instruction_seed stringlengths 120 1.22k | source stringclasses 1
value | embeddings listlengths 1.02k 1.02k | mean_positive_score float64 0.78 0.86 | mean_negative_score float64 0.76 0.83 | difference_score float64 0.02 0.04 |
|---|---|---|---|---|---|
Verify whether the general solution for the differential equation \( \frac{dy}{dx} + \frac{y^2 + y + 1}{x^2 + x + 1} = 0 \) can be expressed as \( x + y + 1 = A(1 - x - y - 2xy) \), where \( A \) is a parameter. If so, determine the conditions under which this holds true. | automath | [
0.0032024241518229246,
0.01002572849392891,
-0.0278429314494133,
-0.06152051314711571,
0.03643036261200905,
-0.03181084990501404,
-0.00861000083386898,
0.09467917680740356,
0.04366707801818848,
-0.02560848370194435,
0.0054310099221765995,
0.022464340552687645,
-0.029066549614071846,
-0.013... | 0.825071 | 0.805543 | 0.019528 |
Let \( P(x) = a_k x^k + a_{k-1} x^{k-1} + \ldots + a_1 x + a_0 \) be a polynomial of degree \( k > 0 \) with \( a_0 \neq 0 \). Define the sequence of functions \( f_n(x) = P\left(\frac{x}{n}\right) \) for \( x \in (0, \infty) \).
What is the limit of \( f_n(x) \) as \( n \) approaches infinity for all \( x \in (0, \i... | automath | [
0.023780042305588722,
0.045367561280727386,
-0.02790611796081066,
-0.023046093061566353,
0.010442260652780533,
-0.023862116038799286,
-0.023090748116374016,
0.07141689956188202,
0.033471833914518356,
-0.0026408883277326822,
-0.01290063001215458,
0.025439167395234108,
-0.025518329814076424,
... | 0.821676 | 0.799738 | 0.021938 |
Given the recurrence relation
$$P(n) = \begin{cases} 1 & \text{if } n = 1, \\ \sum_{k=1}^{n-1} P(k) P(n - k) & \text{if } n \ge 2 \end{cases}$$
demonstrate that \(P(n) \ge c \cdot 2^n\) for some constant \(c > 0\) and for all integers \(n \ge 2\), using substitution or an appropriate method. What is the minimum pos... | automath | [
0.019068829715251923,
0.04665948823094368,
-0.028774302452802658,
-0.017332565039396286,
0.029294107109308243,
-0.0033134822733700275,
-0.007592094596475363,
0.07559741288423538,
0.026226429268717766,
-0.005155711434781551,
-0.007918108254671097,
0.035198986530303955,
-0.01819210685789585,
... | 0.840956 | 0.817102 | 0.023855 |
Given a point \( x = (x_0, y_0) \) in the upper half-plane \( \mathbb{H} \) where \( y_0 > 0 \) and a unit tangent vector \( v \), if the geodesic corresponding to the unit tangent vector at point \( x \) intersects the real line at the point \( (a, 0) \), express \( a \) as a function of \( x_0, y_0 \), and the compon... | automath | [
0.020719138905405998,
0.03529961407184601,
-0.02476545423269272,
-0.03624865040183067,
0.02512386441230774,
0.008656573481857777,
-0.024567201733589172,
0.07986835390329361,
0.045228201895952225,
0.006854705046862364,
0.0006013564998283982,
0.030506407842040062,
-0.019932880997657776,
-0.0... | 0.815338 | 0.796377 | 0.018961 |
Given a two-player zero-sum game represented by the payoff matrices \( A \) and \( B \), where \( A \) has dimensions \( m \times n \) and \( B \) has dimensions \( m \times n \), if the optimal mixed strategy for the row player corresponds to a probability distribution \( x \) where \( x_i \) represents the probabilit... | automath | [
0.01557475421577692,
0.009114185348153114,
-0.055581554770469666,
-0.030433926731348038,
0.03575064241886139,
0.013639576733112335,
-0.009790580719709396,
0.0628589615225792,
0.022212155163288116,
-0.0230324137955904,
0.0030426676385104656,
0.022702090442180634,
-0.010191729292273521,
-0.0... | 0.822305 | 0.797201 | 0.025103 |
Is it possible to compute \( d^2 \) entries of the 3D circular convolution \( \text{conv}(A,B) \) for matrices \( A, B \in \mathbb{R}^{d \times d \times d} \), where \( A_{i,j,k} = 0 \) for all \( i \neq 0 \) and \( B_{i,j,k} = 0 \) for all \( j \neq 0 \), specifically for \( \text{conv}(A,B)_{i,k,d-1} \) with \( i,k =... | automath | [
0.014135968871414661,
0.05259554833173752,
-0.02929443120956421,
-0.022777199745178223,
0.03471120446920395,
0.016239671036601067,
-0.029213860630989075,
0.07230199128389359,
0.03985501080751419,
-0.006152957212179899,
0.02207067422568798,
0.0182809941470623,
-0.016725745052099228,
-0.0184... | 0.80161 | 0.785189 | 0.016421 |
For a monic irreducible polynomial \( f(x) \in \mathbb{Q}[x] \) with degree \( n \), what are the necessary and sufficient conditions for its splitting field to have the same degree over \( \mathbb{Q} \) as \( f \), specifically when considering the Galois group of the polynomial? Additionally, can you provide an examp... | automath | [
0.022391360253095627,
0.03236643224954605,
-0.030344147235155106,
-0.052506234496831894,
0.015574716031551361,
-0.027654945850372314,
-0.01633513905107975,
0.04730142280459404,
0.011693526059389114,
-0.01635442115366459,
0.014987913891673088,
0.009056459181010723,
-0.028117267414927483,
-0... | 0.814587 | 0.795661 | 0.018926 |
Given two closed, convex, non-empty sets \( K_1, K_2 \subseteq \mathbb{R}^n \), what is the expression for the support function of their intersection \( h_{K_1 \cap K_2} \) in terms of the support functions \( h_{K_1} \) and \( h_{K_2} \)? Is it true that
\[
h_{K_1 \cap K_2}(y) = \sup \{ h \leq \min\{h_{K_1}(y), h_{K_... | automath | [
0.010944490320980549,
0.041465286165475845,
-0.05878478288650513,
-0.03228062018752098,
0.02161371521651745,
-0.003207113593816757,
-0.025079965591430664,
0.05965346097946167,
0.05192294344305992,
-0.007623006124049425,
0.01483786478638649,
0.015219390392303467,
-0.019571609795093536,
-0.0... | 0.818914 | 0.797913 | 0.021001 |
Given a covering space \( p : (\widetilde{X},\widetilde{x}_0) \to (X,x_0) \) and a continuous map \( f : (Y,y_0) \to (X,x_0) \) where \( Y \) is path-connected and locally path-connected, under what conditions does there exist a path \(\widetilde{f} : (Y,y_0) \to (\widetilde{X},\widetilde{x}_0)\) which is a lift of \( ... | automath | [
0.00005261697879177518,
0.03755795210599899,
-0.024688251316547394,
-0.03274450823664665,
0.03334471955895424,
-0.016494959592819214,
0.0005164016620256007,
0.0823473334312439,
0.031814590096473694,
-0.006902745924890041,
0.019756190478801727,
0.03716809302568436,
-0.020962579175829887,
-0... | 0.811036 | 0.787727 | 0.023309 |
Let \( (X, d) \) be a metric space and \( (Y, \Sigma, \mu) \) be a measure space. If we define a metric \( d' \) on the space of equivalence classes of measurable sets under the relation \( A \sim B \) iff \( \mu(A \Delta B) = 0 \), where \( \Delta \) denotes the symmetric difference, what conditions must the measure \... | automath | [
0.020849063992500305,
0.012984270229935646,
-0.046321332454681396,
-0.03738434985280037,
0.03499685600399971,
-0.015147936530411243,
-0.041234634816646576,
0.06851768493652344,
0.06212412565946579,
-0.009720401838421822,
-0.00925145111978054,
0.02220103330910206,
-0.0335482619702816,
-0.01... | 0.804648 | 0.783097 | 0.021552 |
Given a matrix \( A \in \mathbb{R}^{m \times n} \) and vectors \( x \in \mathbb{R}^n \) and \( b \in \mathbb{R}^m \), consider the statement \( Ax \geq b \). Under what conditions can we interpret this as a set of \( m \) linear inequality constraints, and how does it relate geometrically to the feasible region defined... | automath | [
0.015248730778694153,
0.030107269063591957,
-0.039365533739328384,
-0.05411699414253235,
0.03976626694202423,
0.0030255932360887527,
-0.030096035450696945,
0.08897025883197784,
0.04108026251196861,
-0.004178850445896387,
-0.008241127245128155,
0.03070824407041073,
-0.037223707884550095,
-0... | 0.812279 | 0.78293 | 0.02935 |
Under what conditions on a metric space \( (X,d) \) and a subset \( E \subseteq X \) would the following equality hold for any point \( x \in E \):
\[
\operatorname{dist}(x, \partial E) = \sup \{ \gamma \geq 0 : B(x, \gamma) \subset E \}
\]
Provide a characterization of such spaces and discuss whether the property ... | automath | [
0.006968052126467228,
0.02636130154132843,
-0.040043190121650696,
-0.0363818034529686,
0.030349168926477432,
-0.009022150188684464,
-0.014481398276984692,
0.07614552974700928,
0.0535396933555603,
-0.0051807682029902935,
0.014424934983253479,
0.04024365171790123,
-0.046350907534360886,
-0.0... | 0.810632 | 0.782861 | 0.027772 |
Let \( R \) be a commutative ring with unity, and let \( M \) be a finitely generated \( R \)-module. If \( \text{Ann}(M) = I \) and the prime ideal \( \mathfrak{p} \) is not in \( \text{Supp}(M) \), determine the largest integer \( n \) such that \( f^n M = 0 \) for some \( f \in R \) satisfying \( f \notin \mathfrak{... | automath | [
0.06371424347162247,
0.035419393330812454,
-0.029175052419304848,
-0.032977357506752014,
0.013856898061931133,
-0.026678940281271935,
-0.01912377029657364,
0.059406403452157974,
0.027589648962020874,
-0.020579082891345024,
0.008761178702116013,
0.01946611888706684,
-0.021810481324791908,
-... | 0.829571 | 0.806171 | 0.0234 |
Given a random process \( X(t) = \cos(t + \Theta) \) where \( \Theta \) can take on four equally likely values \( 0, \pi/2, \pi, \) and \( 3\pi/2 \), determine if this process is first-order stationary. Additionally, for the same process, compute the expected value \( E[X(t)] \) and the autocorrelation function \( R_X(... | automath | [
0.0014993249205872416,
0.02373560331761837,
-0.04779456555843353,
-0.030322695150971413,
0.046463534235954285,
-0.017298676073551178,
-0.025020550936460495,
0.07755789160728455,
0.045894838869571686,
-0.00826151855289936,
0.023806095123291016,
0.006619752384722233,
-0.026186101138591766,
-... | 0.811221 | 0.794643 | 0.016577 |
Consider a vector \( x \in \mathbb{R}^n \) represented as \( x = (x_1, x_2, \ldots, x_n) \) where \( |x_1| \geq |x_2| \geq \cdots \geq |x_n| \). Define the k-norm as \( \|x\|_{[k]} = |x_1| + |x_2| + \ldots + |x_k| \).
If \( x = (3, -1, 2, -4) \), calculate \( \|x\|_{[2]} \) and \( \|x\|_{[3]} \), and verify the inequ... | automath | [
0.007298698648810387,
0.04367999732494354,
-0.04408906027674675,
-0.040662623941898346,
0.024930736050009727,
0.00203736312687397,
-0.025532986968755722,
0.06418851017951965,
0.025730948895215988,
-0.0018570942338556051,
-0.0008564114104956388,
0.03180629387497902,
-0.03569620102643967,
0.... | 0.828098 | 0.801933 | 0.026166 |
Determine the set of complex numbers \( z = x + iy \) such that the inequality \( |z^2| > |z| \Re(z) \) holds true. Express your answer in the form of inequalities involving the real part \( x \) and the imaginary part \( y \). | automath | [
0.030133135616779327,
0.02888948656618595,
-0.03067266196012497,
-0.03742141276597977,
0.04000694304704666,
0.0017232532845810056,
0.002552859717980027,
0.06530965119600296,
0.04721800610423088,
-0.01300825085490942,
0.02658170461654663,
0.043281834572553635,
-0.05516984313726425,
-0.00789... | 0.825584 | 0.808153 | 0.017431 |
Consider the piecewise function defined as follows:
$$
f(x) =
\begin{cases}
x^2 & \text{if } x < 2 \\
\frac{x-3}{\sqrt{x-2}-1} & \text{if } x \geq 2
\end{cases}
$$
1. Determine the limit of \( f(x) \) as \( x \to 2 \) from the left and from the right.
2. Using a single table command, create a four-column table tha... | automath | [
0.03121556155383587,
0.04153919965028763,
-0.02909502014517784,
-0.013945860788226128,
0.020084165036678314,
-0.016612207517027855,
-0.019851703196763992,
0.09019062668085098,
0.024847716093063354,
0.016287202015519142,
0.013417129404842854,
0.028145907446742058,
-0.023512739688158035,
-0.... | 0.82189 | 0.801487 | 0.020403 |
Let \( \lambda \) be a positive integer parameter of a Poisson distribution, where the probability mass function is given by
\[
p_X(k) = P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \text{ for } k \in \mathbb{N}_0.
\]
Determine the largest integer \( k_0 \) such that \( p_X(k_0) \geq p_X(k) \) for all \( k \in \math... | automath | [
0.03219645842909813,
0.024279914796352386,
-0.028433743864297867,
-0.043231815099716187,
0.03671618178486824,
-0.006826263852417469,
-0.018549608066678047,
0.05888383463025093,
0.024565808475017548,
-0.0026075392961502075,
0.005791145842522383,
0.03569330275058746,
-0.04498785734176636,
-0... | 0.829005 | 0.804174 | 0.02483 |
Given the linear programming problem defined by
\[
\begin{array}{cl}
\text{maximize} & c^Tx\\
\text{subject to} & a^Tx\geq 0 \\
& 0\leq x\leq x^{\max}
\end{array}
\]
where \(c,a \in \mathbb{R}^{M\times 1}\) and \(c_i>0\) for all \(i=1,2,\ldots,M\), if \(a_i > 0\) for all \(i\) and \(a^T x^\max \geq 0\), what is ... | automath | [
-0.004995834548026323,
0.029734186828136444,
-0.0586867593228817,
-0.03282054513692856,
0.04369821771979332,
-0.0026925967540591955,
-0.0315101221203804,
0.07442741096019745,
0.046812646090984344,
-0.014963854104280472,
-0.0031448451336473227,
0.013573041185736656,
-0.023274453356862068,
-... | 0.81454 | 0.798161 | 0.01638 |
Consider the boundary value problem defined by the following equations involving partial derivatives:
$$\left\{x^{(2,0)}(u,t) \left(1-\frac{y^{(1,0)}(u,t)^2}{\left(x^{(1,0)}(u,t)^2+y^{(1,0)}(u,t)^2\right)^{3/2}}\right)+\frac{x^{(1,0)}(u,t) y^{(1,0)}(u,t) y^{(2,0)}(u,t)}{\left(x^{(1,0)}(u,t)^2+y^{(1,0)}(u,t)^2\right)^{... | automath | [
0.0060508400201797485,
0.03371667489409447,
-0.05262957513332367,
-0.015627847984433174,
0.047606211155653,
-0.01174517534673214,
-0.02599882148206234,
0.06315393000841141,
0.036312371492385864,
-0.013649523258209229,
-0.0010559542570263147,
0.03601348400115967,
-0.018091993406414986,
-0.0... | 0.834038 | 0.810603 | 0.023435 |
Let \( G \) be a compact Lie group of type \( SU(n) \) and \( H \) be a closed subgroup such that the quotient space \( G/H \) is a smooth manifold. Using the long exact sequence of homotopy groups associated with the fibration \( H \to G \to G/H \), calculate \( \pi_3(G/H) \) when \( G = SU(3) \) and \( H = SU(2) \). | automath | [
0.012998158112168312,
0.03712645545601845,
-0.029476530849933624,
-0.033545177429914474,
0.033299658447504044,
-0.016294186934828758,
-0.028634440153837204,
0.06664799153804779,
0.0330297015607357,
-0.0021423092111945152,
0.017339197918772697,
0.03949753940105438,
-0.029830895364284515,
-0... | 0.814401 | 0.794768 | 0.019633 |
Given a topological space \( X \) that is simply connected, determine the homotopy group \( \pi_2(X) \). If it is known that the homotopy groups \( \pi_n(X) \) are finitely generated for \( n \geq 1 \), can you conclude the structure of \( \pi_2(X) \)? Specifically, is \( \pi_2(X) \) also finitely generated, and if so,... | automath | [
0.011470404453575611,
0.0295664481818676,
-0.032499730587005615,
-0.04001033678650856,
0.01767122559249401,
-0.03231433779001236,
-0.017245659604668617,
0.07003801316022873,
0.04656725376844406,
-0.001094460254535079,
0.009871833957731724,
0.03364350646734238,
-0.029422448948025703,
-0.013... | 0.816824 | 0.796229 | 0.020595 |
Given two smooth \( d \)-dimensional Riemannian manifolds \( M \) and \( N \), consider a function \( f: M \to N \) that acts as a differentiable isometry, meaning that the differential \( df_p \) is an isometry at every point \( p \in M \).
Is it true that if \( f \) is merely differentiable (but not necessarily \( ... | automath | [
0.0058340998366475105,
0.034545764327049255,
-0.038132186979055405,
-0.046194594353437424,
0.03289218246936798,
-0.0014105475274845958,
-0.032824572175741196,
0.08023138344287872,
0.020900720730423927,
0.00039515222306363285,
-0.0039232284761965275,
0.022940313443541527,
-0.03881435096263885... | 0.804394 | 0.785785 | 0.018608 |
Given a set of points \( x_1, x_2, x_3 \) in \( \mathbb{R}^2 \) forming a triangle, if the convex cone generated by these points consists of all linear combinations where coefficients are non-negative, and the convex hull consists of all points inside or on the boundary of the triangle, calculate the area of the convex... | automath | [
0.0012528188526630402,
0.04982641711831093,
-0.0539216473698616,
-0.046515174210071564,
0.0235749464482069,
-0.004793084692209959,
-0.02468062937259674,
0.07358253747224808,
0.038874201476573944,
-0.008276156149804592,
0.010110924020409584,
0.038912333548069,
-0.04005799442529678,
-0.04317... | 0.833478 | 0.807784 | 0.025693 |
Given a finite group \( G \) acting on the set of \( \mathbb{F}_p \)-points of the projective space \( \mathbb{P}^1_{\mathbb{F}_p} \), derive the formula for the number of \( \mathbb{F}_p \)-points in the quotient \( (X/G)(\mathbb{F}_p) \) where \( X = \mathbb{P}^1_{\mathbb{F}_p} \) and the action of \( G = S_2 \) perm... | automath | [
0.022673342376947403,
0.02743404731154442,
-0.03662232309579849,
-0.0340944342315197,
0.009733245708048344,
-0.028680002316832542,
-0.025274064391851425,
0.04528440535068512,
0.03258611261844635,
0.008354452438652515,
-0.01056336797773838,
0.02542915940284729,
-0.035595931112766266,
-0.036... | 0.825261 | 0.808217 | 0.017044 |
Consider two sequences \( u_n \) and \( v_n \) defined as follows:
\[
u_n = \begin{cases}
\frac{1}{n} & \text{if } n \text{ is even} \\
1 & \text{if } n \text{ is odd}
\end{cases}, \quad
v_n = \begin{cases}
1 & \text{if } n \text{ is even} \\
\frac{1}{n} & \text{if } n \text{ is odd}
\end{cases}
\]
What is \(... | automath | [
-0.003645472927019,
0.04345844313502312,
-0.029247654601931572,
-0.02270430698990822,
0.024201350286602974,
-0.005837827455252409,
-0.029232367873191833,
0.07627339661121368,
0.025499017909169197,
-0.013021678663790226,
0.0031748025212436914,
0.022534506395459175,
-0.0169144906103611,
-0.0... | 0.826394 | 0.805082 | 0.021312 |
Given two piecewise functions defined as follows:
\[
f_1(t) =
\begin{cases}
0, & t \leq 0 \\
t^2, & t > 0
\end{cases}, \quad
f_2(t) =
\begin{cases}
t^2, & t \leq 0 \\
0, & t > 0
\end{cases}
\]
Determine whether the functions \( f_1(t) \) and \( f_2(t) \) are linearly independent or dependent on \( \mathbb{R} ... | automath | [
0.01609141007065773,
0.05519900098443031,
-0.05161348357796669,
-0.04388861358165741,
0.051724180579185486,
-0.01394591573625803,
-0.02085956186056137,
0.08530711382627487,
0.03906016796827316,
-0.010532381944358349,
0.005985693074762821,
0.012757466174662113,
-0.008553408086299896,
0.0138... | 0.8077 | 0.791443 | 0.016257 |
Given a closed interval \([a, b]\) in the real numbers, let \(S\) be the set of all continuous functions \(f: [a, b] \to \mathbb{R}\) that are differentiable on the open interval \((a, b)\). If \(f(a) = 0\) and \(f(b) = 0\), what is the maximum possible number of times the function \(f\) can reach a value of zero insid... | automath | [
0.022882403805851936,
0.0241533350199461,
-0.045119453221559525,
-0.017387107014656067,
0.03945396468043327,
0.011485821567475796,
-0.03768254444003105,
0.05292040854692459,
0.029508141800761223,
-0.00024248081899713725,
0.005452388431876898,
0.03130069747567177,
-0.023000529035925865,
-0.... | 0.829356 | 0.806583 | 0.022773 |
Given the function \( f(x) = x^k e^{-x} \), for \( x \geq 0 \), determine the minimum value of \( \alpha + \beta \) such that the Padé approximation \( R_{\alpha \beta}(x) \) for \( e^{-x} \) satisfies the inequality
\[
\sum_{i=0}^{2n-1} (-1)^i \frac{x^{k+i}}{i!} \leq f(x) \leq \varepsilon_n(x) + \sum_{i=0}^{2n-1} (-... | automath | [
0.026811817660927773,
0.03286329284310341,
-0.036587782204151154,
-0.030456138774752617,
0.03623892366886139,
-0.009260743856430054,
-0.003830904373899102,
0.07004427909851074,
0.02943662740290165,
0.002250346355140209,
0.0030520178843289614,
0.024858102202415466,
-0.04773890972137451,
-0.... | 0.829981 | 0.813055 | 0.016926 |
Given \( n \) people living in \( n \) distinct houses, each person is randomly assigned to one of the \( n \) houses, ensuring that no one returns to their original house. Define \( P(n) \) as the probability that after this arrangement, there is still exactly one person per house. If \( !n \) denotes the number of de... | automath | [
0.014911380596458912,
0.0427350252866745,
-0.034556981176137924,
-0.007253575604408979,
0.034684475511312485,
-0.017861397936940193,
-0.0008721001795493066,
0.02605355717241764,
0.022373860701918602,
-0.0002928178873844445,
0.0017015381017699838,
0.036637719720602036,
-0.026473945006728172,
... | 0.825778 | 0.809476 | 0.016303 |
Given a line that passes through the point \( A(-5, -4) \) and intersects the lines \( x + 3y + 2 = 0 \), \( 2x + y + 4 = 0 \), and \( x - y - 5 = 0 \) at points \( B \), \( C \), and \( D \) respectively, if it satisfies the condition
\[
\left( \frac{15}{AB} \right)^2 + \left( \frac{10}{AC} \right)^2 = \left( \frac{... | automath | [
0.023839719593524933,
0.02170202136039734,
-0.04035365954041481,
-0.06274781376123428,
0.06105054169893265,
-0.029831623658537865,
-0.0039709568955004215,
0.06612172722816467,
0.042046837508678436,
-0.005600638221949339,
0.008254198357462883,
0.026966925710439682,
-0.010575168766081333,
-0... | 0.833248 | 0.817058 | 0.01619 |
Given the non-linear vector equation
$$P = a*(V_0*t + P_0) + b*(V_1*t + P_1) + (1-a-b)*(V_2*t + P_2)$$
for known vectors \( P, V_0, V_1, V_2, P_0, P_1, P_2 \), and considering that \( a, b, t \) are variables, determine the value of \( t \) such that the point \( P \) is expressed as a linear combination of the vec... | automath | [
0.019237278029322624,
0.024480480700731277,
-0.0415211021900177,
-0.029444241896271706,
0.04243432357907295,
-0.017039019614458084,
-0.019150611013174057,
0.06297679990530014,
0.03396592661738396,
-0.0029233768582344055,
0.004684163257479668,
0.020222963765263557,
-0.016721930354833603,
-0... | 0.838611 | 0.819579 | 0.019032 |
Given the definition of a function \( f \) from a set \( A \) to a set \( B \) as \( f \subseteq A \times B \) and \( \forall x (x \in A \implies \exists! y ((x, y) \in f)) \), where \( \exists! \) denotes the uniqueness of \( y \), determine the minimum number of distinct ordered pairs that must be included in \( f \)... | automath | [
0.023785198107361794,
0.029332157224416733,
-0.04377472773194313,
-0.023136615753173828,
0.04718481004238129,
-0.013238866813480854,
-0.029070692136883736,
0.04181179404258728,
0.032769281417131424,
0.005458648316562176,
-0.00331133883446455,
0.017370348796248436,
-0.021862735971808434,
-0... | 0.83066 | 0.808181 | 0.022478 |
Consider a set of positive numbers \( S = \{s_1, s_2, \ldots, s_n\} \) such that \( 0 < s_i < 1 \) for each \( i \) and \( \sum_{i=1}^{n} s_i = k \) where \( k > 0 \). If we define a transformation \( t_i = \frac{s_i^p}{\sum_{j=1}^{n} s_j^p} \) for some fixed positive integer \( p \), what can we say about the relation... | automath | [
0.03718743100762367,
0.031891051679849625,
-0.05371900275349617,
-0.013422121293842793,
0.04116901010274887,
-0.016550568863749504,
-0.009111439809203148,
0.052062418311834335,
0.02656366117298603,
-0.002203870564699173,
0.005582191050052643,
0.006610885262489319,
-0.049107618629932404,
-0... | 0.83318 | 0.808911 | 0.024269 |
Let \( \sum a_n \) and \( \sum b_n \) be series with positive terms, where \( \sum b_n \) diverges.
(a) If \( a_n > b_n \) for all \( n \), what can you conclude about \( \sum a_n \)? Provide a justification for your conclusion.
(b) If \( a_n < b_n \) for all \( n \), what can you conclude about \( \sum a_n \)? Prov... | automath | [
0.01195357646793127,
0.029082508757710457,
-0.0301614198833704,
-0.019430436193943024,
0.03675222769379616,
-0.027065260335803032,
-0.016249196603894234,
0.051154907792806625,
0.04803166911005974,
0.0001287823251914233,
-0.014379656873643398,
0.021667838096618652,
-0.06671856343746185,
-0.... | 0.834415 | 0.816321 | 0.018094 |
Let \( A_1, A_2, A_3, \ldots \) be a sequence of subsets of \( \mathbb{R} \) such that the intersection \( A_1 \cap A_2 \cap \ldots \cap A_n \) is nonempty for each \( n \geq 1 \). Is it necessarily true that the intersection \( \bigcap_{n=1}^{\infty} A_n \) is nonempty? Additionally, does your answer differ if each \(... | automath | [
0.03547859936952591,
0.025738198310136795,
-0.016135796904563904,
-0.011615369468927383,
0.01199235301464796,
0.0020116616506129503,
-0.010158621706068516,
0.054110586643218994,
0.04188411310315132,
-0.020774181932210922,
0.010743576101958752,
0.008973637595772743,
-0.02155339904129505,
-0... | 0.833238 | 0.797714 | 0.035524 |
Consider the topological space \( S^1 \vee S^1 \), which is the wedge sum of two circles. Determine all distinct covering spaces of \( S^1 \vee S^1 \) and classify each covering space by identifying the corresponding subgroup of the fundamental group \( \pi_1(S^1 \vee S^1) \cong \mathbb{Z} * \mathbb{Z} \). For each sub... | automath | [
0.014996957033872604,
0.020401539281010628,
-0.021333592012524605,
-0.04135451465845108,
0.04025186970829964,
-0.024495668709278107,
-0.013640720397233963,
0.041356176137924194,
0.03428547829389572,
0.0008825400145724416,
0.012630253098905087,
0.018681194633245468,
-0.018127422779798508,
-... | 0.828925 | 0.795926 | 0.032999 |
Given the set of vectors \( \{(1,0,0)^T, (0,1,1)^T, (1,0,1)^T, (1,2,3)^T\} \), determine whether these vectors form a spanning set for \( \mathbb{R}^3 \). If they do span \( \mathbb{R}^3 \), construct a linear combination of these vectors to express the standard basis vectors \( (1,0,0)^T \), \( (0,1,0)^T \), and \( (0... | automath | [
0.03183421865105629,
0.03317156434059143,
-0.04260166361927986,
-0.05122501030564308,
0.0632147490978241,
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-0.009548167698085308,
0.054155874997377396,
0.03256886452436447,
-0.006546357646584511,
0.011998647823929787,
0.024928998202085495,
-0.04150286316871643,
-0.0313... | 0.824286 | 0.803906 | 0.02038 |
If \( a \) and \( b \) are two arbitrary positive rational numbers expressed in their canonical forms as \( a = \frac{p_1}{q_1} \) and \( b = \frac{p_2}{q_2} \), where \( p_1, p_2 \in \mathbb{Z}_{\ge 0} \) and \( q_1, q_2 \in \mathbb{Z}_{>0} \), what is the simplified expression for \( a + b \) and how does it demonstr... | automath | [
0.021573327481746674,
0.03924647718667984,
-0.0447240024805069,
-0.02801274135708809,
0.04108418896794319,
-0.010715446434915066,
-0.004340918734669685,
0.05672357976436615,
0.04909449443221092,
-0.008834369480609894,
-0.008010323159396648,
0.021575618535280228,
-0.04480753093957901,
-0.03... | 0.828166 | 0.811247 | 0.016919 |
Given a sequence of positive real numbers \(\{a_n\}\) such that the series \(\sum_{n=1}^\infty a_n\) is convergent, determine the convergence of the following series:
1. \(\sum_{n=1}^\infty \frac{a_n}{1 + a_n}\)
2. \(\sum_{n=1}^\infty \frac{a_n^{1/4}}{n^{4/5}}\)
3. \(\sum_{n=1}^\infty n a_n \sin\frac{1}{n}\)
Which of... | automath | [
0.01864486187696457,
0.0505196750164032,
-0.028023753315210342,
-0.029136348515748978,
0.036196038126945496,
-0.003764640772715211,
-0.02271648868918419,
0.06404843926429749,
0.04464241862297058,
0.003432649653404951,
0.005544776562601328,
-0.0044426568783819675,
-0.039965711534023285,
-0.... | 0.824293 | 0.806721 | 0.017572 |
Let \( V = \mathbb{R}^3 \) be represented as the direct sum of two subspaces \( U \) and \( W \), where \( U \) is the span of the vectors \( \mathbf{u} = (1, 0, 0) \) and \( \mathbf{v} = (0, 1, 0) \). Suppose \( W \) is the span of the vector \( \mathbf{w} = (0, 0, 1) \). Determine the dimension of the subspace \( V \... | automath | [
-0.000003899409875884885,
0.027026167139410973,
-0.03681477904319763,
-0.056990716606378555,
0.044668711721897125,
0.008086333982646465,
-0.015267342329025269,
0.06773950159549713,
0.03677972033619881,
0.0052887252531945705,
0.008645571768283844,
0.0380457378923893,
-0.04669218510389328,
-... | 0.822154 | 0.799062 | 0.023092 |
In triangle \( ABC \), let point \( D \) be located such that the cevians \( AM \), \( BN \), and \( CP \) intersect at \( D \) with \( M \), \( N \), and \( P \) being the feet of the cevians from vertices \( A \), \( B \), and \( C \), respectively. Given this configuration, under what conditions on point \( D \) can... | automath | [
0.027076009660959244,
0.010048330761492252,
-0.05658509209752083,
-0.03561368212103844,
0.028771763667464256,
-0.0032912231981754303,
-0.0006830377969890833,
0.05745774880051613,
0.023953678086400032,
0.0006587505340576172,
0.019014280289411545,
0.04106450453400612,
-0.00906599871814251,
-... | 0.83677 | 0.809575 | 0.027195 |
Let \( a_n = n^2 \) and \( \lambda_n = n \) for \( n \geq 1 \). Determine the value of \( \sigma_0 \) such that the Dirichlet series
\[
f(s) = \sum_{n=1}^{\infty} n^2 e^{-n s}
\]
converges for all \( s = \sigma + it \) where \( \sigma > \sigma_0 \). | automath | [
-0.01744055561721325,
0.0329420305788517,
-0.028623607009649277,
-0.033954933285713196,
0.044007230550050735,
-0.013045924715697765,
-0.012884117662906647,
0.065862275660038,
0.030574748292565346,
-0.0035448961425572634,
0.00892317108809948,
0.020664909854531288,
-0.04683665931224823,
-0.0... | 0.831018 | 0.809665 | 0.021353 |
Let \( f : \mathbb{R} \to \mathbb{R} \) be the identity function defined by \( f(x) = x \) for all \( x \in \mathbb{R} \).
1. Determine the values of the domain, range, and co-domain of the function \( f \).
2. If the identity function is restricted to the set \( A = \{1, 2, 3, 4, 5, 6\} \), what is the value of \( ... | automath | [
0.03927397355437279,
0.03960749879479408,
-0.05609894171357155,
-0.02910810522735119,
0.02786834165453911,
-0.011577977798879147,
-0.00005438454172690399,
0.04596220701932907,
0.04256857931613922,
0.004033781588077545,
-0.0006669543799944222,
0.009425267577171326,
-0.03709152713418007,
-0.... | 0.820399 | 0.802296 | 0.018103 |
Let events \( A \) and \( B \) be defined in a probability space with \( P(B) > 0 \). Demonstrate that \( P(A | B) \) satisfies the properties of a probability measure, specifically verifying that for any event \( C \), the following condition holds:
\[
P(C | B) = \frac{P(C \cap B)}{P(B)}
\]
is non-negative and sum... | automath | [
0.00923843216150999,
-0.0011165884789079428,
-0.044710345566272736,
-0.008718577213585377,
0.045457638800144196,
-0.013111066073179245,
-0.005703050177544355,
0.05064307153224945,
0.045372068881988525,
0.01260028313845396,
-0.015581592917442322,
0.027397504076361656,
-0.007323820609599352,
... | 0.813494 | 0.785345 | 0.028149 |
Let $\mathcal{F}$ be a category cofibred in groupoids over a Noetherian ring $\Lambda$ that satisfies the Rim-Schlessinger condition (RS) and both Schlessinger's conditions (S1) and (S2). If $x_0$ is an object of $\mathcal{F}(k)$, what is the maximum possible dimension of the tangent space $T\mathcal{D}\! \mathit{ef}_{... | automath | [
0.006837199907749891,
0.023676825687289238,
-0.03851288929581642,
-0.04062355309724808,
0.0447644405066967,
-0.008285170421004295,
-0.008362123742699623,
0.069855697453022,
0.03967231884598732,
-0.008971148170530796,
-0.011610530316829681,
0.028172602877020836,
-0.027521930634975433,
-0.01... | 0.80179 | 0.783469 | 0.018321 |
Given \( n \) points in the plane defined by their coordinates \((x_i, y_i)\) for \( i = 1, 2, \ldots, n\), where \( x_i \) and \( y_i \) are positive integers, determine which point is closest to the point of origin \((1, 1)\) without directly calculating the Euclidean distance using multiplication, division, or squar... | automath | [
0.031150121241807938,
0.049353379756212234,
-0.024413349106907845,
-0.03400920704007149,
0.03134874999523163,
-0.015602109022438526,
-0.011606094427406788,
0.04509897157549858,
0.0422498993575573,
-0.0125131756067276,
-0.011557547375559807,
0.020662080496549606,
-0.009710398502647877,
-0.0... | 0.834186 | 0.809936 | 0.02425 |
Let \( n = p^2 q \), where \( p \) and \( q \) are distinct prime numbers. How many integers \( m \) satisfy \( 1 \leq m \leq n \) such that \( \gcd(m, n) = 1 \)?
Please express your answer in terms of \( p \) and \( q \), and provide the correct formula to determine the number of such integers. | automath | [
0.04375995323061943,
0.02828926406800747,
-0.03542887419462204,
-0.04715156927704811,
0.03526226431131363,
-0.027729883790016174,
-0.011647695675492287,
0.04019926115870476,
0.030799372121691704,
-0.012227194383740425,
-0.0043196179904043674,
0.02554413117468357,
-0.026171542704105377,
-0.... | 0.837329 | 0.82078 | 0.01655 |
Consider the partial differential equation given by
\[
u_{t} = a_{1} u_{xx} + a_{2} \Phi(t) u_{x}^{2},
\]
with the initial condition
\[
u(0,x) = \log \cosh x,
\]
defined over the domain \((x,t) \in \mathbb{R} \times [0,1]\), where \(\Phi(t)\) is the cumulative distribution function of the standard Gaussian distri... | automath | [
-0.016454894095659256,
0.03486507013440132,
-0.04268261417746544,
-0.036999061703681946,
0.036811064928770065,
-0.002820547902956605,
-0.021408190950751305,
0.07885061204433441,
0.02467157691717148,
0.0006921452004462481,
0.012073843739926815,
0.02597060054540634,
-0.02363695204257965,
0.0... | 0.814174 | 0.794825 | 0.019349 |
Given two finitely generated groups \( B \) and \( C \), if there exists a finitely generated group \( A \) such that \( A \times B \) is isomorphic to \( A \times C \), under what conditions can we guarantee that \( B \) is isomorphic to \( C \)? Furthermore, can you provide an example of finitely generated groups \( ... | automath | [
0.027712078765034676,
0.04357306659221649,
-0.040797967463731766,
-0.029193507507443428,
0.028948040679097176,
-0.03688807412981987,
0.00003244261824875139,
0.07695679366588593,
0.012983082793653011,
-0.012085352092981339,
0.019739370793104172,
0.023141542449593544,
-0.028029177337884903,
... | 0.811226 | 0.788123 | 0.023103 |
Given a finite indexed family of non-empty sets \( S_1, S_2, \ldots, S_n \) where \( |S_i| = k_i \) for each \( i \) from \( 1 \) to \( n \), compute the total number of distinct mapping choices that can be made from the index set \( \{1, 2, \ldots, n\} \) to the union of the sets, assuming we can choose one element fr... | automath | [
0.02814566157758236,
0.03426773101091385,
-0.0342588946223259,
-0.018885023891925812,
0.02993132919073105,
-0.028609219938516617,
-0.01737377792596817,
0.050050780177116394,
0.03384030982851982,
0.0021140766330063343,
0.03056441619992256,
0.026150347664952278,
-0.023821992799639702,
-0.003... | 0.828781 | 0.81244 | 0.016341 |
Let \( (a_n) \) be a sequence such that
\[
\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1.
\]
Provide an example of a sequence \( (a_n) \) that converges and an example of a sequence that diverges, both satisfying this limit condition. Additionally, explain why the statement "If \( \lim_{n \to \infty} \l... | automath | [
0.013203847222030163,
0.054911017417907715,
-0.02403116039931774,
-0.04282939434051514,
0.025275686755776405,
-0.010466234758496284,
-0.025823378935456276,
0.08816732466220856,
0.0161893330514431,
-0.02060757763683796,
0.006529354490339756,
-0.02044934406876564,
-0.025188978761434555,
0.00... | 0.809577 | 0.77815 | 0.031427 |
Given an integer map \( m: \mathbb{N} \to \mathbb{N} \) such that the set \( E_r = m^{-1}(\{r\}) \) is finite for each \( r \in \mathbb{N} \), consider the operator \( M \) defined on the Hilbert space \( \ell^2(\mathbb{N}) \). Under what conditions is the von Neumann algebra \( \mathcal{M} \) generated by \( M^{\star ... | automath | [
0.04104095324873924,
0.02892032265663147,
-0.027277974411845207,
-0.03636492043733597,
0.02874109335243702,
-0.004854205530136824,
-0.019604282453656197,
0.08650439232587814,
0.04248956963419914,
-0.017779650166630745,
0.011139131151139736,
0.029019342735409737,
-0.01052840892225504,
-0.02... | 0.816712 | 0.79928 | 0.017432 |
Given the parametric surface defined by \( r(u, v) = \langle v \sin(u), v \cos(u), v^2 \rangle \), and knowing that the point \( (1, 1, 2) \) lies on this surface, determine:
1. The grid curve on the surface for a constant \( v \).
2. The grid curve on the surface for a constant \( u \).
After finding both curves, ca... | automath | [
0.008190144784748554,
0.047728221863508224,
-0.04189952835440636,
-0.04200239107012749,
0.03842983767390251,
-0.023085149005055428,
-0.011189987882971764,
0.07789278030395508,
0.02157614938914776,
0.011646656319499016,
0.009472659789025784,
0.037473324686288834,
-0.0285273976624012,
0.0090... | 0.814411 | 0.793777 | 0.020634 |
Let \( X \) be a quasi-compact scheme and let \( \mathcal{L} \) be an ample invertible sheaf on \( X \). If \( S = \Gamma_*(X, \mathcal{L}) \), what is the minimum integer \( n_0 \) such that for every quasi-coherent sheaf \( \mathcal{F} \) of finite type on \( X \), the sheaf \( \mathcal{F} \otimes_{\mathcal{O}_X} \ma... | automath | [
0.01137193851172924,
0.047376424074172974,
-0.03537243977189064,
-0.025485938414931297,
0.026716813445091248,
-0.016330871731042862,
-0.0259260106831789,
0.07308357954025269,
0.03530103713274002,
-0.000020063298507011496,
0.008393014781177044,
0.029749738052487373,
-0.023445747792720795,
0... | 0.812048 | 0.792304 | 0.019744 |
Let \( R \) be a Noetherian local ring and \( I \) a proper ideal contained in its maximal ideal \( \mathfrak{m} \). If \( M \) is a finite \( R \)-module and there exists \( n \) such that the length of the module \( M/I M \) is \( n \), what can be inferred about the associated primes of \( M \) given the conditions ... | automath | [
0.0463431291282177,
0.042204927653074265,
-0.021735534071922302,
-0.03582971543073654,
0.02568710967898369,
-0.02641957253217697,
0.002445111284032464,
0.04807833209633827,
0.037790630012750626,
-0.01236596331000328,
-0.0014386093243956566,
0.015448103658854961,
-0.045080870389938354,
-0.0... | 0.814755 | 0.793457 | 0.021298 |
Given \( n \) distinct numbers, we can create \( n! \) permutations. Now, if we introduce a new number \( X \) into this set, which cannot be placed adjacent to \( k \) specified numbers from the original set, how many valid permutations can be formed such that \( X \) is not adjacent to any of these \( k \) restricted... | automath | [
0.022017471492290497,
0.037457410246133804,
-0.029884057119488716,
-0.019792448729276657,
0.0405714213848114,
-0.018757887184619904,
-0.00561132188886404,
0.047899872064590454,
0.02760588936507702,
-0.013156617991626263,
0.0009637857438065112,
0.0216896403580904,
-0.032629791647195816,
-0.... | 0.83472 | 0.81416 | 0.020561 |
Given the boundary value problem defined by the ordinary differential equation
$$A^2 + f f'' = (f')^2, \quad 0 < R < 1/2,$$
with boundary conditions $f(1/2) = -1$ and $f'(1/2) = 0$, and the additional condition $f(0) = 0$, determine the value of the eigenvalue $A$ such that the solution takes the form $f(R) = -\sin{... | automath | [
-0.018989773467183113,
0.02119629643857479,
-0.043953534215688705,
-0.03749489411711693,
0.037004612386226654,
-0.008068897761404514,
0.0077069164253771305,
0.07945472747087479,
0.04647606611251831,
0.004617266822606325,
0.005210453644394875,
0.01589752919971943,
-0.040867097675800323,
-0.... | 0.820649 | 0.804369 | 0.01628 |
Is there a unique pair of integers \( (m, n) \) such that the identity
\[
\sum_{k=1}^n k^m = \left( \sum_{k=1}^n k \right)^n
\]
holds true, aside from the specific case \( m=3 \) and \( n=2 \)? If so, what are the values of \( m \) and \( n \), and if not, what is the reasoning behind this restriction? | automath | [
0.04183231294155121,
0.04439948871731758,
-0.0404638834297657,
-0.03278818354010582,
0.05242070183157921,
-0.013768660835921764,
0.0073967138305306435,
0.06797914952039719,
0.040022291243076324,
-0.02649012766778469,
0.012791764922440052,
0.008197277784347534,
-0.04359735548496246,
-0.0353... | 0.833929 | 0.816537 | 0.017392 |
Let $S$ be a locally Noetherian scheme and let $s \in S$. If the local ring $\mathcal{O}_{S, s}$ is universally catenary, what can be concluded about the local rings at each point of a scheme $X$ that is locally of finite type over $S$? Specifically, determine the relationship between the catenarity of $\mathcal{O}_{X,... | automath | [
0.0018397263484075665,
0.03849046304821968,
-0.03621397912502289,
-0.031660955399274826,
0.0338536873459816,
-0.03867186978459358,
-0.017844034358859062,
0.06378234922885895,
0.018404645845294,
-0.006744183134287596,
0.022434860467910767,
0.052354998886585236,
-0.03843545541167259,
-0.0025... | 0.805717 | 0.780346 | 0.025371 |
Is there a sequence of connected $k$-regular graphs $G_n$ such that the number of vertices $|V_n| \to \infty$ and both of the following conditions hold:
1. The smallest non-zero eigenvalue of the combinatorial Laplacian satisfies $\lambda_1(G_n) = O(|V_n|^{-2})$? If so, what is the smallest constant that can be achie... | automath | [
-0.015884246677160263,
0.05031729117035866,
-0.030905507504940033,
-0.027843603864312172,
0.022698096930980682,
0.017613576725125313,
-0.023358087986707687,
0.05329614132642746,
0.017262207344174385,
-0.033425524830818176,
0.02118241786956787,
0.02549518272280693,
-0.012381421402096748,
-0... | 0.819302 | 0.799864 | 0.019437 |
Given the following truth table for four boolean variables \( A, B, C, \) and \( D \):
\[
\begin{array}{|c|c|c|c|c|}
\hline
A & B & C & D & F \\
\hline
T & T & T & T & T \\
T & T & T & F & T \\
T & T & F & T & F \\
T & T & F & F & F \\
T & F & T & T & T \\
T & F & T & F & T \\
T & F & F & T & F \\
T & F & F & F & F \\... | automath | [
0.04242495447397232,
0.02918105758726597,
-0.063186876475811,
-0.028650542721152306,
0.049291081726551056,
-0.049866169691085815,
-0.004246886819601059,
0.08105718344449997,
0.04635953903198242,
-0.00933169573545456,
0.0075962599366903305,
0.04065769165754318,
-0.009004111401736736,
-0.018... | 0.824406 | 0.806432 | 0.017975 |
Let \( f: \mathbb{R} \to \mathbb{R} \) be defined by
\[
f(x) =
\begin{cases}
-1 & \text{if } x \in \mathbb{Q} \\
1 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q}
\end{cases}
\]
under the usual topology on \(\mathbb{R}\) (denoted \(T_u\)) and the cofinite topology on \(\mathbb{R}\) (denoted \(T_{cof}\)).
1.... | automath | [
0.01554199866950512,
0.04473952203989029,
-0.03065073862671852,
-0.033451423048973083,
0.03271220251917839,
-0.0036012157797813416,
-0.002727887127548456,
0.07163560390472412,
0.03217396140098572,
-0.0022462166380137205,
0.012812371365725994,
0.015776168555021286,
-0.010639801621437073,
-0... | 0.816131 | 0.794264 | 0.021867 |
In a Hausdorff space \( X \), if \( x \) is a point in \( X \) and \( U \) is a neighborhood of \( x \), what can you say about the existence of a neighborhood \( V \) of \( x \) such that the closure \( \overline{V} \) is compact and contained within \( U \)? Specifically, if \( X \) is locally compact, how can you co... | automath | [
0.00021814292995259166,
0.04874761775135994,
-0.037042804062366486,
-0.028058895841240883,
0.04727425426244736,
-0.004271738231182098,
-0.008446707390248775,
0.06831396371126175,
0.04760035499930382,
-0.0016856435686349869,
0.01920723356306553,
0.04189107194542885,
-0.03323408588767052,
-0... | 0.807277 | 0.790452 | 0.016825 |
Given a relation \( R \) on the set \( A = \{ a, b, c, d \} \) defined as \( R = \{ (a,b), (b,a), (c,c), (d,a), (d,b) \} \), determine whether \( R \) is reflexive, symmetric, and transitive. Provide a detailed justification for each property. Additionally, if any property does not hold, give an example to illustrate w... | automath | [
0.06467963010072708,
0.0374702513217926,
-0.02751675248146057,
-0.02958064153790474,
0.03527693450450897,
-0.025140920653939247,
-0.005715282168239355,
0.06783043593168259,
0.03707239776849747,
-0.022244010120630264,
0.014231057837605476,
0.015457113273441792,
-0.03181348368525505,
-0.0259... | 0.808752 | 0.78908 | 0.019672 |
Given the function \( g: \mathbb{R} \to \mathbb{R} \) such that \( g(x) \) is continuous and \( g(x) > 0 \) for all \( x \), and let \( f_X(x) \) be a non-negative function integrated over the entire real line.
Demonstrate that the following equality holds:
\[
\int_0^\infty \int_{\{x : g(x) > t\}} f_X(x) \, dx \, dt... | automath | [
0.020091552287340164,
0.025003083050251007,
-0.030732616782188416,
-0.02304770052433014,
0.033909641206264496,
-0.0034420527517795563,
-0.029802929610013962,
0.07686510682106018,
0.03169311583042145,
0.003711742116138339,
-0.006930545903742313,
0.020946312695741653,
-0.023526784032583237,
... | 0.806942 | 0.784022 | 0.022919 |
Let \( f(x) = \sqrt{x} \) on the interval \([0,4]\). Find the value of \( c \) that satisfies the conclusion of the Mean Value Theorem, and determine if the secant line connecting the endpoints of the interval and the tangent line at the point \((c, f(c))\) are parallel. If they are parallel, provide a justification fo... | automath | [
0.015733765438199043,
0.029485590755939484,
-0.05191134661436081,
-0.04382803291082382,
0.05001705139875412,
0.001496506854891777,
-0.03421417996287346,
0.07088179886341095,
0.03904721140861511,
-0.0050634583458304405,
0.014864807948470116,
0.034643154591321945,
-0.007731654215604067,
-0.0... | 0.830909 | 0.813309 | 0.0176 |
Consider a bounded domain \(\Omega \subset \mathbb{R}^2\) defined as the area between the graphs of the functions \(y = x^2\) and \(y = -x^2\) for \(-1 \leq x \leq 1\). What kind of domain is \(\Omega\) classified as regarding its boundary regularity (e.g., Lipschitz, \(C^1\), or \(C^2\))? Additionally, how does the re... | automath | [
-0.02041606977581978,
0.03854238986968994,
-0.039332736283540726,
-0.039189018309116364,
0.024386268109083176,
-0.008043712936341763,
-0.02037643827497959,
0.05781969428062439,
0.03830673173069954,
-0.006042963825166225,
-0.009573165327310562,
0.038471221923828125,
-0.009067222476005554,
-... | 0.810529 | 0.792463 | 0.018066 |
Consider a metric space \( X \) and let \( A \) be a subset of \( X \) consisting of sequences \( \{ a_n \} \) such that \( a_n = 0 \) for all but finitely many \( n \). If \( A \) intersects every open set in \( X \), what can you conclude about the nature of \( A \) in relation to the topology of \( X \)? Specificall... | automath | [
0.03906281664967537,
0.023467572405934334,
-0.03659816086292267,
-0.01524330023676157,
0.016007578000426292,
-0.004610082134604454,
-0.01938922517001629,
0.04949335381388664,
0.04235386103391647,
-0.015145204961299896,
0.01357662957161665,
0.018565215170383453,
-0.03408239036798477,
-0.015... | 0.822584 | 0.792589 | 0.029994 |
Consider a function \( F: \omega \to \mathcal{W}(F) \) that satisfies the conditions of the theorem. If \( \mathcal{W}(F) \) has a cardinality of \( \kappa \), what is the maximum possible cardinality of the range of \( F \) if \( \kappa \) is infinite and \( \omega \) is countably infinite? | automath | [
0.043981827795505524,
0.022871503606438637,
-0.0323496013879776,
-0.022365793585777283,
0.04015639051795006,
-0.019862251356244087,
-0.0341637022793293,
0.06131020188331604,
0.00533089367672801,
-0.007800105027854443,
0.003716569161042571,
0.04024603217840195,
-0.03457840532064438,
-0.0211... | 0.825872 | 0.801708 | 0.024165 |
Suppose we cover each rational number \( q \) in \( \mathbb{Q} \) with an open interval of the form \( (q - \epsilon_q, q + \epsilon_q) \), where \( \epsilon_q \) is a positive number that may depend on \( q \). If the total length of all these intervals is finite, does their union necessarily cover all of \( \mathbb{R... | automath | [
0.02493038773536682,
0.011754141189157963,
-0.03776431456208229,
-0.034285206347703934,
0.021539464592933655,
-0.019120261073112488,
-0.0040871561504900455,
0.033152174204587936,
0.036654356867074966,
-0.016416890546679497,
0.013731089420616627,
0.02685662917792797,
-0.024654671549797058,
... | 0.836447 | 0.810775 | 0.025672 |
Let \( n \) be a positive integer with the following properties: if \( n = p_1^{k_1} p_2^{k_2} \cdots p_m^{k_m} \) is its prime factorization, determine the relationship between \( m \) (the number of distinct prime factors) and \( \tau(n) \) (the number of divisors of \( n \)). Specifically, can you express \( \tau(n)... | automath | [
0.052370835095644,
0.04205888509750366,
-0.043579619377851486,
-0.03593916818499565,
0.033389925956726074,
-0.01876496709883213,
-0.023968735709786415,
0.06310546398162842,
0.03409562632441521,
0.009728946723043919,
-0.008945352397859097,
0.025706343352794647,
-0.040847402065992355,
-0.014... | 0.826411 | 0.808591 | 0.01782 |
Given the Pythagorean triple \((a, b, c)\) where \(a^2 + b^2 = c^2\) and \(a, b, c\) are natural numbers less than 42, find the number of distinct triples \((x, y, z)\) such that \(x^2 + y^2 = z^2\) and \((z - x)^2 + y^2 = k^2\) for some natural number \(k\). Additionally, give a specific example of such a triple where... | automath | [
0.013208159245550632,
0.0346941202878952,
-0.034686192870140076,
-0.04927448928356171,
0.025974690914154053,
-0.013303409330546856,
-0.005334625020623207,
0.05096310004591942,
0.02873639576137066,
-0.001999989151954651,
0.011505561880767345,
0.024037905037403107,
-0.04104025661945343,
-0.0... | 0.836902 | 0.807668 | 0.029234 |
Given a function \( g \in W^{1,1}(0,1) \cap C^1(0,1) \cap C[0,1] \), which of the following assertions holds true for the class of absolutely continuous functions? Specifically, determine the smallest class of functions for which property 7 holds:
7. For every \( 0 < \theta < 1 \), there exists a \( 0 < \varphi_{\thet... | automath | [
0.0130724236369133,
0.05580224096775055,
-0.016744477674365044,
-0.046571359038352966,
0.04344430938363075,
-0.01793799363076687,
-0.03736203536391258,
0.0796220600605011,
0.024965936318039894,
-0.012423837557435036,
0.013200283981859684,
0.021339774131774902,
-0.02645040862262249,
-0.0057... | 0.808533 | 0.790625 | 0.017908 |
Given the vertices of a quadrilateral \( P(1,3) \), \( Q(6,5) \), \( R(8,0) \), and \( S(3,-2) \), determine whether the quadrilateral is a parallelogram. If it is, calculate the lengths of the sides \( PQ \), \( QR \), \( RS \), and \( SP \) and verify if any specific type of parallelogram (such as a rhombus or rectan... | automath | [
0.01719707064330578,
0.03132753074169159,
-0.03868398070335388,
-0.038392920047044754,
0.03534294292330742,
-0.008229020982980728,
-0.007183808367699385,
0.07843784987926483,
0.018544521182775497,
-0.003020670497789979,
-0.004056123550981283,
0.03622646629810333,
-0.017410732805728912,
-0.... | 0.816566 | 0.797647 | 0.018919 |
Consider the sequences defined by \( f_n = \frac{n^2}{1+n^3} \) and \( g_n = \frac{n^3}{1+n^4} \). Using the discrete L'Hospital's rule, find the limit
\[
\lim_{n \to \infty} \frac{F_n}{G_n}
\]
where \( F_n = \sum_{j=1}^n f_j \) and \( G_n = \sum_{j=1}^n g_j \). What is the value of this limit? | automath | [
0.005845263600349426,
0.03207456320524216,
-0.030403589829802513,
-0.030366409569978714,
0.03138992190361023,
-0.011664423160254955,
-0.03772646561264992,
0.07027801871299744,
0.016690371558070183,
-0.009004160761833191,
0.0003684062394313514,
0.038148798048496246,
-0.04040737450122833,
0.... | 0.822867 | 0.803547 | 0.01932 |
Given the cubic surface defined by the equation $$X_1^3 + ab^2X_2^3 + a^2bX_3^2 - 3abX_1X_2X_3 = 1,$$ where \( a \) and \( b \) are coprime squarefree integers, determine a specific rational parametrization of this surface. Additionally, demonstrate that this parametrization encompasses all rational points on the surfa... | automath | [
0.02481555938720703,
0.04453949257731438,
-0.056837987154722214,
-0.04537394270300865,
0.034564901143312454,
-0.028655383735895157,
0.018288368359208107,
0.057153575122356415,
0.04231588542461395,
-0.01669272407889366,
0.014025985263288021,
0.03176679089665413,
-0.03941482678055763,
-0.020... | 0.834424 | 0.807201 | 0.027223 |
In a directed graph (digraph) with \( n \) vertices that satisfies the following conditions:
1. No self-loops (no edges of the form \( (v, v) \)),
2. If there is an edge \( (u, v) \), there is no edge \( (v, u) \) (no bidirectional edges),
3. Each vertex \( v \) has at most \( d_{\mathrm{in}}(v) \leq 2 \) (maximum in-... | automath | [
0.004199435003101826,
0.01653742417693138,
-0.015340284444391727,
-0.01960284635424614,
0.022930167615413666,
-0.011782670393586159,
-0.04330878704786301,
0.05248448997735977,
0.02581927925348282,
0.002925347303971648,
-0.002197026740759611,
0.029535161331295967,
-0.00600363640114665,
-0.0... | 0.823795 | 0.807531 | 0.016264 |
Consider a \(3 \times 3 \times 3\) cube composed of \(27\) smaller unit cubes. Define a graph where each vertex represents a small cube, and there is an edge between two vertices if the corresponding cubes share a face.
1. How many edges does this graph contain?
2. Is it possible to find a Hamiltonian path that star... | automath | [
0.02194644883275032,
0.02827467955648899,
-0.041280101984739304,
-0.018564796075224876,
0.020026840269565582,
-0.021416902542114258,
-0.0238027423620224,
0.05792590230703354,
0.019668055698275566,
-0.014103211462497711,
0.025550328195095062,
0.01469784788787365,
-0.02488519623875618,
-0.03... | 0.836306 | 0.816287 | 0.020019 |
Let events \( A \) and \( B \) be defined in a certain probability space where \( P(A) = 0.6 \), \( P(B) = 0.4 \), and \( P(A \cap B) = 0.2 \). Determine the probability of event \( A \) given that event \( B \) has occurred, denoted as \( P(A|B) \), and explain how this can be represented using both a Venn diagram and... | automath | [
0.018429530784487724,
0.007036630064249039,
-0.04059665650129318,
-0.019558951258659363,
0.03234963119029999,
-0.020765604451298714,
-0.018286744132637978,
0.0677725300192833,
0.033310044556856155,
0.015399152413010597,
0.006633896846324205,
0.02820068970322609,
0.0061867800541222095,
-0.0... | 0.805455 | 0.787177 | 0.018278 |
Let \( p \) be a prime number, and let \( a \in \Q_p \). For integers \( n < m \), how many disjoint closed \( p^{-m} \)-balls centered at \( a + i p^n \) exist, where \( i \) ranges over all integers \( 0 \leq i < p^{m-n} \)? Provide the total count of these disjoint closed balls. | automath | [
0.032537586987018585,
0.015530198812484741,
-0.03922337666153908,
-0.013399641960859299,
0.02822202816605568,
-0.00923434179276228,
-0.01394103653728962,
0.04374665766954422,
0.03627710044384003,
0.00671883812174201,
0.007155116181820631,
0.035415828227996826,
-0.03823491558432579,
-0.0211... | 0.843862 | 0.825455 | 0.018407 |
Given two planar polygons \( Q \) and \( Q' \) with the same number of vertices \( n > 3 \), where \( Q \) has strictly convex internal angles at three vertices \( v_1, v_2, v_3 \) and \( Q' \) has corresponding vertices \( v_1', v_2', v_3' \), how can one construct a third polygon \( Q'' \) that is entirely contained ... | automath | [
0.022530635818839073,
0.039883408695459366,
-0.038382142782211304,
-0.020530495792627335,
0.00813292432576418,
0.015018224716186523,
-0.011577693745493889,
0.06069277599453926,
0.02502104640007019,
-0.008599619381129742,
0.016829850152134895,
0.03770590201020241,
-0.024550840258598328,
-0.... | 0.826721 | 0.79005 | 0.036671 |
Let $S$ be a scheme, and let $f: X \to Y$ be a morphism of algebraic spaces over $S$. If $\overline{x}$ is a geometric point of $X$ lying over a point $x \in |X|$ and $\overline{y} = f \circ \overline{x}$, how many equivalent conditions can be stated in terms of flatness regarding the morphism $f$ and the map on étale ... | automath | [
0.009100622497498989,
0.024364447221159935,
-0.05247024446725845,
-0.0332208052277565,
0.022207532078027725,
-0.02021561563014984,
-0.016942862421274185,
0.04468751326203346,
0.048393696546554565,
-0.010636824183166027,
0.002658611163496971,
0.03950328379869461,
-0.036342062056064606,
-0.0... | 0.82141 | 0.799101 | 0.022309 |
Let \( f_n(x) = \frac{x^n}{n!} \) for \( n \in \mathbb{N} \) defined on the interval \( I = [0, 1] \). Determine whether the series \( \sum_{n=1}^{\infty} f_n(x) \) converges almost everywhere on \( I \) and find the value of the integral \( \int_0^1 \sum_{n=1}^{\infty} f_n(x) \, dx \). | automath | [
0.01594998873770237,
0.042292915284633636,
-0.036644019186496735,
-0.02322298102080822,
0.03716384246945381,
0.00592048792168498,
-0.008234087377786636,
0.054794225841760635,
0.03672773763537407,
-0.012341353110969067,
-0.0069612786173820496,
0.00969958957284689,
-0.036062706261873245,
-0.... | 0.82067 | 0.7934 | 0.027269 |
Given a perfect object \( E \) in the derived category \( D(\mathcal{O}_X) \) on a scheme \( X \), let \( H^i(E) \) denote the \( i \)-th cohomology sheaf of \( E \). Suppose that \( H^i(E) \) is a perfect \( \mathcal{O}_X \)-module of tor dimension \( \leq 1 \) for all integers \( i \). If the rank of \( H^i(E) \) is ... | automath | [
0.016220828518271446,
0.026272404938936234,
-0.03672109916806221,
-0.03294296935200691,
0.017153210937976837,
-0.011689925566315651,
-0.015963532030582428,
0.05873710662126541,
0.03912287577986717,
-0.010658243671059608,
-0.004863435868173838,
0.032881174236536026,
-0.03883231058716774,
0.... | 0.819821 | 0.802369 | 0.017452 |
Given a positive-definite quadratic form \( Q \) on a symplectic vector space, if the symplectic inverse is defined as \( Q^\sigma \), how would the conditions imposed by Hörmander on the metric \( g_X \) relate to the stability of the symbols \( p \) when treated in terms of the metrics \( g_X \) and \( g_X^\sigma \)?... | automath | [
0.008998658508062363,
0.043343089520931244,
-0.04520007222890854,
-0.01413713674992323,
0.03470754995942116,
-0.0204648245126009,
-0.01298146415501833,
0.07636971771717072,
0.028454696759581566,
-0.013803867623209953,
-0.007538869511336088,
0.01835126429796219,
-0.02170827053487301,
-0.030... | 0.791043 | 0.770922 | 0.02012 |
Let \( C \) be a contour defined by a finite sequence of directed smooth curves \( C_1, \ldots, C_n \) in the complex plane. If each curve \( C_i \) is parameterized by the smooth path \( \gamma_i: [a_i, b_i] \to \mathbb{C} \) and there exists a reparameterization \( \sigma_i: [c_i, d_i] \to \mathbb{C} \) for every \( ... | automath | [
0.028112195432186127,
0.04802569001913071,
-0.028564710170030594,
-0.021403850987553596,
0.03809650242328644,
-0.011385918594896793,
-0.012143556028604507,
0.07297035306692123,
0.02843698114156723,
0.0035973191261291504,
0.015328440815210342,
0.0277088712900877,
-0.037548597902059555,
0.00... | 0.818712 | 0.794421 | 0.024291 |
Let \( R \) be a ring and \( S \) a finitely presented \( R \)-algebra. If the number of generators \( d \) of the ideal \( I \) corresponding to a certain presentation \( S = R[x_1, \ldots, x_n]/I \) is finite, and if there exists a prime \( \mathfrak{q} \subset S \) lying over a prime \( \mathfrak{p} \subset R \) suc... | automath | [
0.03685315325856209,
0.04859336465597153,
-0.04694412648677826,
-0.03611154109239578,
0.016918573528528214,
-0.02577044442296028,
-0.0175875686109066,
0.06165773421525955,
0.029592955484986305,
-0.016303131356835365,
0.011660637333989143,
0.0382828526198864,
-0.037553656846284866,
-0.01061... | 0.815275 | 0.792122 | 0.023153 |
Given the piecewise function defined by
\[
f(x) =
\begin{cases}
\sin x & \text{if } x \leq c \\
ax + b & \text{if } x > c
\end{cases}
\]
where \( a, b, c \) are non-zero real numbers, determine the number of values of \( a \) for which the function \( f \) is continuous at \( x = c \). Choose from the following ... | automath | [
0.015951821580529213,
0.03535155579447746,
-0.04174807667732239,
-0.016173318028450012,
0.027717754244804382,
0.0024682004004716873,
-0.01989199034869671,
0.0814768448472023,
0.024455370381474495,
-0.002871506381779909,
0.018467651680111885,
0.03228671848773956,
-0.013711497187614441,
-0.0... | 0.818853 | 0.800571 | 0.018282 |
Let \( f \) and \( g \) be two circle homeomorphisms for which there exists a continuous, surjective, and monotone map \( h: S^1 \to S^1 \) such that \( f \circ h = h \circ g \). If the rotation number \( \rho(f) \) is given by \( a \) and \( \rho(g) \) is given by \( b \), under the condition that \( a = b \), what is... | automath | [
0.03198842704296112,
0.01967468671500683,
-0.029786987230181694,
-0.019829094409942627,
0.0289499182254076,
-0.02464764378964901,
-0.03811771422624588,
0.040285367518663406,
0.01641646958887577,
-0.005638970527797937,
0.017962664365768433,
0.03226705640554428,
-0.026676565408706665,
-0.010... | 0.815924 | 0.796411 | 0.019513 |
Given two distinct lines in a projective plane, each containing three points of different colors—red, green, and blue—Pappus’ theorem states that the points of intersection of the lines connecting different colored points (red-green, red-blue, and green-blue) are collinear. If we consider the case where the red points ... | automath | [
0.017229516059160233,
0.03875284641981125,
-0.05108831822872162,
-0.05179539695382118,
0.016229234635829926,
-0.009606755338609219,
-0.018215559422969818,
0.054139282554388046,
0.04226044937968254,
-0.0011035222560167313,
0.006328522227704525,
0.022475488483905792,
-0.012688116170465946,
-... | 0.823794 | 0.79191 | 0.031885 |
Consider a rank-2 tensor \( T \) that can be represented as a bilinear map \( T: V \times V \to \mathbb{R} \) in a vector space \( V \) over \( \mathbb{R}^n \). If \( V \) has an inner product giving rise to a natural isomorphism between \( V \) and its dual space \( V^* \), how many distinct matrices are there that re... | automath | [
0.03409309685230255,
0.03170928359031677,
-0.051697809249162674,
-0.029039788991212845,
0.03205621987581253,
0.0034304638393223286,
-0.019628914073109627,
0.07677357643842697,
0.03816966339945793,
-0.0020865213591605425,
0.0015187388053163886,
0.02812153846025467,
-0.02499539405107498,
-0.... | 0.808006 | 0.790292 | 0.017714 |
Given the integral inequality
$$\int_{R_1}|u|^r \leq C\left(\int_{R_1}|\nabla u|^p\right)^{ar/p}\left(\int_{R_1}|u|^q\right)^{(1-a)r/q},$$
for a function $u \in \mathcal{C}^1_c(\mathbb{R}^n)$, let $R_\rho = \{ x \in \mathbb{R}^n \mid \rho \leq |x| < 2\rho \}$. If we define $v(x) = u(\rho x)$ and change variables ac... | automath | [
0.00040022912435233593,
0.04166581481695175,
-0.0329359732568264,
-0.032432008534669876,
0.027597766369581223,
0.002011717762798071,
-0.02608005702495575,
0.09053473174571991,
0.047396812587976456,
-0.0136534059420228,
-0.009498391300439835,
0.024657145142555237,
-0.03469459339976311,
-0.0... | 0.815888 | 0.788109 | 0.027779 |
Let \( E_n \) be the union of finitely many disjoint closed intervals such that the sum of their lengths is at least \( a > 0 \) for each \( n \), and suppose that \( E_1 \supset E_2 \supset \cdots \). What is the measure of \( \bigcap_{n=1}^{\infty} E_n \)? | automath | [
0.014653416350483894,
0.0022277645766735077,
-0.030567435547709465,
-0.02537652663886547,
0.016225025057792664,
0.0038167089223861694,
-0.04324078559875488,
0.04138598591089249,
0.01922253891825676,
-0.006290009245276451,
0.010213479399681091,
0.03311121463775635,
-0.029520081356167793,
-0... | 0.821626 | 0.800631 | 0.020995 |
In the context of an abelian category, consider a short exact sequence \( 0 \to A \xrightarrow{i} B \xrightarrow{q} C \to 0 \). If this sequence is split, can you find specific conditions under which the morphisms \( j: C \to B \) and \( p: B \to A \) exist such that the structure \( (B, i, j, p, q) \) represents the d... | automath | [
0.021020693704485893,
0.03800463676452637,
-0.01727597415447235,
0.0015300875529646873,
0.03386883810162544,
-0.03144022822380066,
-0.02062506414949894,
0.0717763602733612,
0.0263103898614645,
-0.0062269894406199455,
-0.013617393560707569,
0.0061185117810964584,
-0.022324150428175926,
0.00... | 0.816011 | 0.794764 | 0.021246 |
Let \( c_n \) denote the number of compositions of \( n \) that have no two consecutive even parts. Given the recurrence relation
\[
c_n = c_{n-1} + 2c_{n-2} - c_{n-4}
\]
for \( n \geq 5 \) with initial conditions \( c_1 = 1 \), \( c_2 = 2 \), \( c_3 = 4 \), and \( c_4 = 7 \), find a closed-form expression for \( c_... | automath | [
0.017160337418317795,
0.03157344087958336,
-0.046170953661203384,
-0.015412949025630951,
0.02763720601797104,
0.0059976475313305855,
-0.03206140547990799,
0.08806218951940536,
0.02820839360356331,
0.0038116280920803547,
0.027405565604567528,
0.014276123605668545,
-0.030509596690535545,
-0.... | 0.834484 | 0.816363 | 0.018122 |
Let \( P \) be an \( n \times n \) symmetric matrix with \( n \) real eigenvalues. If the distinct eigenvalues of \( P \) are \( \lambda_1, \ldots, \lambda_k \) with corresponding orthonormal eigenvectors \( u_1, \ldots, u_k \), where \( k \leq n \), what is the relationship between the number of distinct eigenvalues \... | automath | [
0.006031888071447611,
0.04217715561389923,
-0.041326362639665604,
-0.04086527228355408,
0.024524949491024017,
0.0028145008254796267,
-0.004037142731249332,
0.05476531758904457,
0.04318193346261978,
0.0047163814306259155,
-0.005869863089174032,
0.04128654673695564,
-0.043467435985803604,
-0... | 0.8087 | 0.790431 | 0.018269 |
Let \( G \) be a graph with \( 100! \) vertices, each labeled by a distinct permutation of the numbers \( 1, 2, 3, \ldots, 100 \). There is an edge between vertices \( u \) and \( v \) if and only if the label of \( u \) can be obtained by swapping two adjacent numbers in the label of \( v \).
Determine the value of ... | automath | [
0.013307345099747181,
0.014394120313227177,
-0.01806837134063244,
-0.025640690699219704,
0.020733458921313286,
-0.016354301944375038,
-0.030542392283678055,
0.044466812163591385,
0.014311626553535461,
-0.008631648495793343,
0.015124875120818615,
0.02378351055085659,
-0.03485748916864395,
0... | 0.829081 | 0.807557 | 0.021524 |
What conditions must a rigid, symmetric monoidal, semisimple category satisfy in order for it to be equivalent to the category of finite-dimensional representations of a semisimple algebraic group, and how can one use the center of the group to deduce properties of the category? | automath | [
0.024006394669413567,
0.013512303121387959,
-0.04624614119529724,
-0.05120377242565155,
0.03559053689241409,
-0.026721851900219917,
-0.02982909418642521,
0.04825509339570999,
0.041681718081235886,
-0.034444842487573624,
-0.0034547985997051,
0.02493203431367874,
-0.03895469009876251,
-0.011... | 0.807005 | 0.788458 | 0.018547 |
Given six points in \(\mathbb{R}^3\) defined as follows:
\[
\begin{aligned}
&points = \{ \\
&\{ \log(1 + \frac{a}{n}), \log(1 + \frac{b}{a+n}), \log(1 + \frac{c}{a+b+n}) \}, \\
&\{ \log(1 + \frac{a}{n}), \log(1 + \frac{b}{a+c+n}), \log(1 + \frac{c}{a+n}) \}, \\
&\{ \log(1 + \frac{a}{b+n}), \log(1 + \frac{b}{n}), \log(... | automath | [
0.012441452592611313,
0.045304909348487854,
-0.05385822802782059,
-0.02818969078361988,
0.04304509982466698,
-0.0053518107160925865,
-0.007689202204346657,
0.06968692690134048,
0.027103370055556297,
-0.005163294263184071,
-0.012341216206550598,
0.017695222049951553,
-0.02289910614490509,
-... | 0.839165 | 0.816202 | 0.022963 |
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