instruction_seed stringlengths 56 2.52k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.77 0.86 | mean_negative_score float64 0.76 0.85 | difference_score float64 0 0.03 |
|---|---|---|---|---|---|
Consider the following simplicial complex:
<img src="https://i.imgur.com/52GubBe.png" width="200"/>
Find the Euler characteristic of this simplicial complex. | tiger_lab_math | [
0.003080032067373395,
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0.0022585690021514893,
0.08199217915534973,
0.03894182667136192,
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0.01354540791362524,
0.01798664964735508,
-0.02080651931464672,
-0.0... | 0.829498 | 0.824592 | 0.004906 |
Consider the function f(x,y) = x^2 + 4y^2 + 4xy - 6x - 10y + 20, where x and y are real numbers. Using the conjugate gradient method, find the minimum value of this function and the values of x and y that achieve this minimum. | tiger_lab_math | [
0.0015707529382780194,
0.03388598561286926,
-0.035517618060112,
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0.048541873693466187,
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0.007956178858876228,
0.030871836468577385,
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-0.00... | 0.818997 | 0.817919 | 0.001078 |
A cylindrical glass has a height of 10 cm and a diameter of 6 cm. What is the maximum volume of water that can be poured into the glass before the water spills over the top? Use the concept of contact geometry in your solution. | tiger_lab_math | [
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Suppose that we are given 40 points equally spaced around the perimeter of a square, so that four of them are located at the vertices and the remaining points divide each side into ten congruent segments. If $P$, $Q$, and $R$ are chosen to be any three of these points which are not collinear, then how many different p... | tiger_lab_math | [
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-0... | 0.84705 | 0.830384 | 0.016666 |
Let A = {1, 2, 3} and B = {4, 5}. Determine if the following equation holds true:
(AxB)' = A' x B'
where ' denotes the complement and x denotes the Cartesian product. | tiger_lab_math | [
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-0.02... | 0.833112 | 0.827945 | 0.005167 |
What is the largest value of the parameter k in the Chirikov-Taylor map equation, x(n+1) = x(n) + y(n) - k*sin(2πx(n)), for which the system has both regular and chaotic regions?
Hint: The Chirikov-Taylor map exhibits an onset of chaotic dynamics at a critical value of the parameter k, known as the Chirikov parameter... | tiger_lab_math | [
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Consider the manifold given by the equation f(x,y,z) = x^2 + y^2 + z^2 = 5, determine the symmetries of the manifold. Specifically, find the number and nature of isometries of the manifold. | tiger_lab_math | [
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-0.... | 0.817123 | 0.806565 | 0.010558 |
Suppose you flip a coin 100 times and record the results. Use Monte Carlo simulation to estimate the probability of getting exactly 50 heads and 50 tails. | tiger_lab_math | [
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-... | 0.805506 | 0.798363 | 0.007143 |
Let's consider the Ikeda Map, which is a discrete-time dynamical system defined as follows:
<x[n+1]> = 1 + u * (<x[n]> * cos(t[n]) - <y[n]> * sin(t[n]))
<y[n+1]> = u * (<x[n]> * sin(t[n]) + <y[n]> * cos(t[n]))
where t[n] = 0.4 - 6/(1 + <x[n]>^2 + <y[n]>^2) and u is a constant parameter.
Suppose that we start wi... | tiger_lab_math | [
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Find all the poles and zeros of the meromorphic function $f(z) = \frac{z^3 - 1}{z^2 + 9}$, and determine their order. | tiger_lab_math | [
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... | 0.835617 | 0.831381 | 0.004236 |
A student wants to select 5 books from a collection of 12 distinct books on the shelf. How many ways can the student select the books given that order does not matter? Use Stirling numbers to solve the problem. | tiger_lab_math | [
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-0.032... | 0.817246 | 0.81615 | 0.001095 |
Consider an undirected graph G with 6 vertices labeled as A, B, C, D, E, and F. The edges of the graph G are as follows:
- AB, AC, AD
- BC, BE
- CD, CE
- DF, EF
Determine whether G is a connected graph or not. If it is connected, find the number of connected components of the graph. If it is not connected, find the c... | tiger_lab_math | [
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In triangle $ABC$, we have that $E$ and $F$ are midpoints of sides $\overline{AC}$ and $\overline{AB}$, respectively. The area of $\triangle ABC$ is 24 square units. How many square units are in the area of $\triangle CEF$? Please write a program to solve it | tiger_lab_math | [
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Here's a precise numerical problem related to the Traveling Salesman Problem for a math student to solve:
A salesman has to visit 5 cities (A, B, C, D, and E) exactly once and return to his starting point. The distances (in kilometers) between the cities are as follows:
- A to B: 10 km
- A to C: 20 km
- A to D: 15 km... | tiger_lab_math | [
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0.03579321503639221,
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-0.... | 0.841497 | 0.832486 | 0.009012 |
A butterfly flaps its wings and causes a small disturbance in the air. This disturbance can lead to drastic changes in a chaotic system, as described by the butterfly effect. Consider the following logistic map equation:
xₙ₊₁ = r * xₙ * (1 - xₙ)
Assume the initial condition, x₀ = 0.1, and set the growth factor, r, eq... | tiger_lab_math | [
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0.02831229940056801,
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-0... | 0.804688 | 0.802936 | 0.001752 |
Consider the following optimization problem:
Minimize f(x,y) = x^2 + 4y^2 - 4x - 8y + 17
subject to the constraint g(x,y) = x^2 + y^2 - 1 = 0
Use the conjugate gradient method to find the minimum value of f(x,y) and the corresponding values of x and y that achieve this minimum. | tiger_lab_math | [
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0... | 0.820146 | 0.812194 | 0.007952 |
a certain university will select 1 of 4 candidates eligible to fill a position in the mathematics department and 2 of 7 candidates eligible to fill 2 identical positions in the computer science department . if none of the candidates is eligible for a position in both departments , how many different sets of 3 candidate... | tiger_lab_math | [
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0.02670159749686718,
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-0.00866... | 0.824353 | 0.817864 | 0.00649 |
Find all the possible Cayley tables for groups of order 30 and determine which ones are isomorphic to either the cyclic group C30, the dihedral group D15, or the non-abelian group of order 30. | tiger_lab_math | [
0.008779540657997131,
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0.024595871567726135,
0.0010847669327631593,
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-0.... | 0.815224 | 0.804322 | 0.010902 |
A ball is thrown straight up into the air with an initial velocity of 30 m/s. Using calculus, find its maximum height and the time it takes to reach that height. Assume that air resistance is negligible. | tiger_lab_math | [
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-0.0... | 0.803075 | 0.801579 | 0.001496 |
In a Cournot duopoly, two companies, A and B, produce identical products. Company A has a total cost function of TC(Qa) = 8Qa and Company B has a total cost function of TC(Qb) = 10Qb. The inverse demand function is P(Q) = 120 - Q, where P is the price and Q is the total quantity produced by both companies. Determine th... | tiger_lab_math | [
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-0.0... | 0.812073 | 0.800935 | 0.011138 |
Let the function W(n) represent the product of the first n prime numbers, where n > 0. If x = W(n) + 1, which of the following must be true?
(i) x is always odd
(ii) x is always prime
(iii) x is never the square of an integer
Answer Choices: (A) ii only (B) iii only (C) i and ii only (D) i and iii only (E) ii and iii o... | tiger_lab_math | [
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0.01914663426578045,
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-0.04... | 0.832441 | 0.830202 | 0.00224 |
Considering the Chirikov-Taylor map given by the equation:
$$
x_{n+1} = x_n + y_n - \frac{K}{2\pi}\sin(2\pi x_n)
$$
$$
y_{n+1} = y_n - \frac{K}{2\pi}\sin(2\pi x_{n+1})
$$
where $K$ is the parameter and $x_n$, $y_n$ are the coordinates of a point on the plane.
Assuming an initial condition of ($x_0$, $y_0$) = (0.7, 0... | tiger_lab_math | [
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0.014646559022367,
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-0.00522... | 0.838907 | 0.822485 | 0.016422 |
The sequence $a_1,$ $a_2,$ $a_3,$ $\dots$ satisfies $a_1 = 19,$ $a_9 = 99,$ and for all $n \ge 3,$ $a_n$ is the arithmetic mean of the first $n - 1$ terms. Find $a_2.$ | tiger_lab_math | [
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-0.000... | 0.845999 | 0.835391 | 0.010608 |
the length of a rectangle is one - fourth of the radius of a circle . the radius of the circle is equal to the side of the square , whose area is 1225 sq . units . what is the area ( in sq . units ) of the rectangle if the rectangle if the breadth is 10 units ? Please write a program to solve it | tiger_lab_math | [
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-0.024... | 0.818513 | 0.817003 | 0.00151 |
In a dataset of 50 students, each having their grades in 8 subjects, perform Principal Component Analysis to determine the most important factors affecting their academic performance. | tiger_lab_math | [
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Convex hexagon $ABCDEF$ has exactly two distinct side lengths. Side $AB$ measures 5 units, and side $BC$ measures 6 units. The perimeter of hexagon $ABCDEF$ is 34 units. How many sides of hexagon $ABCDEF$ have measure 6 units? | tiger_lab_math | [
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-0.041... | 0.825878 | 0.817643 | 0.008234 |
A Mersenne prime is defined to be a prime number of the form $2^n - 1$, where $n$ must itself be a prime. For example, since $2^3 - 1 = 7$, and 3 is a prime number, 7 is a Mersenne prime. What is the largest Mersenne prime less than 200? Let's program in Python in the response. | tiger_lab_math | [
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0.03680451959371567,
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-0.015... | 0.813789 | 0.811895 | 0.001894 |
Suppose a wave equation is given by:
$$\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2}=0$$
where $c$ is a constant. If the initial displacement and velocity of the string are given by:
$$u(x,0)=\sin(ax), \ \frac{\partial u}{\partial t}(x,0)=0$$
where $a$ is a constant, find the solution to ... | tiger_lab_math | [
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0.01770555041730404,
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-0.0... | 0.81264 | 0.803965 | 0.008675 |
A math student is given the following task:
Find the mean curvature of the surface given by the equation $z = x^2 + y^2$ at the point $(1, 1, 2)$. | tiger_lab_math | [
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0.026362378150224686,
-0.01752406731247902,
-0.01023... | 0.807662 | 0.805285 | 0.002377 |
Here is a numerical problem for a math student to solve in regression analysis:
Suppose an economist wants to predict the annual salary of individuals based on their level of education. The economist collects data on 50 individuals and finds that the correlation coefficient between education level and annual salary is... | tiger_lab_math | [
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0.019027091562747955,
-0.026438787579536438,
-0.... | 0.815053 | 0.812282 | 0.00277 |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is 56. Find the area of the polygon.
[asy]
unitsize(0.5 cm);
draw((3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(6,2)--(6,3)--(7,3)--(7,4)--(6,4)--(6,5)--(5,5)--(5,6)--(4,6)--(4,7)--(3,7)... | tiger_lab_math | [
0.02164977602660656,
0.022352542728185654,
-0.029183153063058853,
-0.0420500673353672,
0.02885325811803341,
-0.024015389382839203,
-0.030360769480466843,
0.049612026661634445,
0.027168024331331253,
-0.001364949974231422,
0.02104013040661812,
0.025755159556865692,
-0.014239097014069557,
-0.... | 0.824731 | 0.817193 | 0.007538 |
A rock climber wants to climb a steep cliff. The cliff consists of 7 ledges and she wants to reach the top of the cliff by stepping on each of the ledges at least once. In how many ways can she climb the cliff, if she may not step twice on the same ledge in a row? | tiger_lab_math | [
0.019823063164949417,
0.008672233670949936,
-0.03946170583367348,
-0.03922215849161148,
0.024933960288763046,
-0.015436667017638683,
-0.026449477300047874,
0.04391319677233696,
0.008946085348725319,
-0.027728037908673286,
0.013102510012686253,
0.03613714128732681,
-0.04425504803657532,
-0.... | 0.825883 | 0.819558 | 0.006325 |
Here's a numerical problem related to the Prisoner's Dilemma in Game Theory:
Two individuals, Alex and Ben, are arrested for a crime together. The police interrogate them separately and offer them the following deal:
- If both of them confess, they will each receive a sentence of 2 years in prison.
- If neither of th... | tiger_lab_math | [
0.02086080238223076,
-0.0035507299471646547,
-0.05322505161166191,
-0.030654413625597954,
0.020490242168307304,
-0.012203128077089787,
-0.021481581032276154,
0.055689487606287,
0.04046173766255379,
-0.012039637193083763,
0.010092082433402538,
0.02133040875196457,
-0.033629704266786575,
-0.... | 0.835172 | 0.829264 | 0.005908 |
In right triangle $XYZ$ with $\angle YXZ = 90^\circ$, we have $XY = 24$ and $YZ = 25$. Find $\tan Y$. | tiger_lab_math | [
0.03565702959895134,
0.044535912573337555,
-0.030015237629413605,
-0.0372045636177063,
0.016359711065888405,
-0.018700843676924706,
-0.008833164349198341,
0.03248590603470802,
0.031739842146635056,
-0.024504922330379486,
0.025143735110759735,
0.024954332038760185,
-0.03236270323395729,
0.0... | 0.833038 | 0.829405 | 0.003633 |
A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation $x^2-y^2=47$? Let's write a Python program. | tiger_lab_math | [
0.01986495405435562,
0.009682880714535713,
-0.05606554448604584,
-0.03677089139819145,
0.02721390314400196,
-0.014810982160270214,
-0.012835693545639515,
0.03313665837049484,
0.04661938548088074,
-0.03118244558572769,
0.02221149578690529,
0.04420642927289009,
-0.039715010672807693,
-0.0206... | 0.82983 | 0.825691 | 0.004139 |
Consider the Riemann surface curve defined by the equation y^2 = x^3 - x^2 - 2x. Compute the values of the first five theta functions associated with the Riemann surface at the origin. | tiger_lab_math | [
0.010330152697861195,
0.036652062088251114,
-0.050225500017404556,
-0.05586652085185051,
0.03421351686120033,
-0.02418634295463562,
-0.02691083401441574,
0.06980886310338974,
0.043211162090301514,
-0.021369025111198425,
0.009352941997349262,
0.02022070251405239,
-0.042175453156232834,
-0.0... | 0.825925 | 0.817057 | 0.008868 |
Here's a precise numerical problem for a math student to solve related to Solving optimization problems using differential evolution:
Maximize the function f(x) = x^2 - 4x + 4 in the interval [0, 5] using the Differential Evolution algorithm with a population size of 20, a crossover probability of 0.8 and a mutation f... | tiger_lab_math | [
0.02584710717201233,
0.0005938253598287702,
-0.03622562065720558,
-0.03629527986049652,
0.04288454353809357,
-0.02238783799111843,
-0.046293552964925766,
0.06619806587696075,
0.05275306850671768,
-0.004501510411500931,
0.018097365275025368,
0.02449187822639942,
-0.024193575605750084,
-0.02... | 0.822025 | 0.821068 | 0.000958 |
Here's a question for a math student to solve related to the substitution ciphers:
Using a Caesar Cipher (shift of 3), encrypt the message "MATHEMATICS" into a substitution cipher. | tiger_lab_math | [
0.04517604783177376,
0.027809156104922295,
-0.030897129327058792,
-0.036342374980449677,
0.0435652993619442,
-0.025488758459687233,
-0.02560635656118393,
0.05444175377488136,
0.04689367488026619,
-0.03217048943042755,
0.018751084804534912,
0.0032305570784956217,
-0.048873044550418854,
-0.0... | 0.825157 | 0.823098 | 0.002059 |
Consider a graph G with 8 vertices and the following adjacency matrix:
$$
\begin{matrix}
0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\
1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 1 & 0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\... | tiger_lab_math | [
0.022907953709363937,
0.017851578071713448,
-0.022345442324876785,
-0.038732729852199554,
0.01811850070953369,
-0.013134007342159748,
-0.05038168281316757,
0.0456160269677639,
0.022109463810920715,
-0.026051973924040794,
0.016869960352778435,
0.02054539881646633,
-0.01794261299073696,
-0.0... | 0.833534 | 0.82348 | 0.010054 |
Square A has a perimeter of $24$ cm. Square B has an area equal to one-fourth the area of square A. What is the perimeter of square B?
[asy]
draw((0,0)--(7,0));
draw((7,0)--(7,7));
draw((7,7)--(0,7));
draw((0,7)--(0,0));
draw((11,2)--(11,5));
draw((11,5)--(14,5));
draw((14,5)--(14,2));
draw((14,2)--(11,2));
label("A... | tiger_lab_math | [
0.03495558723807335,
0.05754079669713974,
-0.03909236192703247,
-0.04696822911500931,
0.040653519332408905,
-0.012937342748045921,
-0.037216104567050934,
0.058841772377491,
0.043371640145778656,
-0.0038381072226911783,
0.008804607205092907,
0.03756696358323097,
-0.01592206209897995,
-0.031... | 0.827216 | 0.8205 | 0.006716 |
Let $H$ be the orthocenter of triangle $ABC.$ For all points $P$ on the circumcircle of triangle $ABC,$
\[PA^2 + PB^2 + PC^2 - PH^2\]is a constant. Express this constant in terms of the side lengths $a,$ $b,$ $c$ and circumradius $R$ of triangle $ABC.$ | tiger_lab_math | [
0.017031224444508553,
0.018214767798781395,
-0.03503871709108353,
-0.04653208702802658,
0.03140421211719513,
-0.03642727807164192,
-0.006620393134653568,
0.03640533611178398,
0.04990116506814957,
-0.012897433713078499,
0.009525058791041374,
0.03652063012123108,
-0.022922640666365623,
-0.00... | 0.849894 | 0.820975 | 0.028919 |
Find the radius of convergence and the interval of convergence of the power series $\sum_{n=0}^{\infty} \frac{(x-1)^{n}}{n!}$. | tiger_lab_math | [
0.0014293461572378874,
0.03436450660228729,
-0.034559495747089386,
-0.039725832641124725,
0.026420921087265015,
0.0066682202741503716,
0.0011733499122783542,
0.04926791787147522,
0.03806517273187637,
-0.026064634323120117,
0.009457672014832497,
0.003445542650297284,
-0.03896457701921463,
-... | 0.823328 | 0.819968 | 0.00336 |
Let A = {1, 2, 3} and B = {2, 3, 4}. Find (A ⊗ B) △ B and A ⊗ (B △ B). | tiger_lab_math | [
0.05338576063513756,
0.020533084869384766,
-0.03197091817855835,
-0.02749086543917656,
0.03407717123627663,
-0.011642633005976677,
-0.005903781391680241,
0.04703560844063759,
0.051331471651792526,
-0.015304616652429104,
0.012883704155683517,
0.016932213678956032,
-0.016696777194738388,
-0.... | 0.829184 | 0.826657 | 0.002527 |
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
| tiger_lab_math | [
0.03339137136936188,
0.012040657922625542,
-0.03711516782641411,
-0.029622118920087814,
0.0322452038526535,
-0.016152281314134598,
-0.03149212896823883,
0.046536944806575775,
0.044761572033166885,
-0.006086197216063738,
0.019236857071518898,
0.031871289014816284,
-0.03417627513408661,
-0.0... | 0.848272 | 0.82276 | 0.025511 |
Find the solution of the following boundary value problem using complex analysis:
$$u_{xx}+u_{yy}=0,$$
subject to the boundary conditions:
$$u(x,0)=x^2,$$
$$u(0,y)=y^2,$$
$$u(x,1)=\sin(\pi x).$$
Express the solution in terms of complex variables. | tiger_lab_math | [
-0.01584584452211857,
0.032694172114133835,
-0.03385232388973236,
-0.039044469594955444,
0.04246250540018082,
-0.009260552935302258,
0.006466571241617203,
0.056807368993759155,
0.054275743663311005,
0.010692121461033821,
0.02217397466301918,
0.03269904479384422,
-0.030635559931397438,
-0.0... | 0.823741 | 0.811179 | 0.012561 |
As a bicycle salesperson, David earns a fixed salary of $40 per week plus $6 per bicycle for the first 6 bicycles he sells, $12 per bicycle for the next 6 bicycles he sells, and $18 per bicycle for every bicycle sold after first 12. This week, he earned more than twice as much as he did last week. If he sold x bicycles... | tiger_lab_math | [
0.018436655402183533,
0.024627311155200005,
-0.040122367441654205,
-0.022328097373247147,
0.02951822429895401,
-0.027461810037493706,
-0.0055657182820141315,
0.08301017433404922,
0.0434601865708828,
-0.01700468733906746,
0.03472650796175003,
0.032408393919467926,
-0.034627385437488556,
-0.... | 0.822048 | 0.820545 | 0.001503 |
Let X be a smooth projective curve over a field of characteristic p > 0. Compute the first cohomology group H^1(X, ℤ/pℤ) where p is a prime. | tiger_lab_math | [
0.0005577035481110215,
0.024830840528011322,
-0.022209148854017258,
-0.04316580668091774,
0.032064370810985565,
-0.02718617022037506,
-0.0037068421952426434,
0.0573359839618206,
0.04722846299409866,
-0.028270298615098,
0.01440895814448595,
0.02600061148405075,
-0.023263391107320786,
-0.020... | 0.821523 | 0.806244 | 0.015279 |
Consider a graph G with 7 vertices and the following edges: {(1,2),(1,3),(2,3),(4,5),(4,6),(5,6),(6,7)}. Determine the maximum independent set in G. | tiger_lab_math | [
0.052126798778772354,
0.027074335142970085,
-0.03077896684408188,
-0.030575783923268318,
0.038648493587970734,
-0.03879108652472496,
-0.02631298080086708,
0.02083379589021206,
0.030811185017228127,
-0.038491517305374146,
0.026137031614780426,
0.02286861091852188,
-0.032619647681713104,
0.0... | 0.833791 | 0.826097 | 0.007694 |
Let $z$ and $w$ be complex numbers such that $|z| = 1$ and $|w| = 3$. If $|z+w| = 2$, what is $ \left | \frac{1}{z} + \frac{1}{w} \right|$? | tiger_lab_math | [
0.02351883426308632,
0.03282006457448006,
-0.04482783377170563,
-0.049325235188007355,
0.04202329367399216,
0.009719712659716606,
-0.0013785999035462737,
0.04216064512729645,
0.04420877993106842,
-0.02178763411939144,
0.023023493587970734,
0.02043725736439228,
-0.03472856059670448,
-0.0244... | 0.832425 | 0.830198 | 0.002227 |
Here is a question for the student to solve:
Given a matrix A = [3 2; 0 2], find a diagonal matrix D and an invertible matrix P such that P^-1AP = D. | tiger_lab_math | [
0.036538880318403244,
0.013449608348309994,
-0.040646057575941086,
-0.047249551862478256,
0.03136894479393959,
-0.02744261920452118,
0.00340458145365119,
0.07137412577867508,
0.03821452334523201,
-0.004850639961659908,
0.0167979933321476,
0.01962561346590519,
-0.021748866885900497,
-0.0296... | 0.83056 | 0.822105 | 0.008456 |
Suppose we have a circular disc of radius a, centered at the origin, that has a temperature distribution given by $u(r,\theta,0) = r\cos(\theta)$, where $r$ is the radial distance from the center of the disc and $\theta$ is the polar angle. If the disc is heated at its boundary according to $u(a,\theta,t) = T + \sin(2\... | tiger_lab_math | [
0.022953035309910774,
0.016936933621764183,
-0.04182586446404457,
-0.028997797518968582,
0.005703856237232685,
-0.024923045188188553,
-0.0450429767370224,
0.043607667088508606,
0.032842282205820084,
-0.003333032364025712,
0.027650749310851097,
0.031003089621663094,
-0.023319954052567482,
0... | 0.824492 | 0.805095 | 0.019398 |
Consider the differential equation y'(x) = 2x. Find the particular solution that satisfies the initial condition y(0) = 1. | tiger_lab_math | [
0.007685034070163965,
0.01591719314455986,
-0.028483202680945396,
-0.03909475728869438,
0.028849972411990166,
-0.012655258178710938,
-0.010288076475262642,
0.09153960645198822,
0.04634816199541092,
-0.020671483129262924,
0.024335043504834175,
0.011792481876909733,
-0.02448240853846073,
-0.... | 0.816878 | 0.814749 | 0.002129 |
Let $g$ be the metric defined on the unit sphere $S^2$ given by $g_p(v,w) = \|p \times v \cdot p \times w\|$, where $p$ is a fixed point in $S^2$ and $\times$ denotes the cross product. Find the volume form of the metric $g$ on $S^2$. | tiger_lab_math | [
0.01337183266878128,
0.02664712443947792,
-0.04791519418358803,
-0.04452089965343475,
0.03588000312447548,
-0.03242470324039459,
-0.029125243425369263,
0.06388972699642181,
0.04599175974726677,
0.008146390318870544,
0.01955597661435604,
0.037633027881383896,
-0.012123720720410347,
-0.02337... | 0.828859 | 0.807427 | 0.021432 |
Let $z$ be a complex number satisfying $z^2 + z + 1 = 0.$ Compute
\[z^{97} + z^{98} + z^{99} + z^{100} + z^{101}.\] | tiger_lab_math | [
0.040056731551885605,
0.026308393105864525,
-0.021198084577918053,
-0.052893009036779404,
0.03717873618006706,
-0.0266430601477623,
0.001470278832130134,
0.04621678218245506,
0.05242320895195007,
-0.015058686956763268,
0.0194334015250206,
0.02646947093307972,
-0.04795616865158081,
-0.01856... | 0.847995 | 0.837953 | 0.010043 |
Consider the following Nonlinear Schrödinger equation:
$$i\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2} + |u|^2u = 0$$
where $u(x,t)$ is a complex-valued function.
Suppose we have a soliton wave solution of the form $u(x,t) = A\text{sech}(k(x-vt))e^{iwx}$, where $A$, $k$, $v$, and $w$ ... | tiger_lab_math | [
0.03592060133814812,
0.048186831176280975,
-0.0421738363802433,
-0.03856202960014343,
0.025315728038549423,
-0.01981506496667862,
-0.029213441535830498,
0.0934334397315979,
0.03950219973921776,
-0.0006957666482776403,
0.020437853410840034,
0.01768205314874649,
-0.027867315337061882,
-0.026... | 0.819287 | 0.803458 | 0.015828 |
A control system has the transfer function given by:
$$G(s) = \frac{10}{s(s+1)}$$
Design a proportional controller $K_p$ to obtain a closed-loop system with a 10% overshoot and settling time of 2 seconds. Use the root locus method to sketch the root loci and determine the pole locations for $K_p$. | tiger_lab_math | [
0.020448647439479828,
0.024419456720352173,
-0.0442839115858078,
-0.03779580071568489,
0.03463494032621384,
-0.05395806208252907,
-0.027244960889220238,
0.06685543060302734,
0.042688049376010895,
-0.0031382497400045395,
0.029702678322792053,
0.048639651387929916,
-0.02521635964512825,
0.00... | 0.821821 | 0.814298 | 0.007523 |
Faces $ABC$ and $BCD$ of tetrahedron $ABCD$ meet at an angle of $30^\circ$. The area of face $ABC$ is $120$, the area of face $BCD$ is $80$, and $BC=10$. Find the volume of the tetrahedron.
| tiger_lab_math | [
0.021616749465465546,
0.01672508381307125,
-0.05547310784459114,
-0.037795159965753555,
0.033736418932676315,
-0.03031499683856964,
-0.02829909138381481,
0.05060410499572754,
0.045292820781469345,
0.010397030040621758,
0.011599868535995483,
0.03246219456195831,
-0.03851865977048874,
-0.009... | 0.812573 | 0.805355 | 0.007218 |
Tom cannot afford a normal doctor, so he goes to a discount clinic that is 70% cheaper. It takes two visits, though, instead of 1. A normal doctor charges $200 for a visit. How much money does he save? Let's write a program. | tiger_lab_math | [
0.0016432972624897957,
0.0036973138339817524,
-0.04117443040013313,
-0.037271648645401,
0.037104297429323196,
-0.00035697899875231087,
-0.024476006627082825,
0.0720050036907196,
0.055120088160037994,
-0.019877709448337555,
0.01013828907161951,
0.04514847695827484,
-0.04503751918673515,
-0.... | 0.802122 | 0.798097 | 0.004025 |
Here's a numerical problem for the Beauty Contest Game:
Suppose there are four players in a Beauty Contest Game, and each player is asked to choose a number between 0 and 100. The winner of the game will be the player whose number is closest to 2/3 of the average of all the numbers chosen by everyone. If two players c... | tiger_lab_math | [
0.020122937858104706,
0.0029283862095326185,
-0.047477178275585175,
-0.02810465544462204,
0.029935989528894424,
-0.014939402230083942,
-0.04007696732878685,
0.022764086723327637,
0.02905181795358658,
-0.015498923137784004,
0.01141227688640356,
0.03130246326327324,
-0.03163924440741539,
-0.... | 0.829013 | 0.818556 | 0.010457 |
Consider a simple undirected graph G with n vertices and edge set E. Let the diameter of G be d(G), which is defined as the longest shortest path between any two vertices of G. Given a graph with n = 7 vertices and the following edge set E = {(1,2), (1,3), (3,4), (3,5), (4,6), (5,7)}, determine the diameter of the grap... | tiger_lab_math | [
0.01257285475730896,
0.01584109477698803,
-0.02025236003100872,
-0.038738444447517395,
0.026282502338290215,
-0.023601001128554344,
-0.03369302302598953,
0.02965746447443962,
0.018980011343955994,
-0.016077091917395592,
0.0295831598341465,
0.028076935559511185,
-0.018765315413475037,
-0.01... | 0.817036 | 0.804073 | 0.012963 |
A polynomial with integer coefficients is of the form
\[9x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 15 = 0.\]Find the number of different possible rational roots of this polynomial. | tiger_lab_math | [
0.03326218202710152,
0.031041057780385017,
-0.03235424682497978,
-0.04204676300287247,
0.027336735278367996,
-0.04303992912173271,
-0.0018732808530330658,
0.05477592349052429,
0.056628283113241196,
-0.021386485546827316,
0.021495724096894264,
0.025448007509112358,
-0.03907160088419914,
-0.... | 0.833462 | 0.832393 | 0.001069 |
Suppose an investor has $100,000 to invest and wants to create a portfolio consisting of two assets, asset A and asset B. Asset A has an expected return of 8% and a standard deviation of 12%, while asset B has an expected return of 10% and a standard deviation of 15%. The correlation between the two assets is 0.4. What... | tiger_lab_math | [
0.0239474568516016,
0.004293186590075493,
-0.05334547534584999,
-0.01918167620897293,
0.030093688517808914,
-0.0077698430977761745,
-0.03359534963965416,
0.0555490143597126,
0.047326356172561646,
-0.006550372112542391,
0.019036924466490746,
0.03810058906674385,
-0.026403741911053658,
-0.02... | 0.79622 | 0.793167 | 0.003054 |
[asy] draw((0,0)--(0,3)--(4,0)--cycle,dot); draw((4,0)--(7,0)--(7,10)--cycle,dot); draw((0,3)--(7,10),dot); MP("C",(0,0),SW);MP("A",(0,3),NW);MP("B",(4,0),S);MP("E",(7,0),SE);MP("D",(7,10),NE); [/asy]
Triangle $ABC$ has a right angle at $C, AC=3$ and $BC=4$. Triangle $ABD$ has a right angle at $A$ and $AD=12$. Points $... | tiger_lab_math | [
0.034938231110572815,
0.020871523767709732,
-0.03245418518781662,
-0.04191283881664276,
0.037255216389894485,
-0.026133310049772263,
-0.023225337266921997,
0.04561582952737808,
0.033685196191072464,
0.012275880202651024,
0.02474166639149189,
0.02535482682287693,
-0.004442794714123011,
-0.0... | 0.852165 | 0.829441 | 0.022724 |
Here's a precise numerical problem related to Adverse Selection in Game theory:
Suppose there are 100 used cars for sale in the market, out of which 40 are bad quality and 60 are good quality. The sellers know the quality of their cars, but the buyers are not sure and can only check the condition of the car after they... | tiger_lab_math | [
0.011945761740207672,
-0.015151525847613811,
-0.04465244710445404,
-0.012615430168807507,
0.03542047366499901,
-0.004959422163665295,
-0.02915743552148342,
0.054643772542476654,
0.026591094210743904,
-0.006694620940834284,
0.02342098206281662,
0.016867972910404205,
-0.05032249167561531,
-0... | 0.826441 | 0.813965 | 0.012476 |
Let X be the affine variety defined by the equation x^2 + y^2 - 1 = 0, and let G be the group of matrices {(a,b; -b,a) : a,b ∈ ℝ, a^2 + b^2 = 1}.
Suppose that G acts on X by g · (x,y) = (ax+by,−bx+ay) for all g = (a,b) ∈ G and (x,y) ∈ X.
Find the set of all invariant polynomials on X under this action of G. | tiger_lab_math | [
0.016365163028240204,
0.029969748109579086,
-0.04300319403409958,
-0.039115890860557556,
0.04667084664106369,
-0.04228348284959793,
0.005664866417646408,
0.07115472853183746,
0.03873148933053017,
-0.007009164430201054,
-0.0007655138615518808,
0.0380067341029644,
-0.045565951615571976,
-0.0... | 0.837166 | 0.807991 | 0.029174 |
Suppose that two players, Ann and Bob, are either lazy or diligent. If both are diligent, they will each earn $100. If both are lazy, they will each earn $20. However, if one is diligent and the other is lazy, the lazy player will earn $120 while the diligent player will earn only $50.
Assuming that Ann has a probabil... | tiger_lab_math | [
-0.004973029252141714,
0.011532725766301155,
-0.06278165429830551,
-0.03875314071774483,
0.023726852610707283,
-0.004291601944714785,
-0.011523835361003876,
0.034295521676540375,
0.027991557493805885,
-0.03035118617117405,
0.015657799318432808,
0.03178124502301216,
-0.02660989761352539,
-0... | 0.820809 | 0.818857 | 0.001952 |
Let S be the surface defined by the equation $x^2 + y^2 + z^2 = 5$ in $\mathbb{R}^3$. Find the equations of the tangent planes to S at the points where the surface intersects the coordinate planes. | tiger_lab_math | [
0.01223152969032526,
0.03190194442868233,
-0.057353198528289795,
-0.05301349610090256,
0.044787436723709106,
-0.010325601324439049,
-0.013073456473648548,
0.04580850154161453,
0.02876603789627552,
-0.007607132662087679,
0.022928640246391296,
0.02820742130279541,
-0.03453748673200607,
-0.01... | 0.836691 | 0.818319 | 0.018371 |
Statement 1| The intersection of two subgroups in a group forms a subgroup. Statement 2 | A group with more than one element and with an operation that is commutative and associative is always an abelian group. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.05440545082092285,
0.021695779636502266,
-0.028639864176511765,
-0.02820776216685772,
0.04066678509116173,
-0.03952411562204361,
-0.0018897858681157231,
0.06016990914940834,
0.04230846092104912,
-0.002397178439423442,
0.012460148893296719,
0.009135870262980461,
-0.015944704413414,
-0.018... | 0.809582 | 0.799914 | 0.009668 |
Suppose that we have the Hénon map defined as follows:
x[i+1] = 1 - a * x[i]^2 + y[i]
y[i+1] = b * x[i]
where a = 1.4 and b = 0.3.
Given the initial conditions x[0] = 0.9 and y[0] = 0, calculate the first 5 iterations of the Hénon map and plot the resulting points in the (x, y) plane. What does the res... | tiger_lab_math | [
0.013547454960644245,
0.036586351692676544,
-0.02667645737528801,
-0.0562027208507061,
0.036184825003147125,
-0.04424770176410675,
-0.010033749975264072,
0.06955936551094055,
0.03418685868382454,
-0.018852751702070236,
0.010732235386967659,
0.020198483020067215,
-0.03060687892138958,
-0.02... | 0.83593 | 0.826605 | 0.009325 |
Consider a graph with 7 vertices and 8 edges. Find the size of the maximum independent set in this graph. | tiger_lab_math | [
0.04191716015338898,
0.02616918459534645,
-0.033858008682727814,
-0.03251102566719055,
0.035291045904159546,
-0.028141463175415993,
-0.03218035027384758,
0.03271249309182167,
0.04006168618798256,
-0.04462800174951553,
0.024275315925478935,
0.02350628562271595,
-0.03092985972762108,
-0.0116... | 0.828308 | 0.82454 | 0.003768 |
Here's a precise numerical problem related to the topic of Black-Scholes-Merton equation for option pricing with stochastic volatility:
Suppose that a stock currently trades at $100 per share, with a volatility of 0.20, and that the risk-free interest rate is 5% per annum. You have a European call option on this stock... | tiger_lab_math | [
-0.0029509514570236206,
0.01273959968239069,
-0.03351036459207535,
-0.01708352379500866,
0.025925900787115097,
-0.0179881751537323,
-0.024050969630479813,
0.07139011472463608,
0.04765686020255089,
-0.008832970634102821,
0.016688207164406776,
0.03159519284963608,
-0.026694856584072113,
-0.0... | 0.814295 | 0.804148 | 0.010146 |
Let X be a topological space such that its first cohomology group with coefficients in Z is given by H^1(X;Z) = Z/3Z. Determine the number of connected components of X. | tiger_lab_math | [
0.027432648465037346,
0.014517391100525856,
-0.05084959417581558,
-0.025112440809607506,
0.03161311149597168,
-0.003315467620268464,
-0.000682370737195015,
0.04820030555129051,
0.03584162890911102,
-0.0039759231731295586,
0.022641051560640335,
0.03493013605475426,
-0.03230142965912819,
-0.... | 0.832004 | 0.822971 | 0.009033 |
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)? Let's program in Python in the response. | tiger_lab_math | [
-0.0033828269224613905,
0.03491144999861717,
-0.03625563159584999,
-0.00030829806928522885,
0.0433533750474453,
-0.01486934069544077,
-0.037992678582668304,
0.07772646844387054,
0.022559598088264465,
0.012178917415440083,
0.013400645926594734,
0.0189508069306612,
-0.021799951791763306,
-0.... | 0.820777 | 0.81736 | 0.003417 |
A bag contains 3 tan, 2 pink and 4 violet chips. If the 9 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are drawn in such a way that the 3 tan chips are drawn consecutively, the 2 pink chips are drawn consecutively, and the 4 violet chips are drawn ... | tiger_lab_math | [
-0.0025712656788527966,
0.025740768760442734,
-0.038023706525564194,
-0.0413394570350647,
0.02518942765891552,
-0.009091059677302837,
-0.022033346816897392,
0.045993298292160034,
0.017250342294573784,
0.0004858651082031429,
0.018030034378170967,
0.02176467329263687,
-0.016866806894540787,
... | 0.83545 | 0.833543 | 0.001906 |
In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let's write a Python program. | tiger_lab_math | [
0.03495434299111366,
0.04387194290757179,
-0.038620080798864365,
-0.030926920473575592,
0.033300768584012985,
-0.016004078090190887,
-0.037065811455249786,
0.05043066665530205,
0.03419531509280205,
-0.007296820171177387,
0.021399470046162605,
0.04360026493668556,
-0.030064530670642853,
0.0... | 0.844282 | 0.829023 | 0.01526 |
Given that the graphs of $y=h(x)$ and $y=j(x)$ intersect at $(2,2),$ $(4,6),$ $(6,12),$ and $(8,12),$ there is one point where the graphs of $y=h(2x)$ and $y=2j(x)$ must intersect. What is the sum of the coordinates of that point? | tiger_lab_math | [
0.02358376793563366,
0.027151765301823616,
-0.04321719706058502,
-0.036911122500896454,
0.043637022376060486,
-0.015769880264997482,
-0.02843296155333519,
0.048018865287303925,
0.044704508036375046,
-0.015630818903446198,
0.002413616981357336,
0.02721940353512764,
-0.02704431116580963,
-0.... | 0.842321 | 0.834571 | 0.007749 |
Say that a complex number $z$ is three-presentable if there is a complex number $w$ of absolute value $3$ such that $z = w - \frac{1}{w}$. Let $T$ be the set of all three-presentable complex numbers. The set $T$ forms a closed curve in the complex plane. What is the area inside $T$? | tiger_lab_math | [
0.02253575250506401,
0.037630483508110046,
-0.06435425579547882,
-0.05097922682762146,
0.04670604690909386,
-0.011901002377271652,
-0.007196194026619196,
0.05006511136889458,
0.039272937923669815,
-0.01095018070191145,
0.014030846767127514,
0.04788777977228165,
-0.04337097704410553,
-0.019... | 0.846869 | 0.826507 | 0.020363 |
The legs of a right triangle have lengths $\log_4 27$ and $\log_2 9.$ If the length of the hypotenuse is $h,$ compute $4^h.$ Please respond by writing a program in Python. | tiger_lab_math | [
0.013241459615528584,
0.029708541929721832,
-0.03870842605829239,
-0.03257974237203598,
0.027074052020907402,
-0.0293542742729187,
-0.02156706713140011,
0.04173765704035759,
0.046062782406806946,
-0.013477446511387825,
0.029275061562657356,
0.0406954288482666,
-0.027301635593175888,
-0.024... | 0.834397 | 0.832614 | 0.001782 |
Let M be a two-dimensional manifold with a metric tensor given by
g = [dx^2 + dy^2 + x^2 dy^2]
Calculate the scalar curvature of M at the point (1, 0). | tiger_lab_math | [
-0.0013401081087067723,
0.028663258999586105,
-0.022870462387800217,
-0.06384094804525375,
0.041484057903289795,
-0.016714220866560936,
-0.029450852423906326,
0.06293274462223053,
0.02734302170574665,
-0.01112687774002552,
-0.006384305655956268,
0.025261515751481056,
-0.032664474099874496,
... | 0.81394 | 0.809459 | 0.004481 |
Let X be a smooth projective algebraic curve of degree 3 in the complex projective plane. Write down the intersection cohomology groups IH^k(X) for k = 0, 1, 2, 3. | tiger_lab_math | [
0.025055374950170517,
0.02292204275727272,
-0.030774783343076706,
-0.04534721001982689,
0.025482632219791412,
-0.0012460116995498538,
-0.021976936608552933,
0.044340189546346664,
0.043300140649080276,
-0.015522642992436886,
0.021537652239203453,
0.026844827458262444,
-0.033927783370018005,
... | 0.817692 | 0.797141 | 0.020551 |
Except for the first two terms, each term of the sequence $1000, x, 1000 - x,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $x$ produces a sequence of maximum length? Let's write a Python program... | tiger_lab_math | [
0.023044243454933167,
0.0010579207446426153,
-0.03562202304601669,
-0.02272939495742321,
0.017843151465058327,
-0.03585232049226761,
-0.03593628853559494,
0.050680823624134064,
0.04163343831896782,
-0.02297781966626644,
0.021520022302865982,
0.026081260293722153,
-0.0454074963927269,
-0.00... | 0.825519 | 0.824044 | 0.001475 |
Consider the heat equation in the complex plane for a temperature function u(x,y,t):
∂u/∂t = ∂^2u/∂x^2 + ∂^2u/∂y^2.
Suppose that the temperature distribution of a circular plate is given by:
u(x,y,0) = 10cos(πx)cos(πy)
at time t=0. Also, the boundary condition of the circular plate is given by:
u(a,θ,t) = 0 f... | tiger_lab_math | [
0.0007739068241789937,
0.0241440050303936,
-0.04815811291337013,
-0.032997310161590576,
0.02405879832804203,
-0.019271351397037506,
-0.03926820680499077,
0.06526189297437668,
0.041084468364715576,
-0.0038449913263320923,
0.013215635903179646,
0.03581776097416878,
-0.024918120354413986,
0.0... | 0.81179 | 0.798989 | 0.012802 |
Is the Taylor Series for $f$ at x=5 where $f(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!} absolutely converging? Let's write a program. | tiger_lab_math | [
0.009864440187811852,
0.0388619638979435,
-0.04692636430263519,
-0.018879923969507217,
0.034106601029634476,
-0.021196139976382256,
0.0008531442144885659,
0.06628959625959396,
0.03609307482838631,
-0.024921679869294167,
0.007211232092231512,
0.004360524006187916,
-0.02837706357240677,
-0.0... | 0.833196 | 0.82988 | 0.003316 |
Let $a$ and $b$ be nonnegative real numbers such that
\[\sin (ax + b) = \sin 29x\]for all integers $x.$ Find the smallest possible value of $a.$ | tiger_lab_math | [
0.0006566203664988279,
0.018095726147294044,
-0.047292500734329224,
-0.035005465149879456,
0.02699890173971653,
-0.00211384822614491,
0.002162467921152711,
0.05557052418589592,
0.04361702501773834,
-0.0163879431784153,
0.04147300496697426,
0.0383608415722847,
-0.028655920177698135,
-0.0294... | 0.85612 | 0.846496 | 0.009624 |
A manufacturing company produces two types of products: Product A and Product B. The profit from each unit of Product A is $5, and the profit from each unit of Product B is $8. The company must produce at least 50 units of Product A per day, and cannot produce more than 80 units of Product B per day due to production l... | tiger_lab_math | [
0.020766211673617363,
0.00016287608013954014,
-0.05258157476782799,
-0.002393050119280815,
0.01209936197847128,
-0.03069530427455902,
-0.03793921321630478,
0.09003528952598572,
0.02482444979250431,
-0.005040234420448542,
-0.0035567728336900473,
0.03581976890563965,
-0.016041727736592293,
-... | 0.80551 | 0.782502 | 0.023009 |
One possible question related to geometric topology and knot theory for a math student to solve is:
"What is the minimum number of times that a closed curve in three-dimensional space has to cross itself to form a nontrivial knot?"
Answer: The minimum number of crossings required to form a nontrivial knot depends on ... | tiger_lab_math | [
0.022714167833328247,
0.021968143060803413,
-0.029834281653165817,
-0.0465933158993721,
0.031030597165226936,
0.007598096504807472,
-0.024441340938210487,
0.037261415272951126,
0.012348719872534275,
0.00004912738950224593,
-0.01243223063647747,
0.023855578154325485,
-0.003969603683799505,
... | 0.795883 | 0.782449 | 0.013434 |
Consider the following statements related to abstract algebra: Statement 1 | Every group of order p^2 (where p is a prime number) is abelian. Statement 2 | If two groups G and H are isomorphic, then their center groups Z(G) and Z(H) are also isomorphic. Answer Choices: (A) True, True (B) False, False (C) True, False ... | tiger_lab_math | [
0.04144401475787163,
0.03433232754468918,
-0.02314123697578907,
-0.037813443690538406,
0.03588256239891052,
-0.049560293555259705,
0.007607807405292988,
0.056417316198349,
0.03438218683004379,
-0.019558347761631012,
-0.0031692294869571924,
0.009842991828918457,
-0.015541989356279373,
-0.02... | 0.812183 | 0.804082 | 0.008101 |
The math student is given a data set containing the heights and weights of 50 people. Using multivariate analysis techniques, they are required to determine the correlation coefficient between height and weight. The student should solve the problem precisely and report the correlation coefficient to two decimal places. | tiger_lab_math | [
0.016639096662402153,
0.015914779156446457,
-0.05605814605951309,
-0.04956978186964989,
0.03107292205095291,
-0.008825335651636124,
-0.028800589963793755,
0.054955918341875076,
0.03934333100914955,
-0.015523504465818405,
0.0188558679074049,
0.03862930089235306,
-0.015712715685367584,
0.001... | 0.794045 | 0.789003 | 0.005042 |
Find the Laurent series expansion of the function f(z) = (z^2 + 1)/(z-2) centered at z = 1 and determine the radius of convergence of the series. | tiger_lab_math | [
0.011070044711232185,
0.03218497335910797,
-0.04135429114103317,
-0.04195404052734375,
0.038475580513477325,
-0.004515481181442738,
-0.009341483935713768,
0.03378200903534889,
0.024395059794187546,
-0.010247030295431614,
0.03220510110259056,
0.009194791316986084,
-0.03047339618206024,
-0.0... | 0.818561 | 0.815686 | 0.002874 |
Let $M$ be a smooth manifold with the Riemannian metric $g$ given by $g = dx^2 + 2dydz+ dt^2$. Let $T$ be a rank 2 tensor field on $M$ defined by $T_{ij}(p) = p_1p_2$ at each point $p = (p_1, p_2, p_3, p_4) \in M$.
Determine the covariant derivative of $T$ with respect to the connection associated to the metric $g$ a... | tiger_lab_math | [
0.014419957995414734,
0.037074483931064606,
-0.04815668985247612,
-0.04935368150472641,
0.03390107303857803,
0.005187042523175478,
-0.022984202951192856,
0.06512145698070526,
0.03154655173420906,
0.007250677794218063,
-0.008246749639511108,
0.032009970396757126,
-0.030258871614933014,
-0.0... | 0.810699 | 0.79522 | 0.015479 |
What is the number of rational curves of degree 4 that lie on a quintic surface in $\mathbb{P}^4$ with a fixed intersection number with a fixed plane? | tiger_lab_math | [
0.004576311446726322,
0.03924469277262688,
-0.03254183754324913,
-0.04845503345131874,
0.02905029058456421,
-0.003969294019043446,
-0.013159424066543579,
0.019213035702705383,
0.051724277436733246,
-0.019943686202168465,
0.013723969459533691,
0.017423979938030243,
-0.013394727371633053,
-0... | 0.824489 | 0.822305 | 0.002184 |
Suppose the Belousov-Zhabotinsky reaction is modeled by the system of differential equations:
\begin{align*}
\frac{\mathrm{d} x}{\mathrm{d} t} &= 2y + k(x-y+z) \\
\frac{\mathrm{d} y}{\mathrm{d} t} &= -x+ky \\
\frac{\mathrm{d} z}{\mathrm{d} t} &= -\frac{1}{3}z + xy,
\end{align*}
where $k$ is a constant.
... | tiger_lab_math | [
0.04137562960386276,
0.03166769817471504,
-0.042142316699028015,
-0.04091687873005867,
0.03495216742157936,
-0.015806598588824272,
-0.01235477440059185,
0.0851488783955574,
0.038048386573791504,
-0.019652387127280235,
0.005414808169007301,
0.0259474515914917,
-0.02496621385216713,
-0.00958... | 0.815546 | 0.809049 | 0.006497 |
Let P be a proposition such that P = "n is a multiple of 5". Write the proposition Q that expresses "n is an odd number". Determine the truth value of Q -> ¬P when n = 25. | tiger_lab_math | [
0.01507736835628748,
0.0031440583989024162,
-0.038841452449560165,
-0.04205912724137306,
0.021535687148571014,
-0.03351008519530296,
-0.0032257961574941874,
0.072077676653862,
0.033722080290317535,
-0.007585111074149609,
0.007782470900565386,
0.03986886888742447,
-0.040859948843717575,
-0.... | 0.84053 | 0.832812 | 0.007718 |
Consider a stochastic differential equation:
dX(t) = aX(t)dt + bX(t)dW(t)
where X(t) is a random process, W(t) is a Wiener process or Brownian motion, a and b are constants.
If initial value X(0) = 2, a = 1, b = 2 and the Wiener process has a standard deviation of 0.5, calculate the probability that X(t) exceeds 4 at... | tiger_lab_math | [
0.009711435064673424,
0.002423284575343132,
-0.04983706772327423,
-0.019682230427861214,
0.05031775310635567,
-0.0024059375282377005,
-0.04538281634449959,
0.08628199249505997,
0.0363120511174202,
0.007193760015070438,
0.026173004880547523,
0.03632234036922455,
-0.02474334090948105,
-0.007... | 0.811937 | 0.803336 | 0.0086 |
Statement 1| If G is a finite group and H is a proper subgroup of G, then the order of H is less than the order of G. Statement 2 | The set of all integers under addition is an abelian group. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.034235432744026184,
0.027513273060321808,
-0.01862037181854248,
-0.046184565871953964,
0.04863728955388069,
-0.0492330826818943,
-0.010081516578793526,
0.05935623496770859,
0.04539471119642258,
-0.011513151228427887,
0.004139358177781105,
0.0128846550360322,
-0.034944940358400345,
-0.010... | 0.813413 | 0.811125 | 0.002287 |
Consider the noncommutative polynomial algebra $R = \mathbb{C} \langle x, y \rangle/(yx - qxy - 1)$, where $q = \frac{1}{\sqrt{2}}$.
Find the dimension of the vector space $R^n$ for $n = 2$. | tiger_lab_math | [
0.029627781361341476,
0.04167666658759117,
-0.03690550476312637,
-0.053372323513031006,
0.017959114164114,
-0.0063276286236941814,
-0.017499350011348724,
0.05519963428378105,
0.03413310647010803,
-0.02399148792028427,
0.010972579009830952,
0.02630475163459778,
-0.037951961159706116,
-0.018... | 0.836632 | 0.818269 | 0.018363 |
Let $S = \{2^0,2^1,2^2,\ldots,2^{10}\}$. Consider all possible positive differences of pairs of elements of $S$. Let $N$ be the sum of all of these differences. Find $N.$ | tiger_lab_math | [
0.026831723749637604,
0.01148929726332426,
-0.04205497354269028,
-0.02132151834666729,
0.03443896770477295,
-0.029716268181800842,
-0.03005598671734333,
0.0426098108291626,
0.03842740133404732,
-0.010619711130857468,
0.026692083105444908,
0.03592732548713684,
-0.051927514374256134,
-0.0150... | 0.857203 | 0.838976 | 0.018226 |
Sure, here's a problem for a math student to solve in predicate logic:
Let P(x) denote "x is a prime number" and let Q(x) denote "x is a multiple of 3."
Write the negation of the statement: "There exists a prime number that is also a multiple of 3."
The student must translate this sentence into logical notation usin... | tiger_lab_math | [
0.03145786374807358,
0.004077245015650988,
-0.03952417150139809,
-0.041011542081832886,
0.04128871485590935,
-0.03708050400018692,
-0.02326364256441593,
0.06325232982635498,
0.039488282054662704,
-0.023338116705417633,
0.010599997825920582,
0.02811274118721485,
-0.03276873379945755,
-0.027... | 0.830357 | 0.815588 | 0.01477 |
Suppose that the graph of a certain function, $y=f(x)$, has the property that if it is shifted $20$ units to the right, then the resulting graph is identical to the original graph of $y=f(x)$.
What is the smallest positive $a$ such that if the graph of $y=f\left(\frac x5\right)$ is shifted $a$ units to the right, then... | tiger_lab_math | [
0.002175139496102929,
0.027194775640964508,
-0.05006221681833267,
-0.03777584806084633,
0.03499877080321312,
-0.01618773862719536,
-0.01790057122707367,
0.06876640766859055,
0.04978559911251068,
-0.03405194729566574,
0.0032515430357307196,
0.013410942628979683,
-0.03012661822140217,
-0.004... | 0.851488 | 0.837836 | 0.013652 |
Consider the metric on $\mathbb{R}^2$ given by $g = e^{2y}(dx^2 + dy^2)$. Determine the Levi-Civita connection associated with $g$ at the point $(1,0)$. | tiger_lab_math | [
-0.006267880089581013,
0.02644181065261364,
-0.022766439244151115,
-0.041346922516822815,
0.031742457300424576,
-0.015262284316122532,
-0.012815885245800018,
0.05406039580702782,
0.036708615720272064,
-0.0020633467938750982,
0.010241912677884102,
0.035836681723594666,
-0.03002508543431759,
... | 0.820078 | 0.804327 | 0.015751 |
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