shows the graphs for @xmath115 ( solid red line ) , @xmath116 ( green dash - dotted line ) , @xmath113 ( dotted blue line ) , and @xmath117 ( dashed pink line ) . [ fig : f - s ] ] in order to determine the value of @xmath107 , we study the dimensionless quantity @xmath118 as a function of @xmath119 , where @xmath120 and @xmath121 are invariant under the scale transformation . it should be noted that , for any @xmath107 and @xmath106 , @xmath120 approaches @xmath122 in the ultraviolet limit @xmath123 and @xmath120 approaches the value @xmath124 in the infrared limit @xmath125 . in fig . [ fig : f - s ] , we show graphs of @xmath120 as a function of @xmath121 for several values of @xmath107 . in general , there are two characteristic scales of @xmath121 , the departure scale from @xmath122 and the relaxation scale to @xmath126 , as clearly observed for @xmath127 . when @xmath107 increases , the peak of @xmath120 decreases and eventually vanishes at @xmath128 . in this case , the transition scale between the infrared universal region and the ultraviolet region is simply given by the cross point @xmath129 of the ultraviolet behavior @xmath130 and the infrared behavior @xmath131 . that is , @xmath132 , which gives @xmath133 . thus , we conclude that the value of @xmath107 of the most effective model is @xmath128 . for @xmath75 and the fitted curve , respectively . the inset shows the graphs of @xmath134 and @xmath135 with @xmath136 and @xmath137 . [ fig : diff - nu ] ] next , we determine the value of @xmath106 . from the cross point @xmath129 , we define the transition length scale @xmath138 by @xmath139 , which gives @xmath140 . here , the value of @xmath106 is determined by identifying @xmath141 with @xmath142 . thus , we estimate @xmath142 from the graph of @xmath0 for the noisy ks equation under study . in fig . [ fig : diff - nu ] , we show how @xmath0 approaches @xmath143 . we find that @xmath144 is well fitted to a power - law function of @xmath145 , which does not provide any wavenumber scale . through more detailed analysis , we find a fitting function @xmath146 \label{eq : difference}\end{aligned}\ ] ] with @xmath136 , @xmath137 , @xmath147 , and @xmath148 . from the second term of eq . ( [ eq : difference ] ) , we obtain the characteristic scale @xmath149 . now , from the condition @xmath150 , we obtain @xmath151 . thus , we have arrived at the most effective model for the universal range of the noisy ks equation with @xmath75 , where the collection of bare parameter values of the most effective model , @xmath152 , is determined as @xmath153 . now , the linear decay rate of the disturbance of wavenumber @xmath154 in the universal range is expressed as @xmath155 at an early time . here , we notice that @xmath156 defines one wavenumber scale . since the most effective model has only one wavelength scale @xmath141 , @xmath157 holds . this implies that the linear decay rate @xmath155 is estimated as @xmath158 for @xmath159 . in this manner , @xmath160 can be measured in experiments . indeed , by applying this method to the numerical simulation of the noisy ks equation , the result @xmath161 was obtained @xcite . thus , our theoretical value @xmath162 is in good agreement with the numerical value . and @xmath71 for the most effective model and the noisy ks equation , respectively . the solid ( red ) , dotted ( green ) , and dashed ( blue ) lines represent @xmath0 for @xmath75 , @xmath163 , and the infrared scaling behavior , respectively . the inset shows @xmath71 for these cases . [ fig : repredkpz ] ] _ concluding remarks. _ the main result of this letter is illustrated in fig . [ fig : repredkpz ] . for a given noisy ks equation , we construct the most effective model exhibiting the same infrared universal behavior with just one cross - over wavenumber scale @xmath142 connecting the infrared behavior and the ultraviolet behavior . we emphasize that our theory enables us to calculate the bare viscosity @xmath164 of the effective model in the universal range , which could not be obtained by previous studies . we conclude this letter by presenting a few remarks . the first remark is on the relevant parameter space in the universal range . since @xmath9 is a conserved quantity along the solution of the rg equation , it obviously depends on the initial condition @xmath60 . thus , it is relevant in the universal range . furthermore , @xmath165 is not relevant because @xmath166 approaches zero . at the same time , @xmath167 is a relevant parameter because its value is invariant along the solution trajectory when @xmath168 . finally , in the limit @xmath3 , @xmath169 approaches the universal constant value @xmath170 which is independent of @xmath60 . thus , we can state that @xmath169 is irrelevant , following the argument in @xcite . in other words , @xmath0 and @xmath71 are not independent of each other in the universal range . in summary , the relevant parameter space in the universal range is spanned by the three parameters @xmath171 . however , the parameter @xmath8 can not be negligible because the irrelevant parameter @xmath169 is not zero in the universal range . this is different from many standard rg analysis @xcite . second , we remark that the original yakhot conjecture claims a statistical property of the deterministic ks equation @xcite . here , we discuss the noiseless limit @xmath172 for the noisy ks equation . in this case , we obtain @xmath173 which is not consistent with observations . this implies that the lowest order contribution in loop expansions is not sufficient to yield statistical properties for the small @xmath174 limit . in order to overcome this situation , we have to formulate a non - perturbative calculation . this is an interesting problem for future work . finally , we expect that the concept proposed in this letter will be applied to various systems , although we have studied a specific phenomenon as an example of scale - dependent parameters . the most interesting example may be fluid turbulence . the effective model for the universal range in turbulence is given by a noisy navier - stokes equation , the noise intensity of which exhibits a divergence in the infrared limit , as suggested in @xcite . the analysis of solution trajectories of the rg equation for such a noisy navier - stokes equation may provide fresh insight into the understanding of turbulence . we hope that this letter stimulates the study of whole solutions of rg equations in various research fields . the authors thank k. a. takeuchi , m. itami , and t. haga for useful discussions . the present study was supported by kakenhi ( nos . 25103002 and 26610115 ) . 99 u. frisch , _ turbulence _ , ( cambridge university press , cambridge , 1995 ) . l. f. richardson , proc . r. soc . lond . a. * 110 * , 709 - 737 ( 1926 ) . y. pomeau and p. rsibois , phys . rep . * 19c * 64 ( 1975 ) . d. forster , d. r. nelson , and m. j. stephen phys . rev . a * 16 * , 732 ( 1977 ) . c. dedominicis and p. c. martin , phys . rev . a. * 19 * , 419 - 421 ( 1979 ) j. d. fournier and u. frisch , phys . rev . a * 17 * , 747 ( 1978 ) . j. d. fournier and u. frisch , phys . rev . a * 28 * , 1000 ( 1983 ) . v. yakhot and s. a. orszag , phys . rev . lett . * 57 * , 1722 ( 1986 ) . v. yakhot and s. orszag , j. sci . comput . * 1 * , 3 ( 1986 ) . v. yakhot and m. l. smith , j. sci . comput . * 7 * , 35 ( 1992 ) . g. l. eyink , phys . fluids * 6 * 3063 ( 1994 ) . m. kardar , g. parisi , and y .- c.zhang , phys . rev . lett . * 56 * , 889 ( 1986 ) . l. bertini and g. giacomin , comm . math . phys . * 183 * , 571 ( 1997 ) . t. sasamoto and h. spohn , phys . rev . lett . * 104 * , 230602 ( 2010 ) . k. a. takeuchi and m. sano , phys . rev . lett . * 104 * , 230601 ( 2010 ) . k. a. takeuchi , m. sano , t. sasamoto and h. spohn , sci . rep . * 1 * , 34 ( 2011 ) . g. amir , i. corwin , and j. quastel , comm . pure appl . math . * 64 * 466 ( 2010 ) . p. calabrese and p. ledoussal , phys . rev . lett . * 106 * , 250603 ( 2011 ) . t. imamura and t. sasamoto phys . rev . lett . * 108 * , 190603 ( 2012 ) . m. hairer , annals of mathematics * 178 * 559 , ( 2013 ) . y. kuramoto and t. tsuzuki , prog . . phys . * 55 * , 356 ( 1976 ) . g. i. sivashinsky , acta astron . * 4 * , 1177 ( 1977 ) . y. kuramoto , _ chemical oscillations , waves , and turbulence _ , ( springer , berlin , 1984 ) . v. yakhot , phys . rev . a * 24 * , 642 ( 1981 ) . v. yakhot and z .- s . she , phys . rev . lett . * 60 * , 1840 ( 1988 ) . s. zaleski , physica d * 34 * , 427 ( 1989 ) . k. sneppen , j. krug , m. h. jensen , c. jayaprakash , and t. bohr , phys . rev . a * 46 * , r7351 ( 1992 ) . f. hayot , c. jayaprakash , and c. josserand , phys . rev . e * 47 * , 911 ( 1993 ) . h. sakaguchi , prog . . phys . * 107 * , 879 ( 2002 ) . k. ueno , h. sakaguchi , and m. okamura , phys . rev . e * 71 * , 046138 ( 2005 ) . r. cuerno and k. b. lauritsen , phys . rev . e * 52 * , 4853 ( 1995 ) . see supplemental material for the determination of @xmath28 . we do not employ the rescaling transformation after the the coarse - graining . p. c. martin , e. d. siggia , and h. a. rose , phys . rev . a * 8 * , 423 ( 1973 ) . h. k. janssen , z. phys . b * 23 * , 377 ( 1976 ) . c. de dominicis , j. phys . colloq . * 37 * , c1 - 247 ( 1976 ) . c. de dominicis , phys . rev . b * 18 * , 4913 ( 1978 ) . see supplemental material for the derivation of @xmath175 . e. frey and u. c. tuber , phys . rev . e * 50 * , 1024 ( 1994 ) . l. canet , h. chat , b. delamotte , and n. wschebor , phys . rev . e * 84 * , 061128 ( 2011 ) . see supplemental material for the derivation of @xmath176 . @xmath72 does not show the plateau region in fig . [ fig : nud - ks ] , but this graph quickly converges to @xmath72 with @xmath177 at @xmath178 . thus , the bare parameter is defined as @xmath179 . see supplemental material for the detail . see supplemental material for the time - reversal symmetry of the generalized kpz equation and the derivation of the fluctuation - dissipation relation . see supplemental material for the explicit calculation of the scale transformation . j. polchinski , nucl . phys . b * 231 * , 269 ( 1984 ) . s. weinberg , _ the quantum theory of fields _ , ( cambridge university press , cambridge , 1995 ) . k. g. wilson and j. kogut , phys . rep . * 12 * , 75 ( 1974 ) . in general , by assuming time dependence of the infinitesimal parameter for a continuous symmetry transformation , we can obtain non - trivial identities such as eq . ( [ eq : wtid - shift1 ] ) . this technique , which has been referred to as `` gauging a global symmetry '' , is standard when we derive identities from a continuous global symmetry @xcite . for such a case , the variation of an action under a time - gauged transformation is expressed as @xmath180 , where @xmath181 is a noether charge of the corresponding global symmetry , and @xmath182 is the time - gauged infinitesimal parameter . the noether charge of the shift symmetry is calculated as @xmath183 , which is consistent with eq . ( [ eq : variation of shift ] ) . addtoresetequationsection addtoresetfiguresection

1703.08946

the unitary fermi gas is a basic many - body problem which describes strongly interacting fermions ranging from ultracold atoms near a feshbach resonance @xcite to dilute neutron matter . the properties in the dilute limit are independent of the microscopic details of the interaction potential and share a common universal phase diagram . a quantum critical point ( qcp ) at zero temperature governs the critical behavior in the whole phase diagram as a function of temperature @xmath0 , chemical potential @xmath7 , detuning from the feshbach resonance @xmath8 , and magnetic field @xmath9 @xcite . whereas conventional qcps separate two phases of finite density , in our case the density itself is the order parameter which vanishes for @xmath10 and assumes a finite value for @xmath11 @xcite . in the spin balanced case @xmath12 , and at resonance @xmath13 the fermi gas is unitary and scale invariant . in terms of the thermal length @xmath14 the density equation of state @xmath15 is a universal function which has been measured experimentally @xcite . the unitary fermi gas becomes superfluid at a universal @xmath16 @xcite , see fig . [ fig : phase ] . in this work we focus on the quantum critical regime @xmath17 above the qcp at @xmath12 , @xmath13 and @xmath18 , where @xmath19 is a universal constant . since the thermal length @xmath20 is comparable to the mean particle spacing @xmath21 , quantum and thermal effects are equally important . there is no small parameter , and it is a theoretical challenge to compute the critical properties . recent measurements @xcite and computations @xcite of the equation of state now agree to the percent level . however , a precise determination of transport properties is much more demanding . in order to reliably estimate transport coefficients we perform controlled calculations in a large-@xmath5 expansion @xcite . due to the lack of an intrinsic small parameter we introduce an artificial small parameter , @xmath22 , which organizes the different diagrammatic contributions , or scattering processes , into orders of @xmath22 . the original theory is recovered in the limit @xmath23 . one can perform controlled calculations by including all diagrams up to a certain order in @xmath22 , and these approximations can be systematically improved by going to higher order . this approach is similar to the @xmath24 expansion in the dimension of space . the advantage over perturbation theory is that it is controlled even at strong interaction , while in contrast to quantum monte carlo it works directly in the thermodynamic limit and needs no finite size scaling . we thus obtain new results for the tan contact density @xcite and the transport properties in the quantum critical region . the shear viscosity @xmath25 assumes a universal value at @xmath18 . in kinetic theory @xmath26 is given by the pressure @xmath27 times the viscous scattering time @xmath28 , which is related to the incoherent relaxation time of the gapless critical excitations above the qcp . the entropy density @xmath29 at @xmath18 is exactly proportional to the pressure , @xmath30 , and the viscosity to entropy ratio ( at @xmath23 ) @xmath31 is a universal number _ independent of temperature_. a temperature independent ratio @xmath4 has been found in certain string theories @xcite and is conjectured to hold as a lower bound in other models @xcite . strongly interacting quantum fluids which saturate this bound are called perfect fluids @xcite . among real non - relativistic fluids the unitary fermi gas comes closest to the bound and is almost perfect @xcite , while for graphene the viscosity decreases logarithmically with temperature in the quantum critical regime @xcite . we compare our large-@xmath5 results at @xmath23 @xcite with experimental measurements @xcite and other theoretical approaches , including self - consistent luttinger - ward @xcite and bold diagrammatic monte carlo ( bdmc ) @xcite calculations , see table [ tab : vals ] . .thermodynamic properties and transport coefficients of the unitary fermi gas in the quantum critical region @xmath18 , @xmath17 : density @xmath32 , pressure @xmath27 , entropy density @xmath33 , tan contact density @xmath6 , and shear viscosity @xmath34 , with fermi momentum @xmath35 . large-@xmath5 results extrapolated to @xmath23 . [ cols="<,<,<,<,<",options="header " , ] the excellent agreement between experiment and bdmc provides a reliable reference to assess the accuracy of other methods . we find very good agreement of the pressure @xmath27 with large-@xmath5 ( @xmath36 above bdmc ) and luttinger - ward ( @xmath37 below ) calculations , just slightly outside the error bars , and we find similarly good agreement for the entropy density @xmath33 . from the bdmc equation of state simulations of @xcite , one can extract ( via the pair propagator ) a preliminary value for the contact density @xcite @xmath38 . our large-@xmath5 value is just @xmath39 below the bdmc value , which is remarkable given how simple the calculation is , while the luttinger - ward value lies about @xmath40 above the bdmc value , just inside the error bars . experimental measurements of the contact @xcite yield @xmath41 for the trapped gas at @xmath18 ( @xmath42 ) , which agrees well with trap averaged calculations @xcite . however , knowledge of the trap averaged contact does not allow us to reconstruct the corresponding value for the homogeneous system , so we refrain from a direct comparison . dynamical and transport properties such as @xmath2 are harder to compute than thermodynamic properties , which makes simple approximations all the more valuable : we find that @xmath2 agrees to @xmath40 between large-@xmath5 and luttinger - ward theory , giving a narrow estimate . the viscosity of a trapped gas has been measured experimentally and agrees with trap averaged calculations @xcite , but differs from the viscosity of the homogeneous system . the body of this paper explains how these values are obtained : in section [ sec : ufg ] we review the renormalization group ( rg ) analysis of the unitary fermi gas and its universal phase diagram , in section [ sec : largen ] we perform thermodynamic and transport calculations using the controlled large-@xmath5 expansion , and in section [ sec : lw ] we extract the @xmath18 data from the self - consistent luttinger - ward calculation , before concluding in section [ sec : disc ] . in particular , in appendix [ app : tan ] we give a new derivation of the tan adiabatic and energy relations and show that they are satisfied _ exactly _ in self - consistent luttinger - ward approximations , while appendix [ app : boltz ] provides technical details on the quantum kinetic equation . the interacting two - component fermi gas is described by the action @xmath43 where @xmath44 are grassmann variables representing fermion species @xmath45 of equal mass @xmath46 , and the imaginary time @xmath47 runs up to the inverse temperature @xmath48 ( we use units where @xmath49 ) . @xmath50 is the chemical potential of species @xmath51 , but we will only consider the spin - balanced case @xmath52 . in @xmath53 dimensions the scattering amplitude for small relative momenta @xmath54 can be written in the form @xcite @xmath55 where the scattering length @xmath56 can be varied experimentally by an applied magnetic field , and the effective range @xmath57 depends on the details of the interatomic potential . by fine - tuning to a feshbach resonance @xmath58 the two - particle scattering remains strong at low energy @xmath59 and reaches the unitarity limit @xmath60 independent of @xmath57 . the low - energy properties remain universal at finite density @xmath61 if @xmath57 is much shorter than the mean particle spacing @xmath21 . this condition @xmath62 is realized physically for a dilute gas and near a broad feshbach resonance as in @xmath63li @xcite . a finite @xmath57 regularizes the contact interaction at short distances ( uv ) , and for a sharp momentum cutoff @xmath64 the detuning @xmath8 is related to the bare coupling @xmath65 in by @xmath66 note that the resonance @xmath13 can only be reached for attractive interactions @xmath67 , when a bound state of the interatomic potential is at the continuum threshold . more generally , this can be understood from an rg analysis of the model : at zero temperature and density the running coupling @xmath68 obeys the _ exact _ flow equation @xcite @xmath69 which in @xmath70 has an unstable fixed point at @xmath71 corresponding to the feshbach resonance . for smaller @xmath72 the fermions will form a bec of fermion pairs ; for larger @xmath73 the flow runs toward the attractive fixed point @xmath74 of the free fermi gas ( bcs limit ) . at the feshbach resonance fixed point the detuning @xmath8 is a relevant perturbation with scaling dimension @xmath75=d-2 $ ] . the zero temperature phase diagram exhibits a quantum critical point at the feshbach resonance @xmath13 , zero chemical potential @xmath18 , and zero spin imbalance @xmath12 , where the `` magnetic field '' @xmath9 couples to the difference in chemical potential @xmath76 . this critical point determines a universal phase diagram for finite @xmath0 , @xmath8 , @xmath7 and @xmath9 @xcite . in this work we concentrate on the spin balanced gas @xmath12 at unitarity @xmath13 : the phase diagram for finite @xmath0 and @xmath7 is depicted in fig . [ fig : phase ] . on the lower right for @xmath77 there is a superfluid phase of fermion pairs , while the left part is a normal fermi liquid phase at finite density . the phase transition line @xmath78 @xcite is universal and strictly linear , in contrast to the corresponding phase diagram for a dilute bose gas @xcite . on the left for @xmath79 the fermi liquid crosses over to a dilute classical gas . the line @xmath1 , @xmath10 has zero density ( vacuum ) . here we focus on the high - temperature quantum critical regime @xmath80 , and in the following we compute the thermodynamic and transport properties specifically for the representative value @xmath18 . it is useful to perform a hubbard - stratonovich transformation to decouple the fermion interaction . we introduce a complex field @xmath81 representing a fermion pair and write the bose - fermi action @xmath82 note that the pairing field @xmath83 has a positive gap because @xmath67 near the feshbach resonance . the action has the same critical behavior as the two - channel atom - molecule model at its zero - range fixed point @xcite . one can now proceed by integrating out the fermions to obtain an effective bosonic action for the pairing field @xmath83 . this action has bosonic vertices with any even number @xmath84 of fields which are given by a bare fermion loop with @xmath84 vertex insertions . in contrast to the repulsive fermi gas , where these vertices are irrelevant in the rg sense , for the unitary fermi gas these vertices all have marginal scaling . already the particle - particle loop ( @xmath85 ) , which contributes to the self - energy of the @xmath83 field , changes the bare scaling dimension @xmath86=d/2 $ ] of the @xmath83 field by an anomalous contribution @xmath87 to the true scaling dimension @xmath86= ( d+\eta_\phi)/2=2 $ ] , which is independent of @xmath88 ( for @xmath70 ) . similarly , all higher bosonic vertices @xmath89 are singular for small external frequencies and momenta and scale marginally in the rg sense . there is no small parameter to suppress these higher loop diagrams , and they are _ a priori _ equally important in the infrared ( ir ) . at zero density the @xmath90 particle @xmath91 functions are decoupled from the @xmath92 particle @xmath91 function in eq . , which is therefore exact . nevertheless , there may also be a three - particle resonance ( efimov effect ) in the three - particle @xmath91 function depending on the mass ratio and whether the particles are fermions or bosons @xcite . this changes the ground state from a two - particle to a three - particle bound state and leads to limit cycles in the rg flow @xcite . at finite density all higher bosonic vertices couple back into the self - energy of the @xmath83 field . in order to assess the quantitative importance of these higher vertices , one can introduce an artificial expansion parameter such as the dimension @xmath93 for @xmath70 @xcite or @xmath22 for a large number of fermion flavors @xmath5 @xcite . alternatively , one can use a monte carlo sampling of diagrams @xcite . in this work we perform a large-@xmath5 expansion and compare it with the results from other approaches . we modify the bose - fermi action by introducing @xmath5 identical copies , or flavors , of @xmath94 and @xmath95 fermions , denoted by @xmath96 with @xmath97 and the flavor index @xmath98 . the pairing field @xmath83 is chosen to create an @xmath99 pair of any flavor , and we obtain the action @xcite @xmath100 this action is @xmath101 invariant under rotations in flavor space . the gaussian integral over the fermion field yields the effective bosonic action @xmath102 \\ - \frac{1}{g_0 } |\phi|^2 \bigr\ } \\ = n t\sum_{\omega_m } \sum_{{\mathbfk } } \bigl\ { \sum_\sigma \ln g_{0\sigma}(k,\omega_m ) \\ - \mathcal t^{-1}(k,\omega_m ) |\phi(k,\omega_m)|^2 + \mathcal o(|\phi|^{\geq 4 } ) \bigr\}\end{gathered}\ ] ] with the trace running over the spin index @xmath51 . the bare fermi propagator @xmath103 is given by @xmath104 with dispersion @xmath105 , and the bosonic propagator @xmath106 is given by the regularized @xmath0-matrix in medium , @xmath107 the number of flavors @xmath5 appears only as a global prefactor in the action , hence a controlled loop expansion is possible @xcite . each closed fermion loop contributes a factor of @xmath5 , while each @xmath83 propagator is suppressed by @xmath22 . even though the higher bosonic vertices still have marginal scaling , their contributions to the grand potential are now suppressed quantitatively by powers of @xmath22 . for @xmath108 the system is in the normal phase , and the action has a saddle point at @xmath109 . to order @xmath110 the grand potential reads @xmath111 note that this order of the @xmath22 expansion extrapolated to @xmath23 is exactly the nozires schmitt - rink ( nsr ) theory @xcite . the matsubara frequency summation can be continued analytically to real frequency , @xmath112 \\ - \frac{1}{n } \int_{-\infty}^\infty \frac{d\omega}{\pi } \ , b(\omega ) \ , \delta(k,\omega,\mu,\nu ) \bigr\}\end{gathered}\ ] ] with the scattering phase shift @xmath113 and the bose function @xmath114^{-1}$ ] . specifically in @xmath53 the @xmath0-matrix reads ( in the spin - balanced case @xmath12 ) @xmath115 \,.\end{aligned}\ ] ] the integral is convergent and readily evaluated numerically . using we obtain for the pressure @xmath116 ( equation of state ) at @xmath18 , @xmath13 , @xmath12 and @xmath17 : @xmath117 where @xmath118 since the unitary fermi gas is scale invariant the internal energy density @xmath24 is proportional to the pressure @xcite @xmath119 also the entropy density @xmath120 at unitarity and @xmath18 is proportional to the pressure , @xmath121 the density at @xmath18 to order @xmath110 is @xmath122 where @xmath123 if this order of the @xmath22 expansion is evaluated at @xmath23 ( nsr ) we obtain for the density @xmath124 the ratio of thermal length to mean particle spacing , @xmath125 , is of order unity , hence quantum and thermal fluctuations are equally important in the high - temperature quantum critical region . the density determines the fermi temperature @xmath126 which is useful to compare with data given in terms of the reduced temperature @xmath127 finally , the tan contact density is defined as the total spectral weight ( density ) of the pairing field @xcite @xmath128 at @xmath23 the contact can be expressed in terms of @xmath129 using eq . which yields @xmath130 . this is equivalent to the non - self - consistent @xmath0 matrix result @xcite and agrees with the bdmc calculation within @xmath39 ( see table [ tab : vals ] ) , but it differs from the result in @xcite by a factor of two . note that the tan adiabatic theorem @xcite @xmath131 is fulfilled exactly in the @xmath22 expansion : the change of the pressure with detuning is @xmath132 because the change of scattering phase shift with detuning is @xmath133 , and using eq . we obtain . at @xmath134 the fermions are free : once a shear flow is excited in the infinite system it will continue forever , and the dynamic shear viscosity is @xmath135 the drude weight is proportional to the pressure , in accordance with the viscosity sum rule @xcite . at order @xmath22 the fermions acquire a self - energy correction by scattering off pairing fluctuations , so for large @xmath5 the fermions are almost free quasi - particles with lifetime @xmath136 and an energy shift of the quasi - particle dispersion @xmath137 . in kinetic theory the dynamic viscosity becomes @xmath138 with the viscous scattering time @xmath139 : the @xmath140 function in is broadened to a peak of width @xmath22 and height @xmath5 . note that the high - frequency tail @xmath141 @xcite is not seen in kinetic theory @xcite . in order to compute transport properties for large @xmath5 it is justified to use the quantum boltzmann equation @xcite : ( i ) the fermions propagate as free particles between collisions , up to subleading corrections , ( ii ) the collision integral @xmath142 contains only particle - particle scattering described by the medium @xmath0-matrix @xmath143 [ eq . ] because particle - hole scattering appears at higher orders , and ( iii ) in addition to the collision ( dynamic ) term there is a shift of the dispersion ( kinetic ) term @xmath137 of the same order . however , it is only a subleading correction to the leading real term @xmath144 ( see below ) and can be neglected . based on these considerations we arrive at the boltzmann equation @xcite @xmath145\end{aligned}\ ] ] for the distribution function @xmath146 , where @xmath147 $ ] is the collision integral . for the shear viscosity we consider a velocity field @xmath148 with a small shear gradient @xmath149 , and the local equilibrium distribution @xmath150 with @xmath151 . in the stationary limit the boltzmann equation becomes @xcite @xmath152 \,.\end{aligned}\ ] ] the velocity gradient induces a momentum current density @xmath153 proportional to @xmath149 , with the coefficient given by the shear viscosity @xmath34 . we choose a deviation from the equilibrium distribution , @xmath154 with @xmath155 and @xmath156 , such that the momentum current density is @xmath157 this is equal to the pressure for free fermions ( @xmath134 ) at arbitrary temperature , as can be seen by integrating by parts . we can now replace @xmath158 in and take moments of the boltzmann equation by integrating both sides with @xmath159 . the left - hand side becomes @xmath160 while the right - hand side yields the collision integral @xcite @xmath161 \notag \\ & = \frac 2 t \int \frac{d^3p}{(2\pi)^3 } \ , v_y p_x \int \frac{d^3p_1}{(2\pi)^3 } \ , \int d\omega \ , \frac{d\sigma}{d\omega}\ , |{\mathbfv } - { \mathbfv}_1| \notag \\ & \quad \times f_p^0 f_{p_1}^0 ( 1-f_{p'}^0 ) ( 1-f_{p_1'}^0 ) \notag \\ & \quad \times \bigl [ \varphi({\mathbfp } ) + \varphi({\mathbfp}_1 ) - \varphi({\mathbfp } ' ) - \varphi({\mathbfp}_1 ' ) \bigr]\end{aligned}\ ] ] where fermions with incoming momenta @xmath162 , @xmath163 scatter into outgoing momenta @xmath164 , @xmath165 . it will be convenient to express these momenta in terms of the total momentum @xmath166 and the relative momenta @xmath167 ( @xmath168 ) of the incoming ( outgoing ) particles , with @xmath169 by energy conservation . the occupation numbers give the probability that the incoming states are occupied , and the outgoing states are not . the differential cross section is given by the medium @xmath0-matrix @xmath170 in the vacuum limit the center - of - mass scattering depends only on the relative momentum @xmath54 , @xmath171 but at finite density there is an additional dependence on the total momentum @xmath172 in the medium @xmath0-matrix @xmath173 . in relative coordinates the shear term in eq . is @xcite @xmath174 the collision integral then reads @xmath175\end{aligned}\ ] ] with the @xmath176-wave angular average @xmath177 over the fermi distribution functions derived analytically in appendix [ app : boltz ] ( the @xmath88-wave average @xmath178 contributes only @xmath179 to the integral ) . finally , only two integrals over the radial momenta @xmath172 and @xmath54 have to be performed . in the dilute classical regime the collision integral can be computed analytically , @xmath180 with fugacity @xmath181 , and in the same limit the pressure is @xmath182 . the viscosity is then given by @xcite @xmath183 in the high - temperature quantum critical regime @xmath17 , @xmath18 the collision integral has to be computed with the full medium @xmath0-matrix @xmath184 from eq . , which is done numerically and yields @xmath185 and together with the pressure at leading order in @xmath22 , @xmath186 , we obtain in the quantum critical regime @xmath187 this value is about @xmath188 lower than in the dilute classical limit , which is mostly due to the reduced pressure , while the effects of reduced density and increased medium scattering almost cancel each other in @xmath189 . with the viscous relaxation time @xmath190 and the entropy density @xmath191 we obtain the universal viscosity to entropy ratio independent of temperature , @xmath192 a related computation of the viscosity using the medium @xmath0-matrix has been performed for large attractive interaction @xmath193 which found @xmath194 for @xmath18 at @xmath195 @xcite , slightly larger than our value at @xmath23 . note that we have evaluated @xmath34 using only a single moment of the boltzmann equation , but is has been shown that corrections to @xmath34 from higher moments are less than @xmath196 @xcite . a similar transport calculation using the medium @xmath0-matrix in two dimensions has been performed recently @xcite . the luttinger - ward theory provides a systematic way to obtain self - consistent and conserving approximations , such that the green s functions satisfy all symmetries and conservation laws of the model @xcite . the luttinger - ward functional @xmath197 $ ] can be defined in terms of full fermionic propagators @xmath198 and full bosonic propagators @xmath199 . the exact theory is given by an infinite set of irreducible contributions to the @xmath200 functional which can not be evaluated in practice , so typically one chooses a subclass of diagrams . for the unitary fermi gas a very successful approximation is to use ladder diagrams with full fermionic green s functions @xcite . then the full @xmath0-matrix is given by an expression similar to but with full green s functions , @xmath201 since we are interested in the high - temperature critical region we consider only the expressions valid in the normal phase . the luttinger - ward theory then prescribes that the @xmath94 fermionic self - energy is given by scattering a @xmath95 fermion off pair fluctuations described by the full @xmath0-matrix , @xmath202 and analogously for @xmath203 . the dyson equation determines the full fermionic green s functions @xmath204 this set of equations is solved self - consistently by iteration @xcite . the resulting green s functions in matsubara frequency can be continued analytically to obtain the spectral functions in real frequency , which show substantial broadening near @xmath205 and additional excitations beyond a single quasi - particle peak @xcite . similar features are observed in the spin polarized case @xcite . the pressure @xmath116 is obtained from the grand potential @xcite @xmath206 - \ln \mathcal t(k,\omega_m ) \bigr\}\end{gathered}\ ] ] evaluated using the self - consistent fermion propagator and the full @xmath0-matrix . we extract the high - temperature quantum critical behavior from the existing thermodynamic data @xcite interpolated at @xmath18 . specifically , we make a cubic spline interpolation of @xmath207 and find the solution of @xmath208 at @xmath209 , which implies @xmath210 . furthermore , we find @xmath211 , @xmath212 , and @xmath213 , which can be recast in terms of @xmath20 . these values are summarized in table [ tab : vals ] and are remarkably close to the experimental values . the shear viscosity @xmath214 has been computed in luttinger - ward theory as a function of temperature and frequency @xcite : it has a lorentzian peak at low frequency , followed by a universal tail @xmath215 proportional to the contact density . we make a cubic spline interpolation of @xmath216 and find the root at @xmath217 , which yields @xmath218 . this result is slightly lower than the large-@xmath5 value in eq . . we note that in this self - consistent calculation the minimum of @xmath219 is found at a somewhat lower temperature @xmath220 @xcite . the unitary fermi gas in the high - temperature quantum critical region is a challenging many - body problem . it is strongly interacting , with the density almost twice the non - interacting value at @xmath18 @xcite , and has no small expansion parameter . still , our large-@xmath5 results at the first non - trivial order beyond the free fermi gas are already remarkably close to reliable experimental and theoretical results @xcite . a main result of the present paper is that this is true also for the transport properties @xmath2 once medium effects are included in the quantum kinetic equation . a possible reason for this good agreement is that large-@xmath5 and luttinger - ward approximations satisfy the tan adiabatic and energy relations exactly , as we show in appendix [ app : tan ] . in addition , luttinger - ward theory exactly fulfills the scale invariance of the unitary fermi gas @xcite . for a better comparison between calculations for the homogeneous system and experiments it would be desirable to have local measurements in the spirit of ref . @xcite also for the contact and transport properties , since the comparison of trap averaged quantities is less sensitive to the details of the temperature dependence . a promising step in this direction is to selectively probe atoms near the center of the trap in order to extract the contact density from the tail of the momentum distribution @xcite . i wish to thank lars fritz , subir sachdev , jrg schmalian , richard schmidt , and wilhelm zwerger for fruitful discussions and mark ku , thomas schfer , chris vale , flix werner , and martin zwierlein for sharing their data . consider the fermionic action : a small variation of the quadratic term , @xmath221 , will lead to a change in the grand potential @xmath222 with the trace running over space , time and possibly spin indices . however , this equation is often violated if approximations are made for the full green s function @xmath223 . a unique feature of conserving approximations , which are derived from a luttinger - ward functional @xmath224 $ ] , is that eq . holds exactly even for approximate @xmath225 and @xmath223 @xcite . for the strongly interacting fermi gas it is convenient to start from the bose - fermi action and define a luttinger - ward functional @xmath197 $ ] in terms of both fermionic and bosonic green s functions @xcite . then a variation of the microscopic parameters @xmath226 and/or @xmath227 induces a change of the grand potential @xcite @xmath228 again , this exact equation continues to hold within conserving approximations with full self - consistent propagators @xmath198 and @xmath229 . we will now show that the tan adiabatic theorem @xcite @xmath230 and the tan energy formula @xcite @xmath231 are consequences of and therefore hold not only in the exact theory but in any conserving approximation , including the self - consistent @xmath0-matrix approximation introduced in section [ sec : lw ] . a variation of detuning changes only the bosonic quadratic term @xmath232 in the action , @xmath233 the variation of the grand potential is then @xmath234 with the density of bosons expressed by the tan contact density , @xmath235 inserting into directly yields the adiabatic theorem . in order to derive the energy formula we consider a variation of mass , @xmath236 usually this yields only the kinetic energy ( cf . ( 61 ) in @xcite ) , but in our case the interaction term @xmath237 also depends on mass , so is the full internal energy @xmath238 including the potential term . specifically , @xmath239 and with the momentum distribution function @xmath240 we obtain the internal energy density @xmath241 where the @xmath54 integral extends to the momentum cutoff @xmath242 . the regularization term @xmath243 can be written as @xmath244 , and we arrive at the energy formula . in a similar way the tan pressure relation has been derived in the luttinger - ward theory by an infinitesimal scale transformation on the grand potential @xcite . this concludes our proof that the tan relations are fulfilled exactly in the self - consistent @xmath0-matrix approximation . a useful feature of @xmath53 dimensions is that the angular averages of the distribution functions can be performed analytically . one can write the product of fermi functions in with @xmath169 as @xmath245 [ 1-f(\varepsilon_{{\mathbfq}/2-{\mathbfk } ' } ) ] \\ = \frac{1}{4(\cosh{a}+\cosh{bx})(\cosh{a}+\cosh{bx'})}\end{gathered}\ ] ] with @xmath246 , @xmath247 and @xmath248 , @xmath249 . the angular average over the solid angles of the vectors @xmath250 , @xmath251 and @xmath252 is then @xmath253[1-f ] \\ = \frac{k^4}{15 } \bigl ( i_{\ell=0}^2(q , k ) - i_{\ell=2}^2(q , k ) \bigr ) \end{gathered}\ ] ] where we have defined the @xmath176-wave angular average of the distribution functions @xmath254 with legendre polynomials @xmath255 . the @xmath33-wave average is given by @xmath256 } { \cosh[(a - b)/2]}\end{aligned}\ ] ] while the @xmath88-wave average can be expressed in terms of polylogarithms @xmath257 , @xmath258.\end{gathered}\ ] ] thus , all angular integrations can be done analytically and only the two radial integrations over @xmath172 and @xmath54 in eq . need to be performed numerically .

the thermodynamic and transport properties of the unitary fermi gas at finite temperature @xmath0 are governed by a quantum critical point at @xmath1 and zero density . we compute the universal shear viscosity to entropy ratio @xmath2 in the high - temperature quantum critical regime @xmath3 and find that this strongly coupled quantum fluid comes close to perfect fluidity @xmath4 . using a controlled large-@xmath5 expansion we show that already at the first non - trivial order the equation of state and the tan contact density @xmath6 agree well with the most recent experimental measurements and theoretical luttinger - ward and bold diagrammatic monte carlo calculations .

1204.1980

discovery of charge migration in dna molecules has opened new avenues to investigate various possibilities ranging from its role in the dna oxidative damage and repair @xcite to application of dna in nanoelectronic device developments @xcite . in fact , dna - based molecular electronic devices are expected to operate within the picoseconds range @xcite that can exceed the potential of the present solid state devices . quite expectedly , the dna molecule has become a subject of intense research activities both theoretically @xcite and experimentally @xcite . from all these studies of charge migration in the dna molecule reported as yet , it is clear that there are two mechanisms for transfer of charge depending on the dna structure and transfer parameters : a superexchange charge transfer and the incoherent hopping @xcite . the charge migration leads to the geometry changes in the nucleotides and the surrounding environment , which significantly contribute to the charge migration process . due to the interaction of the @xmath0 orbitals of the nearest neighbor duplexes and insignificant ip difference between them , hole can be distributed over several sites in the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers . this significantly changes the magnitudes of the geometry relaxation of the nucleobases inner - sphere component and environment contribution outer - sphere component . however , the investigation of the transfer parameters , such as orbital overlapping @xcite and activation energy for charge migration i.e. the ip and the reorganization energy @xcite , have been performed mostly for the nucleobases or / and base pairs . the main purpose of our work is to estimate the electronic coupling between the two nearest nucleobases , their charge distribution and inner - sphere reorganization energy , when they are placed within the ( a - t)@xmath1 and ( g - c)@xmath1 oligomer duplexes . all these computations have been performed using accurate quantum - chemical methods . the relatively small reaction free energy in the dna molecule makes the dna hole transfer mechanism qualitatively different from that in most proteins @xcite . the electron transfer in the dna molecule was found to be strongly dependent on the details of the donor and acceptor energies and deviation of their geometries @xcite . the charge transfer in a dna molecule occurs due to the overlapping between the @xmath0-electrons of the carbon and the nitrogen atoms that forms the @xmath4 orbitals between the parallel nucleobases . charge migration in the molecular systems with weakly interacting donors and acceptors , such as between the base pairs in the dna molecule , is described by the standard high - temperature nonadiabatic electron - transfer rate @xmath5 where @xmath6 is the electronic donor - acceptor matrix element , and fc is the franck - condon factor . the electronic donor - acceptor matrix element @xmath6 is defined by the coupling of the orbitals of the donor and the acceptor and depends on the structure of the dna molecule . for the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers the simple expressions for the deviation of the electronic coupling on the sequence number @xmath2 have been generated @xcite . according to these expressions , the value of the electronic coupling decreases with elongation of the oligomers @xcite . in sect . iii a , we simulate the electronic coupling of the nucleobases within the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers using the quantum chemistry methods with the jaguar 6.5 program @xcite . according to the koopmans theorem , the electronic coupling can be estimated as half of the adiabatic state splitting between the homo and the homo-1 of the closed shell neutral system , determined in a hartree - fock self - consistent field . therefore , the rhf/6 - 31@xmath7g@xmath8 have been applied for the electronic coupling calculations . the 6 - 31@xmath7g@xmath8 basis set is appropriate for our purposes . previous investigations indicated that any further extension has little influence on the electron coupling @xcite . the geometries of the separated dna base pairs have been optimized with the rhf/6 - 31@xmath7g@xmath8 bases and in the following , the optimized geometries of the base pairs have been stacked with a twist angle 36@xmath9 and a distance of 3.38 . the stacking of the preliminary optimized geometries allows us to consider the same nucleobases to be ` in resonance ' @xcite within the structures of the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers . the fc factor deals with the influence of the vibronic interaction on the charge propagation and can be expressed as @xmath10 where @xmath11 is the free energy of the reaction , and @xmath12 is the reorganization energy . the interaction of the molecule with the solvent environment is included in the outer - sphere reorganization energy @xmath13 , while the relaxation of the acceptor , the donor and the molecular bridge geometries are included in the inner - sphere reorganization energy @xmath14 . for adding one electron to the positive ion and ( b ) the reorganization energy @xmath15 for removing one electron from the neutral geometry , where @xmath16 and @xmath17 are the hole donor and the hole acceptor . ] the inner - sphere reorganization energy accounts for the low - frequency inter - molecular modes and can be estimated within the quantum chemical approach as @xcite @xmath18 where @xmath19 is the energy of the neutral state in a neutral geometry , @xmath20 is the energy of the neutral state in an ionic geometry , @xmath21 is the energy of the ionic state in an ionic geometry , and @xmath22 is the energy of the ionic state in a neutral geometry . the reorganization energy @xmath15 is the energy to remove an electron from the hole acceptor @xmath17 , while the reorganization energy @xmath23 is the energy to add an electron to the hole donor @xmath24 . the scheme for calculation of the reorganization energy is presented in figure [ fig : fig1 ] . clearly , the vibronic interactions stabilize the geometry of the donor and the acceptor from a non - equilibrium state ( @xmath25 ) to the equilibrium state ( @xmath26 ) . the vertical ionization potential is determined as @xmath27 and differ from the adiabatic @xmath28 by the inner - sphere reorganization energy . the inner - sphere reorganization energy has been evaluated within the unrestricted becke3p86/6 - 311@xmath7g@xmath8 approximation of the dft method . the dft theory was found to be reasonable for this purpose based on a comparison of the results of ref . these results show that the dft theory predicts the magnitude of the inner - sphere reorganization energy with a minimum error when compared to the experimental data @xcite . furthermore , we have also tested the application of the hf method and the dft theory for the vertical ionization potential ( vip ) calculations and have found significant qualitative and quantitative disagreement of the hf with the experimental data @xcite , while the becke3p86 approximation is appropriate for this purpose . at first we consider the system of two stacked duplexes . the results for the highest occupied base orbital ( hobo ) are presented in table [ tab : table2 ] . in the case when the pyrimidine / pyrimidine and purine / purine bases are stacked in one strands , the hobos of the adenine and guanine bases have lowest energy in comparison to the pyrimidine / purine configurations . for the ( a - t)@xmath29 and ( g - c)@xmath29 oligomers the hobos are delocalized over the two intrastrand nucleobases , and therefore , it produces a significant coupling between the @xmath0 orbitals of the stacked pyrimidine / pyrimidine and purine / purine bases . for oligomers where the pyrimidine and the purine bases are stacked in the same strand ( a - t / t - a , g - c / c - g , a - t / c - c ) or in the mixed structures ( a - t / g - c and g - c / a - t ) , for some cases the @xmath0 orbitals are delocalized , but electronic coupling is weak . for others the @xmath0 orbitals are localized mostly on one nucleobase , and we can consider the weak intrastrand and interstrand coupling between the nucleobases as well . the low interstrand coupling for the cases a - t / t - a and g - c / c - g has been observed experimentally @xcite . [ cols="^,^,^,^,^,^,^,^ " , ] further , the hydrogen bonds are the channels for charge transfer between the nucleobases . in the oxidized state the hydrogen bonds participate in the charge transfer between the nucleobases to bring the pairs from the nonequilibrum state , where the charge is localized only on the nucleobase with a lower ip , to the equilibrium state , where the charge is spread over the base pair @xcite . we consider the a - t and g - c base pairs as a single state for the following calculations of @xmath14 . the stacking of the base pairs into the ( g - c)@xmath1 and ( a - t)@xmath1 oligomers leads to a decrease of the inner - sphere reorganization energy @xmath30 and a decrease of the vip as well . the results are presented in figure [ fig : fig2 ] , where the decrease of @xmath14 is seen to occur due to the contribution of the rotation and translation of the base pairs relative to each other and to the spreading of the charge between the pairs . according to our data , with elongation of the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers the twist of the base pairs mostly contributes to the decrease of the geometry relaxation of each nucleobase and in a reduction of the @xmath30 . the decrease of the energies of the adiabatic ip ( see fig . [ fig : fig5 ] ) and the inner - sphere reorganization energy ( see fig . [ fig : fig2 ] ) provide the decrease of the vip , which is the sum of above two components . and the vip values versus the number of pairs in a dna duplex oligomers ( a - t)@xmath1 and ( g - c)@xmath1 performed with ub3p86/6 - 311@xmath7g@xmath8 ] as we mentioned above , @xmath30 depends on the charge distribution over the chain . the electrostatic potential distribution in the ( g - c)@xmath1 and ( a - t)@xmath1 oligomers and respectively the residence of the homo in the oligomer centers provides the localization of the charge on the central guanines and adenines in the oxidized state . we have calculated the charge distribution as the difference between oxidized @xmath32 and neutral states @xmath33 with mulliken population analysis @xcite . in the ( g - c)@xmath1 and ( a - t)@xmath1 sequences the charge is distributed along the chain and is characterized by the low charge density at the dna molecule sides . for example , the density of the atomic partial charge localized on the @xmath2=1 site is lower than that at the chain center by 0.06 coul for the ( a - t)@xmath1 and by 0.25 coul for the ( g - c)@xmath1 sequences . for the ( g - c)@xmath34 sequence our results are in agreement with the data in ref . @xcite . therefore , the charge accumulation in the oligomer centers in the oxidized state produces the maximum geometry relaxation in the center of the dna chain . we have performed an estimation of the geometry relaxation of the separated base pairs @xmath35 within the optimized geometries of the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers , where @xmath36 . the simulation results of @xmath35 for @xmath2=3 and @xmath2=5 are presented in figure [ fig : fig3 ] . clearly , for the ( g - c)@xmath1 sequences the difference of the structure relaxation at the sides of the chain and in the center is significant than that for the ( a - t)@xmath1 sequences . the behavior of these curves repeats primarily the charge distribution in the ( a - t)@xmath1 and ( g - c)@xmath1 sequences . corresponding to the single base pairs within the ( a - t)@xmath1 and the ( g - c)@xmath1 oligomers , where @xmath37 and @xmath2=5 are calculated with ub3p86/6 - 311@xmath7g@xmath8 . ] the difference between the inner - sphere reorganization energy of the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers should provides the larger magnitude of the vibrational coupling constant for the g - c pairs than that for the a - t pairs , and larger for the guanine than that for the adenine ( see table [ tab : table1 ] ) . we have performed accurate quantum - chemical calculations to determine the electron coupling and the inner - sphere reorganization energy for the ( a - t)@xmath1 and ( g - c)@xmath1 dna oligomers , where @xmath36 . the electronic coupling between the two neighbor nucleobases within the same strand decreases exponentially with increasing of the base pairs number @xmath2 participating in the chain formation . the @xmath384 is the sequence number required for an accurate evaluation of the electron coupling in the dna molecule . the orbital distribution in oligomers with the hobo residing on the central nucleobase have been found to be the main reason for charge accumulation on the base pair located close to the chain center . the charge distribution in the chain determines degree of the the geometry relaxation of the base pair during the oxidation process in dependence on their location within the oligomer . therefore , the base pairs in the chain center have stronger geometry distortion during the oxidation process . such results are in good agreement with the theory of polaron formation in the dna molecule , where the maximum structure distortion occurs in the polaron center @xcite . the authors would like to thank dr . e.b . starikov for useful discussion . this work has been supported by the canada research chair program and a canadian foundation for innovation ( cfi ) grant . k.b . beckman , b.n . ames , _ j. biol . chem . _ * 1997 * , _ 272 _ , 19633 - 19636 ; s. loft , h.e . poulsen , _ _ * 1996 * , _ 74 _ , 297 - 312 ; a.p . grollman , m. moriya , _ trends in genetics _ * 1993 * , _ 9 _ , 246 - 249 . m. taniguchi and t. kawai , _ physica e _ * 2006 * , _ 33 _ , 1 - 12 . c. wan , t. fiebig , s. o. kelley , c. r. treadway , j. k. barton , a. h. zewail , _ proc . u.s.a . _ * 1999 * , _ 36 _ , 6014 - 6019 . kelley , j.k . barton , _ science _ * 1999 * , _ 283 _ , 375 - 383 . chem . _ * 1990 * , _ 94 _ , 4877 - 4883 . berashevich , t. chakraborty , _ j. chem . phys . _ * 2007 * , _ 126 _ , 035104 . b. giese , j. amaudrut , k. khler , m. spormann , s. wessely , _ nature _ * 2001 * , _ 412 _ , 318 - 320 . h. sugiyama , i. saito , _ soc . _ * 1996 * , _ 118 _ , 7063 - 7068 . voityuk , n. rsch , _ j. chem . phys . _ * 2002 * , _ 117 _ , 5607 - 5616 . a. troisi , g. orlandi , _ chem . lett . _ * 2001 * , _ 344 _ , 509 - 518 . s. rche , _ phys . lett . _ * 2003 * , _ 91 _ , 108101 . x.f . wang , t. chakraborty , _ phys . _ * 2006 * , _ 97 _ , 106602 ; v.m . apalkov , t. chakraborty , _ phys . * 2006 * , _ 72 _ , 161102 ( r ) . f.d . lewis , r.s . kalgutkar , y. wu , x. liu , j. liu , r. t. hayes , s. e. miller , m. r. wasielewskia , _ j. am . _ * 2000 * , _ 122 _ , 12346 - 12351 . g. b. schuster , _ acc . chem . _ * 2000 * , _ 33 _ , 253 - 260 . s. hess , m. gotz , w.b . davise , m.e . michel - beyerle , _ j. am . soc . _ * 2001 * _ 123 _ , 10046 - 10055 . oneill , j.k . soc . _ * 2004 * , _ 126 _ , 11471 - 11483 . k. senthilkumar , f. c. grozema , c. f. guerra , f. m. bilkelhaupt , f. d. lewis , y. a. berlin , m. a. ratner , l. d. a. siebbeles , _ soc . _ * 2005 * , _ 127 _ , 14894 - 14903 . tavernier , m.d . fayer , _ j. phys b _ * 2000 * , _ 104 _ , 11541 - 11550 . g.s.m . tong , i.v . kurnikov , d.n . beratan , _ j. phys b _ * 2002 * , _ 106 _ , 2381 - 2392 . a. harriman , _ angew . ed . _ * 1999 * , _ 38 _ , 945 - 949 . j. olofsson , s. larsson , _ j. chem b _ * 2001 * , _ 105 _ , 10398 - 10406 . hush , a.s . cheung , _ chem . _ * 1975 * , _ 34 _ , 11 - 13 . starikov , _ philosophical magazine _ * 2005 * , _ 85 _ , 3435 - 3462 . jaguar , version 6.5 . llc : new york , ny , 2005 . marcus , _ j. phys . chem . _ * 1994 * , _ 98 _ , 7170 - 7174 . sevilla , b. besler , a.o . colson , _ j. phys . chem . _ * 1995 * , _ 99 _ , 1060 - 1063 . orlov , a.n . smirnov , y.m . varshavsky , _ tetrahedron lett . _ * 1976 * , _ 48 _ , 4377 - 4378 . oneill , j.k . barton , _ pnas _ * 2002 * , _ 99 _ , 16543 - 16550 . q.zhu , p.r . lebreton , _ _ * 2000 * , _ 122 _ , 12824 - 12834 . baker , d. sept , s. joseph , m.j . holst , j.a . mccammon , _ proc . usa _ * 2001 * , _ 98 _ 10037 - 10041 . besler , k.m . merz , p.a . kollman , _ j. comput . chem . _ * 1990 * , _ 11 _ , 431 - 439 . guerra , f.m . bickelhaupt , j. g. snijders , e. j. baerends , _ chem . j. _ * 1999 * , _ 5 _ , 3581 - 3595 . mulliken , _ * 1955 * , _ 23 _ , 1833 - 1840 . s.s . alexandre , e. artacho , j.m . soler , h. chacham , _ phys . lett . _ * 2003 * , _ 91 _ , 108105 . p. maniadis , g. kalosakas , k.o . rasmussen , a.r . bishop , _ phys . b _ * 2003 * , _ 68 _ , 174304 .

we report on our calculations of the inner - sphere reorganization energy and the interaction of the @xmath0 orbitals within dna oligomers . the exponential decrease of the electronic coupling between the highest and second highest occupied base orbitals of the intrastrand nucleobases in the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers have been found with an increase of the sequence number @xmath2 in the dna structure . we conclude that for realistic estimation of the electronic coupling values between the nucleobases within the dna molecule , a dna chain containing at least four base pairs is required . we estimate the geometry relaxation of the base pairs within the ( a - t)@xmath1 and ( g - c)@xmath1 oligomers ( @xmath3 ) due to their oxidation . the decrease of the inner - sphere reorganization energy with elongation of the oligomer structure participating in the oxidation process have been observed . the maximum degree of geometry relaxation of the nucleobase structures and correspondingly the higher charge density in the oxidized state are found to be located close to the oligomer center . = 1.5truecm

cond-mat0703764

photoemission spectroscopy measures the energy distribution of photo - emitted electrons when materials are irradiated with light @xcite(fig . it is widely used in solid state physics and chemistry for investigating the electronic structure of surface , interface and bulk materials@xcite . recently it has become a prime choice of technique in studying strongly correlated electron systems@xcite , such as high temperature superconductors@xcite . the availability of synchrotron light sources and lasers , combined with the latest advancement of electron energy analyzer , has made a dramatic improvement on the energy resolution of photoemission technique in the last decade ; an energy resolution of @xmath0 5mev or better can now be routinely obtained . these achievements have made it possible to probe intrinsic properties of materials and many - body effects@xcite . for example , measurements of the superconducting gap on the order of 1 mev , as in conventional superconductors@xcite and in some high temperature superconductors@xcite , have been demonstrated . on the other hand , the utilization of pulsed light sources , such as synchrotron light or pulsed lasers , has also brought about concerns of the space charge effect@xcite . when a large number of electrons are generated from a short pulsed source and leave the sample surface , the electrons will first experience a rapid spatial distribution depending on their kinetic energy . then , because of the coulomb interaction , the fast electrons tend to be pushed by the electrons behind them while the slow electrons tend to be retarded by those fast electrons . this energy redistribution will distort the intrinsic information contained in the initial photoelectrons by giving rise to two kinds of effects . one is a general broadening of the energy distribution , due to both acceleration and retardation of electrons in their encounters . the other is a systematic shift in the energy . the space charge broadening of the energy distribution has been known for a long time as a limiting factor in electron monochromators and other electron beam devices@xcite , but it has not been considered in photoemission until very recently@xcite . the main concern there was whether such an effect will set an ultimate limit on further improving the energy resolution of the photoemission technique@xcite . here we report the first experimental observation of the space charge effect in photoemission . in addition , by combining experimental measurement with numerical simulations , we show that the mirror charges ( also known as image charges in the literature ) in the sample also play an important role in the energy shift and broadening . the combined effect of these coulomb interactions gives an energy shift and broadening on the order of 10 mev for a typical third - generation synchrotron light source , which is already comparable or larger than the energy resolution set by the light source and the electron analyzer . the value is also comparable to the many - body effect actively pursued by modern photoemission spectroscopy . these effects , therefore , should be taken seriously in interpreting experimental data and in designing next generation experiments . the experiment was carried out on beamline 10.0.1 at the advanced light source . this is a third - generation synchrotron source which generates pulsed light with a frequency of 500 mhz and a duration of @xmath060 ps . the beamline can generate linearly - polarized bright ultraviolet light with a photon flux on the order of 10@xmath1 photons / second with a resolving power e/@xmath2e of 10,000 ( e is the photon energy and @xmath2e the beamline energy resolution ) . the endstation is equipped with a high resolution scienta 2002 analyzer . the analyzer , together with the chamber , is rotatable with respect to the beam while the sample position is fixed . the measurement geometry is illustrated in the upright inset of fig . 1 . there are two angles to define the direction of electrons entering the analyzer with respect to the sample normal : tilt angle @xmath3 and analyzer rotation angle @xmath4 . we measured the sample current to quantitatively measure the number of electrons escaping from the sample which is proportional to the photon flux . with the pulse frequency of 500mhz at the als , 1 na of the sample current corresponds to 12.5 electrons per pulse . 2a shows a typical photoemission spectrum of polycrystalline gold taken with a photon energy of 35 ev . it consists of a fermi edge drop ( e@xmath5 ) near @xmath030 ev , valence band between 20@xmath030ev and a secondary electron tail extending to lower kinetic energy arising from the inelastic scattering . we chose to measure on gold because the sharp fermi edge at low temperature ( @xmath020 k for all the measurements in the paper ) gives a good measure of both the energy position and width ( fig . the fermi edge is fitted by the fermi - dirac function , _ f_(e)=1/@xmath6 , at zero temperature convoluted with a gaussian with a full - width - at - half - maximum ( fwhm ) @xmath7 . this width @xmath7 includes all the contributions from thermal broadening , analyzer resolution , beamline resolution and others . in photoemission experiments , it is a routine procedure to use fermi level of a metal ( such as gold ) as the energy referencing point for the sample under study because the fermi levels are expected to line up with each other when the metal and the sample are in good electrical contact . the fermi level of the metal is also expected to be dependent only on the photon energy and not on other experimental conditions , such as sample temperature , photon flux etc . it was therefore quite surprising when we first found out that the gold fermi edge shifts position with incident photon intensity ( fig . a systematic measurement reveals that , under some measurement geometries , the fermi level varies linearly with the sample current and the shift can be as high as @xmath020mev within the photon flux range measured ( fig . note that the fermi level energy gets higher with increasing photon flux . this rules out the possibility of sample charging that usually occurs due to poor electrical grounding of the sample . in that case , the fermi level energy would be pushed downward with increasing photon flux . we can also rule out the possibility of the local sample heating due to high photon flux because temperature only affects the fermi edge broadening but will not change the fermi level position . as we estimated , for a photon flux of @xmath010@xmath8 photons / second at a photon energy of 35 ev , the corresponding power is @xmath00.056 mw . the temperature increase with such a small power , spread over an area of 1 mm@xmath9 , is negligible so it also has little effect on the thermal broadening of the fermi edge . the first thing to check is whether this fermi level shift with photon flux is due to instrumental problems , which can be from either the beamline or the electron analyzer . regarding the beamline , the photon flux is usually varied by adjusting the size of the beamline slits . this will change the beamline energy resolution correspondingly but may potentially also cause energy position change . to check whether this is the case , we put a photon blocker in the beamline ( fig . 1 ) so that it can attenuate the photon flux while keeping the photon energy and resolution intact . using the photon blocker , we observed a similar variation of the fermi level with photon flux ( fig . 3a ) , thus ruling out the possibility of beamline problems . we also put an electron blocker ( fig . 1 ) to vary the number of electrons collected by the analyzer . when the photon flux on the sample is fixed , the fermi edge shows little change with the number of electrons entering the analyzer ( fig . this indicates that the energy shift we have observed is not due to problems of the electron analyzer either . therefore , the observed energy shift must be associated with the photoemission process itself . in addition to the energy position shift , there is also an energy broadening associated with increasing photon flux . to observe such an effect , we have to compromise the beamline energy resolution in the way that it has a relatively high photon flux to induce an obvious broadening effect , and a relatively high energy resolution ( @xmath010mev ) in order to resolve the additional broadening from all other contributions . the measurement is made possible by taking the advantage of the photon blocker to fix the contribution from the beamline . the total width increases with increasing photon flux ( inset of fig . 4 ) . taking the width at the lowest photon flux as arising from all the other contributions including the beamline , the analyzer and sample temperature broadening , the photon - induced energy broadening can be extracted after deconvolution . as seen in fig . 4 , it varies with the photon flux with a magnitude comparable to but slightly larger than the energy shift . we have found that the fermi edge shift and broadening are sensitive to the spot size of the beam on the sample ( fig . the spot size is changed by varying the vertical focus of the beamline ; the horizontal beamsize is fixed . it is measured using the transmission mode of the analyzer , calibrated by using samples with known size . as seen from fig . 5a , as the spot size increases , the energy shift gets less sensitive to the change of photon flux , as also seen from the slope change as a function of the spot size ( fig . 6 ) . for comparison , 6 also includes the simulated data over a large range of spot sizes . although the data of energy broadening ( fig . 5b ) is scattered as a result of deconvolution from a relatively large background value , the trend is clear that the broadening gets smaller with increasing spot size . again , for a given beam size , the magnitude of the energy broadening is comparable to but slightly larger than the corresponding energy shift . the fermi edge shift and broadening are also sensitive to the electron emission angle . we set the gold sample at different tilt angles and measured the fermi level position and width as a function of the analyzer angle under various photon flux . as seen in fig . 7 , the fermi level position exhibits a strong variation with the analyzer angle , particularly at high photon flux . the fermi level is higher near smaller analyzer angle and decreases with increasing analyzer angle . when the analyzer angle is close to 90 degrees all the curves with different sample tilt angle and with different sample current tend to approach to a similar position within the experimental error . the overall measured fermi level width basically follows the trend of the energy shift : it becomes smaller with increasing analyzer angle . we also notice that the curves are not symmetrical with respect to the zero analyzer angle . since the surface of the polycrystalline gold we used is not perfectly flat , one possible reason is that the exact angle may be slightly off from the nominal value . another possibility is the presence of a small systematic error . as indicated from fig . 7 , when the sample current is small ( 23 na ) , one can still observe fermi level shift with the analyzer angle which may be due to a systematic error associated with the experimental setup . to gain more insight on the angle dependence , we also measured the energy shift and broadening as a function of the sample current at different analyzer angles ( figs . 8a and b ) . it is interesting to note that , while for small analyzer angles , the energy shift is proportional to the sample current , as we have seen before , it deviates significantly from the straight line for large angles . in this case , the energy shift exhibits linear relation only at high sample current . when the sample current gets smaller , it goes through a minimum , and then gets larger again even with further decreasing of the sample current . one may expect that at zero sample current the energy shift approach zero so that all curves should converge at the zero sample current , as indeed shown by the data in fig . ( the small fermi level scattering at zero sample current may be due to the systematic error as discussed before ) . this implies that , for large analyzer angles , the energy shift can be even negative at some sample current . fitting the high sample current part of the curves in figs . 8a and b with a straight line , we extracted their slopes and plotted them in fig . 8c for two sample tilt angles . the shape of the curves is similar to that in fig . the high sample current part overlaps with each other . when extrapolated to 90 degrees the fermi level shift is approaching zero which is also consistent with the converging of the fermi level at high analyzer angle as seen in fig . 7 . to further investigate the origin of the angle - dependent energy shift and broadening , we measured the gold valence band at different analyzer angles ( fig . the intensity of these spectra are normalized to the photon flux so they are comparable with each other . the shape of the valence band shows no obvious change with the analyzer angle , but their relative intensity changes dramatically . for a quantitative comparison , we integrated the spectral weight over a large energy range ( 5@xmath035 ev ) and the result is shown in the inset of fig . 9 . integration over a smaller energy window such as 25@xmath035 gives essentially the same shape . we have found that the angular variation of the relative valence band intensity and the fermi level shift is identical ( inset of fig . this indicates that the angle dependence of the fermi level is directly related to the angle - dependence of the number of photo - emitted electrons . it is expected that the space charge effect depends on a number of parameters@xcite : ( 1 ) . the number of electrons per pulse ; ( 2 ) . the pulse length ; ( 3 ) . the size and shape of the excitation area ; and ( 4 ) . the energy distribution of the electrons . we have performed numerical simulations using the monte carlo - based technique developed earlier@xcite in order to quantitatively examine our results . this serves first to check whether the observed energy shift and broadening can be entirely attributed to the space charge effect . it then helps to understand the microscopic processes associated with it , such as the time scale of the process . moreover , it can be extended to investigate situations that are difficult or not accessible for the experiments , such as the effect of the electron energy distribution , the effect of the pulse length , and the case of a continuous source , as we will discuss below . in the simulation , a specified number of electrons ( 1 - 100000 ) ( denoted as interaction electrons hereafter ) are started at random positions within the specified source area , at random times during the pulse , and with random energies with some specified distribution . because the acceptance angle of the electron energy analyzer is small , the electrons for which the energy spread and broadening are to be calculated ( denoted as test electrons hereafter ) are started in the forward direction with a specified initial energy but with a random distribution in start position and time . this condition corresponds to the measurement geometry of the analyzer angle @xmath4=0 and the sample tilt angle @xmath3=0 . each test electron is assumed to feel the coulomb force from all interaction electrons within some cut - off distance . the interaction electrons are assumed to move in straight lines defined by their initial conditions , i.e. all mutual interactions between them are neglected . this is legitimate because their position changes are extremely small and random . the energy evolution of a single test electron is followed until all interaction electrons have vanished outside the cut - off distance . then , the process is repeated with a new set of interaction electrons and one new test electron . this procedure is repeated a few thousand times , after which the energy distribution of the test electrons is calculated . for the accuracy of the integration to be of the same order of magnitude as the statistical uncertainty , the cut - off distance has to be at least 1 mm , and for most calculations it was chosen to be 2 mm . the energy distribution can usually be well fitted by a gaussian , although the number of electrons which experience very large shifts is significantly larger than for the gaussian distribution . such extreme outliers are neglected when calculating the width of the distribution . the electrons in the pulse will experience coulomb interaction from all the other electrons at different energies , including the large number of low - energy secondary electrons ( fig . to evaluate the effect of the electron energy distribution on the electrons at the fermi level , we divided the energy range below e@xmath5 into a number of regions , and calculated the contribution from each individual region . the simulated energy shift and broadening from the direct space charge effect are plotted in figs . 10a and 10b , respectively . the energy shift displays a strictly linear relation with the number of electrons in a pulse and the slope as a function of test electron kinetic energy is plotted in fig . 11 . on the other hand , the energy broadening exhibits a nearly linear relation only at large number of electrons ; at small number of electrons it shows a bend . clearly all electrons contribute to the fermi level energy shift and broadening but they contribute differently : the high - energy electrons contribute more than the low - energy ones ( fig . in fact , an electron at a distance _ z _ in front of a conducting metal surface will also experience an attractive force f(z)=-e@xmath9@xmath10(2z)@xmath9 , identical to that produced by a positive ( mirror image ) charge at a distance _ z _ inside the metal@xcite . the basic assumption behind the mirror charge concept is that the charges on the sample surface redistribute themselves in such a way that the surface is always an equipotential surface . whether this assumption is correct on the time scale considered here may be dependent , e.g. , on the conductivity of the sample . in this case , each interaction electron is accompanied by a mirror charge in the sample ( inset of fig . 12a ) , which also interacts with the test electron . the interaction of the test electron with its own mirror charge is not included here because it is always present . in the earlier simulation@xcite , the mirror charges could be neglected when only considering the broadening caused by interaction electrons with energies close to that of the test electron . for the case when the test electron has higher energy than all interaction electrons , this is no longer true , in particular when the energy shifts are also considered . fig . 12a and 12b show simulated energy shift and broadening for different energy ranges by incorporating both the space and mirror charge effects . the energy shift retains a linear variation with the number of test electrons per pulse and the slope is plotted in fig . the contribution from the mirror charge alone can be easily extracted . apparently the mirror charge gives rise to a negative energy shift with increasing number of electrons per pulse . this helps in compensating the positive energy shift from the space charge effect . the combined effect on the energy broadening is more complicated . for the highest energy range of the interaction electrons ( 25 - 30 ev ) , the combined broadening ( fig . 12b ) is larger than that from the space charge effect alone ( fig . but for the lower energy range of the interaction electrons , it is smaller than that from the space charge effect . we have found that the energy shift and broadening occur at very different time scales . as seen from fig . 13 , the energy shift evolves gradually within the first nanosecond . the energy broadening , on the other hand , has already reached its equilibrium value at 100 ps , followed by random fluctuations . this is because the energy shift takes place only after the electrons have spatially sorted themselves according to their energy ; after that the forces are all acting in the same direction . we also note that initially each interaction electron and its mirror charge form a very short dipole , from which the field decreases rapidly with distance . the broadening , on the other hand , is much more of a nearest - neighbor effect , which is strongest when the pulse is dense . detailed study of the energy evolution for individual electrons shows that the random part of the energy change is often dominated by one single event , i.e. , a close encounter with another electron . since the energy shift continues to grow over a time that is comparable to the interval between pulses , we have also checked whether it can be affected by remaining slow electrons from the previous pulse : we have found that this contribution is completely negligible . since a time - continuous light source , such as discharge lamps , is widely used as a lab source for photoemission , it is important to check whether similar effects still exist in that case . for a continuous light source , because there will be no spatial redistribution of the electrons according to their energy , one might expect the contribution to the energy shift from the space charge to be close to zero , while the mirror charge will give a negative shift . the broadening can be expected to be of the same order of magnitude as that from a pulsed source with the same number of electrons per unit time . to simulate a continuous source , we first start with a pulsed source , varying the pulse length while keeping the number of electrons per unit time constant , and try to extrapolate to infinite length to approximate a continuous source . we have considered a typical case of helium i radiation ( photon energy 21.12ev ) on polycrystalline gold , and varied the sample current during the pulse from 0.15 to 50 electrons / ps . 14 shows the energy shift and broadening for different sample currents as a function of the pulse length . when scaled by the sample current , all energy shift curves overlap with each other because the shift is proportional to the current for all pulse lengths ( fig . the energy shift shows non - monotonic dependence with the pulse length , owing to the competition between the direct space charges and mirror charges . when the pulse length is short , the space charge dominates which gives positive energy shift . when the pulse length is long enough , the effect from mirror charges dominates which leads to a negative energy shift . eventually it asymptotically reaches a value that can be taken for a continuous source . the shift is -0.7mev for 1.5@xmath11@xmath1 electrons / second and can get significant when the photon flux is larger . the energy broadening ( fig . 14b ) , on the other hand , does not scale with the sample current , particularly at longer pulse length . it also exhibits a non - monotonic variation with the pulse length , reaching a maximum around 10@xmath12 ps and then decreases with further increasing of the pulse length . if we assume an asymptotic behavior following the drop , the broadening for the continuous source is close to 1 mev for a sample current of 1.5@xmath1310@xmath1 electrons / second . as we have seen from both the experimental measurements and the simulation , the energy shift and broadening depend on many parameters , such as the number of electrons per pulse , the pulse length , the spot size on the sample , the emission angle of electrons and the photon energy used . moreover , it is material - specific . this is first because it depends on the shape of the valence band , i.e , the energy distribution of photoelectrons . seocnd , for metals and insulators , the effect of mirror charge may vary significantly . with so many factors coming into play simultaneously , it is hard to exhaust all the possibilities and a proper approach to take is to measure or simulate on an individual basis . as shown in figs . 4 and 5 , the measured energy shift is proportional to the sample current and the broadening is nearly linear at high sample current and shows a bend at lower sample current . qualitatively speaking , both observations are consistent with the simulated results from either the space charge effect ( fig . 10 ) or combined space and mirror charge effects ( fig . 12 ) . after obtaining the contribution for each individual energy range from the space charge effect ( fig . 10 ) , we calculated the overall energy shift from the measured valence band ( fig . 2a ) as a weighted sum of the contributions from the different energy ranges . we also used a model where the energy distribution is approximated by a rectangular shape corresponding to the valence band and a triangular distribution of the secondary electrons ; the obtained results are similar . it was found that the value for the energy shift obtained from the space charge effect alone is much higher than that measured from experiment . for example , for the spot size of [email protected] mm , the calculated energy shift is 0.175 mev / na , much higher than the measured 0.055 mev / na . the large discrepancy indicates that the space charge effect alone can not account for the observed energy shift . this prompted us in identifying the mirror charge effect that should be present for metals such as gold . after considering both effects(fig . 12 ) , the calculated energy shift becomes quantitatively consistent with the experiment , as seen in fig . 6 even for different spot sizes . considering that there are no adjustable parameters in the simulation , this level of agreement is striking . this indicates that we have captured the main contributors to the energy shift effect . the quantitative comparison between the measurement and the simulation has made us able to identify the mirror charge effect that was not included before@xcite . for the area dependence ( fig . 6 ) , we note that the size of the spot on the sample relative to the distance an electron travels during the pulse is important . depending on the relative ratio , the space charge effect may exhibit different dependence on the spot size . if the light spot is much larger than the electron travelling distance ( for 30 ev electron , the travelling distance is @xmath00.2 mm within 60 ps ) , the shape of electron spatial distribution is basically flat . the space charge effect is expected to be proportional to the number of electrons / area . when the spot size gets smaller , one will get increasingly important edge effects , because electrons that move outside the spot will not be compensated by electrons coming from the outside . in the limit where the spot is very small , the spatial distribution of electrons is a half - sphere . the average distance between electrons will be defined by their time interval rather than by the distance between the points where they started . so in that case , the effect may become independent of the spot size . on the other hand , there are cases where the simulation deviates from measurements . we found that the measured broadening is larger than the values calculated from the simulation . as shown in fig . 6 , from the simulation , the broadening is smaller than the shift whereas from the measurement ( fig . 4 and 5 ) , the broadening is comparable or slightly larger than the shift . the reason for this discrepancy is not clear yet and probably more sophisticated simulations are needed to address the discrepancy . we note the broadening can be larger than the shift when the energy of the interaction electrons is close to that of the test electron ( energy range 25@xmath030 ev in fig . 12b ) which is probably due to the longer average interaction times . however , in the case of gold , because the fraction of electrons in the range close to the fermi edge is very small ( fig . 2a ) , this contribution is small to the overall broadening . the angular dependence of the energy shift ( fig . 7 ) can be well attributed to angle - dependent number of electrons at different emission angles ( fig . 9 ) which is probably associated with the linear polarization of the synchrotron light . however , to understand the negative energy shift for high analyzer angles at lower sample current , more simulation is also needed . the observation of space and mirror charge effects has important implications in photoemission experiments as well as the future development of the technique . these findings first ask for particular caution in interpreting photoemission data . one immediate issue is the electron energy referencing in photoemission spectroscopy . in photoemission community it is a routine procedure to use the fermi level of a metal as a reference . this is usually realized by measuring the fermi level from a metal ( such as gold ) which is electrically connected to the sample under measurement . it is true that the intrinsic fermi level of the sample is lined up with that of the metal , but the measured fermi level has an offset from the space and mirror charge effects . this offset can be different between the sample and the metal because it is not only material - specific , but also depends on many other factors . when the effect on the energy shift is strong , using the fermi level from a metal as a reference becomes unreliable . another related issue is the fermi level instability during measurement . because the photon flux usually changes with time for many synchrotron light sources due to the finite life - time of electrons in the storage ring , the fermi level is always changing with time during measurements . as we have shown before , this can give rise to an fermi level uncertainty on the order of 10 mev for a typical experimental setting using a third - generation synchrotron light source . this is comparable or larger than many energy scales which are actively pursued in many - body problems in the condensed matter physics@xcite . measurement with an energy precision of 1 mev is necessary , for example , when the superconducting gap in some conventional metals as well as in some high temperature superconductors is on the order of 1 mev@xcite . in this case , an uncertainty or shift on the order of 10 mev definitely poses a big problem . to resolve the fermi level referencing problem , one can always minimize the space charge effect by reducing the photon flux , or increasing the spot size . apparently this is not desirable , particularly when a high photon flux is necessary to take data with a good statistics and a high efficiency . given that the fermi level referencing to a metal is no longer reliable , one may use an internal reference from the sample under measurement . this internal reference can be obtained from _ priori _ knowledge or measurements with negligible space charge effect . for example , in high temperature cuprate materials , the ( 0,0 ) to ( @xmath14,@xmath14 ) nodal direction can be used as an internal reference to locate the fermi level because it has been shown that the superconducting gap and pseudogap approaches zero along this direction except for slightly doped samples@xcite . as for the fermi level instability with time , since the energy shift exhibits a linear relation with the photon flux , it is possible to make corrections by recording the sample current or photon flux . ideally , this problem can be minimized if the synchrotron light source is operated at a constant or quasi - constant photon flux ( top - off " ) mode . in addition to the fermi level uncertainty , the energy broadening is another serious issue facing the photoemission technique . since most physical properties of materials are dictated by electronic excitations within an energy range of @xmath0k@xmath15 t near the fermi level ( k@xmath15 is the boltzman constant and t a temperature ) , to probe the intrinsic electronic properties , the energy resolution has to be comparable or better than k@xmath15 t , which is 0.8 mev for 10 k. therefore , there is a strong scientific impetus to improve the photoemission technique to even higher energy resolution ( sub - mev ) , accompanied by high photon flux and small beam size . the space and mirror charge effects should be taken into account seriously in the future development of new light sources and electron energy analyzers . the high photon flux and small spot size will enhance the space and mirror charge effects ; the resultant energy broadening can be well beyond the resolution from the electron analyzer and the light source . with the increasing demand of high energy resolution , it is important to investigate how to alleviate or remove the space charge effect . for example , it is interesting to study whether applying a bias voltage between the sample and the electron detector will affect the space charge effect . on the other hand , in addition to putting more effort on improving the performance of the light sources , it is very important to put emphasis on enhancing the capabilities of the electron energy analyzer . one aspect is to further increase the sensitivity of electron detection by using new electron detection schemes . the other aspect is to keep improving the analyzer throughput . note that even for the state - of - the - art display electron analyzer , using angle - resolved mode , only less than 1@xmath16 of electrons are collected during measurements while all the rest of electrons emitted over 2@xmath14 solid angle from the sample surface are wasted . a new scheme needs to be explored on how to record large solid angle at the same time when maintaining high energy resolution . it is apparent that much work needs to be done and we hope our identification of the coulomb effects can stimulate more work along this direction . s. huefner , photoemission spectroscopy : principles and applications ( springer - verlag , berlin , 1995 ) . angle - resolved photoemission : theory and current applications , edited by s. d. kevan , ( elsevier , the netherlans , 1992 ) . special issue of science * 288 * , no . 5465 , ( 2000 ) . special issue of j. electron spectroscopy and related phenomena , * 117 - 118 * 1(2001 ) . a. damascelli , z. hussain and z .- x . shen , rev . modern phys . * 75 * , 473(2003 ) . a. chainani et al . , phys . lett . * 85 * , 1966(2001 ) . n. p. armitage et al . lett . * 86 * , 1126(2001 ) ; t. sato et al . , science * 291 * , 1517(2001 ) . b. wannberg , p. baltzer and s. shin , preprint ( 2000 ) . h. boersch , z. physik * 139 * , 115 ( 1954 ) . u. hofer et al . , science * 277 * , 1480 ( 1997 ) ; p. m. echenique and j. b. pendry , progress in surf . * 32 * , 111 ( 1989 ) . we thank a. fujimori , j. bozek and s. sodergren for stimulating discussions . the experiment was performed at the als of lbnl , which is operated by the doe s office of bes , division of material science , with contract de - fg03 - 01er45929-a001 . the division also provided support for the work at ssrl with contract de - fg03 - 01er45929-a001 . the work at stanford was supported by nsf grant dmr-0304981 and onr grant n00014 - 98 - 1 - 0195-p0007 , and the work at colorado was supported by nsf grant dmr 0402814 and doe grant de - fg02 - 03er46066 . in the xy horizontal plane and parallel to y axis . the sample normal is in the xz plane and its angle with respect to the z axis is referred to as @xmath3 . the analyzer is rotatable and the lens axis is in the yz plane . the angle of the lens axis with respect to the z axis is referred to as @xmath4 . ] is 45 degrees and the analyzer angle @xmath4 is 0 . the beam spot size is @xmath0 [email protected] mm@xmath9 . the photon flux corresponding to 150 na sample current is @xmath05@xmath1310@xmath8 photons / second . the inset shows the measured overall fermi edge width as a function of the sample current , which includes all contributions including the beamline , the analyzer and the temperature broadening . the net broadening resulting from pulsed photons is obtained by deconvolution of the measured data , taking the width at low photon flux as from all the other contributions . ] is 45 degrees and the analyzer angle @xmath4 is 0 . effect of beam size on the energy broadening . the broadening at high sample current can be approximated as a straight line ; the solid lines also act as a guide to the eye . ] for different sample tilt angles ( a).@xmath3=22 , ( b).@xmath3=37 and ( c).@xmath3=52 . the curves in each panel represent different sample currents ( sc ) under a given beamline resolution ( de ) . for any given curve the sample current is nearly a constant . ] . the sample tilt angle is @xmath3=37 degrees . in the inset shows the integrated spectral weight over the entire energy range of 5@xmath035 ev as a function of the analyzer angle @xmath4 ( black solid square ) . for comparison , the fermi level as a function of the analyzer angle measured under similar condition is also plotted ( blue circle ) . ] 0.42 mm . for the energy shift , all curves overlap with each other , indicating that the energy shift is proportional to the electron current . but for the energy broadening , they do not strictly overlap with each other , particularly at longer pulse length . ]

we report the observation and systematic investigation of the space charge effect and mirror charge effect in photoemission spectroscopy . when pulsed light is incident on a sample , the photoemitted electrons experience energy redistribution after escaping from the surface because of the coulomb interaction between them ( space charge effect ) and between photoemitted electrons and the distribution of mirror charges in the sample ( mirror charge effect ) . these combined coulomb interaction effects give rise to an energy shift and a broadening which can be on the order of 10 mev for a typical third - generation synchrotron light source . this value is comparable to many fundamental physical parameters actively studied by photoemission spectroscopy and should be taken seriously in interpreting photoemission data and in designing next generation experiments . key words : space charge , mirror charge , photoemission , fermi level shift , fermi level broadening .

cond-mat0410006

elucidating the mechanism responsible for electro - weak symmetry breaking is one of the most important tasks of future collider based particle physics . experimental and theoretical indications of a light higgs boson make the precision study of the properties of higgs bosons one of the major physics motivations of a linear collider ( lc ) . both the higgs boson of the standard model ( sm ) and those of extended models will be copiously produced in @xmath0 collisions in various production mechanisms . a large variety of different decay modes can be observed with low backgrounds and high efficiency . these measurements allow us to extract the fundamental parameters of the higgs sector with high precision . the series of ecfa / desy workshops aims at a comprehensive study of the physics case , a determination of the achievable precisions on higgs observables as well as on a fruitful cross - talk between theory , physics simulations and detector layout . a future linear collider offers also the option of photon - photon collisions from back - scattered laser light . the physics potential and progress in higgs physics at a photon collider is discussed elsewhere in these proceedings @xcite . in @xmath0 collisions , the sm higgs boson is predominantly produced through the higgs - strahlung process , @xmath1 @xcite and through the vector boson fusion processes @xmath2 @xcite . the sm production cross - sections are precisely known including full electro - weak corrections at the one - loop level . for a recent review of the theoretical calculations see e.g. @xcite . recently the full one - loop corrections to the ww - fusion process have been calculated @xcite . the radiatively corrected cross - sections for higgs - strahlung and ww - fusion are shown in fig . [ fig : zhxsec ] . for higgs - strahlung the corrections are positive for small higgs masses and negative for large higgs masses and are of @xmath3 . for ww - fusion the corrections are of similar size but always negative . + with the higgs boson being responsible for mass generation its couplings to massive sm particles are proportional to their masses : @xmath4 , @xmath5 . thus higgs bosons decay preferentially into the heaviest kinematically possible final states . state - of - the - art branching ratio calculations including electro - weak and qcd corrections @xcite are coded in the program hdecay @xcite for the sm and its minimal supersymmetric extension , the mssm . branching ratios of the neutral higgs bosons in the mssm can be also calculated with program feynhiggsdecay @xcite . the sm higgs branching ratios in the mass range relevant to a lc are shown in fig . [ fig : hbr ] . a variety of leading - order monte carlo generators exist which are commonly used for higgs studies in @xmath0 collisions . they are pythia @xcite , herwig @xcite , hzha @xcite , comphep @xcite , and whizard @xcite . comphep and whizard offer the possibility of generating the complete @xmath6 and ( in the case of whizard ) also @xmath7 processes including their interference with sm backgrounds . beamstrahlung was simulated in most analyses presented below using the parameterization circe @xcite . the vast majority of experimental analyses in this summary includes the simulation of complete sm backgrounds . the effects of limited detector acceptance and resolution have been incorporated using the parametric detector simulation program simdet @xcite which is based on the detector performance specified for the tesla detector in the tdr . a comparative study of different event generators and of different fast detector simulation programs was carried out in @xcite . most analyses which involve tagging of heavy quarks use a realistic event - wise neural - net algorithm based on zvtop @xcite which was first used at the sld detector . a detailed simulation ( brahms @xcite ) of the tesla tdr detector based on geant3 along with a reconstruction program is available and can be used for comparative studies . the anchor of a model - independent precision analysis of higgs boson properties at a lc is the measurement of the total cross - section for the higgs - strahlung process , @xmath1 . z bosons can be selected in @xmath8 and @xmath9 decays . from energy - momentum conservation the invariant mass recoiling against the @xmath10 candidate can be calculated . through a cut on the recoil mass , higgs bosons can be selected independent of their decay mode , allowing for a model - independent measurement of the effective hz coupling , @xmath11 . once @xmath11 is known , all other higgs couplings can be determined absolutely . the total higgs - strahlung cross - section can be measured with an accuracy of 2.5% for @xmath12 gev and @xmath13 gev for 500 fb@xmath14 @xcite . assuming that the uncertainty scales with the square root of the cross - section and that the selection purity and efficiency is independent of the center - of - mass energy , one can obtain an accuracy between 1.2 % and 10% for @xmath15 gev , for an integrated luminosity of @xmath16 fb@xmath17 gev at a center - of - mass energy corresponding to the maximum of the cross - section for a given higgs mass . the relative error is shown in fig . [ fig : recoil ] together with the optimal center - of - mass energy as a function of the higgs mass . the importance of a precise and model - independent determination of @xmath11 has e.g. recently been discussed in the context of supersymmetric models @xcite and in the context of models with higher higgs field representations , as well as in the context of extra - dimensional models @xcite . the measurements of differential production cross - sections and decay angular distributions provide access to the discrete quantum numbers of the higgs boson : @xmath18 . in the tdr , the measurement of the @xmath19-dependence of the higgs - strahlung cross - section close to the production threshold was exploited to determine the spin of the higgs boson . the spin can also be determined from the invariant mass of the off - shell @xmath10 boson in the decay @xmath20 for @xmath21 . this method is independent of the higgs production process and thus potentially applicable also in @xmath22 and gg collisions . the invariant mass distribution for @xmath23 gev is shown in fig . [ fig : hzz ] . for @xmath24 above @xmath25 , azimuthal correlations of the two z boson decay planes can be exploited to gain sensitivity to higgs boson spin and cp @xcite . the cp quantum number , like the spin , can be determined from both higgs boson production and decay @xcite . in the tdr , the sensitivity of the angular distribution of the @xmath10 recoiling against the @xmath26 in higgs - strahlung was exploited . recently a method has been proposed which makes use of the transverse spin correlation in @xmath27 decays . the spin correlations between the two @xmath28 leptons is probed through angular correlations of their decay products . in particular , events from @xmath29 and from @xmath30 can be used . the angle between the decay planes of the two @xmath31 mesons from either @xmath28 decay provides a suitable observable @xcite . while this angle can be determined in the laboratory frame , ideally it is evaluated in the higgs boson rest frame , which can be approximately reconstructed using @xmath28 lifetime information @xcite . preliminary results including detector simulation have shown that from a sample of 1 ab@xmath14 of higgs - strahlung events at @xmath32 gev , a statistical separation between a cp - even and a cp - odd higgs boson of eight standard deviations may be achieved assuming production cross section and branching ratio as for @xmath33 ( see fig . [ fig : andreas ] , note that background is not yet taken into account ) @xcite . the precise measurement of higgs boson decay branching ratios is one of the key tasks in lc higgs physics . in the tesla tdr as well as in all other regional lc studies @xcite analyses have been performed to investigate the expected precisions on the branching ratio determination . for a light higgs boson with @xmath34 gev , a large variety of higgs decay modes can be measured . the hadronic decays into @xmath35 and @xmath36 are disentangled via the excellent capabilities of a lc vertex detector . progress has been achieved recently in the level of detail at which the algorithms to tag b- and c - quarks are implemented into the simulation . although these studies are not finished , it looks conceivable that the results of the tdr study will essentially be confirmed @xcite . there are two different methods to extract branching ratios from the observed events : 1 . measure the topological cross - section for a given final state , e.g. @xmath37 and divide by the total measured higgs strahlung cross - section ( as obtained from the recoil mass measurement ) @xcite . 2 . select a sample of unbiased @xmath38 events ( events in the recoil mass peak ) and determine the fraction of events corresponding to a given @xmath39 decay within this sample . the latter method was first applied to higgs branching ratio studies in @xcite . since in this approach binomial ( or in principle multi - nomial ) statistics can be applied , smaller errors of the branching ratios can be inferred for the same number of events than from a rate measurement . although only relying on events with @xmath40 , the latter method yields errors very similar to those of the tdr method @xcite . the achievable precision for the both methods for a sm higgs boson of 120 gev from a sample of 500 @xmath41 is shown in table [ tab : higgsbr ] . a possible combination of both methods is currently being investigated . while for the hadronic higgs decays , there is a sizable overlap , for the @xmath42 decay a significant improvement may be expected from combination . besides the decays into @xmath43 , @xmath44 and @xmath22 further decay modes have been studied . the very rare decay @xmath45 might be detectable in ww - fusion events at @xmath46 gev for @xmath12 gev . a measurement of the muon yukawa coupling with approximately 15% relative accuracy may be obtained from a sample of 1 ab@xmath14 . here , the logarithmic rise of the signal cross - section with @xmath47 is of advantage . a precision measurement of the @xmath45 branching ratio however can only be performed at even higher luminosity or at higher energy @xcite . the expected signal is shown in fig . [ fig : htomu ] . [ cols="<,>,>,>,>,>,>,>,>,>",options="header " , ] [ tab : heavysusy ] since at the tree level and in the decoupling limit the heavy neutral mssm higgs bosons decouple from the @xmath10 , the mass reach for their discovery at a lc is limited to approximately @xmath48 from the pair production process . it has been investigated during the workshop , how single production mechanisms could extend the mass reach of an @xmath0 lc . in particular , the ww - fusion process @xmath49 has been investigated @xcite . its tree level cross - section is proportional to @xmath50 . depending on the susy parameters , radiative corrections might increase the cross - section for @xmath49 , possibly allowing discovery beyond the pair production kinematic limit for certain choices of the mssm parameters . using left - polarized electron beams and right - polarized positron beams the cross - section can further be enhanced . a particular scenario where this is the case has been chosen in @xcite ( @xmath51 350 gev , @xmath52 gev , @xmath53 gev and large stop mixing ) . cross - section contours for this scenario are shown in fig . [ fig : heavymssm ] . charged higgs bosons can be pair - produced at the lc via @xmath54 if @xmath55 . a complete simulation of this process for the decay @xmath56 has been performed for @xmath46 gev , 1 ab@xmath14 , and @xmath57 gev @xcite . the expected signal and background are shown in fig . [ fig : hphm ] . the mass resolution is approximately 1.5% . a 5@xmath58 discovery will be possible for @xmath59 gev . since in pair production the mass reach for charged higgs bosons is limited to @xmath48 , also the rare processes of single charged higgs production may be considered . the dominant processes for single charged higgs production are @xmath60 , and @xmath61 . their cross - sections have been calculated at leading order in @xcite . qcd corrections to @xmath62 have recently become available @xcite and are sizable . in general , parameter regions for which the production cross - section exceeds 0.1 fb are rather small for charged higgs masses beyond the pair production threshold . cross - section contours for @xmath63 gev and 800 gev are shown in fig . [ fig : singlecharged ] . + at tree level , the mssm higgs sector only depends on @xmath64 and @xmath65 . thus , if @xmath65 would be measured , @xmath64 could in principle be uniquely determined from the observed higgs properties . in particular , the coupling of @xmath66 to down - type fermions is directly proportional to @xmath64 . therefore this coupling which appears in the rate of the @xmath67 and @xmath68 processes , as well as in the total decay width @xmath69 can be used to extract @xmath64 in principle . this has been studied in @xcite . due to the large radiative corrections the predictions for the observables also depend on other susy parameters ( in particular the sfermion masses and mixings ) which are fixed in this analysis . therefore the resulting errors ( see [ fig : tanbeta ] ) are only valid if all other susy parameters , were precisely known . a different approach to @xmath64 determination has been proposed in @xcite . in a scenario where all susy particles are light compared to the center - of - mass energy , the dependence of the cross - section for charged higgs production on @xmath70 in the 1 tev domain can be compared to the logarithmic sudakov expansion of the cross - section . in particular , it has been shown , that the first coefficient of the expansion depends only on @xmath64 . a complete study of susy parameter determination in the full mssm is only possible when studies of the higgs sector are combined with information on sparticle production . within more constrained susy models which assume specific susy breaking schemes higgs observables alone can lead to significant constraints @xcite . as an example , the nuhm ( non - universal higgs mass ) model has been considered in @xcite . the nuhm model assumes unification of sfermion masses and mixing terms as well as unification of gaugino mass terms at a high scale . however , in contrast to the msugra ( minimal supergravity ) model , both @xmath71 and @xmath65 are free parameters . in fig . [ fig : cmssmsensi ] , the deviation of branching ratios of the lightest higgs boson from the sm is shown for the nuhm scenario as a function of @xmath65 . the deviation is plotted in terms of standard deviations of the prospective measurement error at the lc as described in the tdr . it can be seen that in particular @xmath72 and @xmath73 provide good sensitivity to @xmath65 while the dependence on @xmath71 is only weak . as a caveat , the values of @xmath64 as well as the other model parameters are fixed in this study and thus have to be allowed to vary freely in the study or assumed to be known from elsewhere in order to translate the plotted deviations into expected errors on the parameter measurements . another study utilizes the ratio @xmath74 @xcite . at tree level , in the mssm , this ratio is constant since both b quarks and @xmath28 leptons are down - type fermions , coupling proportionally to @xmath75 to the @xmath76 . a precise measurement of this ratio is therefore sensitive to the difference of the radiative corrections to these two decays . in particular at large @xmath77 these corrections become relevant , allowing to gain sensitivity to the value of @xmath77 itself if all other susy parameters are fixed . the ratio of @xmath78 as a function of @xmath77 is shown in fig . [ fig : bbttsensi ] . + in the mssm the higgs potential is invariant under the cp transformation at tree level . however , it is possible to break cp symmetry in the higgs sector by radiative corrections , especially by contributions from third generation scalar - quarks @xcite . such a scenario is theoretically attractive since it provides a possible solution to the cosmic baryon asymmetry @xcite . in a cp violating scenario the three neutral higgs bosons , h@xmath79 , h@xmath80 , h@xmath81 , are mixtures of the cp even and cp odd higgs fields . consequently , they all couple to the z boson and to each other . these couplings may be very different from those of the cp conserving case . in the cp violating scenario the higgs - strahlung processes @xmath82 ( @xmath83 ) and pair production processes @xmath84 ( @xmath85 ) may all occur , with widely varying cross - sections . in a case study , for @xmath86 gev and @xmath87 , the sensitivity of the observable higgs masses @xmath88 , @xmath89 and of the observed cross - section for @xmath90 to the real and imaginary part of the trilinear coupling @xmath91 has been analyzed . under the assumption that the other susy parameters are known , the complex phase of @xmath91 may be extracted from these observables @xcite . clearly , further studies are needed in order to extract cp - violating susy parameters from the higgs sector . if a light higgs boson is discovered at the lhc but no additional particles are seen at the lhc or the lc , it is important to search for small deviations of the higgs boson potential from the sm predictions to probe new physics scales . if the reason for such small deviations is beyond - sm physics at large scales @xmath92 , the effective operator approach can be chosen to parameterize the low - energy behavior of such models . recently , operators of dimension six have been studied , which involve only the higgs field and which are not severely constrained by precision electro - weak data @xcite . these operators are @xmath93 which lead to a lagrangian @xmath94 in @xcite , it has been shown that the parameter @xmath95 can be measured to an accuracy of @xmath96 corresponding to a scale @xmath97 tev , from 1 @xmath98 of data at 500 ( 800 ) gev through the measurement of the production cross - sections from higgs - strahlung and ww / zz - fusion for @xmath12 gev . the parameter @xmath99 modifies the form of the higgs potential and thus the higgs pair production cross - section . with the same integrated luminosity , for @xmath12 gev , @xmath99 can be measured to @xmath100 at 500 ( 800 ) gev corresponding to a scale @xmath101 1 tev . the prospects for the exploration of general two higgs doublet models ( 2hdm ) at a lc have been discussed e.g. in @xcite . during the workshop , a 2hdm scenario has been discussed in which the lightest cp - even higgs boson has absolute values of the tree level couplings to fermions and massive gauge bosons exactly as in the sm and the other higgs bosons are heavy ( @xmath102 ) @xcite . within the 2hdm such a scenario can be realized differently from the sm in two ways : ( a ) the tree level couplings have the same sign as in the sm or ( b ) either up - type or down - type fermions have opposite sign couplings as in the sm . the only possibility to distinguish such a scenario from the sm is through loop - induced processes , in particular through the loop - induced @xmath103 and @xmath104 couplings . depending on @xmath105 the effect can be large enough to be distinguishable from the sm at the lc ( and lhc ) from higgs branching ratio measurements or at a photon collider through the @xmath106 process ( see fig . [ fig:2hdm ] ) . the addition of a higgs singlet field defines the next - to - minimal mssm ( nmssm ) . this addition is theoretically motivated mainly since it allows a naturally small @xmath71 parameter . if the associated peccei - quinn symmetry were unbroken , it would lead to a massless cp odd higgs boson which is ruled out . the lc phenomenology of the model depends on how strong this symmetry is broken . the higgs spectrum of the nmssm consists of three cp - even and two cp - odd neutral higgs bosons and two charged higgs bosons . the complete lc phenomenology has recently been reviewed in @xcite . as an example , the masses of the neutral and charged higgs bosons and the coupling of the cp - even higgs bosons to the z are shown in fig . [ fig : nmssm ] as a function of @xmath65 ( defined as the top left parameter of the cp - odd higgs mixing matrix , see @xcite ) . it can be seen that in a large portion of the parameter space , all three cp - even higgs bosons would have significant couplings to the z , thus significant higgs - strahlung cross - sections at the lc . models which postulate the existence of additional space dimensions in order to explain the hierarchy between the electro - weak and the planck scale have been discussed extensively in recent years . their common feature is that the apparent weakness of gravity in our 4-dimensional world is a result of its dilution in the extra dimensions . two scenarios , that of large extra dimensions ( add ) @xcite and that of warped extra dimensions ( rs ) @xcite have been discussed in particular . the classic signatures involve deviations of sm processes like @xmath107 and @xmath108 from the virtual exchange of towers of ( add ) @xcite or single @xcite kaluza - klein ( kk ) excitations of gravitons , or their real emission together with sm fermions or gauge bosons @xcite . these modes have been studied experimentally e.g. in the tesla tdr . more recently , also the impact of extra dimensions on the higgs boson phenomenology has been studied . in the add scenario , two effects have been analyzed : \1 . a modification of the quasi - resonant @xmath109 production process through interference of the sm amplitude with the imaginary part of the graviton / graviscalar kk exchange amplitude @xcite . in order to yield a significant modification , a large total higgs width is needed ( i.e. large @xmath24 ) , which implies on the other hand a large center - of - mass energy . while the graviscalar contribution only modifies the normalization of the cross - section ( by few percent for @xmath110 1 tev , @xmath111 500 gev and 2 extra dimensions at a fundamental planck scale of 1 tev ) , a significant change of the angular distribution is expected from the spin-2 graviton exchange . \2 . a modification of the process @xmath112 and the existence of the process @xmath113 which is absent at tree level in the sm @xcite . for a 1 tev lc and @xmath111 120 gev , a sizable correction to @xmath112 both in normalization and angular distribution is expected for fundamental planck scale up to a few tev . furthermore , the cross - section for @xmath113 exceeds 0.1 fb for a fundamental planck scale below approximately 2 tev . in @xcite , expected 5 @xmath114 discovery limits on the fundamental planck scale of 8801560 ( 16402850 ) gev have been derived at @xmath110 500 ( 1000 ) gev for 63 extra dimensions . in the rs scenario , the influence on the higgs sector might be much more drastic . besides the spin-2 kk graviton excitations , graviscalar excitations , called radions , are predicted @xcite . they are predicted to couple to sm particles through the trace of the energy - momentum tensor , i.e. up to the trace anomaly of qcd , very similar to the higgs boson . the lightest radion might in fact be lighter than the lightest graviton excitation and thus the discovery channel for the model . higgs boson and radion may exhibit kinetic mixing , which leads to a modification of both higgs boson and radion properties , in particular their couplings to gauge bosons and fermions . for a review of the radion phenomenology , see e.g. @xcite . the radion sector is governed by 3 parameters : the strength of the radion - matter interactions described by an energy scale @xmath115 , the mass of physical radion , @xmath116 , and the radion - higgs mixing parameter @xmath117 . in fig . [ fig : radionprop ] , the effective couplings squared of the higgs boson and the radion ( relative to those of a sm higgs boson ) are shown for the choice @xmath118 tev , and three values of the radion mass ( 20 , 55 , 200 gev ) as a function of @xmath117 . large deviations of the higgs couplings from their sm values are expected if there is large radion higgs mixing present . the radion itself has couplings which are reduced by a factor @xmath119 with respect to those of a sm higgs in the case of no mixing , which requires high luminosity for direct discovery . the sensitivity of the trilinear higgs coupling to radion admixtures has been studied as well in @xcite . + the lc capability of precisely measuring the higgs branching ratios @xmath120 and @xmath121 has been exploited in @xcite . in fig . [ fig : battaglia ] , the regions where the lc would observe larger than 2.5@xmath58 deviations of the higgs branching ratios due to radion higgs mixing is shown together with the regions where the lhc can observe the higgs bosons . in particular the regions in which the lhc might be blind to the higgs boson are well covered by the lc . a study of the sensitivity of the ww - fusion channel to radion effects has also been presented at the workshop @xcite . a higgs boson with sm - like properties will most likely be discovered at the large hadron collider lhc . in recent years , the potential of the lhc to make measurements of higgs boson properties has been investigated . for a recent summary of the atlas studies , see @xcite . in most cases the capabilities of a lc are superior to those of the lhc as far as higgs physics is concerned . in particular , _ no model - independent measurements of higgs boson couplings are possible at the lhc . however , there are cases where the synergy of both colliders is vital and rewarding . examples are in the determination of the top yukawa coupling , in the mass reach for heavy susy higgs bosons , and in lhc measurements on third generation squark properties in order to constrain the interpretation of a supersymmetric higgs sector . these examples are currently been worked out in more detail in a world - wide lhc / lc study group @xcite . _ the precision study of higgs bosons is at the core of the physics program of a future linear collider . in the course of the extended ecfa / desy study 2001 - 2003 this physics case has been developed further : the precision of theoretical calculations has been improved , the implication of new theoretical models has been investigated and the experimental studies of the lc sensitivity have been extended and improved . the studies are vital for the preparation of the worldwide lc project and will be continued both in the three regions america , asia , and europe and in worldwide workshops . in europe , the study will continue in the framework of a new ecfa study . major goals of this new study are to continue to incorporate new theoretical ideas and to improve the precision of theoretical predictions . on the experimental side , a more detailed study of systematic limitations , impact of machine conditions and in particular dependence of the precision on specific detector properties are of utmost importance . i would like to warmly thank all contributors to the higgs working group for their huge efforts to make the workshop a success . in particular the work of my co - convenors m. battaglia , a. djouadi , e. gross , and b. kniehl is greatly acknowledged . j.r.ellis , m.k.gaillard and d.v.nanopoulos , nucl . b * 106 * ( 1976 ) 292 ; b.l.ioffe and v.a.khoze , sov . j. part . * 9 * ( 1978 ) 50 ; b.w.lee , c.quigg and h.b.thacker , phys . * 38 * ( 1977 ) 883 . d.r.jones and s.t.petcov , phys . b * 84 * ( 1979 ) 440 ; r.n.cahn and s.dawson , phys . b * 136 * ( 1984 ) 196 ; g.l.kane , w.w.repko and w.b.rolnick , phys . b * 148 * ( 1984 ) 367 ; g.altarelli , b.mele and f.pitolli , nucl . b * 287 * ( 1987 ) 205 ; w.kilian , m.kramer and p.m.zerwas , phys . b * 373 * ( 1996 ) 135 , hep - ph/9512355 . g.belanger , f.boudjema , j.fujimoto , t.ishikawa , t.kaneko , k.kato and y.shimizu , phys . lett . b * 559 * ( 2003 ) 252 , hep - ph/0212261 ; nucl . * 116 * ( 2003 ) 353 , hep - ph/0211268 ; + f.jegerlehner and o.tarasov , nucl . phys . proc . * 116 * ( 2003 ) 83 , hep - ph/0212004 ; + a.denner , s.dittmaier , m.roth , m.m.weber , _ electroweak radiative corrections to single higgs - boson production in e+ e- annihilation , phys . b * 560 * ( 2003 ) 196 , hep - ph/0301189 , lc - th-2003 - 008 ( 2003 ) . _ a.djouadi , j.kalinowski and m.spira , comput . commun . * 108 * ( 1998 ) 56 . s.heinemeyer , w.hollik and g.weiglein , _ decay widths of the neutral cp - even mssm higgs bosons in the feynman - diagrammatic approach , eur . j. c * 16 * ( 2000 ) 139 , hep - ph/0003022 , lc - th-2001 - 064 . _ g.corcella _ et al . _ , jhep 01 ( 2001 ) 010 , hep - ph/0011363 ; hep - ph/0107071 , hep - ph/0201201 , hep - ph/0210213 . + s.moretti , _ herwig : an event generator for @xmath123 linear colliders , hep - ph/0209209 , lc - tool-2002 - 009 . _ g.r . bower , t. pierzchala , z. was , m. worek , phys . b * 543 * ( 2002 ) 227 , hep - ph/0204292 ; + m.worek , _ higgs cp from h / a @xmath122 tau tau decay , acta phys . b34 ( 2003 ) 4549 , hep - ph/0305082 , lc - phsm-2003 - 050 ( 2003 ) . _ k.desch , z.was and m.worek , _ measuring the higgs boson parity at a linear collider using the tau impact parameter and tau @xmath122 rho nu decay , eur . j. c * 29 * ( 2003 ) 491 , hep - ph/0302046 , lc - phsm-2003 - 003 ( 2003 ) . _ t.abe _ et al . _ [ american linear collider working group collaboration ] , _ linear collider physics resource book for snowmass 2001 , hep - ex/0106055 ( part 1 ) , hep - ex/0106056 ( part 2 ) , hep - ex/0106057 ( part 3 ) , and hep - ex/0106058 ( 2001 ) . _ t.kuhl , _ higgs branching ratios with simdet4+zvtop : first results , talk given at the ecfa / desy workshop on physics and detectors for a future linear collider , amsterdam , spring 2003 , lc note in preparation . _ y.you , w.g.ma , h.chen , r.y.zhang , s.yan-bin and h.s.hou , phys . lett . b * 571 * ( 2003 ) 85 , hep - ph/0306036 ; + g.belanger _ et al . _ , phys . b * 571 * ( 2003 ) 163 , hep - ph/0307029 ; + a.denner , s.dittmaier , m.roth and m.m.weber , hep - ph/0307193 ; s.heinemeyer , w.hollik and g.weiglein , phys . rev . * d58 * , 091701 ( 1998 ) [ hep - ph/9803277 ] ; phys . b440 * , 296 ( 1998 ) [ hep - ph/9807423 ] ; eur . j. * c9 * , 343 ( 1999 ) [ hep - ph/9812472 ] ; m.carena , h.e.haber , s.heinemeyer , w.hollik , c.e.m.wagner and g.weiglein , nucl . phys . * b580 * , 29 ( 2000 ) [ hep - ph/0001002 ] ; r .- j.zhang , phys . lett . * b447 * , 89 ( 1999 ) [ hep - ph/9808299 ] ; j.r.espinosa and r .- j.zhang , jhep * 0003 * , 026 ( 2000 ) [ hep - ph/9912236 ] ; nucl b586 * , 3 ( 2000 ) [ hep - ph/0003246 ] ; j.r.espinosa and i.navarro , nucl . phys . * b615 * , 82 ( 2001 ) [ hep - ph/0104047 ] ; g.degrassi , p.slavich and f.zwirner , nucl . * b611 * , 403 ( 2001 ) [ hep - ph/0105096 ] ; a.brignole , g.degrassi , p.slavich and f.zwirner , nucl . * b631 * , 195 ( 2002 ) [ hep - ph/0112177 ] ; nucl . * b643 * , 79 ( 2002 ) [ hep - ph/0206101 ] ; a.dedes , g.degrassi and p.slavich , nucl . b * 672 * ( 2003 ) 144 , hep - ph/0305127 ; computer programs developed for computing the radiatively - corrected higgs masses , feynhiggs and feynhiggsfast , are described respectively in s.heinemeyer , w.hollik and g.weiglein , comput . commun . * 124 * , 76 ( 2000 ) [ hep - ph/9812320 ] ; and cern - th-2000 - 055 [ hep - ph/0002213 ] . p.h.chankowski , s.pokorski and j.rosiek , _ nucl . * b 423 * ( 1994 ) 437 , hep - ph/9303309 ; _ nucl . * b 423 * ( 1994 ) 497 ; + v.driesen and w.hollik , _ z. phys . _ * c 68 * ( 1995 ) , 485 , hep - ph/9504335 ; + v.driesen , w.hollik and j. rosiek , _ z. phys . _ * c 71 * ( 1996 ) 259 , hep - ph/9512441 . s.heinemeyer , w.hollik , j.rosiek and g.weiglein , _ eur . j. _ * c 19 * ( 2001 ) 535 , hep - ph/0102081 ; + s.heinemeyer and g.weiglein , _ nucl . phys . proc . * 89 * ( 2000 ) 210 . t.hahn , s.heinemeyer and g.weiglein , _ mssm higgs - boson production at the linear collider : dominant corrections to the w w fusion channel , nucl . b * 652 * ( 2003 ) 229 , hep - ph/0211204 , lc - th-2002 - 018 . + t.hahn , s.heinemeyer , g.weiglein , _ very heavy mssm higgs - boson production at the lc , nucl . suppl . * 116 * ( 2003 ) 336 , lc - th-2002 - 019 ( 2002 ) . _ _ t.klimkovich , _ experimental study of heavy susy higgs bosons at the lc , talk given at the ecfa / desy workshop on physics and detectors for a future linear collider , amsterdam , spring 2003 , lc note in preparation . a.raspereza , _ cp properties of higgs bosons and @xmath124 selection , talk given at the ecfa / desy workshop on physics and detectors for a future linear collider , prague , autumn 2002 , + a.raspereza , t.klimkovich , t.kuhl , k.desch , lc note in preparation . s.kanemura , s.moretti and k.odagiri , jhep * 0102 * ( 2001 ) 011 , hep - ph/0012030 . + h.e.logan and s.f.su , phys . d * 67 * ( 2003 ) 017703 , hep - ph/0206135 . + s.moretti , _ detection of heavy charged higgs bosons at future linear colliders via @xmath125 production , hep - ph/0209210 , lc - th-2002 - 010 ( 2002 ) . + o.brein , hep - ph/0209124 . _ s.heinemeyer , eur . j. c * 22 * ( 2001 ) 521 , hep - ph/0108059 . + m.frank , s.heinemeyer , w.hollik , g.weiglein , _ the higgs - boson masses of the complex mssm : a complete one - loop calculation , hep - ph/0212037 , lc - th-2002 - 016 ( 2002 ) . _

this report summarizes the progress in the study of higgs physics at a future linear electron positron collider at center - of - mass energies up to about 1000 gev and high luminosity . after the publication of the tesla technical design report @xcite , an extended ecfa / desy study on linear collider physics and detectors was performed . the paper summarizes the status of the studies with main emphasis on recent results obtained in the course of the workshop .

hep-ph0311092

the investigation of @xmath1-conjugated polymers is in many ways affected by the structural disorder in these systems . in contrast , the ladder - type poly - p - phenylenes ( lppp ) @xcite offer the opportunity to study large , rod - like chains of planarised phenylene units . as a consequence , the @xmath1-system might spread out over an entire polymer and a vibronic resolution of the @xmath2 transition is discernible @xcite . in order to deduce some characteristics of the polymeric films @xcite , like the effective conjugation length , several oligomers have been synthesized in the past to study the low - lying electronic excited states of the polymer @xcite . photoconduction in lppp films @xcite has been measured as a function of the energy of the exciting light , too . a typical small plateau of the photocurrent occurs between the absorption edge and its steep rise at higher energies and extends in this case over 1.6 ev . this behavior of the photocurrent which does not follow directly the absorption features is sometimes called `` abatic '' . one possible explanation for this effect rests on the interpretation of the electronic excited states for the individual molecules . excited states of @xmath1-conjugated molecules are usually described as coulomb bound electron - hole pairs . this physical picture originates from solid - state physics of ( organic ) semi - conductors . therefore , these molecular states are often referred to as excitons , although they have to be clearly distinguished from the extended band states in a crystal . a reasonable estimate of the exciton binding energy in conjugated polymers has been determined , e.g. , by scanning tunneling spectroscopy measurements @xcite which have lead to a value of about @xmath3 ev . excited states with a smaller value , and larger electron - hole distance , respectively , should be more susceptible to the separation via an external electric field . following this idea , the conjecture has been brought forward that such a state is responsible for the steep rise of the photocurrent in poly - phenylene - vinylene ( ppv ) @xcite . later on , another explanation has followed based on the excess photon energy which is converted to the vibrational heat bath @xcite . the latter proposal is now widely accepted . in order to test these concepts for excited states of @xmath1-conjugated systems , several oligomers of the lppp type with up to eleven phenylene rings are investigated in this work . the study of oligomers instead of an ( infinite ) polymer follows the above mentioned approach and allows the direct comparison with experiment . the main difference to the experiments in condensed phases is the restriction to single chains in the vacuum . as the experimentally used molecules are computationally too demanding one has to replace the large aliphatic substituents attached to lppp by hydrogen ( see figure [ fig:1 ] and table [ tab:0 ] ) . this should have only negligible effects on the optical properties , which are governed by the frontier orbitals of @xmath1-symmetry . these aliphatic substituents are only necessary for the better solubility of the polymer , or to prohibit the formation of aggregates in the film ( @xmath4 = methyl in figure [ fig:1 ] ) . since the systems studied here reach the size of the effective conjugation length proposed for lppp ( about 14 phenylene rings @xcite ) , ab - initio or density functional methods are not applicable , and one has to assent to less expensive semiempirical methods ( am1 , indo / s ; see below ) . thus , the wave functions of the corresponding ground states are the indo ( intermediate neglect of differential overlap ) slater determinants @xmath5 @xcite . for the excited states @xmath6 ( see equation [ phicisdef ] ) , the indo / s expansion is used in the spin - singlet sector . the excited states with dominant oscillator strengths will be addressed as @xmath7 for the first excited state , @xmath8 for the intermediate excited state and @xmath9 for the high energy , `` frenkel - type '' state . the electronic ground state will be denoted as @xmath10 , the semiempirical approach is briefly described . in sect . [ geometryspectra ] , the results for the geometric structure of the oligomers and their spectra are presented . the main part of this article , sect . [ interpretation ] , focuses on the development of a general interpretation scheme for excited - state wave functions . its application to indo / s wave functions leads in a straightforward way to the interpretation of the excited states as bound electron - hole pairs . a short conclusion closes the presentation . although it is not feasible to calculate the higher oligomers by first - principle methods , the oligomer with two phenylene rings ( @xmath11 ) has been calculated at mp2/6 - 31 g * level @xcite(mller - plesset pertubation theory of second order ) . the results are used as a reference for the semiempirical methods . following a procedure of brdas _ et al . _ @xcite , one has to choose a semiempirical method which yields reliable geometric structures . in the present case the austin semiempirical parametrization ( am1 ) @xcite gives an almost exact agreement with the mp2 results ( cf . @xmath12 for the bond lengths in table [ tab:1 ] ) . this method will therefore be used to calculate the geometric structure of the ground states . note , however , that the pm3 method @xcite yields better results for zero - point vibrational energies ( zpe ) . the am1 and mp2 calculations have been carried out on a ibm / sp2 computer using the gaussian94 ( rev . d4 ) @xcite program package . all minima are characterized by diagonalization of the hessian matrix , whereby the zero - point energies ( zpe ) have been obtained , too . the latter will be given unscaled throughout the paper . in the next step , excited singlet states are calculated using zerner s indo / s method @xcite based on the minimum am1 structures from the electronic ground states . thus it is clear that geometric relaxation effects in the excited state are neglected . the active ci space consists of the 22 highest occupied and the 22 lowest unoccupied molecular orbitals . this is the biggest active space possible within the used program package and it contains orbitals of @xmath13 symmetry for some oligomers as well . as expected , the dominant configuration state functions in the wave function describe @xmath1@xmath14 excitations . the calculation of the spectra have been accomplished on a pentiumiii pc using the program package cache 3.1 @xcite . in order to get a more realistic view of the calculated line spectra , gaussian peaks are least - square fitted to the indo / s oscillator strengths . this procedure masks transitions with moderate oscillator strengths which are close in energy to a dominant transition . this is especially the case for the high - energy region of the spectrum , in which strong @xmath15 transitions cover others . the hidden states are not visible in the optical spectrum , and consist of a number of important configurations sensitive to the size of the active ci space ( figure [ fig:2 ] ) . the remaining part of the spectra comprises only features with one transition , except in two cases , where two almost identical states are close together . in these cases only one state will be discussed ( the @xmath2 transition for the @xmath11 oligomer and the @xmath16 transition for the trimer ; see discussion below ) . all structures have been optimized without any symmetry constraints . therefore , the geometries only fit to certain point groups within crystallographic accuracy ( @xmath17 0.001 , @xmath18 ) . the resulting point groups are equivalent to ones suggested by the simple lewis - type structure , see figure [ fig:1 ] . oligomers with an even number of phenylene rings adopt @xmath19 symmetry . those with an odd number adopt @xmath20 symmetry . the c c bond lengths range between about 1.380 and about 1.500 . figure [ fig:1.5 ] shows the mp2 result for the @xmath21 oligomer as an example . as the smallest c c bond distance in a methylene bridge is 1.503 long , they are assigned to single bonds . as a consequence , the hydrogen atoms of the methylene bridge do not participate in the conjugated system , although this would be allowed . this supports the validity of neglecting the aliphatic substituents . in figure [ fig:2 ] the ( vibronically unresolved ) calculated spectra of lppp oligomers are plotted . a comparison with measured fluorescence spectra of different oligomers @xcite shows qualitative agreement , i.e. , they show a broad @xmath2 transition which is shifting to lower energies with increasing system size . as one would expect for the @xmath7 state , the homo lumo excitation is the dominant one . homo ( lumo ) refers to the highest occupied ( lowest unoccupied ) molecular orbital in the indo ground state . a second interesting feature can be seen in the calculations for the oligomers with five and more rings . it will be referred to as the @xmath22 transition , because it is energetically in the middle between the @xmath7 state and the high energy region . in the @xmath8 state the dominant determinants are built by the substitution of the homo by the lumo+2 and the substitution of the homo@xmath231 by the lumo+1 . the position of the @xmath16 transition also shifts to lower energies with increasing oligomer length . this is not discernible in the experimental spectra because of the small corresponding oscillator strengths . finally , both sets of spectra show a steep rise at around 5.40 ev which is dominated by a transition at approximately 5.85 ev for every oligomer . this transition will be called the @xmath15 transition , as the @xmath9 state shows a high degree of localization ( i.e. , a `` frenkel '' state ) , which will become clear later in the discussion . the @xmath9 state is determined by several configuration state functions , where low - lying occupied molecular orbitals are exchanged by orbitals with high energy . in figure [ fig:3a ] the energy of all three optically important transitions is drawn as a function of the reciprocal number of phenylene rings in the molecule . the transitions of the @xmath7 and the @xmath8 states show a strong linear dependence , whereas the energy of the @xmath15 transitions hardly changes with system size , in agreement with the experimental observation . despite this qualitative agreement , a quantitative comparison with the experimental values for the effective conjugation length @xcite is not possible , since the theoretical curve leads to an unphysical negative value . the main reason for this error is the neglect of the polarization energy in the calculations which stabilizes the excited states in the condensed phases in the experiments . moreover , the linear extrapolations for the transitions to the @xmath8 and the @xmath7 states suggest a crossing of the two energies which must not be taken for granted at this stage . from the electronic dipole transition moments @xmath24 for the @xmath2 transitions , one obtains an estimate for the radiative fluorescence lifetimes @xmath25 for every oligomer , using [ lifetime ] @xmath26 here , @xmath27 is the elementary charge , @xmath28 is the position vector of the _ i_-th particle of charge @xmath29 , @xmath30 is planck s constant , @xmath31 is the speed of light , and @xmath32 is the energy of the transition , all quantities measured in cgs units . the lifetimes @xmath25 show a linear dependence on the reciprocal number of phenylene rings , see figure [ fig:3b ] . here , the value of several hundred picoseconds for the large oligomers is in quantitative agreement with the experimental value of about 300 ps for the polymer , measured in an organic matrix @xcite . under the assumption of a linear relation between @xmath25 and the inverse of the oligomer length , one finds from the data in table [ tab:3 ] that a value of @xmath33 corresponds to an effective conjugation length of about 20 phenylene rings , in reasonable agreement with the value of 14 rings estimated from other optical experiments @xcite . for an interpretation the wave functions for the ground state and the excited states need to be related to experimentally observable quantities . the optical absorption at frequency @xmath34 is proportional to the real part of the optical conductivity , as given by the kubo formula @xcite @xmath35= \frac{{\rm im\ , } \left [ \chi_{jj}(\omega > 0)\right ] } { \omega } \ ; , \end{aligned}\ ] ] where @xmath36 is the current current correlation function , @xmath37 here , @xmath38 is the hamilton operator of the system , @xmath39 are its eigenstates with energies @xmath40 ( @xmath41 ) , @xmath42 , and @xmath43 is positive infinitesimal . therefore , the real part of the optical conductivity may be written as @xmath44= \frac{\pi}{\omega}\sum\limits_{\left|\phi_s\right\rangle } \left|\left\langle\phi_s\left|\hat{\jmath } \right|\phi_0\right\rangle\right|^2 \;\delta\left(\omega-(e_s - e_0)\right ) \ ; . \label{resigmadelta}\ ] ] the current operator is defined by @xmath45 where @xmath46 , @xmath47 are creation and annihilation operators for electrons with spin @xmath48 in the molecular orbitals @xmath49 , and @xmath50 is the matrix element between the corresponding one - particle states . equation ( [ resigmadelta ] ) is readily interpreted . the absorption of a photon with energy @xmath34 induces an excitation between the ground state @xmath51 and the excited states @xmath39 with energy @xmath52 . the amplitude for this absorption process @xmath53 determines the oscillator strength , @xmath54 . for a further analysis of the amplitudes @xmath55 , the current operator is expressed in terms of the field operators , @xmath56 which create / annihilate an electron with spin @xmath13 at @xmath57 . from the completeness of the molecular orbitals @xmath58 one readily obtains @xmath59 with @xmath60 therefore , the amplitudes can be cast into the form @xmath61 where the particle - hole amplitudes @xmath62 are given by the overlap between the excited state @xmath63 and the ground state @xmath64 with an electron at @xmath65 and a hole at @xmath66 ( if @xmath67 ) . it is thus seen that the quantities @xmath68 allow to address the question in how far a given excited state @xmath63 may be viewed as an electron - hole excitation of the ground state , and they directly enter the kubo formula for the optical conductivity . note that the analysis of ( [ amplitude ] ) does not require the full spatial dependence of the many - particle wave functions @xmath63 or @xmath64 as in @xcite or the density density correlation function of the ground state as in @xcite . the matrix elements for the current operator , @xmath69 in ( [ matrixelement ] ) , do not change much over atomic distances . therefore , it is usually sufficient to introduce the coarse - grained densities for the electron - hole content of @xmath39 with respect to @xmath70 , @xmath71 where @xmath72 is the step function for the ( atomic ) volume @xmath73 around atom @xmath74 . @xmath75 is the overlap density between the excited @xmath76 and the ground state @xmath70 with an electron - hole pair around the nuclei at @xmath74 and @xmath77 , respectively . equation ( [ amplitude ] ) is exact and applies to all quantum - mechanical systems . it is currently under investigation for the numerical analysis of quantum lattice systems with the density matrix renormalisation group ( dmrg ) method @xcite . in the next subsection , the present case of indo / s wave functions will be studied in more detail . as described earlier , the ground state is approximated as an indo slater determinant @xmath78 \psi_{\sigma}^{\dagger}(\vec{x}_1 ) \ldots \psi_{\sigma}^{\dagger}(\vec{x}_n ) where the molecular orbitals @xmath79 ( @xmath80 ) are expressed as a linear combination of spatial atomic orbitals ( aos ) @xmath81 @xmath82 the atomic orbitals are centered at certain nuclei such that @xmath83 comprises the orbital type ( @xmath84 ) and its position @xmath85 . the spin singlet excited - state wave functions are described as linear combinations of singly - excited configuration state functions [ phicisdef ] @xmath86 where the indices @xmath87 and @xmath88 refer to the active space of occupied and virtual molecular orbitals , respectively . with these approximations for @xmath51 and @xmath39 , it is obvious that the coordinates of the added electron enter ( [ amplitude ] ) only through the virtual mos @xmath89 whereas the hole is described by the eliminated , occupied mos @xmath90 , @xmath91 to make further progress , one has to address the coarse - grained densities @xmath75 in ( [ pij ] ) , @xmath92 in the last step the indo approximation @xcite and the orthogonality of the atomic orbitals on the same atom were used , i.e. , @xmath93 recall that @xmath94 denote ao coefficients , and @xmath95 are the ci coefficients of the cis wave function . in this case , one may verify that @xmath96 , if one takes into account the indo approximation and the normalization of @xmath97 in ( [ phicisdef ] ) , @xmath98 therefore , @xmath99 may be viewed as a discrete probability function for finding a hole around the nucleus @xmath74 and an electron around the nucleus @xmath77 . this simplification is only valid in the indo / s approximation . a check can be made , whether the values of @xmath99 carry significant weight corresponding to aos of @xmath13 symmetry . it turns out that this weight is vanishingly small , and one can sum all contributions of the aos of an atom without losing information . one has to verify that the detected electron - hole correlations for an excited state really stem from the electron - hole interaction rather than from a coincidence of the motion of two independent particles confined to a small molecule . therefore , the quantity @xmath100 has been investigated according to the scheme ( [ amplitude ] ) . here , @xmath101 is the indo wave function where the homo is substituted by the lumo . this corresponds to the pure `` band - like '' excitation , i.e. , in a cis state representation this configuration state function has the coefficient @xmath102 for @xmath103 , @xmath104 , and all other coefficients are set to zero . given the @xmath99 an expectation value @xmath105 of the distance of the electron - hole pair can be calculated with the help of the bond lengths @xmath106 the standard deviation @xmath107 is equally accessible , @xmath108 in small molecules , the confinement of the oligomers will determine the distance of the electron - hole pair . in larger molecules , if the interaction between hole and electron is weak , @xmath109 and @xmath107 will increase with system size since the particles move essentially independently through the molecule . on the other hand , the interaction between electron and hole may keep them together at a fixed distance even though the size of the molecule increases . such a bound electron - hole pair may lead to a constant value of @xmath109 and @xmath107 for every system size . note , however , @xmath109 and @xmath107 only contain an information about the overall extension of an excitation but may fail to describe the localization of an electron - hole pair onto segments of the molecule ; an example of this situation is given below . in order to get a more pictorial way of the electron - hole pair distribution , one may concentrate on a quasi one - dimensional chain of carbon atoms ; figure [ fig:4a ] shows how those chains are chosen . when @xmath110 is plotted for this chain a bound electron - hole pair will show large values along the diagonal of the plot and vanishingly small values in the off - diagonal regions . unbound pairs will lead to the opposite situation . the two - dimensional distribution @xmath110 can be further smoothed into @xmath111 which is solely a function of the electron - hole distance . the choice of @xmath112 gives smooth curves as a function of @xmath88 . @xmath113 gives the most concise description of the electron - hole excitation . the @xmath114 matrix has been calculated for the states , whose transitions are of significant oscillator strength in the spectra : the @xmath7 , the @xmath8 , and the @xmath9 state . first , the state @xmath7 is discussed . as seen from figure [ fig:4b ] , no bound state is formed for only two phenylene rings . in fact , the molecular confinement is dominant up to a chain length of four phenylene rings . for five phenylene rings , see figure [ fig:4c ] , and larger systems , a bound electron - hole pair is discernible . the shape of @xmath115 in the plot along the one - dimensional chain does not change for systems larger then five phenylene rings . the @xmath7 state clearly corresponds to a strongly bound electron - hole pair . figure [ fig:5a ] shows the average electron - hole separation and the corresponding standard deviation for the @xmath7 state as a function of system size . they saturate for more than five phenylene rings , @xmath116 . this is in reasonable agreement with the experimental value of 7 obtained from electro - absorption measurements @xcite . some results of electron energy - loss spectroscopy , however , seem to hint at a totally different behavior of the excited states : they are supposed always to increase with increasing length of the oligomer @xcite . at present , the reason for this discrepancy is not clear . the saturation behavior for @xmath117 and the apparently large saturation value for the standard deviation @xmath118 are readily understood from the `` smoothed '' probability distribution @xmath119 , see ( [ smoothedp ] ) . as seen in figure [ fig:5b ] , @xmath120 decreases with increasing system size , the whole distribution broadens , and develops a maximum around @xmath121 . for oligomers with five and more phenylene rings the distribution does not change significantly . since at the same time the system grows , the values of @xmath117 and @xmath122 saturate . the relatively large values of @xmath122 are due to the location of the maximum of the distribution at a finite value of @xmath123 . finally , figure [ fig:5c ] shows the probability distribution @xmath124 for the largest oligomer with eleven phenylene rings ( @xmath125 ) . next , the state @xmath8 is addressed . brdas and coworkers have located a state in the calculated spectrum of poly - p - phenylene - vinylene ( ppv ) , which they assign to a charge transfer state . in this context this state is equivalent to an unbound electron - hole pair . the optical transition to this charge transfer state is in the same region as the steep increase of the photocurrent in the respective polymeric film . as a result , they conclude that the population of this state is responsible for this `` abatic '' behavior of the photocurrent . the @xmath8 state obtained in the present calculation lies in the same region as the charge - transfer state in ppv , that is between the @xmath7 and the high - energy states . the question is , whether the @xmath8 state is a charge transfer state or not . as seen in figure [ fig:6a ] , the @xmath8 state shows almost constant values of @xmath126 and @xmath127 . a rise in these values occurs for the @xmath125 oligomer which hints at unbound states for larger systems . this idea , however , is not supported by a more detailed look at the graphs for @xmath128 as a function of oligomer length , see figure [ fig:6b ] , and the results for @xmath129 for the largest oligomer , see figure [ fig:6c ] . the curves are qualitatively the same as for the corresponding @xmath7 states . for example , compare figures [ fig:5c ] and [ fig:6c ] : @xmath115 and @xmath129 are essentially zero in the off - diagonal regions . similarly , the curves @xmath128 as a function of oligomer length resemble those of @xmath119 , compare figures [ fig:5b ] and [ fig:6b ] . the only difference to the @xmath7 state is the localization of the excitation to three parts of the molecule in @xmath8 . because of this , one can not interpret this state as an unbound elctron - hole pair or ~ exciton , and consequently no explanation for the rise of the photocurrent of lppp at about 4.0 ev can be given at this level of theory . this is in line with the assumption that local heating due to excess energy is the reason for the behavior of the photocurrent @xcite and not the nature of the optically accessible states . lastly , for the @xmath9 state , the mean electron - hole separation does not saturate but grows with system size , see fig . [ fig:7a ] . from such an analysis one might conclude a delocalized state . on the contrary , for the @xmath9 state the degree of localization is _ higher _ than in the other two excited states , and the electron - hole pair is actually restricted to every single phenylene ring . as can be seen from figures [ fig:7b ] and [ fig:7c ] , the apparent lack of convergence in @xmath130 and @xmath131 relates to the fact that isolated elctron - hole pairs on chain segments can be linearly superimposed , i.e. , an extended state can be formed which , nevertheless , does not contribute to the conductivity . as for the states @xmath7 and @xmath8 , there is no spreading of @xmath132 into off - diagonal regions with increasing system size , and the smoothed distribution function @xmath133 displays the same trends as before . so far , every state investigated can be regarded as a bound electron - hole pair with different degrees of localization . they are molecular analogues of the excitons in the physics of semiconductors . in order to see the differences to a pure `` band excitation '' , the analysis is repeated for @xmath101 , see ( [ homolumo ] ) . as expected , and confirmed in figures [ fig:8a ] , [ fig:8b ] , and [ fig:8c ] , the motion of electron and hole in the molecule is uncorrelated . in contrast to the excitonic cases before , the almost linear increase of the mean electron - hole separation and its variance as a function of system size in figure [ fig:8a ] is accompanied by a broadening and flattening of the smoothed distribution function @xmath134 , see figure [ fig:8b ] . as seen from figure [ fig:8c ] there is considerable weight in the off - diagonal region in the probability distribution @xmath135 , which actually looks like a half sphere . hence , the excitonic behavior of the states @xmath7 , @xmath8 , and @xmath9 is genuine , and not just a coincidence in the uncorrelated motion of an electron and a hole in a restricted geometry . the ground and singlet excited states of various oligomers of lppp have been described with semiempirical methods . a qualitative agreement was achieved with the experimental absorption spectra , especially for the measurements of the fluorescence lifetime , and the mean electron - hole distance . a new analysis of the excited states has been given . as in optical absorption experiments , the overlap matrix elements between excited states and the ground state with an electron - hole pair are studied as a function of their respective positions on the oligomers . excited states with high oscillator strengths are found to be bound electron - hole pairs . therefore , no explanation of the abatic onset of the photocurrent in lppp films can be given at this level of a microscopic theory . this work has been made possible with the kind technical support of prof . frenking , and the financial support of the graduiertenkolleg `` optoelektronik mesoskopischer halbleiter '' as well as the aurc and the marsden fund in wellington . r. ditchfield , w. j. hehre and j. a. pople , _ j. chem . phys . _ * 54 * , 724 ( 1971 ) . + w. j. hehre , r. ditchfield and j. a. pople , _ j. chem . phys . _ * 56 * , 2257 ( 1972 ) . + p. c. hariharan and j. a. pople , _ mol . * 27 * , 209 ( 1974 ) . + m. s. gordon , _ chem . lett . _ * 76 * , 163 ( 1980 ) . + p. c. hariharan and j. a. pople , _ theo . . acta _ * 28 * , 213 ( 1973 ) . + g. a. petersson , a. bennett , t. g. tensfeldt , m. a. al - laham , w. a. shirley and j. mantzaris , _ j. chem . phys . _ * 89 * , 2193 ( 1988 ) . gaussian 94 , revision d4 ; m.j . frisch , g.w . trucks , h.b . schlegel , g.e . scuseria , m.a . robb , j.r . cheeseman , v.g . zakrzewski , j.a . montgomery , jr . stratmann , j.c . burant , s. dapprich , j.m . millam , a.d . daniels , k.n . kudin , m.c . strain , o. farkas , j. tomasi , v. barone , m. cossi , r. cammi , b. mennucci , c. pomelli , c. adamo , s. clifford , j. ochterski , g.a . petersson , p.y . ayala , q. cui , k. morokuma , d.k . malick , a.d . rabuck , k. raghavachari , j.b . foresman , j. cioslowski , j.v . ortiz , a.g . baboul , b.b . stefanov , g. liu , a. liashenko , p. piskorz , i. komaromi , r. gomperts , r.l . martin , d.j . fox , t. keith , m.a . al - laham , c.y . peng , a. nanayakkara , c. gonzalez , m. challacombe , p.m.w . gill , b. johnson , w. chen , m.w . wong , j.l . andres , c. gonzalez , m. head - gordon , e.s . replogle and j.a . pople , gaussian , inc . , pittsburgh pa , 1998 . j. e. ridley and m. c. zerner , _ theo . . acta _ * 32 * , 111 ( 1973 ) . + m. c. zerner , _ semiempirical molecular orbital methods _ in _ reviews of computational chemistry _ , edited by k. b. lipkowitz and d. b. boyd ( vch publishing , new york , 1991 ) , vol . 2 , pp . 313 - 365 . m. rohlfing and s. g. louie , _ phys . lett . _ * 82 * , 1959 ( 1999 ) . van der horst , p. a. bobbert , p. h. l. de jong , m. a. j. michels , g. brocks and p. j. kelly , _ phys . rev . b _ * 61 * , 15817 ( 2000 ) .

ground state properties and excited states of ladder - type paraphenylene oligomers are calculated applying semiempirical methods for up to eleven phenylene rings . the results are in qualitative agreement with experimental data . a new scheme to interpret the excited states is developed which reveals the excitonic nature of the excited states . the electron - hole pair of the @xmath0-state has a mean distance of approximately 4 .

cond-mat0103320