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 |
|---|---|---|---|---|---|
Consider the set of functions defined as
\[
f_i(x, y) = \frac{\sqrt{(x_i - x)^2 + (y_i - y)^2}}{S_i}
\]
for \(i = 1, 2, \ldots, n\), where \(S_i > 0\) is a constant for each function and \((x_i, y_i)\) are distinct points in the plane. What is the minimum value of
\[
\max_{1 \leq i \leq n} f_i(x, y)
\]
as \( (x,... | automath | [
0.007254057098180056,
0.04539577662944794,
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0.04550881311297417,
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-0.031852997839450836,
0.0623854324221611,
0.03853825852274895,
-0.00676028523594141,
0.0075202686712145805,
0.024489793926477432,
-0.015562554821372032,
-0.002... | 0.832066 | 0.801089 | 0.030976 |
Given a block matrix
$$\mathbf{H}=\begin{array}{cc}\mathbf{A}&\mathbf{B}\\\mathbf{B}^H&\mathbf{A}\end{array}$$
where $\mathbf{A}$ is a diagonal matrix of size $n \times n$ with real entries and $\mathbf{B}$ is any $n \times n$ matrix, consider the matrix
$$\mathbf{G}=\mathbf{H}^H(\mathbf{H}\mathbf{H}^H+\alpha\math... | automath | [
0.031222311779856682,
0.0347478911280632,
-0.04312629625201225,
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0.0344160720705986,
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-0.01620481163263321,
0.07457619905471802,
0.027346445247530937,
-0.014411370269954205,
0.013402153737843037,
0.011221693828701973,
-0.055826906114816666,
-0.0076... | 0.824575 | 0.799217 | 0.025358 |
Suppose we have a volume containing two conductors, each carrying charges \( Q_1 = 5 \, \text{C} \) and \( Q_2 = -3 \, \text{C} \), insulated from each other. If the region surrounding the conductors has a charge density \( \rho = 2 \, \text{C/m}^3 \), and if the surface of each conductor is at a different constant pot... | automath | [
0.0013897267635911703,
0.03149909898638725,
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0.027925627306103706,
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0.05433952063322067,
0.027584992349147797,
-0.010560473427176476,
0.0180340763181448,
0.01732562854886055,
-0.012767350301146507,
-0.02... | 0.804978 | 0.788583 | 0.016395 |
Given the following statements about Alex and chocolate:
1. \( \forall x (C(x) \implies L(a,x)) \): For all objects \( x \), if \( x \) contains chocolate, then Alex likes \( x \).
2. \( \forall x (C(x) \land L(a,x)) \): For all objects \( x \), \( x \) contains chocolate and Alex likes \( x \).
Is statement 2 a corr... | automath | [
0.032558828592300415,
0.025075607001781464,
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0.025165749713778496,
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0.012064865790307522,
0.038587599992752075,
-0.038273874670267105,
-0... | 0.812618 | 0.794328 | 0.01829 |
Is it possible for a rational Bézier curve of degree \(n\) to coincide exactly with the sine function \(y = \sin(x)\) over the interval \([0, \frac{\pi}{2}]\)? If so, what is the minimum degree \(n\) for which this is true, and what would the control points be? | automath | [
0.008597595617175102,
0.03989768400788307,
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0.021292462944984436,
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0.021562989801168442,
0.018503380939364433,
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Let \( A \in \mathbb{C}^{m \times m} \) be a matrix that satisfies the monic polynomial \( p(t) = (t - \mu_1)^{n_1}(t - \mu_2)^{n_2} \cdots (t - \mu_k)^{n_k} \) such that \( p(A) = 0 \). If \( \sigma(A) \) denotes the set of eigenvalues of \( A \), what is the maximum size of any Jordan block associated with the eigenv... | automath | [
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0.027203978970646858,
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0.003056357381865382,
0.029038265347480774,
-0.03772487863898277,
0.00... | 0.82359 | 0.801083 | 0.022507 |
Given two types of particles modeled as anisotropic shapes, one type being prolate spheroids with axes of lengths \(1\), \(1\), and \(10\) and the other type being oblate spheroids with axes of lengths \(10\), \(10\), and \(1\), calculate the probability that two randomly placed prolates will overlap in a cubic volume ... | automath | [
-0.011310252360999584,
0.031007900834083557,
-0.028714211657643318,
-0.01482347585260868,
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0.020769372582435608,
0.03218203783035278,
-0.015831714496016502,
-0... | 0.816379 | 0.788914 | 0.027466 |
Given a concrete category \(\mathcal{C}\) with a forgetful functor \(U:\mathcal{C}\to \mathsf{Set}\), let \(\mathcal{I}\) be a full subcategory of \(\mathcal{C}\) that is finally dense in \(\mathcal{C}\). Is it always true that \(\mathcal{I}\) is dense in \(\mathcal{C}\)? If not, can you provide a specific example of a... | automath | [
0.02068454585969448,
0.044266194105148315,
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0.03858422115445137,
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0.02076711133122444,
0.02661905623972416,
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0.012... | 0.799952 | 0.782766 | 0.017186 |
In the context of compact Hausdorff spaces, how can you characterize the uniformly continuous functions defined on a dense subset \( A \) of a compact Hausdorff space \( X \), specifically in terms of the topology of \( X \)? Can you provide an example illustrating the extension of such functions and state any notable ... | automath | [
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0.05751641467213631,
0.04317612200975418,
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0.024606604129076004,
0.02115842141211033,
-0.03735489025712013,
-0.00... | 0.796977 | 0.777819 | 0.019158 |
Given three real numbers \( x_1, x_2, x_3 \), express the maximum value of these numbers using mathematical notation. Specifically, write an equation that defines \( r \) as the maximum of the set \( \{x_1, x_2, x_3\} \). Additionally, demonstrate how this can be expressed using properties of supremum if applicable. | automath | [
0.03993769735097885,
0.0218372605741024,
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0.028756147250533104,
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0.062326934188604355,
0.025609269738197327,
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0.011181044392287731,
0.04274386540055275,
-0.04152533784508705,
-0.... | 0.817257 | 0.797188 | 0.020069 |
Given the series defined as
\[
S_{n} = \frac{1}{n(n+1)} + \frac{1}{(n+1)(n+2)} + \ldots + \frac{1}{(2n-1)2n}
\]
for each positive integer \( n \), calculate
\[
S_{4}
\]
and verify whether \( S_{n} = \frac{1}{2n} \) holds true for \( n = 4 \). What is the value of \( S_{4} \)? | automath | [
0.013985348865389824,
0.049489472061395645,
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-0.04269041866064072,
0.044973570853471756,
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0.06675680726766586,
0.03517911210656166,
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0.010322608053684235,
0.01934313401579857,
-0.015805331990122795,
-0.... | 0.835298 | 0.818854 | 0.016444 |
Provide an example of a continuous bijection from the space \( l_2 \) onto a subset of \( l_2 \) such that its inverse is everywhere discontinuous. Explain the significance of this example in topology, particularly regarding the role of compactness in relation to continuous bijections and homeomorphisms. What does this... | automath | [
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0.047743022441864014,
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0.022078849375247955,
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0.02359168231487274,
0.011469588615000248,
-0.02427499368786812,
0.00... | 0.81255 | 0.790761 | 0.021789 |
Let \( R \) be a Noetherian ring and \( I \) an ideal in \( R \). If \( M \) is a finite \( R \)-module, express the condition for the intersection \( N = \bigcap_{n \geq 0} I^n M \) to be zero in terms of the prime ideals of \( R \) that contain \( I \). How can one demonstrate that \( N \) is zero if \( I \) is conta... | automath | [
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0.024490954354405403,
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0.01471927110105753,
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0.01078311912715435,
0.06344213336706161,
0.04438282921910286,
-0.018442077562212944,
-0.0006250576116144657,
-0.00429906789213419,
-0.030776308849453926,
-0.... | 0.815408 | 0.786552 | 0.028856 |
Let \( p(n) = a_d n^d + a_{d-1} n^{d-1} + \ldots + a_1 n + a_0 \) be a degree \( d \) polynomial with \( a_d > 0 \). For what values of the constant \( k \) does the asymptotic notation \( p(n) \) satisfy the relations \( p(n) = O(n^k) \), \( p(n) = \Omega(n^k) \), and \( p(n) = \Theta(n^k) \)? Provide the conditions o... | automath | [
0.021207116544246674,
0.02971418760716915,
-0.04697265475988388,
-0.03798398748040199,
0.025512540712952614,
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-0.02615540288388729,
0.03868892416357994,
0.027766067534685135,
0.002495262771844864,
-0.004846890456974506,
0.012900866568088531,
-0.02315443754196167,
-0.00... | 0.830367 | 0.810009 | 0.020358 |
Let \( X = [0, 1] \) with the standard Euclidean topology, and let \( E = \{0\} \). Consider the sequence of Dirac measures \( \delta_{1/n} \) for \( n = 1, 2, 3, \ldots \) as \( n \to \infty \).
What is the limit of \( \delta_{1/n}(E) \) as \( n \) approaches infinity, and does this limit equal \( \delta_0(E) \)? | automath | [
0.0020890538580715656,
0.025356408208608627,
-0.01126452162861824,
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0.014628587290644646,
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0.06561052054166794,
0.0380416139960289,
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-0.00984389241784811,
0.022675782442092896,
-0.03557536378502846,
-0.... | 0.816014 | 0.790597 | 0.025417 |
Given a set \( S \) that has a topology \( \tau \) and an induced neighborhood space \( \mathcal{N} \), consider the mapping \( \phi: (S, \tau) \to (S, \mathcal{N}) \) defined by \( \phi(x) = x \) for all \( x \in S \) and \( \phi(U) = U \) for all \( U \in \tau \).
1. If \( S = \{ a, b \} \) and \( \tau = \{\emptyse... | automath | [
0.032217495143413544,
0.03123931959271431,
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0.038199421018362045,
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0.005688867997378111,
0.03693248704075813,
-0.03649059683084488,
-0.02... | 0.825061 | 0.800358 | 0.024702 |
If \( Q_1 \) is the set of all natural numbers \( \mathbb{N} \) and \( Q_2 \) is the set representing the Halting Problem \( H \), what can be concluded about the countability of the set \( Q_1 \setminus Q_2 \)? Specifically, is \( Q_1 \setminus Q_2 \) countably enumerable, and if not, provide a rigorous explanation of... | automath | [
0.042038992047309875,
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0.012244551442563534,
0.016918443143367767,
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-0.... | 0.818672 | 0.798953 | 0.019718 |
Given the intervals $\displaystyle (1, 3)$, $\displaystyle [1, 3)$, and $\displaystyle (1, 3]$, determine the number of elements in each interval if we define these intervals over the set of all real numbers $\displaystyle \mathbb{R}$. Additionally, identify any potential confusion that might arise from the notation us... | automath | [
0.047777239233255386,
0.01915411464869976,
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0.044619642198085785,
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0.057960186153650284,
0.02297772839665413,
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0.010432346723973751,
0.024807723239064217,
-0.05114284157752991,
-0... | 0.821925 | 0.799479 | 0.022445 |
Consider the parabola defined by the equation \( y = \frac{x^2}{100} \). Determine all points \( (a, b) \) on this parabola such that the tangent line at \( (a, b) \) passes through the point \( (100, 50) \). How many such points exist, and what are their coordinates? | automath | [
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0.03529879078269005,
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0.03167108818888664,
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-0.0... | 0.833659 | 0.815754 | 0.017905 |
Consider a pairwise geodesic distance function \( d(x, y) \) defined on a submanifold \( \mathcal{M} \subseteq \mathbb{R}^n \). Given that this function satisfies the conditions \( |\nabla_x d(x, y)|^2 = 1 \) and \( |\nabla_y d(x, y)|^2 = 1 \) almost everywhere, derive the necessary condition that \( |\nabla d|^2 = 2 \... | automath | [
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0.04612381011247635,
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0.017017632722854614,
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0.008610665798187256,
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0.021760331466794014,
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-0... | 0.815584 | 0.790648 | 0.024936 |
Let \( \Gamma \) be a closed piecewise-regular curve in the complex plane defined by the function \( z(\tau) \) for \( \alpha \leq \tau \leq \beta \), and let \( c \) be a point not on \( \Gamma \). If the integral
\[
\int_\Gamma \frac{1}{z - c} \, dz = 3\pi i,
\]
what is the winding number \( n(\Gamma, c) \)? | automath | [
0.006679119076579809,
0.04427764564752579,
-0.043312907218933105,
-0.030120698735117912,
0.02152188867330551,
0.014006395824253559,
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0.0403243787586689,
0.032187335193157196,
0.004959609359502792,
-0.0009365776204504073,
0.04667633771896362,
-0.03229678422212601,
-0.0... | 0.825293 | 0.809114 | 0.016178 |
Let $\mathbf{C}$ be a category with two objects $A$ and $B$, and let $\mathbf{D}$ be the category of sets. If there are three functors $F, G, H: \mathbf{C} \to \mathbf{D}$ that satisfy the following conditions:
1. $F$ sends both $A$ and $B$ to the set $X$.
2. $G$ sends $A$ to the set $Y$ and $B$ to the set $Z$.
3. $H$... | automath | [
0.026822758838534355,
0.01659051701426506,
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0.035289183259010315,
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0.0015254417667165399,
0.01228327676653862,
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-0.... | 0.826765 | 0.805289 | 0.021475 |
Let \( X_1, X_2, X_3, X_4 \) be four independent random variables drawn from an exponential distribution with parameter \( \lambda = 1 \). Calculate the probability \( P(S \geq 2 \text{ and } \bar{X} < 1) \), where \( S \) is the sample standard deviation and \( \bar{X} \) is the sample mean. Is this probability greate... | automath | [
0.004997671581804752,
0.0314774364233017,
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0.04539007321000099,
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0.02054716646671295,
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-0.010186383500695229,
0.029126038774847984,
-0.03904039412736893,
-0.00... | 0.817212 | 0.797989 | 0.019224 |
Let \( n \) be an integer greater than 1 and \( x > -1 \). Determine the bounds for the expression \( (1+x)^n - 1 \) using the given inequalities. Specifically, evaluate \( (1+x)^3 - 1 \) at \( x = 0.5 \) and compare it to the bounds given by the inequalities \( 3x \) and \( \frac{3x}{1 - 2x} \). What is the relationsh... | automath | [
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0.03617445006966591,
-0.024631479755043983,
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0.03710666671395302,
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0.07903018593788147,
0.044236693531274796,
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0.010618403553962708,
0.04057340696454048,
-0.0573577806353569,
-0.02... | 0.825738 | 0.807778 | 0.01796 |
Let \( f(x) = \sum_{i,j \geq 0} a_{ij} x^i (1-x)^j \), where the coefficients \( a_{ij} \) are non-negative and bounded by 1, and consider the convergence of this series on the interval \([0,1]\). Is it possible to find values of \( c \) and \( d \) within the interval \( (0,1) \) such that \( f(x) \) is analytic on \(... | automath | [
0.0280095636844635,
0.04316520318388939,
-0.04108491539955139,
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0.030753785744309425,
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0.07300403714179993,
0.03806224465370178,
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0.003764112014323473,
0.006271986290812492,
-0.04829525202512741,
-0.0... | 0.83478 | 0.812044 | 0.022735 |
Given two vectors \( u = (1, 2, 3) \) and \( v = (4, 5, 6) \) in \( \mathbb{R}^3 \), calculate the following:
1. The dot product \( \langle u, v \rangle \).
2. The norms \( \|u\| \) and \( \|v\| \).
3. Verify whether the Cauchy-Schwarz inequality \( \langle u, v \rangle^2 \leq \|u\|^2 \|v\|^2 \) holds for these vector... | automath | [
0.02145341783761978,
0.047280602157115936,
-0.042835742235183716,
-0.027063747867941856,
0.04607463255524635,
0.004824392963200808,
-0.013458583503961563,
0.07372720539569855,
0.029798347502946854,
-0.0076654343865811825,
0.005328901577740908,
0.025485394522547722,
-0.020338118076324463,
-... | 0.820564 | 0.798408 | 0.022157 |
Given two vectors \( A \) and \( B \) of length \( n \), representing samples from a bivariate distribution, describe a method to transform these vectors into two new vectors \( A' \) and \( B' \) such that \( A' \) and \( B' \) are as independent as possible. Additionally, if the Spearman correlation coefficient of th... | automath | [
0.01738419197499752,
0.028631895780563354,
-0.04416203498840332,
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0.028909677639603615,
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0.06522712856531143,
0.03494961932301521,
-0.009857487864792347,
-0.00033063406590372324,
0.022818386554718018,
-0.027678852900862694,
... | 0.813296 | 0.796543 | 0.016754 |
Can you provide an example of an embedded torus in \(\mathbb{E}^3\) that has no umbilics and where at least one family of its principal curves winds densely around the torus? Additionally, explain why nontrivial solutions \(f_2\) and \(f_3\) do not exist in this case. | automath | [
0.015360865741968155,
0.03137725219130516,
-0.0073095280677080154,
-0.03681635856628418,
0.023011306300759315,
-0.02402443252503872,
-0.020388418808579445,
0.05878254398703575,
0.0014980193227529526,
-0.001758959493599832,
0.02733534574508667,
0.023998450487852097,
-0.02696860022842884,
-0... | 0.811465 | 0.794539 | 0.016926 |
In a certain population, each child in a family has an equal probability of being male or female, and the sex of each child is independent of the others.
1. If a randomly chosen family has 4 children, what is the probability that there are exactly 2 males and 2 females?
2. Now consider an alternate model where the f... | automath | [
0.0035388567484915257,
0.011097775772213936,
-0.047419849783182144,
-0.024776112288236618,
0.042964912950992584,
-0.040079034864902496,
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0.06963679939508438,
0.03157833218574524,
-0.009362072683870792,
0.0025942346546798944,
0.017093153670430183,
-0.019552962854504585,... | 0.813534 | 0.795239 | 0.018295 |
Let \( R \) be a unital ring, and let \( M \) be a finitely generated left \( R \)-module. Suppose that \( R \) is a Principal Ideal Domain (PID). What is the minimal number of generators needed for \( M \) if \( M \) can be expressed as a direct sum of cyclic modules, where each cyclic module is of the form \( R/(p^k)... | automath | [
0.02662563882768154,
0.02255702018737793,
-0.03171135485172272,
-0.03279583156108856,
0.02113228663802147,
-0.03202172741293907,
-0.02336397022008896,
0.03741499409079552,
0.02786390483379364,
-0.010247955098748207,
-0.0002906220033764839,
0.022272754460573196,
-0.021787306293845177,
-0.01... | 0.823475 | 0.806152 | 0.017322 |
Construct a smooth function \( f : [a, a+1] \to [q, p] \) where \( a \in \mathbb{Z} \) and \( q, p \in \mathbb{Q} \) such that \( f \) has critical values at both \( q \) and \( p \). What are the necessary conditions on the choices of \( q \) and \( p \) for this function to exist? Provide an example where these condi... | automath | [
0.034103963524103165,
0.03844788670539856,
-0.03284846246242523,
-0.02767099067568779,
0.0428253598511219,
0.004775099456310272,
-0.024851985275745392,
0.07187147438526154,
0.003361224429681897,
-0.012344270944595337,
0.012449104338884354,
0.027179701253771782,
-0.027667105197906494,
-0.02... | 0.836811 | 0.813124 | 0.023687 |
Let \( S \) be a subset of \( \mathbb{R} \). Consider the following statements:
1. The closure of the interior of \( S \) is denoted as \( \overline{S^o} \).
2. The interior of the closure of \( S \) is denoted as \( (\overline{S})^o \).
Give an example of a subset \( S \) of \( \mathbb{R} \) such that \( \overline{... | automath | [
0.03128749504685402,
0.03641080856323242,
-0.03877246379852295,
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0.031521353870630264,
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0.0024117776192724705,
0.013714803382754326,
0.03218887001276016,
-0.04870925471186638,
-0.0022... | 0.830755 | 0.800989 | 0.029766 |
Given the sequence of functions \(f_n(x) = n \sin\left(\frac{x}{n}\right)\) defined for \(x \in [0, 2\pi]\), demonstrate that this sequence converges pointwise to a function \(f(x) = x\). Specifically, find the limit \(\lim_{n \to \infty} f_n(x)\) for \(x\) in the interval \([0, 2\pi]\), and verify the pointwise conver... | automath | [
0.002531162928789854,
0.06420803815126419,
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0.009635086171329021,
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0.06113733723759651,
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0.019408266991376877,
0.010253004729747772,
0.023302551358938217,
-0.018948981538414955,
0.00... | 0.806881 | 0.783874 | 0.023007 |
Consider the functions \( f(x) = x \) and
\[
g(x) =
\begin{cases}
1/x & \text{if } x > 1 \\
1 & \text{otherwise}
\end{cases}
\]
Evaluate
\[
\lim_{x \to \infty} f(x) g(x)
\]
and confirm whether
\[
\lim_{x \to \infty} f(x) g(x) \neq \lim_{x \to \infty} g(x) f(x).
\]
What do you find for these limits, and what d... | automath | [
0.0146888867020607,
0.05860153213143349,
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0.0304205771535635,
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0.08349087089300156,
0.02268863655626774,
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0.013510975055396557,
-0.021449241787195206,
-0.0... | 0.811046 | 0.793183 | 0.017863 |
Is there a smooth function \( O: M_n^+ \setminus \{0\} \to SO_n \) such that for every non-zero matrix \( A \in M_n^+ \), where \( M_n^+ \) is the set of \( n \times n \) real matrices with non-negative determinants, the polar decomposition \( A = O(A)P(A) \) holds with \( P(A) = \sqrt{A^TA} \)? | automath | [
0.018798179924488068,
0.023297937586903572,
-0.058363158255815506,
-0.029372477903962135,
0.03090495988726616,
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0.05854814872145653,
0.029718494042754173,
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0.006520226132124662,
0.013878853991627693,
-0.022966166958212852,
... | 0.815567 | 0.794315 | 0.021252 |
In the group \(( \mathbb{Q} \setminus \{ 0 \} \times \mathbb{Q}, \circ )\) defined by \((a,b) \circ (c,d) = (ac, ad + b)\), consider an infinite cyclic subgroup \(H\) that does not contain any element of the form \((1,q)\) for nonzero rational \(q\). If \(H\) is generated by some element \((a,b)\), find two specific ra... | automath | [
0.027805287390947342,
0.03403722867369652,
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0.042390331625938416,
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0.00513651967048645,
0.02187829650938511,
-0.03752580285072327,
-0.02... | 0.843394 | 0.818274 | 0.025121 |
Let \( f : (0,1) \rightarrow \mathbb{R} \) be defined by
\[
f(x) = \lim_{n\to\infty} \cos^n\left(\frac{1}{n^x}\right).
\]
Determine the value of \( f(x) \) at the point \( x = \frac{1}{2} \) and verify if this point is indeed the only point of discontinuity of the function \( f \). | automath | [
0.014511874876916409,
0.043301451951265335,
-0.03128614276647568,
-0.034665998071432114,
0.027457764372229576,
0.00062672229250893,
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0.07256694883108139,
0.04249701276421547,
0.024632934480905533,
0.006815101020038128,
-0.0011697408044710755,
-0.004726841114461422,
-0... | 0.823031 | 0.801209 | 0.021822 |
Construct a discrete-time random process \( \{Z_n : n \in \mathbb{N}\} \) that is wide-sense stationary (WSS) but not strictly stationary. Provide a clear definition of the process and verify its wide-sense stationarity by showing that its mean and covariance are time-invariant. Additionally, explain why it fails to be... | automath | [
0.008761363103985786,
0.005192790646106005,
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0.03709740191698074,
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0.06988506764173508,
0.026525842025876045,
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0.02939845621585846,
0.023138316348195076,
-0.014211560599505901,
-0.0... | 0.799815 | 0.781394 | 0.01842 |
Consider the function \( f: \mathbb{R}^3 \to \mathbb{R}^3 \) given by \( f(x,y,z) = \left( ye^x + \sin(\pi y)\cos(z), \cos(yz), z^2 \right) \). At the point \( (0, 1, \pi/2) \), evaluate the Jacobian matrix \( Df \) of \( f \) and determine its determinant. Based on this determinant, conclude whether the function \( f ... | automath | [
0.016952361911535263,
0.047382231801748276,
-0.043858423829078674,
-0.032875899225473404,
0.031205106526613235,
0.0008267750381492078,
0.010248741135001183,
0.07199271023273468,
0.029886426404118538,
0.02853151224553585,
0.01618986763060093,
0.03466147929430008,
-0.048058442771434784,
-0.0... | 0.821413 | 0.801774 | 0.019639 |
Consider the function \( z = \frac{x^2 - y^2}{x^2 + y^2} \). Compute the partial derivatives \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \). Then, evaluate the limits of these partial derivatives as \( (x, y) \) approaches \( (0, 0) \) along the paths \( y = kx \) for various non-zero const... | automath | [
0.005801631137728691,
0.05612732470035553,
-0.03592177480459213,
-0.025922611355781555,
0.0406661182641983,
0.006007118616253138,
0.006437805015593767,
0.058230865746736526,
0.037159405648708344,
-0.01698312722146511,
0.003399000270292163,
0.029317140579223633,
-0.029101097956299782,
-0.00... | 0.809328 | 0.787392 | 0.021936 |
Given that \( A \) is a \( 4 \times 4 \) matrix defined as
\[
A =
\begin{bmatrix}
16 & 2 & 3 & 13 \\
5 & 11 & 10 & 8 \\
9 & 7 & 6 & 12 \\
4 & 14 & 15 & 1
\end{bmatrix}
\]
and \( B \) is a \( 4 \times 4 \) matrix with real entries, describe the conditions on \( B \) under which the matrix equation \( AX = B \) c... | automath | [
0.030641678720712662,
0.021590039134025574,
-0.043450210243463516,
-0.029784785583615303,
0.06214490905404091,
-0.027300214394927025,
-0.013859724625945091,
0.07766223698854446,
0.03657180443406105,
0.005128985736519098,
0.004393596667796373,
0.02773892879486084,
-0.010567792691290379,
-0.... | 0.82174 | 0.798692 | 0.023049 |
Let \( R \) be a commutative ring with unity, and let \( S = \{2, x\} \subset \mathbb{Z}[x] \). What is the ideal \( \langle S \rangle \) generated by \( S \) in \( \mathbb{Z}[x] \), and is this ideal generated by a single element? If so, identify that element. If not, how do the generators of the ideal relate to the e... | automath | [
0.049753587692976,
0.03785402327775955,
-0.04035678505897522,
-0.03232381492853165,
0.03005947731435299,
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0.000638021738268435,
0.06664615869522095,
0.04462452977895737,
-0.021777469664812088,
0.017885282635688782,
0.014152479358017445,
-0.01970929093658924,
-0.021763... | 0.813402 | 0.793283 | 0.020118 |
Solve the equation $$2\cos^2 3\theta = \cos 3\theta$$ for $$0 \leq \theta < 2\pi$$ and provide all distinct solutions for $$\theta$$ in that interval. | automath | [
0.02499145269393921,
0.03218964859843254,
-0.04112386330962181,
-0.048460863530635834,
0.028705410659313202,
-0.023426761850714684,
-0.009639061987400055,
0.0660804808139801,
0.043320633471012115,
-0.014091253280639648,
0.029028235003352165,
0.00005392477396526374,
-0.014279008843004704,
-... | 0.826135 | 0.804752 | 0.021383 |
Let \( G(x, x') = \frac{1}{S_n} \frac{x - x'}{\| x - x' \|^n} \) be the Green's function for the gradient in \( n \)-dimensional space, where \( S_n \) is the surface area of the unit sphere. If we consider the integral
\[
F(x) = \int_V dV' G(x, x') M(x'),
\]
where \( M(x') \) is a continuous function defined over vo... | automath | [
-0.004237242508679628,
0.040034323930740356,
-0.04196222499012947,
-0.04858812317252159,
0.007765514776110649,
-0.0337698757648468,
-0.044134125113487244,
0.06722524762153625,
0.04199321195483208,
-0.015472163446247578,
0.023449843749403954,
0.03782910481095314,
-0.028973659500479698,
-0.0... | 0.810812 | 0.790393 | 0.020419 |
Given a cartesian diagram of sheaves on $(\mathit{Sch}/S)_{fppf}$ as follows:
$$
\xymatrix{ E \ar[r]_ a \ar[d]_ f & F \ar[d]^ g \\ H \ar[r]^ b & G }
$$
where $E = H \times_G F$, and knowing the properties of the morphisms $g$ and $a$ as described in the lemma, what can be concluded about the properties of the morphis... | automath | [
0.00869848020374775,
0.03526538982987404,
-0.05339070409536362,
-0.042619697749614716,
0.026700841262936592,
-0.04546119645237923,
-0.013003088533878326,
0.0853400006890297,
0.0449654795229435,
-0.011407965794205666,
0.013168909586966038,
0.04606799781322479,
-0.017731182277202606,
0.00372... | 0.811715 | 0.792506 | 0.01921 |
Let \( \mathbf{X} \) be the category of sets and \( \mathbf{A} \) be the category of groups with the usual forgetful functor \( U : \mathbf{A} \to \mathbf{X} \). If \( G \) and \( H \) are groups and \( f: G \to H \) is a function of sets, under what conditions can we classify \( f \) as a group homomorphism, and how d... | automath | [
0.03303958848118782,
0.022934861481189728,
-0.046565745025873184,
-0.03394056484103203,
0.04521529749035835,
-0.03441064804792404,
-0.00415743188932538,
0.06837908178567886,
0.03549036756157875,
-0.019940286874771118,
-0.004965505097061396,
0.018225526437163353,
-0.027989113703370094,
0.00... | 0.80583 | 0.789166 | 0.016663 |
Given the Frenet frame equations for a natural parameter curve \( r: (a,b) \rightarrow \mathbb{R}^n \) with curvature conditions as stated in the theorems for \( n=2 \) and \( n=3 \), derive the curvature formulas for a spiral line defined in \( \mathbb{R}^n \) in the form \( (a \cos t, a \sin t, b_1 t, b_2 t, \ldots, ... | automath | [
-0.007798412814736366,
0.025607919320464134,
-0.03187357261776924,
-0.05747625604271889,
0.029352499172091484,
-0.00024766524438746274,
-0.030261054635047913,
0.06436570733785629,
0.029287025332450867,
-0.0021365080028772354,
-0.006065621972084045,
0.023847466334700584,
-0.03727292641997337,... | 0.806825 | 0.786284 | 0.020541 |
Given a twice-differentiable convex function \( f: \mathbb{R} \rightarrow \mathbb{R} \), with a maximum second derivative \( C = \max_{x} f''(x) \), can you find a specific convex function for which the quadratic upper bound \( f(x) \leq f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2}{2}C \) holds, and also determine the co... | automath | [
-0.0025059219915419817,
0.04609213396906853,
-0.04175642132759094,
-0.025120900943875313,
0.02719084918498993,
0.00874798558652401,
-0.0263817198574543,
0.08014784008264542,
0.03874129801988602,
-0.0006904933252371848,
0.0009624231606721878,
0.014405079185962677,
-0.033213723450899124,
-0.... | 0.815299 | 0.796239 | 0.019061 |
Given the function \( S(z) \) whose saddle point \( z^0 \) maximizes \( \mathop{\rm Re} S(z) \) in a bounded manifold \( \gamma \), and assuming \( \Delta_S(z^0) = \det S^{\prime\prime}(z^0) \neq 0 \), determine the asymptotic behavior of the integral
\[
F(\lambda) = \int_{\gamma} f(z) e^{\lambda S(z)} dz
\]
as \( \l... | automath | [
0.020004866644740105,
0.028826475143432617,
-0.024831339716911316,
-0.031314264982938766,
0.007599066011607647,
-0.0064605651423335075,
-0.020601535215973854,
0.05774705484509468,
0.03385045751929283,
-0.0022936402820050716,
0.017481070011854172,
0.019283685833215714,
-0.04986555874347687,
... | 0.814105 | 0.79518 | 0.018925 |
Is it possible for a globally continuous, nowhere differentiable parameterized curve \( c : \mathbb{R} \to \mathbb{R}^k \) to have finite length over some neighborhood \( (x_0 - \epsilon, x_0 + \epsilon) \) for any \( \epsilon > 0 \)? If so, how does the dimension \( k \) influence this possibility? | automath | [
0.000023665652406634763,
0.036198921501636505,
-0.02946736291050911,
-0.03130671754479408,
0.02602143958210945,
-0.021051185205578804,
-0.030400704592466354,
0.0697210505604744,
0.017623066902160645,
-0.019783996045589447,
-0.008195671252906322,
0.0443827360868454,
-0.03691135719418526,
-0... | 0.809819 | 0.789107 | 0.020712 |
Let $\mathcal{X}$ be an algebraic stack, and let $|\mathcal{X}|$ denote its underlying topological space. If $|\mathcal{X}|$ is locally quasi-compact, what can we conclude about the existence of a scheme $U$ such that there is a surjective, open, continuous map from $U$ to $|\mathcal{X}|$? Specifically, what properties... | automath | [
0.01639736257493496,
0.024447264149785042,
-0.03183257579803467,
-0.027528248727321625,
0.030775748193264008,
-0.013333404436707497,
-0.019110921770334244,
0.06656331568956375,
0.036083146929740906,
-0.027168618515133858,
0.004350487608462572,
0.040943995118141174,
-0.026033543050289154,
-... | 0.811513 | 0.784067 | 0.027446 |
In a metric space defined by a distance function \( d \), suppose we have an entourage \( E \) such that for any points \( x \) and \( y \) in the space, \( (x, y) \in E \) implies \( d(x, y) < \epsilon \) for some fixed \( \epsilon > 0 \). If \( \epsilon = 0.1 \) and there are exactly \( 5 \) points \( \{x_1, x_2, x_3... | automath | [
0.014300353825092316,
0.025468574836850166,
-0.043108392506837845,
-0.03225052356719971,
0.029634827747941017,
-0.005350963212549686,
-0.04786256328225136,
0.04549090564250946,
0.03712546080350876,
0.005487028043717146,
0.006438387557864189,
0.03441513329744339,
-0.027793223038315773,
-0.0... | 0.828925 | 0.80629 | 0.022635 |
Given independent random variables \( X \) and \( Y \), along with a \( \sigma \)-algebra \( \mathcal{F} \) such that \( X \) is also independent of \( \mathcal{F} \), is it true that
\[
\mathbb{E}(XY \mid \mathcal{F}) = \mathbb{E}(X) \cdot \mathbb{E}(Y \mid \mathcal{F})?
\]
If this statement is false, can you provi... | automath | [
0.019307177513837814,
0.013476534746587276,
-0.046521253883838654,
-0.03623659908771515,
0.041925132274627686,
-0.012776380404829979,
-0.006850601173937321,
0.06842631101608276,
0.0379129983484745,
-0.019590681418776512,
0.0016415356658399105,
0.027944836765527725,
-0.0520019568502903,
-0.... | 0.811164 | 0.791281 | 0.019882 |
Given an isosceles right triangle \( \triangle ABC \) with vertices \( A(0, 0) \), \( B(a, 0) \), and \( C(0, a) \) for some \( a > 0 \), define another isosceles right triangle \( \triangle DEF \) that is inscribed within \( \triangle ABC \). The distances from the sides of \( \triangle ABC \) to the respective sides ... | automath | [
0.03988748416304588,
0.02321479842066765,
-0.029311561957001686,
-0.02384297177195549,
0.025176674127578735,
-0.023188654333353043,
-0.012643538415431976,
0.039034850895404816,
0.04799925535917282,
0.012364241294562817,
0.025358660146594048,
0.0251646526157856,
-0.02194719947874546,
-0.012... | 0.831846 | 0.80799 | 0.023856 |
Let \( M \subseteq \mathbb{R}^n \) be a compact smooth \( k \)-manifold and \( f: M \to \mathbb{R} \) be a continuous function. Define the integral of \( f \) over \( M \) as
\[
\int_M f \, dV = \int_{U^{\text{int}}} (f \circ \phi) \, \text{Vol}(D\phi)
\]
where \( \phi: U \to V \) is a chart and \( U \) is bounded.
... | automath | [
0.02742418646812439,
0.04539771378040314,
-0.03272238001227379,
-0.03506855666637421,
0.02843829244375229,
-0.014328740537166595,
-0.01814964786171913,
0.06342530995607376,
0.05061258375644684,
0.0022492629941552877,
0.007381770294159651,
0.031933631747961044,
-0.009430713951587677,
-0.008... | 0.803554 | 0.772052 | 0.031503 |
Given an equilateral triangle inscribed in a circle, find the probability that a randomly chosen chord of the circle, selected by the method of choosing two random points on the circumference, is longer than the side length of the triangle. What is this probability, and how does it compare to the probabilities obtained... | automath | [
0.0034524661023169756,
0.0012051198864355683,
-0.039141613990068436,
-0.026026170700788498,
0.01674974523484707,
-0.027273667976260185,
-0.01748306304216385,
0.04378831386566162,
0.029521187767386436,
0.003038628026843071,
0.015628846362233162,
0.028773203492164612,
-0.034300316125154495,
... | 0.835652 | 0.811355 | 0.024296 |
Given $n \leq 10^3$ vectors $v_i \in \mathbb{R}^2$ such that $\|v_i\| \leq 1$ for all $i$, and a fixed vector $w \in \mathbb{R}^2$, is there a subset $S \subseteq \{1, 2, \ldots, n\}$ that minimizes the expression $\| w + \sum_{i \in S} v_i \|^2$? If so, describe a polynomial-time approximation algorithm for this probl... | automath | [
-0.008835854940116405,
0.03878115117549896,
-0.04620349407196045,
-0.0421120785176754,
0.041555002331733704,
-0.0072717973962426186,
-0.027088571339845657,
0.05714844912290573,
0.03400394693017006,
-0.007435136009007692,
0.022084195166826248,
0.021965721622109413,
-0.015729177743196487,
-0... | 0.830583 | 0.807758 | 0.022826 |
Given an isosceles trapezoid where both legs and one base each have a length of 1, find the angle \( \theta \) (measured from the base to a leg) that maximizes the area of the trapezoid. What is the maximum area obtained when the angle \( \theta \) is set to this optimal value? | automath | [
0.043204016983509064,
0.018805401399731636,
-0.04679356887936592,
-0.032839588820934296,
0.025000683963298798,
-0.039731670171022415,
-0.02038094773888588,
0.04253663122653961,
0.012103388085961342,
-0.017234401777386665,
0.03890787810087204,
0.036333583295345306,
-0.03879508748650551,
-0.... | 0.812489 | 0.795675 | 0.016814 |
Given a set \( A \subseteq \mathbb{R} \), describe a scenario in which \( A \) is neither open nor closed. Provide an example of such a set and explain why it meets these criteria. Additionally, determine whether the set \( A \) has any limit points, and if so, quantify how many limit points \( A \) has. | automath | [
0.040472012013196945,
0.01228831335902214,
-0.021864978596568108,
-0.026427997276186943,
0.017138930037617683,
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-0.03047231212258339,
0.0626276507973671,
0.030118240043520927,
-0.014836997725069523,
0.009091302752494812,
0.027469847351312637,
-0.04001009091734886,
-0.... | 0.819812 | 0.797471 | 0.022341 |
In the vector space \(\mathbb{R}^3\), consider the following subsets:
1. \(S_1 = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}\)
2. \(S_2 = \{(1, 0, 0), (0, 1, 0)\}\)
3. \(S_3 = \{(1, 1, 0), (1, 0, 1), (0, 1, 1)\}\)
For each set \(S_i\) (where \(i = 1, 2, 3\)), determine if it is linearly independent and if it spans \(\mathb... | automath | [
0.024913739413022995,
0.04761265218257904,
-0.054255545139312744,
-0.04699248448014259,
0.0429103821516037,
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-0.026365717872977257,
0.05390440300107002,
0.03125446289777756,
-0.006679440848529339,
0.023899242281913757,
0.011600818485021591,
-0.041542503982782364,
-0.0... | 0.824683 | 0.800224 | 0.024459 |
Consider a 2SAT instance defined by the implications \( x_1 \rightarrow x_2 \), \( x_2 \rightarrow \lnot x_3 \), \( x_3 \rightarrow x_1 \), and \( \lnot x_2 \rightarrow \lnot x_1 \).
1. Construct the compatibility graph for this 2SAT instance.
2. Determine whether the compatibility graph is bipartite and justify you... | automath | [
0.019822144880890846,
0.02639925293624401,
-0.05171963945031166,
-0.02261156216263771,
0.02171269617974758,
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-0.011739645153284073,
0.06770223379135132,
0.03014354407787323,
-0.011069252155721188,
0.011168226599693298,
0.011497288011014462,
-0.01441730186343193,
-0.01... | 0.827799 | 0.800973 | 0.026826 |
Let \( (X, d_1) \) and \( (X, d_2) \) be two metrics on a set \( X \). Suppose it is given that the open balls defined by both metrics satisfy the following conditions:
1. For every \( r > 0 \), the open ball \( B_{d_1}(x, r) \) is an open set in the topology induced by \( d_2 \).
2. For every \( s > 0 \), the open b... | automath | [
0.014850064180791378,
0.020213790237903595,
-0.05030756816267967,
-0.033262308686971664,
0.023148344829678535,
0.009599282406270504,
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0.05787050351500511,
0.04776141792535782,
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-0.005420126020908356,
0.028147244825959206,
-0.03263064846396446,
-0... | 0.816608 | 0.784027 | 0.032581 |
Is it possible to construct an uncountable ring whose spectrum is countably infinite, and what are the general conditions on cardinalities \( \kappa \) and \( \lambda \) for the existence of a ring \( R \) such that \( |R| = \kappa \) and \( |Spec(R)| = \lambda \)? | automath | [
-0.0003259403456468135,
0.04158056154847145,
-0.01908409595489502,
-0.01311249565333128,
0.0213216133415699,
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0.05987919494509697,
0.009642567485570908,
-0.02061554603278637,
0.01919965259730816,
0.024954261258244514,
-0.033579085022211075,
-0.0... | 0.810687 | 0.793957 | 0.016729 |
Is the subgroup \( H \) of \( \mathrm{GL}_n(\mathbb{R}) \) defined as \( H = \{ A \in \mathrm{GL}_n(\mathbb{R}) \mid \exists \text{ continuous } f:[0,1] \to \mathrm{GL}_n(\mathbb{R}) \text{ such that } f(0) = A \text{ and } f(1) = I \} \) a normal subgroup of \( \mathrm{GL}_n(\mathbb{R}) \)? | automath | [
0.0080098332837224,
0.02361760288476944,
-0.04138679429888725,
-0.03599194437265396,
0.05192169174551964,
-0.02818959951400757,
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0.060735464096069336,
0.026732878759503365,
-0.006457771174609661,
0.007697474677115679,
0.02310674451291561,
-0.041089389473199844,
-0.015... | 0.813712 | 0.795252 | 0.01846 |
Is it possible for a real sequence \( (x_n) \) to satisfy the property that for every \( \epsilon > 0 \), there exists a \( b \in \mathbb{R} \) such that \( |x_n - b| < \epsilon \) for almost all \( n \), yet the sequence does not converge? If so, provide an example. If not, explain why this condition implies convergen... | automath | [
0.00043047318467870355,
0.03145242854952812,
-0.030462658032774925,
-0.02752874419093132,
0.023267000913619995,
-0.01221164409071207,
-0.010591871105134487,
0.08700012415647507,
0.014551954343914986,
-0.0059782653115689754,
0.0018970441306009889,
0.013187633827328682,
-0.0411946102976799,
... | 0.827053 | 0.80266 | 0.024392 |
Consider the function \( F: \mathbb{R}_{\geq 0}^2 \to \mathbb{R}_{\geq 0} \) defined by
\[
F(x, y) = \frac{xy}{x+y} + xy.
\]
If we tropicalize this identity by replacing multiplication \( \times \) with \( + \), division \( / \) with \( - \), addition \( + \) with \( \min \), and constant \( 1 \) with \( 0 \), what ... | automath | [
0.01354081742465496,
0.028271222487092018,
-0.02873995155096054,
-0.022098224610090256,
0.020305447280406952,
-0.021421754732728004,
0.0016012502601370215,
0.05737975612282753,
0.02151850238442421,
0.0065213767811656,
0.008229738101363182,
0.012943579815328121,
-0.040630683302879333,
-0.01... | 0.821417 | 0.800157 | 0.02126 |
In a Pólya urn model, we start with \( a \) azure balls and \( b \) black balls. After \( n \) draws, let \( Y_n \) represent the number of azure balls drawn and \( Z_n \) the number of black balls drawn.
1. Derive the expected proportion of azure balls \( M_n \) after \( n \) draws in terms of \( a \) and \( b \).
2... | automath | [
0.007040065247565508,
0.018097499385476112,
-0.04244788736104965,
-0.02116306871175766,
-0.002084544626995921,
0.0038493985775858164,
-0.03541528061032295,
0.05470336973667145,
0.024927498772740364,
-0.004256643354892731,
0.012200936675071716,
0.05058015510439873,
-0.020831499248743057,
0.... | 0.825422 | 0.806168 | 0.019255 |
Let \( R \) be a normed division ring with a submultiplicative norm \( \norm{\, \cdot \,}_R \). If \( R \) is finite-dimensional and has dimension \( n \), what is the maximum number of distinct multiplicative norms on \( R \) that satisfy the property \( \norm{rs}_R = \norm{r}_R \norm{s}_R \) for all \( r, s \in R \)? | automath | [
0.007994812913239002,
0.030104627832770348,
-0.036463215947151184,
-0.030579976737499237,
0.0329664945602417,
-0.0098823057487607,
-0.01569494977593422,
0.048246968537569046,
0.035460956394672394,
-0.019388321787118912,
-0.004651108756661415,
0.027225375175476074,
-0.03885319456458092,
-0.... | 0.832363 | 0.810691 | 0.021672 |
Let \( V \) be a proper subspace of \( W \) over a field \( \mathbb{F} \), and let \( x, y, z \in W \). Determine whether the following statements hold true:
(a) If \( x, y, z \) are linearly independent in \( V \), must they also be linearly independent in \( W \)?
(b) If \( x, y, z \) are linearly independent in \(... | automath | [
0.020017573609948158,
0.03644973039627075,
-0.03413150832056999,
-0.04987793043255806,
0.061844442039728165,
-0.025497721508145332,
0.013598538935184479,
0.047441039234399796,
0.03212675079703331,
-0.01780206896364689,
0.01538586150854826,
0.020365262404084206,
-0.035988207906484604,
-0.02... | 0.819766 | 0.798313 | 0.021454 |
Given a countable amenable group \( G \) and a positive definite function \( \gamma: G \to \mathbb{C} \), does the Cesaro limit
\[
L = \lim_{N\to\infty}\frac{1}{|F_N|}\sum_{g \in F_N} \gamma(g)
\]
necessarily satisfy \( L \geq 0 \), and if so, what is the reasoning behind this conclusion? | automath | [
0.011753859929740429,
0.034882962703704834,
-0.037913404405117035,
-0.01604187674820423,
0.04133695363998413,
-0.020890923216938972,
-0.02097664214670658,
0.04635684937238693,
0.022103721275925636,
-0.008954343385994434,
-0.007638565730303526,
0.028638750314712524,
-0.04034220427274704,
-0... | 0.816315 | 0.792836 | 0.023479 |
Consider independent and identically distributed random variables \( X_1, X_2, \ldots, X_n \) drawn from a Gamma distribution with parameters \( \alpha = \mu \) and \( \beta = 1 \). Let \( \bar{X}_n = \frac{X_1 + X_2 + \cdots + X_n}{n} \) denote the sample mean. Define the statistic
\[
T_n = \frac{\sqrt{n}(\bar{X}_n -... | automath | [
0.0020208610221743584,
0.03516456112265587,
-0.04824915900826454,
-0.029823221266269684,
0.037149425595998764,
-0.020013708621263504,
-0.01658974029123783,
0.05723549425601959,
0.035009488463401794,
-0.005181291606277227,
-0.002638406353071332,
0.020916592329740524,
-0.04700934886932373,
0... | 0.816507 | 0.794425 | 0.022082 |
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on a ringed topos $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$. If $\mathcal{F}$ is generated by finitely many global sections, what can be said about the existence of a global finite presentation of $\mathcal{F}$? Specifically, if it is known that $... | automath | [
0.02533690445125103,
0.027336474508047104,
-0.05034802481532097,
-0.03249279037117958,
0.015522868372499943,
-0.03962796553969383,
-0.01306396909058094,
0.05253275856375694,
0.035711854696273804,
-0.018847867846488953,
0.017688268795609474,
0.03839894011616707,
-0.03123348020017147,
0.0205... | 0.800503 | 0.775161 | 0.025342 |
Suppose we have two sets of independent and identically distributed exponential random variables \(X_1, X_2, \ldots, X_n\) and \(Y_1, Y_2, \ldots, Y_n\) with parameter \(1\). Define \(Z_i = \min(X_i, Y_i)\) for each \(i\) and let \(U = \max(Z_1, Z_2, \ldots, Z_n)\). What is the probability density function of \(U\)? | automath | [
0.006425606552511454,
0.02538861334323883,
-0.048262327909469604,
-0.024674568325281143,
0.037610091269016266,
-0.014570648781955242,
-0.024290725588798523,
0.03996482864022255,
0.02488069236278534,
-0.01452973484992981,
-0.005864214617758989,
0.032335832715034485,
-0.024982888251543045,
0... | 0.824002 | 0.803511 | 0.020491 |
In spherical coordinates, determine the limits for the angles \( \theta \) and \( \phi \) required to cover the entire surface of a unit sphere. Specifically, provide the range for \( \theta \) and \( \phi \) when \( \theta \) represents the azimuthal angle (longitude) and \( \phi \) denotes the polar angle (co-latitud... | automath | [
-0.0012429592898115516,
0.010456958785653114,
-0.03216511383652687,
-0.05242377892136574,
0.006003673188388348,
-0.040114518254995346,
-0.028805367648601532,
0.02792121283710003,
0.03341938182711601,
-0.02024627849459648,
0.0027773911133408546,
0.0403510183095932,
-0.02216203697025776,
-0.... | 0.799907 | 0.777742 | 0.022165 |
Consider an ordered metric space \( X \) where \( f: X \to X \) is a monotone continuous mapping that satisfies the property of being order-contractive. If we know that the fixed point equation \( x = f(x) \) has an upper solution \( u \) such that for all \( x \in X \), \( x \leq u \), what can we conclude about the e... | automath | [
0.016418013721704483,
0.019092783331871033,
-0.039662476629018784,
-0.039247337728738785,
0.010821240022778511,
-0.009037354961037636,
-0.005722351372241974,
0.05694592744112015,
0.036212701350450516,
0.008303671143949032,
-0.0008709279354661703,
0.015574625693261623,
-0.017145343124866486,
... | 0.808509 | 0.782793 | 0.025716 |
Consider the sequences of independent random variables defined as follows:
1. \( P(X_n = 1) = \frac{1}{n^2}, \, P(X_n = 0) = 1 - \frac{1}{n^2} \)
2. \( P(X_n = n) = \frac{1}{n^2}, \, P(X_n = 0) = 1 - \frac{1}{n^2} \)
3. \( P(X_n = n) = \frac{1}{n}, \, P(X_n = 0) = 1 - \frac{1}{n} \)
For each sequence, determine wheth... | automath | [
-0.0035665580071508884,
0.04058872163295746,
-0.04320989549160004,
-0.032709941267967224,
0.047851864248514175,
-0.020657187327742577,
-0.01526677142828703,
0.07425681501626968,
0.008684162981808186,
-0.007305411621928215,
0.0037701025139540434,
0.02184080332517624,
-0.03944303095340729,
0... | 0.818499 | 0.795489 | 0.023009 |
Let triangle \( ABC \) have sides \( AB = 7 \), \( BC = 5 \), and \( AC = 8 \). Triangle \( PQR \) has sides \( PQ = 7 \), \( QR = 5 \), and \( PR = 8 \). Are triangles \( ABC \) and \( PQR \) congruent? If so, identify the criterion used for establishing their congruence. | automath | [
0.040116582065820694,
0.023676307871937752,
-0.018996771425008774,
-0.038891248404979706,
0.025233253836631775,
-0.01027457881718874,
-0.003030196763575077,
0.050605837255716324,
0.013745005242526531,
0.0005314305890351534,
-0.007313997950404882,
0.024071183055639267,
-0.03395180404186249,
... | 0.826206 | 0.805446 | 0.02076 |
Suppose \( z_n = r_n e^{i x_n} \) converges to \( z = r e^{i x} \) where \( r_n \) and \( r \) are positive real numbers. If the distance \( |z_n - z| < \epsilon \) for some \( \epsilon > 0 \), find an explicit bound in terms of \( \epsilon \) on the difference \( |x_n - x| \) as \( n \) approaches infinity. What is th... | automath | [
0.022309767082333565,
0.04027944430708885,
-0.020905055105686188,
-0.014654593542218208,
0.014333625324070454,
-0.016762750223279,
-0.029004547744989395,
0.034567248076200485,
0.03260613977909088,
-0.0027725088875740767,
0.01695927232503891,
0.012964471243321896,
-0.042010992765426636,
-0.... | 0.8285 | 0.811936 | 0.016564 |
Given a separable Banach space \( E \), a right-continuous stochastic process \( X = (X_t) \) with stationary independent increments, and the transition function defined by \( P_t(x, A) = \mathbb{P}(X_t + x \in A) \), under what conditions can \( P_t(x, A) \) be shown to be a transition kernel, specifically verifying t... | automath | [
0.02522898279130459,
0.029460273683071136,
-0.027030624449253082,
-0.023190399631857872,
0.038324251770973206,
-0.009402396157383919,
-0.02211347594857216,
0.07826977223157883,
0.04012858495116234,
0.002903376007452607,
0.017490766942501068,
0.011846384964883327,
-0.028346704319119453,
0.0... | 0.805506 | 0.781828 | 0.023679 |
In the context of a monoidal category \((\mathcal{C}, \otimes)\) and a monoid \((A, m, e)\) that belongs to the center of \((\mathcal{C}, \otimes)\), define the functor \(F_A(-) = A \otimes (-)\).
What are the specific coherence conditions required to ensure that \(F_A\) is a monoidal functor, particularly focusing o... | automath | [
0.030970454216003418,
0.027027511969208717,
-0.027722610160708427,
-0.008959047496318817,
0.025563707575201988,
-0.0145303038880229,
-0.019552327692508698,
0.08258626610040665,
0.026229728013277054,
0.00613974267616868,
-0.014908382669091225,
0.023913972079753876,
-0.02503526210784912,
0.0... | 0.790163 | 0.771398 | 0.018765 |
Given a ring \( R \) and an \( R \)-module \( M \), suppose we have a functor \( F: R\text{-Mod} \to \text{Ab} \) satisfying the universal property of the tensor product, meaning there exists an isomorphism \( F(N) \cong M \otimes_R N \) for every \( R \)-module \( N \). Can \( F \) and the functor \( (M \otimes_R -) \... | automath | [
0.04464372619986534,
0.024270355701446533,
-0.011853205040097237,
-0.01790313795208931,
0.03200233727693558,
-0.02209140919148922,
-0.010987299494445324,
0.06826014816761017,
0.025524551048874855,
-0.0031194547191262245,
-0.002618292812258005,
0.011886881664395332,
-0.026405345648527145,
-... | 0.799505 | 0.779183 | 0.020323 |
Given a set \( X \) with a subbasis \( S \subset \mathcal{P}(X) \) such that the union of all sets in \( S \) equals \( X \), what is the minimum number of elements \( n \) in \( S \) such that the collection of all finite intersections of elements of \( S \) can yield at least \( m \) distinct open sets in the topolog... | automath | [
0.04297928139567375,
0.0009346673614345491,
-0.051920078694820404,
-0.026225481182336807,
0.023134594783186913,
-0.015527741052210331,
-0.028490612283349037,
0.04682598263025284,
0.04308764636516571,
-0.004279801622033119,
0.028049718588590622,
0.030175894498825073,
-0.015980452299118042,
... | 0.838691 | 0.813577 | 0.025114 |
Consider the metric space \((\mathbb{R}, d)\) where the distance function \(d\) is defined as follows:
1. For all \(x, y \in \mathbb{R}\) where \(x \neq 0\) and \(y \neq 1\), define \(d(x,y) = |x - y|\).
2. Define \(d(1,x) = |x|\) for all \(x \neq 0,1\).
3. Define \(d(0,y) = |y - 1|\) for all \(y \neq 0,1\).
4. Set \(... | automath | [
0.009786761365830898,
0.04249056056141853,
-0.047234080731868744,
-0.028908614069223404,
0.02498270943760872,
-0.005952067673206329,
-0.026921862736344337,
0.06023387983441353,
0.048571567982435226,
-0.006115410942584276,
-0.007980503141880035,
0.03352150321006775,
-0.033826179802417755,
-... | 0.825549 | 0.806699 | 0.01885 |
Given two independent and identically distributed (iid) discrete random variables \(X\) and \(Y\), let \(P(|X-Y|=0) = P(X=Y)\) and for any positive integer \(x\), let \(P(|X-Y|=x)\) be the probability that the absolute difference between \(X\) and \(Y\) is \(x\). Under what conditions does the inequality
\[
2P(X=Y) \... | automath | [
-0.004542530048638582,
0.02936270646750927,
-0.04458656907081604,
-0.03299984335899353,
0.01941673271358013,
0.003593301633372903,
-0.024325398728251457,
0.080900177359581,
0.04645543918013573,
-0.020663823932409286,
0.004765999969094992,
0.01440385915338993,
-0.03881967067718506,
0.013409... | 0.819917 | 0.799111 | 0.020806 |
Let \( S \) be a connected, oriented, simply connected surface in \( \mathbb{R}^3 \), and let \( \mathrm{I} \) and \( \mathrm{I\!I} \) be the first and second fundamental forms of \( S \).
What are the necessary conditions that the quadratic forms \( \mathrm{I} \) and \( \mathrm{I\!I} \) must satisfy for an immersion... | automath | [
0.02441594935953617,
0.03763645887374878,
-0.060718826949596405,
-0.04047061875462532,
0.024999264627695084,
-0.00030854434589855373,
-0.0032270574010908604,
0.06800737231969833,
0.019351035356521606,
-0.010885754600167274,
0.03243251517415047,
0.018321752548217773,
-0.022934312000870705,
... | 0.815036 | 0.785986 | 0.02905 |
Consider the boundary value problem defined by the differential equation
\[
-u''(x) = f(x)u - g(x), \quad u(0) = u(1) = 0,
\]
where \( f(x) \) and \( g(x) \) are positive functions within the interval \( [0, 1] \). Let \( F = \max_{x \in [0, 1]} f(x) \) and \( G = \max_{x \in [0, 1]} g(x) \). If it is known that the... | automath | [
-0.01271942537277937,
0.017605653032660484,
-0.041911154985427856,
-0.021647542715072632,
0.042038630694150925,
-0.0126469312235713,
-0.007885009981691837,
0.07749493420124054,
0.044918324798345566,
-0.01092755887657404,
0.0009757858933880925,
0.034796275198459625,
-0.029346134513616562,
-... | 0.809242 | 0.784604 | 0.024637 |
Given two sequences of real numbers \( (a_k) \) and \( (b_k) \) for \( k = 1, 2, \ldots, n \), demonstrate that the following inequality holds:
\[
\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right).
\]
Under what conditions does equality occur? | automath | [
0.018807366490364075,
0.032522279769182205,
-0.04151581972837448,
-0.044015467166900635,
0.03527039289474487,
-0.014152602292597294,
-0.021774999797344208,
0.08919858187437057,
0.03512964025139809,
-0.017515461891889572,
-0.007559503428637981,
0.024253515526652336,
-0.051665112376213074,
0... | 0.839589 | 0.809313 | 0.030276 |
Given a cone \( C \subset \mathbb{R}^n \) with a vertex at the origin, aperture \( \theta \), and height \( h \), determine the lower bound for the expression
\[
\sup_{P \in \mathcal{P}} \frac{Vol_n(P)}{Vol_n(C)}
\]
where \( \mathcal{P} \) is the set of all parallelotopes that are completely contained in \( C \) and... | automath | [
0.025677654892206192,
0.047768447548151016,
-0.02702713944017887,
-0.022258788347244263,
0.028014667332172394,
-0.007172787096351385,
-0.033931173384189606,
0.06785636395215988,
0.04250729829072952,
0.004247206263244152,
0.011247309856116772,
0.036817990243434906,
-0.025516757741570473,
-0... | 0.834917 | 0.807425 | 0.027492 |
Given a convex quadrilateral with sides of lengths \( a, b, c, d \), where the angles opposite these sides are \( \alpha, \beta, \gamma, \omega \) respectively, if it is known that:
1. \( \alpha = \frac{\beta}{2} + 90^\circ \)
2. The quadrilateral is convex and adheres to the angle sum property.
Determine the angles ... | automath | [
-0.0067606475204229355,
0.025142794474959373,
-0.05443726107478142,
-0.022780997678637505,
0.02060752920806408,
-0.011359212920069695,
0.016231762245297432,
0.0656619742512703,
0.03707949072122574,
-0.0006977937882766128,
-0.003058310365304351,
0.021738098934292793,
-0.049795106053352356,
... | 0.823256 | 0.801737 | 0.021519 |
Given the statements:
a) $\exists !\,x\;P(x) \implies \exists x\;P(x)$
b) $\forall x\;P(x) \implies \exists !x\;P(x)$
c) $\exists!x\;\neg P(x) \implies \neg\forall x\;P(x)$
Determine the truth values of each statement as clearly as possible, and provide a brief justification for each answer. | automath | [
0.02975817210972309,
0.018901629373431206,
-0.03733063489198685,
-0.01801510713994503,
0.04080160707235336,
-0.04668748751282692,
0.010252472944557667,
0.0771501362323761,
0.051700737327337265,
-0.020389975979924202,
0.004227453842759132,
0.03657655790448189,
-0.02762596309185028,
-0.03138... | 0.817957 | 0.798914 | 0.019042 |
If \( \sin(\sqrt{x}) \) were a periodic function with period \( T \neq 0 \), then there would exist distinct integers \( n \) and \( m \) such that the condition \( \frac{m^2}{n^2} = 2 \) holds true. What can you conclude about the nature of the integers \( m \) and \( n \)? | automath | [
0.015404102392494678,
0.04433094337582588,
-0.06058802828192711,
-0.04223277419805527,
0.05233987420797348,
-0.01681806519627571,
-0.024185379967093468,
0.07366441935300827,
0.030443724244832993,
0.0007395309512503445,
0.016751326620578766,
0.02512822486460209,
-0.03179784491658211,
-0.036... | 0.834277 | 0.812741 | 0.021536 |
Let \( X_1, X_2, X_3, \ldots \) be a sequence of independent and identically distributed (i.i.d.) random variables, each having mean \( 1 \). Let \( N \) be a geometric random variable with the probability mass function \( \mathbb{P}(N = k) = \frac{1}{2^k} \) for \( k = 1, 2, 3, \ldots \), independent of the \( X_i \)'... | automath | [
0.00849457923322916,
0.03178839385509491,
-0.05815332010388374,
-0.03579746186733246,
0.03475480154156685,
0.0034955902956426144,
-0.023882420733571053,
0.05449662357568741,
0.024553686380386353,
0.0019438976887613535,
-0.00006313277845038101,
0.02230020985007286,
-0.03608869016170502,
-0.... | 0.829398 | 0.807698 | 0.0217 |
Evaluate the integral
\[
E_n(a_1, \ldots, a_n; t) = \int_{-\infty}^{\infty} \frac{x^2 \cos(tx)}{\prod_{k=1}^{n}(x^2 + a_k^2)} \, dx
\]
for distinct non-zero real numbers \( a_1, a_2, \ldots, a_n \) and a non-negative real parameter \( t \). Provide a closed form of the result in terms of \( a_1, a_2, \ldots, a_n \) ... | automath | [
0.03191174566745758,
0.028779087588191032,
-0.031115995720028877,
-0.015241595916450024,
0.016687151044607162,
0.0036265505477786064,
-0.026874138042330742,
0.06856804341077805,
0.032036349177360535,
-0.008107916451990604,
0.011235702782869339,
0.02383394166827202,
-0.01786123774945736,
-0... | 0.821853 | 0.798662 | 0.023191 |
Given that a set of points \( X \) in the plane satisfies the condition \( |X| > \frac{(k-1)(k'-1)}{2} \), can we establish a specific range of values for \( k \) and \( k' \) such that the statement \( C(X; 2, k') \cup C(X; 3, 3) \cup C(X; k, 2) \) must hold? Specifically, what are the smallest values of \( k \) and \... | automath | [
0.020125268027186394,
0.05213466286659241,
-0.055885642766952515,
-0.01652297005057335,
0.038308851420879364,
-0.008912408724427223,
-0.005429498385637999,
0.0623830109834671,
0.03730093315243721,
-0.010234815068542957,
0.007471626624464989,
0.04079611971974373,
-0.014051481150090694,
-0.0... | 0.830371 | 0.806662 | 0.023709 |
In a separable Hilbert space, consider a complete orthonormal sequence \((e_n)\) and an element \(x\) in that space. If it is known that Bessel's inequality \(\sum_n |\langle x, e_n \rangle |^2 \leq \|x\|^2\) holds, what is the necessary condition for Parseval's identity \(\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2\)... | automath | [
0.03454747796058655,
0.041923895478248596,
-0.030571306124329567,
-0.03823390230536461,
0.029281293973326683,
-0.019463500007987022,
-0.007064648438245058,
0.07398806512355804,
0.036877844482660294,
-0.012705594301223755,
-0.002183184027671814,
0.014326240867376328,
-0.03131392225623131,
-... | 0.826786 | 0.797284 | 0.029502 |
Given a set of distinct integers \( x_1, x_2, \ldots, x_N \) presented by Elsie, which may be one of the following: \( A \), \( B \), \( C \), \( A+B \), \( B+C \), \( C+A \), or \( A+B+C \) for some positive integers \( A \leq B \leq C \), how many distinct triples \( (A, B, C) \) can be consistent with the integers p... | automath | [
0.03182559832930565,
0.012070052325725555,
-0.04089735075831413,
-0.014841590076684952,
0.03701229393482208,
-0.0262011531740427,
-0.029573753476142883,
0.04346974566578865,
0.038585543632507324,
-0.008181856945157051,
0.018478669226169586,
0.02632811851799488,
-0.0234643816947937,
-0.0239... | 0.841506 | 0.820377 | 0.021129 |
Is the space \( X = \bigcup_{n\in \mathbb{N}} \left( \{1/n\} \times \mathbb{R} \cup \mathbb{R} \times \{1/n\} \right) \cup \{(0, 0)\} \) path connected? If it is, provide an explicit path from the point \( (0,0) \) to any point \( (1/k, 0) \) for \( k \in \mathbb{N} \). If not, explain why it is not path connected. | automath | [
0.013445991091430187,
0.03707195073366165,
-0.04008905217051506,
-0.02195069007575512,
0.02438521571457386,
0.0031400304287672043,
-0.011062700301408768,
0.05888756737112999,
0.0331244021654129,
0.009479782544076443,
0.00806734524667263,
0.03247247636318207,
-0.013902613893151283,
-0.03723... | 0.833157 | 0.811253 | 0.021905 |
Given the parabola defined in the rotated $(z,w)$ axes by the equation \((z-\frac{\sqrt{2}}{2})^2 = -\frac{\sqrt{2}}{2}(w-\frac{\sqrt{2}}{2})\), and the transformation equations \(z = \frac{x+y}{\sqrt{2}}\) and \(w = \frac{y-x}{\sqrt{2}}\), derive the equation of the parabola in the original $(x,y)$ coordinates and ver... | automath | [
0.023888438940048218,
0.0453062430024147,
-0.036952029913663864,
-0.044537581503391266,
0.0424676351249218,
-0.00000826083942229161,
-0.010426641441881657,
0.07311145216226578,
0.03205262869596481,
-0.011239332146942616,
-0.007825012318789959,
0.023176079615950584,
-0.021200228482484818,
-... | 0.825269 | 0.805958 | 0.019311 |
Let \( G \) be an Abelian group equipped with two different \( \mathcal{L}_{\infty} \) topologies \( \mathcal{L}_{1} \) and \( \mathcal{L}_{2} \). If \( (G, \mathcal{L}_{1}) \) is separable and \( (G, \mathcal{L}_{2})^{\wedge} \) is metrizable, what is the cardinality of the subset \( B \) whose closure in the compact ... | automath | [
0.013606240041553974,
0.04837479442358017,
-0.039366815239191055,
-0.02262219227850437,
0.030962713062763214,
-0.026165032759308815,
-0.036383967846632004,
0.045133139938116074,
0.021311208605766296,
0.0011873655021190643,
-0.009061245247721672,
0.02949162945151329,
-0.013683457858860493,
... | 0.82329 | 0.802427 | 0.020862 |
Given a set of points sampled from a 2-dimensional manifold embedded in \(\mathbb{R}^3\), assume you estimate the mean curvature at a point \(p\) on the manifold using a local neighborhood defined by a radius \(r\). Suppose the point cloud consists of 100 noisy samples that are uniformly distributed within a disk of ra... | automath | [
-0.01974533312022686,
0.03309997543692589,
-0.045256130397319794,
-0.05058891698718071,
0.040271103382110596,
-0.004448038060218096,
-0.03484804928302765,
0.053846921771764755,
0.016975294798612595,
-0.010649872943758965,
-0.0010963008971884847,
0.040514785796403885,
-0.033976368606090546,
... | 0.816848 | 0.79757 | 0.019278 |
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