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 a curve \(\beta(s)\) that is parametrized by a differentiable function \(s\). If the tangent vector is given by \(T = \frac{\beta'(s)}{|\beta'(s)|}\), can the Frenet-Serret frame still be defined for \(N\) and \(B\) in the same way as for natural parametrization? Specifically, what conditions must the curve sa... | automath | [
0.008904008194804192,
0.021195465698838234,
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0.042810771614313126,
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0.029674453660845757,
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0.02043449692428112,
-0.042585477232933044,
-0.... | 0.811773 | 0.795073 | 0.0167 |
Given \( n \) points \( a_1, a_2, \ldots, a_n \) on the surface of a unit sphere, where each point \( a_i \) is represented by normalized coordinates derived from integers \( x, y, z \), calculate the expected value \( E[f] \) where \( f \) is defined as \( 1 \) if there exists a hemisphere that contains \( a_0, a_1, \... | automath | [
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0.011989534832537174,
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0.018254775553941727,
0.04449953883886337,
-0.028577445074915886,
-... | 0.827289 | 0.807101 | 0.020188 |
Consider a function \( f : \mathbb{R}^p \rightarrow \mathbb{R} \) given by \( f(x) = x^T A x \) where \( A \) is a fixed \( p \times p \) real symmetric matrix. What is the expression for the derivative \( \frac{\partial f(x)}{\partial x} \) in terms of \( A \) and \( x \), and how does this relate to the form \( (A + ... | automath | [
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0.018297644332051277,
-0.056393422186374664,
-0... | 0.808524 | 0.790109 | 0.018415 |
Given Shamir's Secret Sharing scheme with parameters \( k = 2 \) and \( n = 4 \), can arbitrary points \( (x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4) \) in the plane be chosen such that they all lie on the same linear polynomial and represent valid shares of a secret? If so, what conditions must be satisfied for the... | automath | [
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0.017415741458535194,
0.02586073987185955,
-0.0369386225938797,
-0.02338... | 0.829677 | 0.805327 | 0.02435 |
Let \( A \) be a \( 3 \times 3 \) matrix that projects any vector \( \vec{x} \) in \( \mathbb{R}^3 \) onto a non-zero vector \( \vec{v} \). Determine the eigenvalues of \( A \) and their corresponding algebraic multiplicities. | automath | [
0.0343640074133873,
0.026307184249162674,
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0.0028978583868592978,
0.03948390483856201,
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-0.0... | 0.814362 | 0.797767 | 0.016595 |
Let \( A \) and \( B \) be \( 2 \times 2 \) matrices with entries \( A = \begin{pmatrix} a_{00} & a_{01} \\ a_{10} & a_{11} \end{pmatrix} \) and \( B = \begin{pmatrix} b_{00} & b_{01} \\ b_{10} & b_{11} \end{pmatrix} \). Compute \( S_1 = \text{Trace}(AB) \) and \( S_2 = \text{Trace}(A) \cdot \text{Trace}(B) \). Under w... | automath | [
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0.009136777371168137,
0.02020498551428318,
0.014244113117456436,
-0.031331636011600494,
-0.01... | 0.822996 | 0.800927 | 0.022069 |
Given a convex region \( C \) in the plane, if we partition \( C \) into \( n \) convex pieces such that the least width among the pieces is maximized, what is the minimum possible least width of these pieces in terms of the diameter of \( C \)? More specifically, can this minimum least width be achieved by a partition... | automath | [
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0.0368952713906765,
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0.024604447185993195,
0.02707173116505146,
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-0.0... | 0.828657 | 0.811743 | 0.016914 |
Consider a rational function \( f(x) \) of the form:
\[
f(x) = \frac{ax+b}{x+c}
\]
Given that \( f(x) \) has a horizontal asymptote at \( y = -4 \), a vertical asymptote at \( x = 3 \), and an x-intercept at \( (1, 0) \):
1. Find the values of \( a \), \( b \), and \( c \) such that the function satisfies all the gi... | automath | [
0.03102654218673706,
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0.018615959212183952,
0.0044097681529819965,
-0.029032623395323753,
... | 0.825436 | 0.806177 | 0.019259 |
Given a set of \( N \) points in a two-dimensional plane, each with integer coordinates, determine the maximum number of points that can lie on a single straight line. For example, if \( N = 7 \) and the points are \( (1, 2) \), \( (2, 3) \), \( (3, 4) \), \( (4, 5) \), \( (5, 6) \), \( (1, 1) \), and \( (1, 0) \), wha... | automath | [
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0.034446753561496735,
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0.045505717396736145,
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-0.0... | 0.836775 | 0.815626 | 0.021149 |
Let \( M \) be a smooth surface in \( \mathbb{R}^3 \) and consider a family of immersions defined by \( f(t) = f_0 + tuN_0 \), where \( u: M \to \mathbb{R} \) is a smooth function. If the principal curvatures of the surface \( M \) at the point of immersion \( f_0 \) are denoted by \( \kappa_1 \) and \( \kappa_2 \), wh... | automath | [
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0.005613118875771761,
0.028390370309352875,
0.00025078258477151394,
-0.03846016153693199,
-0.0... | 0.808824 | 0.787964 | 0.02086 |
If we have a sum of \( n \) independent Poisson random variables, each with a mean \( \lambda \), and we set \( n\lambda = 1 \) while allowing \( n \) to approach infinity, what can we conclude about the applicability of the Central Limit Theorem to this sum? Specifically, does this scenario imply that the sum will not... | automath | [
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0.002321920357644558,
0.022267935797572136,
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-0.0... | 0.812195 | 0.794827 | 0.017368 |
Given a topological space \(X\) that can be covered by an open cover \(X = \bigcup_{i \in I} U_i\), where \(|I| = n\) for some integer \(n\), and given glueing data \((\mathcal{F}_i, \varphi_{ij})\) such that each \(\mathcal{F}_i\) is a sheaf of sets over \(U_i\), what is the minimum number of open sets \(U \subset X\)... | automath | [
0.025203697383403778,
0.011081705801188946,
-0.03946195915341377,
-0.029527075588703156,
0.026721393689513206,
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0.06509880721569061,
0.0319807305932045,
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0.01335016917437315,
0.032490380108356476,
-0.02084202505648136,
0.008... | 0.823904 | 0.803251 | 0.020652 |
Given the linear map \( f: \mathbb{R}^2 \to \mathbb{R}^3 \) defined by \( f(x, y) = (x, 3x + 8y, x + 12y) \), determine whether there exists a linear map \( g: \mathbb{R}^3 \to \mathbb{R}^2 \) such that the matrix representation of the composition \( f \circ g \) with respect to the standard bases of \(\mathbb{R}^3\) a... | automath | [
0.016359109431505203,
0.03363565728068352,
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0.050737809389829636,
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0.06802555918693542,
0.014929389581084251,
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0.011490315198898315,
0.023871034383773804,
-0.019931096583604813,
-0.... | 0.81819 | 0.801192 | 0.016998 |
Let \( A \) be a unique factorization domain (UFD). Consider \( a, b \in A \) such that \( \gcd(a, b) = 1 \) and \( c \in A \) for which \( a \mid bc \). Can we conclude that \( a \mid c \) without employing prime factorization, and if so, what is the argument that shows this is true? | automath | [
0.025016365572810173,
0.036019839346408844,
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0.04416760802268982,
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0.009747005999088287,
0.06319522857666016,
0.03335325047373772,
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-0.006514410488307476,
0.015590133145451546,
-0.007724864408373833,
-... | 0.828372 | 0.809166 | 0.019207 |
What is the relationship between the statement that for any infinite set \( A \), the cardinality \( |A \times A| = |A| \) and the Axiom of Choice, particularly in the context of sets with cardinality \( 2^{\aleph_0} \)? Can you provide an example of a set where this equality holds without invoking the Axiom of Choice,... | automath | [
0.02155773714184761,
0.03654494136571884,
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0.020532047376036644,
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0.06324034929275513,
0.01475547719746828,
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0.01297165360301733,
0.005855980794876814,
-0.026132170110940933,
-0.01... | 0.81531 | 0.792944 | 0.022365 |
In a topological space \( T = \struct{S, \tau} \), if it is given that for every pair of distinct points \( x, y \in S \) there exist open sets \( U \) and \( V \) such that \( x \in U \), \( y \in V \), and \( U \cap V = \emptyset \), what can be concluded about the relationship between closed neighborhoods of points ... | automath | [
0.03790251538157463,
0.031460195779800415,
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0.008594082668423653,
0.00967391300946474,
0.040231719613075256,
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-0.... | 0.811671 | 0.781908 | 0.029763 |
Suppose \( f \) is an analytic function in the annulus \( 1 < |z| < 2 \) and is uniformly approximated by a sequence of polynomials on every compact subset of this annulus. What can be concluded about the nature of the singularity of \( f \) at \( z = 0 \), and how does this information allow you to extend \( f \) to a... | automath | [
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0.0121186263859272,
0.01277419924736023,
0.024090580642223358,
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-0.... | 0.823846 | 0.801808 | 0.022039 |
Let \( x \) and \( y \) be real numbers. Determine the conditions under which the inequality \( x^2 - y^2 > 0 \) holds true. Given the following statements:
(1) \( x^2 - y^2 > x - y \)
(2) \( x^2 - y^2 < x + y \)
Is the inequality \( x^2 - y^2 > 0 \) true based on the information from the two statements? Provide a d... | automath | [
0.030336005613207817,
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0.05713639408349991,
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0.0031300398986786604,
0.04030672088265419,
-0.052020881325006485,
-0.... | 0.827715 | 0.805748 | 0.021967 |
Given an abelian group \( A \) with a fixed element \( e \in A \), let \( R(A, e) \) be the constructed commutative ring as described. What necessary conditions must \( A \) and \( e \) satisfy for \( R(A, e) \) to be a nontrivial ring (meaning \( R(A, e) \neq 0 \)), and provide an example of an abelian group and an el... | automath | [
0.04096270725131035,
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0.009548710659146309,
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-... | 0.813278 | 0.792903 | 0.020376 |
Given a positive integer \( n \) that is not a perfect square, determine whether there exist infinitely many integer solutions \( (x, y) \) to the equation \( x^2 - n y^2 = k \) for any fixed integer \( k \). If so, describe a method to generate such solutions. | automath | [
0.029692357406020164,
0.028828993439674377,
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0.026068510487675667,
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0.055488038808107376,
0.048373881727457047,
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0.030837006866931915,
0.03359697386622429,
-0.03716784343123436,
-0.02... | 0.841883 | 0.821039 | 0.020844 |
Consider the measurable space \((S, \Sigma)\) where \(\Sigma\) is a \(\sigma\)-algebra and \(\mu: \Sigma \rightarrow [0, \infty]\) is a measure on this space. If \(\mu\) is a finite measure and \(\mu(S) = 10\), what can you conclude about the measure of any measurable subset \(A \in \Sigma\) for which \(\mu(A) \geq 5\)... | automath | [
0.02732440084218979,
0.03749188780784607,
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0.03536402806639671,
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0.05007965490221977,
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0.01145176775753498,
0.01899433508515358,
-0.04610619321465492,
-0.0092... | 0.822163 | 0.797597 | 0.024566 |
Given a number field \(K\) with a ring of integers \(\mathcal{O}_{K}\) and an order \(\mathcal{O}=\mathbb{Z}[\beta_{1},\ldots,\beta_{n}] \subseteq \mathcal{O}_{K}\), how can one determine the conditions under which \(\mathcal{O}=\mathcal{O}_{K}\) holds, particularly in relation to the linear independence of the reducti... | automath | [
0.019674794748425484,
0.047506868839263916,
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0.026826174929738045,
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0.002025134861469269,
0.015035531483590603,
-0.02602049708366394,
0.... | 0.823618 | 0.801462 | 0.022155 |
Given a Monte Carlo algorithm \( A \) for a problem \( \Pi \) with expected running time \( T(n) \) and correctness probability \( \gamma(n) \), derive the expected running time of a Las Vegas algorithm that guarantees a correct answer and runs in at most \( \frac{T(n) + t(n)}{\gamma(n)} \), where \( t(n) \) is the ver... | automath | [
0.019755056127905846,
0.020762797445058823,
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0.04606890678405762,
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0.0709405317902565,
0.024172397330403328,
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0.019311808049678802,
0.04673019051551819,
-0.0031678772065788507,
-0.... | 0.8194 | 0.799699 | 0.019702 |
Let \( X \) be a countably infinite subset of \( \mathbb{N} \). Define a function \( f: \mathbb{N} \to X \) that maps each positive integer \( n \) to the \( n \)-th smallest element of \( X \). What is the cardinality of the set \( \mathcal{P}(X) \), the power set of \( X \)? Provide a detailed explanation of your rea... | automath | [
0.03366462141275406,
0.04149177297949791,
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0.027008360251784325,
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0.014488598331809044,
0.03255289047956467,
-0.0341729111969471,
-0.0... | 0.842648 | 0.824388 | 0.01826 |
Given a prime number \( p \) and a positive integer \( n \), can we always construct an irreducible polynomial \( p(x) \) of degree \( n \) over the finite field \( GF(p) \) such that the degree of \( p(x) - x^n \) is less than or equal to \( \frac{n}{2} \)? If so, what can be said about the existence of a tighter boun... | automath | [
0.02723436802625656,
0.026669563725590706,
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0.0252011027187109,
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0.04320194199681282,
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0.003990784753113985,
0.03322324901819229,
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-0.012... | 0.827503 | 0.802442 | 0.025061 |
Let \( a, b, c \) be real numbers with \( a = -5 \), \( b = 10 \), and \( c = -7 \). Calculate \( |a + b + c| \) and compare it with \( |a| + |b| + |c| \). Does the triangle inequality hold in this case? | automath | [
0.03900996223092079,
0.043407540768384933,
-0.03710072860121727,
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0.04446995258331299,
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0.07790907472372055,
0.027177883312106133,
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-0.006080032791942358,
0.04636760056018829,
-0.04455512389540672,
-0.... | 0.844243 | 0.82645 | 0.017793 |
Given a Fourier series expansion of a periodic function \( f(x) \) on the interval \([-L, L]\), can we determine the coefficients \( a_n \) and \( b_n \) of the series such that they correspond to a specific non-elementary function \( f(x) \)? If \( f(x) \) is represented by its Fourier series \( f(x) = a_0 + \sum_{n=1... | automath | [
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0.0390779972076416,
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0.02382318675518036,
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0.05288467928767204,
0.031169703230261803,
0.011797170154750347,
0.01445549726486206,
0.025175895541906357,
-0.033658940345048904,
-0.0... | 0.822382 | 0.804869 | 0.017513 |
Let \( A \) and \( B \) be symmetric matrices of size \( n \times n \). Consider the function
\[
f(x) = \frac{x^T A x}{x^T B x}
\]
where \( x \) is a nonzero vector in \( \mathbb{R}^n \).
Calculate the gradient \( \nabla f(x) \) with respect to \( x \) and determine the critical points of \( f(x) \). In the case w... | automath | [
0.028805969282984734,
0.03963283821940422,
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0.02250031568109989,
-0.0036553004756569862,
-0.018825478851795197,
0.022134454920887947,
-0.03402615338563919,
-0... | 0.82012 | 0.792259 | 0.027861 |
Let $X$ and $Y$ be schemes, with a completely decomposed morphism $f : X \to Y$. If $Y'$ is another scheme and the fibred product $X' = X \times_Y Y'$, how many points in $X'$ map uniquely to a given point $y' \in Y'$ corresponding to some point $y \in Y$ such that the residue fields $\kappa(x)$ and $\kappa(y)$ are equ... | automath | [
0.013324880972504616,
0.021274734288454056,
-0.044105980545282364,
-0.034769125282764435,
0.02907695807516575,
-0.023029383271932602,
-0.004971737042069435,
0.06020285561680794,
0.041463516652584076,
-0.005251530557870865,
-0.010696325451135635,
0.03820021077990532,
-0.0330740362405777,
0.... | 0.840497 | 0.823348 | 0.01715 |
Given two one-sheet hyperboloids defined by the equations
\[
\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2} - \sqrt{(x-x_2)^2+(y-y_2)^2+(z-z_2)^2} = d_{1,2}
\]
and
\[
\sqrt{(x-x_1)^2+(y-y_1)^2+(z-z_1)^2} - \sqrt{(x-x_3)^2+(y-y_3)^2+(z-z_3)^2} = d_{1,3}
\]
for fixed points \( P_1(x_1, y_1, z_1) \), \( P_2(x_2, y_2, z_2) \),... | automath | [
0.025781182572245598,
0.04051881283521652,
-0.036190010607242584,
-0.050757504999637604,
0.02145509235560894,
-0.0008574770181439817,
-0.03807923197746277,
0.04979630559682846,
0.04224950075149536,
-0.022208670154213905,
0.028037622570991516,
0.02602103166282177,
-0.005921672098338604,
-0.... | 0.828592 | 0.804557 | 0.024036 |
Consider the function \( f(x, y) = \frac{x - y}{(x + y)^2} \) defined on the square \( [0, 1] \times [0, 1] \). Determine whether the function is Lebesgue integrable over this region, and compute the value of the double integral \( \iint_{[0, 1] \times [0, 1]} f(x, y) \, d\lambda^2(x, y) \). | automath | [
0.016635894775390625,
0.034936413168907166,
-0.0467706136405468,
-0.030812514945864677,
0.03938066214323044,
0.01145415659993887,
-0.01934582181274891,
0.08095314353704453,
0.04371322691440582,
0.006188518367707729,
0.011941997334361076,
0.028564587235450745,
-0.035588596016168594,
-0.0212... | 0.822443 | 0.799305 | 0.023138 |
Let \( S \) be the set of upper triangular matrices with \( 1 \)'s on the diagonal in \( V = \mathbb{Q}^{n \times n} \). Define \( B \) as the subspace spanned by \( S \). What is the dimension of the subspace \( B \) for any positive integer \( n \), and can you describe the form of the matrices in \( B \)? | automath | [
0.03083088807761669,
0.014030334539711475,
-0.055429838597774506,
-0.03701169043779373,
0.032718390226364136,
-0.026692191138863564,
0.00001566468199598603,
0.04549306258559227,
0.03566376119852066,
0.02064807526767254,
0.025197217240929604,
0.04300602152943611,
-0.039045825600624084,
-0.0... | 0.832558 | 0.809855 | 0.022703 |
Let \( f \) be a continuous function defined on the interval \([a, b]\), and let \( g \) be a function defined as follows:
\[
g(x) =
\begin{cases}
1 & \text{if } x \in [a, b] \\
0 & \text{otherwise}
\end{cases}
\]
Is it true that
\[
\int_{p}^{q} f(x) g(x) \, dx = \int_a^b f(x) \, dx
\]
for any \( p < a < b < q... | automath | [
0.019626500084996223,
0.04297564551234245,
-0.04011448845267296,
-0.032243672758340836,
0.04541854187846184,
-0.009757823310792446,
-0.022582082077860832,
0.08164939284324646,
0.02181585691869259,
0.00009191969729727134,
0.012935206294059753,
0.005229647271335125,
-0.03360236808657646,
-0.... | 0.81532 | 0.789063 | 0.026257 |
Given a positive integer \( n \), let \( S \) be the affine surface defined as the spectrum of the subring of \( \mathbb{C}[x,y] \) generated by the monomials \( x, xy, \ldots, xy^n \). Determine the following properties of the surface \( S \):
1. Is \( S \) normal?
2. Does \( S \) have rational singularities?
3. What... | automath | [
0.009749941527843475,
0.042208921164274216,
-0.034589625895023346,
-0.023503368720412254,
0.03171084448695183,
-0.035180915147066116,
-0.011185985058546066,
0.05502694472670555,
0.0482785739004612,
0.004655847791582346,
0.026099886745214462,
0.022125856950879097,
-0.03431669622659683,
-0.0... | 0.822487 | 0.79128 | 0.031207 |
Given a player who wins the \( n \)th game with probability \( \frac{1}{\sqrt{n}} \), independently of other games, calculate the probability that the player will accumulate at least \( A \) dollars after playing \( n \) games if the player earns a dollar for every two consecutive wins. How does this probability behave... | automath | [
0.00013646369916386902,
0.02399410307407379,
-0.0378330759704113,
-0.026130756363272667,
0.0146019970998168,
-0.009972828440368176,
-0.026721032336354256,
0.04685666784644127,
0.027463233098387718,
-0.022348741069436073,
0.004147520288825035,
0.02850286476314068,
-0.02582990936934948,
-0.0... | 0.830845 | 0.812755 | 0.01809 |
Given the function \( f(x) = \frac{1}{x^2} \), determine the maximum possible value of \( \delta \) such that for every \( x_0 > 0 \), if \( |x - x_0| < \delta \), then \( |f(x) - f(x_0)| < \epsilon \) for \( \epsilon > 0 \). Express your answer in terms of \( x_0 \) and \( \epsilon \). | automath | [
0.017806705087423325,
0.027053434401750565,
-0.031102154403924942,
-0.032297514379024506,
0.0316692590713501,
-0.03259778022766113,
-0.024721313267946243,
0.07606900483369827,
0.03288339078426361,
-0.01721428520977497,
0.005539864767342806,
0.021672267466783524,
-0.022028591483831406,
-0.0... | 0.821473 | 0.803718 | 0.017754 |
Consider two independent random variables \(X\) and \(Y\) with uniform probability density functions over the interval \([-1, 1]\). What is the probability that \(X^2 + Y^2 > 1\)?
Additionally, let \(X\) be a random variable with the probability density function \(p_X(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\) an... | automath | [
-0.018656091764569283,
0.027394425123929977,
-0.05136619135737419,
-0.03347768262028694,
0.04890407994389534,
0.0067282733507454395,
-0.02181141823530197,
0.05503964051604271,
0.03244243562221527,
0.0036245188675820827,
-0.0003272461472079158,
0.031057389453053474,
-0.029288925230503082,
0... | 0.820753 | 0.798003 | 0.02275 |
Consider the recursion defined by \( x_{n+1} = f(x_n) \) where \( f(x) = p^k \cdot \sqrt[k+1]{x} \). For which values of \( p \) and \( k \) does this recursion converge to \( p^{k+1} \) for any initial value \( x_0 \)? Additionally, can you derive the fixed points of \( f(x) \) explicitly in terms of \( p \) and \( k ... | automath | [
0.019521642476320267,
0.047823913395404816,
-0.046497780829668045,
-0.03383013606071472,
0.02429605834186077,
-0.009536899626255035,
0.002197404857724905,
0.0723915547132492,
0.03145141154527664,
-0.00019100916688330472,
0.0021335172932595015,
0.007479651365429163,
-0.0322093702852726,
-0.... | 0.838555 | 0.816345 | 0.02221 |
Consider a field \( K \) and an integer \( n \). Let \( B(n, K) \) denote the Borel subgroup of the general linear group \( GL(n, K) \), which consists of all upper-triangular invertible matrices over \( K \).
1. If \( K \) is the finite field \( \mathbb{F}_{q} \) where \( q = p^k \) for some prime \( p \) and positi... | automath | [
0.021056311205029488,
0.045526422560214996,
-0.04528091475367546,
-0.04005623981356621,
0.03931440785527229,
-0.03800027817487717,
-0.010648458264768124,
0.037454139441251755,
0.03873443230986595,
0.0062902942299842834,
0.01957378163933754,
0.03568769246339798,
-0.049874577671289444,
0.001... | 0.828311 | 0.806843 | 0.021468 |
Let \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) be a polynomial function of degree \( n \). If \( f \) is observed at two distinct points \( x_1, x_2 \in \mathbb{R} \) with \( f(x_1) = 2 \) and \( f(x_2) = 5 \), what is the minimum possible value of \( n \) such that there exists at least one real pol... | automath | [
0.026582030579447746,
0.031497664749622345,
-0.04853802174329758,
-0.03819288685917854,
0.016004357486963272,
-0.007235343102365732,
-0.02678847126662731,
0.07301821559667587,
0.03162519633769989,
-0.0015534507110714912,
-0.004831086844205856,
0.01892443187534809,
-0.017241498455405235,
-0... | 0.837684 | 0.814834 | 0.02285 |
Consider a sequence of random variables \((X_n)\) that converges almost surely to a random variable \(X\). Let \(p_n\) and \(p\) be the probability density functions (pdfs) of \(X_n\) and \(X\), respectively. If \(f\) is in \(L^\infty[0,1]\), for which measures of convergence, if any, does the statement \(\int f p_n \t... | automath | [
-0.013393520377576351,
0.03919154405593872,
-0.043048348277807236,
-0.017910433933138847,
0.019788077101111412,
0.012286416254937649,
-0.025748668238520622,
0.03870055824518204,
0.025749793276190758,
0.009438673034310341,
-0.00843050330877304,
0.015461992472410202,
-0.007357423193752766,
-... | 0.812002 | 0.792049 | 0.019953 |
Consider a dataset of size \( n \), where each observation \( y_i \) follows a binomial distribution given by the parameters \( p_i \) and \( n \). Assume that each probability of success \( p_i \) is independently drawn from a Beta distribution with parameters \( a \) and \( b \).
If the parameters \( \kappa, a, b \... | automath | [
0.010608826763927937,
0.020655622705817223,
-0.05768651142716408,
-0.041290029883384705,
0.02712043933570385,
-0.010388366878032684,
-0.02542453072965145,
0.04525810480117798,
0.02761433832347393,
-0.008568904362618923,
0.009180012159049511,
0.028092700988054276,
-0.03478796035051346,
0.01... | 0.821954 | 0.804029 | 0.017925 |
Consider the function \( u(x) = |x| \) defined on \( \mathbb{R}^2 \) such that \( u(x) = |x| \) for \( |x| \leq 1 \) and \( u(x) = 0 \) for \( |x| > 1 \). Analyze whether \( u(x) \) belongs to the Sobolev space \( W^{1,3}(\mathbb{R}^2) \). If so, demonstrate why it should have a continuous representative, and if not, p... | automath | [
-0.008820653893053532,
0.05366988107562065,
-0.049884989857673645,
-0.04805547744035721,
0.03690062090754509,
0.0008800834184512496,
-0.018956119194626808,
0.08260084688663483,
0.02349090576171875,
-0.004563467111438513,
0.009749204851686954,
0.03761753439903259,
0.0018800209509208798,
-0.... | 0.800135 | 0.775682 | 0.024453 |
Consider a curve \( \mathcal{L} \) defined by the equation \( y = mx + b \) where \( m \) and \( b \) are constants.
1. If the curve is a straight line, what is the value of the gradient \( m \)?
2. If the gradient \( m \) is constant and equal to \( k \), determine the equation of the curve \( \mathcal{L} \).
Afte... | automath | [
0.00020145940652582794,
0.035125602036714554,
-0.04155336320400238,
-0.04798489063978195,
0.045198921114206314,
-0.025893228128552437,
-0.02302568405866623,
0.07218697667121887,
0.028237083926796913,
-0.011189024895429611,
-0.013493664562702179,
0.02268802933394909,
-0.03719668090343475,
-... | 0.826426 | 0.807762 | 0.018664 |
Given $R$-modules $M$, $N$, and $P$, define a map $\varphi : M \otimes_R N \longrightarrow (M \otimes_R N) \otimes_R P$ by $\varphi(m \otimes n) = m \otimes (n \otimes p)$ for a fixed $p \in P$. If $m \otimes n = m' \otimes n'$ for some $m, m' \in M$ and $n, n' \in N$, under what conditions can we conclude that $\varph... | automath | [
0.04155992716550827,
0.033119991421699524,
-0.02730853296816349,
-0.04489656537771225,
0.02101030759513378,
-0.03684094548225403,
0.005842671729624271,
0.06399700045585632,
0.0317118875682354,
-0.0068716383539140224,
0.0052397954277694225,
0.014711177907884121,
-0.026833640411496162,
-0.01... | 0.813823 | 0.791747 | 0.022076 |
Let \( k \) be a field and \( n \) a positive integer. Define the left ideals of \( M_n(k) \) generated by the column vectors of size \( n \).
What is the number of distinct column left ideals in \( M_2(k) \), and can you explicitly describe the structure of \( M_2(k) \) as a direct sum of these column left ideals? | automath | [
0.027616605162620544,
0.018318679183721542,
-0.0410810261964798,
-0.04299008101224899,
0.03223703056573868,
-0.02040739171206951,
-0.005054967012256384,
0.05782981216907501,
0.040065109729766846,
-0.004432712681591511,
0.0061685554683208466,
0.029700879007577896,
-0.029920747503638268,
-0.... | 0.831142 | 0.814667 | 0.016475 |
Given the identity
$$\sum_{k=1}^{n}{\sigma_0(k^2)} = \sum_{k=1}^{n}{\left\lfloor \frac{n}{k}\right\rfloor 2^{\omega(k)}},$$
where $\sigma_0(k^2)$ is the number of divisors of $k^2$, and $\omega(k)$ denotes the number of distinct prime divisors of $k$, can you compute the value of
$$\sum_{k=1}^{10}{\sigma_0(k^2)}$$... | automath | [
0.01949078030884266,
0.046777933835983276,
-0.03821396827697754,
-0.03620810806751251,
0.024058248847723007,
-0.038227636367082596,
0.0032778913155198097,
0.062420234084129333,
0.01700044795870781,
-0.011673097498714924,
0.007177335675805807,
0.011535833589732647,
-0.041054219007492065,
-0... | 0.839938 | 0.823187 | 0.016751 |
Given two measure spaces \((\Omega_1, \mathcal{F}_1, \mu_1)\) and \((\Omega_2, \mathcal{F}_2, \mu_2)\) with \(\sigma\)-finite measures, if we define the product measure \(\mu\) on the product space \(\Omega = \Omega_1 \times \Omega_2\) such that \(\mu(E_1 \times E_2) = \mu_1(E_1) \mu_2(E_2)\) for all \(E_1 \in \mathcal... | automath | [
0.010221784934401512,
0.04225830361247063,
-0.0416153147816658,
-0.02793973870575428,
0.025695009157061577,
0.005418853368610144,
-0.03426603600382805,
0.07043483853340149,
0.03303339704871178,
0.0011327860411256552,
-0.003852818626910448,
0.03728371858596802,
-0.0353107824921608,
-0.01753... | 0.816619 | 0.790684 | 0.025935 |
Consider the equation \( \sqrt{1+x^{2}y^{2}} = 2xy \). If we square both sides to eliminate the square root, we obtain the equation \( 1 + x^2y^2 = 4x^2y^2 \).
Determine the derivative \( \frac{dy}{dx} \) implicitly for the original equation, and then for the squared equation. Compare the results. Do the two derivati... | automath | [
0.010282468982040882,
0.051681019365787506,
-0.04402518644928932,
-0.03823818266391754,
0.04289110377430916,
-0.024971339851617813,
-0.0055051143281161785,
0.082788847386837,
0.03371766582131386,
-0.01891009695827961,
0.0016902269562706351,
0.013244099915027618,
-0.04700146242976189,
-0.01... | 0.817173 | 0.797606 | 0.019566 |
Let \( H \) be a five-dimensional subspace of a ten-dimensional vector space \( V \). Determine whether the following statements are true or false:
1. Every set containing seven vectors from \( H \) must be linearly dependent.
2. If \( \mathcal{B} \) is a basis of \( \mathbb{R}^{n} \) and \( A \) is the matrix whose c... | automath | [
0.03420339152216911,
0.04395357146859169,
-0.037315983325242996,
-0.04262175410985947,
0.05808552727103233,
-0.029410861432552338,
-0.004170294851064682,
0.06420469284057617,
0.02849189192056656,
-0.013825398869812489,
-0.005303172394633293,
0.024218028411269188,
-0.03929644823074341,
-0.0... | 0.812623 | 0.787121 | 0.025502 |
Let \( X_1, X_2, \ldots, X_n \) be random variables drawn from a Beta distribution with shape parameters \( 3 \) and \( 1 \). Define the cumulative mean at time \( t \) as \( y_t = \frac{X_1 + X_2 + \cdots + X_t}{t} \).
Given the function
\[
\phi^{0.5} (x, y) = \begin{cases}
0.5x & \text{if } x < y \\
x & \text{i... | automath | [
-0.014432022348046303,
0.025957776233553886,
-0.04792587831616402,
-0.03002886287868023,
0.030498381704092026,
-0.005998098757117987,
-0.030642928555607796,
0.08032836765050888,
0.03111804462969303,
-0.023517969995737076,
-0.0004354646953288466,
0.03578990325331688,
-0.04785074666142464,
0... | 0.817135 | 0.800312 | 0.016823 |
Consider the implications \( A: P \rightarrow Q \) and \( B: Q \rightarrow R \), where \( P, Q, R \) are propositions that can be either true (T) or false (F).
1. If \( P \) is false, what can be concluded about \( A \) and its truth value?
2. If \( Q \) is true, what can be concluded about the truth value of \( R \)... | automath | [
0.05142437666654587,
0.019984550774097443,
-0.051171641796827316,
-0.014386790804564953,
0.021595995873212814,
-0.034808795899152756,
-0.004189897794276476,
0.08221130073070526,
0.03195130079984665,
-0.0009018123964779079,
0.013262004591524601,
0.03193997964262962,
-0.029895693063735962,
-... | 0.813378 | 0.79513 | 0.018248 |
Find all integers \( n > 1 \) such that \( \frac{2^n + 1}{n^2} \) is an integer, and determine if \( n = 3 \) is the only such solution. | automath | [
0.035982538014650345,
0.02862493321299553,
-0.018590427935123444,
-0.03852568194270134,
0.048106323927640915,
-0.03189271315932274,
0.004014941863715649,
0.06882546842098236,
0.03238207474350929,
-0.018782148137688637,
0.021567799150943756,
0.0221237912774086,
-0.015692926943302155,
-0.037... | 0.848942 | 0.826471 | 0.022471 |
Let \( T: \mathbb{R}^n \rightarrow \mathbb{R}^n \) be a linear transformation represented by a matrix \( M \in \mathrm{Mat}_{n,n}(\mathbb{R}) \). If \( T \) is self-dual, what is the necessary condition that the entries of \( M \) must satisfy? Provide an example of a \( 2 \times 2 \) matrix that represents a self-dual... | automath | [
0.009183607995510101,
0.029146987944841385,
-0.04867109656333923,
-0.07067140191793442,
0.04981791600584984,
-0.008183404803276062,
-0.013238588348031044,
0.07219910621643066,
0.012278771959245205,
-0.003624335862696171,
-0.008869536221027374,
0.0052653285674750805,
-0.032220955938100815,
... | 0.813144 | 0.790782 | 0.022362 |
Given a topological space \( T \), let \( \mathcal{P}_{closed}(T) \) denote the space of closed subsets of \( T \) equipped with the topology generated by the subbasis consisting of sets of the form \( \{ F \in \mathcal{P}_{closed}(T) : F \cap S \neq \varnothing \} \) for open sets \( S \subseteq T \). If \( T \) is a ... | automath | [
0.02714022994041443,
0.044959232211112976,
-0.03869394212961197,
-0.01920725405216217,
0.04843601956963539,
-0.017698820680379868,
-0.010273772291839123,
0.06454939395189285,
0.03142119199037552,
-0.0028855232521891594,
0.019949378445744514,
0.030326073989272118,
-0.02565794624388218,
-0.0... | 0.792071 | 0.765631 | 0.02644 |
Let \( X \) be a compact metric space and let \( \{ f_n: X \to \mathbb{R} \} \) be a sequence of bounded continuous functions such that the product \( \prod_{n=1}^{\infty} f_n \) converges uniformly to a function \( f \). If \( f_n(x) \geq \frac{1}{2} \) for all \( n \) and \( x \in X \), what is the minimum value of \... | automath | [
0.00454127648845315,
0.04254256561398506,
-0.04874075576663017,
-0.0224420465528965,
0.03606879338622093,
0.007554336916655302,
-0.04431380704045296,
0.06678168475627899,
0.03645554929971695,
0.0011524263536557555,
-0.009615916758775711,
0.015231247060000896,
-0.01397839654237032,
-0.01452... | 0.827688 | 0.802933 | 0.024754 |
In a metric space \((M, \rho)\), consider a subset \(S \subseteq M\). If \(S\) is expressed as \(S = \bigcup_{\alpha \in A} U_\alpha\), where each \(U_\alpha\) is an open set, how can we construct a specific \(x_0 \in S\) for which there exists a radius \(r_0 > 0\) such that the open ball \(B(x_0, r_0) \subseteq S\)? W... | automath | [
0.01662287302315235,
0.03045508824288845,
-0.04593968018889427,
-0.01499549113214016,
0.028860557824373245,
0.0038765191566199064,
-0.031146595254540443,
0.05986449867486954,
0.05414494127035141,
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0.01939328759908676,
0.026174984872341156,
-0.02761394903063774,
-0.0245... | 0.818706 | 0.788779 | 0.029927 |
In rectangle \(ABCD\), where \(AB=12\) and \(BC=10\), points \(E\) and \(F\) lie such that \(BE=9\), \(DF=8\), \(\overline{BE} \parallel \overline{DF}\), \(\overline{EF} \parallel \overline{AB}\), and line \(BE\) intersects segment \(\overline{AD}\). If the length \(EF\) can be expressed in the form \(m\sqrt{n}-p\), wh... | automath | [
0.026140613481402397,
0.011012888513505459,
-0.04568895325064659,
-0.05683702230453491,
0.02409305050969124,
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0.05359333008527756,
0.05150629207491875,
-0.0031058229506015778,
0.014890773221850395,
0.05136815458536148,
-0.035549432039260864,
-0.... | 0.849175 | 0.830133 | 0.019042 |
Let \( R = M_2(A) \) be the algebra of \( 2 \times 2 \) matrices over an algebra \( A \), and consider the \( G \)-grading defined by the group \( G = \mathbb{Z}/2\mathbb{Z} \). If \( R_0 \) consists of matrices where the sum of the indices is even and \( R_1 \) consists of matrices where the sum of the indices is odd,... | automath | [
0.020121969282627106,
0.024855706840753555,
-0.03528638929128647,
-0.03968549519777298,
0.026212142780423164,
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0.05443650111556053,
0.02015962265431881,
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0.0056905061937868595,
-0.04784378409385681,
-0.... | 0.823856 | 0.805849 | 0.018008 |
Let \( 1 \rightarrow A \xrightarrow{a} B \xrightarrow{c} C \rightarrow 1 \) be a short exact sequence of topological groups where \( A \) and \( C \) are compact Hausdorff spaces, and \( a \) is a homeomorphism onto its image, with \( c \) a topological quotient map. If the groups are finite, what can be concluded abou... | automath | [
0.04018086567521095,
0.047541819512844086,
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0.03202306851744652,
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0.04919389262795448,
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0.013409746810793877,
0.009727125987410545,
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-0.02... | 0.796425 | 0.766672 | 0.029753 |
Given a sequence of points \( x_n \) in \( \mathbb{R}^2 \) defined by \( x_n = \frac{1}{n} e_1 \) if \( n \) is odd and \( x_n = \frac{1}{n} e_2 \) if \( n \) is even, is it possible to construct a differentiable path \( \gamma(t) \) such that \( \gamma(0) = 0 \) and \( \gamma(t_n) = x_n \) for a sequence \( t_n \) tha... | automath | [
-0.0062812939286231995,
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0.012293875217437744,
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... | 0.824291 | 0.799949 | 0.024342 |
Let \( M \) be a smooth manifold of dimension \( n \). Determine the necessary and sufficient conditions under which every \( k \)-form \( \omega \) on \( M \) can be expressed as a finite sum of wedge products of \( k \)-1-forms, specifically in the form \( \omega = \sum_{i=1}^m \alpha_1^{(i)} \wedge \alpha_2^{(i)} \w... | automath | [
0.013333836570382118,
0.04334261640906334,
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0.04077932611107826,
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0.0059584565460681915,
0.036497313529253006,
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-0... | 0.812736 | 0.783748 | 0.028987 |
Let \( X \) be a set with \( n = 100 \) elements, and let \( S_1, S_2, \ldots, S_{100} \) be subsets of \( X \) such that \( |S_i \cap S_j| \leq 1 \) for all \( 1 \leq i < j \leq 100 \). What is the maximum possible size of any subset \( S_i \), and does at least one such subset satisfy \( |S_i| \leq \sqrt{100} + 1 \)? | automath | [
0.023914940655231476,
0.014857690781354904,
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0.0180608369410038,
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0.028618013486266136,
0.03395296260714531,
-0.039183929562568665,
-0.013... | 0.836506 | 0.80332 | 0.033187 |
Given a right circular cone with a height of \( h = 120 \) cm and a radius of \( r = 60 \) cm, water is poured into the cone at a constant rate of \( x \) cm³/s.
1. What is the rate at which the water level rises when the height of the water is 60 cm, expressed in terms of \( x \)?
2. Calculate the rate of change of... | automath | [
-0.003448872361332178,
0.0610961951315403,
-0.046430282294750214,
-0.029224367812275887,
0.028584100306034088,
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0.01581171154975891,
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0.01341292355209589,
0.023522762581706047,
-0.028375277295708656,
-0... | 0.80461 | 0.788112 | 0.016498 |
Given the recurrence relation \( T(n) = aT(n/b) + f(n) \), where \( a = 1 \) and \( b = 2 \), explain how the regularity condition \( af(n/b) \leq cf(n) \) for some constant \( c < 1 \) and sufficiently large \( n \) impacts the application of Case 3 of the Master Theorem. If the regularity condition is not satisfied, ... | automath | [
0.016134880483150482,
0.038347452878952026,
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0.036954741925001144,
0.030844319611787796,
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0.009439739398658276,
0.011832017451524734,
0.010967995040118694,
0.012514752335846424,
-0.015680598095059395,
0.0... | 0.813199 | 0.796638 | 0.016561 |
Let \( a \) and \( b \) be real numbers such that \( |b| < |a| \), and let \( n \) be a positive integer. Using the generalized binomial theorem, find the first three non-zero terms of the expansion of \( (a+b)^{-n} \). | automath | [
0.03217653930187225,
0.0214528851211071,
-0.0343918576836586,
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0.030948350206017494,
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0.05257507786154747,
0.03278941661119461,
0.0007779087754897773,
0.012906518764793873,
0.031159257516264915,
-0.06666100025177002,
-0.035111... | 0.843206 | 0.82266 | 0.020547 |
Given two series \( \sum_{n=1}^{\infty} a_n \) and \( \sum_{n=1}^{\infty} b_n \) with positive terms such that \( \lim_{n \to \infty} \frac{a_n}{b_n} = L \), where \( L \) is a finite positive number. If \( \sum_{n=1}^{\infty} b_n \) converges, can you determine a constant \( C \) such that for sufficiently large \( n ... | automath | [
-0.011680304072797298,
0.041341569274663925,
-0.038831260055303574,
-0.023522334173321724,
0.023901576176285744,
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0.05505071580410004,
0.03935004398226738,
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-0.011116613633930683,
0.012145196087658405,
-0.03226589784026146,
0... | 0.836296 | 0.812325 | 0.023971 |
If the polynomial \( p_n(z) = (n+1)z^n + nz^{n-1} + nz + (n+1) \) has roots that all lie on the unit circle, what can you state about the roots of \( p_n(z) \) as \( n \) approaches infinity? Specifically, describe the behavior of the distribution of the roots in relation to the unit circle \( S^1 \) as \( n \to \infty... | automath | [
0.028841756284236908,
0.041109420359134674,
-0.03952905908226967,
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0.021015621721744537,
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0.0006383122527040541,
0.02833310328423977,
0.012707540765404701,
0.005845564417541027,
0.024098249152302742,
0.02934606559574604,
-0.02529218979179859,
-0.0... | 0.822585 | 0.80326 | 0.019325 |
Evaluate the integral
\[
\int_0^{\pi/2} \frac{1}{1+\tan^\alpha{x}}\,\mathrm{d}x
\]
for all positive integers \(\alpha\) and verify its value. | automath | [
0.01946917362511158,
0.04105573520064354,
-0.03485821560025215,
-0.0159946046769619,
0.027392785996198654,
-0.0152335399761796,
-0.009527657181024551,
0.03959793224930763,
0.062299780547618866,
-0.002254227176308632,
0.009006950072944164,
0.009949566796422005,
-0.02550920471549034,
-0.0213... | 0.820287 | 0.803763 | 0.016524 |
Let $S$ be a scheme, and consider an algebraic space $X$ over $S$. Let $\mathcal{I} \subset \mathcal{O}_X$ and $\mathcal{J}$ be quasi-coherent sheaves of ideals on $X$. If the blowing up of $X$ in $\mathcal{I}$ is denoted by $b: X' \to X$, and the blowing up of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$ is denoted by... | automath | [
0.033833060413599014,
0.03615696728229523,
-0.04024877771735191,
-0.026057062670588493,
0.019346771761775017,
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-0.01569376327097416,
0.08958762884140015,
0.035447362810373306,
-0.01846349984407425,
0.019635071977972984,
0.01936136931180954,
-0.02971913479268551,
-0.001... | 0.81335 | 0.791932 | 0.021419 |
Given the orthogonality property of Legendre polynomials, specifically the relation
\[
\int_{-1}^1 P_n(x) P_m(x) \, dx = 0 \quad \text{for } n \neq m,
\]
find the value of the integral
\[
\int_{-1}^1 (P_n(x))^2 \, dx
\]
in terms of \( n \) and verify that it equals
\[
\frac{2}{2n+1}.
\] | automath | [
0.020732713863253593,
0.03769531846046448,
-0.037582311779260635,
-0.04898122698068619,
0.03569791093468666,
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0.0015361669939011335,
0.05285182595252991,
0.036462076008319855,
-0.008153640665113926,
-0.002155713038519025,
0.024646200239658356,
-0.015524688176810741,
-... | 0.826479 | 0.805979 | 0.0205 |
Let \( R \) be a Dedekind domain, and consider invertible ideals \( I \) in \( R \). If \( I \) can be factored into prime ideals in two distinct ways as follows:
\[
I = \mathfrak{p}_1 \mathfrak{p}_2 \cdots \mathfrak{p}_m = \mathfrak{q}_1 \mathfrak{q}_2 \cdots \mathfrak{q}_n,
\]
where \( \mathfrak{p}_i \) and \( \mat... | automath | [
0.05261286348104477,
0.03493192046880722,
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0.020231636241078377,
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0.05299585685133934,
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0.0016787465428933501,
0.011614102870225906,
-0.036032337695360184,
-... | 0.822545 | 0.80007 | 0.022475 |
Given two convex polyhedra \( P \) and \( Q \) represented by their lists of vertices, edges, and faces, is there a polynomial-time algorithm to determine if \( P \) can be placed inside \( Q \) using only translations and rotations? If so, provide a brief description of the algorithm; if not, explain why such an algor... | automath | [
0.012447617016732693,
0.028373073786497116,
-0.031241390854120255,
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0.02142501249909401,
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0.07206703722476959,
0.021320726722478867,
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0.019417298957705498,
0.03562239184975624,
-0.018112052232027054,
-0... | 0.818216 | 0.794462 | 0.023755 |
Given two vectors \( \mathbf{v_1} = (x_1, y_1) \) and \( \mathbf{v_2} = (x_2, y_2) \) in 2D, derive a formula for the area of the triangle formed by these two vectors when placed with a common origin. Then, extend this to vectors \( \mathbf{V_1} = (x_1, y_1, z_1) \) and \( \mathbf{V_2} = (x_2, y_2, z_2) \) in 3D, and p... | automath | [
0.03354864940047264,
0.03897418454289436,
-0.045165177434682846,
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0.036720097064971924,
0.0074214572086930275,
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0.06406359374523163,
0.03376022353768349,
-0.009535522200167179,
0.01663467474281788,
0.02292039804160595,
-0.015348962508141994,
-0.02... | 0.803094 | 0.783452 | 0.019642 |
Given a smooth, continuous function \( f: \mathbb{R} \to \mathbb{R} \) that satisfies the conditions \( f(0) = 0 \), \( f(x) > 0 \) for all \( x > 0 \), and passes through the point \( (x_1, y_1) \) for some \( x_1 > 0 \), determine if it is possible to explicitly construct such a function in closed form. If so, provid... | automath | [
0.01849539764225483,
0.051271919161081314,
-0.038204070180654526,
-0.03720609471201897,
0.03765181452035904,
0.006222967989742756,
-0.021298019215464592,
0.09030426293611526,
0.03000638633966446,
-0.00862875021994114,
0.008151660673320293,
0.021225517615675926,
-0.028590073809027672,
-0.01... | 0.820383 | 0.795504 | 0.02488 |
Let \( Y_1, Y_2, \ldots, Y_n \) be a sequence of i.i.d. random variables with finite mean \( \mu \) and variance \( \sigma^2 \). According to the Weak Law of Large Numbers (WLLN), as \( n \to \infty \), the sample average \( \overline{Y}_n = \frac{1}{n}\sum_{i=1}^n Y_i \) converges in probability to \( \mu \).
1. If ... | automath | [
-0.011684590019285679,
0.041613638401031494,
-0.031230749562382698,
-0.04434293508529663,
0.022959982976317406,
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0.06318728625774384,
0.006400801241397858,
-0.005398154724389315,
-0.00946826208382845,
0.02140211872756481,
-0.03269467130303383,
0... | 0.811002 | 0.794102 | 0.0169 |
Let \( U, V, W \) be finite dimensional real vector spaces. Consider the linear transformations \( T: U \rightarrow V \), \( S: V \rightarrow W \), and \( P: W \rightarrow U \) such that:
1. \( \text{range}(ST) = \text{nullspace}(P) \),
2. \( \text{nullspace}(ST) = \text{range}(P) \),
3. \( \text{rank}(T) = \text{rank... | automath | [
0.021401725709438324,
0.04582632705569267,
-0.03684190660715103,
-0.05752403661608696,
0.05500291660428047,
-0.002123239217326045,
0.0021601077169179916,
0.06282179057598114,
0.030273133888840675,
-0.019130170345306396,
0.0026298740413039923,
0.0027258661575615406,
-0.025097444653511047,
-... | 0.819682 | 0.798233 | 0.021449 |
Suppose \( x \in \mathbb{N} \) is fixed. For which pairs of natural numbers \( (y, z) \) do the following conditions hold:
1. \( x^2 = y^2 + z \)
2. \( y \neq x \)
Identify the smallest natural number \( x \) such that no pairs \( (y, z) \) satisfy both conditions simultaneously, and explain your reasoning. | automath | [
0.033665724098682404,
0.032438021153211594,
-0.038584597408771515,
-0.03177584335207939,
0.04395661503076553,
-0.0120299207046628,
0.009530828334391117,
0.04545479640364647,
0.03517014905810356,
-0.015256835147738457,
0.017020074650645256,
0.005807161796838045,
-0.041810933500528336,
-0.03... | 0.846024 | 0.825476 | 0.020548 |
Consider the inequality \( \frac{n-\alpha}{n-\beta} \geq \left(\frac{\beta}{\alpha}\right)^{\delta} \) where \( n, \alpha, \beta \) are integers such that \( n > \beta + \alpha \), \( \beta > \alpha \), and \( \alpha > 0 \). For what smallest real number \( \delta \) does this inequality hold for all integers \( n \), ... | automath | [
0.025268564000725746,
0.04422004520893097,
-0.036064762622117996,
-0.026805808767676353,
0.02677828073501587,
0.0050166575238108635,
-0.02841031365096569,
0.07869483530521393,
0.04903681203722954,
-0.03247227892279625,
-0.009684179909527302,
0.03678591549396515,
-0.042109787464141846,
-0.0... | 0.840728 | 0.81802 | 0.022708 |
Given the formula $$\mathscr{R}(x) = \forall y[\overline{\mathrm{Neg}}(x,y)\to\forall z(\overline{\mathrm{Prov}}(z,y)\to\exists w< z\overline{\mathrm{Prov}}(w,x)],$$ describe the implications of proving the sentence $$\psi$$ such that $$\vdash\psi\leftrightarrow\neg\mathscr{R}(\overline{g(\psi)})$$ in the context of th... | automath | [
0.02499566040933132,
0.027549244463443756,
-0.033404260873794556,
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0.006036440376192331,
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-0.005787919275462627,
0.06767187267541885,
0.028095006942749023,
-0.008753259666264057,
0.0032480179797858,
0.023588240146636963,
-0.012107999995350838,
-0... | 0.82186 | 0.795489 | 0.026371 |
In a parallelogram \(ABCD\), let angles \(A = a^\circ\), \(B = b^\circ\), and let \(C = x^\circ\). Given that the angles of a parallelogram satisfy the property that opposite angles are equal and adjacent angles sum to \(180^\circ\), can the angle \(x\) be expressed solely as an algebraic combination of \(a\) and \(b\)... | automath | [
0.024053897708654404,
0.029344985261559486,
-0.030795445665717125,
-0.04295557364821434,
0.015144658274948597,
-0.004550818353891373,
-0.015044229105114937,
0.059330251067876816,
0.046912867575883865,
-0.001165695139206946,
-0.011015593074262142,
0.016992028802633286,
-0.03186081349849701,
... | 0.830793 | 0.805629 | 0.025165 |
Let \( X \) and \( Y \) be topological spaces, and let \( A \subset X \) and \( B \subset Y \) be non-empty subsets. Define a function \( f: A \times B \to X \times Y \) by \( f(a, b) = (a, b) \). If \( (A, \mathcal{T}_A) \) is the subspace topology on \( A \) and \( (B, \mathcal{T}_B) \) is the subspace topology on \(... | automath | [
0.03849896788597107,
0.02825973369181156,
-0.03586890175938606,
-0.019463540986180305,
0.03111320547759533,
0.01120883971452713,
-0.012602564878761768,
0.06755570322275162,
0.035434044897556305,
-0.00035901073715649545,
0.010373099707067013,
0.030334515497088432,
-0.03008894808590412,
-0.0... | 0.810992 | 0.781615 | 0.029377 |
Suppose a two-stage robust optimization problem minimizes the total cost given by \( c^T x_1 + d^T x_2 \) subject to the constraints \( A_1(\xi) + A_2(\xi)x_2 \geq b \) for all \( \xi \in \Xi \), with \( x_1 \) and \( x_2 \) subject to various constraints, including potential integrality constraints. If the uncertainty... | automath | [
0.006150545086711645,
0.043625637888908386,
-0.04335038736462593,
-0.022737134248018265,
0.02808297611773014,
0.00570213096216321,
-0.02385191060602665,
0.07309003919363022,
0.03020252101123333,
-0.011747980490326881,
0.0012455555843189359,
-0.0005075819208286703,
-0.017321467399597168,
0.... | 0.801715 | 0.783527 | 0.018187 |
Let \( E \) be a non-empty subset of \( \mathbb{R} \) with the property that the set of isolated points of \( E \) is at most countable. Let \( f: E \to \mathbb{R} \) be a monotonic function. Is the function \( f \) differentiable at almost every point of \( E \)? If so, what is the measure of the set of points in \( E... | automath | [
0.033522579818964005,
0.021249840036034584,
-0.029156824573874474,
-0.02958500199019909,
0.01844179444015026,
-0.03356877341866493,
-0.02099085971713066,
0.051458388566970825,
0.0370972640812397,
-0.0005200220621190965,
0.00034096158924512565,
0.039178960025310516,
-0.04651718959212303,
0.... | 0.816929 | 0.791761 | 0.025168 |
Given the sequence of integers \( a_1, a_2, \ldots, a_n \) such that each \( a_i \) is between \( 1 \) and \( 3 \cdot 10^{18} \), determine a valid rearrangement of the sequence such that each number in the rearrangement is either double the previous number or one-third of the previous number, starting from any number ... | automath | [
0.038002338260412216,
0.03644721955060959,
-0.012111743912100792,
-0.0174668300896883,
0.021217962726950645,
0.0010451339185237885,
-0.02465033158659935,
0.07572917640209198,
0.027594026178121567,
-0.01540721207857132,
0.014445946551859379,
0.00828306470066309,
-0.046034205704927444,
-0.02... | 0.835293 | 0.817641 | 0.017652 |
Let \( x = (x_1, x_2) \) and \( y = (y_1, y_2) \), and consider the inner product defined by
\[
\langle x, y \rangle = 3x_1y_1 - x_1y_2 - x_2y_1 + 3x_2y_2.
\]
Determine whether this defines an inner product by verifying the properties of inner products. In particular, check if the following four conditions hold:
1.... | automath | [
0.029878055676817894,
0.04333513230085373,
-0.03863045200705528,
-0.027322618290781975,
0.039115261286497116,
-0.015438626520335674,
0.0012309877201914787,
0.08247373253107071,
0.020961161702871323,
-0.004620436113327742,
0.0029347133822739124,
0.03420541062951088,
-0.03318221867084503,
-0... | 0.822337 | 0.79818 | 0.024157 |
Given a finite set \( A \) of size \( n \) and a positive integer \( k \), determine the number of distinct \( k \)-combinations that can be formed from \( A \). Then, describe a way to efficiently assign a unique natural number to each \( k \)-combination, ensuring the assignment is injective and follows a specified o... | automath | [
0.04180005192756653,
0.034303054213523865,
-0.033393148332834244,
-0.013689297251403332,
0.030935212969779968,
-0.02741279825568199,
-0.015003747306764126,
0.03786741942167282,
0.026399899274110794,
-0.008246410638093948,
0.01645207405090332,
0.03130226582288742,
-0.03147750720381737,
0.00... | 0.826361 | 0.808859 | 0.017501 |
Let \( S_i \) be a topological space with \( |S_i| = n_i \) for each \( i \in I \), where \( I \) is a finite index set with \( |I| = k \). If \( \tau_i \) is the topology on \( S_i \) and we define the synthetic basis \( \mathcal B \) for the product space \( S = \prod_{i \in I} S_i \), what is the maximum number of o... | automath | [
0.04736023396253586,
0.022038035094738007,
-0.053664375096559525,
-0.013593269512057304,
0.029005220159888268,
-0.023386262357234955,
-0.0412222258746624,
0.05901021882891655,
0.037816938012838364,
-0.005863475147634745,
0.01643804833292961,
0.03500344231724739,
-0.012831238098442554,
-0.0... | 0.822677 | 0.801247 | 0.02143 |
Find a sequence of functions \( f_n: [0, \infty) \to \mathbb{R} \) that converges uniformly to a function \( f \), such that \( f_n^2 \) does not converge uniformly to \( f^2 \). Provide the explicit forms of \( f_n \) and \( f \), and justify your answer by showing that for any given \( \epsilon > 0 \), there exists a... | automath | [
0.0034575334284454584,
0.045977555215358734,
-0.03584161773324013,
-0.04927567392587662,
0.023780744522809982,
-0.0017592513468116522,
-0.03805295005440712,
0.07468617707490921,
0.010355604812502861,
0.01001480408012867,
0.014127403497695923,
0.007379158399999142,
-0.017913440242409706,
-0... | 0.828564 | 0.800911 | 0.027652 |
Let \( A \subseteq E \subseteq B \subset \mathbb{R}^d \) where \( A \) and \( B \) are measurable sets with \( m(A) = m(B) < \infty \). If it is shown that \( m^*(B \backslash E) = 0 \) and for every open set \( O \supseteq E \) that contains \( B \), \( m^*(O \backslash E) < \varepsilon \) for any \( \varepsilon > 0 \... | automath | [
0.01850414089858532,
0.026441851630806923,
-0.023883212357759476,
-0.04011102765798569,
0.036242783069610596,
-0.008024693466722965,
-0.014069324359297752,
0.059985484927892685,
0.0724438950419426,
-0.00016282785509247333,
0.010788955725729465,
0.02467728964984417,
-0.028993990272283554,
-... | 0.81602 | 0.78364 | 0.03238 |
Consider an ordered set $\Phi$ consisting of all pairs of integers $(r, s)$ such that $1 \leq r < s \leq k$. Define the size of $\Phi$ as $n = \binom{k}{2}$. If you create an $n \times n$ matrix $M[n]$ where the entry at the $(i, j)$-th position is given by a function $f(i,j)$ with $i$ and $j$ taken from the ordered se... | automath | [
0.037981126457452774,
0.041699301451444626,
-0.050842203199863434,
-0.026427924633026123,
0.026348024606704712,
-0.02507578767836094,
-0.014397405087947845,
0.03944728896021843,
0.030081667006015778,
0.009371155872941017,
0.006396738346666098,
0.025225427001714706,
-0.038537971675395966,
-... | 0.834201 | 0.812451 | 0.02175 |
Do the sets \( A = \{ x \in \mathbb{R} : 2 \leq x \leq 3 \} \) and \( B = \{ x \in \mathbb{R} : 2 \leq x \leq 5 \} \) have the same cardinality? If so, demonstrate a bijection between the two sets. If not, explain why they do not share the same cardinality. | automath | [
0.01393867377191782,
0.02119438163936138,
-0.03499704971909523,
-0.02302774041891098,
0.02656969428062439,
-0.00534440390765667,
-0.013972840271890163,
0.06245541200041771,
0.02544384077191353,
-0.009145190007984638,
-0.0048807477578520775,
0.019256936386227608,
-0.028599750250577927,
-0.0... | 0.829778 | 0.805154 | 0.024624 |
In the context of paired experiments versus unpaired random sampling, if we assume a scenario where treatment outcomes $Y^T$ and $Y^C$ are uncorrelated across the entire sample, demonstrate mathematically whether the variance of the estimator $\mathcal{E}_2$ (derived from paired samples) will always be less than, equal... | automath | [
-0.003976058680564165,
0.026127343997359276,
-0.024946879595518112,
-0.04487098008394241,
0.036187801510095596,
-0.022134458646178246,
-0.013549325056374073,
0.08346863090991974,
0.03827402740716934,
-0.020861009135842323,
0.013659353367984295,
0.028052810579538345,
-0.024149000644683838,
... | 0.809461 | 0.791148 | 0.018313 |
What is the asymptotic behavior of the product
\[
\prod_{k=1}^n \frac{k^\alpha}{\lambda + k^\alpha}
\]
as \( n \to \infty \), specifically proving that for \( \lambda > 0 \) and \( 0 < \alpha < 1 \),
\[
\prod_{k=1}^n \frac{k^\alpha}{\lambda + k^\alpha} \sim \exp\left(-\frac{\lambda}{1-\alpha}n^{1-\alpha}+o(n^{1-\a... | automath | [
0.006910128053277731,
0.039917025715112686,
-0.02567255310714245,
-0.034706518054008484,
0.027499500662088394,
-0.014509500935673714,
-0.02736291103065014,
0.07738868147134781,
0.02129286341369152,
0.00159380619879812,
-0.009397448040544987,
0.01793207973241806,
-0.04509197548031807,
-0.02... | 0.823549 | 0.806711 | 0.016838 |
Let \( R \) be a commutative ring, and let \( X \) and \( Y \) be schemes over \( R \) such that \( X \) is quasi-compact with affine diagonal. Given an \( R \)-linear exact functor \( F : \mathit{QCoh}(\mathcal{O}_X) \to \mathit{QCoh}(\mathcal{O}_Y) \) that commutes with arbitrary direct sums, what properties must the... | automath | [
0.009961344301700592,
0.05358599126338959,
-0.04090424254536629,
-0.02312692627310753,
0.024799028411507607,
-0.016385670751333237,
-0.01704864390194416,
0.07312792539596558,
0.05260663479566574,
-0.00285930372774601,
0.006758387200534344,
0.011092671193182468,
-0.021181825548410416,
0.012... | 0.801024 | 0.78039 | 0.020634 |
Given a model \( V \) of ZF(C), under what conditions can we assert that \( V \) includes the class of all sets, and in what ways does this relate to the properties and definitions of inner models? Furthermore, can you provide a quantitative example of an inner model that demonstrates distinct properties from its class... | automath | [
0.026059728115797043,
0.017117321491241455,
-0.014913869090378284,
-0.025874046608805656,
0.03315858170390129,
-0.031140079721808434,
-0.0001512733579147607,
0.036219749599695206,
0.024488231167197227,
-0.026915570721030235,
0.008299445733428001,
0.03658410534262657,
-0.03985109552741051,
... | 0.805732 | 0.789032 | 0.016701 |
Given a function \( f: \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^2 \), determine if \( f \) is continuous everywhere on \( \mathbb{R} \) according to the continuity criterion stated in the theorem, and provide a specific \(\epsilon\)-neighborhood around \( f(2) \) for a chosen \(\epsilon = 0.5\). What is the ... | automath | [
0.01455457229167223,
0.028319967910647392,
-0.033307261765003204,
-0.03513606637716293,
0.03026672638952732,
-0.016498951241374016,
-0.032813891768455505,
0.06961959600448608,
0.018416935577988625,
-0.0026033646427094936,
-0.013688981533050537,
0.02475849539041519,
-0.018851691856980324,
-... | 0.821347 | 0.798683 | 0.022664 |
Let \( V \) be a vector space over the real numbers, and let \( v_1 = (1, 2) \), \( v_2 = (2, 4) \), and \( v_3 = (3, 6) \). Determine whether the vectors \( v_1, v_2, \) and \( v_3 \) are linearly independent, and if not, express one of the vectors as a linear combination of the others. | automath | [
0.025584189221262932,
0.03940245881676674,
-0.02794226072728634,
-0.04118288680911064,
0.06571616977453232,
-0.017292983829975128,
-0.005300824996083975,
0.06642571091651917,
0.03399321809411049,
-0.01019993331283331,
0.005906819365918636,
0.021829135715961456,
-0.020564880222082138,
-0.03... | 0.823254 | 0.805169 | 0.018084 |
Let \( p \) and \( q \) be two distinct points in the Poincaré disk \( D \) given by their complex coordinates \( z_p \) and \( z_q \). Using the Poincaré metric, determine the explicit equation of the geodesic connecting the points \( p \) and \( q \) within the disk, expressed in terms of a parameter \( t \). Additio... | automath | [
0.013653344474732876,
0.03219098225235939,
-0.024665415287017822,
-0.04370466619729996,
0.007681585382670164,
0.0004491946310736239,
-0.020992811769247055,
0.04119163751602173,
0.05490695685148239,
0.0015781631227582693,
0.001683756010606885,
0.03216753527522087,
-0.016865713521838188,
-0.... | 0.811185 | 0.788814 | 0.022371 |
Is there a bi-invariant metric on $GL_n(\mathbb{R})$ that generates the standard topology from the subspace topology inherited from $\mathbb{R}^{n^2}$? If such a metric exists, provide a specific example or demonstrate a contradiction to its existence. | automath | [
-0.003911369014531374,
0.011853919364511967,
-0.04947876185178757,
-0.03978068754076958,
0.015244657173752785,
-0.008545157499611378,
-0.01554352231323719,
0.046876877546310425,
0.030547862872481346,
-0.007583282887935638,
0.005880267359316349,
0.033156007528305054,
-0.03799264878034592,
-... | 0.811903 | 0.794999 | 0.016904 |
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