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 a topological quantum field theory over a closed oriented manifold of dimension 2. Let the partition function of this theory be given by:
Z(M) = ∑p,q,r (1 + (-1)^p + (-1)^q + (-1)^r)χ(M; p,q,r)
where χ(M; p,q,r) denotes the Euler characteristic of the moduli space of flat G-bundles on M with holonomy in the... | tiger_lab_math | [
0.02649666927754879,
0.02295810356736183,
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0.02706787921488285,
0.0021012513898313046,
-0.034495189785957336,
0.046147044748067856,
0.030602773651480675,
0.00442360108718276,
-0.0024389750324189663,
0.023027507588267326,
-0.0328100249171257,
-0.... | 0.797315 | 0.78558 | 0.011735 |
On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling ... | tiger_lab_math | [
0.011132723651826382,
0.025357171893119812,
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0.022437505424022675,
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0.08333440124988556,
0.041533492505550385,
-0.0201733261346817,
-0.0003258945362176746,
0.016627147793769836,
-0.009508789516985416,
... | 0.848682 | 0.831644 | 0.017039 |
A point $P$ is chosen at random in the interior of equilateral triangle $ABC$. What is the probability that $\triangle ABP$ has a greater area than each of $\triangle ACP$ and $\triangle BCP$? | tiger_lab_math | [
0.009107954800128937,
0.010308493860065937,
-0.037525344640016556,
-0.03327294439077377,
0.024623122066259384,
-0.022146426141262054,
-0.020428452640771866,
0.04514995962381363,
0.03831961378455162,
-0.012053732760250568,
0.005597306415438652,
0.04252445325255394,
-0.04160286486148834,
-0.... | 0.836059 | 0.819433 | 0.016627 |
Point $D$ lies on side $AC$ of equilateral triangle $ABC$ such that the measure of angle $DBC$ is $45$ degrees. What is the ratio of the area of triangle $ADB$ to the area of triangle $CDB$? Express your answer as a common fraction in simplest radical form. | tiger_lab_math | [
0.03254495561122894,
0.002923061605542898,
-0.03776983171701431,
-0.057060398161411285,
0.0183649193495512,
-0.020749220624566078,
0.005117019172757864,
0.05659782886505127,
0.04881782457232475,
-0.007212498225271702,
0.022841567173600197,
0.026170596480369568,
-0.026374543085694313,
-0.01... | 0.831003 | 0.825623 | 0.00538 |
Let M be a 3-dimensional manifold with the standard Euclidean metric, and let ω = dx ∧ dy + dy ∧ dz + dz ∧ dx be a 2-form on M. Calculate the Hodge star of ω with respect to the metric on M. | tiger_lab_math | [
-0.002051380928605795,
0.013446296565234661,
-0.033151108771562576,
-0.06622426211833954,
0.0381678007543087,
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-0.03017948754131794,
0.07322467118501663,
0.04294060170650482,
-0.018380407243967056,
0.0007441487978212535,
0.045285213738679886,
-0.023422876372933388,
-0... | 0.815219 | 0.803172 | 0.012046 |
In a combinatorial game, two players take turns removing exactly 1 or 2 stones from a pile. The player who removes the last stone wins.
If there is a starting pile of 7 stones, who has the winning strategy? | tiger_lab_math | [
0.05060604587197304,
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0.0604109950363636,
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0.019000742584466934,
0.02970278076827526,
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-0.018... | 0.824407 | 0.819811 | 0.004596 |
Let
\[f(x) = x^3 + 6x^2 + 16x + 28.\]The graphs of $y = f(x)$ and $y = f^{-1}(x)$ intersect at exactly one point $(a,b).$ Enter the ordered pair $(a,b).$ | tiger_lab_math | [
0.04588699713349342,
0.02192719839513302,
-0.05628030374646187,
-0.042826466262340546,
0.0436604805290699,
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0.06416042149066925,
0.04738660901784897,
-0.011401161551475525,
0.018146224319934845,
0.02268272265791893,
-0.021022802218794823,
-0.03174... | 0.842319 | 0.826418 | 0.015901 |
A triangle has sides of lengths 7, 10, and 13. Find the measure of the largest angle in the triangle using the law of cosines. | tiger_lab_math | [
0.03496363013982773,
0.02506537362933159,
-0.025634625926613808,
-0.04888693988323212,
0.02938983216881752,
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-0.02668660134077072,
0.03046569414436817,
0.008124676533043385,
-0.01043619029223919,
0.04286768287420273,
0.034791912883520126,
-0.03520901873707771,
-0.0048... | 0.82308 | 0.821502 | 0.001578 |
Consider the following graph:
1. Is there an Eulerian path in this graph? If yes, find one.
2. Is there a Hamiltonian path in this graph? If yes, find one.
3. What is the maximum number of edges that can be added to this graph without creating a Hamil... | tiger_lab_math | [
0.003817020682618022,
0.012037865817546844,
-0.019581031054258347,
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0.03531615808606148,
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-0.016171637922525406,
0.07171183824539185,
0.024174761027097702,
-0.013430608436465263,
0.012383423745632172,
0.00996322464197874,
-0.017491942271590233,
-0... | 0.822916 | 0.811434 | 0.011483 |
Consider the conformal mapping $f(z)=\frac{z-2i}{z+i}$ from the upper half-plane $\mathbb{H}$ onto the unit disk $\mathbb{D}$ in the complex plane. Using this mapping, find the image under $f$ of the half-line $\mathbb{R}^+ = \{x+0i: x>0\}$ in the upper half-plane $\mathbb{H}$. | tiger_lab_math | [
0.01349851954728365,
0.03216004744172096,
-0.03411242738366127,
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0.019678233191370964,
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0.005062807351350784,
0.06538442522287369,
0.046828072518110275,
0.0011341215576976538,
0.014844407327473164,
0.01058186311274767,
-0.00660356879234314,
-0.01... | 0.824173 | 0.798651 | 0.025522 |
Here's a numerical problem related to the chaotic dynamics of the Henon-Heiles system:
Consider the Henon-Heiles potential given by
$V(x,y) = \frac{1}{2}(x^2+y^2) + x^2y -\frac{1}{3}y^3$
1. Find the equations of motion for this system.
2. Choose an initial condition $(x_0,y_0)$ and use numerical methods to integra... | tiger_lab_math | [
0.02061316929757595,
0.020726051181554794,
-0.02365254983305931,
-0.03149234130978584,
0.04328424111008644,
-0.02031836286187172,
-0.03478160873055458,
0.07275009155273438,
0.036058373749256134,
-0.00960493367165327,
0.011981199495494366,
0.024352265521883965,
-0.02264484576880932,
-0.0374... | 0.817511 | 0.804667 | 0.012844 |
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$?
$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286$
| tiger_lab_math | [
0.026337960734963417,
0.0024794999044388533,
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0.01741260103881359,
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0.001079533714801073,
0.03954966366291046,
0.02956470474600792,
-0.013124021701514721,
0.014003674499690533,
0.053448621183633804,
-0.03712836280465126,
-0.0... | 0.850493 | 0.846559 | 0.003934 |
an equilateral triangle t 2 is formed by joining the mid points of the sides of another equilateral triangle t 1 . a third equilateral triangle t 3 is formed by joining the mid - points of t 2 and this process is continued indefinitely . if each side of t 1 is 60 cm , find the sum of the perimeters of all the triangles... | tiger_lab_math | [
0.02659614570438862,
0.014738724566996098,
-0.02485586330294609,
-0.03709407150745392,
0.03392067179083824,
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0.04890871047973633,
0.028693292289972305,
-0.017260011285543442,
0.027174703776836395,
0.04701518639922142,
-0.03882306069135666,
-0.031... | 0.82702 | 0.811917 | 0.015103 |
Let's say we have an elliptic curve defined by the equation y² = x³ - 7x +10 over the field Z17.
Now suppose that point A(7,12) is on the curve and we want to find the result of 27A where 27 is an integer and A is point. Use Double and Add algorithm to solve it. What is the resulting point in affine coordinates? | tiger_lab_math | [
0.01314232312142849,
0.03023619018495083,
-0.020202413201332092,
-0.047481469810009,
0.033665843307971954,
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0.05018322914838791,
0.027769362553954124,
-0.01859302818775177,
-0.0000833548983791843,
0.043349459767341614,
-0.05277298390865326,
-0.01... | 0.83245 | 0.82834 | 0.00411 |
Oliver is trying to decide between two different sized boxes of macaroni and cheese. He wants to know which is the better value, so instead of just looking at the overall price, he looks at the price per ounce. The larger box is 30 ounces and the smaller box is 20 ounces. The larger box costs $4.80. The smaller box cos... | tiger_lab_math | [
0.0053289495408535,
0.018107961863279343,
-0.04717991501092911,
-0.018577737733721733,
0.036505796015262604,
-0.01762664131820202,
-0.035765331238508224,
0.05772177502512932,
0.05326889827847481,
-0.006779084447771311,
0.02772311121225357,
0.01464332826435566,
-0.05631180480122566,
-0.0021... | 0.801966 | 0.801157 | 0.000809 |
Consider a closed symplectic 2-form $\omega$ on $\mathbb{R}^4$, given by $\omega = dx\wedge dy + dz\wedge dw$.
Determine whether the following vector fields are symplectic with respect to $\omega$:
(a) $X = x \frac{\partial}{\partial y} + y \frac{\partial}{\partial x} + z \frac{\partial}{\partial w} + w \frac{\parti... | tiger_lab_math | [
0.017602970823645592,
0.03919236734509468,
-0.03163696080446243,
-0.040672075003385544,
0.033325113356113434,
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-0.004222812131047249,
0.06285849213600159,
0.02646862156689167,
-0.013619639910757542,
0.004662611056119204,
0.03208678588271141,
-0.01828000321984291,
-0.0... | 0.816006 | 0.80052 | 0.015486 |
A company produces two products, X and Y. The profit for product X is $5 per unit and for product Y is $7 per unit. The production constraints demand that production of product X can't exceed 500 units while production of product Y can't exceed 400 units. The production process also requires that for every unit of X pr... | tiger_lab_math | [
0.01934320107102394,
0.009461876004934311,
-0.0384303480386734,
-0.007436092477291822,
0.012626098468899727,
-0.01814081519842148,
-0.041911572217941284,
0.08133044838905334,
0.03793402761220932,
-0.003839976852759719,
-0.005407552234828472,
0.021567760035395622,
-0.010900718159973621,
-0.... | 0.806431 | 0.783307 | 0.023123 |
Suppose we have a moduli space M_g of genus g algebraic curves. Let C be a curve with equation y^2=x^3+2x.
What is the dimension of the space of all first-order deformations of C in M_1?
(Hint: The space of first-order deformations can be described as the tangent space at C to the moduli space M_1 of elliptic curve... | tiger_lab_math | [
0.014336672611534595,
0.02707788534462452,
-0.05571147799491882,
-0.06543286889791489,
0.03387391194701195,
-0.024294903501868248,
-0.021216215565800667,
0.05310353636741638,
0.03196171671152115,
-0.023151805624365807,
-0.013715368695557117,
0.023880034685134888,
-0.038407083600759506,
-0.... | 0.829842 | 0.820795 | 0.009047 |
Sure, here's a numerical problem for a math student to solve in the context of solving optimization problems using tabu search:
A company has to assign 10 employees to 5 different tasks in a way that maximizes productivity. Each employee has a different skill set that makes them more suitable for certain tasks. Each t... | tiger_lab_math | [
0.023371022194623947,
0.01216026023030281,
-0.05425010621547699,
-0.01692449115216732,
0.033340491354465485,
-0.022404154762625694,
-0.02702426165342331,
0.04768575355410576,
0.03279869630932808,
-0.0015280537772923708,
0.003453948302194476,
0.009992039762437344,
-0.031125275418162346,
-0.... | 0.83431 | 0.825993 | 0.008318 |
Given a matrix A =
\begin{pmatrix}
2 & 1\\
1 & 2
\end{pmatrix}
Determine a matrix P where $P^{-1}AP$ is a diagonal matrix. | tiger_lab_math | [
0.024709053337574005,
0.024730369448661804,
-0.0335114449262619,
-0.04650530964136124,
0.0352102667093277,
-0.01967730186879635,
-0.005443258676677942,
0.08064243197441101,
0.02144041657447815,
0.005519460886716843,
0.010250808671116829,
0.020188970491290092,
-0.02972506359219551,
-0.01046... | 0.823207 | 0.809086 | 0.014121 |
A company produces two products, X and Y. The profit per unit of X and Y are $8 and $10, respectively. The production process of X requires 2 hours of machine time and 1 hour of man time, while production process of Y requires 1 hour of machine time and 3 hours of man time. The company has 200 hours of machine time and... | tiger_lab_math | [
0.024227064102888107,
0.014848447404801846,
-0.05650871992111206,
-0.0029550448525696993,
0.012788889929652214,
-0.013753487728536129,
-0.04081819951534271,
0.08623045682907104,
0.03662872314453125,
-0.006223081145435572,
0.009335830807685852,
0.03734045848250389,
-0.016381502151489258,
-0... | 0.818469 | 0.808827 | 0.009642 |
Determine the geodesic equation for the surface of a sphere with radius of 5 meters. Find the shortest path (geodesic) between two points on the sphere located at (1, 0, 0) and (-1, 0, 0) in terms of the polar angle $\theta$. | tiger_lab_math | [
0.0005147476331330836,
0.016714734956622124,
-0.03969164192676544,
-0.04397675767540932,
0.0331253744661808,
-0.02828000672161579,
-0.047021184116601944,
0.047985099256038666,
0.048790715634822845,
-0.005117709282785654,
0.016590962186455727,
0.020622866228222847,
-0.008113292045891285,
-0... | 0.808028 | 0.806899 | 0.001129 |
miller street begins at baker street and runs directly east for 4.5 kilometers until it ends when it meets turner street . miller street is intersected every 250 meters by a perpendicular street , and each of those streets other than baker street and turner street is given a number beginning at 1 st street ( one block ... | tiger_lab_math | [
0.03148965165019035,
0.0171990767121315,
-0.013211164623498917,
-0.0024895595852285624,
0.023986687883734703,
0.004928655922412872,
-0.033086709678173065,
0.027303548529744148,
0.0609903484582901,
0.0011467479635030031,
0.026777850463986397,
0.03762643411755562,
-0.020674824714660645,
-0.0... | 0.791282 | 0.789538 | 0.001744 |
For the gala luncheon planning committee, Margaret wants to serve chicken salad sandwiches using mini croissants. She wants there to be enough food that each person on the committee can have 2 sandwiches each. Her bakery offers 12 minis croissants for $8.00. There are 24 people on the committee. How much will she sp... | tiger_lab_math | [
0.008394998498260975,
0.02916332334280014,
-0.03716057538986206,
-0.03842517361044884,
0.02613360807299614,
-0.022767940536141396,
-0.018527960404753685,
0.03815522789955139,
0.036475859582424164,
-0.0332462452352047,
0.015778087079524994,
0.025623885914683342,
-0.04157049581408501,
-0.027... | 0.810563 | 0.808207 | 0.002355 |
The six faces of a three-inch wooden cube are each painted red. The cube is then cut into one-inch cubes along the lines shown in the diagram. How many of the one-inch cubes have red paint on at least two faces? [asy]
pair A,B,C,D,E,F,G;
pair a,c,d,f,g,i,j,l,m,o,p,r,s,u,v,x,b,h;
A=(0.8,1);
B=(0,1.2);
C=(1.6,1.3);
D=... | tiger_lab_math | [
0.040295861661434174,
0.05107564479112625,
-0.04682864993810654,
-0.01826840080320835,
0.04172981157898903,
-0.028778858482837677,
-0.021771596744656563,
0.05027354881167412,
0.000009066084203368519,
-0.00731954351067543,
0.016233228147029877,
0.03698253631591797,
-0.007970430888235569,
-0... | 0.826242 | 0.821932 | 0.00431 |
Find the number of distinct numbers in the list
\[\left\lfloor \frac{1^2}{1000} \right\rfloor, \ \left\lfloor \frac{2^2}{1000} \right\rfloor, \ \left\lfloor \frac{3^2}{1000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{1000} \right\rfloor.\] | tiger_lab_math | [
0.030248921364545822,
0.0461813285946846,
-0.03927517309784889,
-0.028781097382307053,
0.03088458813726902,
-0.031300295144319534,
-0.012996161356568336,
0.038248348981142044,
0.051146000623703,
-0.013408047147095203,
0.01389264315366745,
0.010380360297858715,
-0.03551064059138298,
-0.0193... | 0.821601 | 0.818339 | 0.003262 |
Consider the temporal logic formula Fp, where p is a proposition. Let S = {1,2,3,4} and R = {<1,2>, <2,3>, <3,4>, <4,1>}. Determine if the model <S, R, V> satisfies Fp, where V(p)={1,4}. | tiger_lab_math | [
0.02418532408773899,
-0.0029949271120131016,
-0.024790121242403984,
-0.029202502220869064,
0.04669296368956566,
-0.0394885279238224,
-0.013694624416530132,
0.0652407556772232,
0.028292810544371605,
-0.0071524945087730885,
0.02335357293486595,
0.04318724200129509,
-0.011509887874126434,
-0.... | 0.818637 | 0.81381 | 0.004827 |
Let X and Y be two affine algebraic varieties in $\mathbb{A}^3$ defined by $X: x^2+y^2+z^2=1$ and $Y: x+y+z=0$. Find the degree of the intersection of X and Y. | tiger_lab_math | [
0.030277300626039505,
0.013174104504287243,
-0.044593293219804764,
-0.044378623366355896,
0.027296392247080803,
-0.01590123027563095,
-0.004861234687268734,
0.0550573356449604,
0.042711690068244934,
-0.019455518573522568,
0.024603616446256638,
0.02281389944255352,
-0.035305559635162354,
-0... | 0.834338 | 0.816854 | 0.017484 |
Find the differential structure of the surface S defined by the equation z = x^3 + y^3 - 3xy for all (x, y) in the xy-plane. | tiger_lab_math | [
-0.002015636535361409,
0.010837477631866932,
-0.06496523320674896,
-0.05080875754356384,
0.035110052675008774,
-0.02706257626414299,
-0.015717195346951485,
0.05616747587919235,
0.040866006165742874,
-0.013606014661490917,
0.03492823988199234,
0.03796892613172531,
-0.04654940962791443,
-0.0... | 0.827184 | 0.824008 | 0.003176 |
Consider the rectangular wave function defined as:
$$f(x)= \begin{cases} 1, &\mbox{if } 0<x<\pi/2 \\ -1, &\mbox{if } \pi/2<x<\pi \end{cases} $$
Find the Fourier series for $f(x)$ and evaluate the Fourier coefficients. | tiger_lab_math | [
0.009774724952876568,
0.05440517142415047,
-0.05041144788265228,
-0.03482820466160774,
0.024909045547246933,
-0.01930997334420681,
-0.02203793078660965,
0.06892184168100357,
0.0424017459154129,
0.031481385231018066,
0.034253284335136414,
0.026213228702545166,
-0.04014870524406433,
-0.02156... | 0.817424 | 0.811292 | 0.006131 |
If triangle $ABC$ has sides of length $AB = 6,$ $AC = 5,$ and $BC = 4,$ then calculate
\[\frac{\cos \frac{A - B}{2}}{\sin \frac{C}{2}} - \frac{\sin \frac{A - B}{2}}{\cos \frac{C}{2}}.\] | tiger_lab_math | [
0.027893031015992165,
0.04447118565440178,
-0.04008541628718376,
-0.02695072628557682,
0.04270089045166969,
-0.023707440122961998,
-0.03134804219007492,
0.03493306413292885,
0.03196537867188454,
-0.0051276590675115585,
0.02469584159553051,
0.033341094851493835,
-0.0051026782020926476,
-0.0... | 0.829185 | 0.818547 | 0.010638 |
Below is the graph of $y = a \sin (bx + c)$ for some positive constants $a,$ $b,$ and $c.$ Find the smallest possible value of $c.$
[asy]import TrigMacros;
size(300);
real f(real x)
{
return 2*sin(4*x + pi/2);
}
draw(graph(f,-pi,pi,n=700,join=operator ..),red);
trig_axes(-pi,pi,-3,3,pi/2,1);
layer();
rm_trig_labe... | tiger_lab_math | [
0.01367306336760521,
0.045844756066799164,
-0.03887506201863289,
-0.030836619436740875,
0.04113585874438286,
-0.01910821534693241,
-0.02014208398759365,
0.06734297424554825,
0.03380119055509567,
0.0033041173592209816,
0.025078902021050453,
0.036547690629959106,
-0.008548048324882984,
-0.01... | 0.838617 | 0.831143 | 0.007474 |
The quantity $\tan 7.5^\circ$ can be expressed in the form
\[\tan 7.5^\circ = \sqrt{a} - \sqrt{b} + \sqrt{c} - d,\]where $a \ge b \ge c \ge d$ are positive integers. Find $a + b + c + d.$ | tiger_lab_math | [
0.03239845111966133,
0.05454299598932266,
-0.04246313497424126,
-0.04181818291544914,
0.03122406080365181,
-0.014746918343007565,
-0.02815975435078144,
0.04599538445472717,
0.03504975140094757,
-0.007099027279764414,
-0.002137449337169528,
0.020686233416199684,
-0.038473114371299744,
-0.02... | 0.830326 | 0.824189 | 0.006137 |
Suppose that two individuals are considering starting a business venture together. Each person must decide whether to put a high level of effort (H) or a low level of effort (L) into the venture. If both individuals exert a high level of effort, they will achieve a profit of $10,000 each. If either individual exerts a ... | tiger_lab_math | [
-0.0018266664119437337,
-0.0011978576658293605,
-0.05438091978430748,
-0.016239184886217117,
0.03218203783035278,
0.002004325622692704,
-0.0067045786418020725,
0.061831798404455185,
0.031119342893362045,
-0.029186861589550972,
0.011731129139661789,
0.013764088973402977,
-0.024473298341035843... | 0.806005 | 0.790625 | 0.01538 |
Consider the following boundary value problem:
y''(x) + 4y(x) = 0,
y(0) = 0,
y'(1) + y(1) = 2.
Use the Fourier transform to determine the solution to this problem. | tiger_lab_math | [
0.01373319048434496,
0.03144117444753647,
-0.032377228140830994,
-0.04305355250835419,
0.03929629549384117,
-0.0067670708522200584,
-0.00933112483471632,
0.06780049949884415,
0.05846443772315979,
0.001328346785157919,
0.00640651723369956,
0.021267781034111977,
-0.024363664910197258,
-0.024... | 0.82262 | 0.81625 | 0.00637 |
A group of 6 students organized a fundraiser to go to Nicaragua for the summer. For them to go on the trip, each of them needs at least $450. On top of that, they need $3000 for their miscellaneous collective expenses. On the first day of their fundraiser, they receive $600. On the second day, they receive $900, and on... | tiger_lab_math | [
0.007134449668228626,
0.009980046190321445,
-0.044058334082365036,
-0.03351972997188568,
0.024062881246209145,
-0.026242850348353386,
-0.023686537519097328,
0.05895502492785454,
0.025660371407866478,
-0.014930706471204758,
0.032680489122867584,
0.027538638561964035,
-0.04328018054366112,
-... | 0.824013 | 0.815034 | 0.008978 |
Find the truth value of the following statement for the given domain:
∀x ∈ {1, 2, 3}, P(x): "x + 1 > 2"
Is the statement true or false? | tiger_lab_math | [
0.028793996199965477,
0.019091345369815826,
-0.03931887075304985,
-0.03403899446129799,
0.028378218412399292,
-0.03103064000606537,
0.0022868860978633165,
0.09498490393161774,
0.04205343499779701,
-0.02244975045323372,
0.0016960594803094864,
0.043320197612047195,
-0.012833974324166775,
-0.... | 0.815638 | 0.814841 | 0.000796 |
A Sierpinski arrowhead curve is created by taking an equilateral triangle and cutting out the middle third of each side, then forming two new equilateral triangles in the spaces that are left. If we start with an equilateral triangle with a side length of 6 units, how many line segments will there be in the resulting S... | tiger_lab_math | [
0.010736474767327309,
-0.010371629148721695,
-0.0352046824991703,
-0.037424176931381226,
0.03274169936776161,
-0.019442778080701828,
-0.02243092656135559,
0.06984809041023254,
0.038757797330617905,
-0.031651269644498825,
0.0023160502314567566,
0.0320674367249012,
-0.015326768159866333,
-0.... | 0.820115 | 0.817119 | 0.002996 |
From a circular piece of paper with radius $BC$, Jeff removes the unshaded sector shown. Using the larger shaded sector, he joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius 12 centimeters and of volume $432\pi$ cubic centimeters. What is the number of degrees in the measure of angle $ABC$ of th... | tiger_lab_math | [
0.012522886507213116,
0.030578598380088806,
-0.036436572670936584,
-0.03482017293572426,
0.015300913713872433,
-0.017637183889746666,
-0.024490930140018463,
0.0349685437977314,
0.030957529321312904,
0.0014566826866939664,
0.022768476977944374,
0.030521636828780174,
-0.017860157415270805,
-... | 0.832122 | 0.824073 | 0.008049 |
A math student is asked to use the Schwarz reflection principle to find the analytic continuation of the function $f(z)=\sqrt{z-1}$ to the upper half plane.
The student is given the following conditions:
- $f(z)$ is defined in the upper half of the complex plane, $\{z: \text{Im}(z)>0\}$.
- $f(z)$ is real on the real a... | tiger_lab_math | [
0.013430019840598106,
0.057395875453948975,
-0.026207618415355682,
-0.04418528079986572,
0.03218020498752594,
0.02499438263475895,
0.005458517000079155,
0.07305771112442017,
0.03381269797682762,
0.0003968454257119447,
0.005599519703537226,
0.012879472225904465,
-0.025330450385808945,
-0.00... | 0.807292 | 0.786436 | 0.020856 |
A group G of order 10 is given such that every element of G except the identity element has order 2. Determine whether this group is isomorphic to the cyclic group C10, the dihedral group D5, or the non-abelian group of order 10. Justify your answer. | tiger_lab_math | [
0.016092801466584206,
0.014541787095367908,
-0.02491062879562378,
-0.03266188129782677,
0.008492427878081799,
-0.06042667105793953,
-0.008594764396548271,
0.054149799048900604,
0.0294052567332983,
-0.01508418656885624,
0.026496903970837593,
0.0032564185094088316,
-0.04193202406167984,
-0.0... | 0.811 | 0.808582 | 0.002418 |
Sure, here's a numerical optimization problem for a math student to solve using cuckoo search algorithm:
A manufacturer has a factory that produces two types of products: Product A and Product B. It takes 2 hours to produce one unit of Product A and 3 hours to produce one unit of Product B. The factory has 40 hours of... | tiger_lab_math | [
0.012902442365884781,
0.00883406214416027,
-0.04246437922120094,
-0.004695836920291185,
0.016030438244342804,
-0.011647917330265045,
-0.04555182158946991,
0.05145690217614174,
0.04959632456302643,
-0.01752767153084278,
-0.0004477048059925437,
0.035758454352617264,
-0.024499209597706795,
-0... | 0.819966 | 0.80947 | 0.010496 |
Consider the Hill cipher encryption of a plaintext message using the key matrix:
K = [ 7 8 ; 11 11 ]
Encrypt the message "MATH" by representing each letter by a number (A=0, B=1, ..., Z=25) and grouping them into pairs. Then, solve for the ciphertext using modular arithmetic where the modulo is 26.
Hint: You can use... | tiger_lab_math | [
0.04277727007865906,
0.030070358887314796,
-0.020840005949139595,
-0.04534780606627464,
0.060146890580654144,
-0.06568536162376404,
-0.014846001751720905,
0.04670397937297821,
0.020873574540019035,
-0.03302957862615585,
0.019151289016008377,
0.013914168812334538,
-0.040272220969200134,
-0.... | 0.815062 | 0.813246 | 0.001816 |
Let A = {1, 2} , B = {3}, C = {4}.
Determine (A × B) ∪ C and A × (B ∪ C) and verify that they are not equal. | tiger_lab_math | [
0.03682425990700722,
0.03407565504312515,
-0.028041264042258263,
-0.03465607017278671,
0.04454251378774643,
-0.013984295539557934,
-0.0012043486349284649,
0.05930716544389725,
0.03676167502999306,
-0.019955016672611237,
0.009239458478987217,
0.02157740108668804,
-0.027323951944708824,
-0.0... | 0.83703 | 0.825018 | 0.012012 |
A student needs to solve the following question related to the Lorenz system of equations:
Assume the Lorenz system is given by the following equations:
$x' = \sigma(y - x)$
$y' = x(\rho - z) - y$
$z' = xy - \beta z$
where $\sigma = 10$, $\rho = 28$, and $\beta = \frac{8}{3}$. Starting with the initial condition... | tiger_lab_math | [
-0.0008940091938711703,
0.023604122921824455,
-0.053082358092069626,
-0.04546653851866722,
0.0349377878010273,
-0.0017854402540251613,
-0.025845183059573174,
0.07059863209724426,
0.02623508870601654,
-0.030273454263806343,
0.014935340732336044,
0.021271413192152977,
-0.047013796865940094,
... | 0.820257 | 0.813289 | 0.006968 |
Suppose $f(z)$ is an entire function of order 1 such that $f(0) = 1,$ and $|f(z)|\leq e^{2|\text{Re}(z)|}$ for all $z\in \mathbb{C}.$ Determine the value of $f(i).$ | tiger_lab_math | [
0.022858425974845886,
0.046271912753582,
-0.041339267045259476,
-0.026037460193037987,
0.02498876117169857,
-0.02153899520635605,
-0.00047328704386018217,
0.05391867458820343,
0.04172820225358009,
-0.0004526088305283338,
0.01399813313037157,
0.010840796865522861,
-0.04044710472226143,
0.00... | 0.836723 | 0.818353 | 0.01837 |
Suppose you want to encrypt the message "MATH" using a Hill cipher with a 2x2 key matrix [5, 7; 2, 3]. Convert the letters of the message to their corresponding numerical values (A=0, B=1, etc.) and then group them into pairs: (12,0) and (19,7). Find the encrypted message using the formula C = K * P, where C is the cip... | tiger_lab_math | [
0.04194151610136032,
0.03204566240310669,
-0.0289057195186615,
-0.04634464532136917,
0.050753600895404816,
-0.051547158509492874,
-0.009130822494626045,
0.05041832849383354,
0.02623755671083927,
-0.023151656612753868,
0.01119832880795002,
0.017441360279917717,
-0.042452000081539154,
-0.005... | 0.795019 | 0.79134 | 0.003679 |
How many different positive values of $x$ will make this statement true: there are exactly $2$ positive two-digit multiples of $x$. Let's write a program. | tiger_lab_math | [
0.01514057070016861,
0.008409364148974419,
-0.04832742363214493,
-0.019353331997990608,
0.035621315240859985,
-0.029295621439814568,
-0.027214890345931053,
0.061628907918930054,
0.05277165770530701,
-0.03405905142426491,
0.012454948388040066,
0.04831300303339958,
-0.021375998854637146,
-0.... | 0.847661 | 0.841959 | 0.005702 |
Let $x$ and $y$ be complex numbers such that
\[\frac{x + y}{x - y} + \frac{x - y}{x + y} = 1.\]Find
\[\frac{x^4 + y^4}{x^4 - y^4} + \frac{x^4 - y^4}{x^4 + y^4}.\] Let's write a program. | tiger_lab_math | [
0.021998783573508263,
0.03453388437628746,
-0.04240003600716591,
-0.04440348967909813,
0.054660968482494354,
-0.016317185014486313,
-0.013540990650653839,
0.05967378616333008,
0.0638766661286354,
-0.020832501351833344,
0.013288822956383228,
0.03262711688876152,
-0.03676768019795418,
-0.020... | 0.847702 | 0.838184 | 0.009518 |
Suppose that the complex function $f(z) = z^2 + iz + 1$ maps a unit circle centered at the origin in the $w$-plane to a region $R$ in the $z$-plane. Find the image of $-1$ under this mapping. | tiger_lab_math | [
0.015209714882075787,
0.03312811255455017,
-0.030511220917105675,
-0.05065196752548218,
0.051067326217889786,
-0.00020610319916158915,
0.003486947389319539,
0.048303477466106415,
0.03520357981324196,
0.005041060969233513,
0.027614416554570198,
0.0130930719897151,
-0.02620132640004158,
-0.0... | 0.829685 | 0.817884 | 0.011801 |
Find the number of distinct numbers in the list
\[\left\lfloor \frac{1^2}{1000} \right\rfloor, \ \left\lfloor \frac{2^2}{1000} \right\rfloor, \ \left\lfloor \frac{3^2}{1000} \right\rfloor, \ \dots, \ \left\lfloor \frac{1000^2}{1000} \right\rfloor.\] | tiger_lab_math | [
0.030248921364545822,
0.0461813285946846,
-0.03927517309784889,
-0.028781097382307053,
0.03088458813726902,
-0.031300295144319534,
-0.012996161356568336,
0.038248348981142044,
0.051146000623703,
-0.013408047147095203,
0.01389264315366745,
0.010380360297858715,
-0.03551064059138298,
-0.0193... | 0.821601 | 0.818339 | 0.003262 |
Calculate the sum of the geometric series $1+\left(\frac{1}{3}\right)+\left(\frac{1}{3}\right)^2 + \left(\frac{1}{3}\right)^3 + \dots$. Express your answer as a common fraction. Let's write a program. | tiger_lab_math | [
0.008612235076725483,
0.032816097140312195,
-0.04501725360751152,
-0.042552683502435684,
0.040826465934515,
-0.027587667107582092,
-0.015579340979456902,
0.05911089852452278,
0.04626224935054779,
-0.02866024523973465,
0.01948101818561554,
0.017266780138015747,
-0.04227238893508911,
-0.0237... | 0.830459 | 0.828675 | 0.001785 |
Michael and Thomas are selling their lego collections. They agree to split any money they earn. They sell them based on how many circles are on top. Each circle costs 1 cent. They earned $5 each after selling 100 single pieces, 45 double pieces, 50 triple pieces and a number of quadruple pieces. How many quadruple piec... | tiger_lab_math | [
0.000710108841303736,
0.00869207177311182,
-0.04359279200434685,
-0.022038612514734268,
0.03328326344490051,
-0.014139426872134209,
-0.03508198261260986,
0.024458475410938263,
0.019343094900250435,
-0.01985105499625206,
0.030471419915556908,
0.006958482321351767,
-0.036699578166007996,
-0.... | 0.817208 | 0.816052 | 0.001156 |
Let M be a 3-dimensional manifold with a closed 2-form ω, such that ω ∧ ω ≠ 0. Calculate the Hodge star operator of ω at a point p in M. | tiger_lab_math | [
0.010890636593103409,
0.01915637031197548,
-0.034441027790308,
-0.05279993265867233,
0.03813156858086586,
-0.0035006047692149878,
-0.023337211459875107,
0.07379224896430969,
0.04202146828174591,
-0.011844598688185215,
0.004032129887491465,
0.048043228685855865,
-0.02954636514186859,
-0.043... | 0.820316 | 0.813551 | 0.006765 |
Let X be a smooth projective hypersurface of degree d in P^3. Compute the Hodge numbers h^{1,1}(X) and h^{2,0}(X) in terms of d. | tiger_lab_math | [
0.006108134984970093,
0.024387244135141373,
-0.040179021656513214,
-0.044188257306814194,
0.034341126680374146,
-0.020792096853256226,
-0.02252146787941456,
0.0462334081530571,
0.047222673892974854,
-0.02318701334297657,
0.01012177113443613,
0.034373391419649124,
-0.035524528473615646,
-0.... | 0.823297 | 0.808126 | 0.015171 |
Here's a question:
Suppose we have a computer program that takes a positive integer n as input and outputs the nth prime number. Is this program computable? That is, is there an algorithm that always correctly outputs the nth prime for any input n? If so, describe such an algorithm. If not, explain why not. | tiger_lab_math | [
0.03765987604856491,
0.027113493531942368,
-0.011688797734677792,
-0.0004858886240981519,
0.03660958260297775,
-0.05149718001484871,
-0.0025750435888767242,
0.06628302484750748,
0.03557993471622467,
-0.035337746143341064,
0.02540457993745804,
0.02048657089471817,
-0.04006904736161232,
-0.0... | 0.810039 | 0.806327 | 0.003712 |
What is the maximum number of iterations required to determine if a complex number c = a + bi is part of the Mandelbrot set, where a and b range from -2 to 2? Use a precision of 1000 iterations for each point in the complex plane. | tiger_lab_math | [
0.026763595640659332,
0.01924975775182247,
-0.03518418222665787,
-0.02949359081685543,
0.024246472865343094,
-0.015567086637020111,
-0.015911929309368134,
0.04041064530611038,
0.03775506839156151,
-0.02408607117831707,
0.0049596806056797504,
0.04132966697216034,
-0.022123586386442184,
-0.0... | 0.831072 | 0.828447 | 0.002625 |
Consider the function f(z) = (z^2 - 3z + 2)/(z - 1)^3. Determine the order of the zero of f(z) at z = 1 and the location and order of the poles of f(z) on the complex plane. | tiger_lab_math | [
0.04507679492235184,
0.03313027322292328,
-0.05247480049729347,
-0.04822658747434616,
0.03135599195957184,
-0.010166844353079796,
-0.004499765578657389,
0.034941889345645905,
0.02846180833876133,
0.005254044663161039,
0.016072211787104607,
0.02384227328002453,
-0.04808428883552551,
-0.0266... | 0.821239 | 0.817396 | 0.003842 |
Let A = {1, 2, 3, 4} and B = {2, 3, 4, 5}. Find (A Δ B) ∪ C where C = {3, 4, 5, 6} and Δ symbolizes the symmetric difference operator. | tiger_lab_math | [
0.03489820659160614,
0.017380045726895332,
-0.04745374619960785,
-0.022457102313637733,
0.0445571206510067,
-0.012077339924871922,
-0.026289740577340126,
0.06254182010889053,
0.061848875135183334,
-0.00982520543038845,
0.0025631834287196398,
0.03889886289834976,
-0.03428524360060692,
-0.01... | 0.83994 | 0.837032 | 0.002908 |
Find the residue of the function f(z) = (z^5 + 2z^3 + 3z) / (z^2 - 1) at each of its poles, and use the residue theorem to evaluate the integral of f(z) over the circle |z| = 3. | tiger_lab_math | [
0.028562141582369804,
0.024629099294543266,
-0.030195381492376328,
-0.04071998596191406,
0.047847308218479156,
-0.014686556532979012,
-0.0030456730164587498,
0.03821529820561409,
0.03164595738053322,
0.006449783220887184,
0.024881213903427124,
0.024526648223400116,
-0.044893424957990646,
-... | 0.826291 | 0.82311 | 0.003181 |
Given the function f(x) = x^2 on the interval [-π, π], find its Fourier series expansion and use it to compute the Fourier sine series of the boundary value problem:
u''(x) + u(x) = f(x), with u(-π) = 0 and u(π) = 0. | tiger_lab_math | [
-0.007978289388120174,
0.03150258958339691,
-0.05044766142964363,
-0.026814179494976997,
0.03603868559002876,
-0.02152782306075096,
0.004561164882034063,
0.06347677856683731,
0.06578875333070755,
0.010177545249462128,
0.025745192542672157,
0.03317546099424362,
-0.03346671536564827,
-0.0261... | 0.819503 | 0.815699 | 0.003804 |
Consider the boundary value problem given by the differential equation:
$$\frac{d^2u(x)}{dx^2} - 4u(x) = 0\quad ,\quad 0<x<\pi$$
with boundary conditions:
$$u(0) = 0\quad ,\quad u(\pi) = 2$$
Find the solution to the boundary value problem. | tiger_lab_math | [
-0.016751501709222794,
0.01698850840330124,
-0.035758208483457565,
-0.023693008348345757,
0.03877377137541771,
-0.01965554989874363,
-0.008306355215609074,
0.07170481234788895,
0.07504010200500488,
0.011364860460162163,
0.007945438846945763,
0.03532630577683449,
-0.011488333344459534,
-0.0... | 0.814213 | 0.803297 | 0.010915 |
Consider a graph G with 10 vertices and 18 edges. Determine if there exists a perfect matching in G. If there is one, provide the set of edges that forms the perfect matching. | tiger_lab_math | [
0.02246551774442196,
0.013300403952598572,
-0.030490120872855186,
-0.030448006466031075,
0.03247823938727379,
-0.04118415340781212,
-0.025150693953037262,
0.029863428324460983,
0.027303600683808327,
-0.031122669577598572,
0.011581355705857277,
0.010937061160802841,
-0.019894182682037354,
-... | 0.824288 | 0.815356 | 0.008932 |
Determine all solutions for the equation: 2sin(2x) + 1 = 0 in the interval [0, 2π]. | tiger_lab_math | [
0.021384287625551224,
0.028105271980166435,
-0.016698896884918213,
-0.04883719980716705,
0.03784364089369774,
-0.01032383181154728,
-0.013486607000231743,
0.06619846820831299,
0.0622086226940155,
-0.008842459879815578,
0.03789485618472099,
0.01959674246609211,
-0.01782362349331379,
-0.0323... | 0.826776 | 0.82453 | 0.002245 |
Find the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Please write a program to solve it | tiger_lab_math | [
0.019942328333854675,
0.032719582319259644,
-0.05202553793787956,
-0.031483329832553864,
0.0167336855083704,
-0.009690904058516026,
-0.02058793231844902,
0.05394669994711876,
0.03262142091989517,
-0.037845179438591,
0.012834952212870121,
0.043561842292547226,
-0.022733185440301895,
-0.0307... | 0.855011 | 0.843017 | 0.011994 |
Consider a surface given by the equation z = x^2 + y^2.
1. Determine the first-order partial derivatives of this function with respect to x and y.
2. Compute the second-order partial derivatives of z with respect to x and y.
3. Use the values obtained in step 2 to calculate the Gaussian curvature of the surface at th... | tiger_lab_math | [
0.001892932690680027,
0.045501962304115295,
-0.03157353773713112,
-0.045409828424453735,
0.043915968388319016,
0.0018087716307491064,
-0.0031980809289962053,
0.04873280227184296,
0.024620261043310165,
-0.012298856861889362,
0.010596523992717266,
0.033021293580532074,
-0.03671819716691971,
... | 0.809789 | 0.801767 | 0.008022 |
In the diagram below, $AB = AC = 115,$ $AD = 38,$ and $CF = 77.$ Compute $\frac{[CEF]}{[DBE]}.$
[asy]
unitsize(0.025 cm);
pair A, B, C, D, E, F;
B = (0,0);
C = (80,0);
A = intersectionpoint(arc(B,115,0,180),arc(C,115,0,180));
D = interp(A,B,38/115);
F = interp(A,C,(115 + 77)/115);
E = extension(B,C,D,F);
draw(C--B... | tiger_lab_math | [
0.04922160878777504,
0.005918512120842934,
-0.035696785897016525,
-0.05921357125043869,
0.025512637570500374,
-0.04245689511299133,
-0.007492639124393463,
0.06763286143541336,
0.042727675288915634,
0.013419770635664463,
0.02168363146483898,
0.0205696988850832,
-0.024493729695677757,
-0.001... | 0.837924 | 0.831715 | 0.006209 |
Let G be a group of order 9. Find the number of elements of order 3 in G, and use this information to determine whether G is isomorphic to the cyclic group C9 or the non-abelian group of order 9. | tiger_lab_math | [
0.009295439347624779,
-0.0077168308198452,
-0.04139275848865509,
-0.03546982631087303,
0.02200622297823429,
-0.05887025594711304,
-0.0034989325795322657,
0.04815591871738434,
0.02799120731651783,
-0.01179249957203865,
0.0296773761510849,
0.01675516366958618,
-0.041227180510759354,
-0.00328... | 0.819896 | 0.813415 | 0.00648 |
In a game of Stag Hunt, two players can either choose to hunt a stag or a hare. If both players hunt a stag, they each receive 5 points. If one player hunts a stag and the other hunts a hare, the player who hunted the hare gets one point and the other gets nothing. If both players hunt a hare, they each get 2 points.
... | tiger_lab_math | [
-0.004709220491349697,
0.011491836048662663,
-0.054688096046447754,
-0.034262336790561676,
0.02401803247630596,
-0.019081635400652885,
-0.029287168756127357,
0.027930671349167824,
0.03446769714355469,
-0.022739261388778687,
0.019437594339251518,
0.023866668343544006,
-0.024240246042609215,
... | 0.820574 | 0.812672 | 0.007902 |
Since it is Maurice's turn to host this year’s neighborhood cookout, he goes to the local wholesale club to buy ground beef. The wholesale club sells ground beef in 5-pound packages. Maurice wants to make one 2-pound burger for each person that attends, so he purchases 4 packages of ground beef. How many people can Mau... | tiger_lab_math | [
0.032106664031744,
0.02843116596341133,
-0.030636318027973175,
-0.036122336983680725,
0.021583160385489464,
-0.0338415764272213,
-0.012993945740163326,
0.02178061008453369,
0.04405149817466736,
-0.003951250575482845,
0.008104576729238033,
0.03906455263495445,
-0.051501743495464325,
-0.0214... | 0.804776 | 0.802906 | 0.00187 |
A manufacturing company produces two types of products: Product A and Product B. The profit per unit for Product A is $10, and the profit per unit for Product B is $15. Each unit of Product A requires 3 labor hours and 2 machine hours, while each unit of Product B requires 4 labor hours and 1 machine hour. The company ... | tiger_lab_math | [
0.00711526395753026,
0.005591230932623148,
-0.06031109392642975,
-0.007689740043133497,
0.014841509982943535,
-0.013010667636990547,
-0.05500587821006775,
0.0768025740981102,
0.025193531066179276,
-0.007423689588904381,
0.0029579410329461098,
0.03608706593513489,
-0.018244734033942223,
-0.... | 0.805674 | 0.802721 | 0.002953 |
The line with equation $y = x$ is an axis of symmetry of the curve with equation
\[y = \frac{px + q}{rx + s},\]where $p,$ $q,$ $r,$ $s$ are all nonzero. Which of the following statements must hold?
(A) $p + q = 0$
(B) $p + r = 0$
(C) $p + s = 0$
(D) $q + r = 0$
(E) $q + s = 0$
(F) $r + s = 0$ | tiger_lab_math | [
0.014198335818946362,
0.013805693946778774,
-0.04128352925181389,
-0.046630147844552994,
0.04097204655408859,
-0.03134237974882126,
-0.004658552352339029,
0.0895453616976738,
0.05077890679240227,
-0.0349026694893837,
-0.0030384131241589785,
0.03763745725154877,
-0.022637030109763145,
-0.02... | 0.836321 | 0.830493 | 0.005828 |
Let X be the algebraic variety defined as the locus of zeros of the polynomial system f(x,y,z) = {x^3 - y^2 - z^2, x^2 - y^3 - z^2}. Determine the singular points of X in the complex projective space P^2(C). | tiger_lab_math | [
0.00903661735355854,
0.036274466663599014,
-0.05654563754796982,
-0.04737105965614319,
0.019921477884054184,
-0.025549819692969322,
-0.0090670520439744,
0.03780737891793251,
0.045364219695329666,
-0.0023772341664880514,
0.009426035918295383,
0.03623717278242111,
-0.028842855244874954,
-0.0... | 0.835171 | 0.816637 | 0.018534 |
A manufacturing company produces two products A and B, and the cost of producing each product is as follows:
- The cost of producing one unit of A is $20.
- The cost of producing one unit of B is $30.
The company wants to maximize its profit while ensuring that its production does not exceed 500 units of A and 400 un... | tiger_lab_math | [
0.01927180215716362,
0.008158626966178417,
-0.07257683575153351,
-0.010854484513401985,
0.019622107967734337,
-0.009187900461256504,
-0.03952084481716156,
0.08189167082309723,
0.0379163883626461,
-0.015980549156665802,
0.007239813916385174,
0.03321221098303795,
-0.019962254911661148,
-0.03... | 0.82743 | 0.819291 | 0.008139 |
What is the smallest positive integer $n$ such that $2n$ is a perfect square and $3n$ is a perfect cube? Please write a program to solve it | tiger_lab_math | [
0.02484864927828312,
0.047065410763025284,
-0.05388401448726654,
-0.04034911468625069,
0.031120719388127327,
-0.009168627671897411,
-0.02256014384329319,
0.04410562664270401,
0.0334940068423748,
-0.03301816061139107,
0.03573521226644516,
0.03789850324392319,
-0.029774174094200134,
-0.03343... | 0.849262 | 0.844816 | 0.004446 |
Let C be the positively oriented circle centered at -1+i and radius 2. Evaluate the integral ∮_C (z^2 + 4z)/(z+2) dz. | tiger_lab_math | [
0.025144604966044426,
0.02587282285094261,
-0.041671376675367355,
-0.01617162488400936,
0.04245666787028313,
-0.019103672355413437,
-0.023608649149537086,
0.02559898979961872,
0.051713209599256516,
-0.002293575555086136,
0.03170452266931534,
0.01930530183017254,
-0.022393127903342247,
-0.0... | 0.825281 | 0.821814 | 0.003467 |
Consider the circle $S^1$ and a loop $\gamma:S^1\to S^1$ that winds around the circle $n$ times in a counterclockwise direction. Determine the fundamental group $\pi_1(S^1/\gamma , [1])$, where $S^1/\gamma$ denotes the quotient space obtained by identifying each point $x\in S^1$ with its image $\gamma(x)$ and $[1]$ is ... | tiger_lab_math | [
0.0001560795644763857,
0.027467889711260796,
-0.019871778786182404,
-0.039387382566928864,
0.02842305786907673,
-0.04107188433408737,
-0.011649446561932564,
0.05079901963472366,
0.03768875449895859,
-0.0005055353394709527,
0.02559032291173935,
0.018325351178646088,
-0.03943271562457085,
-0... | 0.830233 | 0.797148 | 0.033085 |
Consider the function $f(x,y) = x^3 + 3x^2y - 3xy^2 - y^3$ on $\mathbb{R}^2$.
(a) Find all critical points of $f$, and determine whether they are local maxima, local minima, or saddle points.
(b) Compute the Morse complex for $f$, where the critical points are the vertices and the edges correspond to the ascending a... | tiger_lab_math | [
0.02683061733841896,
0.031817104667425156,
-0.04297299683094025,
-0.031699758023023605,
0.02529081515967846,
-0.015490446239709854,
-0.02391168288886547,
0.05592330917716026,
0.019407695159316063,
0.0007038008770905435,
0.01307872124016285,
0.036501988768577576,
-0.03391648456454277,
-0.02... | 0.835846 | 0.810669 | 0.025177 |
On the period-doubling route to chaos, a logistic map is defined as x_n+1 = r*x_n*(1-x_n), where x_0 = 0.2 and r = 3.5. Compute the first ten iterates of this map and plot them on a graph. What is the first value of r at which the logistic map becomes chaotic? | tiger_lab_math | [
0.013635652139782906,
0.00934437196701765,
-0.022100962698459625,
-0.03652609884738922,
0.01614956185221672,
-0.026450952515006065,
-0.035919755697250366,
0.06785156577825546,
0.00041742672328837216,
-0.014462241902947426,
0.0018579564057290554,
0.024236472323536873,
-0.035973742604255676,
... | 0.819838 | 0.81652 | 0.003318 |
A roadwork company is paving a newly constructed 16-mile road. They use a mixture of pitch and gravel to make the asphalt to pave the road. Each truckloads of asphalt uses two bags of gravel and five times as many bags of gravel as it does barrels of pitch to make. It takes three truckloads of asphalt to pave each mile... | tiger_lab_math | [
0.027102051302790642,
0.01995493657886982,
-0.03276975452899933,
-0.032403189688920975,
0.03757106512784958,
-0.02972867339849472,
-0.044148243963718414,
0.07100893557071686,
0.03813119977712631,
-0.02009730413556099,
0.014222364872694016,
0.03438063710927963,
-0.021241571754217148,
-0.032... | 0.824694 | 0.823731 | 0.000963 |
Sure, here is a stochastic optimization question for the math student to solve:
A delivery company has three trucks that can each carry a maximum of 500 packages. The company needs to deliver 2000 packages to four different locations, but the demand at each location is uncertain and follows a normal distribution with ... | tiger_lab_math | [
0.01875988207757473,
0.010995293036103249,
-0.05871383100748062,
-0.024645723402500153,
0.023718252778053284,
-0.029419200494885445,
-0.04257729649543762,
0.03600147366523743,
0.03918912261724472,
-0.02376522682607174,
0.012022429145872593,
0.025743160396814346,
-0.020796366035938263,
-0.0... | 0.82622 | 0.818193 | 0.008027 |
Suppose two players are deciding on how to split a prize of $100. Player 1 can choose to put in effort or shirk, and Player 2 cannot observe Player 1's effort level. If Player 1 puts in effort, the probability of winning the prize increases from 50% to 75%. However, exerting effort reduces Player 1's utility by $20 due... | tiger_lab_math | [
0.012524344958364964,
0.007911321707069874,
-0.04347192868590355,
-0.018029430881142616,
0.019280653446912766,
-0.00880028773099184,
-0.009151758626103401,
0.05764390155673027,
0.04522198066115379,
-0.03209392726421356,
0.034696679562330246,
0.029894033446907997,
-0.037917863577604294,
-0.... | 0.815016 | 0.803867 | 0.011149 |
Using the Ikeda map, which is defined as:
x(n+1) = 1 + a(x(n) cos(t(n)) - y(n) sin(t(n)))
y(n+1) = b(x(n) sin(t(n)) + y(n) cos(t(n)))
where a = 0.9 and b = 0.4, and starting with x = 0.1 and y = 0.1, determine the values of x and y after 100 iterations of the map, using an initial value for the angle t of 1.... | tiger_lab_math | [
0.026933327317237854,
0.02031799592077732,
-0.03248900547623634,
-0.03482663258910179,
0.03875305876135826,
-0.02093547023832798,
-0.013276545330882072,
0.056150585412979126,
0.04288262128829956,
-0.018970713019371033,
0.015812521800398827,
0.03076421655714512,
-0.03480715677142143,
-0.001... | 0.83517 | 0.833223 | 0.001947 |
Suppose you have the following system of linear equations:
$x + 2y - 3z = 4$
$2x - y + z = -1$
$3x + 4y - 2z = 7$
However, due to measurement errors, the values in the system are not exact but have a small amount of noise. For example, the first equation could be $1.9x + 2.1y - 2.8z = 4.2$ instead of $x + 2y - ... | tiger_lab_math | [
0.003376147709786892,
0.03674055263400078,
-0.04780694097280502,
-0.05554746836423874,
0.06078549474477768,
-0.01521104946732521,
-0.01304478757083416,
0.04980083182454109,
0.010192684829235077,
-0.013241235166788101,
0.03275448456406593,
0.032345887273550034,
-0.030105479061603546,
-0.027... | 0.812062 | 0.803206 | 0.008856 |
Assuming the student has knowledge of homotopy and group theory, here is a question they could solve:
Let X be a path-connected topological space with the fundamental group π_1(X,x_0) isomorphic to the cyclic group of order 4. Determine the number of homotopy classes of continuous maps f: S^1 → X that induce a nontriv... | tiger_lab_math | [
0.019668497145175934,
0.01798068732023239,
-0.023713575676083565,
-0.04269831255078316,
0.03211734816431999,
-0.012525590136647224,
-0.00990108959376812,
0.04422878101468086,
0.04304177314043045,
-0.0036884695291519165,
0.02395765297114849,
0.016258325427770615,
-0.011280958540737629,
-0.0... | 0.834621 | 0.815885 | 0.018736 |
Let the metric of a 2-dimensional Riemannian manifold be given by $ds^2=9x^2dy^2+dx^2$, where $x$ and $y$ are the coordinates of the manifold. Find the Levi-Civita connection of this metric at $p=(1,0)$. | tiger_lab_math | [
-0.010698189958930016,
0.016334686428308487,
-0.033463187515735626,
-0.04362676665186882,
0.015573551878333092,
-0.0018009617924690247,
-0.0060301111079752445,
0.049951229244470596,
0.038347505033016205,
0.0006997205200605094,
0.006376223638653755,
0.03471773862838745,
-0.035432904958724976,... | 0.822939 | 0.804953 | 0.017986 |
Consider a vector bundle over a topological space $X$ where the fiber over each point in $X$ is $\mathbb{R}^3$. Suppose that this vector bundle is trivial, i.e., there exists a homeomorphism between $X\times\mathbb{R}^3$ and the vector bundle. If the rank of the vector bundle is $2$, find the Chern class of this vector... | tiger_lab_math | [
0.018886175006628036,
0.0341821163892746,
-0.025571202859282494,
-0.042534325271844864,
0.021016236394643784,
-0.0034077835734933615,
-0.01686973124742508,
0.05825314298272133,
0.02464067004621029,
-0.0005001968238502741,
0.0039346651174128056,
0.031040837988257408,
-0.037769898772239685,
... | 0.828163 | 0.802915 | 0.025248 |
the maximum numbers of students among them 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and same number of pencils is ? Let's program in Python in the response. | tiger_lab_math | [
0.00963942613452673,
0.0085368063300848,
-0.03133754804730415,
-0.031559694558382034,
0.02184997871518135,
-0.04913463816046715,
-0.022287707775831223,
0.034422338008880615,
0.019062425941228867,
-0.0327829085290432,
0.035810019820928574,
0.042609937489032745,
-0.01887180097401142,
-0.0130... | 0.818163 | 0.813472 | 0.00469 |
Seven people arrive to dinner, but the circular table only seats six. If two seatings such that one is a rotation of the other are considered the same, then in how many different ways can we choose six people and seat them at the table? Let's write a Python program. | tiger_lab_math | [
0.01645476371049881,
0.02085978351533413,
-0.028198232874274254,
-0.010632595978677273,
0.03366389125585556,
-0.03035847842693329,
-0.018581995740532875,
0.025661973282694817,
0.03312768414616585,
-0.02103043720126152,
0.0225245151668787,
0.028800921514630318,
-0.027960367500782013,
-0.015... | 0.83177 | 0.824842 | 0.006928 |
Consider the following optimization problem:
Minimize the function f(x) = 2x^2 - 8x + 7, subject to the constraint 0 ≤ x ≤ 5, using tabu search algorithm.
Find the optimal value of f(x) and the value of x that minimizes the function using tabu search algorithm. | tiger_lab_math | [
0.029111824929714203,
0.03069278411567211,
-0.052404724061489105,
-0.03835059329867363,
0.040572572499513626,
-0.025316957384347916,
-0.016534660011529922,
0.07354886829853058,
0.046373628079891205,
-0.0065444465726614,
-0.006377519574016333,
0.008024596609175205,
-0.023037360981106758,
-0... | 0.816376 | 0.814328 | 0.002049 |
In triangle $ABC,$ the side lengths $a,$ $b,$ and $c$ satisfy $a + 2b + 2c = a^2$ and $a + 2b - 2c = -3.$ Find the largest angle of the triangle, in degrees.
Note: $a$ is the side length opposite $\angle A,$ etc. | tiger_lab_math | [
0.04033762589097023,
0.013022593222558498,
-0.03654303401708603,
-0.04516121745109558,
0.010208173654973507,
-0.015670763328671455,
-0.019425509497523308,
0.03955090418457985,
0.02563650719821453,
-0.01544015109539032,
0.030081607401371002,
0.04778455197811127,
-0.0427444651722908,
-0.0049... | 0.843866 | 0.822339 | 0.021526 |
Suppose a company produces two types of products, X and Y, each with a different profit margin. They have limited resources and can only produce a total of 500 units of these products. Product X requires 2 units of resource A and 5 units of resource B to produce, while product Y requires 7 units of resource A and 3 uni... | tiger_lab_math | [
0.008726434782147408,
0.025917157530784607,
-0.052754104137420654,
-0.01818218268454075,
0.011354084126651287,
-0.007877014577388763,
-0.02173244021832943,
0.08809822052717209,
0.028654426336288452,
-0.02076023817062378,
0.00004250612255418673,
0.04273906350135803,
-0.01969398930668831,
-0... | 0.81729 | 0.805279 | 0.012011 |
A stream cipher uses a key stream $\{K\}=\{k_1,k_2,k_3,\dots,k_n\}$ generated by a Linear Feedback Shift Register (LFSR) to encrypt plaintext $P=\{p_1,p_2,p_3,\dots,p_n\}$. If the degree of the LFSR is $k=4$, and its connection polynomial is $x^4+x+1$, and the initial state is $S_0=1010$, what is the ciphertext produce... | tiger_lab_math | [
0.0037099304609000683,
0.029017915949225426,
-0.03724132478237152,
-0.052289437502622604,
0.04466072842478752,
-0.02329191565513611,
-0.03698807209730148,
0.07454914599657059,
0.02679891139268875,
-0.011722848750650883,
0.02265428565442562,
0.014959869906306267,
-0.03320198133587837,
-0.01... | 0.819693 | 0.816117 | 0.003576 |
Consider a two-dimensional surface with metric tensor given by:
g = [ dx^2 + 2xy dydx + y^2 dy^2 ]
Calculate the Christoffel symbols for this metric. | tiger_lab_math | [
0.021009963005781174,
0.02349192462861538,
-0.04797254130244255,
-0.06889348477125168,
0.04165925085544586,
-0.036924466490745544,
-0.023376232013106346,
0.05931020900607109,
0.03857460618019104,
-0.014731175266206264,
0.014142416417598724,
0.04526170343160629,
-0.02548433467745781,
-0.010... | 0.818676 | 0.812923 | 0.005754 |
Consider the function f: R^2 -> R given by f(x,y) = x^2 + y^2. Identify all critical points of f and classify each as a minimum, maximum, or saddle point. | tiger_lab_math | [
0.02964167855679989,
0.021398689597845078,
-0.0348009392619133,
-0.0376054123044014,
0.03118116967380047,
-0.025838829576969147,
-0.021170636638998985,
0.06797824054956436,
0.024161092936992645,
-0.008922589011490345,
0.012805487029254436,
0.02488449215888977,
-0.035098474472761154,
-0.022... | 0.824269 | 0.818638 | 0.005631 |
Sure, here's an example:
A healthcare researcher is studying the survival rate of patients with a rare disease. She analyzes the data of 50 patients and finds that the probability of surviving for more than 2 years after diagnosis is 0.4. What is the probability that at least 20 patients will survive for more than 2 y... | tiger_lab_math | [
0.011491147801280022,
0.0013406167272478342,
-0.04073710739612579,
-0.024708082899451256,
0.024164944887161255,
-0.020486686378717422,
-0.021370884031057358,
0.062116581946611404,
0.0341644361615181,
-0.02064463309943676,
0.001909349113702774,
0.03255689516663551,
-0.010164442472159863,
-0... | 0.805492 | 0.804395 | 0.001097 |
Compute $\arccos (\sin 2).$ All functions are in radians. Let's write a program. | tiger_lab_math | [
0.03227216377854347,
0.04986521601676941,
-0.04390862584114075,
-0.013961896300315857,
0.032645776867866516,
-0.02178826928138733,
-0.02160048671066761,
0.04408351331949234,
0.05254349112510681,
-0.00602918304502964,
0.026325533166527748,
0.03545650839805603,
-0.01657794415950775,
-0.02471... | 0.823871 | 0.821117 | 0.002754 |
Consider a symplectic manifold M of dimension 4 with a symplectic form $\omega$. Suppose that there exists a Hamiltonian function $H \in C^{\infty}(M)$ with only two critical points, both of which are non-degenerate. If one of the critical points of H has $H(p)=2$ and the other has $H(q)=8$, find the number of closed o... | tiger_lab_math | [
0.017484096810221672,
0.022407392039895058,
-0.03547664359211922,
-0.029550306499004364,
0.01895901933312416,
-0.0009770775213837624,
-0.035440050065517426,
0.05988876149058342,
0.039436861872673035,
-0.007903344929218292,
-0.010655701160430908,
0.02543637529015541,
-0.01867228001356125,
-... | 0.833181 | 0.807405 | 0.025775 |
In the metric space $\mathbb{R}$, with the usual metric $d(x,y)=|x-y|$, consider the sequence $(x_n)_{n \geq 1}$ defined by $x_n= \frac{1}{n}$. Determine whether the sequence is convergent or divergent. | tiger_lab_math | [
-0.010798795148730278,
0.023619670420885086,
-0.03235337883234024,
-0.03695148974657059,
0.024210125207901,
-0.0036486878525465727,
-0.01867840625345707,
0.0760369747877121,
0.04837852716445923,
-0.010365298017859459,
0.0034996934700757265,
0.0016227146843448281,
-0.03489522635936737,
-0.0... | 0.823455 | 0.803994 | 0.019461 |
"statement 1": Susan , Daisy and Tim need to be seated in 3 identical chairs in straight line so that Susan is seated always left to Tim, "statement 2" :Susan , Daisy and Tim need to be seated in less than 5 identical chairs in straight line so that Susan is seated always left to Tim, "options: " Entailment or contrad... | tiger_lab_math | [
0.0203850194811821,
-0.0011117741232737899,
-0.0528518483042717,
-0.04258663207292557,
0.04699213057756424,
-0.024322981014847755,
-0.0127567695453763,
0.05909088999032974,
0.025368064641952515,
-0.012122459709644318,
0.021638423204421997,
0.02986770309507847,
-0.01176645327359438,
-0.0194... | 0.809572 | 0.804851 | 0.004721 |
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