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 |
|---|---|---|---|---|---|
Suppose a company has three factories located at coordinates $(0,0)$, $(10,5)$, and $(6,8)$, respectively. The company needs to transport goods from these factories to three warehouses located at coordinates $(5,5)$, $(10,10)$, and $(1,9)$, respectively. However, there is a chance that a route may be disrupted due to u... | tiger_lab_math | [
-0.009961094707250595,
0.01718692108988762,
-0.040149372071027756,
-0.02523047849535942,
0.03492994233965874,
-0.001879749819636345,
-0.041988201439380646,
0.06004191190004349,
0.027356624603271484,
-0.012933507561683655,
0.008320032618939877,
0.026793353259563446,
-0.023168904706835747,
-... | 0.821539 | 0.797153 | 0.024386 |
Consider the Hénon map given by the equations:
x<sub>n+1</sub> = 1 - a(x<sub>n</sub><sup>2</sup>) + y<sub>n</sub>
y<sub>n+1</sub> = bx<sub>n</sub>
where a = 1.4 and b = 0.3. For the initial condition (x<sub>0</sub>, y<sub>0</sub>) = (0, 0) find the first four iterations (i.e, n = 1, 2, 3, 4) of the Hénon map. | tiger_lab_math | [
0.0329640656709671,
0.039420485496520996,
-0.02819155342876911,
-0.05275551974773407,
0.04949786886572838,
-0.026504553854465485,
-0.025274304673075676,
0.07841779291629791,
0.024789700284600258,
-0.01207505352795124,
0.013853819109499454,
0.02074507623910904,
-0.032764285802841187,
-0.027... | 0.827423 | 0.812142 | 0.01528 |
"statement 1": What is the minimum number of packages of Aero flight tees Bill must purchase to ensure that he has at least 10 golf tees for each member of his foursome , if he will buy no more than 3 packages of the generic golf tees ?, "statement 2" :What is the minimum number of packages of Aero flight tees Bill mu... | tiger_lab_math | [
0.018856454640626907,
0.016256557777523994,
-0.056464456021785736,
-0.03321437910199165,
0.052891191095113754,
-0.025371942669153214,
-0.013183221220970154,
0.05019430071115494,
0.018602045252919197,
-0.011095404624938965,
-0.0007184932474046946,
0.02873433567583561,
-0.031116750091314316,
... | 0.813039 | 0.811932 | 0.001106 |
A manufacturing company needs to transport products from 5 warehouses to 8 retailers. The company wants to minimize the total transportation cost while meeting the demand of the retailers. The transportation cost from each warehouse to each retailer is given in the table below. The company must transport all the produc... | tiger_lab_math | [
0.033410150557756424,
0.021360188722610474,
-0.0476168729364872,
-0.013851461932063103,
0.04167577996850014,
-0.012321497313678265,
-0.04291068762540817,
0.04870348796248436,
0.031368423253297806,
-0.0028953126166015863,
0.006885289680212736,
0.03375673666596413,
-0.011218097060918808,
-0.... | 0.800857 | 0.796778 | 0.004079 |
For each positive integer,the quantitySn is defined such thatSn+3=(Sn)^2 -S(n+1).in addition ,S2=1.then which of the following must be true.
Answer Choices: (A) a) S4 +S3 +1=0 (B) b) (S1) ^2 +S4 +2=0 (C) c) ifS1=-1thenS4 is also equal to-1. (D) d) ifS4 =-7thenS1=-2. (E) e) ifS1 =-3thenS4=-7 | tiger_lab_math | [
0.025096680968999863,
0.023325316607952118,
-0.04540687054395676,
-0.04009085148572922,
0.04029783234000206,
-0.046388108283281326,
-0.01866416446864605,
0.04880703240633011,
0.02513047866523266,
-0.01907355710864067,
0.014370075426995754,
0.04236593842506409,
-0.048454686999320984,
-0.034... | 0.839038 | 0.835844 | 0.003195 |
Consider a two-player game where each player can choose to play either A or B. If both players choose A, each gets 2 units of payoff, if both players choose B, each gets 3 units of payoff, and if the players choose differently, each gets 0 units of payoff.
Write down the payoff matrix and find the Nash equilibrium of ... | tiger_lab_math | [
0.05427275970578194,
-0.014542057178914547,
-0.043480996042490005,
-0.012318898923695087,
0.02425793744623661,
-0.03317366912961006,
-0.022564971819519997,
0.06688370555639267,
0.04699647054076195,
-0.03521000221371651,
0.0320582315325737,
0.029714038595557213,
-0.04058300703763962,
-0.027... | 0.823148 | 0.812423 | 0.010725 |
Consider the function $f:\mathbb{R}^2 \to \mathbb{R}$ given by $f(x,y) = x^2+y^2$. Find all critical points of $f$ and classify them as local maxima, local minima, or saddle points using Morse theory. | tiger_lab_math | [
0.018560465425252914,
0.02482716739177704,
-0.04665420204401016,
-0.03693627193570137,
0.034408994019031525,
-0.015804175287485123,
-0.026200275868177414,
0.05551024153828621,
0.02517865039408207,
-0.0013916569296270609,
0.013163993135094643,
0.029354682192206383,
-0.02536713145673275,
-0.... | 0.831879 | 0.813352 | 0.018527 |
The hypotenuse of a right triangle measures $6\sqrt{2}$ inches and one angle is $45^{\circ}$. What is the number of square inches in the area of the triangle? | tiger_lab_math | [
0.005647583864629269,
0.03599579632282257,
-0.0451531819999218,
-0.05052660033106804,
0.03481138497591019,
-0.0014170889044180512,
-0.015110247768461704,
0.055371612310409546,
0.04224392771720886,
-0.019262071698904037,
0.030238280072808266,
0.03275534138083458,
-0.027161473408341408,
-0.0... | 0.826075 | 0.823803 | 0.002273 |
Suppose two players, Alice and Bob, are playing the matching pennies game. Alice chooses "heads" or "tails" by placing a penny on her left or right hand, and Bob guesses which hand it's under. If Bob guesses correctly, he wins the penny from Alice; if he guesses wrongly, he gives Alice a penny. If both players randomly... | tiger_lab_math | [
0.005071722902357578,
0.011040403507649899,
-0.046870891004800797,
-0.016567911952733994,
0.01776271127164364,
-0.018363377079367638,
-0.018189746886491776,
0.03309138864278793,
0.04933922365307808,
-0.023728232830762863,
0.021164514124393463,
0.03339735046029091,
-0.027393944561481476,
-0... | 0.828852 | 0.823585 | 0.005267 |
A space curve in three-dimensional space is given by the equation:
r(t) = (2t, 3t^2, 4t^3)
Find the curvature and torsion of this curve at the point where t = 1. | tiger_lab_math | [
0.013598019257187843,
0.03946225717663765,
-0.033586300909519196,
-0.0487041175365448,
0.04021219536662102,
-0.011008623987436295,
-0.01886567659676075,
0.06236065924167633,
0.023935643956065178,
-0.026395436376333237,
0.008745545521378517,
0.019794663414359093,
-0.03738955780863762,
-0.01... | 0.831601 | 0.824783 | 0.006818 |
The optimization problem is to minimize the function f(x) = x^2 + 1 using the firefly algorithm. Use an initial population size of 10, a maximum iteration count of 50 and a light absorption coefficient of 0.5. Find the minimum value of f(x) and the corresponding value of x. | tiger_lab_math | [
-0.0000682975587551482,
0.01853993348777294,
-0.04308830574154854,
-0.04260563477873802,
0.043908994644880295,
-0.04935986176133156,
-0.02694912999868393,
0.08632219582796097,
0.04965055733919144,
-0.010520408861339092,
0.006410966627299786,
-0.00024305997067131102,
-0.028750013560056686,
... | 0.818243 | 0.817083 | 0.00116 |
The graphs of two functions, $p(x)$ and $q(x),$ are shown here on one set of axes: [asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3;
real who... | tiger_lab_math | [
0.011364759877324104,
0.030441375449299812,
-0.03786642104387283,
-0.03917037695646286,
0.026637734845280647,
-0.021416760981082916,
-0.01856520026922226,
0.07765645533800125,
0.061870377510786057,
-0.008947636932134628,
0.018530000001192093,
0.028673946857452393,
-0.015820929780602455,
0.... | 0.813919 | 0.804379 | 0.00954 |
Suppose there are two players, John and Jane, playing a game of chess, and they both know that they have a 50% chance of winning. John is asked to predict whether he will win or lose, and then he must tell Jane what he predicted. However, John can lie about his prediction. If John predicts he will win, then he gets a r... | tiger_lab_math | [
0.004441045690327883,
0.004522127564996481,
-0.06177360191941261,
-0.020474281162023544,
0.016521746292710304,
-0.014204893261194229,
-0.013945378363132477,
0.056706588715314865,
0.04151757061481476,
-0.029916711151599884,
0.0285943653434515,
0.025357943028211594,
-0.0452583022415638,
-0.0... | 0.815761 | 0.794951 | 0.020809 |
Here is a precise numerical problem on Cluster analysis:
A group of 100 customers were surveyed and their annual income and age were recorded. Apply k-means clustering algorithm to group these customers into 3 clusters based on their income and age.
The recorded data is given in the following table:
| Customer | Ann... | tiger_lab_math | [
0.016514379531145096,
0.021641256287693977,
-0.045048195868730545,
-0.040912188589572906,
0.03393998369574547,
-0.00027653411962091923,
-0.022308126091957092,
0.045569829642772675,
0.034939054399728775,
0.01926233060657978,
0.025929845869541168,
0.008332950994372368,
-0.03336251527070999,
... | 0.804793 | 0.797513 | 0.00728 |
The region shown is bounded by the arcs of circles having radius 4 units, having a central angle measure of 60 degrees and intersecting at points of tangency. The area of the region can be expressed in the form $a\sqrt{b}+c\pi$ square units, where $\sqrt{b}$ is a radical in simplest form. What is the value of $a + b + ... | tiger_lab_math | [
0.026036560535430908,
0.03784327581524849,
-0.04212527722120285,
-0.030119638890028,
0.03546379506587982,
-0.030046548694372177,
-0.02068919502198696,
0.0576234869658947,
0.04679180681705475,
0.008932635188102722,
0.02346341498196125,
0.037061698734760284,
-0.025039957836270332,
-0.0331620... | 0.835827 | 0.820444 | 0.015383 |
Suppose that f(z) is a complex-valued function that is analytic in the closed unit disc $\overline{D}$ (the set of complex numbers z with |z| <= 1). Assume that |f(z)| <= 5 for all z in $\overline{D}$. What is the maximum value that |f(0)| can attain? | tiger_lab_math | [
0.017823614180088043,
0.02730819396674633,
-0.04911399632692337,
-0.029186032712459564,
0.031788911670446396,
0.004227441269904375,
-0.004703804384917021,
0.048074569553136826,
0.04477035254240036,
0.007224082015454769,
0.025152824819087982,
0.023788297548890114,
-0.03234340623021126,
-0.0... | 0.827143 | 0.812904 | 0.014239 |
A study compares the scores of three different teaching methods (A, B, and C) on a test. The scores are normally distributed and the sample sizes for each group are equal.
Method A: 70, 80, 75, 85, 90
Method B: 60, 70, 75, 80, 85
Method C: 65, 75, 80, 85, 90
Calculate the ANOVA table and determine if there is a s... | tiger_lab_math | [
0.013361290097236633,
-0.002105764113366604,
-0.0466466061770916,
-0.034230347722768784,
0.029388118535280228,
-0.04201650992035866,
-0.00003011403896380216,
0.08658462017774582,
0.03507940098643303,
-0.012602187693119049,
-0.0031116597820073366,
0.02281983196735382,
-0.038462817668914795,
... | 0.79669 | 0.790505 | 0.006185 |
Consider the metric on $\mathbb{R}^3$ given by $$g = dx^2 + dy^2 + e^{2z}dz^2.$$ Determine the Levi-Civita connection of this metric. | tiger_lab_math | [
0.0010927689727395773,
0.029998254030942917,
-0.03413495421409607,
-0.04777328670024872,
0.025495050475001335,
-0.011984817683696747,
-0.012927609495818615,
0.04092850908637047,
0.030962254852056503,
-0.007788400165736675,
0.015170550905168056,
0.03332637995481491,
-0.03428630903363228,
-0... | 0.820347 | 0.80972 | 0.010628 |
Consider the complex function $f(z) = \frac{z+2}{(z-1)(z+3)^2}$. Determine the order of the pole at $z=-3$ and the order and residue of the pole at $z=1$. | tiger_lab_math | [
0.03769850730895996,
0.03804195299744606,
-0.037158284336328506,
-0.04677271470427513,
0.021378541365265846,
-0.02330050617456436,
-0.0016858591698110104,
0.03991859406232834,
0.024821104481816292,
-0.007982594892382622,
0.0283514354377985,
0.033913154155015945,
-0.053247202187776566,
-0.0... | 0.835471 | 0.832096 | 0.003375 |
Consider the following rational function:
f(x, y) = (4x^2 + 4xy + y^2)/(3x^2 + 3y^2)
1. Find all the points (x, y) in the affine plane such that f(x,y) = 1.
2. Determine whether there exists a rational function g(x,y) such that f = g^2, and explain your reasoning. | tiger_lab_math | [
0.03497663140296936,
0.04194946587085724,
-0.036731839179992676,
-0.04776681214570999,
0.05541296303272247,
-0.03723369911313057,
-0.007692073006182909,
0.051890406757593155,
0.03683171421289444,
-0.0027235988527536392,
0.021105563268065453,
0.026583867147564888,
-0.03628621995449066,
-0.0... | 0.836684 | 0.822077 | 0.014606 |
Let G be a group of 20 elements. Determine whether G is isomorphic to the cyclic group C20, the dihedral group D10, or the non-abelian group of order 20, and justify your answer. | tiger_lab_math | [
0.009329472668468952,
0.016075609251856804,
-0.041548628360033035,
-0.031944390386343,
0.026971355080604553,
-0.0668385848402977,
-0.012918266467750072,
0.049897514283657074,
0.0346507765352726,
-0.01890576258301735,
0.019658850505948067,
0.02113383635878563,
-0.05156043916940689,
-0.00840... | 0.812215 | 0.811426 | 0.000789 |
A plumber is trying to decide which of three different jobs he should take to make the most money. The first job involves fixing three toilets and three sinks at an apartment complex. The second involves two toilets and five sinks at a different apartment complex. The third involves fixing one toilet, two showers an... | tiger_lab_math | [
0.001305450452491641,
0.026907114312052727,
-0.04291082173585892,
-0.023227090016007423,
0.04356355592608452,
-0.010275958105921745,
-0.04625852033495903,
0.048459213227033615,
0.0232352577149868,
-0.007645926438271999,
0.028755448758602142,
0.04095592722296715,
-0.038519229739904404,
-0.0... | 0.833524 | 0.824518 | 0.009006 |
The task for the student is to calculate the fundamental group of the space consisting of a torus with a point removed.
To write the problem numerically, we can ask:
What is the fundamental group of a torus with one point removed? | tiger_lab_math | [
0.041426606476306915,
0.03138919547200203,
-0.0251279529184103,
-0.046289943158626556,
0.0428168848156929,
-0.027846064418554306,
-0.005697225220501423,
0.05044087767601013,
0.05124436691403389,
0.006235918495804071,
0.012350835837423801,
0.03369218483567238,
-0.04704921692609787,
-0.02836... | 0.82296 | 0.817574 | 0.005386 |
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 | [
0.03411035239696503,
0.010513345710933208,
-0.029299085959792137,
-0.023744603618979454,
0.05479123815894127,
-0.03508034721016884,
-0.020902302116155624,
0.059097595512866974,
0.0393543615937233,
-0.0056798155419528484,
0.0134140495210886,
0.029505910351872444,
-0.04404851421713829,
-0.02... | 0.831988 | 0.828548 | 0.00344 |
Consider a manifold with the following metric tensor in standard spherical coordinates:
$ds^2 = dr^2 + r^2 d\theta^2 + r^2\sin^2(\theta) d\phi^2$
Calculate the scalar curvature of this manifold. | tiger_lab_math | [
-0.009454395622015,
0.02670256234705448,
-0.03790966048836708,
-0.0620124377310276,
0.036733273416757584,
-0.01976788602769375,
-0.015996543690562248,
0.05544239282608032,
0.03617158532142639,
-0.01696649380028248,
-0.004201182164251804,
0.024159178137779236,
-0.036539554595947266,
-0.0061... | 0.816619 | 0.805489 | 0.01113 |
Here's a precise numerical problem for a math student to solve in Epistemic logic:
"Alice claims that she knows a number between 1 and 100. Bob also claims that he knows a number between 1 and 100. However, they don't know each other's number. Later, Alice claims that she doesn't know Bob's number. If Alice is telling... | tiger_lab_math | [
0.027451565489172935,
0.002635400975123048,
-0.03215155005455017,
-0.035388387739658356,
0.022819288074970245,
-0.027492618188261986,
-0.009359643794596195,
0.040706224739551544,
0.03117717243731022,
-0.0259034913033247,
0.020268013700842857,
0.03688433766365051,
-0.024508019909262657,
-0.... | 0.836512 | 0.826353 | 0.010158 |
A fluid has a velocity field given by $u(x,y,z) = 3y^2z$ and a density distribution given by $\rho(x,y,z) = 2 + \cos(x+y)$. Find the Fourier transform of the velocity field in the $x$ and $y$ directions. | tiger_lab_math | [
0.020355181768536568,
0.029591407626867294,
-0.0438251793384552,
-0.0294724740087986,
0.02181350253522396,
-0.019072625786066055,
-0.042749304324388504,
0.04827754572033882,
0.02591663971543312,
0.0019276741659268737,
0.03405788540840149,
0.028386281803250313,
-0.019073378294706345,
-0.026... | 0.815903 | 0.814307 | 0.001596 |
A piece of string fits exactly once around the perimeter of a square whose area is 144. Rounded to the nearest whole number, what is the area of the largest circle that can be formed from the piece of string? | tiger_lab_math | [
0.024053726345300674,
0.03966241702437401,
-0.02573506347835064,
-0.04287080466747284,
0.026654062792658806,
-0.027697771787643433,
-0.022972481325268745,
0.0218745656311512,
0.024064358323812485,
-0.009358899667859077,
0.027687590569257736,
0.035389143973588943,
-0.022897804155945778,
-0.... | 0.829559 | 0.82769 | 0.00187 |
A manufacturing company produces two products, A and B. The profit per unit for A is $5 while for B it's $4. The company has a total of 200 units of raw materials and 300 hours of production time available. Product A takes 2 units of raw materials and 3 hours of production time, while product B requires 1 unit of raw m... | tiger_lab_math | [
0.012724497355520725,
0.007490754593163729,
-0.04472147673368454,
-0.016958149150013924,
0.016527332365512848,
-0.020052732899785042,
-0.06856821477413177,
0.08100330084562302,
0.022147266194224358,
-0.014867715537548065,
0.006657462101429701,
0.031942322850227356,
-0.007819474674761295,
-... | 0.810377 | 0.806292 | 0.004085 |
When a class of math students lined up for a class picture on Picture Day, they found that when they stood in rows of four there was one person left over. When they formed rows of five students, there were two extra students, and when they lined up in rows of seven, there were three students left over. What is the fewe... | tiger_lab_math | [
0.019786642864346504,
0.001818373566493392,
-0.04460718482732773,
-0.04877897724509239,
0.042062386870384216,
-0.032712314277887344,
-0.03557010740041733,
0.04442239925265312,
0.045648783445358276,
-0.023364270105957985,
0.008150231093168259,
0.011167296208441257,
-0.018197426572442055,
-0... | 0.825852 | 0.823329 | 0.002524 |
Consider a Stackelberg duopoly game where the demand function is P = 100 - Q, where P is the price and Q is the quantity. Firm 1 (the leader) has a constant marginal cost of $20, while firm 2 (the follower) has a constant marginal cost of $40. Determine the profit-maximizing quantity and price for both firms in the Sta... | tiger_lab_math | [
0.01785980351269245,
0.004879822954535484,
-0.05428372696042061,
-0.03697042167186737,
0.02545831725001335,
0.0019840486347675323,
-0.009525999426841736,
0.06682775169610977,
0.050489578396081924,
-0.03980530425906181,
0.02001604624092579,
0.03486905246973038,
-0.04320184513926506,
-0.0185... | 0.808961 | 0.796251 | 0.012711 |
Let $\alpha$ and $\beta$ be conjugate complex numbers such that $\frac{\alpha}{\beta^2}$ is a real number and $|\alpha - \beta| = 2 \sqrt{3}.$ Find $|\alpha|.$ | tiger_lab_math | [
0.01492382027208805,
0.041570231318473816,
-0.05424180626869202,
-0.027089502662420273,
0.024855272844433784,
-0.008113750256597996,
-0.0053495378233492374,
0.055945079773664474,
0.04331677034497261,
-0.026842201128602028,
0.012578191235661507,
0.03521116077899933,
-0.03697977215051651,
-0... | 0.848603 | 0.825954 | 0.022648 |
Let X be an elliptic curve defined by the equation y^2 = x^3 + ax + b over the finite field F_p, where p is a prime and a, b are elements of F_p. Let P be a point on X with order n, i.e., nP = 0. If n is an odd prime and Q is a point on X with the property that 2Q = P, then what is the order of Q in terms of n? | tiger_lab_math | [
0.009228610433638096,
0.02407902479171753,
-0.04199334233999252,
-0.05410999059677124,
0.0070735989138484,
-0.030996549874544144,
-0.0024112947285175323,
0.04207415506243706,
0.03575712814927101,
-0.020718414336442947,
-0.0026391444262117147,
0.037271928042173386,
-0.0552917905151844,
-0.0... | 0.838805 | 0.827207 | 0.011598 |
A construction company wants to connect 6 cities in a region with minimum cost. The distance (in kilometers) between each pair of cities is given below:
| | City 1 | City 2 | City 3 | City 4 | City 5 | City 6 |
|-----|--------|--------|--------|--------|--------|--------|
|City 1| 0 | 4 | 2 | 999999 ... | tiger_lab_math | [
0.0038488979917019606,
0.021405497565865517,
-0.031590256839990616,
-0.04855269566178322,
0.03728943690657616,
-0.01835930161178112,
-0.02568732388317585,
0.04183357208967209,
0.023961424827575684,
-0.018367795273661613,
0.031172290444374084,
0.036028698086738586,
-0.019340161234140396,
-0... | 0.819043 | 0.815703 | 0.00334 |
Suppose that $a$ is inversely proportional to $b$. Let $a_1,a_2$ be two nonzero values of $a$ such that $\frac{a_1}{a_2}=\frac{2}{3}$. Let the corresponding $b$ values be $b_1,b_2$. If $b_1,b_2$ are nonzero, find the value of $\frac{b_1}{b_2}$. Please respond by writing a program in Python. | tiger_lab_math | [
0.00529760867357254,
0.039516761898994446,
-0.039478421211242676,
-0.028824007138609886,
0.021945221349596977,
-0.006511926185339689,
0.005044910591095686,
0.06344446539878845,
0.03928644210100174,
-0.005029106512665749,
0.022373199462890625,
0.04157369211316109,
-0.047555457800626755,
-0.... | 0.839124 | 0.83045 | 0.008674 |
Let $M$ be a complex manifold of complex dimension 2, whose Chern classes are given by $c_1(M)=3H$ and $c_2(M)=7H^2$, where $H$ is the hyperplane class of $\mathbb{CP}^2$. Find the values of the other Chern classes $c_3(M), c_4(M),$ and $c_5(M)$. | tiger_lab_math | [
0.0169044379144907,
0.03060939535498619,
-0.030483238399028778,
-0.0595170296728611,
0.03238769993185997,
-0.010146861895918846,
-0.011681688018143177,
0.04515421390533447,
0.04675007611513138,
-0.016260601580142975,
0.012157736346125603,
0.0282739344984293,
-0.03543506935238838,
-0.024003... | 0.822973 | 0.8081 | 0.014873 |
Sherlock Holmes and Dr. Watson recover a suitcase with a three-digit combination lock from a mathematician turned criminal. Embedded in the suitcase above the lock is the cryptic message "AT SEA BASE. SEAS EBB SEA: BASS. "
Dr. Watson comments, "This probably isn't about ocean fish. Perhaps it is an encrypted message. ... | tiger_lab_math | [
0.042715225368738174,
0.016825752332806587,
-0.03512701019644737,
-0.04959387704730034,
0.03160642832517624,
-0.015607067383825779,
-0.021082419902086258,
0.0434577614068985,
0.02614218182861805,
-0.0009401552379131317,
0.026363641023635864,
0.03397654369473457,
-0.03410553187131882,
-0.02... | 0.822439 | 0.812832 | 0.009607 |
A furniture manufacturer produces tables at a cost of $100 each. The manufacturer currently sells the tables at a price of $200 each and sells 100 tables per month. The demand for tables decreases by 10% for each $10 increase in price. What is the optimal price that the manufacturer should charge to maximize its profit... | tiger_lab_math | [
0.0258546844124794,
0.005247604567557573,
-0.06660837680101395,
-0.013171513564884663,
0.028836941346526146,
-0.027777275070548058,
-0.029524561017751694,
0.05894855782389641,
0.048018112778663635,
-0.002405059989541769,
0.006136057432740927,
0.017906323075294495,
-0.0519544780254364,
-0.0... | 0.805511 | 0.801245 | 0.004265 |
Statement 1 | The order of a cyclic group is equal to the number of its elements. Statement 2 | A group of order 28 always has a subgroup of order 14. Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True | tiger_lab_math | [
0.026261966675519943,
0.010105462744832039,
-0.03499303758144379,
-0.043645478785037994,
0.03757854923605919,
-0.03980445861816406,
-0.011307383887469769,
0.042618487030267715,
0.031204909086227417,
-0.017120910808444023,
0.012051181867718697,
0.008469469845294952,
-0.027238618582487106,
-... | 0.814147 | 0.808117 | 0.00603 |
Here is a precise numerical problem on Residue calculus and its applications for the student to solve:
Find the residue of the function f(z) = (z^2 + 2) / (z^3 + 3z) at its poles and use the Residue theorem to evaluate the integral ∮γ f(z) dz over the positively oriented closed contour γ defined by the unit circle |z|... | tiger_lab_math | [
0.04821336641907692,
0.015981659293174744,
-0.03540024906396866,
-0.03437358886003494,
0.04870181530714035,
-0.008653360418975353,
-0.0034632980823516846,
0.04392608255147934,
0.03165639191865921,
-0.0013220008695498109,
0.026939237490296364,
0.02003152295947075,
-0.04451528564095497,
-0.0... | 0.825422 | 0.815641 | 0.009781 |
Consider a graph with 8 vertices and the following degree sequence: {3, 3, 3, 2, 2, 1, 1, 1}. Can this sequence be realized by a simple undirected graph? If so, draw such a graph. If not, explain why not. | tiger_lab_math | [
0.004139349330216646,
0.02955295704305172,
-0.013569866307079792,
-0.03454011678695679,
0.024634724482893944,
-0.020944422110915184,
-0.02792433090507984,
0.06976139545440674,
0.02281751111149788,
-0.02169557474553585,
0.026542190462350845,
0.024900594726204872,
-0.02080964855849743,
-0.02... | 0.821203 | 0.809546 | 0.011657 |
Let A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7} be two sets. Find the symmetric difference of A and B using the given formula:
(A ∆ B) ∩ C = (A ∩ C) ∆ (B ∩ C) for any set C. | tiger_lab_math | [
0.054101403802633286,
0.01535499282181263,
-0.04140772297978401,
-0.018986603245139122,
0.0371539443731308,
-0.0359799824655056,
-0.02911173738539219,
0.05237213894724846,
0.06488759815692902,
-0.02403406612575054,
0.012425483204424381,
0.024572519585490227,
-0.042311202734708786,
-0.01444... | 0.829631 | 0.822746 | 0.006885 |
In the game Deal or No Deal, participants choose a box at random from a set of $26,$ one containing each of the following values: \begin{tabular}{|c|c|}\hline\$.01&\$1,000\\\hline\$1&\$5,000\\\hline\$5&\$10,000\\\hline\$10&\$25,000\\\hline\$25&\$50,000\\\hline\$50&\$75,000\\\hline\$75&\$100,000\\\hline\$100&\$200,000\\... | tiger_lab_math | [
0.01564217358827591,
0.000734701519832015,
-0.038891784846782684,
-0.03705504909157753,
0.01804780587553978,
-0.008123461157083511,
-0.0299516711384058,
0.04126507788896561,
0.04091518372297287,
-0.02501845359802246,
0.005673198029398918,
0.021643495187163353,
-0.024097885936498642,
-0.029... | 0.826693 | 0.817809 | 0.008884 |
Triangle $ABC$ has sides of $6$ units, $8$ units, and $10$ units. The width of a rectangle, whose area is equal to the area of the triangle, is $4$ units. What is the perimeter of this rectangle, in units? | tiger_lab_math | [
0.029989786446094513,
0.018566057085990906,
-0.04663196578621864,
-0.03124195709824562,
0.029688948765397072,
-0.030926650390028954,
-0.04340605437755585,
0.04396744444966316,
0.04231792315840721,
-0.013161602430045605,
0.022380584850907326,
0.034502170979976654,
-0.02945478819310665,
-0.0... | 0.82774 | 0.820331 | 0.007409 |
Given the complex function $f(z) = \frac{1}{z^2-1}$, find the singularities of $f(z)$ and classify them as removable, pole or essential singularities. | tiger_lab_math | [
0.02893306501209736,
0.04245540872216225,
-0.01686473935842514,
-0.04606270790100098,
0.028338845819234848,
-0.026585828512907028,
0.005695100873708725,
0.05470428615808487,
0.03377756103873253,
0.012358903884887695,
0.0202137790620327,
0.0075079756788909435,
-0.02598409727215767,
-0.00561... | 0.817211 | 0.811844 | 0.005367 |
How many different sets of natural numbers are there that satisfy the following second-order logic sentence?
∃X ∀Y (Y ∈ X ↔ ∃Z (Y = 2Z))
Here, ∃ means "there exists", ∀ means "for all", ∈ means "is a member of", ↔ means "if and only if", and Z is a variable that can take on any value. | tiger_lab_math | [
0.03731610253453255,
-0.012447038665413857,
-0.0403154119849205,
-0.03463474661111832,
0.0269461702555418,
-0.03036096692085266,
-0.023626646026968956,
0.036922283470630646,
0.04410824179649353,
-0.02703426592051983,
0.0183451846241951,
0.02906416729092598,
-0.030493853613734245,
-0.010710... | 0.812639 | 0.81115 | 0.001489 |
A disease is spreading through a population of 1000 people, and the rate of new infections is proportional to the number of infected individuals and the number of susceptible individuals. The disease has an incubation period of 2 days and an infectious period of 5 days. Assume that there are initially 50 infected indiv... | tiger_lab_math | [
0.02296135015785694,
0.006191650405526161,
-0.04904576763510704,
-0.026232924312353134,
0.04124744236469269,
-0.03990074619650841,
-0.023287957534193993,
0.06560584902763367,
0.019459333270788193,
-0.026637759059667587,
0.020446360111236572,
0.027840811759233475,
-0.021419936791062355,
-0.... | 0.805823 | 0.798181 | 0.007642 |
The legs of a right triangle have lengths $\log_4 27$ and $\log_2 9.$ If the length of the hypotenuse is $h,$ compute $4^h.$ Let's write a Python program. | tiger_lab_math | [
0.0147150419652462,
0.027442632243037224,
-0.03564434498548508,
-0.02797624096274376,
0.026117686182260513,
-0.03330710157752037,
-0.02566765993833542,
0.0394451767206192,
0.04819345846772194,
-0.016386888921260834,
0.026963787153363228,
0.04271729663014412,
-0.025566117838025093,
-0.02201... | 0.840208 | 0.83506 | 0.005149 |
company c sells a line of 20 products with an average retail price of $ 1,200 . if none of these products sells for less than $ 400 , and exactly 10 of the products sell for less than $ 1,000 , what is the greatest possible selling price of the most expensive product ? Let's write a Python program to solve it. | tiger_lab_math | [
0.022437548264861107,
0.016991667449474335,
-0.048968929797410965,
-0.014751510694622993,
0.03717118129134178,
-0.025850465521216393,
-0.03633909672498703,
0.04090641811490059,
0.03294742852449417,
-0.023601394146680832,
0.024284040555357933,
0.03653910756111145,
-0.03968565911054611,
-0.0... | 0.812335 | 0.807365 | 0.00497 |
Here is a problem from Modal Logic:
Consider a model M with three worlds: w1, w2, and w3. The accessibility relation R is defined such that R(w1, w2), R(w1, w3), and R(w2, w3). At each world, the proposition p is true if and only if it is at least as true as q. Determine whether the following formula is true in the mo... | tiger_lab_math | [
0.03230780363082886,
0.012554064393043518,
-0.017334815114736557,
-0.04683369770646095,
0.03262075036764145,
-0.042569175362586975,
-0.03234808146953583,
0.07618547976016998,
0.0181271955370903,
-0.03401310369372368,
-0.0006682205712422729,
0.0384334921836853,
-0.006147152744233608,
-0.026... | 0.810441 | 0.802547 | 0.007894 |
Here is a precise numerical problem for a math student to solve in relation to Cournot competition in Game theory:
Consider two firms, A and B, producing identical goods. The total cost of producing q units of the good for each firm is given by C(q) = 10q. The inverse demand function for the good is given by P = 100 -... | tiger_lab_math | [
0.03530625253915787,
0.012957085855305195,
-0.04147979989647865,
-0.018958397209644318,
0.01139187440276146,
-0.00008563767187297344,
-0.03586779534816742,
0.060126908123493195,
0.028205759823322296,
-0.028158769011497498,
0.012522545643150806,
0.031526751816272736,
-0.043478939682245255,
... | 0.828319 | 0.809832 | 0.018487 |
Maximize the objective function f(x,y) = 3x + 2y subject to the constraints:
- x + y ≤ 5
- x + 2y ≤ 8
- x ≥ 0
- y ≥ 0
- The matrix [[1,x,y],[x,4,y],[y,y,z]] is positive semi-definite.
Find the maximum value of f(x,y) and the corresponding values of x and y that achieve the maximum. | tiger_lab_math | [
0.03289474919438362,
0.018055612221360207,
-0.053964853286743164,
-0.023645997047424316,
0.03486684709787369,
-0.0039021603297442198,
-0.04077637568116188,
0.0528118759393692,
0.03391622006893158,
-0.004389016889035702,
0.010376008227467537,
0.035088084638118744,
-0.03274531289935112,
-0.0... | 0.818194 | 0.80724 | 0.010954 |
Consider the surface given by the equation z = x^2 + y^2. Find the sectional curvature at the point (1,1,2) in the direction of the vector v = 3i + 4j + 2k. | tiger_lab_math | [
0.011218643747270107,
0.03231656178832054,
-0.04301098361611366,
-0.044696781784296036,
0.04311719164252281,
-0.0034143298398703337,
-0.007952763698995113,
0.06627684086561203,
0.02121121622622013,
-0.009475878439843655,
0.027606867253780365,
0.03293872997164726,
-0.03049646131694317,
0.00... | 0.825282 | 0.818792 | 0.00649 |
Let $\mathbf{f}(u,v,w) = (u^2, uv, u+w)$ be a parametrization of a surface $S$ on the manifold $\mathbb{R}^3$. Compute the exterior derivative of the 1-form $\omega = 2u\,du + v\,dv + w\,dw$ on the manifold $S$. | tiger_lab_math | [
-0.0021331224124878645,
0.04109211266040802,
-0.059795595705509186,
-0.044690728187561035,
0.03279715031385422,
-0.010545454919338226,
-0.017454352229833603,
0.0762374997138977,
0.03096895106136799,
-0.0007352869724854827,
0.014794153161346912,
0.04730970785021782,
-0.04108768329024315,
-0... | 0.818919 | 0.806053 | 0.012865 |
Suppose $f(z)$ is an analytic function on the lower half plane $\text{Im}(z) < 0$ with real part $\text{Re}(z)>0$ such that it extends continuously to the real axis and satisfies $f(z) = \overline{f(\overline{z})}$ for all $z$ on the real axis. If $f(1+i) = 2$, find $f(2+i)$. | tiger_lab_math | [
0.027340734377503395,
0.033739037811756134,
-0.04296862334012985,
-0.03229768946766853,
0.027332322672009468,
0.0028013382107019424,
-0.000862005865201354,
0.06767396628856659,
0.03101932816207409,
0.0113410335034132,
0.01970817707479,
0.014747785404324532,
-0.0419108085334301,
-0.00035265... | 0.837694 | 0.812389 | 0.025305 |
Let X = {1, 2, 3, 4, 5} be a set and let τ be the topology on X given by τ = {∅, {1}, {1, 2, 3, 4, 5}}. Determine whether the space (X, τ) is connected and path-connected. | tiger_lab_math | [
0.03725506365299225,
0.024369854480028152,
-0.05158120393753052,
-0.0220323633402586,
0.04895060881972313,
-0.019378986209630966,
-0.02079818956553936,
0.05211152508854866,
0.044430240988731384,
0.004083413165062666,
0.020493710413575172,
0.024431414902210236,
-0.024529723450541496,
-0.011... | 0.81974 | 0.806164 | 0.013577 |
A company produces two products: chairs and tables. The production of each chair requires 2 hours of labor and 3 units of raw material, while the production of each table requires 3 hours of labor and 5 units of raw material. The company has 200 hours of labor and 300 units of raw material available per day. The chairs... | tiger_lab_math | [
0.025937581434845924,
0.012142354622483253,
-0.06277353316545486,
0.002069731242954731,
0.020267391577363014,
-0.0066911447793245316,
-0.025500981137156487,
0.09038439393043518,
0.03132305666804314,
-0.009017626754939556,
0.013679630123078823,
0.029643764719367027,
-0.0192033089697361,
-0.... | 0.809714 | 0.798433 | 0.011281 |
Laura is training for a triathlon, but she doesn't feel like swimming. She bikes $20$ miles at $2x+1$ miles per hour, spends five minutes parking her bike, changing to her running shoes, and catching a drink of water, and then runs $5$ miles at $x$ miles per hour. Her total workout lasts $110$ minutes. How fast did Lau... | tiger_lab_math | [
0.02805519662797451,
0.01885392516851425,
-0.04762743040919304,
-0.019621074199676514,
0.027002334594726562,
-0.008859464898705482,
-0.039316046983003616,
0.06425833702087402,
0.05052194371819496,
-0.017092451453208923,
0.01215589139610529,
0.016334736719727516,
-0.024884557351469994,
-0.0... | 0.824215 | 0.822503 | 0.001712 |
A delicious circular pie with diameter $12\text{ cm}$ is cut into three equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $l^2$? Let's write a Python program to solve it. | tiger_lab_math | [
-0.006491811014711857,
0.02600288949906826,
-0.04054153338074684,
-0.04587825387716293,
0.03578534349799156,
-0.008016319945454597,
-0.03155585750937462,
0.036345019936561584,
0.033808913081884384,
-0.013795072212815285,
0.026570510119199753,
0.036445897072553635,
-0.02043461985886097,
-0.... | 0.844511 | 0.834055 | 0.010456 |
Consider the complex projective plane algebraically defined by the equation $x^3 + y^3 + z^3 = 0$. Determine the Hodge numbers $h^{p,q}$ of this projective plane. | tiger_lab_math | [
0.023875469341874123,
0.03207392990589142,
-0.04052754491567612,
-0.05357543006539345,
0.03789509832859039,
-0.020213009789586067,
-0.008485094644129276,
0.03258154168725014,
0.04448149353265762,
-0.025359857827425003,
0.01683410443365574,
0.02402975782752037,
-0.0369369313120842,
-0.04024... | 0.829235 | 0.815399 | 0.013836 |
If $\mathbf{a},$ $\mathbf{b},$ and $\mathbf{c}$ are unit vectors, then find the largest possible value of
\[\|\mathbf{a} - \mathbf{b}\|^2 + \|\mathbf{a} - \mathbf{c}\|^2 + \|\mathbf{b} - \mathbf{c}\|^2.\]Note: A unit vector is a vector of magnitude 1. Let's write a Python program to solve it. | tiger_lab_math | [
0.03320670500397682,
0.04766962677240372,
-0.054452914744615555,
-0.03969726711511612,
0.03949107602238655,
-0.015085075050592422,
-0.029092323035001755,
0.06356899440288544,
0.03331895172595978,
-0.0192891675978899,
0.01387257780879736,
0.025458987802267075,
-0.03262919560074806,
-0.02825... | 0.827646 | 0.810125 | 0.01752 |
A telecommunications company needs to set up towers in three different locations to provide coverage for a specific region. The company has six possible locations where they can set up the towers, but only three towers are available. Each tower has a limited range of coverage and will cost a certain amount to set up.
... | tiger_lab_math | [
0.010831625200808048,
0.038379330188035965,
-0.06737497448921204,
-0.032229483127593994,
0.040160611271858215,
-0.0067038764245808125,
-0.028729325160384178,
0.06557845324277878,
0.03175761178135872,
-0.019829830154776573,
0.018700530752539635,
0.03978145122528076,
-0.028325289487838745,
-... | 0.823843 | 0.79708 | 0.026763 |
Sure, here's an example:
A manufacturing company needs to determine the minimum cost of producing 1000 units of a product. The cost is modeled by the function C(x) = 10000 + 200x + 0.5x^2, where x is the number of units produced. Using particle swarm optimization, what is the minimum cost of producing 1000 units? | tiger_lab_math | [
0.004190743900835514,
0.019731251522898674,
-0.041457220911979675,
-0.01784157007932663,
0.021599141880869865,
-0.04191471263766289,
-0.035192616283893585,
0.042746130377054214,
0.037265270948410034,
-0.01149249542504549,
0.016551267355680466,
0.02398819476366043,
-0.012052752077579498,
-0... | 0.805366 | 0.803797 | 0.001569 |
Sure, here's an example problem:
A town has a population of 10,000 people. At the beginning of a flu outbreak, 100 people are infected. The rate at which the flu spreads is proportional to the product of the number of infected people and the number of susceptible people. The rate constant is 0.001. What is the maximum... | tiger_lab_math | [
0.02630920149385929,
0.023774057626724243,
-0.037718359380960464,
-0.026376638561487198,
0.03006652742624283,
-0.04347554221749306,
-0.02125602588057518,
0.06386515498161316,
0.018610045313835144,
-0.017855888232588768,
0.010398313403129578,
0.05009560286998749,
-0.03315476328134537,
-0.02... | 0.818557 | 0.814234 | 0.004323 |
Consider a Riemannian manifold with the following metric:
$ds^2 = \frac{4}{5}(dx^1)^2 + \frac{4}{5}(dx^2)^2 + \frac{1}{5}(dx^3)^2$
Find the Levi-Civita connection for this metric at the point $(0,0,0)$. | tiger_lab_math | [
0.002477743662893772,
0.03332311287522316,
-0.025912459939718246,
-0.04155375063419342,
0.025393448770046234,
-0.0035519772209227085,
-0.009798914194107056,
0.0543152391910553,
0.031828541308641434,
-0.002429808024317026,
-0.00003252245733165182,
0.026626700535416603,
-0.02481854148209095,
... | 0.834858 | 0.821144 | 0.013713 |
Sure, here's a question for the student to solve:
Consider two players who are deciding whether to cooperate or defect in a Prisoner's Dilemma game. The payoffs matrix for this game is as follows:
| | Player 2 cooperates | Player 2 defects |
|---------|-------------------|------------------|
| Player 1 cooper... | tiger_lab_math | [
0.035863615572452545,
-0.005069380160421133,
-0.04205207899212837,
-0.03209559991955757,
0.01061693113297224,
-0.025623392313718796,
-0.023478535935282707,
0.05862194672226906,
0.0386759489774704,
-0.028588535264134407,
0.02105174958705902,
0.02490440383553505,
-0.038548193871974945,
-0.03... | 0.829175 | 0.820497 | 0.008678 |
Suppose there are 5 farmers who share a water source, and they can choose to irrigate their land for one or two months. Each farmer gets a profit of $1000/month for irrigation, but if all 5 of them irrigate for 2 months, the water will be depleted and they will suffer a loss of $2000 each. What is the optimal decision ... | tiger_lab_math | [
0.01781131699681282,
0.02618330344557762,
-0.061998579651117325,
-0.01633761264383793,
0.016309194266796112,
-0.045792169868946075,
-0.02780936472117901,
0.05957552418112755,
0.01705251634120941,
-0.023544693365693092,
0.017436204478144646,
0.006640006322413683,
-0.01702805608510971,
-0.01... | 0.816266 | 0.810832 | 0.005434 |
Consider two algebraic curves given by the equations $y=x^2+1$ and $y=-2x+3$. Find the number of intersection points between the two curves. How many solutions does Bezout's theorem predict for this situation? | tiger_lab_math | [
0.02623043954372406,
0.007772534154355526,
-0.055058322846889496,
-0.045831240713596344,
0.024491438642144203,
-0.010315991938114166,
-0.02526843175292015,
0.04676051437854767,
0.0491495206952095,
-0.03166290372610092,
0.0029066053684800863,
0.027644211426377296,
-0.016290046274662018,
-0.... | 0.840362 | 0.831017 | 0.009345 |
Sure, here's a question for the math student to solve:
In a group of volunteers, each member can choose to either donate $5 or refrain from donating. If at least half of the group donates, the entire group will receive a prize of $50. Otherwise, the prize will not be awarded. Assume that there are seven volunteers in ... | tiger_lab_math | [
0.01829221285879612,
0.004144859500229359,
-0.04764493927359581,
-0.025382859632372856,
0.030809249728918076,
-0.03649769723415375,
-0.016576018184423447,
0.027847448363900185,
0.034019485116004944,
-0.015208101831376553,
0.0076314364559948444,
0.016650954261422157,
-0.031693484634160995,
... | 0.826613 | 0.817636 | 0.008977 |
Let C be the algebraic curve defined by the equation y^2 = x^3 - 3x + 1, and let J(C) be its Jacobian variety. Find the dimension of J(C) as a vector space over the field of complex numbers. | tiger_lab_math | [
0.022904422134160995,
0.01480991579592228,
-0.05787452682852745,
-0.0413164384663105,
0.03040180541574955,
-0.036938928067684174,
-0.012946543283760548,
0.06553567200899124,
0.03438520058989525,
-0.009836429730057716,
0.005356972571462393,
0.042741984128952026,
-0.04933128505945206,
-0.012... | 0.836231 | 0.823863 | 0.012367 |
The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base $10$, where digit a is not zero. Find the three-digit number $abc$.
| tiger_lab_math | [
0.02768142893910408,
0.003919216804206371,
-0.028420340269804,
-0.037151336669921875,
0.04988199844956398,
-0.025725625455379486,
-0.019809888675808907,
0.0351065918803215,
0.027795124799013138,
-0.012634580954909325,
0.02474549226462841,
0.018494486808776855,
-0.043635591864585876,
-0.028... | 0.845522 | 0.835525 | 0.009997 |
Let X be a complex manifold given by the equation x^2 + y^2 + z^2 = 1 in C^3, and suppose that X is equipped with the standard orientation. Compute the first and second Chern classes of X. | tiger_lab_math | [
0.015268146991729736,
0.025998881086707115,
-0.026555465534329414,
-0.0477699413895607,
0.024091655388474464,
-0.014806794933974743,
-0.009395236149430275,
0.03879210725426674,
0.03343106433749199,
-0.011672375723719597,
0.017740342766046524,
0.02081175707280636,
-0.04729507118463516,
-0.0... | 0.823016 | 0.807767 | 0.015249 |
Let
\[f(a,b) = \left\{
\renewcommand{\arraystretch}{3}
\begin{array}{cl}
\dfrac{ab - a + 2}{2a} & \text{if $a + b \le 3$}, \\
\dfrac{ab - b - 2}{-2b} & \text{if $a + b > 3$}.
\end{array}
\renewcommand{\arraystretch}{1}
\right.\]Find $f(2,1) + f(2,4).$ Please respond by writing a program in Python. | tiger_lab_math | [
0.030731268227100372,
0.03555699437856674,
-0.03805980831384659,
-0.04969068989157677,
0.03993932157754898,
-0.006050752941519022,
-0.021473834291100502,
0.07237053662538528,
0.02196248434484005,
0.010540401563048363,
0.023072702810168266,
0.04227761551737785,
-0.019302519038319588,
-0.015... | 0.825945 | 0.818401 | 0.007544 |
Consider a surface S defined by the equation z = 3x^2 + 2y^2. Find the total Gaussian curvature of the surface over the entire domain of x between -1 to 1 and y between -2 to 2, by using the Gauss-Bonnet Theorem. | tiger_lab_math | [
0.005756445694714785,
0.014391477219760418,
-0.04361451789736748,
-0.05402306839823723,
0.044449225068092346,
-0.019896095618605614,
-0.012390709482133389,
0.05583072826266289,
0.011457271873950958,
-0.01674380525946617,
0.022259047254920006,
0.057309992611408234,
-0.023693960160017014,
-0... | 0.824424 | 0.814246 | 0.010178 |
Let $M$ be a smooth manifold of dimension 3 with coordinates $(x,y,z)$, and let there be a metric $g$ defined on $M$ given by:
$$ g = dx^2 + dy^2 + \frac{1}{2}e^{2z}dz^2 $$
Calculate the Cartan connection of $M$ with respect to this metric $g$. | tiger_lab_math | [
0.023291658610105515,
0.03685837611556053,
-0.03841662406921387,
-0.05274767428636551,
0.03817024081945419,
0.0002517347747925669,
-0.021987801417708397,
0.042234327644109726,
0.032965369522571564,
0.002118837321177125,
0.003645499236881733,
0.042742952704429626,
-0.045035410672426224,
-0.... | 0.823412 | 0.807568 | 0.015843 |
Here's a numerical problem for a math student to solve related to Maxwell's equations:
A circular loop of wire has a radius of 5 cm and is placed in a uniform magnetic field of 0.5 T. If the loop has a total resistance of 10 ohms, find the current induced in the loop when the magnetic field changes at a rate of 200 T/... | tiger_lab_math | [
0.015429090708494186,
0.029773859307169914,
-0.0563042089343071,
-0.057218361645936966,
0.03398039937019348,
-0.0325450524687767,
-0.0499611534178257,
0.06854990869760513,
0.037601541727781296,
0.0003784699074458331,
0.006430876906961203,
0.008425649255514145,
-0.018441561609506607,
-0.045... | 0.811787 | 0.800188 | 0.011599 |
Suppose you have a portfolio of four stocks with initial investments of $10,000 in Alphabet Inc. (GOOGL), $15,000 in Amazon.com, Inc. (AMZN), $7,000 in Microsoft Corporation (MSFT), and $8,000 in Facebook, Inc. (FB). Given the annualized mean returns and standard deviations for each stock as follows:
| Stock | Mean Re... | tiger_lab_math | [
0.011961063370108604,
0.00563328992575407,
-0.0578920915722847,
-0.04371098801493645,
0.03794650733470917,
-0.03293316066265106,
-0.03234002739191055,
0.05356273800134659,
0.03470325097441673,
-0.008298852480947971,
0.014544054865837097,
0.02473253197968006,
-0.02692977525293827,
-0.030332... | 0.812334 | 0.804376 | 0.007958 |
Consider a population of 100,000 individuals in a town. A new infectious disease has just emerged and it is spreading rapidly. The rate of spread of the disease can be modeled using the differential equation:
dS/dt = -0.001SI
where S represents the number of susceptible individuals and I represents the number of infec... | tiger_lab_math | [
0.010761669836938381,
0.013048338703811169,
-0.04886965453624725,
-0.027366112917661667,
0.044886514544487,
-0.03999845311045647,
-0.03629937767982483,
0.07656319439411163,
0.035829637199640274,
-0.014126776717603207,
0.019087666645646095,
0.04368913546204567,
-0.028734814375638962,
-0.012... | 0.810453 | 0.809118 | 0.001335 |
In Computability Theory, let's consider the subset of natural numbers that are perfect squares. Write a program that takes in a natural number as input and outputs 'Yes' if the number is a perfect square, or 'No' if it is not. Use the Python programming language to write this program. | tiger_lab_math | [
0.026651151478290558,
0.018072593957185745,
-0.035869188606739044,
-0.015765968710184097,
0.0387662798166275,
-0.02902897633612156,
-0.005282741971313953,
0.04682254046201706,
0.038097843527793884,
-0.02490357682108879,
0.025390619412064552,
0.024444030597805977,
-0.031559474766254425,
-0.... | 0.814035 | 0.801342 | 0.012694 |
There are 6 equally spaced points A,B,C,D,E and F marked on a circle with radius R. How many convex pentagons of distinctly different areas can be drawn using these points as vertices?
Answer Choices: (A) 6P5 (B) 1 (C) 5 (D) None of these (E) Cannot be determined | tiger_lab_math | [
0.025353724136948586,
0.03863152489066124,
-0.036397822201251984,
-0.025340918451547623,
0.025809556245803833,
-0.03697272762656212,
-0.010818371549248695,
0.04402992129325867,
0.043266527354717255,
-0.02096976526081562,
0.02377038076519966,
0.05388195440173149,
-0.026132766157388687,
-0.0... | 0.831827 | 0.827105 | 0.004722 |
Sure, here's a precise numerical problem for a math student to solve -
The maximum value of a multi-variable function f(x,y,z), based on the following constraints is to be computed using the Grey Wolf Optimizer algorithm.
Constraints:
2x + 4y + 6z <= 20
x + 2y + 4z <= 10
x,y,z >= 0
Function:
f(x,y,z) = 3x^2 - 5y... | tiger_lab_math | [
0.0346716083586216,
0.030299359932541847,
-0.04603688418865204,
-0.039078354835510254,
0.03565886989235878,
-0.009073402732610703,
-0.03565511852502823,
0.03627808764576912,
0.03427832946181297,
-0.009304920211434364,
0.018861442804336548,
0.04268568009138107,
-0.04545633867383003,
-0.0154... | 0.833203 | 0.826005 | 0.007198 |
Suppose $f(z) = z^2 + 2z + 2$ is a complex valued function defined on the complex plane. Find the maximum value of $|f(z)|$ on the circle $|z| = 2$. | tiger_lab_math | [
0.021348200738430023,
0.03632165864109993,
-0.04593544825911522,
-0.0376315638422966,
0.0415264368057251,
-0.010792501270771027,
-0.0092497942969203,
0.03833155706524849,
0.02639281563460827,
-0.005194062367081642,
0.028955239802598953,
0.030359340831637383,
-0.020137082785367966,
-0.00454... | 0.829261 | 0.81439 | 0.014871 |
Circle $A$ is in the interior of circle $B$. The diameter of circle $B$ is 16 cm. What is the diameter of circle $A$ for which the ratio of the shaded area to the area of circle $A$ is 3:1?
[asy]size(101);
import graph;
pair B = (0,0);
pair A = (1,-3);
filldraw(Circle(B,10)^^Circle(A,5),gray(.6)+fillrule(1),linewidth(... | tiger_lab_math | [
0.006319570355117321,
0.03192920237779617,
-0.03771437332034111,
-0.05298649147152901,
0.043124694377183914,
-0.019080456346273422,
-0.030895475298166275,
0.04211167246103287,
0.020495593547821045,
-0.012456605210900307,
0.03221867233514786,
0.04026027396321297,
-0.011208757758140564,
-0.0... | 0.830536 | 0.825682 | 0.004854 |
A researcher wants to test a new fertilizer on tomato plants to see if it will increase their yield. She sets up an experiment where she randomly selects 20 tomato plants and applies the new fertilizer to 10 of them, while leaving the other 10 without any fertilizer. After harvesting the tomatoes, the researcher discov... | tiger_lab_math | [
0.0030975413974374533,
0.013045428320765495,
-0.040879830718040466,
-0.034221235662698746,
0.02554435282945633,
-0.03900633379817009,
-0.00920937955379486,
0.08775235712528229,
0.02634340338408947,
-0.021456314250826836,
-0.00828680582344532,
0.02712840400636196,
-0.037633687257766724,
0.0... | 0.79944 | 0.791398 | 0.008041 |
Three positive integers $a,$ $b,$ and $x$ form an O'Hara triple $(a,b,x)$ if $\sqrt{a}+\sqrt{b}=x.$ For example, $(1,4,3)$ is an O'Hara triple because $\sqrt{1}+\sqrt{4}=3.$
If $(36,25,x)$ is an O'Hara triple, determine the value of $x.$ | tiger_lab_math | [
0.037326499819755554,
0.04164273664355278,
-0.04649597033858299,
-0.038531139492988586,
0.037694282829761505,
-0.017541127279400826,
-0.028792506083846092,
0.05351362004876137,
0.04664379730820656,
0.0030696196481585503,
0.02224782668054104,
0.04401703178882599,
-0.0496884323656559,
-0.042... | 0.844339 | 0.823114 | 0.021224 |
Let $a,$ $b,$ $c$ be distinct integers, and let $\omega$ be a complex number such that $\omega^3 = 1$ and $\omega \neq 1.$ Find the smallest possible value of
\[|a + b \omega + c \omega^2|.\] | tiger_lab_math | [
0.02358105778694153,
0.042248714715242386,
-0.05889444425702095,
-0.034947093576192856,
0.0458374060690403,
-0.02104162983596325,
-0.013848355039954185,
0.05499018356204033,
0.04228904843330383,
-0.018219521269202232,
0.017692184075713158,
0.029762567952275276,
-0.038028083741664886,
-0.04... | 0.86088 | 0.846748 | 0.014132 |
Let X be a smooth, projective algebraic variety of dimension 3 over the complex field, with Hodge numbers h^{p,q}. Suppose the intersection cohomology groups IH^{k}(X) (with integer coefficients) are given by:
IH^{0}(X) = 0,
IH^{1}(X) = H^{3}(X, Z),
IH^{2}(X) = H^{2}(X, Z),
IH^{3}(X) = 0,
IH^{4}(X) = H^{1}(X, Z).
Det... | tiger_lab_math | [
0.030963469296693802,
0.021220050752162933,
-0.03897092491388321,
-0.0455346442759037,
0.03834562748670578,
-0.013839389197528362,
-0.021916227415204048,
0.042463503777980804,
0.04885757714509964,
-0.024788016453385353,
0.016114963218569756,
0.03458359092473984,
-0.03392673656344414,
-0.03... | 0.803782 | 0.780403 | 0.023379 |
It can be shown that for any positive integer $n,$
\[\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} F_{n + 1} & F_n \\ F_n & F_{n - 1} \end{pmatrix},\]where $F_n$ denotes the $n$th Fibonacci number.
Compute $F_{784} F_{786} - F_{785}^2.$ Please respond by writing a program in Python. | tiger_lab_math | [
0.042995695024728775,
0.04294707626104355,
-0.037916216999292374,
-0.04909999296069145,
0.04188145324587822,
-0.038837235420942307,
-0.02919023297727108,
0.054162442684173584,
0.03497615456581116,
-0.01291078794747591,
0.011933520436286926,
0.013127057813107967,
-0.05159001797437668,
-0.01... | 0.838045 | 0.819487 | 0.018558 |
Here is a precise numerical problem for a math student to solve related to the Volunteer's Dilemma in Game Theory:
Suppose three individuals are each deciding whether or not to volunteer for a fundraising event. If no one volunteers, each person earns zero dollars. If one person volunteers, they earn $100 and the oth... | tiger_lab_math | [
0.026591677218675613,
-0.006772881373763084,
-0.04115492105484009,
-0.026521379128098488,
0.020235469564795494,
-0.004903719294816256,
-0.020499959588050842,
0.03539455682039261,
0.04613412916660309,
-0.03671842813491821,
0.019261326640844345,
0.017342517152428627,
-0.036978960037231445,
-... | 0.830056 | 0.820978 | 0.009078 |
The radius $r$ of a circle inscribed within three mutually externally tangent circles of radii $a$, $b$ and $c$ is given by
\[\frac{1}{r} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + 2 \sqrt{\frac{1}{ab} + \frac{1}{ac} + \frac{1}{bc}}.\]What is the value of $r$ when $a = 4$, $b = 9$ and $c = 36$?
[asy]
unitsize(0.15 cm... | tiger_lab_math | [
0.015177038498222828,
0.05447697639465332,
-0.041198261082172394,
-0.031986054033041,
0.03656306862831116,
-0.021974464878439903,
-0.026770349591970444,
0.03508654609322548,
0.02946748025715351,
0.00022785237524658442,
0.022285589948296547,
0.026565995067358017,
-0.016266150400042534,
-0.0... | 0.824707 | 0.8196 | 0.005107 |
Consider a 3-dimensional surface represented by the equation z = x^2 + y^2. Find the scalar curvature of this surface at the point (1, 1, 2). | tiger_lab_math | [
-0.0034560998901724815,
0.03400738537311554,
-0.042066313326358795,
-0.052994802594184875,
0.04520611837506294,
-0.003103886963799596,
-0.014552772045135498,
0.06515970081090927,
0.010684521868824959,
-0.02982143498957157,
0.015679754316806793,
0.03350810706615448,
-0.03998440504074097,
-0... | 0.824817 | 0.815714 | 0.009103 |
Suppose a quantum mechanical system is described by the wavefunction:
ψ(x) = A(exp(-ax^2) + exp(-bx^2))
where A, a and b are constants.
The Fourier transform of this wavefunction is given by:
Φ(k) = √(π/(a+b)) * exp(-k^2/(4(a+b)))
Find the constant A such that the normalization condition is satisfied:
∫|ψ(x)|^2 ... | tiger_lab_math | [
0.017990147694945335,
0.016526728868484497,
-0.037367478013038635,
-0.032292190939188004,
0.0412437841296196,
-0.031079059466719627,
-0.012281973846256733,
0.04992329701781273,
0.04399976134300232,
-0.006680564489215612,
0.021289169788360596,
0.02683810330927372,
-0.007170728873461485,
-0.... | 0.820894 | 0.811497 | 0.009396 |
A bakery sells cakes for $20 each. The cost of making one cake is $10. The owner wants to maximize the profit by finding the optimal price to charge for a cake. Using optimization techniques, calculate the price that will give the bakery the highest profit. | tiger_lab_math | [
0.01424973551183939,
0.024890616536140442,
-0.05444424971938133,
-0.010997610166668892,
0.020970243960618973,
-0.014990131370723248,
-0.04257950931787491,
0.046710506081581116,
0.0452265627682209,
-0.012053420767188072,
0.006280144676566124,
0.01210983470082283,
-0.03948855772614479,
-0.01... | 0.808735 | 0.796841 | 0.011894 |
In a Trust Game, one player (the "truster") is given $10 and can choose to transfer any amount of it to another player (the "trustee"). The amount transfered is then tripled and given to the trustee. The trustee can then choose to return any portion of the tripled amount to the truster. What is the optimal strategy for... | tiger_lab_math | [
0.009189198724925518,
0.007652990520000458,
-0.039201654493808746,
-0.021388843655586243,
0.016037320718169212,
-0.020402826368808746,
-0.0055237216874957085,
0.04562696814537048,
0.03438745066523552,
-0.023636160418391228,
0.025358794257044792,
0.030044155195355415,
-0.05431065335869789,
... | 0.795808 | 0.783079 | 0.012729 |
Suppose
$$a(2+i)^4 + b(2+i)^3 + c(2+i)^2 + b(2+i) + a = 0,$$where $a,b,c$ are integers whose greatest common divisor is $1$. Determine $|c|$. | tiger_lab_math | [
0.036538586020469666,
0.039169181138277054,
-0.04181830957531929,
-0.039031676948070526,
0.04032161459326744,
-0.026136035099625587,
-0.01079379953444004,
0.06412921100854874,
0.04826977849006653,
-0.018197979778051376,
0.005615439731627703,
0.019894475117325783,
-0.03175697103142738,
-0.0... | 0.85401 | 0.832022 | 0.021988 |
Let M be a 3-dimensional Riemannian manifold with metric given by:
$$ ds^2 = 4dx^2 + 9dy^2 + 16dz^2$$
Determine the Ricci curvature of M at the point $(1,2,3)$. | tiger_lab_math | [
-0.0021741429809480906,
0.024374468252062798,
-0.0293582696467638,
-0.05964288488030434,
0.03168286755681038,
-0.0018841699929907918,
-0.01212912518531084,
0.05026789382100105,
0.027563149109482765,
-0.010542969219386578,
0.012499850243330002,
0.03574056550860405,
-0.03597887605428696,
-0.... | 0.829231 | 0.82073 | 0.008501 |
Suppose we have two fuzzy sets A and B defined on a universe of discourse X = [0, 100] as follows:
A(x) = {0.3/30, 0.6/60, 0.9/90} and B(x) = {0.2/50, 0.4/70, 0.8/90}
Find the maximum value of the fuzzy set A ∩ B in terms of the membership function. | tiger_lab_math | [
0.017060179263353348,
0.0034923874773085117,
-0.030519556254148483,
-0.012468624860048294,
0.027008021250367165,
-0.026622865349054337,
-0.0209941565990448,
0.0334882028400898,
0.051865383982658386,
-0.020928142592310905,
0.013095542788505554,
0.03706439584493637,
-0.01768760196864605,
0.0... | 0.8274 | 0.817669 | 0.009731 |
Stephanie is moving into a new apartment. She needs to figure out how many pieces of silverware she should purchase. She needs spoons, butter knives, steak knives, and forks. For herself she figures 5 of each would be sufficient. But in case she has guests, she wants to have 10 extra pieces of each type. Then she reali... | tiger_lab_math | [
0.015517584048211575,
0.02130059152841568,
-0.05214715749025345,
-0.024415984749794006,
0.050251975655555725,
-0.021146200597286224,
-0.03467441722750664,
0.058506812900304794,
0.0456792414188385,
0.0036982151214033365,
0.01167347002774477,
0.03479362651705742,
-0.04711710661649704,
-0.012... | 0.844901 | 0.8394 | 0.005502 |
Consider the following graph with 7 vertices and 11 edges:
```
1 -- 2 -- 3 -- 4 -- 5
| | / | \ | |
6 -- 7 8 9 -- 10
| |
11 12
```
1. Determine if the graph has an Eulerian path or circuit.
2. If the graph has a Hamiltonian path or circuit, find one. If it does... | tiger_lab_math | [
0.014315825887024403,
0.007307248655706644,
-0.017288409173488617,
-0.03200168162584305,
0.02570931427180767,
-0.044560469686985016,
-0.03312494978308678,
0.06591243296861649,
0.02041991427540779,
-0.003948004450649023,
0.010719766840338707,
0.028118440881371498,
-0.017961887642741203,
-0.... | 0.820078 | 0.811939 | 0.00814 |
Let A = {1, 2, 3} and B = {4, 5, 6}. Find the values in the set (AxB)∪(AxC) if C = {7, 8, 9}. | tiger_lab_math | [
0.041105903685092926,
0.02234436199069023,
-0.04449833557009697,
-0.017450863495469093,
0.034579019993543625,
-0.03665460646152496,
-0.007218156009912491,
0.044785305857658386,
0.0614931657910347,
-0.03325134888291359,
0.017806747928261757,
0.030431918799877167,
-0.04022783413529396,
-0.00... | 0.846775 | 0.841304 | 0.005471 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.