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 partial differential equation:
u_t = ku_xx
The boundary conditions are:
u(x,0) = x(1-x), 0 ≤ x ≤ 1
u(0,t) = 0, u(1,t) = 0, t > 0
Using the Fourier Series method, find the solution to this for values of k=1, x=0.5 and t=2. | tiger_lab_math | [
-0.003661088878288865,
0.038463614881038666,
-0.04142380878329277,
-0.0210472010076046,
0.0379239097237587,
-0.011057432740926743,
-0.006827040109783411,
0.08065371215343475,
0.047033246606588364,
0.014434631913900375,
0.025113755837082863,
0.03493121638894081,
-0.028669804334640503,
-0.01... | 0.82357 | 0.811117 | 0.012453 |
Jane bought pens for her three students. Each student received 62 red pens and 43 black pens. They pooled their pens together and took them from the pool whenever they needed them. After the first month, they had taken a total of 37 pens from the pool. After the second month, they had taken another 41 pens from the poo... | tiger_lab_math | [
0.007236623205244541,
0.032540641725063324,
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0.0239648949354887,
-0.04947895556688309,
-0.026714075356721878,
0.03432965278625488,
0.030930642038583755,
-0.012247491627931595,
0.04192790389060974,
0.03008597530424595,
-0.044292088598012924,
-0.02... | 0.821884 | 0.821038 | 0.000846 |
In the diagram, the smaller circles touch the larger circle and touch each other at the center of the larger circle. The radius of the larger circle is $6.$ What is the area of the shaded region?
[asy]
size(100);
import graph;
filldraw(Circle((0,0),2),mediumgray);
filldraw(Circle((-1,0),1),white);
filldraw(Circle((1,0... | tiger_lab_math | [
0.020216748118400574,
0.0386631041765213,
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0.02927406318485737,
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0.040354400873184204,
0.031531091779470444,
-0.021038701757788658,
0.028193458914756775,
0.027431782335042953,
-0.017420366406440735,
-0.0... | 0.828035 | 0.822879 | 0.005156 |
Let $M$ be a 3-dimensional manifold and $X=(x_1^2 + x_2^2)\partial/\partial x_1+3x_3\partial/\partial x_2$ be a vector field on $M$. Determine the Lie derivative of the vector field $X$ with respect to the vector field $Y=\partial/\partial x_1+2\partial/\partial x_2$ at the point $(1,1,1)\in M$. | tiger_lab_math | [
0.014847740530967712,
0.027051858603954315,
-0.04611540585756302,
-0.060074858367443085,
0.03719485178589821,
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-0.018858958035707474,
0.07523863762617111,
0.029242292046546936,
-0.005533785559237003,
0.008641625754535198,
0.033300984650850296,
-0.05714505910873413,
-0.... | 0.820365 | 0.810939 | 0.009426 |
Lorenzo put three full cans of thumbtacks into his pocket and went off to work at the corkboard factory. It was Lorenzo's job, as head of quality control, to test every cork board for its ability to hold thumbtacks. He placed one thumbtack from each of the three cans of thumbtacks into every board tested. At the end ... | tiger_lab_math | [
0.01902923732995987,
0.024706516414880753,
-0.046216852962970734,
-0.0364244282245636,
0.03888900578022003,
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0.058766670525074005,
-0.04105253517627716,
-... | 0.811091 | 0.810241 | 0.00085 |
Consider the reaction-diffusion equation for pattern formation:
u_t = Du_xx + f(u)
where D is the diffusivity coefficient and f(u) is a nonlinear function of u.
Suppose the reaction term is given by f(u) = -u^3 + 4u, and the initial condition is u(x,0) = 2sin(pi*x/L). Find the solution u(x,t) using the method of se... | tiger_lab_math | [
-0.010177872143685818,
0.02605164609849453,
-0.05681883916258812,
-0.020634973421692848,
0.037785276770591736,
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0.1011299416422844,
0.02981559932231903,
0.008858217857778072,
0.022952256724238396,
0.021925050765275955,
-0.01239680964499712,
-0.008... | 0.818964 | 0.812756 | 0.006208 |
In Algebraic Geometry, consider a Calabi-Yau threefold defined by the equation $x^3+y^3+z^3+w^3=0$. Find the volume of this Calabi-Yau manifold by computing the integral of the holomorphic three-form over the manifold. | tiger_lab_math | [
0.03916482999920845,
0.021779779344797134,
-0.04510703682899475,
-0.04580503702163696,
0.030255625024437904,
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0.050442811101675034,
0.05026108771562576,
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0.01078720111399889,
0.026817165315151215,
-0.04317300766706467,
-0.004... | 0.824491 | 0.809935 | 0.014556 |
Find the product of all real values of $r$ for which $\frac{1}{2x}=\frac{r-x}{7}$ has exactly one real solution. | tiger_lab_math | [
0.018371671438217163,
0.039570871740579605,
-0.03328751027584076,
-0.03762485459446907,
0.03168235719203949,
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0.005955683533102274,
0.08281894773244858,
0.049012165516614914,
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0.005932452157139778,
0.024699466302990913,
-0.011290513910353184,
-0.02... | 0.845264 | 0.834165 | 0.011099 |
Here's a possible numerical problem for the student to solve:
Consider the following two-dimensional Poisson equation with Dirichlet boundary conditions:
$\nabla^2 u(x,y) = - 4\pi\sin(2\pi x)\cos(2\pi y)$
on the unit square $\Omega = [0,1]\times[0,1]$. Use the finite element method with linear triangular elements to... | tiger_lab_math | [
0.013637194409966469,
0.02029755339026451,
-0.031257208436727524,
-0.053344421088695526,
0.03990781679749489,
-0.021000178530812263,
-0.02667950466275215,
0.06211710348725319,
0.015587826259434223,
0.02185780555009842,
0.030310627073049545,
0.04165366291999817,
-0.0005269119283184409,
-0.0... | 0.83553 | 0.818888 | 0.016641 |
Suppose we have an entire function defined as $f(z) = z^3 + 2z^2 + 3z + 4$. Find the sum of the residues of $f(z)$ at its zeros. | tiger_lab_math | [
0.035882651805877686,
0.03097047097980976,
-0.038710806518793106,
-0.0348600335419178,
0.047284964472055435,
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0.008010567165911198,
0.03782108053565025,
0.0388651117682457,
-0.00797139573842287,
0.010984686203300953,
0.021089576184749603,
-0.050606124103069305,
-0.0108... | 0.84461 | 0.84353 | 0.00108 |
A manufacturer makes shirts that are available in three sizes: small, medium, and large. The manufacturer has the following rules for producing these shirts:
- For each small shirt produced, the manufacturer needs to produce one medium shirt to go with it.
- For each medium shirt produced, the manufacturer needs to pr... | tiger_lab_math | [
0.012612906284630299,
0.019096894189715385,
-0.05496858060359955,
-0.03956400603055954,
0.025840651243925095,
-0.010417486540973186,
-0.051427263766527176,
0.06720743328332901,
0.017690127715468407,
-0.020139027386903763,
-0.0005784531240351498,
0.03304988145828247,
-0.03881584480404854,
-... | 0.839379 | 0.836964 | 0.002415 |
George flips an unfair coin 7 times. The coin has a $\frac{1}{4}$ probability of coming up heads and a $\frac{3}{4}$ probability of coming up tails. What is the probability that he flips exactly 2 tails? Please write a program to solve it | tiger_lab_math | [
0.010853821411728859,
0.03566402941942215,
-0.029425980523228645,
-0.03969264402985573,
0.02158920094370842,
-0.009204810485243797,
-0.03457200899720192,
0.056052275002002716,
0.008718803524971008,
-0.02665727771818638,
0.01056023221462965,
0.03196168318390846,
-0.003646652912721038,
-0.01... | 0.83127 | 0.828987 | 0.002283 |
An $\textit{abundant number}$ is a positive integer, the sum of whose distinct proper factors is greater than the number. (The $\textit{proper factors}$ of a number are all of its factors except the number itself.) How many numbers less than 25 are abundant numbers? Please respond by writing a program in Python. | tiger_lab_math | [
0.03226199373602867,
0.02352410927414894,
-0.040994156152009964,
-0.02604348212480545,
0.02976366877555847,
-0.036238182336091995,
-0.022988678887486458,
0.03587879613041878,
0.028256097808480263,
-0.01950642839074135,
0.004462307319045067,
0.03799222409725189,
-0.05320519208908081,
-0.016... | 0.826649 | 0.825484 | 0.001166 |
Determine the analytic continuation of $f(z)=\sqrt{z+1}$ from the half plane $\{z: \text{Re}(z)>0\}$ to the whole complex plane. | tiger_lab_math | [
0.012794080190360546,
0.05375475808978081,
-0.0246709156781435,
-0.05621003732085228,
0.02307257242500782,
0.012012066319584846,
-0.003623569617047906,
0.06028643995523453,
0.04066424071788788,
-0.009410849772393703,
0.021548530086874962,
0.013128947466611862,
-0.037713367491960526,
-0.005... | 0.820317 | 0.80883 | 0.011487 |
Here's a precise numerical problem that a math student can solve:
Consider a transfer function with the form:
H(s) = 4s / (s^2 + 4s + 3)
(a) Use the partial fraction decomposition method to write H(s) as a sum of simpler fractions.
(b) Use the Fourier transform to analyze the stability of the system described by th... | tiger_lab_math | [
0.029505783692002296,
0.03731277957558632,
-0.04181478172540665,
-0.033993810415267944,
0.055652916431427,
-0.029934849590063095,
-0.0037656824570149183,
0.06342551857233047,
0.05513541027903557,
0.004821180831640959,
0.023333655670285225,
0.017822610214352608,
-0.02394801937043667,
-0.023... | 0.827121 | 0.815235 | 0.011887 |
Evaluate $\left\lceil\sqrt{2}\,\right\rceil+\left\lceil\sqrt{22}\,\right\rceil+\left\lceil\sqrt{222}\,\right\rceil$. Let's write a program. | tiger_lab_math | [
0.0031881267204880714,
0.03392673656344414,
-0.03256068751215935,
-0.04862618446350098,
0.04036923870444298,
-0.01463073119521141,
-0.038556378334760666,
0.07516279816627502,
0.0646190270781517,
-0.018240652978420258,
0.040026579052209854,
0.03284970298409462,
-0.04794663190841675,
-0.0237... | 0.832265 | 0.829417 | 0.002847 |
There are infinitely many lines passing through the point (-1,1) in the plane. Let C be the curve defined by the equation y^3 + 3x^2y - 3xy^2 - x^3 = 0. Find the equations of all lines that intersect C exactly three times. | tiger_lab_math | [
0.020991504192352295,
0.019726237282156944,
-0.05038384720683098,
-0.05067218840122223,
0.03974411264061928,
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0.05488736554980278,
0.028369169682264328,
-0.02046641707420349,
0.005652484484016895,
0.03043525479733944,
-0.025116214528679848,
-0.0... | 0.856655 | 0.837697 | 0.018959 |
Consider a graph G with 5 vertices and edges connecting all possible pairs of vertices except for two vertices that are not connected. Find the chromatic polynomial of G. | tiger_lab_math | [
0.017542313784360886,
0.017360970377922058,
-0.0390426404774189,
-0.037807971239089966,
0.017496418207883835,
-0.048533156514167786,
-0.02276718243956566,
0.03546614944934845,
0.02462702989578247,
-0.02178511768579483,
0.03165465220808983,
0.03541120886802673,
-0.002763445023447275,
-0.002... | 0.831858 | 0.82066 | 0.011197 |
If the LCM of two integers c, d (where d> c and c>1) is c*d, then which of the following can not be true?
Answer Choices: (A) Both c and d can be consecutive integers. (B) Both c and d can be Prime Numbers. (C) All Prime factors of c must be Prime factors of d. (D) c and d do not have a common factor. (E) c and d do no... | tiger_lab_math | [
0.014936271123588085,
0.03370475769042969,
-0.02522146888077259,
-0.03994302079081535,
0.04148789867758751,
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0.07169234752655029,
0.03084282949566841,
-0.035356272011995316,
0.012060010805726051,
0.023599982261657715,
-0.028900455683469772,
-0.039... | 0.819854 | 0.81476 | 0.005093 |
Let $a_1,$ $a_2,$ $\dots,$ $a_{12}$ be positive real numbers such that $a_1 + a_2 + \dots + a_{12} = 1.$ Find the minimum value of
\[\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_{12}}.\] | tiger_lab_math | [
0.020914636552333832,
0.03317141905426979,
-0.03526009991765022,
-0.039559345692396164,
0.04565662518143654,
-0.010212922468781471,
-0.013939699158072472,
0.055795926600694656,
0.04391278699040413,
-0.023397846147418022,
-0.003390450496226549,
0.012395424768328667,
-0.030438806861639023,
-... | 0.846773 | 0.830993 | 0.01578 |
Consider a simple undirected graph G with seven vertices and ten edges. Determine the minimum number of edges whose removal from G will disconnect G. | tiger_lab_math | [
0.019688332453370094,
0.021613772958517075,
-0.027549372985959053,
-0.037113796919584274,
0.04260442778468132,
-0.04708269611001015,
-0.03231678903102875,
0.03799152374267578,
0.029939237982034683,
-0.02419895865023136,
0.02305689826607704,
0.014433231204748154,
-0.01568029820919037,
-0.01... | 0.828103 | 0.825051 | 0.003052 |
Evaluate the following integral using residue theorem:
∫(z^2+1)/(z^4+1) dz, where the path of integration is the unit circle |z|=1, traversed in the counterclockwise direction. | tiger_lab_math | [
0.028659608215093613,
0.023141371086239815,
-0.010428376495838165,
-0.049201514571905136,
0.045305415987968445,
-0.008755269460380077,
-0.00704045407474041,
0.041025612503290176,
0.050734180957078934,
0.004322769120335579,
0.03188600391149521,
0.009443609975278378,
-0.021059049293398857,
-... | 0.819496 | 0.816933 | 0.002562 |
In the figure, the area of square $WXYZ$ is $25 \text{ cm}^2$. The four smaller squares have sides 1 cm long, either parallel to or coinciding with the sides of the large square. In $\triangle
ABC$, $AB = AC$, and when $\triangle ABC$ is folded over side $\overline{BC}$, point $A$ coincides with $O$, the center of sq... | tiger_lab_math | [
0.0287203099578619,
0.03470873460173607,
-0.04668883606791496,
-0.05925329029560089,
0.02841666340827942,
-0.021896397694945335,
-0.015296845696866512,
0.04335937649011612,
0.017235370352864265,
0.006850218866020441,
0.010530905798077583,
0.014757939614355564,
-0.021645834669470787,
-0.013... | 0.830568 | 0.818414 | 0.012153 |
Sarah has an unlimited number of identical red, blue, and green tiles. How many ways can she arrange these tiles in a line such that each group of tiles of the same color contains at least 4 tiles? | tiger_lab_math | [
0.01375487633049488,
0.02670533023774624,
-0.031011216342449188,
-0.023503093048930168,
0.023066094145178795,
-0.01137526798993349,
-0.04461965337395668,
0.05844080075621605,
0.029948774725198746,
-0.0222586989402771,
0.0312604084610939,
0.03578012064099312,
-0.015019596554338932,
-0.03552... | 0.83313 | 0.825588 | 0.007542 |
Consider the algebraic variety given by the equation $y^2 = x^3 - x^2$. Find the singular points of this variety. | tiger_lab_math | [
0.015124808065593243,
0.029866307973861694,
-0.044784415513277054,
-0.042352624237537384,
0.025538919493556023,
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0.0015592308482155204,
0.07617869973182678,
0.05682725831866264,
-0.03352885693311691,
0.012950625270605087,
0.031717173755168915,
-0.02400069124996662,
-0.... | 0.837466 | 0.828573 | 0.008893 |
Suppose there are 10 players in a Public Goods Game. Each player decides how much money they want to contribute to a public project from their initial endowment of $100. If the total contribution from all players is less than $500, the project fails and everyone loses their contribution. If the total contribution is eq... | tiger_lab_math | [
0.018347270786762238,
-0.009955613873898983,
-0.04234998673200607,
-0.016640204936265945,
0.02888008952140808,
-0.02518807165324688,
-0.02487163431942463,
0.04439177364110947,
0.030747678130865097,
-0.031432412564754486,
0.014227568171918392,
0.02069699764251709,
-0.04468683898448944,
-0.0... | 0.812216 | 0.806724 | 0.005491 |
Using predicate logic, express the following sentence mathematically: "All cats are mammals." Then, write the negation of the statement and determine its truth value using a truth table. | tiger_lab_math | [
0.009397044777870178,
0.014297228306531906,
-0.02609100192785263,
-0.026883795857429504,
0.04380222037434578,
-0.060002487152814865,
-0.013294494710862637,
0.06095252186059952,
0.059138569980859756,
-0.02750188671052456,
-0.006149626336991787,
0.030140230432152748,
-0.025455785915255547,
-... | 0.809107 | 0.80248 | 0.006626 |
A local school is hosting a math fair, and they need to create a schedule of events where each teacher presents at the same time slot across different rooms. There are 6 teachers, and they have to present in 4 time slots; each slot will have exactly 3 teachers presenting. How many different schedules are possible for t... | tiger_lab_math | [
0.03801362216472626,
0.014368555508553982,
-0.038806576281785965,
-0.044154275208711624,
0.041031546890735626,
-0.0051474906504154205,
-0.05581112951040268,
0.046429578214883804,
0.047286506742239,
-0.02120138332247734,
0.011689144186675549,
0.04002068564295769,
-0.02036001905798912,
-0.03... | 0.828702 | 0.821076 | 0.007625 |
Evaluate $\sqrt{2 -\!\sqrt{2 - \!\sqrt{2 - \!\sqrt{2 - \cdots}}}}$. Let's write a Python program to solve it. | tiger_lab_math | [
0.009552440606057644,
0.03433987870812416,
-0.04333430901169777,
-0.04070543870329857,
0.04531322419643402,
-0.019576821476221085,
-0.02131810411810875,
0.056282129138708115,
0.056129708886146545,
-0.02807699888944626,
0.016904110088944435,
0.028001314029097557,
-0.04626002162694931,
-0.02... | 0.832639 | 0.831589 | 0.00105 |
In triangle $XYZ$, we have $\angle Z = 90^\circ$, $XY = 10$, and $YZ = \sqrt{51}$. What is $\tan X$? Let's write a Python program. | tiger_lab_math | [
0.022180939093232155,
0.027429543435573578,
-0.03577565401792526,
-0.03600456938147545,
0.034238457679748535,
-0.017622098326683044,
-0.02776467241346836,
0.030589915812015533,
0.035505663603544235,
-0.02200145460665226,
0.03170846775174141,
0.022099491208791733,
-0.047818563878536224,
0.0... | 0.829489 | 0.8277 | 0.001789 |
A study was conducted to find out if there is any significant difference in the average time taken to solve math problems between three different teaching methods: Lecture, Discussion, and Simulation. The data collected is shown in the table below. Can you use ANOVA to determine if there is a statistically significant ... | tiger_lab_math | [
0.031251199543476105,
0.001603951444849372,
-0.036356959491968155,
-0.046603016555309296,
0.02623651921749115,
-0.022425435483455658,
-0.0172878410667181,
0.09186315536499023,
0.035879433155059814,
-0.006519279442727566,
0.008910339325666428,
0.0228828527033329,
-0.025700459256768227,
-0.0... | 0.794033 | 0.790477 | 0.003556 |
Suppose you have the following recursive equation:
$x_{n+1}=rx_n(1-x_n)$
where $0 \leq x_0 \leq 1$ and $0 \leq r \leq 4$. Find the value of $r$ at which the doubling process of the period reaches 3 from 2. | tiger_lab_math | [
0.02529142051935196,
0.017377685755491257,
-0.04699358716607094,
-0.044852036982774734,
0.01930694282054901,
-0.007573675364255905,
-0.01141586247831583,
0.0908844843506813,
0.022817466408014297,
-0.020363952964544296,
0.017869247123599052,
0.02723456360399723,
-0.041282374411821365,
-0.01... | 0.854844 | 0.844508 | 0.010336 |
The polynomial $f(x)=x^4+ax^3+bx^2+cx+d$ has real coefficients, and $f(2i)=f(2+i)=0$. What is $a+b+c+d$? | tiger_lab_math | [
0.03873458504676819,
0.03490578755736351,
-0.05812843516469002,
-0.03948752209544182,
0.024915466085076332,
-0.03151540085673332,
0.004136661067605019,
0.07172691076993942,
0.051645200699567795,
-0.016497144475579262,
0.005598391406238079,
0.02540084905922413,
-0.023548495024442673,
-0.046... | 0.842066 | 0.837373 | 0.004694 |
Consider the following undirected graph G with 7 vertices and 8 edges:
```
1 -- 2 -- 3 -- 4
| |
5 6
| |
7 8
```
Find all cut vertices and bridges present in the graph G. | tiger_lab_math | [
0.015927912667393684,
0.027591461315751076,
-0.015588326379656792,
-0.03090226650238037,
0.03428128734230995,
-0.03180690482258797,
-0.028022324666380882,
0.04320782050490379,
0.020768241956830025,
-0.016851672902703285,
0.012913309037685394,
0.016428787261247635,
-0.021151334047317505,
-0... | 0.816647 | 0.81033 | 0.006317 |
A double pendulum consists of two masses attached to each other by rigid, weightless rods. Consider a double pendulum with mass 1 and mass 2 weighing 1 kilogram and 2 kilograms respectively, attached to each other by rods of length 1 meter and 2 meters respectively. If the double pendulum is initially at rest at a star... | tiger_lab_math | [
0.020337242633104324,
0.01479883398860693,
-0.027296647429466248,
-0.04976160451769829,
0.02650740183889866,
-0.03600633889436722,
-0.012965095229446888,
0.050908636301755905,
0.018575144931674004,
-0.003414253005757928,
0.01766558177769184,
0.03542269021272659,
-0.03140376880764961,
-0.02... | 0.78656 | 0.785749 | 0.000811 |
A company is designing a rectangular storage container with a volume of 1000 cubic meters. The base is to be a square. Find the dimensions of the container that will minimize its surface area. | tiger_lab_math | [
0.014615106396377087,
0.03288397565484047,
-0.055085185915231705,
-0.02686912938952446,
0.03147372975945473,
-0.03156215324997902,
-0.044167015701532364,
0.06116339936852455,
0.04215671494603157,
-0.0063799466006457806,
0.014339148998260498,
0.029855597764253616,
-0.026169022545218468,
-0.... | 0.80476 | 0.801613 | 0.003147 |
Determine the equation of the line that passes through points P(3,4) and Q(5,1) and intersects the elliptic curve E: y^2 = x^3 + x + 1 at a third point R. Find the coordinates of point R. | tiger_lab_math | [
0.025879930704832077,
0.023695457726716995,
-0.03603785112500191,
-0.06279557943344116,
0.03151068836450577,
-0.03884367644786835,
-0.004502916242927313,
0.055053576827049255,
0.03596360981464386,
-0.013223256915807724,
0.018118126317858696,
0.03109409287571907,
-0.029996328055858612,
-0.0... | 0.831499 | 0.828916 | 0.002582 |
In a Public Goods Game, there are 5 players who each have an initial amount of $10. They can choose to contribute any amount between $0 and $5 to a common pot. The total contributions are multiplied by a factor of 1.5 and then divided equally among all the players, regardless of how much they contributed. If a player d... | tiger_lab_math | [
0.0042265779338777065,
0.008004684932529926,
-0.03729607164859772,
-0.013723079115152359,
0.031110260635614395,
-0.032068151980638504,
-0.028600243851542473,
0.03592102229595184,
0.036475542932748795,
-0.029403336346149445,
0.005167408846318722,
0.01919059455394745,
-0.041243113577365875,
... | 0.813568 | 0.803845 | 0.009723 |
Sure! Here's a specific problem for a math student studying graphing quadratic equations:
1. Graph the quadratic equation y = 2x^2 + 4x - 3 and find the vertex, axis of symmetry, and y-intercept of the parabola. | tiger_lab_math | [
0.052123844623565674,
0.027403298765420914,
-0.042402904480695724,
-0.04453926160931587,
0.03859299421310425,
-0.03496542200446129,
-0.023228026926517487,
0.06453842669725418,
0.04842311143875122,
-0.02518422342836857,
0.018061567097902298,
0.025993509218096733,
-0.020608650520443916,
-0.0... | 0.823931 | 0.816353 | 0.007578 |
Find the Laurent series expansion of the function $f(z) = \frac{1}{z^3 - z^2}$ centered at $z=0$ and determine the radius of convergence of the series. | tiger_lab_math | [
0.0011101816780865192,
0.03415600210428238,
-0.042378928512334824,
-0.03672393038868904,
0.03768080100417137,
0.010829142294824123,
-0.006344035733491182,
0.037667665630578995,
0.028037451207637787,
-0.01880672574043274,
0.03172791376709938,
0.005507288966327906,
-0.037456464022397995,
0.0... | 0.823578 | 0.817165 | 0.006414 |
Let L = {+, -, 0, 1} be a first-order language and let M be an L-structure, where M consists of a nonempty set A and interpretations of the symbols +, -, 0, and 1 as usual. Suppose that M satisfies the following conditions:
1. For all a,b,c in A, a+b = b+a, a + (b+c) = (a+b) + c, and there exists an element 0 in A suc... | tiger_lab_math | [
0.022140687331557274,
0.018518447875976562,
-0.03265456482768059,
-0.027254855260252953,
0.05273430049419403,
-0.035204846411943436,
-0.026284201070666313,
0.06611385941505432,
0.024505266919732094,
-0.018133247271180153,
-0.002850291086360812,
0.02779645100235939,
-0.048861872404813766,
-... | 0.825754 | 0.810466 | 0.015288 |
Suppose Alice wants to send a message to Bob using digital signatures. Alice's private key is 3 and her public key is (5, 20) on the elliptic curve y^2 = x^3 + 2x + 2 over the prime field GF(23). The message is "HELLO". To sign the message, Alice calculates the hash value using SHA-256 as 38762CF7 07DCA0BBCDAD728AEB6FA... | tiger_lab_math | [
0.004877177998423576,
0.04344141110777855,
-0.036520835012197495,
-0.05118197202682495,
0.02056940644979477,
-0.03678439185023308,
-0.023454027250409126,
0.04834059625864029,
0.02202240377664566,
-0.02714032307267189,
0.029764918610453606,
0.017465192824602127,
-0.027901509776711464,
-0.01... | 0.828451 | 0.820745 | 0.007706 |
Suppose Chris and Alex are planning a date night. Chris prefers to go to the movies while Alex wants to go to a concert. They must decide where to go together, but they value spending time together more than going to their preferred event alone. In this case, the payoff for going to their preferred event alone is 0, an... | tiger_lab_math | [
0.015878280624747276,
-0.008049633353948593,
-0.06335832178592682,
-0.023123130202293396,
0.02090740017592907,
-0.019340533763170242,
-0.017018994316458702,
0.060774412006139755,
0.03319346904754639,
-0.03456810861825943,
0.008137880824506283,
0.019602902233600616,
-0.030918080359697342,
-... | 0.812666 | 0.798006 | 0.01466 |
Consider a network flow problem modeling the transportation of goods from a source to a destination. Let the source have a supply of 500 units, the destination have a demand of 800 units, and there be three intermediate nodes with the following capacities:
- Node 1: can send at most 300 units
- Node 2: can send at mos... | tiger_lab_math | [
0.036826241761446,
0.007126958109438419,
-0.03198441118001938,
-0.022412804886698723,
0.02168406918644905,
-0.020211396738886833,
-0.04671229422092438,
0.055234394967556,
0.017348291352391243,
-0.006624662317335606,
0.02303501032292843,
0.03613627701997757,
-0.036059726029634476,
-0.019222... | 0.816734 | 0.803603 | 0.013131 |
The first three stages of a pattern are shown below, in which each line segment represents a toothpick. If the pattern continues such that at each successive stage, three toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangement for the 250th stage? [asy]
size(150);
d... | tiger_lab_math | [
0.01929631642997265,
0.021259209141135216,
-0.03786281496286392,
-0.05412837490439415,
0.03832627832889557,
-0.008945264853537083,
-0.031926047056913376,
0.08253703266382217,
0.03649207577109337,
-0.008220840245485306,
0.013765763491392136,
0.02932768687605858,
-0.024613700807094574,
-0.01... | 0.829919 | 0.82635 | 0.003569 |
Camille goes to the Ice Cream Palace with her husband and two children. Each one orders a different ice cream. Camille orders a banana split, her husband orders a waffle bowl, her daughter orders a single cone, and her son orders a double cone. The cone has 1 scoop of ice cream, the banana split has 3 times as many sco... | tiger_lab_math | [
0.00326071516610682,
0.03430933505296707,
-0.018248800188302994,
-0.03805183246731758,
0.03641999512910843,
-0.006292680744081736,
-0.036557987332344055,
0.05633766949176788,
0.04033788666129112,
-0.029263125732541084,
0.031979117542505264,
0.033804040402173996,
-0.03392661735415459,
-0.02... | 0.832797 | 0.829658 | 0.003138 |
A network has four nodes (A, B, C, and D), and the following table represents the capacities of the directed edges between them:
|From/To|A|B|C|D|
|-------|-|-|-|-|
|A|0|7|4|0|
|B|0|0|5|6|
|C|0|0|0|4|
|D|0|0|0|0|
Find the maximum flow that can be sent from node A to node D. | tiger_lab_math | [
0.03035370446741581,
0.025592543184757233,
-0.02717164345085621,
-0.01208025123924017,
0.02197432890534401,
-0.031735148280858994,
-0.041968826204538345,
0.050306882709264755,
0.04569879546761513,
-0.004948245361447334,
0.025467649102211,
0.03793951869010925,
-0.0189494788646698,
-0.021053... | 0.818619 | 0.811266 | 0.007353 |
Here's a precise numerical problem for a math student to solve related to intuitionistic logic:
Mary and Tom are playing a game. Mary says, "If I win three games in a row, then I will win the whole competition." However, Tom is good at breaking the rules. He believes that Mary will not win the next game. If we assume ... | tiger_lab_math | [
0.026308832690119743,
0.011517503298819065,
-0.06642746925354004,
-0.02724313922226429,
0.03853585571050644,
-0.00621522031724453,
-0.02111353911459446,
0.04533392935991287,
0.03313959017395973,
-0.029684342443943024,
0.0023397186305373907,
0.029501618817448616,
-0.017986750230193138,
-0.0... | 0.812762 | 0.792159 | 0.020604 |
A company wants to minimize the cost of transporting goods across a network of five cities. The cost of transporting goods between cities is given by the matrix below:
| City | A | B | C | D | E |
|-------|---|---|---|---|---|
| **A** | 0 | 4 | 2 | 5 | 3 |
| **B** | 4 | 0 | 1 | 7 | 4 |
| **C** | 2 | 1 | 0 | 6 | 1 |
|... | tiger_lab_math | [
0.01647448167204857,
0.0150031428784132,
-0.020119141787290573,
-0.025114692747592926,
0.03761086240410805,
-0.03121635876595974,
-0.03482623025774956,
0.06236971914768219,
0.029383746907114983,
-0.028018126264214516,
0.021348992362618446,
0.03250864893198013,
-0.018237590789794922,
-0.010... | 0.817221 | 0.815798 | 0.001423 |
Consider the surface given by the equation x² + y² - z³ = 0. Find the principal curvatures at the point (1,1,1). | tiger_lab_math | [
-0.004243304952979088,
0.031550146639347076,
-0.0349515825510025,
-0.05563024431467056,
0.03848171979188919,
0.0024571798276156187,
0.0006286257412284613,
0.06204408407211304,
0.010418818332254887,
-0.015079786069691181,
0.013355814851820469,
0.027389947324991226,
-0.029992882162332535,
-0... | 0.822043 | 0.812293 | 0.00975 |
Let X be a topological space with the property that every pair of distinct points can be separated by disjoint open sets. Suppose that X has exactly 7 points. How many different topologies can X have? | tiger_lab_math | [
0.038373202085494995,
0.008187715895473957,
-0.04688983038067818,
-0.01672527939081192,
0.026229802519083023,
-0.016810696572065353,
-0.03178296983242035,
0.038528259843587875,
0.04925254359841347,
-0.022336358204483986,
0.004817150998860598,
0.03492981940507889,
-0.04005761072039604,
-0.0... | 0.831734 | 0.818947 | 0.012787 |
Prudence was starting a cupcake business. She figured that each cupcake cost $0.75 to make. The first 2 dozen that she made burnt and she had to throw them out. The next 2 came out perfectly and she ended up eating 5 cupcakes right away. Later that day she made 2 more dozen cupcakes and decided to eat 4 more. If sh... | tiger_lab_math | [
-0.005285191815346479,
0.019489074125885963,
-0.028054604306817055,
-0.02787376381456852,
0.01574929617345333,
0.0020935945212841034,
-0.03133828565478325,
0.073898084461689,
0.026162156835198402,
-0.00804123654961586,
0.024149587377905846,
0.01717083528637886,
-0.02906743250787258,
-0.044... | 0.827933 | 0.826727 | 0.001206 |
Suppose two players A and B are playing a game where they need to simultaneously choose a number between 0 and 10. If their choices sum up to less than 10, player B gets 2 points and player A gets 1 point. If their choices sum up to 10 or more, player A gets 2 points and player B gets 1 point. If their choices are the ... | tiger_lab_math | [
0.016523947939276695,
0.004071333911269903,
-0.04384518042206764,
-0.018431562930345535,
0.022304367274045944,
-0.017613975331187248,
-0.012216860428452492,
0.048539839684963226,
0.030142778530716896,
-0.03199618682265282,
0.007837827317416668,
0.03544892743229866,
-0.054697245359420776,
-... | 0.818179 | 0.814792 | 0.003387 |
A factory produces two types of products: Product A and Product B. The factory has 10 machines and can produce either Product A or Product B, but not both at the same time. It takes 2 hours to produce one unit of Product A and 3 hours to produce one unit of Product B. Each machine can produce a maximum of 6 units per d... | tiger_lab_math | [
0.03524940833449364,
0.015322950668632984,
-0.04838300496339798,
0.010886503383517265,
0.011322549544274807,
-0.018557101488113403,
-0.05992788448929787,
0.10114743560552597,
0.023422863334417343,
-0.004337755497545004,
0.007552524097263813,
0.0389210507273674,
-0.015403250232338905,
-0.03... | 0.809898 | 0.802941 | 0.006957 |
A particle moves randomly in a fluid through a process governed by a stochastic differential equation given by:
dX(t) = -2X(t)dt + 3dB(t)
where B(t) is a Brownian motion.
Given that the particle starts at position X(0) = 1, find the probability that the particle crosses the point X = 2 before it crosses the point X ... | tiger_lab_math | [
0.005386875476688147,
0.01767089031636715,
-0.05329982191324234,
-0.02839585766196251,
0.03988790139555931,
-0.024880900979042053,
-0.05330212041735649,
0.07301850616931915,
0.02315879985690117,
0.0038989160675555468,
0.025915930047631264,
0.02884921059012413,
-0.0326247401535511,
-0.01891... | 0.813399 | 0.811624 | 0.001775 |
Find the solution of the partial differential equation $\frac{\partial^2u}{\partial x^2} - 4\frac{\partial^2u}{\partial y^2} = x$ subject to the boundary conditions $$u(x,0) = u(x,1) = 0,\quad\text{and}\quad u(0,y) = \sin \pi y$$ using Fourier series. | tiger_lab_math | [
-0.013327608816325665,
0.0369325615465641,
-0.034924402832984924,
-0.037775587290525436,
0.04062344878911972,
-0.008900417014956474,
-0.005342954769730568,
0.05714329704642296,
0.07606589794158936,
0.013308319263160229,
0.028532177209854126,
0.03754807636141777,
-0.013364330865442753,
-0.0... | 0.821479 | 0.808581 | 0.012898 |
A curry house sells curries that have varying levels of spice. Recently, a lot of the customers have been ordering very mild curries and the chefs have been having to throw away some wasted ingredients. To reduce cost and food wastage, the curry house starts monitoring how many ingredients are actually being used and c... | tiger_lab_math | [
-0.000028855065465904772,
0.04009109362959862,
-0.03491981327533722,
-0.03129997104406357,
0.02735290117561817,
-0.03193856403231621,
-0.02140876278281212,
0.04919958859682083,
0.041243404150009155,
-0.035077065229415894,
0.014775172807276249,
0.031982842832803726,
-0.03140698000788689,
-0... | 0.820121 | 0.817775 | 0.002347 |
Consider the logistic map function given by the equation `x_{n+1} = r*x_n*(1-x_n)` , where `x_n` is the population density at time `n` and `r` is the growth rate parameter.
For a given `r = 3.2`, calculate the first 10 population densities starting with `x_0 = 0.3`.
Next, using the values of `x_9` and `x_10` as the... | tiger_lab_math | [
0.021280597895383835,
0.023194922134280205,
-0.03611968830227852,
-0.033167921006679535,
0.027407800778746605,
-0.03767034411430359,
-0.048811834305524826,
0.07350751012563705,
0.013194936327636242,
-0.0015303910477086902,
0.005517973098903894,
0.028089964762330055,
-0.024193288758397102,
... | 0.810232 | 0.805521 | 0.004711 |
Consider a sequence of numbers defined by the recurrence relation: $a_n = 2a_{n-1} - 3a_{n-2}$, where $a_0 = 1$ and $a_1 = 2$.
Find the value of $a_4$. | tiger_lab_math | [
0.02646706812083721,
0.034267108887434006,
-0.03630605712532997,
-0.028653796762228012,
0.04225941002368927,
-0.009719212539494038,
-0.002002433408051729,
0.07725364714860916,
0.036405060440301895,
-0.024104546755552292,
0.01801249384880066,
0.02520197071135044,
-0.0438365712761879,
0.0047... | 0.843105 | 0.838158 | 0.004947 |
Here's a question for a math student to solve:
Consider a knot that is a trefoil knot, i.e., a knot with three crossings. Calculate its Alexander polynomial. | tiger_lab_math | [
0.04893212392926216,
0.012047138065099716,
-0.042381931096315384,
-0.05093338340520859,
0.02225497178733349,
-0.032506175339221954,
-0.008696619421243668,
0.03918486461043358,
0.032086677849292755,
-0.001211895840242505,
0.02197393961250782,
0.035921163856983185,
-0.03037358447909355,
-0.0... | 0.830248 | 0.821142 | 0.009106 |
A factory needs to produce 300 units of a product using 5 machines. Each machine has a different production rate and to balance out their utilization, the production planner decides that each machine should produce a different number of units. Machine 1 produces 25 units per hour, machine 2 produces 30 units per hour, ... | tiger_lab_math | [
0.0161154642701149,
0.010545236989855766,
-0.057008858770132065,
-0.011842952109873295,
0.020720714703202248,
-0.0014487826265394688,
-0.049829572439193726,
0.09105663001537323,
0.036507461220026016,
0.0021800389513373375,
-0.005191896576434374,
0.01108284667134285,
-0.018344447016716003,
... | 0.7929 | 0.787486 | 0.005414 |
If the height of an equilateral triangle QBC is equal in length to the hypotenuse of an isosceles right triangle DEF, a side of ΔQBC is how many times as long as a leg of the right triangle DEF?
Answer Choices: (A) (√2) / (√3) (B) (√3) / (√2) (C) √2 (D) (2√2) / (√3) (E) 2/ (√3) | tiger_lab_math | [
0.020957181230187416,
-0.009563855826854706,
-0.04232572764158249,
-0.02995295450091362,
0.023209024220705032,
-0.025923259556293488,
-0.005377206020057201,
0.04802801087498665,
0.02943159081041813,
0.0004662646970245987,
0.025946924462914467,
0.03926984220743179,
-0.03641057014465332,
-0.... | 0.825188 | 0.823397 | 0.001791 |
Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\] | tiger_lab_math | [
0.01683107018470764,
0.016767241060733795,
-0.033177610486745834,
-0.038023095577955246,
0.027411431074142456,
-0.018282169476151466,
-0.022622399032115936,
0.04200275242328644,
0.034875769168138504,
-0.01970343478024006,
0.031782377511262894,
0.03213999420404434,
-0.04195563122630119,
-0.... | 0.840974 | 0.838204 | 0.00277 |
Consider the system of linear equations:
\begin{align*}
2x + 3y + z &= 7 \\
x + 4y + 2z &= 4 \\
3x + y + 4z &= 4
\end{align*}
Using the Jacobi iterative method, approximate the solution to this system starting with initial guesses $x=0$, $y=0$, and $z=0$. Perform iterations until the absolute error between two succes... | tiger_lab_math | [
0.033664241433143616,
0.036200981587171555,
-0.03765161335468292,
-0.046582214534282684,
0.03916982561349869,
-0.024287940934300423,
-0.009532286785542965,
0.057270292192697525,
0.022032348439097404,
-0.0051899682730436325,
0.023600038141012192,
0.016977153718471527,
-0.028226390480995178,
... | 0.821171 | 0.81138 | 0.009791 |
Let $X_0, X_1, X_2, \ldots$ be drawn i.i.d. from $p(x)$, and $x\in\{1,2,3,\ldots,100\}. Let $N$ be the waiting time to the next occurrence of $X_0$. Compute $E(N)$. Let's program in Python in the response. | tiger_lab_math | [
0.008384753949940205,
-0.0034543690271675587,
-0.03876766934990883,
-0.018207591027021408,
0.01679733768105507,
-0.02091689221560955,
-0.038789428770542145,
0.050808582454919815,
0.03941123187541962,
-0.01463170163333416,
0.033951468765735626,
0.04095064848661423,
-0.02580944262444973,
0.0... | 0.829101 | 0.823509 | 0.005592 |
Consider a smooth manifold with a metric given by $g=dx^2+dy^2+dz^2$. Determine the Ricci curvature of the manifold at the point $(1,2,3)$. | tiger_lab_math | [
-0.0005357197369448841,
0.04019094631075859,
-0.031467411667108536,
-0.056945957243442535,
0.02920737862586975,
-0.0038632582873106003,
-0.015518072992563248,
0.045005038380622864,
0.0240107923746109,
-0.01941159926354885,
0.004811631515622139,
0.03195785731077194,
-0.0365692637860775,
-0.... | 0.831457 | 0.814989 | 0.016468 |
Consider the following optimization problem:
Minimize 𝑓(𝑥, 𝑦) = −sin(𝑥)sin(𝑦)exp[−(𝑥+𝑦)^2],
where −10 ≤ 𝑥, 𝑦 ≤ 10. Use the firefly algorithm to find the minimum value of 𝑓(𝑥, 𝑦) and the corresponding values of 𝑥 and 𝑦. | tiger_lab_math | [
-0.010076071135699749,
0.01143092755228281,
-0.03924739360809326,
-0.04431186988949776,
0.045636896044015884,
-0.026310360059142113,
-0.026317784562706947,
0.06763140857219696,
0.046040650457143784,
-0.009456317871809006,
0.022675974294543266,
0.021992316469550133,
-0.04267161339521408,
-0... | 0.821511 | 0.820242 | 0.001268 |
Thomas wants to throw a party for his best friend Casey. He needs to order enough chairs for all the party guests to sit in, one for each guest. First he orders 3 dozen chairs for 3 dozen guests he invites. Then he finds out that 1/3 of the guests want to bring a guest of their own, so Thomas needs to order more chairs... | tiger_lab_math | [
0.02679995633661747,
0.025540666654706,
-0.03742649778723717,
-0.027185741811990738,
0.027481788769364357,
-0.029370548203587532,
-0.026028279215097427,
0.03893471136689186,
0.02512459270656109,
-0.0003709597513079643,
0.030741654336452484,
0.04027347266674042,
-0.040972668677568436,
-0.00... | 0.832849 | 0.824455 | 0.008394 |
A math student is given a surface defined by the equation z = 6 - x^2 - y^2. They are asked to find the mean curvature of the surface at the point (1,1,4). | tiger_lab_math | [
0.010131158865988255,
0.0337325744330883,
-0.05021034926176071,
-0.06240275129675865,
0.05076222866773605,
-0.0005315012531355023,
-0.036711592227220535,
0.044971682131290436,
0.02527168020606041,
-0.027905523777008057,
0.011125984601676464,
0.03006245568394661,
-0.016606023535132408,
-0.0... | 0.828592 | 0.817843 | 0.010749 |
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exact... | tiger_lab_math | [
0.022874804213643074,
0.00020085823780391365,
-0.03557080030441284,
-0.0322149395942688,
0.02093403972685337,
-0.03516852483153343,
-0.007266842760145664,
0.04479501023888588,
0.027234744280576706,
-0.012879102490842342,
0.024066755548119545,
0.03823911398649216,
-0.035670965909957886,
-0.... | 0.840477 | 0.830341 | 0.010136 |
Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time $t = 0$ to $t = \infty.$ Rational Man drives along the path parameterized by
\begin{align*}
x &= \cos t, \\
y &= \sin t,
\end{align*}and Irrational Man drives along the path parameterized by
\begin{align*}
x &= ... | tiger_lab_math | [
0.009144789539277554,
0.036972641944885254,
-0.020565006881952286,
-0.0168690737336874,
0.04073801636695862,
-0.007158062420785427,
-0.031149230897426605,
0.05575283616781235,
0.05433646962046623,
-0.020205441862344742,
0.005427555646747351,
0.020295508205890656,
-0.011655859649181366,
-0.... | 0.830289 | 0.809379 | 0.02091 |
Consider the function $f(x) = \begin{cases} 1 &\text{if } 0 \leq x < \pi \\ -1 &\text{if } \pi \leq x < 2\pi \end{cases}$. Determine the Fourier series of $f(x)$ and sketch the first 5 non-zero terms of the series. | tiger_lab_math | [
0.010976780205965042,
0.03453768417239189,
-0.043729037046432495,
-0.034610409289598465,
0.03263642638921738,
-0.01860642433166504,
-0.018281003460288048,
0.06284312903881073,
0.03696330264210701,
0.019481997936964035,
0.0281439870595932,
0.014313279651105404,
-0.02782587520778179,
-0.0192... | 0.826788 | 0.817254 | 0.009535 |
Forty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four cards. Four cards are drawn from the box at random and without replacement. Let $p$ be the probability that all four cards bear the same number. Let $q$ be the probability that three of the cards ... | tiger_lab_math | [
0.012422421015799046,
0.02915932796895504,
-0.05397932976484299,
-0.02482307143509388,
0.03569979965686798,
-0.010261774063110352,
-0.012651519849896431,
0.045191410928964615,
0.0265060942620039,
-0.02682376094162464,
0.022640559822320938,
0.02863280661404133,
-0.03163854405283928,
-0.0202... | 0.842699 | 0.831858 | 0.010841 |
A company A and a company B are competing in a Stackelberg competition. Company A is the leader and company B is the follower. The total market demand is P=100-Q, where Q is the quantity produced by both companies. The cost function for company A is C_A=10Q and for company B is C_B=15Q. Determine the quantity of output... | tiger_lab_math | [
0.013003631494939327,
0.023802386596798897,
-0.06106160208582878,
-0.03494785353541374,
0.047124508768320084,
-0.0016580980736762285,
-0.027270400896668434,
0.0796319991350174,
0.04128081724047661,
-0.030519595369696617,
0.0038807133678346872,
0.03117009624838829,
-0.031907957047224045,
-0... | 0.810067 | 0.80498 | 0.005086 |
A math teacher wants to know if her students study consistently throughout the semester or only before exams. She randomly selects 10 students from her class and records the number of hours each student studies per week. The data is shown below:
5, 3, 7, 6, 8, 2, 4, 6, 10, 1
Using inductive logic, what percentage of ... | tiger_lab_math | [
0.009107882156968117,
-0.0006869958015158772,
-0.06699225306510925,
-0.034305620938539505,
0.036196935921907425,
-0.038328107446432114,
-0.015533518977463245,
0.05361315980553627,
0.025831686332821846,
-0.016824273392558098,
0.006225030403584242,
0.011272654868662357,
-0.024808885529637337,
... | 0.816751 | 0.81025 | 0.006502 |
Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \cap B| = |B \cap C| = |C \cap A| = 1$ and $A \cap B \cap C = \emptyset$. For example, $(\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $N$ be the number of minimally intersecting ordered triples of sets for which each... | tiger_lab_math | [
0.04169294610619545,
0.01204897090792656,
-0.032309893518686295,
-0.014803506433963776,
0.037481147795915604,
-0.0032557197846472263,
-0.027368322014808655,
0.04995274916291237,
0.03613956645131111,
-0.006897966843098402,
0.02759448066353798,
0.004779202863574028,
-0.024979714304208755,
-0... | 0.844731 | 0.81222 | 0.032511 |
a cylindrical container with 6 meters diameter and a height of 8 meters is filled to capacity with water . if the water is then used to fill several smaller cylinders ( 2 meters radius and 5 meters height ) , how many smaller cylinders can be filled to capacity before the larger cylinder becomes empty ? Let's write a p... | tiger_lab_math | [
0.03056439571082592,
0.04569650813937187,
-0.024670690298080444,
-0.021590569987893105,
0.028966326266527176,
-0.049175772815942764,
-0.04741237685084343,
0.06488365679979324,
0.02218768373131752,
-0.019515208899974823,
0.008981384336948395,
0.04154207929968834,
-0.03927420452237129,
-0.02... | 0.821197 | 0.816237 | 0.00496 |
A town with a population of 100,000 people is experiencing an outbreak of a contagious disease. After analyzing the data, epidemiologists determine that the rate of new infections is proportional to the number of people who are infected at any given time, with a rate constant of 0.03. Assuming no one ever recovers or d... | tiger_lab_math | [
0.03456942364573479,
0.019096624106168747,
-0.04491136595606804,
-0.028699198737740517,
0.034411463886499405,
-0.0371270552277565,
-0.025038160383701324,
0.05722621828317642,
0.018729858100414276,
-0.024952441453933716,
0.014410585165023804,
0.038905005902051926,
-0.017723584547638893,
-0.... | 0.813945 | 0.806262 | 0.007683 |
In the game of Chicken, Player A and Player B drive their cars towards each other. The first player to swerve loses the game, but if neither player swerves, they both crash and lose. In a particular game, Player A has a 70% chance of swerving and Player B has a 60% chance of swerving. What is the probability that neith... | tiger_lab_math | [
-0.005725427530705929,
-0.0034095074515789747,
-0.04021388292312622,
-0.024237319827079773,
0.023446664214134216,
0.00814809463918209,
-0.008792724460363388,
0.05431552231311798,
0.038604624569416046,
-0.047314804047346115,
0.0048538451083004475,
0.024247078225016594,
-0.023036619648337364,
... | 0.816807 | 0.814696 | 0.002111 |
A factory produces two types of products: Product A and Product B. To make one unit of Product A, it takes 3 hours of assembly time and 2 hours of packaging time. To make one unit of Product B, it takes 4 hours of assembly time and 3 hours of packaging time. The factory has a total of 180 assembly hours and 120 packagi... | tiger_lab_math | [
0.03296251967549324,
0.0014222662430256605,
-0.05254354700446129,
0.002108584390953183,
0.011475801467895508,
-0.014140087179839611,
-0.05999196693301201,
0.0781538337469101,
0.02952997200191021,
0.00029439173522405326,
0.018264440819621086,
0.03660960495471954,
-0.010662305168807507,
-0.0... | 0.806604 | 0.80321 | 0.003393 |
Here is a mathematics question related to Game Theory:
In a two-player game, Player 1 can choose between Strategy A and Strategy B, while Player 2 can choose between Strategy C and Strategy D. If Player 1 chooses Strategy A and Player 2 chooses Strategy C, then Player 1 wins $5 and Player 2 wins $2. If Player 1 choose... | tiger_lab_math | [
0.0022126082330942154,
0.011442232877016068,
-0.03731832653284073,
-0.011192777194082737,
0.021594712510704994,
-0.014994935132563114,
-0.015204938128590584,
0.05393416807055473,
0.0391283854842186,
-0.041010964661836624,
0.002116343704983592,
0.028180520981550217,
-0.026278870180249214,
-... | 0.807061 | 0.796727 | 0.010334 |
Compute $$\sum_{n=1}^{\infty} \frac{3n-1}{2^n}.$$ Please write a program to solve it | tiger_lab_math | [
0.025976525619626045,
0.04667335003614426,
-0.03353377804160118,
-0.049257371574640274,
0.03177168592810631,
-0.017805518582463264,
-0.01655907742679119,
0.05148827284574509,
0.028155425563454628,
-0.04528682678937912,
0.0276747178286314,
0.03904716297984123,
-0.034652844071388245,
-0.0230... | 0.848042 | 0.847186 | 0.000856 |
Two lines in a plane are given by the equations $3x + 4y = 4$ and $6x + 8y = 7$. Use Bezout's theorem to find the number of common points of these lines. | tiger_lab_math | [
0.024724500253796577,
0.013785547576844692,
-0.043978091329336166,
-0.05921546369791031,
0.042354244738817215,
-0.008415505290031433,
-0.021996987983584404,
0.03616391122341156,
0.054068513214588165,
-0.018897125497460365,
0.00015546315989922732,
0.044903289526700974,
-0.01793046109378338,
... | 0.837781 | 0.827286 | 0.010495 |
In a certain mathematical activity, we have five cards with five different prime numbers on them. We will distribute these five cards among three envelope: all could go in any envelope, or they could be broken up in any way among the envelopes. Then in each envelop, we find the product of all the cards in that envelope... | tiger_lab_math | [
0.03367282822728157,
0.00429077772423625,
-0.018715249374508858,
-0.025197675451636314,
0.04203496873378754,
-0.03627157583832741,
-0.01872038096189499,
0.06241871789097786,
0.042955685406923294,
-0.008649650029838085,
0.008858646266162395,
0.0390145443379879,
-0.0429636687040329,
-0.02068... | 0.831472 | 0.828512 | 0.00296 |
What is the Cramer-Rao lower bound on $E_\theta(\hat{\theta}(X)-\theta)^2$, where $\hat{\theta}(X)$ is an unbaised estimator of $\theta$ for the distribution family $f_\theta(x)=\theta e^{-\theta x}$, $x \geq 0$? (a) $\theta$. (b) $\theta^2$. (c) $\theta^{-1}$. (d) $\theta^{-2}$. Let's program in Python in the response... | tiger_lab_math | [
0.010895333252847195,
0.03488423675298691,
-0.040763970464468,
-0.03367138281464577,
0.03839336708188057,
-0.01827441155910492,
-0.024673176929354668,
0.06443142890930176,
0.03465810790657997,
-0.025370528921484947,
0.012800469994544983,
0.021485069766640663,
-0.03989456221461296,
-0.00335... | 0.824063 | 0.82284 | 0.001223 |
company c sells a line of 25 products with an average retail price of $ 1,200 . if none of these products sells for less than $ 400 , and exactly 12 of the products sell for less than $ 1,000 , what is the greatest possible selling price of the most expensive product ? Let's program in Python in the response. | tiger_lab_math | [
0.02888106182217598,
0.012260927818715572,
-0.04540987312793732,
-0.015772158280014992,
0.036620303988456726,
-0.028344744816422462,
-0.030946414917707443,
0.04073562100529671,
0.027185924351215363,
-0.02220173552632332,
0.024011388421058655,
0.03319348394870758,
-0.036015287041664124,
-0.... | 0.810569 | 0.809258 | 0.001311 |
The quadratic $10x^2+100x+1000$ can be written in the form $a(x+b)^2+c$, where $a$, $b$, and $c$ are constants. What is $a+b+c$? Let's write a Python program to solve it. | tiger_lab_math | [
0.01601313427090645,
0.030273081734776497,
-0.04564272612333298,
-0.021124858409166336,
0.03258710727095604,
-0.04340029135346413,
-0.011765954084694386,
0.06915689259767532,
0.054903317242860794,
-0.023644154891371727,
0.0074482448399066925,
0.03396008908748627,
-0.03912167623639107,
-0.0... | 0.828649 | 0.82681 | 0.00184 |
Statement 1| The order of an element in a group is always less than or equal to the order of the group. Statement 2 | All cyclic groups are abelian. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.03594892844557762,
0.021011337637901306,
-0.029888730496168137,
-0.04742669686675072,
0.034085821360349655,
-0.05344779044389725,
0.005673195701092482,
0.05269527807831764,
0.034628503024578094,
-0.014770432375371456,
0.008874263614416122,
0.005684370175004005,
-0.032330840826034546,
-0.... | 0.805099 | 0.800275 | 0.004824 |
The numbers 2, 3, 5, 7, 11, 13 are arranged in a multiplication table, with three along the top and the other three down the left. The multiplication table is completed and the sum of the nine entries is tabulated. What is the largest possible sum of the nine entries?
\[
\begin{array}{c||c|c|c|}
\times & a & b & c \... | tiger_lab_math | [
0.040370404720306396,
0.030513033270835876,
-0.04107135534286499,
-0.040567755699157715,
0.05297401547431946,
-0.012289228849112988,
-0.016872363165020943,
0.0517624169588089,
0.02573498710989952,
-0.00455164210870862,
0.008953631855547428,
0.03988925740122795,
-0.05068383365869522,
-0.014... | 0.829649 | 0.82088 | 0.00877 |
Let X be a smooth projective algebraic variety defined over a field k with the function field k(X). Suppose there exists a birational map f : X --> Y , where Y is another algebraic variety. Let K(X) be the field of rational functions on X and K(Y) be the field of rational functions on Y. If dim(k(X)) = 3, dim(K(Y)) = 4... | tiger_lab_math | [
0.023334909230470657,
0.029131365939974785,
-0.04645491763949394,
-0.04153672233223915,
0.02984464354813099,
-0.010046864859759808,
-0.017901811748743057,
0.03788938373327255,
0.05568331852555275,
-0.01965022273361683,
0.004180860240012407,
0.04468810558319092,
-0.03812563046813011,
-0.002... | 0.819721 | 0.804032 | 0.015689 |
Using the Riemann mapping theorem, find a conformal equivalent of the unit disc to the region between the curves $y=x^3$ and $y=x$. | tiger_lab_math | [
0.007401973940432072,
0.019426442682743073,
-0.041363246738910675,
-0.04823608696460724,
0.029127396643161774,
-0.005819138139486313,
-0.00843900442123413,
0.05186585336923599,
0.04403970018029213,
-0.017354000359773636,
0.01757454127073288,
0.03159831464290619,
-0.03448347747325897,
-0.01... | 0.830184 | 0.817448 | 0.012735 |
Here's a numerical problem for a math student to solve:
Consider the Duffing oscillator with the equation of motion:
d^2x/dt^2 + delta*dx/dt + alpha*x + beta*x^3 = gamma*cos(omega*t)
where delta = 0.1, alpha = -1, beta = 1, gamma = 0.3, and omega = 1.
a) Find the steady-state solution for the equation of motion. ... | tiger_lab_math | [
0.03786764666438103,
0.014258849434554577,
-0.046143967658281326,
-0.044164858758449554,
0.036243781447410583,
-0.035148829221725464,
-0.018639210611581802,
0.07159045338630676,
0.023214444518089294,
-0.005721605382859707,
0.00994484219700098,
0.026315346360206604,
-0.04128697142004967,
-0... | 0.830791 | 0.827105 | 0.003686 |
A manufacturing company produces two types of products, A and B, with a profit of $10 and $15 per unit, respectively. The company has three machines that can produce these products with the following hourly capacities:
- Machine 1 can produce 100 units of A or 50 units of B per hour.
- Machine 2 can produce 80 units... | tiger_lab_math | [
0.013745443895459175,
0.014293399639427662,
-0.06125175952911377,
-0.0024868291802704334,
0.014522247947752476,
-0.01785094290971756,
-0.054526180028915405,
0.08566208183765411,
0.024597391486167908,
-0.008122826926410198,
0.004473881330341101,
0.033097926527261734,
-0.028582612052559853,
... | 0.798778 | 0.784708 | 0.01407 |
How about this:
A traveling salesman has to visit 5 different cities. The distances between these cities are as follows:
- City 1 to City 2: 5 km
- City 1 to City 3: 9 km
- City 1 to City 4: 7 km
- City 1 to City 5: 8 km
- City 2 to City 3: 4 km
- City 2 to City 4: 3 km
- City 2 to City 5: 6 km
- City 3 to City 4: ... | tiger_lab_math | [
0.015237277373671532,
0.04038956016302109,
-0.030996527522802353,
-0.03663571923971176,
0.02715742588043213,
-0.016093309968709946,
-0.026701243594288826,
0.0437425896525383,
0.0287599079310894,
-0.029848333448171616,
0.017910804599523544,
0.030939392745494843,
-0.002817020285874605,
-0.01... | 0.832114 | 0.828451 | 0.003663 |
Let A = {1, 2, 3} , B = {2, 3, 4} and C = {3, 4, 5}. Find (A △ B) △ C, where △ denotes the symmetric difference. | tiger_lab_math | [
0.046213019639253616,
0.020569415763020515,
-0.04433974623680115,
-0.019676964730024338,
0.03846364468336105,
-0.0159074068069458,
-0.023781664669513702,
0.05940581485629082,
0.05419430881738663,
-0.01854965090751648,
0.01240165252238512,
0.020562482997775078,
-0.02523396536707878,
-0.0095... | 0.833078 | 0.830408 | 0.00267 |
Consider the matrix A = [3 -1; 2 5]. Using power iteration method, find the largest eigenvalue and its corresponding eigenvector of A. Start with the initial guess of x1 = [1; 1]. Repeat the process until the eigenvalue estimate changes by less than 0.01%. | tiger_lab_math | [
0.029951879754662514,
0.024428730830550194,
-0.01636119931936264,
-0.043812841176986694,
0.04580651968717575,
-0.018738102167844772,
-0.0022835435811430216,
0.07402768731117249,
0.028383392840623856,
-0.014776247553527355,
0.02561515010893345,
0.025129420682787895,
-0.042107827961444855,
0... | 0.81339 | 0.809878 | 0.003512 |
Alec is running for Class President. He thinks that if he can get three-quarters of the class to vote for him then there is no chance anyone else can beat him. Half of the class have already said they will vote for him but out of the remaining students, only 5 have said they are thinking about voting for him. He survey... | tiger_lab_math | [
0.014904872514307499,
0.008805536665022373,
-0.03065965138375759,
-0.042596422135829926,
0.02540658414363861,
-0.05402053892612457,
-0.022290389984846115,
0.041062187403440475,
0.010482569225132465,
-0.003643334610387683,
0.007065277546644211,
0.03762023150920868,
-0.04153981804847717,
-0.... | 0.813309 | 0.808557 | 0.004751 |
Suppose the following system of differential equations models the Belousov-Zhabotinsky reaction:
$\frac{dx}{dt}=a(y-x)-p(x^2-y^2)+F,$
$\frac{dy}{dt}=bx-xz-x,$
$\frac{dz}{dt}=xy-cz,$
where $a = 2.5,$ $b=0.01,$ $p=5,$ $c=0.1,$ $F=0.5.$
Given the initial conditions $x(0)=1,$ $y(0)=2,$ and $z(0)=3,$ find the ... | tiger_lab_math | [
0.029814472422003746,
0.018726611509919167,
-0.04987121373414993,
-0.04000815376639366,
0.030801616609096527,
-0.013317440636456013,
-0.01665770262479782,
0.08738141506910324,
0.03839024901390076,
-0.019301077350974083,
-0.003672471269965172,
0.031220797449350357,
-0.032684896141290665,
-0... | 0.83195 | 0.826605 | 0.005344 |
The asymptotes of a hyperbola are $y = x + 1$ and $y = 3 - x.$ Also, the hyperbola passes through $(3,3).$ Find the distance between the foci of the hyperbola. | tiger_lab_math | [
0.023680103942751884,
0.0015994170680642128,
-0.04388565570116043,
-0.040186330676078796,
0.03201363608241081,
-0.037181563675403595,
-0.03750009462237358,
0.05077941343188286,
0.034517571330070496,
-0.029099030420184135,
0.03686705604195595,
0.025826333090662956,
-0.046070851385593414,
-0... | 0.829364 | 0.825533 | 0.003831 |
Let X be a complex algebraic curve defined as X = V(y^2 - x^3 + 1) and let F be the sheaf of holomorphic functions on X. Find H^1(X, F). | tiger_lab_math | [
0.013901873491704464,
0.01955519989132881,
-0.04147773236036301,
-0.0506536066532135,
0.03773007541894913,
-0.024795573204755783,
-0.0069289254024624825,
0.07252226769924164,
0.0514419823884964,
-0.010515467263758183,
0.01065831072628498,
0.03569081053137779,
-0.036147113889455795,
-0.0043... | 0.829927 | 0.817627 | 0.0123 |
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