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TIGER-Lab 110

How many people at least shall we include in one group, such that there must exist two different people in this group whose birthdays are in the same month?

13

integer

Consider Convolutional Neural Network D2 which takes input images of size 32x32 with 1 colour channels. The first layer of D2 uses 4 filters of size 5x5, a stride of 2, and zero-padding of width 1. Consider CNN D2 which takes input images of size 32x32 with 1 colour channels. The first layer of D2 uses 4 filters of size 5x5, a stride of 2, and zero-padding of width 1. What would be the total size of the flattened output vector from each filter?

25

integer

Consider an additive white Gaussian noise channel with an expected output power constraint $P=2$. Thus $Y = X + Z$, $Z \sim N(0, 1)$, $Z$ is independent of $X$, and $E(Y)^2 \leq 2$. Find the channel capacity in bits.

0.5

float

Assume the half-life of the proton is 10^33 years. How many decays per year would you expect in a tank of water containing 350,000 liters of water?

0.08

float

Two bicycle tires are set rolling with the same initial speed of 3.5 m/s on a long, straight road, and the distance each travels before its speed is reduced by half is measured. One tire is inflated to a pressure of 40 psi and goes 18.1 m; the other is at 105 psi and goes 92.9 m. What is the coefficient of rolling friction for each? Assume that the net horizontal force is due to rolling friction only.

[0.0259, 0.00505]

list of float

Find the size of angle MBD in the figure below.

72

integer

Let a undirected graph G with edges E = {<0,3>, <1,3>, <2,3>}, which <A,B> represent Node A is connected to Node B. What is the minimum vertex cover of G? Represent the vertex cover in a list of ascending order.

[3]

list of integer

A train pulls out of the station at constant velocity. The received signal energy thus falls off with time as $1/i^2$. The total received signal at time $i$ is $Y_i = \frac{1}{i}X_i + Z_i$ where $Z_1, Z_2, \ldots$ are i.i.d. drawn from $N(0,1)$. The transmitter constraint for block length $n$ is $\frac{1}{n}\sum_{i=1}^n x_i^2(w) \leq 2 $ for $w \in \{1,2,\ldots, 2^{nR}\}$. Use Fano's inequality to find the capacity for this channel.

0.0

float

Adding a row to a channel transition matrix does not decrease capacity. True or False?

True

bool

Find the curvature for f(x) = \sqrt{4x - x^2}, x = 2.

0.5

float

Find the absolute minimum value of the function $f(x,y)=x^2+y^2$ subject to the constraint $x^2+2*y^2=1$.

0.5

float

What is the number of labelled forests on 8 vertices with 5 connected components, such that vertices 1, 2, 3, 4, 5 all belong to different connected components?

320

integer

Compute $\int_C dz / (z * (z-2)^2)dz$, where C: |z - 2| = 1. The answer is Ai with i denoting the imaginary unit, what is A?

-0.3926

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 ST?

14

integer

Let f = u(z) + iv(z) be an entire function in complex plane C. If |u(z)| < M for every z in C, where M is a positive constant, is f is a constant function?

True

bool

Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were 71,76,80,82,and 91. What was the last score Mrs. Walter entered?

80

integer

In a group of 10 people, each of whom has one of 3 different eye colors, at least how many people must have the same eye color?

4

integer

A group of 9 people is split into 3 committees of 3 people. Committees are identical besides of members. In how many ways can this be done?

280

integer

An observer S who lives on the x-axis sees a flash of red light at x = 1210 m, then after 4.96 µs, a flash of blue at x = 480 m. Use subscripts R and B to label the coordinates of the events. What is the measured time interval (in µs) between these flashes?

4.32

float

Given image \begin{tabular}{|llll|} \hline 7 & 1 & 6 & 0 \\ 3 & 3 & 7 & 6 \\ 6 & 6 & 5 & 7 \\ \hline \end{tabular} , and the bit-depth of the image is 4. Suppose you want to use the thresholding technique to segment the image. What is the appropriate threshold value based on the histogram of the image? Follow the following rule when you do thresholding or grouping: pixel $(i, j) \in$ Group A pixels if $g(i, j) \leq$ current threshold $\mathrm{T}$; pixel $(i, j) \in$ Group B pixels otherwise, where $g(i, j)$ is the intensity value of pixel $(i, j)$.

4

integer

George is seen to place an even-money $100,000 bet on the Bulls to win the NBA Finals. If George has a logarithmic utility-of-wealth function and if his current wealth is $1,000,000, what must he believe is the minimum probability that the Bulls will win?

0.525

float

The distortion rate function $D(R)=\min_{p(\hat{x}|x):I(X;\hat{X})\leq R} E(d(X,\hat{X}))$ is nonincreasing. True or False?

True

bool

In complex analysis, define U^n={(z_1, \cdots, z_n): |z_j|<1, j=1, \cdots, n} and B_n={(z_1, \cdots, z_n): \sum_{j=1}^n |z_j|^2<1 }. Are they conformally equivalent in C^n? Here C^n is the d-dimensional complex space. Return 1 for yes and 0 for no.

0.0

float

In how many ways can 8 people be seated at 2 identical round tables? Each table must have at least 1 person seated.

13068

integer

Suppose H=L^2[0,1]. Operator $A: u(t) \mapsto t\times u(t)$ is a map from H to H. Then A is a bounded linear operator. Then the spectrum of A is: (a) [0,1], (b) [0,1/2], (c) [1/2, 1], (d) none of the above. Which one is correct?

(a)

option

For matrix A = [[3, 1, 1], [2, 4, 2], [1, 1, 3]], what are its eigen values?

[2, 6]

list of integer

for a given function f(x)=x^2*sin(x). Is there a value $x$ between 10pi and 11pi such that $f'(x) = 0$?

True

bool

Is cos(\pi/8) equal to (\sqrt{2+\sqrt{2}})/2?

True

bool

Does f (x) = x2 + cx + 1 have a real root when c=0?

False

bool

Is there exist a holomorphic function $f$ on the unit disk $B(0,1)$ (boundary excluded) such that $f(B(0,1))=C$? Here C is the complex space. Return 1 for yes and 0 for no.

1.0

float

Find the solutions y of the differential equation y'=(t^2+3y^2)/2ty with y(1) = 1. What is y(2)?

3.464

float

Consider that the following two signals: $x(t)$ and $v(t)$ $$ x(t)=\left\{\begin{array}{cc} 1 & 0 \leq t \leq 3 \\ 0 & \text { otherwise } \end{array} \quad v(t)=\left\{\begin{array}{cc} 1 & 0 \leq t \leq 2 \\ 0 & \text { otherwise } \end{array}\right.\right. $$ Let $y(\tau)=\int_{-\infty}^{\infty} x(\tau-t) v(t) d t$. Let $\tau=2.5$. Determine $y(\tau)$.

2

integer

Assume a temperature of 300 K and find the wavelength of the photon necessary to cause an electron to jump from the valence to the conduction band in germanium in nm.

1850.0

float

Given that the spacing between vibrational energy levels of the HCl molecule is 0.36 eV, calculate the effective force constant in N/m.

490.0

float

Let $X$ be uniformly distributed over $\{1, 2, \ldots, 256\}$. We ask random questions: Is $X\in S_1$? Is $X\in S_2$? ... until only one integer remains. All $2^256$ subsets of $\{1, 2, \ldots, 256\}$ are equally likely. How many deterministic questions are needed to determine $X$?

8

integer

Water stands 12.0 m deep in a storage tank whose top is open to the atmosphere. What are the gauge pressures at the bottom of the tank? (Unit: 10 ^ 5 Pa)

1.18

float

Find the maximum entropy density $f$, defined for $x\geq 0$, satisfying $E(X)=\alpha_1$, $E(\ln{X})=\alpha_2$. Which family of densities is this? (a) Exponential. (b) Gamma. (c) Beta. (d) Uniform.

(b)

option

In the figure, what is the magnitude of the potential difference across the $20 \Omega$ resistor? Answer in unit of W (3 sig.fig.).

7.76

float

One is given a communication channel with transition probabilities $p(y|x)$ and channel capacity $C=max_{p(x)}I(X;Y)$. If we preprocesses the output by forming $Y=g(Y)$ the capacity will not improve. True or False?

True

bool

Consider the matrix of A=[[1, 4], [4, 1]], is this a positive definite matrix?

False

bool

If polygon ACDF is similar to polygon VWYZ, AF = 12, CD = 9, YZ = 10, YW = 6, and ZV = 3y-1, find y.

3

integer

A young couple has made a non-refundable deposit of the first month's rent (equal to $1, 000) on a 6-month apartment lease. The next day they find a different apartment that they like just as well, but its monthly rent is only $900. They plan to be in the apartment only 6 months. Should they switch to the new apartment?

0.0

float

If at the beginning of each month a deposit of $500 is made in an account that pays 8% compounded monthly, what will the final amount be after five years?

36983.35

float

Using Taylor's Approximation Theorem to show: What is $\lim_{x \to 0} \frac{e^\frac{x^4}{2}-\cos(x^2)}{x^4}$

1.0

float

what is the value of $2/\pi*\prod_{k=1}^{\infty} \frac{(2*k)^2}{(2*k-1)(2*k+1)}$?

1.0

float

When 30! is computed, it ends in 7 zeros. Find the digit that immediately precedes these zeros.

8

integer

What is the determinant of the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

-3

integer

A cascade of $n$ identical independent binary symmetric channels each with raw error probability $p$, and $0<p<1$. What is the capacity of the cascade when $n$ goes to infinity?

0.0

float

How many integers between 1 (included) and 100 (included) are divisible by either 2, 3, or 5?

74

integer

Given the following circuit (with all current and voltage values in rms), find the value of $V_C$ in the unit of V.

14.5

float

In a Gigabit Ethernet LAN, the average size of a frame is 1000 bytes. If a noise of 2ms occurs on the LAN, how many frames are destroyed?

250

integer

The cross section for a 2.0-MeV neutron (a typical energy for a neutron released in fission) being absorbed by a U-238 nucleus and producing fission is 0.68 barn. For a pure U-238 sample of thickness 3.2 cm, what is the probability of a 2.0-MeV neutron producing fission?

0.1

float

Calculate the Hamming pairwise distances and determine the minimum Hamming distance among the following codewords: 000000,010101,101010,110110

3

integer

The spontaneous fission activity rate of U-238 is 6.7 fissions/kg s. A sample of shale contains 0.055% U-238 by weight. Calculate the number of spontaneous fissions in one day in a 106-kg pile of the shale by determining the number of fissions.

320000000.0

float

Assuming we are underground, and the only thing we can observe is whether a person brings an umbrella or not. The weather could be either rainy or sunny. Assuming the P(rain)=0.6 and P(sunny)=0.4. Assuming the weather on day $k$ is dependent on the weather on day $k-1$. We can write the transition probability as P(sunny $\mid$ sunny) = P(rain $\mid$ rain) = 0.7. The person has 60\% chance to bring an umbrella when the weather is rainy, and 40\% chance to bring an umbrella when the weather is sunny, i.e. P(umbrella $\mid$ rain) = 0.6 and P(umbrella $\mid$ sunny) = 0.4. If we observe that the person (1) brought an umbrella on day 1, (2) did not bring an umbrella on day 2, (3) brought an umbrella on day 3, (4) did not bring an umbrella on day 4. What are the most likely weather from day 1 to day 4? Return the answer as a list of binary values, where 1 represents rain and 0 represents sunny.

[1, 1, 1, 1]

list of integer

Consider a two-layer fully-connected neural network in which the hidden-unit nonlinear activation functions are given by logistic sigmoid functions. Does there exist an equivalent network in which the hidden unit nonlinear activation functions are given by hyperbolic tangent functions?

True

bool

Please solve x^3 + 2*x = 10 using newton-raphson method.

1.8474

float

A firm in a perfectly competitive industry has patented a new process for making widgets. The new process lowers the firm's average cost, meaning that this firm alone (although still a price taker) can earn real economic profits in the long run. Suppose a government study has found that the firm's new process is polluting the air and estimates the social marginal cost of widget production by this firm to be SMC = 0.5q. If the market price is $20, what should be the rate of a government-imposed excise tax to bring about optimal level of production?

4

integer

A radioactive sample contains two different isotopes, A and B. A has a half-life of 3 days, and B has a half-life of 6 days. Initially in the sample there are twice as many atoms of A as of B. In how many days will the ratio of the number of atoms of A to B be reversed?

12.0

float

Let h(x) = (x^{-1/2} + 2x)(7 - x^{-1}). What is h'(x) when x = 4?

13.609

float

If $u(x, y) = 4x^3y - 4xy^3$, is there a function v(x, y) such that u(x,y) + iv(x,y) is an analytical function?

True

bool

How many paths are there from the origin (0,0) to the point (10,10) on a grid such that the path only moves up or right and does not cross the diagonal line y = x?

16796

integer

Estimate the PEG ratio for a firm that has the following characteristics: Length of high growth = five years Growth rate in first five years = 25% Payout ratio in first five years = 20% Growth rate after five years = 8% Payout ratio after five years = 50% Beta = 1.0 Risk-free rate = T-bond rate = 6% Cost of equity = 6% + 1(5.5%) = 11.5% Risk premium = 5.5% What is the estimated PEG ratio for this firm?

1.15

float

You throw a ball from your window $8.0 \mathrm{~m}$ above the ground. When the ball leaves your hand, it is moving at $10.0 \mathrm{~m} / \athrm{s}$ at an angle of $20^{\circ}$ below the horizontal. How far horizontally from your window will the ball hit the ground? Ignore air resistance. (Unit: m)

9.2

float

A group of 10 people is split into 3 different committees of 3, 4, and 3 people, respectively. In how many ways can this be done?

4200

integer

The root of the equation x = (1 / 2) + sin x by using the iteration method: x_{k+1} = 1/2 + sin(x_k), x_0 = 1 correct to o six decimals is x = 1.497300. Determine the number of iteration steps required to reach the root by linear iteration. If the Aitken ∆2-process is used after three approximations are available, how many iterations are required?

3

integer

Let a undirected graph G with edges E = {<2,6>,<2,8>,<2,5>,<6,5>,<5,8>,<6,10>,<10,8>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 2 to node 10? Represent the path as a list.

[2, 8, 10]

list of integer

A Chord based distributed hash table (DHT) with 26 address space is used in a peer- to-peer file sharing network. There are currently 10 active peers in the network with node ID N1, N11, N15, N23, N31, N40, N45, N51, N60, and N63. Show all the target key (in ascending order, ignore the node's identifier itself) for N1.

[2, 3, 5, 9, 17, 33]

list of integer

Across what potential difference in V does an electron have to be accelerated to reach the speed v = 1.8 x 10^7 m/s? Calculate this relativistically.

924.0

float

Use Stoke's Theorem to evaluate $\iint_S curl \vec{F} \cdot d \vec{r}$ where $\vec{F} = z^2 \vec{i} - 3xy \vec{j} + x^3y^3 \vec{k}$ and $S$ is the part of $z = 5 - x^2 - y^2$ above the plane $z$=1. Assume that S is oriented upwards.

0.0

float

./mingyin/mdp.png shows a rectangular gridworld representation of a simple finite MDP. The cells of the grid correspond to the states of the environment. At each cell, four actions are possible: north, south, east, and west, which deterministically cause the agent to move one cell in the respective direction on the grid. Actions that would take the agent off the grid leave its location unchanged, but also result in a reward of $-1$. Other actions result in a reward of $0$, except those move the agent out of the special states A and B. From state A, all four actions yield a reward of +10 and take the agent to A'. From state B, all actions yield a reward of +5 and take the agent to B'. Suppose the discount gamma=0.9. The state-value function of a policy $\pi$ is defined as the expected cumulative reward of $\pi$ given the current state. What is the state-value of state A if the policy is random (choose all four directions with equal probabilities)? What is the state-value of state A under the optimal policy? Return the answer of the two questions using a list.

[8.8, 24.4]

list of float

Derive the solution y = f(t) to the following IVP. $ty' - 2y = t^5sin(2t) - t^3 + 4t^4$, where $y(\pi) = 3\pi^4/2$. What is y(t) when $t=pi/2$.

19.095

float

Consider Convolutional Neural Network D2 which takes input images of size 32x32 with 1 colour channels. The first layer of D2 uses 4 filters of size 5x5, a stride of 2, and zero-padding of width 1. The dimensions of the resulting activation map for each filter in this first layer will be k x k. What is the value of k?

15

integer

Let X_2,X_3,... be independent random variables such that $P(X_n=n)=P(X_n=-n)=1/(2n\log (n)), P(X_n=0)=1-1/(n*\log(n))$. Does $n^{-1}\sum_{i=2}^n X_i$ converges in probability? Does $n^{-1}\sum_{i=2}^n X_i$ converges in almost surely? Return the answers of the two questions as a list.

[1, 0]

list of integer

For any triangle ABC, we have sin(A) + sin(B) + sin(C) $\le$ 3\sqrt(3)/2, is this true or false?

True

bool

Let $X_0, X_1, X_2, \ldots$ be drawn i.i.d. from $p(x)$, and $x\in\{1,2,3,\ldots,100\}. Let $N$ be the waiting time to the next occurrence of $X_0$. Compute $E(N)$.

100.0

float

Let {N(t), t \in [0, \infty)} be a Poisson process with rate of $\lambda = 4$. Find it covariance function $C_N(t1, t2) for t1, t2 \in [0, \infy)$. What is C_N(2, 4)?

8

integer

Let P[0,1] denotes all the polynomials on the interval [0,1]. Define the distance \rho(p, q)=\int_0^1|p(x)-q(x)| dx. Is (P[0,1],\rho) a complete space? Return 1 for yes and 0 for no.

0.0

float

Does r(t) = [8 - 4t^3, 2 + 5t^2, 9t^3] parametrize a line?

False

bool

x=0.3168. what is the value of $x*\prod_{n=1}^\infty(1-\frac{x^2}{n^2 \pi^2})/sin(x)$?

1.0

float

Use Euler's Method to calculate the approximation of y(0.2) where y(x) is the solution of the initial-value problem that is as follows. y''+xy'+y=0 and y(0)=2 and y'(0) = 3.

2.58

float

Suppose g(x) is the horizontal asymptote of function f(x) = (\sqrt{36 x^2 + 7}) / (9x + 4). What are possible values of g(2023)?

[0.6667, -0.6667]

list of float

Is the conditional entropy $H(X_0|X_n)$ non-decreasing with n for any Markov chain?

True

bool

If T_1 and T_2 are stopping times with respect to a filtration F. Is T_1+T_2 stopping time? Is max(T_1, T_2} stopping time? Is min(T_1, T_2} stopping time? Answer 1 for yes and 0 for no. Return the answers of the three questions as a list.

[1, 1, 1]

list of integer

Given the following equation: x - e^{-x} = 0. determine the initial approximations for finding the smallest positive root. Use these to find the root correct to three decimal places with Regula-Falsi method.

0.567

float

Does the following series $\sum_{i=0}^{\infty} \frac{n-1}{n^3+1}$ converge?

1.0

float

Given 2 colors whose HSI representations are given as follows: (a) $(pi, 0.3,0.5)$, (b) $(0.5 pi, 0.8,0.3)$, which color is brighter?

(a)

option

If p is a prime number and a is an integer, what is (a^p - a) mod p?

0

integer

An investor is looking to purchase a security for $100 with an initial margin of 50% (meaning the investor is using $50 of his money to purchase the security and borrowing the remaining $50 from a broker). In addition, the maintenance margin is 25%. At what price of the security will the investor receive a margin call?

66.67

float

Point charges q1=50μC and q2=−25μC are placed 1.0 m apart. What is the force on a third charge q3=20μC placed midway between q1 and q2?

53.94

float

compute the integral $\iint_{\Sigma} x^3 dy*dz +y^3 dz*dx+z^3 dx*dy$, where is the outward of the ellipsoid x^2+y^2+z^2/4=1. Round the answer to the thousands decimal.

30.15928896

float

Two argon atoms form the molecule $Ar_2$ as a result of a van der Waals interaction with $U_0 = 1.68 \times 10 ^ {-21}$ J and $R_0 = 3.82 \times 10 ^ {-10}$ m. Find the frequency of small oscillations of one Ar atom about its equilibrium position. (Unit: 10^11 Hz)

5.63

float

Determine values of the real numbers a, b, and c to make the function $x^2 + ay^2 + y + i(bxy + cx)$ by an analytical function of the complex variable of $x+iy$? Return your answer as a list [a, b, c].

[-1, 2, -1]

list of integer

Let S be the set of integers between 1 and 2^40 that contain two 1’s when written in base 2. What is the probability that a random integer from S is divisible by 9?

0.1705

float

Represent the contour of the object shown in the figure in a clockwise direction with a 4-directional chain code. Use the left upper corner as the starting point. The answer need to be normalized with respect to the orientation of the object. Represent the answer as a list with each digit as a element.

[1, 0, 1, 1, 3, 0, 1, 1, 3, 1, 1, 3]

list of integer

Let $f(x) = 1/x$ on $(0, 1]$ and $f(x) = 3$ if $x = 0$. Is there a global maximum on interval $[0, 1]$?

False

bool

If A and B are both orthogonal square matrices, and det A = -det B. What is det(A+B)? Return the numerical value.

0.0

float

The Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory collides gold ions onto other gold ions head on. The energy of the gold ions is 100 GeV per nucleon. What is the center-of-mass energy of the collision in TeV?

39.4

float

Find the fraction of the standard solar flux reaching the Earth (about 1000 W/m^2) available to a solar collector lying flat on the Earth’s surface at Miami (latitude 26°N) at noon on the winter solstice.

0.656

float

Is the Fourier transform of the signal $x_1(t)=\left\{\begin{array}{cc}\sin \omega_0 t, & -\frac{2 \pi}{\omega_0} \leq t \leq \frac{2 \pi}{\omega_0} \\ 0, & \text { otherwise }\end{array}\right.$ even?

False

bool