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TIGER-Lab 110

How many ways are there to color the vertices of a cube with two colors, up to rotation?

23

integer

In Chord, assume the size of the identifier space is 16. The active nodes are N3, N6, N8 and N12. Show all the target key (in ascending order, ignore the node's identifier itself) for N6.

[7, 8, 10, 14]

list of integer

Fig. Q2 shows a 1st-order noise shaper. The input is bounded by 0 v and 1 v. A constant 0.4 v input is fed into the noise shaper. The output is a periodic pattern sequence. What is the period of the sequence?

5

integer

The stock of the CCC Corporation is currently valued at $12 and is assumed to possess all the properties of geometric Brownian motion. It has an expected annual return of 15%, an annual volatility of 20%, and the annual risk-free is 10%. Using a binomial lattice, determine the price of a call option on CCC stock maturing in 10 monthes time with a strike price of $14 (Let the distance between nodes on your tree be 1 month in length).

53.0

float

Evaluate $\int_c 1 / (z^ + 4)^2 dz$ over the contour. This contour is a circle centered at (0, i) with a diameter of 3 on the (Re, Im) plane, the contour goes counter-clockwise.

0.19634

float

The difference equation of a digital system is given by $$ y[n]=8 x[n]+2 x[n-1]-x[n-2], $$ where $x[n]$ and $y[n]$ are, respectively the current samples of the input and the output signals of the system. Determine if the system is a FIR.

True

bool

$H(X_n|X_0)$ is a concave function of n for a stationary Markov process. True or False?

True

bool

There are only three active stations in a slotted Aloha network: A, B and C. Each station generates a frame in a time slot with the corresponding probabilities p_A=0.2, p_B=0.3 and p_C=0.4 respectively. What is the normalized throughput of the system?

0.452

float

A bird is lost in a 3 by 3 by 3 cubical maze. The bird flies from room to room going to adjoining rooms with equal probability through each of the walls. To be specific, the corner rooms have 3 exits. What is the entropy rate of this random walk? Use base 2 logarithm and return the entropy rate in bits.

2.03

float

If the sum-product algorithm is run on a factor graph with a tree structure (no loops), then after a finite number of messages have been sent, there will be no pending messages. True or false?

True

bool

Suppose the Markov Chain satisfies the diagram ./mingyin/diagram.png What is the period of state 0? What is the period of state 1? Return the two answers as a list.

[2, 2]

list of integer

Let {N(t), t \in [0, \infty)} be a Poisson process with rate of $\lambda = 4$ and $X_1$ be the first arrival time. Given N(t) = 1, then what is $P(X_1 <= t / 2)$?

0.5

float

A $200-cm^3$ glass flask is filled to the brim with mercury at 20°C How much mercury overflows when the temperature of the system is raised to 100°C. The coefficient of linear expansion of the glass is $0.40 \times 10^{-5} K^{-1}. (Unit: cm^3)

2.7

float

Suppose a monopoly market has a demand function in which quantity demanded depends not only on market price (P) but also on the amount of advertising the firm does (A, measured in dollars). The specific form of this function is Q = (20 - P)(1 + 0.1A - 0.01A^2). The monopolistic firm's cost function is given by C = 10Q + 15 + A. Suppose there is no advertising (A = 0). What output will the profit-maximizing firm choose?

5

integer

A single firm monopolizes the entire market for widgets and can produce at constant average and marginal costs of AC = MC = 10. Originally, the firm faces a market demand curve given by Q = 60 - P. Calculate the profit-maximizing price for the firm.

35

integer

If four points are picked independently at random inside the triangle ABC, what is the probability that no one of them lies inside the triangle formed by the other three?

0.6667

float

Consider a file with a size of 350 Kbytes storing in a web server. Client A sends a request to the server to retrieve the file from a remote location. It is known that the link capacity between client A and the server is 10 Mbps and the round trip time (RTT) between the server and client is fixed at 20ms. Assume that the segment size is 20 Kbytes and the client has a receiver buffer of 200Kbytes. Assume that the window size (W) is fixed at 2. How long (in ms) does client A take to receive the whole file from the server after sending a request?

352

integer

In how many ways can 3 students be selected from a class of 20 to form a study group?

1140

integer

Find the size of angle x in the figure.

24

integer

Fig.Q3 shows an excerpt of the transmission phase of a TCP connection. Assume the length of the IP header is 20 bytes. What is the ACK number at message 6?

839

integer

Determine the period of the following signal, $$ x_1(t)=\cos (3 \pi t)-4 \cos (5 \pi t-0.5 \pi) $$

2

integer

Suppose a stock has the following information. It is listed on the London stock exchange and operates throughout Europe. The yield on a UK 10 year treasury is 2.8%. The stock in question will earn 8.6% as per historical data. The Beta for the stock is 1.4, i.e., it is 140% volatile to the changes in the general stock market. What is the expected rate of return?

10.92

float

Use divergence therem to evaluate $\iint_S \vec{F} \cdot d \vec{S}$ where $\vec{F} = xy \vec{i} - \frac{1}{2}y^2\vec{j} + z\vec{k}$ and the surface $S$ consists of the three surfaces, $z=4 - 3*x^2 - 3y^2, 1 \le z \le 1$ on the sides and $z=0$ on the bottom.

7.853

float

A pure lead bar 10 cm long is maintained with one end at T &=300 K and the other at 310 K. The thermoelectric potential difference thus induced across the ends is 12.8 micro-volts. Find the thermoelectric power for lead in this temperature range in V/K. (Note: Q varies nonlinearly with temperature, but over this narrow temperature range, you may use a linear approximation.)

1.28e-06

float

Find the sum of $\sum_{n=1}^{\infty} (1/e^n + 1/(n*(n+1)))$

1.581

float

Lore Ltd. estimates that its dividend growth will be 13% per year for the next five years. It will then settle to a sustainable, constant, and continuing rate of 5%. Let’s say that the current year’s dividend is $14 and the required rate of return (or discount rate) is 12%. What is the current fair value of Lore Ltd. stock?

291.45

float

You wish to put a 1000-kg satellite into a circular orbit 300 km above the earth's surface. How much work must be done to the satellite to put it in orbit? The earth's radius and mass are $R_E}=$ $6.38 \times 10^6 m$ and $m_E=5.97 \times 10^{24} kg$. (Unit: 10^10 J)

3.26

float

In triangle RST, X is located on the side RS, Y is located on the side RT, Z is located on the side ST, and XY and XZ are midsegments of △RST. If the length of side XY is 7, the length of side RT is 13, and the measure of angle YXZ is 124°, what is the length of side XZ?

6.5

float

Let h(x) = 1/(\sqrt{x} + 1). What is h''(x) when x = 1?

0.125

float

If z = \frac{1 + e^{-2x}}{x + tan(12x)}, what's the derivative of $\frac{\partial z}{\partial x}$ at $x = 1$.

-153.59

float

Determine the AC power gain for the common-emitter amplifier in the figure. Assume that $\beta_{ac} = 100$, the internal emitter resistance $r_e = 12.3 \Omega$.

33540

integer

Compute the double integrals over indicated rectangles $\iint\limits_{R}{{2x - 4{y^3}\,dA}}$, $R = [-5,4] \times [0, 3]

-756

integer

How many labeled trees are there on 6 vertices?

1296

integer

The position of a point for any time t (t>0) s defined by the equations: x=2t, y=ln(t), z = t^2. Find the mean velocity of motion between times t=1 and t=10.

11.25

float

Is function f defined by $f(z) = \int_0^{\infy} |e^{zt}| / (t+1) dt$ analytical on the left plane D: Re(z) < 0

True

bool

A random variable $X$ takes on $m$ values and has entropy $H(X)$. An instantaneous ternary code is found for this source, with an average length $L=H_3(X)$ that achieves the entropy bound. Then $m$ must be odd. True or False?

True

bool

Suppose the graph of a polynomial f(t) = a + bt + ct^2 passes through points (1, -1), (2, 3), and (3, 13). What is f(-1)?

9

integer

Suppose the codeword that we use to describe a random variable X always starts with a symbol chosen from the set {7, 8, 9} , followed by binary digits {0, 1}. Thus we have a ternary code for the first symbol and binary thereafter. Give the optimal uniquely decodeable code (minimum expected number of symbols) for the probability distribution $p = (16/69, 15/69, 12/69, 10/69, 8/69, 8/69)$.

[7, 8, 9, 70, 80, 90]

list of integer

H(z) = $\int_0^1 e^{-z^2 t^2} dt$, what is H'(1)?

-0.3789

float

matrix $A=(\begin{array}{rrrr} -2 & -1 & -1 & -1 \ 2 & 1 & 3 & 2 \ 1 & 1 & 0 & 1 \ -1 & -1 & -2 & -2 \end{array})$. Suppose f is the minimal polynomial of A. What is f(99)? Return the numeric without explanation.

990000.0

float

The planet Pluto (radius 1180 km) is populated by three species of purple caterpillar. Studies have established the following facts: 1. A line of 5 mauve caterpillars is as long as a line of 7 violet caterpillars. 2. A line of 3 lavender caterpillars and 1 mauve caterpillar is as long as a line of 8 violet caterpillars. 3. A line of 5 lavender caterpillars, 5 mauve caterpillars and 2 violet caterpillars is 1 m long in total. 4. A lavender caterpillar takes 10 s to crawl the length of a violet caterpillar. 5. Violet and mauve caterpillars both crawl twice as fast as lavender caterpillars. How many years would it take a mauve caterpillar to crawl around the equator of Pluto?

23.0

float

Find the smallest positive integer that leaves a remainder of 3 when divided by 5, a remainder of 4 when divided by 7, and a remainder of 2 when divided by 9.

263

integer

Three years ago, Fred invested $10,000 in the shares of ABC Corp. Each year, the company distributed dividends to its shareholders. Each year, Fred received $100 in dividends. Note that since Fred received $100 in dividends each year, his total income is $300. Today, Fred sold his shares for $12,000. What is the holding period return of his investment?

0.23

float

If z = arctan(e^{1 + (1 + x)^2}), what's the derivative of $\frac{\partial z}{\partial x}$ at x = 0.

0.3017

float

Electrons used to produce medical x rays are accelerated from rest through a potential difference of 25,000 volts before striking a metal target. Calculate the speed of the electrons in m/s.

90000000.0

float

You are asked to determine the price of a European put option on a stock. Assuming the Black-Scholes framework holds, you are given: (i) The stock price is $100. (ii) The put option will expire in 6 months. (iii) The strike price is $98. (iv) The continuously compounded risk-free interest rate is r = 0.055. (v) δ = 0.01 (vi) σ = 0.50. What is the price of the put option?

11.9

float

Consider the set S:= {2^{-m} + n^{-1}: m, n \in N}. What is the maximum of S?

1.5

float

For any triangle ABC, we have cos(A)cost(B)cos(C) $\leq$ 1/8, is this true or false?

True

bool

Fig 1(a) and 1(b) show the situation of a reference frame and a current block for block matching motion estimation. The size of searching window is 14x2 while the block size is 2x2. The numbers within the squares are the pixel values. Determine the optimum motion vector.

[-4, 0]

list of integer

You are interviewing two investment managers. Mr. Wong shows that the average return on his portfolio for the past 10 years has been 14%, with a standard deviation of 8% and a beta of 1.2. Ms. Petrov shows that the average return on her portfolio for the past 10 years has been 16%, with a standard deviation of 10% and a beta of 1.6. You know that over the past 10 years, the US Treasury security rate has averaged 2% and the return on the S&P 500 has averaged 11%. By measuring Jensen’s alpha, Mr. Wong has done the better job. Is this correct? Answer True or False.

True

bool

We know that $y'=(x+y) / 2$, we also know that $y(x=0) = 2, y(x=0.5) = 2.636, y(x=1) = 3.595, y(x=1.5) = 4.9868$, what is the value of y(2) using Adams bashforth predictor method.

6.8731

float

James (mass 90.0 kg) and Ramon (mass 60.0 kg) are 20.0 m apart on a frozen pond. Midway between them is a mug of their favorite beverage. They pull on the ends of a light rope stretched between them. When James has moved 6.0 m toward the mug, how far has Ramon moved? (Unit: m)

1.0

float

Does 2^x +1/x = -4 have a solution?

True

bool

Passengers on a carnival ride move at constant speed in a horizontal circle of radius 5.0 m, making a complete circle in 4.0 s. What is their acceleration? (Unit: m/s^2))

12

integer

A basketball team has 12 players, including 5 guards and 7 forwards. How many different starting lineups can be formed that include 3 guards and 2 forwards?

210

integer

Photoelectrons may be emitted from sodium (phi = 2.36 eV) even for light intensities as low as 10^-8 W/m^2. Calculate classically how much time (in seconds) the light must shine to produce a photoelectron of kinetic energy 1.00 eV. Return the numeric value.

463000000.0

float

\lim_{x \to 1}(1/(x - 1) - c/(x^3 - 1)) exists. What is the value of c?

3

integer

Find the entropy rate of the Markov chain associated with a random walk of a king on the 3 by 3 chessboard. Use base 2 logarithm and return the entropy rate in bits.

2.24

float

Find the sum of $\sum_{n=1}^{\infty} \frac{2}{n^2 + 4n + 3}$

0.8333

float

ABCD is a Quadrilateral. E is the midpoint of BC. F is the midpoint of AD. Area of ABG=9 and Area of GEHF=21. What is the Area of CHD?

12

integer

Is the Fourier transform of the signal x(t)=(1-e^{-|t|})[u(t+1)-u(t-1)] even?

True

bool

What's the maximum number of edges in a simple planar graph with 30 vertices?

84

integer

What is the smallest number of standard deviations from the mean that we must go if we want to ensure that we have at least 50% of the data of a distribution?

1.4

float

In how many ways can a group of 6 people be divided into 2 teams? Notice that members in each team are ordered.

1800

integer

Calculate the de Broglie Wavelength, in nm, of an electron with kinetic energy 50 eV.

0.17

float

Let V be the space spanned by functions cos(2x) and sin(2x). Find the determinant of the linear transformation D(f) = f' from V to V.

4

integer

Is the transformation T(M) = [[1, 2], [3, 4]]M from R^{2*2} to R^{2*2} an isomorphism?

True

bool

Consider a horizontal strip of N+2 squares in which the first and the last square are black and the remaining N squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color this neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let $w(N)$ be the expected number of white squares remaining. What is the limit of $w(N)/N$ as $N$ goes to infinity?

0.36787944

float

The difference equation of a digital system is given by $$ y[n]-y[n-1]=2 x[n-1]-x[n-2], $$ where $x[n]$ and $y[n]$ are, respectively the current samples of the input and the output signals of the system. Determine if the system is a stable system.

False

bool

How many triangles are there whose sides are all integers and whose maximum side length equals 11?

36

integer

Use euler's method to find the solution to the differential equation $\frac{\partial y}{\partial x} = 3x + 4y$ at $x=1$ with the initial condition y(0) = 0 and step size $h=0.25$. What is y(1)?

2.0625

float

Given $V_s$ = 5V, $R_1$ = 1kΩ, $R_2$ = 2.2kΩ, $R_3$ = 2.2kΩ, $R_4$ = 1.5kΩ, and $R_L$ = 4.7kΩ. Determine the voltage and current across $R_L$. Answer in unit of V (3 sig.fig.).

1.06

float

Calculate the Fermi energy for copper in eV.

7.03

float

What is the RC time constant of the circuit in seconds?

3800.0

float

Calculate the total capacitive reactance in the figure. Answer in unit of Ohm (3 sig.fig.).

3.18

float

In the figure, given $V_{S1} = V_{S2} = V_{S3} = 5V$, and $R_1 = R_2 = R_3 = 100\Omega$. Find the voltage values with reference to ground $V_A, V_B, V_C, V_D$ in the figure. Represent the answer in a list $[V_A, V_B, V_C, V_D]$ (in 3 sig.fig.) in the unit of V.

[-5.0, -8.33, -6.66, 0.0]

list of float

In a particular semiconductor device, electrons that are accelerated through a potential of 5 V attempt to tunnel through a barrier of width 0.8 nm and height 10 V. What fraction of the electrons are able to tunnel through the barrier if the potential is zero outside the barrier?

4.1e-08

float

The marginal distribution for the variables $x_s$ in a factor $f_s(x_s)$ in a tree-structured factor graph, after running the sum-product message passing algorithm, can be written as the product of the message arriving at the factor node along all its links, times the local factor $f_s(x_s)$. True or false?

True

bool

What is the coefficient of $x^2y^5$ for the formula $(x + 2y)^7$?

672

integer

Suppose V is a finite-dimensional vector space on F. $M1={a_1,a_2,a_3}$ is a basis of V, $M2={b_1,b_2,b_3}$ is another basis of V. Suppose the coordinates of b_1,b_2,b_3 under M1 are $c_1=(1,1,-1),c_2=(1,-1,1),c_3=(-1,1,1)$. Suppose the coordinate of $d\in V$ under M1 is (1,3,5). What is the coordinate of d under M2? Return the three coordinate values as a list.

[2, 3, 4]

list of integer

Assuming $x$ and $y$ are both 2-d random variable. The covariance matrix of $x=((1,2),(2,3),(3,3),(4,4))$, $y=((3,4),(1,5),(5,3),(3,3))$ is $Cov$. What is summation of the eigenvalue of $Cov$?

2.767

float

A model rocket follows the trajectory c(t) = (80t, 200t - 4.9t^2) until it hits the ground, with t in seconds and distance in meters. Find the rocket's maximum height in meters.

2041

integer

If $x=4*cost(t)$ and $y=8*sin(x)$, what is $y{''}_{xx}$ at t=pi/3?

-4.0

float

A surveyor uses a steel measuring tape that is exactly 50.000 m long at a temperature of 20°C. The markings on the tape are calibrated for this temperature. When it is 35°C, the surveyor uses the tape to measure a distance. The value that she reads off the tape is 35.794 m. What is the actual distance? (Unit: m)

35.8

float

How many ways are there to color the faces of a cube with three colors, up to rotation?

57

integer

Which of the following matrices takes any vector $v$ and projects it onto the space spanned by the columns of $\Phi$? (a) $(\Phi^T\Phi)^{-1}$. (b) $\Phi(\Phi^T\Phi)^{-1}$. (c) $\Phi(\Phi^T\Phi)^{-1}\Phi^T$. (d) $\Phi^T(\Phi^T\Phi)^{-1}\Phi^T$.

(c)

option

How many trees are there on 5 labeled vertices?

125

integer

A box contains 4 red, 3 green, and 2 blue balls. Balls are identical besides of their colors. In how many ways can we choose 4 balls, if at least 2 are red?

6

integer

A steel rod 2.0 m long has a cross-sectional area of $0.30 cm ^ 2$. It is hung by one end from a support, and a 550-kg milling machine is hung from its other end. Determine the stress on the rod and the resulting strain and elongation. (Unit: mm)

1.8

float

Consider a $21 \times 17$ rectangular region. This region is to be tiled using tiles of the two types shown in ./mingyin/square1.png (The dotted lines divide the tiles into $1\times 1$ squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping. What is the minimum number of tiles required to tile the region?

99

integer

G = Q, and G is under the operation a * b = a + b + 3. Is G a group?

True

bool

what is the value of \int_a^b \frac{dx}{\sqrt{(x-a)(b-x)}}? Round the answer to the thousands decimal.

3.1415926

float

A remote database contains 30 seconds of color motion-video. The video sequence is of the format (352 ́288 pixels) with RGB digitization at 30 frames per second. Find the the data rate for this motion-video in Mbits/s (3 sig. fig.).

69.6

float

What is (sin(2x) / x)^(1+x) when x is approaching 0?

2.0

float

The open mapping theorem can be proved by (a) Baire category theorem; (b) Cauchy integral theorem; (c) random graph theorem; (d) None of the above. Which option is correct?

(a)

option

Is differential equation $sin(t)y' + t^2e^yy' - y' = -ycos(t) - 2te^y$ exact or not?

True

bool

Comet Halley moves in an elongated elliptical orbit around the sun (Fig. 13.20). Its distances from the sun at perihelion and aphelion are $8.75 \times 10^7 km$ and $5.26 \times 10^9 km$, respectively. The orbital period is X * 10^9 s. What is X?

2.38

float

A robotic lander with an earth weight of 3430 N is sent to Mars, which has radius $R_M=3.40 \times 10^6 m$ and mass $m_M=6.42 \times$ $10^{23} kg$. Find the acceleration there due to gravity. (Unit: $m/s^2$)

3.7

float

Use divergence therem to evaluate $\iint_S \vec{F} \cdot d \vec{S}$ where $\vec{F} = sin(\pi x) \vec{i} + (z y^3)\vec{j} + (z^2 + 4x)\vec{k}$ and $S$ is the suface of the box with $-1 \le x \le 2, 0 \le y \le 1$ and $1 \le z \le 4$. Note that all six sides of the box are included in $S$.

67.5

float

Denote m(\cdot) to be Lebesgue measure. Given a point set E. Suppose for any closed set F and open set G with F \subset E \subset G, it holds $\sup _F {m(F)}<\inf _G {m(G)}$. Is set E Lebesgue measurable? Answer 1 for yes and 0 for no. Return the number

0.0

float