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 |
|---|---|---|---|---|---|
In a country, there are three major political parties: Party A, Party B, and Party C. A survey of a random sample of 1000 voters was conducted, and it was found that 540 voters support Party A, 360 voters support Party B, and 100 voters support Party C. What is the probability that a randomly selected voter from this c... | tiger_lab_math | [
0.006103079300373793,
0.005006375256925821,
-0.05789684131741524,
-0.03222452849149704,
0.030987050384283066,
-0.025568347424268723,
-0.0048195007257163525,
0.03181877359747887,
0.02566848322749138,
0.006604397669434547,
0.013390219770371914,
0.028555355966091156,
-0.008398892357945442,
-0... | 0.812026 | 0.809067 | 0.002958 |
for any integer k > 1 , the term “ length of an integer ” refers to the number of positive prime factors , not necessarily distinct , whose product is equal to k . for example , if k = 24 , the length of k is equal to 4 , since 24 = 2 × 2 × 2 × 3 . if x and y are positive integers such that x > 1 , y > 1 , and x + 3 y ... | tiger_lab_math | [
0.033282577991485596,
0.003641616553068161,
-0.04115679860115051,
-0.03793197125196457,
0.03650997579097748,
-0.03918084874749184,
-0.029884429648518562,
0.0352003239095211,
0.03926291689276695,
-0.024037698283791542,
0.009153496474027634,
0.03207024931907654,
-0.04431655630469322,
-0.0069... | 0.815936 | 0.812357 | 0.003579 |
Consider the nonlinear differential equation:
y'' + sin(y) = 0
with the initial conditions:
y(0) = 1 and y'(0) = 0.
Find the value of y(π/2) accurate to two decimal places using the fourth-order Runge-Kutta method with a step size of h=0.1. | tiger_lab_math | [
0.0051628402434289455,
0.027116423472762108,
-0.0247250497341156,
-0.031029675155878067,
0.04024175554513931,
0.0004699442652054131,
-0.014185458421707153,
0.06710872799158096,
0.04783516749739647,
-0.0034725128207355738,
0.018349995836615562,
0.022109821438789368,
-0.021160544827580452,
-... | 0.818251 | 0.817022 | 0.001228 |
A line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ -2 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \end{pmatrix}.\]A second line is parameterized by
\[\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -8 \\ 12 \end{pmatrix} + u \begin{pmatrix} 1 \\ 3 \end{pmatrix}.\]If $\theta$ is... | tiger_lab_math | [
0.02775351144373417,
0.025887934491038322,
-0.04686852917075157,
-0.036992572247982025,
0.043099019676446915,
-0.02881782315671444,
-0.018109463155269623,
0.06855229288339615,
0.04424825310707092,
-0.0013669192558154464,
0.02429225854575634,
0.04269587621092796,
-0.018027203157544136,
-0.0... | 0.819146 | 0.813672 | 0.005474 |
Consider the following algebraic curve in the complex plane: $C: x^2 + y^2 = 1$. Find the equation of the image of $C$ under the rational map $\varphi: \mathbb{C}^2 \to \mathbb{C}^2$ defined by $\varphi(x, y) = (\frac{x}{1-y}, \frac{y}{1-x})$. | tiger_lab_math | [
0.016753509640693665,
0.03940105810761452,
-0.037548694759607315,
-0.047938279807567596,
0.041386980563402176,
-0.028582971543073654,
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0.06897453963756561,
0.055634401738643646,
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0.007738215383142233,
0.01781509630382061,
-0.038828205317258835,
-... | 0.838805 | 0.810244 | 0.028561 |
Consider a source X with a distortion measure $d(x, \hat{x})$ that satisfies the following property: all columns of the distortion matrix are permutations of the set $\{d_1, d_2, \ldots, d_m\}$. The function $\phi(D) = \max_{b:\sum_{i=1}^m p_i d_i \leq D} H(p)$ is concave. True or False? | tiger_lab_math | [
0.008777913637459278,
0.018143687397241592,
-0.06183161959052086,
-0.028873879462480545,
0.04020165652036667,
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0.07903757691383362,
0.03608555719256401,
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0.012946479953825474,
0.03147527202963829,
-0.03020484372973442,
-0.0... | 0.832347 | 0.811773 | 0.020574 |
Consider a simplicial complex with 5 vertices and 10 edges, where each edge is shared between exactly 2 vertices. What is the maximum possible number of faces in this simplicial complex? | tiger_lab_math | [
0.03217460587620735,
0.029218167066574097,
-0.03178589418530464,
-0.03437964990735054,
0.018589068204164505,
-0.01553544495254755,
-0.022462381049990654,
0.04280367121100426,
0.0278693325817585,
-0.028115171939134598,
0.016000933945178986,
0.029199987649917603,
-0.019335592165589333,
-0.00... | 0.830965 | 0.823135 | 0.007829 |
Let $f(x) = \displaystyle \frac{1}{ax+b}$ where $a$ and $b$ are nonzero constants. Find all solutions to $f^{-1}(x) = 0$. Express your answer in terms of $a$ and/or $b$. | tiger_lab_math | [
0.01938535086810589,
0.020393850281834602,
-0.0455126017332077,
-0.03687507286667824,
0.03585551679134369,
-0.021072380244731903,
0.026141511276364326,
0.08899248391389847,
0.05027063563466072,
-0.005572402384132147,
0.019772451370954514,
0.011967918835580349,
-0.016688145697116852,
-0.025... | 0.82778 | 0.817684 | 0.010096 |
A delivery driver has to make deliveries to 4 different locations (A, B, C, and D) in a city. The distance between each location is shown in the table below:
| Location | A | B | C | D |
|----------|--------|--------|--------|--------|
| A | 0 | 9 | 12 | 7 |
| B | 9... | tiger_lab_math | [
-0.00422493414953351,
0.017962686717510223,
-0.044885434210300446,
-0.026185117661952972,
0.023500818759202957,
-0.02169504389166832,
-0.03614978492259979,
0.0321379117667675,
0.048112526535987854,
-0.00799438264220953,
0.019351901486516,
0.03377871960401535,
-0.021652843803167343,
-0.0079... | 0.822441 | 0.816028 | 0.006413 |
The length of each side of square A is increased by 100 percent to make square B. If the length of the side of square B is increased by 80 percent to make square C, by what percent is the area of square C greater than the sum of the areas of squares A and B?
Answer Choices: (A) 75% (B) 80% (C) 100% (D) 150% (E) 159.2% | tiger_lab_math | [
0.018036631867289543,
0.0016653207130730152,
-0.0456610843539238,
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0.027284450829029083,
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-0.011751066893339157,
0.0804276093840599,
0.043362028896808624,
-0.010770655237138271,
-0.0036248567048460245,
0.04632331058382988,
-0.06461399048566818,
-0... | 0.818708 | 0.817453 | 0.001255 |
In a community of 100 farmers, each has a plot of land to cultivate. The total land area is 500 acres. If any farmer exceeds a certain limit of pesticide use, it is assumed that the entire community would suffer negative effects of pollution. The maximum pesticide use on one acre of land that the community deems safe i... | tiger_lab_math | [
0.015381130389869213,
0.012644956819713116,
-0.03567460551857948,
-0.024930382147431374,
0.01807040348649025,
-0.05872839689254761,
-0.032620254904031754,
0.05782681703567505,
0.02651207521557808,
-0.015339536592364311,
-0.014448486268520355,
0.03403890132904053,
-0.022695334628224373,
-0.... | 0.793295 | 0.781495 | 0.011801 |
Let A = {1, 2, 3} and B = {2, 3, 4}. Find the intersection of (A x B) ∩ (B x A) and (B x A) ∩ (A x B), and verify that the Cartesian product of two sets is not commutative with respect to intersection. | tiger_lab_math | [
0.0617908239364624,
0.021556099876761436,
-0.0319579541683197,
-0.020978888496756554,
0.031008904799818993,
-0.01613645814359188,
-0.014268103986978531,
0.05880746245384216,
0.054061297327280045,
-0.0202032383531332,
0.025466367602348328,
0.02258886769413948,
-0.012604227289557457,
-0.0273... | 0.830939 | 0.813448 | 0.017491 |
A company needs to transport a product through a network of transportation routes. Each route has a maximum capacity limit, and the company needs to determine how much of the product should be sent through each route in order to minimize cost. The network flow problem is given by the following table:
| Route | Maximum... | tiger_lab_math | [
0.028645623475313187,
0.00932857021689415,
-0.030274761840701103,
-0.009787875227630138,
0.027072155848145485,
-0.017546283081173897,
-0.03565331548452377,
0.05573868378996849,
0.018314989283680916,
-0.004398786462843418,
0.0202452652156353,
0.024458371102809906,
-0.011952877044677734,
-0.... | 0.809458 | 0.799414 | 0.010045 |
Consider a probability density $p_x(x)$ defined over a continuous variable x, and suppose that we make a nonlinear change of variable using $x = g(y)$. The location $\hat{y}$ of the maximum of the density in $y$ is not in general related to the location $\hat{x}$ ̂of the maximum of the density over x by the simple func... | tiger_lab_math | [
0.019382460042834282,
0.0255513247102499,
-0.06034255027770996,
-0.01635504513978958,
0.03984496742486954,
-0.010637164115905762,
-0.039477694779634476,
0.0583888441324234,
0.020540399476885796,
-0.025163473561406136,
0.0007075628964230418,
0.020457938313484192,
-0.024204125627875328,
-0.0... | 0.818228 | 0.798187 | 0.020041 |
The owner of a Turkish restaurant wanted to prepare traditional dishes for an upcoming celebration. She ordered ground beef, in four-pound packages, from three different butchers. The following morning, the first butcher delivered 10 packages. A couple of hours later, 7 packages arrived from the second butcher. Finally... | tiger_lab_math | [
0.029347483068704605,
0.016461873427033424,
-0.029483137652277946,
-0.041561808437108994,
0.01865285076200962,
-0.014473131857812405,
-0.012508024461567402,
0.031509656459093094,
0.0438978485763073,
-0.003670346923172474,
0.006267371121793985,
0.053016215562820435,
-0.05674535036087036,
-0... | 0.826871 | 0.825514 | 0.001357 |
Determine the solution to the boundary value problem:
$\frac{\partial ^2u}{\partial x^2} + \frac{\partial ^2u}{\partial y^2} = 0, \quad 0<x<1, 0<y<1,$
with boundary conditions:
$u(0, y) = u(1, y) = u(x, 0) = \sin(2\pi x), \quad u(x, 1) = 0$ | tiger_lab_math | [
-0.014896971173584461,
0.040424928069114685,
-0.044235486537218094,
-0.03190341964364052,
0.04777710139751434,
0.0038986108265817165,
-0.006068084854632616,
0.06481176614761353,
0.05862261727452278,
0.003209258196875453,
0.012513603083789349,
0.03665568307042122,
-0.015415243804454803,
-0.... | 0.821635 | 0.797534 | 0.024101 |
Suppose you have a graph with 10 vertices and 15 edges. Find the maximum independent set for this graph. | tiger_lab_math | [
0.032199110835790634,
0.027045678347349167,
-0.025416549295186996,
-0.0193711519241333,
0.03814267739653587,
-0.03523373603820801,
-0.02742825821042061,
0.04168069735169411,
0.032997265458106995,
-0.0431676059961319,
0.027791952714323997,
0.013514373451471329,
-0.03725501522421837,
-0.0005... | 0.825766 | 0.823114 | 0.002652 |
If $\left( r + \frac{1}{r} \right)^2 = 3,$ then find $r^3 + \frac{1}{r^3}.$ | tiger_lab_math | [
0.02087290771305561,
0.03789807856082916,
-0.033967867493629456,
-0.03671184554696083,
0.031616948544979095,
-0.01019369624555111,
-0.0041789161041378975,
0.06624740362167358,
0.04193928465247154,
-0.02495146170258522,
0.01028131041675806,
0.02819599024951458,
-0.03709561377763748,
-0.0393... | 0.83199 | 0.820078 | 0.011912 |
A Duffing oscillator is described by the differential equation:
$\ddot{x} + 2\gamma\dot{x} + (\alpha x + \beta x^3) = \gamma \cos(\omega t)$
where $\alpha = -1$, $\beta = 1$, $\gamma = 0.3$, and $\omega = 1.2$. If $x(0) = 0$ and $\dot{x}(0) = 0.4$, find the maximum amplitude of the oscillations of $x(t)$ over the int... | tiger_lab_math | [
0.031233271583914757,
0.025779731571674347,
-0.038032516837120056,
-0.035970211029052734,
0.029194656759500504,
-0.037268564105033875,
-0.020283754914999008,
0.09320216625928879,
0.03064725361764431,
-0.01775442063808441,
0.013072149828076363,
0.034717198461294174,
-0.044222794473171234,
-... | 0.816803 | 0.810735 | 0.006069 |
A game of Nim is played with 4 piles of stones, with 3 stones in the first pile, 5 stones in the second pile, 6 stones in the third pile, and 7 stones in the fourth pile. If the players alternate turns, and can only remove stones from one pile at a time, with the option to remove any number of stones up to the maximum ... | tiger_lab_math | [
0.029254529625177383,
0.006658068858087063,
-0.03012261912226677,
-0.028956230729818344,
0.01866156794130802,
-0.03876369446516037,
-0.02229515090584755,
0.052200425416231155,
0.026618996635079384,
-0.005159744992852211,
0.029650188982486725,
0.027278656139969826,
-0.061841484159231186,
-0... | 0.821336 | 0.812134 | 0.009202 |
In the diagram, $\angle PQR=\angle PRQ$. If $QR=5$ and $PR=7$, what is the perimeter of $\triangle PQR$? [asy]
draw((0,0)--(2.5,7.43)--(5,0)--cycle);
label("5",(2.5,0),S);
label("$Q$",(0,0),SW);
label("$R$",(5,0),SE);
label("$P$",(2.5,7.43),N);
label("7",(4.2,3.7));
[/asy] | tiger_lab_math | [
0.03017316944897175,
0.032812707126140594,
-0.039809249341487885,
-0.053550805896520615,
0.020084884017705917,
-0.006421507801860571,
-0.035097699612379074,
0.053261898458004,
0.018080834299325943,
-0.0009196772007271647,
0.014160485938191414,
0.024793988093733788,
-0.03008856438100338,
-0... | 0.830016 | 0.821434 | 0.008582 |
Consider the following 4-bit block cipher using the S-box shown below:
S-box:
```
0 1 2 3 4 5 6 7 8 9 A B C D E F
0 3 C A F 8 E 5 9 B 1 7 D 2 0 4 6
1 5 2 F C 4 9 B 1 7 A E 3 D 0 8 6
2 2 6 E F C 1 9 A B ... | tiger_lab_math | [
0.04217872768640518,
0.041561391204595566,
-0.021966030821204185,
-0.03276136517524719,
0.031545624136924744,
-0.04648870229721069,
-0.0300943311303854,
0.05752309411764145,
0.04755639284849167,
0.0020920392125844955,
0.02626759186387062,
0.01990082673728466,
-0.030700448900461197,
-0.0033... | 0.805686 | 0.800674 | 0.005012 |
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overlin... | tiger_lab_math | [
0.029429489746689796,
0.03440570458769798,
-0.030714886263012886,
-0.04298118129372597,
0.03321585804224014,
-0.04167721047997475,
-0.020408814772963524,
0.05314961075782776,
0.0400497168302536,
0.013381031341850758,
0.035474274307489395,
0.03931212052702904,
-0.00726295867934823,
-0.03077... | 0.841479 | 0.81965 | 0.021829 |
Let the Ricci curvature of a 3-dimensional manifold be given as follows:
$R_{ij} = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 4 & 1 \\ 0 & 1 & 4 \end{pmatrix} $
Determine the scalar curvature of this manifold. | tiger_lab_math | [
-0.0051698158495128155,
0.036725543439388275,
-0.01767612434923649,
-0.051469847559928894,
0.03616201877593994,
-0.005495504010468721,
-0.013205223716795444,
0.07288262993097305,
0.02104000747203827,
-0.0015320833772420883,
0.0010671999771147966,
0.027124371379613876,
-0.056171633303165436,
... | 0.812024 | 0.804981 | 0.007043 |
Sure, here's a specific problem related to the Fourier transform and image processing:
"A digital image is represented as a 512 x 512 matrix of pixels which has undergone zero-padding to increase its resolution. Apply the discrete Fourier transform to the image and find the amplitude of the frequency component at row ... | tiger_lab_math | [
0.011315571144223213,
0.019074562937021255,
-0.04150001332163811,
-0.029494594782590866,
0.013687756843864918,
-0.02377011440694332,
-0.005428801290690899,
0.053682729601860046,
0.031273022294044495,
-0.008732772432267666,
0.026845404878258705,
0.0068262191489338875,
-0.03279312327504158,
... | 0.814568 | 0.813312 | 0.001256 |
Let
\[x^8 - 98x^4 + 1 = p(x) q(x),\]where $p(x)$ and $q(x)$ are monic, non-constant polynomials with integer coefficients. Find $p(1) + q(1).$ | tiger_lab_math | [
0.014677340164780617,
0.023150093853473663,
-0.0352105014026165,
-0.04309505596756935,
0.02258012443780899,
-0.01734314300119877,
-0.005629397463053465,
0.06744672358036041,
0.05507221817970276,
-0.00513651454821229,
0.0043600029312074184,
0.021800506860017776,
-0.03704167902469635,
-0.020... | 0.847233 | 0.833441 | 0.013792 |
Here is a numerical problem for a math student to solve in solving systems of linear equations using Gaussian elimination:
Solve the following system of linear equations using Gaussian elimination:
x + 2y + z = 3
2x - y + 3z = 1
3x - y - z = 2 | tiger_lab_math | [
0.026114782318472862,
0.013234030455350876,
-0.04601132497191429,
-0.07606542110443115,
0.04676355421543121,
-0.011677193455398083,
-0.021081462502479553,
0.04818616434931755,
0.02956949733197689,
-0.018302425742149353,
0.019703585654497147,
0.02082747220993042,
-0.02915159799158573,
-0.02... | 0.827232 | 0.822391 | 0.004841 |
In a centipede game with 10 rounds, the payoffs at each round are as follows:
Player 1 receives $1 in round 1 and each subsequent round, the payoff increases by $1.
Player 2 receives $0 in round 1 and each subsequent round, the payoff decreases by $1.
What is the optimal strategy for each player and what is the final o... | tiger_lab_math | [
-0.008472195826470852,
0.0208656694740057,
-0.037111103534698486,
-0.019101809710264206,
0.02108113281428814,
-0.04010844975709915,
-0.03206552192568779,
0.0470510832965374,
0.03605111688375473,
-0.02423994056880474,
0.029798002913594246,
0.016805993393063545,
-0.057223279029130936,
-0.014... | 0.817196 | 0.812631 | 0.004565 |
In the diagram, the area of triangle $ABC$ is 27 square units. What is the area of triangle $BCD$?
[asy]
draw((0,0)--(32,0)--(9,15)--(0,0));
dot((0,0));
label("$A$",(0,0),SW);
label("6",(3,0),S);
dot((6,0));
label("$C$",(6,0),S);
label("26",(19,0),S);
dot((32,0));
label("$D$",(32,0),SE);
dot((9,15));
label("$B$",(9,... | tiger_lab_math | [
0.01724693551659584,
0.020830756053328514,
-0.05127331241965294,
-0.051779963076114655,
0.02886905148625374,
-0.03078307770192623,
-0.022396039217710495,
0.03737206757068634,
0.04608723893761635,
-0.005992971360683441,
0.02248283103108406,
0.03575817123055458,
-0.023849589750170708,
-0.023... | 0.827267 | 0.824533 | 0.002734 |
A basketball team won 40 games in their last competition. In the next competition, they won 5/8 times as many games as they won in their first competition. In the third competition, they won the same number of games as the sum of the first and second competition winnings. Calculate the total number of wins they had in ... | tiger_lab_math | [
0.007941360585391521,
0.011230614967644215,
-0.043044332414865494,
-0.04619818553328514,
0.04412965476512909,
-0.015467518009245396,
-0.03948260098695755,
0.045998409390449524,
0.029626648873090744,
-0.003417045809328556,
-0.004687043838202953,
0.02895805425941944,
-0.04322811961174011,
-0... | 0.824314 | 0.821752 | 0.002561 |
In the graph of $\frac{x^2+3x+2}{x^3+x^2-2x}$, let $a$ be the number of holes in the graph, $b$ be the number of vertical asympotes, $c$ be the number of horizontal asymptotes, and $d$ be the number of oblique asymptotes. Find $a+2b+3c+4d$. | tiger_lab_math | [
0.009930395521223545,
0.02494114451110363,
-0.06263310462236404,
-0.02229197695851326,
0.024584641680121422,
-0.03138824552297592,
-0.02526525966823101,
0.06282446533441544,
0.02851821482181549,
-0.021775849163532257,
-0.0037850940134376287,
0.019604139029979706,
-0.028416365385055542,
-0.... | 0.850488 | 0.833773 | 0.016715 |
There were students from three different schools taking part in a debate organized by the town's mayor on a stage. The first school had twice as many students as the second school, and the second school had 40 more students than the third school. If the third school had 200 students, and after the debate was over, ever... | tiger_lab_math | [
0.04847123473882675,
0.007793533615767956,
-0.05895170569419861,
-0.04722139611840248,
0.038274217396974564,
-0.036585185676813126,
-0.03608165681362152,
0.05033913254737854,
0.026095790788531303,
-0.021239718422293663,
0.015003674663603306,
0.051455359905958176,
-0.048356544226408005,
-0.... | 0.820436 | 0.819567 | 0.000869 |
Consider a simple graph with 10 vertices and 15 edges. Determine whether this graph is planar or not by applying a suitable planarity testing algorithm, and justify your answer. | tiger_lab_math | [
0.00718925055116415,
0.019038276746869087,
-0.014601628296077251,
-0.03819148615002632,
0.04214099422097206,
-0.02307899110019207,
-0.012255263514816761,
0.05176420882344246,
0.022529209032654762,
-0.03065592609345913,
0.016105806455016136,
0.015619918704032898,
-0.014941169880330563,
-0.0... | 0.807976 | 0.800859 | 0.007117 |
Consider the algebraic surface defined by the equation $y^2z^3 + 3x^2z^3 - x^3 = 0$.
Find the equation of the tangent plane to this surface at the point $(1,-1,-1)$. | tiger_lab_math | [
0.017198894172906876,
0.03249721974134445,
-0.05882586911320686,
-0.057738665491342545,
0.041275445371866226,
-0.018049677833914757,
-0.0048986379988491535,
0.06122704967856407,
0.03453890606760979,
-0.006732056848704815,
0.005289990920573473,
0.034552257508039474,
-0.0331583172082901,
-0.... | 0.834492 | 0.823231 | 0.011261 |
rectangular tile each of size 25 cm by 65 cm must be laid horizontally on a rectangular floor of size 150 cm by 390 cm , such that the tiles do not overlap and they are placed with edges jutting against each other on all edges . a tile can be placed in any orientation so long as its edges are parallel to the edges of f... | tiger_lab_math | [
0.020903613418340683,
0.023819835856556892,
-0.029387637972831726,
-0.017835957929491997,
0.027711110189557076,
-0.028530701994895935,
-0.0218721441924572,
0.046272482722997665,
0.025332439690828323,
-0.005884841550141573,
0.03429832309484482,
0.03746405243873596,
-0.04295525699853897,
-0.... | 0.812473 | 0.806247 | 0.006226 |
Consider an epidemic in a population of 100,000 people. The number of infected people at time t (in days) is given by the differential equation:
dy/dt = (0.03)y(1000-y)
where y(0) = 100 and y(t) represents the number of infected people at time t.
What is the maximum number of infected people during the epidemic, and... | tiger_lab_math | [
0.03485829755663872,
0.008771849796175957,
-0.04962065443396568,
-0.03132137656211853,
0.033709097653627396,
-0.03832735866308212,
-0.034501295536756516,
0.06499694287776947,
0.033481452614068985,
-0.005339816678315401,
0.028160303831100464,
0.05782673880457878,
-0.02400006167590618,
-0.00... | 0.816061 | 0.814951 | 0.001109 |
Given a 2-dimensional manifold with a metric tensor given by
```
g = dx^2 + dy^2 + 2y dx dy
```
Calculate the scalar curvature of the manifold at the point (1,1). | tiger_lab_math | [
-0.002771493513137102,
0.02930247038602829,
-0.02065480872988701,
-0.06929288804531097,
0.03800833225250244,
-0.022781159728765488,
-0.025227805599570274,
0.06616755574941635,
0.02579871192574501,
-0.0034573625307530165,
0.002867677714675665,
0.03187740221619606,
-0.03322405740618706,
-0.0... | 0.812803 | 0.809354 | 0.003449 |
Consider the following partial differential equation:
u_t = 2u_xx - u
with initial condition u(x,0) = sin(πx) and boundary conditions u(0,t) = u(1,t) = 0. Use finite element method with linear elements to approximate the solution of the PDE at t = 1.
What is the maximum error between the exact solution and the nume... | tiger_lab_math | [
0.008733128197491169,
0.029282651841640472,
-0.03463863953948021,
-0.045975565910339355,
0.039631810039281845,
-0.022823836654424667,
-0.008972405456006527,
0.07572109252214432,
0.008980914950370789,
0.0014607408083975315,
0.029788052663207054,
0.04194139316678047,
-0.01913284696638584,
-0... | 0.816947 | 0.815116 | 0.001831 |
Consider the variety in $\mathbb{P}^3$ defined by the equations
$$
\begin{aligned}
x_0^2+x_1^2+x_2^2+x_3^2&=0\\
x_0x_2-x_1x_3 &= 0
\end{aligned}
$$
Find its minimal resolution, that is, a resolution of singularities with the smallest number of blown up points. How many points do we need to blow up? | tiger_lab_math | [
-0.0031914215069264174,
0.025295009836554527,
-0.0370279997587204,
-0.03263749182224274,
0.034240689128637314,
-0.023105401545763016,
-0.014932583086192608,
0.06788013875484467,
0.03637709096074104,
-0.0036397615913301706,
-0.005164719186723232,
0.034313373267650604,
-0.014741524122655392,
... | 0.845637 | 0.825722 | 0.019915 |
A wire antenna of length 10 meters is excited with a current of 2 amps. The current distribution along the antenna is sinusoidal. Suppose that the antenna is surrounded by a medium with a refractive index of 1.5. At a distance of 100 meters from the antenna, determine the magnitude and phase of the electric field of th... | tiger_lab_math | [
0.025313332676887512,
0.040920719504356384,
-0.03763974830508232,
-0.05402142554521561,
0.02439369261264801,
-0.03658721223473549,
-0.028560388833284378,
0.07467091083526611,
0.022138718515634537,
-0.0009664957760833204,
0.010888753458857536,
0.013480211608111858,
-0.02801624871790409,
-0.... | 0.794986 | 0.790568 | 0.004418 |
Determine the Fourier series expansion of the function f(x) = x on the interval -π < x < π, and use it to solve the partial differential equation ∂u/∂t = ∂^2u/∂x^2 with the initial condition u(x,0) = 2x. | tiger_lab_math | [
-0.005952463950961828,
0.021814599633216858,
-0.050419826060533524,
-0.03332648053765297,
0.0349762961268425,
-0.02509968914091587,
-0.00577573711052537,
0.06869028508663177,
0.0592958889901638,
-0.00015482644084841013,
0.025978123769164085,
0.024815451353788376,
-0.025953685864806175,
-0.... | 0.818788 | 0.814149 | 0.004639 |
Let $f(n)$ be the integer closest to $\sqrt[4]{n}.$ Find $\sum_{k=1}^{1995}\frac 1{f(k)}.$ Please write a program to solve it | tiger_lab_math | [
0.019989877939224243,
0.045364443212747574,
-0.04612565040588379,
-0.042209018021821976,
0.023785781115293503,
-0.019686786457896233,
-0.010068976320326328,
0.04216175898909569,
0.02453027479350567,
-0.011924433521926403,
0.02696082927286625,
0.024878185242414474,
-0.027430105954408646,
-0... | 0.852174 | 0.846819 | 0.005355 |
Consider the wave equation $u_{tt} = c^2 u_{xx}$ with initial conditions $u(x,0) = \sin(2\pi x)$ and $u_t(x,0) = 0$. Assume that $c = 2$ and $L = 1$, where $L$ is the length of the interval $[0, L]$. Find the expression for $u(x,t)$ in terms of the Fourier series. | tiger_lab_math | [
0.005729402881115675,
0.04988889768719673,
-0.0580262616276741,
-0.02798706293106079,
0.04115404188632965,
-0.01893775351345539,
-0.027517663314938545,
0.0885317474603653,
0.046589143574237823,
0.0055656395852565765,
0.03912987560033798,
0.020360099151730537,
-0.02775486186146736,
-0.02516... | 0.823097 | 0.808571 | 0.014526 |
Here is a numerical problem for a math student to solve related to the Chirikov-Taylor map:
Suppose we have a Chirikov-Taylor map with the parameter value of k = 2. The initial condition is x(0) = 0.1 and p(0) = 0.3, and we iterate the map for 500 steps. What are the final values of x and p?
(Note: The Chirikov-Taylo... | tiger_lab_math | [
0.03769790753722191,
0.016774943098425865,
-0.04992128536105156,
-0.04398510977625847,
0.04642244800925255,
-0.014565532095730305,
-0.02233659289777279,
0.07812570780515671,
0.033824145793914795,
-0.0027044471353292465,
0.003501950530335307,
0.02094799093902111,
-0.03352014347910881,
-0.01... | 0.831699 | 0.819969 | 0.01173 |
Using the Fourier transform, calculate the probability density of an electron in a one-dimensional infinite square well with a width of 2 nm and an energy level of 15 eV. The potential function is zero within the well and infinite outside of it. The mass of the electron is 9.11 x 10^-31 kg and Planck's constant is 6.62... | tiger_lab_math | [
0.01557860430330038,
0.052556298673152924,
-0.032442186027765274,
-0.04540979489684105,
0.039435259997844696,
-0.0405055470764637,
-0.03757619857788086,
0.0497584268450737,
0.021683992817997932,
0.00042610778473317623,
0.02371070720255375,
0.025043776258826256,
-0.02172953449189663,
-0.023... | 0.803379 | 0.801921 | 0.001458 |
What is the Euler characteristic of the simplicial complex formed by gluing together two tetrahedra along a common face?
Hint: The Euler characteristic, denoted as $\chi$, of a simplicial complex is defined as the alternating sum of the number of k-simplices. That is, $\chi = \sum_{k=0}^n (-1)^k b_k$, where $b_k$ is t... | tiger_lab_math | [
0.007304310332983732,
0.030273238196969032,
-0.01329251378774643,
-0.05757373198866844,
0.03612745180726051,
0.0028090684209018946,
-0.023917444050312042,
0.07295522838830948,
0.015883540734648705,
0.006851083133369684,
0.011563864536583424,
0.04587088152766228,
-0.02438448928296566,
0.002... | 0.810082 | 0.80186 | 0.008222 |
Suppose that the stock price of a company follows a stochastic differential equation given by
$$dS = \mu S dt + \sigma S dB$$
where $S$ is the stock price, $\mu$ is the drift coefficient, $\sigma$ is the diffusion coefficient, $t$ is time, and $dB$ is a Brownian motion. If the current stock price is $S_0=100$, $\mu=0.0... | tiger_lab_math | [
-0.011250922456383705,
0.004967735148966312,
-0.04853317514061928,
-0.041488438844680786,
0.03175146505236626,
-0.014322251081466675,
-0.041549067944288254,
0.07475777715444565,
0.04199831560254097,
-0.0031925239600241184,
0.019091572612524033,
0.03029645048081875,
-0.04552474990487099,
-0... | 0.818466 | 0.812559 | 0.005907 |
In Perfectville, the streets are all $20$ feet wide and the blocks they enclose are all squares of side length $400$ feet, as shown. Sarah runs around the block on the $400$-foot side of the street, while Sam runs on the opposite side of the street. How many more feet than Sarah does Sam run for every lap around the ... | tiger_lab_math | [
0.017887147143483162,
0.05009724944829941,
-0.04767238721251488,
-0.0194646455347538,
0.0460628978908062,
-0.03723382577300072,
-0.03607171028852463,
0.05109553411602974,
0.03805258870124817,
-0.016076406463980675,
0.012078465893864632,
0.032647114247083664,
-0.02794596366584301,
-0.019036... | 0.836187 | 0.828046 | 0.008141 |
Consider the elliptic curve $y^2 = x^3 + 3x^2 + x$ over the field $\mathbb{Q}$ of rational numbers. Find the rational points on this curve. | tiger_lab_math | [
0.0058891307562589645,
0.03256799653172493,
-0.03742507845163345,
-0.05170620605349541,
0.027174655348062515,
-0.043079011142253876,
0.004289698321372271,
0.044033508747816086,
0.040776874870061874,
-0.04293075203895569,
0.011507253162562847,
0.027218760922551155,
-0.041972219944000244,
-0... | 0.846788 | 0.832277 | 0.01451 |
Consider the partial differential equation:
(1 + x^2) ∂u/∂x + (2xy) ∂u/∂y = 0
Find the general solution of this PDE using the method of characteristics. | tiger_lab_math | [
0.00513097457587719,
0.015825485810637474,
-0.03099009580910206,
-0.04992974177002907,
0.044942449778318405,
-0.029377853497862816,
0.0036560287699103355,
0.06956726312637329,
0.04300156980752945,
-0.018488207831978798,
0.004628408700227737,
0.043534792959690094,
-0.010690310038626194,
-0.... | 0.81863 | 0.808742 | 0.009888 |
Consider two firms, A and B, that are competing in a Cournot market with demand given by P(Q) = 96 - 4Q, where Q = qA + qB. Both firms have the same constant marginal cost of $10 per unit. Firm A currently produces 20 units of output. How much output should firm B produce to maximize their profit? | tiger_lab_math | [
0.009334220550954342,
0.015488166362047195,
-0.05958028510212898,
-0.028100045397877693,
0.014958127401769161,
-0.010447106324136257,
-0.024172890931367874,
0.064247265458107,
0.02908707782626152,
-0.027251632884144783,
0.0010625607101246715,
0.03703805059194565,
-0.04492998495697975,
-0.0... | 0.813275 | 0.810715 | 0.00256 |
Consider the following partial differential equation:
$\frac{\partial u} {\partial t} = k\frac{\partial^2 u} {\partial x^2} $
where $k$ is a constant and $u(x,t)$ is the unknown function.
Using a finite element method with a time-discretization scheme, solve the above equation on the domain $[0,1]$ with homogeneous ... | tiger_lab_math | [
-0.0038346939254552126,
0.04407583922147751,
-0.03739936277270317,
-0.033701278269290924,
0.05290565639734268,
-0.000008183227691915818,
-0.005552995018661022,
0.09305363148450851,
0.03639505058526993,
0.0140928840264678,
0.025775404646992683,
0.038069482892751694,
-0.004896909464150667,
-... | 0.8088 | 0.789385 | 0.019415 |
A square and a regular heptagon are coplanar and share a common side $\overline{AD}$, as shown. What is the degree measure of exterior angle $BAC$? Express your answer as a common fraction.
[asy]
for(int i=0; i <=7; ++i) {
draw(dir(360*i/7+90)--dir(360*(i+1)/7+90));
}
pair A = dir(360*3/7+90);
pair F = dir(360*4/7+9... | tiger_lab_math | [
0.0009305429994128644,
0.02040642872452736,
-0.048260290175676346,
-0.0629444494843483,
0.027484288439154625,
-0.005904588382691145,
-0.007732801139354706,
0.06883557885885239,
0.028998417779803276,
-0.010090765543282032,
0.015194561332464218,
0.03185553103685379,
-0.04282933846116066,
-0.... | 0.836046 | 0.823094 | 0.012952 |
Suppose a group of 10 farmers owns a common land of 100 acres where they can graze their cattle. Each farmer has a maximum capacity of 15 cows per acre. The total profit from one cow per year is $1,000. However, each additional cow that grazes on the land beyond its limit causes $1,000 of environmental degradation to t... | tiger_lab_math | [
0.008144625462591648,
0.019013425335288048,
-0.042433325201272964,
-0.023910287767648697,
0.007900937460362911,
-0.04716945067048073,
-0.022497693076729774,
0.06247389316558838,
0.016740070655941963,
-0.026965873315930367,
-0.007728189695626497,
0.03284080699086189,
-0.0344068817794323,
-0... | 0.807876 | 0.801744 | 0.006132 |
Sure, here's a numerical problem for a math student to solve:
Suppose the population of a city is currently 50,000 and its growth rate is proportional to the current population with a proportionality constant of 0.02. Write a differential equation to model the city's population growth and use it to predict the populat... | tiger_lab_math | [
0.01587887480854988,
0.019392330199480057,
-0.03673434630036354,
-0.04236957058310509,
0.03129126876592636,
-0.021551448851823807,
-0.029814615845680237,
0.051513612270355225,
0.05499234423041344,
-0.011668278835713863,
0.009633732959628105,
0.032388415187597275,
-0.039780061691999435,
-0.... | 0.817804 | 0.816783 | 0.001021 |
Consider a regular tetrahedron with side length 2. Find the total surface area of the tetrahedron after it has been subdivided into 8 smaller congruent tetrahedra using a regular triangular pyramid as a base, each of which shares an edge with the original tetrahedron. | tiger_lab_math | [
0.00933076348155737,
0.030789857730269432,
-0.033254776149988174,
-0.044503118842840195,
0.030121272429823875,
-0.028515012934803963,
-0.03895886614918709,
0.06486824154853821,
0.024606460705399513,
-0.009080643765628338,
0.014037667773663998,
0.020153023302555084,
-0.045340895652770996,
-... | 0.822326 | 0.809719 | 0.012607 |
Here's a combinatorial optimization problem for a math student to solve:
Suppose you have a set of 10 integers, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, and you want to select a subset of size 5 such that the sum of the numbers in the subset is minimized. What is the smallest possible sum you can achieve? | tiger_lab_math | [
0.027094785124063492,
0.01588720828294754,
-0.05278276652097702,
-0.032991040498018265,
0.039554744958877563,
-0.015518060885369778,
-0.03217252716422081,
0.04328141361474991,
0.037921249866485596,
-0.022007174789905548,
0.009912383742630482,
0.010238885879516602,
-0.043043069541454315,
-0... | 0.839391 | 0.831343 | 0.008048 |
A telecommunications company wants to test different combinations of three factors - A, B, and C - to determine the best combination that minimizes signal interference. Each factor has two possible levels: high (+1) and low (-1).
Using an orthogonal array, design an experiment to test all possible combinations of the... | tiger_lab_math | [
0.01617523841559887,
0.007831361144781113,
-0.057016514241695404,
-0.03810642659664154,
0.036037083715200424,
-0.004192075692117214,
-0.015804734081029892,
0.06994666904211044,
0.03066098317503929,
-0.001221120823174715,
-0.004569037351757288,
0.0370628722012043,
-0.033922065049409866,
-0.... | 0.825451 | 0.81615 | 0.009301 |
Here is a numerical problem related to robust optimization in mathematics:
A manufacturer produces two types of smartphones, A and B. The profit from each phone is $30 for type A and $50 for type B. The manufacturer has a total of 200 hours of labor available per day for production. Each unit of type A requires 2 hour... | tiger_lab_math | [
0.025823542848229408,
0.011971259489655495,
-0.051052749156951904,
-0.02503257803618908,
0.024508828297257423,
-0.004389592446386814,
-0.04097292199730873,
0.07022838294506073,
0.02719203382730484,
-0.011412757448852062,
-0.00607023760676384,
0.030108675360679626,
-0.010407392866909504,
-0... | 0.823484 | 0.800071 | 0.023413 |
Let \[f(x) = \left\{
\begin{array}{cl} ax+3, &\text{ if }x>2, \\
x-5 &\text{ if } -2 \le x \le 2, \\
2x-b &\text{ if } x <-2.
\end{array}
\right.\]Find $a+b$ if the piecewise function is continuous (which means that its graph can be drawn without lifting your pencil from the paper). Please respond by writing a program ... | tiger_lab_math | [
0.01014839019626379,
0.03195784613490105,
-0.04474663734436035,
-0.027744505554437637,
0.027703523635864258,
-0.011760782450437546,
-0.015602605417370796,
0.08293236047029495,
0.027338717132806778,
-0.007680233102291822,
0.027308586984872818,
0.046413108706474304,
-0.013401266187429428,
-0... | 0.824611 | 0.816563 | 0.008049 |
To raise funds for her local soup kitchen, Didi enlisted the help of her family, friends, and neighbors. They donated 10 same-size cakes that she sliced into 8 slices per cake and started selling a slice for $1. A local business owner was so impressed by Didi's efforts that she offered to donate 50 cents for each slice... | tiger_lab_math | [
0.0009640270145609975,
0.023929936811327934,
-0.02704751305282116,
-0.03193000704050064,
0.01219921838492155,
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-0.02550671249628067,
0.04690016806125641,
0.033879272639751434,
-0.012702913023531437,
0.025420015677809715,
0.0211802925914526,
-0.05824188143014908,
-0.02... | 0.80406 | 0.796254 | 0.007806 |
Consider the partial differential equation:
$\frac{\partial^2 u}{\partial t^2} = 4 \frac{\partial^2 u}{\partial x^2}$
with the boundary conditions $u(0,t) = u(\pi,t) = 0$ and the initial conditions $u(x,0) = \sin(2x)$ and $\frac{\partial u}{\partial t}(x,0) = 0$.
Use Fourier series to find the solution to this parti... | tiger_lab_math | [
0.004589663352817297,
0.044166840612888336,
-0.04370873048901558,
-0.0297858864068985,
0.04048905894160271,
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-0.0060663241893053055,
0.07150377333164215,
0.06766054779291153,
0.007349719293415546,
0.030351705849170685,
0.029080452397465706,
-0.01801084168255329,
-0.03... | 0.823828 | 0.807474 | 0.016354 |
Let M be a connected 2-dimensional manifold which is an embedded in the Euclidean space $R^4$. If M has genus 3 (i.e. three handles), what is the Euler characteristic of M? | tiger_lab_math | [
0.003263693070039153,
0.02526123635470867,
-0.022665053606033325,
-0.061489738523960114,
0.03334596008062363,
-0.00028620113153010607,
-0.019606491550803185,
0.07080474495887756,
0.018379298970103264,
-0.0016914240550249815,
0.009074314497411251,
0.059633467346429825,
-0.020715242251753807,
... | 0.819918 | 0.818201 | 0.001717 |
A factory produces two types of products, Product A and Product B. The profit obtained from selling Product A is $6 per unit, and the profit obtained from selling Product B is $8 per unit. The factory can produce a maximum of 200 units per week due to resource constraints. An order for at least 120 units of Product A m... | tiger_lab_math | [
0.0070423828437924385,
0.024372920393943787,
-0.04689403995871544,
0.012744840234518051,
0.003391620935872197,
-0.02148323878645897,
-0.05596785992383957,
0.08192429691553116,
0.0408649817109108,
-0.015368682332336903,
-0.005048678256571293,
0.048303212970495224,
-0.008186714723706245,
-0.... | 0.797895 | 0.783055 | 0.01484 |
Consider a graph G, which has 8 vertices and 11 edges. Determine whether the graph has any cut vertices or bridges. If so, identify them and explain why they are cut vertices or bridges. | tiger_lab_math | [
0.011577912606298923,
0.011035025119781494,
-0.009506238624453545,
-0.026310674846172333,
0.03229563310742378,
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0.04773814603686333,
0.02527817152440548,
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0.016631558537483215,
0.01253120880573988,
-0.05189632251858711,
-0... | 0.801639 | 0.798757 | 0.002882 |
Consider a surface given by the equation f(x,y,z) = x^3 - 3xy^2 + 3yz^2 - z^3. Calculate the sectional curvature of the surface at the point (-1,1,1) in the direction of the vectors (1,0,1) and (0,−1,1). | tiger_lab_math | [
0.013452786020934582,
0.03190051019191742,
-0.04601278528571129,
-0.05604715272784233,
0.03947405144572258,
-0.0030362107791006565,
-0.008545836433768272,
0.06078352406620979,
0.00546244578436017,
-0.002248616423457861,
0.027283083647489548,
0.03575444594025612,
-0.04198095202445984,
-0.00... | 0.8196 | 0.813282 | 0.006317 |
There exist vectors $\mathbf{a}$ and $\mathbf{b}$ such that
\[\mathbf{a} + \mathbf{b} = \begin{pmatrix} 6 \\ -3 \\ -6 \end{pmatrix},\]where $\mathbf{a}$ is parallel to $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix},$ and $\mathbf{b}$ is orthogonal to $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}.$ Find $\mathbf{b}.$ | tiger_lab_math | [
0.03640488162636757,
0.0522681400179863,
-0.04050096496939659,
-0.034050654619932175,
0.05584658682346344,
0.020207660272717476,
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0.07702330499887466,
0.027106286957859993,
-0.01434833463281393,
0.010491223074495792,
0.03363360837101936,
-0.02170415222644806,
-0.03365... | 0.838231 | 0.822946 | 0.015285 |
Consider the following optimization problem:
Minimize the function f(x) = x^2 - 5x + 6 using cuckoo search algorithm over the interval [0,6].
Use an initial population size of 15 cuckoos, a maximum number of generations of 1000, a step size of 0.01, and a probability of cuckoo eggs being laid in a new nest of 0.25.... | tiger_lab_math | [
0.0003946085926145315,
0.038313254714012146,
-0.051557332277297974,
-0.02338229864835739,
0.041257213801145554,
-0.03153946250677109,
-0.038122568279504776,
0.06567180156707764,
0.04418577626347542,
-0.017326347529888153,
-0.0005875012720935047,
0.024721425026655197,
-0.007185147609561682,
... | 0.821136 | 0.81447 | 0.006666 |
A point $(x,y)$ is a distance of 12 units from the $x$-axis. It is a distance of 10 units from the point $(1,6)$. It is a distance $n$ from the origin. Given that $x>1$, what is $n$? | tiger_lab_math | [
0.034439872950315475,
0.01774703711271286,
-0.0500289611518383,
-0.036516718566417694,
0.041318606585264206,
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-0.024906853213906288,
0.043590396642684937,
0.05810682103037834,
-0.011551053263247013,
0.02017221972346306,
0.03680044412612915,
-0.04117148369550705,
-0.01... | 0.843548 | 0.839611 | 0.003937 |
The equation $\sin^2 x + \sin^2 2x + \sin^2 3x + \sin^2 4x = 2$ can be reduced to the equivalent equation
\[\cos ax \cos bx \cos cx = 0,\]for some positive integers $a,$ $b,$ and $c.$ Find $a + b + c.$ | tiger_lab_math | [
0.021137230098247528,
0.030537959188222885,
-0.0569421760737896,
-0.04003400355577469,
0.04305761307477951,
-0.01135586854070425,
-0.009073678404092789,
0.07217055559158325,
0.0523918978869915,
0.008027993142604828,
0.016783474013209343,
0.01975913532078266,
-0.03011562116444111,
-0.034755... | 0.835724 | 0.819129 | 0.016595 |
Consider the following partial differential equation:
𝑢𝑡 = 𝑢𝑥𝑥 + 2𝑢𝑦𝑦 + sin(𝑥)cos(𝑦) on the rectangular domain [0, π] × [0, 1].
Apply the finite element method using linear triangular elements with 𝑅 = {(𝑥, 𝑦) ∈ ℝ^2 | 𝑥 ∈ [0, 𝜋], 𝑦 ∈ [0, 1]} to find an approximation to the solution 𝑢(𝑥, 𝑦) with �... | tiger_lab_math | [
0.007801548577845097,
0.034050196409225464,
-0.03385084122419357,
-0.05461915209889412,
0.04048950597643852,
-0.01605246588587761,
-0.0030716490000486374,
0.07607021927833557,
0.021769646555185318,
0.02320866659283638,
0.04109634459018707,
0.05653655156493187,
-0.017549386247992516,
-0.010... | 0.811102 | 0.792661 | 0.018441 |
Using the finite element method, calculate the solution to the following partial differential equation:
∂²u/∂x² + ∂²u/∂y² - 4u = f(x,y)
where f(x,y) = e^(-x) * sin(y) on the square region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 with boundary conditions:
u(0,y) = u(1,y) = u(x,0) = 0
u(x,1) = e^(-x) * sin(1)
Use a uniform grid with h... | tiger_lab_math | [
-0.005223501473665237,
0.031659118831157684,
-0.03606713190674782,
-0.05711124837398529,
0.042553164064884186,
-0.021802078932523727,
-0.008600370027124882,
0.08476969599723816,
0.03548102453351021,
0.014389709569513798,
0.02465951256453991,
0.04585830867290497,
-0.024462100118398666,
-0.0... | 0.816968 | 0.809269 | 0.007699 |
there is a rectangular prism made of 1 in cubes that has been covered in tin foil . there are exactly 128 cubes that are not touching any tin foil on any of their sides . if the width of the figure created by these 128 cubes is twice the length and twice the height , what is the measure m in inches of the width of the ... | tiger_lab_math | [
0.012835209257900715,
0.03475997596979141,
-0.043493255972862244,
-0.044212792068719864,
0.03911099582910538,
-0.034262463450431824,
-0.038178812712430954,
0.05902230739593506,
0.026772242039442062,
-0.003180439118295908,
0.018926095217466354,
0.03935455158352852,
-0.004175812937319279,
-0... | 0.831897 | 0.827247 | 0.00465 |
"Modulo $m$ graph paper" consists of a grid of $m^2$ points, representing all pairs of integer residues $(x,y)$ where $0\le x, y <m$. To graph a congruence on modulo $m$ graph paper, we mark every point $(x,y)$ that satisfies the congruence. For example, a graph of $y\equiv x^2\pmod 5$ would consist of the points $(0,0... | tiger_lab_math | [
0.02793637476861477,
0.028845615684986115,
-0.029757967218756676,
-0.04940253868699074,
0.048191554844379425,
-0.03993634134531021,
-0.03431394696235657,
0.0646183043718338,
0.04926123097538948,
-0.030027195811271667,
0.003989631310105324,
0.027928248047828674,
-0.020778000354766846,
-0.00... | 0.817253 | 0.80135 | 0.015903 |
Oliver is trying to decide between two different sized boxes of macaroni and cheese. He wants to know which is the better value, so instead of just looking at the overall price, he looks at the price per ounce. The larger box is 30 ounces and the smaller box is 20 ounces. The larger box costs $4.80. The smaller box cos... | tiger_lab_math | [
0.0053289495408535,
0.018107961863279343,
-0.04717991501092911,
-0.018577737733721733,
0.036505796015262604,
-0.01762664131820202,
-0.035765331238508224,
0.05772177502512932,
0.05326889827847481,
-0.006779084447771311,
0.02772311121225357,
0.01464332826435566,
-0.05631180480122566,
-0.0021... | 0.801966 | 0.801157 | 0.000809 |
Let A = {2, 3, 4, 5} and B = {3, 5, 7, 11} be two sets. Use intuitionistic logic to determine whether the following statement is true or false: "There exists an element in A that is not in B." | tiger_lab_math | [
0.03756902739405632,
0.006668411195278168,
-0.03149815648794174,
-0.028223633766174316,
0.04890240728855133,
-0.03381304815411568,
0.001101129106245935,
0.07673899829387665,
0.041768401861190796,
-0.030898770317435265,
0.019248098134994507,
0.0221930630505085,
-0.016361085698008537,
-0.022... | 0.816265 | 0.808693 | 0.007572 |
A fluid flows through a straight circular pipe with a radius of 0.02 m. The fluid has a density of 1000 kg/m^3 and a viscosity of 0.01 Pa*s. The maximum velocity of the fluid is 2 m/s. Determine the pressure drop (in Pa/m) along the length of the pipe if the length of the pipe is 10 m. Use the Navier-Stokes equation to... | tiger_lab_math | [
-0.016612010076642036,
0.046795811504125595,
-0.034567419439554214,
-0.023922011256217957,
0.03290639817714691,
-0.04545985534787178,
-0.04840322583913803,
0.06943760067224503,
0.028699826449155807,
0.021130817010998726,
0.01939287967979908,
-0.008049525320529938,
-0.012041616253554821,
-0... | 0.794678 | 0.792305 | 0.002372 |
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
$\textbf{(A)}\ 2128 \qquad\textbf{(B)}\ 2148 \qquad\textbf{(C)}\ 2160 \qquad\textbf{(D)}\ 2200 \qquad\textbf{(E)}\ 2300$
Please respond by writing a pr... | tiger_lab_math | [
0.01057257130742073,
0.025064561516046524,
-0.0488576777279377,
-0.0304556954652071,
0.02282806672155857,
-0.02821502462029457,
-0.032859861850738525,
0.028609955683350563,
0.03172275051474571,
-0.0006484872428700328,
0.02201469987630844,
0.04712073877453804,
-0.023916149511933327,
-0.0144... | 0.838858 | 0.82836 | 0.010499 |
The values of $a$, $b$, $c$ and $d$ are 1, 2, 3 and 4, but not necessarily in that order. What is the largest possible value of the sum of the four products $ab$, $bc$, $cd$ and $da$? Let's write a program. | tiger_lab_math | [
0.0394832082092762,
0.025131627917289734,
-0.046556416898965836,
-0.02142396569252014,
0.04780241474509239,
-0.008421947248280048,
-0.022224638611078262,
0.04833231493830681,
0.04751357063651085,
-0.02070583403110504,
0.020963672548532486,
0.04850029572844505,
-0.03849831223487854,
-0.0556... | 0.830021 | 0.828591 | 0.00143 |
The diagonal of a particular square is 5 inches. The diameter of a particular circle is also 5 inches. By how many square inches is the area of the circle greater than the area of square? Express your answer as a decimal to the nearest tenth. [asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((2,0)--(0,2));
draw(c... | tiger_lab_math | [
0.007008879445493221,
0.047904543578624725,
-0.043828386813402176,
-0.0450131818652153,
0.033319611102342606,
-0.016819598153233528,
-0.026245424523949623,
0.055388953536748886,
0.02158285863697529,
-0.008352439850568771,
0.03255787119269371,
0.029748445376753807,
-0.04128317907452583,
-0.... | 0.830981 | 0.828925 | 0.002056 |
cubes with each side one inch long are glued together to form a larger cube . the larger cube ' s face is painted with red color and the entire assembly is taken apart . 26 small cubes are found with no paints on them . how many of unit cubes have at least one face that is painted red ? Let's write a program. | tiger_lab_math | [
0.018368827179074287,
0.038849834352731705,
-0.03385591134428978,
-0.016842368990182877,
0.03357991576194763,
-0.03886354714632034,
-0.024001678451895714,
0.05207188054919243,
0.01547336857765913,
-0.011539764702320099,
0.01965627446770668,
0.03462450951337814,
-0.019801782444119453,
-0.02... | 0.813445 | 0.810203 | 0.003242 |
Sure! Here's a precise numerical problem for a math student to solve:
You are managing a warehouse with 200 items that have varying demand rates. The demand for each item follows a normal distribution with a mean of 40 and a standard deviation of 15. You can only store up to 250 items in the warehouse, and each item c... | tiger_lab_math | [
0.032220158725976944,
0.006233179476112127,
-0.04881138354539871,
-0.019573571160435677,
0.03463609144091606,
-0.017632093280553818,
-0.03858625888824463,
0.03656267002224922,
0.028879720717668533,
-0.018954560160636902,
0.009590549394488335,
0.021410329267382622,
-0.026881473138928413,
-0... | 0.822388 | 0.808856 | 0.013533 |
Three numbers, $a_1\,$, $a_2\,$, $a_3\,$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 1000\}\,$. Three other numbers, $b_1\,$, $b_2\,$, $b_3\,$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $p\,$ be the probability that, after a suitable rotat... | tiger_lab_math | [
0.0025727548636496067,
0.03132268041372299,
-0.046705324202775955,
-0.017158495262265205,
0.038388267159461975,
-0.00032815500162541866,
-0.02355554886162281,
0.03491879627108574,
0.026013340801000595,
0.000709241081494838,
-0.009748481214046478,
0.03263477608561516,
-0.02529427595436573,
... | 0.846561 | 0.823579 | 0.022982 |
Let $f(z) = \frac{z}{z^2 + 1}$ be a meromorphic function defined on the complex plane. Find the poles and zeroes of $f(z)$ and state their orders. | tiger_lab_math | [
0.01212222408503294,
0.028830939903855324,
-0.03875076770782471,
-0.05317402631044388,
0.03513695299625397,
-0.00573973823338747,
0.0012196081224828959,
0.03854835405945778,
0.03602816164493561,
-0.002558161271736026,
0.02534482628107071,
0.019188672304153442,
-0.03355029597878456,
-0.0173... | 0.829175 | 0.817377 | 0.011798 |
"statement 1": Susan , John , Daisy , Tim and Kim need to be seated in more than 4 identical chairs in straight line so that Susan is seated always left to Tim, "statement 2" :Susan , John , Daisy , Tim and Kim need to be seated in 5 identical chairs in straight line so that Susan is seated always left to Tim, "option... | tiger_lab_math | [
0.023864522576332092,
0.009721828624606133,
-0.047231726348400116,
-0.038715213537216187,
0.04435057193040848,
-0.03087679296731949,
-0.013558776117861271,
0.05282771959900856,
0.02383948490023613,
-0.009686651639640331,
0.02160453610122204,
0.028495263308286667,
-0.00928893405944109,
-0.0... | 0.814152 | 0.806725 | 0.007426 |
Consider a graph with 6 vertices and 9 edges. Determine if it has a perfect matching, and if so, identify one. | tiger_lab_math | [
0.02265499345958233,
0.0035371368285268545,
-0.03665870428085327,
-0.026364319026470184,
0.02868402749300003,
-0.04210298880934715,
-0.011461835354566574,
0.024537328630685806,
0.03509897738695145,
-0.04074135422706604,
0.03255664184689522,
0.011594750918447971,
-0.009326819330453873,
-0.0... | 0.818759 | 0.812746 | 0.006013 |
Consider the function $f(x,y) = x^2 + y^2 - x^3 + xy^2$.
(a) Find all critical points of $f(x,y)$.
(b) Determine the index of each critical point found in part (a).
(c) Use Morse theory to calculate the number of critical points of $f(x,y)$ with index less than or equal to 1. | tiger_lab_math | [
0.013087950646877289,
0.0309061910957098,
-0.038395170122385025,
-0.029091212898492813,
0.034842513501644135,
-0.018620066344738007,
-0.04520221799612045,
0.06177959591150284,
0.020611533895134926,
-0.005975071340799332,
0.003922926727682352,
0.029789593070745468,
-0.032771997153759,
-0.01... | 0.849918 | 0.831882 | 0.018036 |
Consider the wave equation given by $\frac{\partial^2 u}{\partial t^2} = 4\frac{\partial^2 u}{\partial x^2}$, where $u(x,t)$ is a complex-valued function. Assume that at $t=0$, we have $u(x,0) = \cos(x)$ and $\frac{\partial u}{\partial t}(x,0) = 0$. Solve for $u(x,t)$ using complex analysis techniques. Find $u(\frac{3\... | tiger_lab_math | [
0.024851402267813683,
0.05297026410698891,
-0.05219034105539322,
-0.03184836357831955,
0.041400596499443054,
-0.015914659947156906,
-0.007730241399258375,
0.08280401676893234,
0.037463270127773285,
0.008539140224456787,
0.04206022247672081,
0.018848123028874397,
-0.01792469061911106,
-0.02... | 0.816488 | 0.803478 | 0.01301 |
The quantity
\[\frac{\tan \frac{\pi}{5} + i}{\tan \frac{\pi}{5} - i}\]is a tenth root of unity. In other words, it is equal to $\cos \frac{2n \pi}{10} + i \sin \frac{2n \pi}{10}$ for some integer $n$ between 0 and 9 inclusive. Which value of $n$? | tiger_lab_math | [
0.024669701233506203,
0.04496688395738602,
-0.04570480063557625,
-0.025395113974809647,
0.026620017364621162,
-0.03069106675684452,
-0.01846727542579174,
0.06732585281133652,
0.03318675234913826,
0.007701558526605368,
0.015554402954876423,
0.013757418841123581,
-0.030120646581053734,
0.006... | 0.825852 | 0.820895 | 0.004957 |
a square is drawn inside a right - angled triangle with the two perpendicular sides as 12 cm and 8 cm . what is the side of the largest possible square that can be drawn ? Please respond by writing a program in Python. | tiger_lab_math | [
0.032290056347846985,
0.044734224677085876,
-0.026238571852445602,
-0.03598897531628609,
0.02981383539736271,
-0.010058018378913403,
-0.02903258055448532,
0.05391755700111389,
0.03519129008054733,
-0.004050761461257935,
0.034033942967653275,
0.04848487302660942,
-0.025436047464609146,
-0.0... | 0.817965 | 0.808083 | 0.009882 |
Find the number of rational points on the elliptic curve defined by the equation: $y^2 = x^3 - 5x + 7$ in $\mathbb{Q}$. | tiger_lab_math | [
0.007701698690652847,
0.026053791865706444,
-0.04478572681546211,
-0.05253968387842178,
0.027252331376075745,
-0.029231799766421318,
-0.005204048473387957,
0.042555417865514755,
0.04713044688105583,
-0.04151501879096031,
0.0074991388246417046,
0.03491764888167381,
-0.039163701236248016,
-0... | 0.841761 | 0.840877 | 0.000884 |
A manufacturing company produces two types of products, A and B. The company has a production capacity of 500 units per day with a labor force of 50 workers. Product A requires 2 hours of labor and 3 hours of machine time, while product B requires 5 hours of labor and 2 hours of machine time. The profit margin of produ... | tiger_lab_math | [
0.018239492550492287,
0.008319215849041939,
-0.06695742160081863,
-0.006090148352086544,
0.018603889271616936,
-0.01047526765614748,
-0.04433848708868027,
0.09206142276525497,
0.03182203695178032,
-0.010550097562372684,
-0.0015530935488641262,
0.034871749579906464,
-0.005974729545414448,
-... | 0.804937 | 0.803459 | 0.001478 |
Consider an m * n matrix A and an n * m matrix B (with n != m) such that AB = I_m. Are the columns of A linearly independent? Please respond by writing a program in Python. | tiger_lab_math | [
0.02584042213857174,
0.021332792937755585,
-0.045611120760440826,
-0.04543794319033623,
0.04902570694684982,
-0.03987609222531319,
0.0075302571058273315,
0.05340888351202011,
0.02521916665136814,
-0.03381723165512085,
0.010346440598368645,
0.0178105216473341,
-0.036035262048244476,
-0.0080... | 0.812427 | 0.810829 | 0.001598 |
Consider the following optimization problem:
minimize f(x) = -x^2 + 5x - 6
using the simulated annealing method, where x is a real number.
Find the optimal solution x* using simulated annealing algorithm with the following parameters:
Initial temperature: T = 100
Temperature reduction factor: alpha = 0.99
N... | tiger_lab_math | [
0.010344256646931171,
0.016369862481951714,
-0.04018212854862213,
-0.0299335066229105,
0.03767719119787216,
-0.03613459691405296,
-0.018219230696558952,
0.08569643646478653,
0.034338004887104034,
-0.00692157493904233,
0.010669238865375519,
0.019501851871609688,
-0.018795860931277275,
-0.00... | 0.808427 | 0.806219 | 0.002207 |
The points $P,$ $Q,$ and $R$ are represented by the complex numbers $z,$ $(1 + i) z,$ and $2 \overline{z},$ respectively, where $|z| = 1.$ When $P,$ $Q$, and $R$ are not collinear, let $S$ be the fourth vertex of the parallelogram $PQSR.$ What is the maximum distance between $S$ and the origin of the complex plane? L... | tiger_lab_math | [
0.029581138864159584,
0.03380398824810982,
-0.04617645964026451,
-0.044751644134521484,
0.027435461059212685,
-0.006641372572630644,
-0.018949788063764572,
0.03818043693900108,
0.047059476375579834,
-0.010818486101925373,
0.030815839767456055,
0.040287140756845474,
-0.02032431773841381,
-0... | 0.844206 | 0.823102 | 0.021104 |
For the graph of a certain quadratic $y = ax^2 + bx + c$, the vertex of the parabola is $(3,7)$ and one of the $x$-intercepts is $(-2,0)$. What is the $x$-coordinate of the other $x$-intercept? | tiger_lab_math | [
0.02648787945508957,
0.02961801178753376,
-0.046288710087537766,
-0.038185473531484604,
0.03365844860672951,
-0.017307966947555542,
-0.00938455667346716,
0.06855617463588715,
0.049071818590164185,
-0.03717074915766716,
0.011852181516587734,
0.027137408033013344,
-0.035342879593372345,
-0.0... | 0.83538 | 0.832754 | 0.002626 |
Find the cross product of two vectors given their magnitudes and the angle between them.
Vector A has a magnitude of 5 units and points towards the positive x-axis.
Vector B has a magnitude of 4 units and points towards the positive y-axis.
The angle between them is 60 degrees.
What is the magnitude and direction... | tiger_lab_math | [
0.03308564051985741,
0.01535396184772253,
-0.03892097994685173,
-0.03511175140738487,
0.0291490126401186,
-0.010873869061470032,
-0.028629718348383904,
0.05513961613178253,
0.038507845252752304,
-0.010043077170848846,
-0.006192999891936779,
0.032075654715299606,
-0.01648763380944729,
-0.02... | 0.803313 | 0.802242 | 0.001072 |
Here is a precise numerical problem for a math student to solve:
Suppose there are 100 players in a beauty contest game, with each player choosing a number between 0 and 100. The winner of the game is the player whose number is closest to (2/3) times the average of all the numbers chosen. What number should you choose... | tiger_lab_math | [
0.03341957554221153,
0.008946463465690613,
-0.05729355290532112,
-0.02715398371219635,
0.034951772540807724,
-0.010726665146648884,
-0.04121160879731178,
0.024876749143004417,
0.03131654113531113,
-0.018682662397623062,
0.012917328625917435,
0.027145616710186005,
-0.03251083940267563,
-0.0... | 0.841228 | 0.830827 | 0.010401 |
My friend and I both have the same math homework one day. I work at a rate of $p$ problems per hour and it takes me $t$ hours to finish my homework. My friend works at a rate of $2p-4$ problems per hour and it only takes him $t-2$ hours to finish his homework. Given that $p$ and $t$ are positive whole numbers and I do ... | tiger_lab_math | [
0.03908802196383476,
0.0035245921462774277,
-0.053399067372083664,
-0.026988016441464424,
0.02573833428323269,
-0.010919357649981976,
-0.02806401252746582,
0.04580320045351982,
0.030948568135499954,
-0.01168159581720829,
0.0028830370865762234,
0.02127619832754135,
-0.02338528446853161,
-0.... | 0.846989 | 0.844188 | 0.002801 |
Consider a graph G with 10 vertices and 25 edges. Find out whether there exists a Hamiltonian cycle in G or not. If yes, then provide a Hamiltonian cycle for the graph G. | tiger_lab_math | [
0.007340366952121258,
-0.00000694659956934629,
-0.024045391008257866,
-0.03235013410449028,
0.017701750621199608,
-0.05772649124264717,
-0.029545815661549568,
0.06609804928302765,
0.01151502225548029,
-0.028795966878533363,
0.022390514612197876,
0.008674069307744503,
-0.036697354167699814,
... | 0.813311 | 0.809112 | 0.004198 |
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