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 equation $$\varphi(\varphi(a))=\varphi(\varphi(b))$$ where $\varphi$ is the Euler totient function and $\varphi(a) \neq \varphi(b)$. If $a = 2^{k-1}3^2$ and $b = 2^{k+1}$ for $k \geq 2$, compute the explicit values of $\phi(\phi(a))$ and $\phi(\phi(b))$ for $k = 2$ and $k = 3$, and verify if they are equal... | automath | [
0.04600238427519798,
0.045935045927762985,
-0.04080057889223099,
-0.04755646735429764,
0.025734473019838333,
-0.013690875843167305,
-0.0049880025908350945,
0.07500509917736053,
0.02116716280579567,
-0.01257823035120964,
0.020063847303390503,
0.02592398412525654,
-0.02182025834918022,
-0.02... | 0.856571 | 0.826255 | 0.030316 |
Let \(X_n\) be a sequence of random variables defined by
\[
P(X_n = 0) = 1 - \left(\frac{1}{2}\right)^n, \quad P(X_n = 1) = \left(\frac{1}{2}\right)^n \text{ for } n = 1, 2, 3, \ldots
\]
What is the limiting behavior of \( E(X_n) \) as \( n \) approaches infinity, and does \( E(X_n) \) converge almost surely and in ... | automath | [
-0.010324639268219471,
0.041191574186086655,
-0.04050051420927048,
-0.020398421213030815,
0.005423265974968672,
-0.008236466906964779,
-0.01498547475785017,
0.06462205946445465,
0.02227318473160267,
-0.011651325970888138,
-0.0108101861551404,
0.02998296171426773,
-0.03633501008152962,
-0.0... | 0.818617 | 0.798521 | 0.020095 |
Let \( A \) be an \( n \times n \) real matrix such that for all vectors \( v \) satisfying \( 1'v = 0 \), the condition \( v'Av \geq 0 \) holds. What is the minimum dimension \( n \) such that it is possible for such a matrix \( A \) to have a rank of \( n-1 \) where its first row and column contain arbitrary real num... | automath | [
0.022409703582525253,
0.020671017467975616,
-0.03584643825888634,
-0.038664452731609344,
0.04170046001672745,
-0.004301147535443306,
-0.010718461126089096,
0.06366799026727676,
0.032515980303287506,
-0.015284513123333454,
-0.020330224186182022,
0.016067199409008026,
-0.02923295646905899,
-... | 0.82793 | 0.798664 | 0.029266 |
Let \( \mathfrak{g} \) be a Lie algebra over \( \mathbb{C} \) and \( \mathfrak{h} \) be an ideal of \( \mathfrak{g} \). Consider the canonical homomorphism \( \pi: \mathfrak{g} \to \mathfrak{g/h} \) defined by \( \pi(x) = x + \mathfrak{h} \) for all \( x \in \mathfrak{g} \).
Compute the image under \( \pi \) of the e... | automath | [
0.04395950213074684,
0.04787546396255493,
-0.02836165577173233,
-0.03773186355829239,
0.04793958365917206,
-0.03162115439772606,
-0.004084398038685322,
0.07180482894182205,
0.05665863677859306,
-0.00027099475846625865,
0.0037574509624391794,
0.015501110814511776,
-0.03465479984879494,
-0.0... | 0.813672 | 0.79073 | 0.022942 |
Given the Cauchy-Riemann equations:
1. \( \frac{\partial u}{\partial x}(x_0, y_0) = \frac{\partial v}{\partial y}(x_0, y_0) \)
2. \( \frac{\partial u}{\partial y}(x_0, y_0) = -\frac{\partial v}{\partial x}(x_0, y_0) \)
determine the conditions under which the functions \( u(x, y) \) and \( v(x, y) \) are differentiab... | automath | [
0.01538369245827198,
0.051970358937978745,
-0.029115652665495872,
-0.038507502526044846,
0.043380022048950195,
-0.000712998618837446,
-0.0005471101612783968,
0.06796492636203766,
0.03319954872131348,
-0.00013314954412635416,
0.004766198806464672,
0.02015502378344536,
-0.03674563765525818,
... | 0.807541 | 0.78829 | 0.019251 |
Given that the divergence of a vector field \(\mathbf{A}\) is zero if and only if \(\mathbf{A}\) can be expressed as the curl of another vector field \(\mathbf{B}\), and that the curl of \(\mathbf{A}\) is zero if and only if \(\mathbf{A}\) can be expressed as the gradient of a scalar function \(\varphi\), what are the ... | automath | [
-0.007882325910031796,
0.021911010146141052,
-0.020352540537714958,
-0.04700283333659172,
0.04100365936756134,
0.0020774155855178833,
-0.040151942521333694,
0.06428495794534683,
0.013308471068739891,
-0.00493610929697752,
-0.0081244595348835,
0.003116789273917675,
-0.027653077617287636,
-0... | 0.814834 | 0.797854 | 0.016979 |
Let \( k \) be a field and \( \mathcal{F} \) be a coherent sheaf on \( \mathbf{P}^n_k \) with a Hilbert polynomial \( P \in \mathbf{Q}[t] \). If \( n \) is fixed and \( P(d) = \dim_k H^0(\mathbf{P}^n_k, \mathcal{F}(d)) \) holds for all \( d \gg 0 \), what is the value of \( P(0) \) if \( P(t) \) is defined as \( P(t) =... | automath | [
0.023889444768428802,
0.03893152251839638,
-0.04631821811199188,
-0.027308322489261627,
0.030852042138576508,
-0.025829458609223366,
-0.0011154067469760776,
0.06795307993888855,
0.0420355387032032,
-0.003554154420271516,
-0.012260040268301964,
0.03248239681124687,
-0.017825359478592873,
0.... | 0.82203 | 0.803171 | 0.018859 |
Given the function \( f : \mathbb{N} \times \mathbb{N} \to \mathbb{N} \) defined by a bijection that maps pairs of natural numbers to natural numbers, determine the number of distinct elements in the image of \( f \) if \( f(a, b) = a + b \). What is the maximum value of \( f(a, b) \) for \( 1 \leq a, b \leq n \)? | automath | [
0.049145475029945374,
0.019371356815099716,
-0.04412835091352463,
-0.019130639731884003,
0.036005742847919464,
-0.018362177535891533,
-0.013645408675074577,
0.06294510513544083,
0.028252525255084038,
0.01654122583568096,
0.005801638122648001,
0.02922958880662918,
-0.03801942989230156,
-0.0... | 0.83178 | 0.814315 | 0.017465 |
Given the function \( f(x) = x \) for \( 0 \leq x \leq 2\pi \) with periodic extension, its Fourier series is given by
\[
\sum_{n \neq 0, n \in \mathbb{Z}} -\frac{1}{in} e^{inx}.
\]
If we differentiate this series termwise, we obtain
\[
-\sum_{n \neq 0} e^{inx},
\]
which diverges. However, if we use the manipulat... | automath | [
-0.001890435814857483,
0.05568404495716095,
-0.034146204590797424,
-0.02959735319018364,
0.01897021196782589,
-0.026044046506285667,
0.0005295185837894678,
0.06224537268280983,
0.017867712303996086,
-0.0013517100596800447,
0.00870010256767273,
0.02781848981976509,
-0.032885413616895676,
-0... | 0.823232 | 0.801432 | 0.0218 |
Let vectors \(\vec{v}\), \(\vec{w}\), and \(\vec{u}\) be in \(\mathbb{R}^n\). If \(\vec{v}\) is orthogonal to both \(\vec{w}\) and \(\vec{u}\), calculate the inner product \(\vec{v} \cdot (k\vec{w} + m\vec{u})\) for any scalars \(k\) and \(m\). What can be concluded about the orthogonality of \(\vec{v}\) with the vecto... | automath | [
0.025601213797926903,
0.0565556138753891,
-0.045053135603666306,
-0.04198604077100754,
0.05196850001811981,
0.013414821587502956,
-0.007651824504137039,
0.07505422830581665,
0.026817061007022858,
-0.015257960185408592,
-0.014648093841969967,
0.026284605264663696,
-0.029406290501356125,
-0.... | 0.814686 | 0.796262 | 0.018423 |
Let \( X_n \) be independent random variables defined as follows: \( X_1 = 1 \), and for \( n \geq 2 \),
\[
P(X_n=n) = n^{-2}, \quad P(X_n=1) = P(X_n=0) = \frac{1}{2}(1-n^{-2}).
\]
Calculate the limiting distribution of
\[
\frac{1}{\sqrt{n}}\left(\sum_{m=1}^{n} X_m - \frac{n}{2}\right)
\]
as \( n \to \infty \). Wha... | automath | [
-0.017341500148177147,
0.06304175406694412,
-0.04694619029760361,
-0.03832247853279114,
0.02248396724462509,
-0.0005178119172342122,
-0.02319139614701271,
0.057464130222797394,
0.030330820009112358,
-0.004401772748678923,
-0.00927392765879631,
0.021890724077820778,
-0.03163805976510048,
-0... | 0.827191 | 0.807283 | 0.019908 |
Let \( A = \{ a, b, c, d \} \) be a partially ordered set with the following order relation: \( a < b \), \( a < c \), \( b < d \), \( c < d \). Determine the supremum (join) and infimum (meet) of the subset \( S = \{ a, b \} \). Also, specify whether the partially ordered set \( A \) is a join-semilattice, a meet-semi... | automath | [
0.030872369185090065,
0.0247823316603899,
-0.042459890246391296,
-0.011515658348798752,
0.010486900806427002,
0.0031016080174595118,
-0.028370851650834084,
0.045236580073833466,
0.043230053037405014,
0.01985183358192444,
0.019539207220077515,
0.01742159202694893,
-0.029178911820054054,
-0.... | 0.817823 | 0.796485 | 0.021338 |
Is the function \( d(\Gamma_1, \Gamma_2) \) defined as the revenue of the first player, where both players employ optimal strategies in a game involving simple graphs with countably many vertices, a metric on the set of isomorphism classes of these graphs, specifically fulfilling the triangle inequality \( d(\Gamma_1, ... | automath | [
0.0005915078800171614,
0.016930662095546722,
-0.04892341420054436,
-0.017185643315315247,
0.020867088809609413,
0.00037966875243000686,
-0.02116980031132698,
0.0563022755086422,
0.04282057657837868,
-0.029755914583802223,
0.019899191334843636,
0.02561865746974945,
-0.009214803576469421,
-0... | 0.817382 | 0.792719 | 0.024663 |
Let \( X \) be a compact metric space and \( Y \) be a separable metric space. Define the space of continuous functions \( \mathcal{C}(X,Y) \) equipped with the compact-open topology. How can you construct a dense countable subset of \( \mathcal{C}(X,Y) \) given a dense countable subset of \( Y \)? Additionally, explai... | automath | [
0.02602389082312584,
0.04610762372612953,
-0.039636898785829544,
-0.019670579582452774,
0.034320104867219925,
-0.01693109981715679,
-0.03325788676738739,
0.0587017685174942,
0.053680866956710815,
-0.006477852817624807,
0.011601349338889122,
0.02105870470404625,
-0.0363098680973053,
-0.0027... | 0.813739 | 0.787579 | 0.02616 |
Let \( E \subset \mathbb{R}^n \) be the set defined by \( E = (0, 1) \cup \{2\} \). Determine which of the points \( 0, 1, 0.5, \) and \( 2 \) are limit points, isolated points, and interior points of the set \( E \). Provide a justification for your classification of each point. | automath | [
0.03005676157772541,
0.022109778597950935,
-0.016299329698085785,
-0.02955184504389763,
0.020846107974648476,
-0.04064410179853439,
-0.014348960481584072,
0.03971704840660095,
0.039591189473867416,
0.0017517829546704888,
0.008838269859552383,
0.03842610493302345,
-0.042475879192352295,
-0.... | 0.827832 | 0.80771 | 0.020122 |
Consider a complex-valued function \( f : \mathbb{C} \times \mathbb{R} \to \mathbb{C} \) such that \( f(z,t) = f^R(z,t) + i f^I(z,t) \), where \( f^R, f^I : \mathbb{C} \times \mathbb{R} \to \mathbb{R} \). Given that the partial derivative of \( f \) with respect to \( t \) is expressed as \( \frac{\partial f(z,t)}{\par... | automath | [
0.024682335555553436,
0.05570721998810768,
-0.03350580111145973,
-0.031671855598688126,
0.03130237013101578,
0.006808794103562832,
-0.01282345037907362,
0.06296402961015701,
0.02890140563249588,
0.0007171593606472015,
0.012657242827117443,
0.015092129819095135,
-0.05254579335451126,
0.0064... | 0.812156 | 0.791003 | 0.021153 |
Consider the overdetermined system of partial differential equations given by:
\[
\begin{align*}
u_{xx} &= y^2, \\
u_{yy} &= x^2.
\end{align*}
\]
Determine the general form of the solution \( u(x,y) \) that satisfies both equations, and state whether a solution exists. If so, express \( u(x,y) \) explicitly. If not, ... | automath | [
-0.016269052401185036,
0.013473416678607464,
-0.0458877757191658,
-0.03207121044397354,
0.03970978036522865,
-0.00955051276832819,
-0.004349976312369108,
0.08906159549951553,
0.034690529108047485,
-0.010523557662963867,
0.02275034226477146,
0.04170531779527664,
-0.026633603498339653,
-0.02... | 0.824457 | 0.800563 | 0.023893 |
Given a vector space \( \mathbb{R}^n \) equipped with the standard inner product defined by \( (\mathbf{a}, \mathbf{b}) = \sum_{i=1}^n a_i b_i \), where \( \mathbf{a} = (a_1, a_2, \ldots, a_n) \) and \( \mathbf{b} = (b_1, b_2, \ldots, b_n) \), calculate the cosine of the angle \( \theta \) between the vectors \( \mathb... | automath | [
0.033762991428375244,
0.04350012540817261,
-0.05472179502248764,
-0.03809100762009621,
0.041325926780700684,
0.010128946974873543,
-0.029014751315116882,
0.059629421681165695,
0.021784670650959015,
0.007309922948479652,
0.012679197825491428,
0.018729805946350098,
-0.02466180920600891,
-0.0... | 0.821363 | 0.800654 | 0.020709 |
Given a morphism \( f: X \to Y \) of algebraic varieties over \( \mathbb{C} \), under the assumptions that \( f \) is bijective on \( \mathbb{C} \)-points, \( X \) is connected, and \( Y \) is normal, can it be concluded that \( f \) is an isomorphism? If so, provide a brief justification for your conclusion, and if no... | automath | [
0.029620742425322533,
0.03187631443142891,
-0.03723453730344772,
-0.01016930304467678,
0.03200548514723778,
-0.044135987758636475,
0.0016822569305077195,
0.07068155705928802,
0.04799853637814522,
-0.006966886110603809,
0.006268141325563192,
0.02271738275885582,
-0.015099427662789822,
-0.00... | 0.816785 | 0.797092 | 0.019692 |
Given an orientable surface \( F \) with a foliation \( \mathcal{F} \) that has only \( k \)-prong saddle singularities for \( k \geq 3 \), under the condition that \(\alpha\) and \(\beta\) are transversal multi-curves that are isotopic in \( F \), what are the necessary conditions on the foliation \( \mathcal{F} \) to... | automath | [
0.0015630860580131412,
0.023271741345524788,
-0.040123749524354935,
-0.03969071805477142,
0.03328188508749008,
-0.008433808572590351,
-0.018993886187672615,
0.07289182394742966,
0.01451961137354374,
0.00006477040005847812,
0.009668859653174877,
0.021155888214707375,
-0.013657622039318085,
... | 0.809417 | 0.785423 | 0.023994 |
Given a finite set \( S \) with \( |S| = 10 \), let \( A = \{0, 1, 2, 3, 4\} \), \( B = \{1, 3, 5, 7, 9\} \), and \( C = \{4, 5, 6, 7\} \). Calculate the value of \( |A \cup B \cup \overline{C}| \) where \( \overline{C} \) is the complement of \( C \) in \( S \). | automath | [
0.03094889037311077,
0.041855067014694214,
-0.04794041067361832,
-0.006883508060127497,
0.03385670483112335,
-0.027859505265951157,
-0.023202234879136086,
0.058314550668001175,
0.04642514884471893,
-0.017280826345086098,
0.017298419028520584,
0.04003537818789482,
-0.039740417152643204,
-0.... | 0.836215 | 0.819611 | 0.016603 |
Given the recurrence relation defined by \( x_n = \cos(x_{n-1}) \sin(x_{n-2}) \) for \( n \geq 2 \), with initial conditions \( x_0 = 2 \) and \( x_1 = 1.4 \), confirm by mathematical induction that the sequence \( \{x_n\} \) satisfies \( 0 \leq x_n \leq 1 \) for all integers \( n \geq 2 \). What is \( x_2 \) calculate... | automath | [
0.005596609320491552,
0.032570917159318924,
-0.03644796460866928,
-0.025019554421305656,
0.0371263325214386,
0.003868347965180874,
-0.007142066955566406,
0.08141182363033295,
0.02293800190091133,
-0.005495602730661631,
0.007417250424623489,
0.019732285290956497,
-0.014648091048002243,
-0.0... | 0.829465 | 0.809169 | 0.020296 |
Consider a semilinear evolution equation of the form
$$u_t(t) = Au(t) + F(u(t)),$$
where \( A \) is the infinitesimal generator of a \( C_0 \)-semigroup \( S(t) \) that satisfies \( \|S(t)\| \leq M \) for some \( M > 0 \) and for all \( t \geq 0 \). Given that \( F: H \rightarrow H \) is locally Lipschitz, can we co... | automath | [
-0.02449181117117405,
0.02301010489463806,
-0.04225146025419235,
-0.029362991452217102,
0.043713316321372986,
0.0014853960601612926,
-0.02829565294086933,
0.10479916632175446,
0.0459396168589592,
-0.0032628222834318876,
0.022615380585193634,
0.024756312370300293,
-0.03958476334810257,
-0.0... | 0.800978 | 0.777748 | 0.02323 |
Let \( X = \mathbb{R}^3 \) and define an equivalence relation \( \sim \) on \( X \) by \( (x,y,z) \sim (x',y',z') \) if and only if \( z = z' \). What is the quotient space \( \mathbb{R}^3 / \sim \), and is it homeomorphic to another well-known topological space? If so, identify that space. | automath | [
0.019759340211749077,
0.013699210248887539,
-0.06189194694161415,
-0.037745241075754166,
0.027113046497106552,
0.008100375533103943,
0.0025049427058547735,
0.06884664297103882,
0.02157805860042572,
-0.010175678879022598,
0.014404023066163063,
0.0319141149520874,
-0.05027338117361069,
-0.03... | 0.816609 | 0.799123 | 0.017486 |
Given two n-dimensional vectors \( a = (a_1, a_2, \ldots, a_n) \) and \( b = (b_1, b_2, \ldots, b_n) \) over the field \( \mathbb{Z}_2 \), where \( \langle a, b \rangle = a_1 \cdot b_1 + a_2 \cdot b_2 + \cdots + a_n \cdot b_n \), what is the value of \( \langle (1,1), (1,1) \rangle \) and does this satisfy the positive... | automath | [
0.026739856228232384,
0.03898635506629944,
-0.050136320292949677,
-0.021081216633319855,
0.03903591260313988,
-0.005066304001957178,
-0.007026982493698597,
0.051294200122356415,
0.03242205083370209,
-0.014071356505155563,
-0.0160392913967371,
0.0336226150393486,
-0.027334069833159447,
-0.0... | 0.830996 | 0.807906 | 0.02309 |
Given an annular ring with outer radius \( b \) and inner radius \( a \), if the maximum number of blue circles that can be painted in this annular space is \( n \), where each circle touches both the inner and outer edges of the ring and has a radius of \( \frac{b-a}{2} \), what is the formula for the unpainted area i... | automath | [
0.0005313801229931414,
0.048990581184625626,
-0.037829212844371796,
-0.027402160689234734,
0.03869455307722092,
-0.01099813636392355,
-0.0367877259850502,
0.06781423091888428,
0.030339058488607407,
0.011518662795424461,
0.020445993170142174,
0.03517509251832962,
-0.004023871384561062,
-0.0... | 0.826631 | 0.810098 | 0.016534 |
Let \( X_1, X_2, \ldots, X_n \) be independent uniform random variables uniformly distributed on the interval [0, 1]. Let \( S_n = X_1 + X_2 + \ldots + X_n \). As \( n \) approaches infinity, determine the limit distribution of the standardized random variable
\[
Z_n = \frac{S_n - \mathbb{E}[S_n]}{\sqrt{\text{Var}(S_... | automath | [
-0.0005680531030520797,
0.03667334094643593,
-0.048452459275722504,
-0.04391717165708542,
0.029791507869958878,
-0.012605154886841774,
-0.03791357949376106,
0.04420926794409752,
0.028681419789791107,
-0.01834130845963955,
0.010716722346842289,
0.03375813364982605,
-0.02713964506983757,
0.0... | 0.816081 | 0.793842 | 0.022239 |
Let \( F \) be a polynomial in \( K[X_1, X_2, \ldots, X_n] \) that describes a \( K \)-hypersurface \( H \) in an algebraically closed field \( L \). Given \( F = c \cdot F_1^{\alpha_1} \cdot F_2^{\alpha_2} \cdots F_s^{\alpha_s} \), where \( F_i \) are unassociated irreducible polynomials over \( K \) and \( c \in K^\t... | automath | [
0.03956387937068939,
0.05138939619064331,
-0.03275521099567413,
-0.04266401380300522,
0.031980764120817184,
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-0.016350243240594864,
0.06907804310321808,
0.047447219491004944,
-0.005520266480743885,
0.003124399809166789,
0.026101525872945786,
-0.03749118745326996,
0.00... | 0.822168 | 0.788275 | 0.033893 |
Given a family of topological spaces \((X_\alpha)_{\alpha \in J}\), can you determine under what conditions the product space \(X = \prod_{\alpha \in J} X_\alpha\) is connected? Specifically, if we know that \(X\) is connected, what can we assert about the connectedness of each \(X_\alpha\) in terms of the index set \(... | automath | [
0.019066432490944862,
0.04683469235897064,
-0.03828278183937073,
-0.01344226859509945,
0.02520953118801117,
0.007474909070879221,
-0.017844952642917633,
0.054199859499931335,
0.04007641598582268,
0.006563148461282253,
0.005384651944041252,
0.03168383240699768,
-0.0339207760989666,
-0.02384... | 0.807368 | 0.782995 | 0.024373 |
Consider the set \( S = \mathbb{R} \setminus \mathbb{Q} \), which is dense in the real numbers \(\mathbb{R}\) with respect to the usual topology. In what topology can \( S \) be made dense while also containing at least one isolated point? Provide an example of such a topology and identify an isolated point within that... | automath | [
0.05109342932701111,
0.03745504096150398,
-0.046762485057115555,
-0.038591790944337845,
0.022686230018734932,
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0.04838000237941742,
0.042588964104652405,
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0.002480465918779373,
0.026316963136196136,
-0.04625895246863365,
-0.... | 0.823309 | 0.802534 | 0.020774 |
Is it possible to find a pair of unitary matrices \( U \) and \( V \) in \( \mathbb{U}_n \) such that they \( \omega \)-commute (i.e., \( UV = \omega VU \) for some root of unity \( \omega \neq 1 \)) and both satisfy the conditions \( \|I_n - U\| < 1 \) and \( \|I_n - V\| < 1 \) in the operator norm? If such matrices e... | automath | [
-0.000040154223825084046,
0.03280960023403168,
-0.04103299602866173,
-0.027549246326088905,
0.04551820084452629,
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0.06773663312196732,
0.01261586882174015,
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0.012011905200779438,
0.018171485513448715,
-0.020247234031558037,
... | 0.806236 | 0.785513 | 0.020723 |
Let \( F \) be a field and let \( S \) be a finite collection of irreducible polynomials in \( F[x] \). Given that \( K \) is the splitting field of \( S \) over \( F \), determine the degree of the extension \( [K : F] \) in terms of the degrees of the polynomials in \( S \). Specifically, if \( S = \{ p_1(x), p_2(x),... | automath | [
0.02900102734565735,
0.025723319500684738,
-0.035462718456983566,
-0.03937806189060211,
0.021752068772912025,
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0.04713074490427971,
0.044635109603405,
-0.008004031144082546,
0.011251182295382023,
0.032740555703639984,
-0.039705727249383926,
0.00... | 0.821391 | 0.801318 | 0.020074 |
Consider a set \( S \) consisting of \( n \) red points and \( n \) blue points in the plane. Each possible non-crossing geometric bichromatic perfect matching corresponds to a vertex in a graph \( G \), where two vertices are connected by an edge if the corresponding matchings are neighbors. What is the maximum possib... | automath | [
0.016207924112677574,
0.01329728215932846,
-0.047362715005874634,
-0.03932889178395271,
0.013342100195586681,
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0.027773670852184296,
0.036903128027915955,
-0.019403310492634773,
0.029077956452965736,
0.034640584141016006,
0.0011147443437948823,
... | 0.827508 | 0.799349 | 0.028159 |
Suppose \( f_r(x) = \frac{e^{r\alpha} e^{ix\alpha}}{1 + e^r e^{ix}} \) where \( 0 < \alpha < 1 \). Determine \( \lim_{r \to \infty} \int_0^{2\pi} f_r(x) \, dx \) and verify if it is valid to interchange the limit and the integral. What conditions must be met for this interchange to hold? | automath | [
0.010457327589392662,
0.04559037461876869,
0.0026372214779257774,
-0.011347074061632156,
0.006265048868954182,
0.010346521623432636,
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0.058899201452732086,
0.04784883186221123,
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0.006493883673101664,
0.023267598822712898,
-0.006336845923215151,
-... | 0.817255 | 0.798769 | 0.018486 |
Construct a piecewise polynomial with the following specifications: a cubic polynomial defined on the interval [0, 4], a quadratic polynomial on the interval [4, 10], and a quartic polynomial on the interval [10, 15]. If the cubic polynomial is given by \( f_1(x) = x^3 - 3x^2 + 2 \) for \( 0 \leq x < 4 \), the quadrati... | automath | [
0.0422319620847702,
0.03414783626794815,
-0.04724881798028946,
-0.03385588154196739,
0.0427892841398716,
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-0.013605885207653046,
0.07287215441465378,
0.025183677673339844,
-0.00006829837366240099,
0.02488092891871929,
0.030909212306141853,
-0.030657324939966202,
-0.01... | 0.816832 | 0.79877 | 0.018062 |
Let \( R \) be a ring and \( x, y \in R \) be nonzerodivisors. Under what conditions will the sequence
\[
0 \to R \xrightarrow{(-1, 1)} R[x/y] \oplus R[y/x] \xrightarrow{(1, 1)} R[x/y, y/x] \to 0
\]
be exact? Additionally, if \( R \) is integrally closed in \( R_x \), what can be concluded about the intersection \( ... | automath | [
0.04041341319680214,
0.0566277839243412,
-0.025255441665649414,
-0.012816359288990498,
0.02669183351099491,
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0.060337550938129425,
0.05125132203102112,
-0.02157948911190033,
0.003287023166194558,
0.004720794968307018,
-0.027036679908633232,
-0... | 0.828357 | 0.806918 | 0.021439 |
Consider a finite group \( G \) generated by \( n+1 \) elements \( x_1, x_2, \ldots, x_{n+1} \) with relations of the form \( x_{n+1}^{-1} x_{\sigma(1)} x_{\sigma(2)} \cdots x_{\sigma(n)} = e \), where \( \sigma \) varies over a set of \( n! - k + 1 \) permutations. What is the maximum number of distinct products that ... | automath | [
0.03385591134428978,
0.03769756108522415,
-0.04268445819616318,
-0.026132196187973022,
0.028394926339387894,
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0.06653528660535812,
0.019278056919574738,
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0.006631957367062569,
0.02513561211526394,
-0.036938998848199844,
-0.... | 0.82368 | 0.803938 | 0.019742 |
Consider the polynomials \( f(x) = kx^2 + 3x - 11 \) and \( g(x) = mx^3 - 2x^2 + 3x - 9 \). Given that the sum of the roots of \( f \) is equal to the product of the roots of \( g \), and that \( g \) has at least one root on the imaginary axis, find the values of \( k \) and \( m \) such that the conditions are satisf... | automath | [
0.040013302117586136,
0.034679070115089417,
-0.03724398836493492,
-0.07144461572170258,
0.04360204190015793,
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0.0076437173411250114,
0.06268897652626038,
0.02404223009943962,
0.009348498657345772,
0.01762406900525093,
0.02570863626897335,
-0.0380648598074913,
-0.026600... | 0.819075 | 0.799818 | 0.019257 |
Given an equilateral triangle with a side length of \( s \), inscribed around six circles that are mutually tangent to each other and tangent to the sides of the triangle, if the radius of each circle is \( r \), express the percentage of the area of the triangle that is not shaded (not covered by the circles) in terms... | automath | [
0.006714830175042152,
0.02653338760137558,
-0.055222101509571075,
-0.03388649597764015,
0.03531375899910927,
-0.031930387020111084,
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0.05726877599954605,
0.02261737547814846,
-0.004089998081326485,
0.014671636745333672,
0.02718942053616047,
-0.02825578674674034,
-0.01... | 0.831411 | 0.814404 | 0.017006 |
Given a finite-dimensional vector space with a norm \( ||\cdot|| \) that satisfies the parallelogram law, what is the relationship between this norm and the existence of an inner product? Specifically, if the inner product \( \langle \cdot, \cdot \rangle \) can be uniquely determined from this norm using the polarizati... | automath | [
0.00642833486199379,
0.037167008966207504,
-0.03831416741013527,
-0.02454618737101555,
0.02952493168413639,
0.008137039840221405,
-0.0225205197930336,
0.0729028731584549,
0.02250605635344982,
-0.017433350905776024,
0.0005696144653484225,
0.033755913376808167,
-0.01950664073228836,
-0.02762... | 0.810435 | 0.78893 | 0.021505 |
Let \( T = (S, \tau) \) be a topological space, and let \( K \subseteq S \) be a subset with the subspace topology \( \tau_K \). If \( H \) is a compact subset of \( K \) under the topology \( T' = (K, \tau_K) \), how would the existence of an open cover of \( H \) in \( T \) that has a finite subcover demonstrate that... | automath | [
0.03247779235243797,
0.039552733302116394,
-0.04064111411571503,
-0.05292834714055061,
0.035197801887989044,
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0.06834220886230469,
0.035836439579725266,
0.012587953358888626,
0.027985699474811554,
0.01447323802858591,
-0.03403929993510246,
-0.009... | 0.805828 | 0.785112 | 0.020716 |
Let \( K \) be a number field and \( \mathcal{O}_K \) its ring of integers. If \( P \) is a prime ideal in \( \mathcal{O}_K \), demonstrate that the norm \( N_{\mathbb{Q}}^K(P) \) can be expressed as \( p^n \) for some prime \( p \) and a positive integer \( n \). Furthermore, if \( [K:\mathbb{Q}] = d \), identify the ... | automath | [
0.0476374626159668,
0.02761797048151493,
-0.02206728607416153,
-0.02409193478524685,
0.021109577268362045,
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-0.01111084595322609,
0.04112632945179939,
0.03962846100330353,
-0.004342370666563511,
0.007113305851817131,
0.018458805978298187,
-0.03635973483324051,
-0.0096... | 0.829455 | 0.798151 | 0.031304 |
In the context of linear programming, consider the following primal and dual problems:
Primal Problem:
\[
\begin{aligned}
\min \quad & x_1 + x_2 \\
\text{s.t.} \quad & x_1 = 1 \\
& x_1 + x_2 = 1 \\
& x_1 \geq 0, \quad x_2 \text{ is free.}
\end{aligned}
\]
Dual Problem:
\[
\begin{aligned}
\max \quad & y_1 + y_2 \\
\t... | automath | [
-0.013457739725708961,
0.01744290255010128,
-0.02918066456913948,
-0.04919031262397766,
0.03876147046685219,
0.015199406072497368,
0.006103074178099632,
0.082527756690979,
0.02877160534262657,
0.000560615852009505,
0.007862970232963562,
0.0037143444642424583,
-0.01900133118033409,
-0.01721... | 0.809136 | 0.782284 | 0.026852 |
Given the quadratic form \( f(x, y) = ax^2 + bxy + cy^2 \) associated with the symmetric matrix \( A = \begin{bmatrix} a & \frac{b}{2} \\ \frac{b}{2} & c \end{bmatrix} \), find the angle \( \theta \) such that the transformation to the new coordinates, expressed via trigonometric functions, is given by the relation \( ... | automath | [
0.027333373203873634,
0.054417096078395844,
-0.05437837168574333,
-0.038293782621622086,
0.01635872945189476,
-0.004639202728867531,
-0.008480548858642578,
0.07527611404657364,
0.034083444625139236,
-0.013364051468670368,
0.005347070749849081,
0.013532510958611965,
-0.03911372646689415,
-0... | 0.811851 | 0.789091 | 0.02276 |
Let \( \mu(\Omega) < \infty \) and \( 1 \leq p_1 \leq p_2 < \infty \). If \( f_n \to f \) in \( L^{p_2} \) as \( n \to \infty \), what upper bound can be established for \( \| f_n - f \|_{p_1} \) in terms of \( \| f_n - f \|_{p_2} \)? Specifically, determine a constant \( C \) such that:
\[
\| f_n - f \|_{p_1} \leq C ... | automath | [
0.007955307140946388,
0.03710947558283806,
-0.03649650886654854,
-0.00734600517898798,
0.019990231841802597,
-0.0005572009831666946,
-0.04757491126656532,
0.057264234870672226,
0.007717135827988386,
-0.013019065372645855,
-0.008000175468623638,
0.02916458435356617,
-0.03057660348713398,
0.... | 0.827381 | 0.808724 | 0.018657 |
Consider the function \( f(x, y) = 6x^2 + 8y^2 + 2xy + 3 \). Given the triangular region \( D \) defined by the vertices \( (0,0) \), \( (1,0) \), and \( (0,1) \), identify the coordinates of the point at which \( f(x, y) \) attains its absolute maximum value on the boundary of \( D \). Additionally, what is the absolu... | automath | [
0.033763181418180466,
0.03497621789574623,
-0.043173667043447495,
-0.0333750881254673,
0.013434537686407566,
-0.020937541499733925,
-0.01715734973549843,
0.06571348011493683,
0.0339149609208107,
0.002794214291498065,
0.007690004538744688,
0.03443705290555954,
-0.023437995463609695,
-0.0173... | 0.826622 | 0.807817 | 0.018804 |
Given a simple undirected graph \( G \) with edge set \( E \) and vertex set \( V \), determine the smallest integer \( k \) such that there exists a collection \( K \) of cliques in \( G \) satisfying \( |K| = k \) and such that every edge in \( E \) is covered by at least one clique in \( K \). Additionally, if \( E ... | automath | [
-0.004652755800634623,
0.027898062020540237,
-0.01692306436598301,
-0.03461267426609993,
0.021931292489171028,
-0.03164149820804596,
-0.01829945109784603,
0.051676418632268906,
0.013445860706269741,
-0.013073374517261982,
0.04079928994178772,
0.01362756546586752,
-0.009610801935195923,
-0.... | 0.825431 | 0.796967 | 0.028463 |
Let \( G \) be a finite group of order \( n \), and let \( H \) be a subgroup of \( G \) with order \( m \). Assuming Lagrange's Theorem holds, express the number of distinct left cosets of \( H \) in \( G \) in terms of \( n \) and \( m \). Additionally, verify if such an expression holds true for infinite groups, and... | automath | [
0.01686261035501957,
0.030023666098713875,
-0.03184196725487709,
-0.03931263089179993,
0.01711834967136383,
-0.010801896452903748,
-0.024991560727357864,
0.047622036188840866,
0.010334830731153488,
-0.02846849523484707,
0.019089708104729652,
0.009398913010954857,
-0.02583843283355236,
-0.0... | 0.831292 | 0.803132 | 0.02816 |
Given the homogeneous linear partial differential equation \( u_x - u_y = 0 \), determine the general solution for \( u(x, y) \) in terms of arbitrary functions. Specifically, can you express the general solution in the form \( u(x, y) = f(x+y) \) where \( f \) is an arbitrary function? Additionally, how does this resu... | automath | [
-0.02635321207344532,
0.032537054270505905,
-0.037795644253492355,
-0.04965111240744591,
0.03648070618510246,
-0.02004736103117466,
-0.018120640888810158,
0.0750662237405777,
0.04607928544282913,
-0.00375190912745893,
0.011606181971728802,
0.025255899876356125,
-0.030246663838624954,
-0.01... | 0.812808 | 0.795465 | 0.017344 |
Let \( f(x) \) be a polynomial of degree \( n \) and let \( x_0, x_1, \ldots, x_n \) be \( n + 1 \) distinct points in the interval \([a, b]\). Define the Lagrange interpolating polynomial \( P(x) \) that passes through these points.
What is the maximum interpolation error \( E(x) = |f(x) - P(x)| \) when \( f(x) \) i... | automath | [
0.02407136559486389,
0.03551802039146423,
-0.02837337739765644,
-0.04587209224700928,
0.006434644106775522,
-0.018529480323195457,
-0.012934185564517975,
0.051730815321207047,
0.01517954096198082,
-0.003396882675588131,
0.023174192756414413,
0.02658374048769474,
-0.028261438012123108,
-0.0... | 0.829037 | 0.808593 | 0.020444 |
Given a prime \( p \) and a positive integer \( k \), let \( f, h \in \mathbb{Z}[x] \) be polynomials such that \( h \) divides \( f \mod p^k \) in \( \mathbb{Z}/p^k\mathbb{Z}[x] \) and \( h \mod p \) is irreducible in \( \mathbb{F}_p[x] \).
How many irreducible factors \( h_0 \) of \( f \) in \( \mathbb{Z}[x] \) sat... | automath | [
0.04233097285032272,
0.03313806280493736,
-0.04082103446125984,
-0.02437211200594902,
0.03291293606162071,
-0.01573636569082737,
0.002955470932647586,
0.03616105392575264,
0.049929678440093994,
-0.011240893043577671,
-0.00979843083769083,
0.02583850733935833,
-0.022716443985700607,
-0.0229... | 0.833628 | 0.817401 | 0.016227 |
Given a scheme \( X \) and a closed subscheme \( Z \), under what conditions can the blowup \( \mathrm{Bl}_Z(X) \) of \( X \) at \( Z \) be shown to be isomorphic to another blowup \( \mathrm{Bl}_{Z'}(X) \) for some closed subscheme \( Z' \) whose support lies in \( X \setminus U \), where \( U \) is an open subscheme ... | automath | [
0.012856332585215569,
0.029279259964823723,
-0.0312587171792984,
-0.02391931600868702,
0.04177190735936165,
0.006734071299433708,
-0.00356105575338006,
0.07515738159418106,
0.030207451432943344,
-0.017197804525494576,
0.01878214254975319,
0.02236172929406166,
-0.021375557407736778,
-0.0044... | 0.811014 | 0.792901 | 0.018113 |
Let \( d \) be a positive integer, and consider the unit \( d \)-cube with vertices in \( \{0, 1\}^d \). Given two parallel hyperplanes defined by \( x_1 + x_2 + \ldots + x_d = a \) and \( x_1 + x_2 + \ldots + x_d = b \) where \( a < b \), what is the maximum possible diameter of the connected components of the subgrap... | automath | [
0.01616799458861351,
0.040299560874700546,
-0.044502295553684235,
-0.0180678628385067,
0.027386628091335297,
0.011362245306372643,
-0.03069007396697998,
0.03882870450615883,
0.04264932498335838,
-0.01005582232028246,
0.0131476279348135,
0.04116936773061752,
-0.006244297605007887,
-0.008684... | 0.834556 | 0.808848 | 0.025708 |
Let \( N \) be a positive integer and consider \( N \) independent random variables \( X_1, X_2, \ldots, X_N \) such that \( X_i \sim \Gamma(1, \frac{1}{N}) \) for \( i \in [N] \). Define the random variable \( Y = \sum_{i=1}^{N} X_i \).
1. What is the distribution of \( Z = \frac{X_i}{Y} \) for a given \( i \) when ... | automath | [
-0.007754363119602203,
0.029092395678162575,
-0.02878735028207302,
-0.03230193629860878,
0.040853243321180344,
0.0008134149247780442,
-0.005963590461760759,
0.04421096667647362,
0.025868764147162437,
-0.0097398916259408,
0.002225865377113223,
0.006921461317688227,
-0.046279482543468475,
-0... | 0.824978 | 0.802067 | 0.022911 |
Let \( S \) be the Stone–Čech compactification \( \beta \mathbb{N} \) excluding a point \( x \in \beta \mathbb{N} \setminus \mathbb{N} \), and let \( \mu \) be the Borel measure defined on \( S \) that assigns mass \( 2^{-n} \) to each point \( n \in \mathbb{N} \). Consider the infinite disjoint family \( \mathcal{H} \... | automath | [
0.04140134155750275,
0.029582349583506584,
-0.025315120816230774,
-0.008216490969061852,
0.04059474542737007,
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0.03906611353158951,
0.05048498511314392,
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0.020019453018903732,
0.009673413820564747,
-0.03708505630493164,
-0.0... | 0.819278 | 0.78927 | 0.030008 |
Let \( n \) be a positive integer.
(a) How many ordered triples \( (a,b,c) \) exist such that \( 1 \leq a,b,c \leq n \), and how is this number expressed in terms of \( n \) using the equation \( n^3 = n + 3n(n - 1) + 6 \binom{n}{3} \)?
(b) What is the combinatorial interpretation of the equation \( \binom{n+2}{3} =... | automath | [
0.035121701657772064,
0.008022882044315338,
-0.056023839861154556,
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0.025592483580112457,
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0.0386681891977787,
0.02171345241367817,
0.017152821645140648,
0.003768820548430085,
0.042534273117780685,
-0.049641869962215424,
-0.0... | 0.836155 | 0.814952 | 0.021202 |
Let \( S \) be a scheme and \( X \) an algebraic space over \( S \) with an effective Cartier divisor \( D \). If \( U = X \setminus D \), what can be said about the mapping properties of the morphism \( U \to X \), and how does it relate to the concept of scheme-theoretic density in algebraic geometry? Specifically, u... | automath | [
0.029013395309448242,
0.02897404506802559,
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0.025820624083280563,
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0.00668988237157464,
0.040013302117586136,
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-0... | 0.800927 | 0.782597 | 0.01833 |
Let \( \kappa \) be the closed-partition number, defined as the size of the smallest nontrivial partition of the unit interval \([0,1]\) into closed sets. If it is consistent with ZFC that \( \kappa = \omega_1 \) while the Continuum Hypothesis (CH) fails, what can we conclude about the contractibility of co-finite topo... | automath | [
0.027090633288025856,
0.02882743626832962,
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-0.013322137296199799,
0.034117113798856735,
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0.018327737227082253,
0.011076466180384159,
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-0.... | 0.799289 | 0.778009 | 0.02128 |
Evaluate the integral
\[
I_n = \int \frac{x^2}{(a^2 + x^2)^n} \, dx
\]
and express it in terms of \( I_{n-1} = \int \frac{1}{(a^2 + x^2)^{n-1}} \, dx \). What is the resulting formula for \( I_n \) in terms of \( I_{n-1} \) and verify if the expression
\[
I_n = \frac{1}{2n-2} \left(-\frac{x}{(a^2+x^2)^{n-1}} + I_{... | automath | [
0.01917441003024578,
0.043250493705272675,
-0.03904823586344719,
-0.03310844302177429,
0.032763268798589706,
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-0.006973433308303356,
0.07540913671255112,
0.027735857293009758,
-0.014656711369752884,
0.003978518769145012,
0.028313711285591125,
-0.04011917486786842,
-0.... | 0.822047 | 0.79807 | 0.023977 |
Given a 3x3 symmetric matrix \( M \) with known entries \( M_{ij} \) and a vector \( x = (x_0, x_1, x_2) \) such that
\[
M_{ij} = \sum_{m=0}^2 x_m \epsilon_i^m \sum_{n=0}^2 x_n \epsilon_j^n,
\]
where \( \epsilon_i^m \) are almost Kronecker delta symbols (small if \( i \neq m \) and nearly 1 otherwise), what is the m... | automath | [
0.030343636870384216,
0.022730540484189987,
-0.0502953939139843,
-0.02564951218664646,
0.040719639509916306,
0.0006009054486639798,
-0.04405239596962929,
0.06717108935117722,
0.02827421762049198,
-0.012499879114329815,
-0.001997555373236537,
0.03104991465806961,
-0.04983080178499222,
-0.03... | 0.819639 | 0.800837 | 0.018802 |
Let \( A \) be an \( n \times n \) complex matrix with the property that there exists an invertible matrix \( S \) such that \( SAS^{-1} = A^{-1} \). If all the eigenvalues of \( A \) are integers and \( \det(A) > 0 \), what are the possible values of \( n \) for which \( 1 \) is definitely an eigenvalue of \( A \)? | automath | [
0.04530893266201019,
0.02567264996469021,
-0.03421039879322052,
-0.04456261917948723,
0.0460297167301178,
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0.007201377302408218,
0.06837759912014008,
0.044002946466207504,
-0.004104778170585632,
0.016165634617209435,
0.03539538383483887,
-0.05349113047122955,
-0.03278... | 0.822725 | 0.801553 | 0.021172 |
Given a set of \( M = 3 \) vectors in \( \mathbb{Z}^2 \):
\[
v_1 = (1,1), \quad v_2 = (2,0), \quad v_3 = (0,2),
\]
determine the volume of the parallelepiped formed by the linearly independent subset of these vectors. Which linear combinations among the vectors can give rise to the same lattice structure, and what i... | automath | [
0.043353091925382614,
0.036791760474443436,
-0.03872445225715637,
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0.05541681870818138,
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0.055544473230838776,
0.02942907065153122,
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0.016683192923665047,
0.027620121836662292,
-0.024067962542176247,
-0.0... | 0.827622 | 0.804104 | 0.023518 |
Consider the metric space \( \mathbb{R}^2 \) with the standard Euclidean distance. Let \( A = \{(4,y) : y \in \mathbb{R}\} \) be a vertical line at \( x = 4 \) and let \( B = \{(x,y) : \sqrt{x^2 + y^2} < 1\} \cup A \). Find a point \( x \in \mathbb{R}^2 \) such that \( \text{dist}(x,A) > \text{dist}(x,B) + \text{diam}(... | automath | [
0.0065768021158874035,
0.034692276269197464,
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-0.03704050928354263,
0.040297120809555054,
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0.05410056188702583,
0.04644980654120445,
0.0003494102566037327,
0.00430673873052001,
0.028920037671923637,
0.003808919107541442,
-0.0... | 0.835539 | 0.806319 | 0.02922 |
Given a list of \( N \) integers where each integer is either 1, 2, or 3, and assuming that the integers are arranged in such a way that their counts are as follows: there are \( a \) occurrences of 1, \( b \) occurrences of 2, and \( c \) occurrences of 3. If we query for the counts of 1's, 2's, and 3's within the int... | automath | [
0.033066414296627045,
0.02006513439118862,
-0.055694252252578735,
-0.02151775360107422,
0.018845580518245697,
-0.009041211567819118,
-0.030368492007255554,
0.05561016499996185,
0.013166693970561028,
0.00938383862376213,
0.015075252391397953,
0.035125233232975006,
-0.02621457725763321,
-0.0... | 0.83438 | 0.817296 | 0.017084 |
Given the sequence \( x \in \ell^1(\mathbb{Z}) \) defined by \( x(0) = 1 \), \( x(1) = -1 \), \( x(2) = -1 \), and \( x(n) = 0 \) for all other integers \( n \), calculate the value of \( \|x^* \ast x\|_1 \) as expressed in the problem and compare it to \( \|x\|_1^2 \). What does this imply about the relationship betwe... | automath | [
0.022952426224946976,
0.03761693835258484,
-0.03123994544148445,
-0.04196799546480179,
0.036081962287425995,
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0.001301875920034945,
0.06545737385749817,
0.03752445429563522,
-0.010988852940499783,
-0.000048014320782385767,
0.024700172245502472,
-0.03965236619114876,
-... | 0.838026 | 0.820817 | 0.017209 |
Let \(\mathbf{X}\) and \(\mathbf{Y}\) be two non-zero, non-constant vectors in \(\mathbb{R}^n\) such that they are linearly independent. Given that the covariance between \(\mathbf{X}\) and \(\mathbf{Y}\) is non-zero, can you construct an example of such vectors in \(\mathbb{R}^2\) and compute their covariance? What sp... | automath | [
0.0038473287131637335,
0.03505678102374077,
-0.04493868723511696,
-0.040443338453769684,
0.03724837675690651,
0.010839534923434258,
0.0015596760204061866,
0.07051217555999756,
0.04206683859229088,
-0.03607212007045746,
-0.00515437126159668,
0.007719268091022968,
-0.02651437744498253,
-0.00... | 0.807867 | 0.786589 | 0.021278 |
In an experiment with \( n \) subjects, each subject \( i \) has \( V_i \) valid responses and \( C_i \) of those are correct. Let \( \text{Total Correct} = \sum_{i=1}^{n} C_i \) and \( \text{Total Valid} = \sum_{i=1}^{n} V_i \). Define the average proportion correct across all subjects as \( P = \frac{\text{Total Corr... | automath | [
0.01913149282336235,
0.030050629749894142,
-0.04930806905031204,
-0.030210161581635475,
0.03584599867463112,
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-0.00023460687953047454,
0.0542273111641407,
0.031995732337236404,
0.0020953163038939238,
0.0025054998695850372,
0.014631460420787334,
-0.017643529921770096,
... | 0.826925 | 0.808997 | 0.017927 |
Given a linear programming problem in standard form:
\[
\begin{align*}
\text{maximize } & c^T x \\
\text{subject to } & Ax \leq b \\
& x \geq 0
\end{align*}
\]
where \( c \in \mathbb{R}^n \), \( A \in \mathbb{R}^{m \times n} \), and \( b \in \mathbb{R}^m \), what is the relationship between the optimal value of the p... | automath | [
-0.014281881973147392,
0.01457071490585804,
-0.057583026587963104,
-0.03284579887986183,
0.0342155396938324,
0.013241722248494625,
-0.008760601282119751,
0.0663565844297409,
0.04722912237048149,
-0.014636371284723282,
0.0014636931009590626,
0.017893366515636444,
-0.022886153310537338,
0.00... | 0.817049 | 0.790956 | 0.026093 |
Given an array of integers \( A \) of size \( n \) with elements ranging from \( 1 \) to \( 10^6 \), define the "good value" of an element \( A[i] \) as the count of indices \( j < i \) such that \( A[j] \) is divisible by \( A[i] \).
Develop an efficient algorithm to compute the maximum good value across all element... | automath | [
0.03721743822097778,
0.02182183787226677,
-0.03408506140112877,
-0.020711436867713928,
0.024443376809358597,
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0.058786485344171524,
0.024129319936037064,
-0.02740025892853737,
0.020014120265841484,
0.027507521212100983,
-0.020292725414037704,
0.0... | 0.814862 | 0.795219 | 0.019643 |
Given the expression $$\|\Sigma_{n(p)} - \Sigma \| = O_P\left( \frac{\log p}{n(p)}\right),$$ can you find a constant \( C > 0 \) such that for sufficiently large \( p \), the probability
\[
\mathbb{P}\left( \|\Sigma_{n(p)} - \Sigma \| > C \frac{\log p}{n(p)}\right)
\]
decreases to zero, and under what conditions do... | automath | [
-0.003107108874246478,
0.035279933363199234,
-0.041327133774757385,
-0.005393075291067362,
0.0172346830368042,
-0.007952186278998852,
-0.009408474899828434,
0.06798940896987915,
0.022186478599905968,
-0.008118316531181335,
0.0017220586305484176,
0.009916846640408039,
-0.02385712042450905,
... | 0.822139 | 0.805162 | 0.016977 |
Let \( f_n(x) = e^{-n|1 - \sin(x)|} \) for \( x \in [a, b] \). Determine the set of points \( x \) in the interval \([a, b]\) where \( f_n(x) \) does not converge to 0 as \( n \to \infty \). Specifically, characterize these points and explain why the convergence to 0 in measure occurs for the sequence of functions \( f... | automath | [
0.00777595117688179,
0.03920457139611244,
-0.037686459720134735,
-0.04038163274526596,
0.033109236508607864,
0.013255352154374123,
-0.02484806999564171,
0.06964853405952454,
0.0357971265912056,
0.017147228121757507,
0.009280799888074398,
0.01651003211736679,
-0.03062424436211586,
-0.009137... | 0.807473 | 0.783384 | 0.024089 |
Let $R$ be a Noetherian local ring, and let $I \subset R$ be an ideal of definition. Given a short exact sequence of finite $R$-modules $0 \to M' \to M \to M'' \to 0$, if the degree of the numerical polynomial $\chi_{I, M'}$ is 3, what can we conclude about the degrees of the numerical polynomials $\chi_{I, M}$ and $\c... | automath | [
0.036679502576589584,
0.023224584758281708,
-0.030800100415945053,
-0.03082527592778206,
0.0193091481924057,
-0.01800713874399662,
-0.015336284413933754,
0.040510792285203934,
0.0352708138525486,
-0.03499171882867813,
0.011769382283091545,
0.04144023358821869,
-0.05366180092096329,
-0.0136... | 0.83113 | 0.81184 | 0.01929 |
Let \( X_1, X_2, \ldots, X_n \) be independent and identically distributed random variables that follow an exponential distribution with rate parameter \( \lambda \). Define \( n = 2m + 1 \) for some integer \( m \).
1. What is the asymptotic expectation of the sample median \( X_{(m+1)} \) as \( n \) approaches infi... | automath | [
0.0035404281225055456,
0.030794011428952217,
-0.04715333133935928,
-0.04597669839859009,
0.0165061317384243,
-0.009766720235347748,
-0.04539769887924194,
0.0677432268857956,
0.013909866102039814,
-0.02319241501390934,
0.010143290273845196,
0.017165306955575943,
-0.042394816875457764,
-0.00... | 0.820357 | 0.802977 | 0.01738 |
Let \( X \) be a non-compact Riemann surface. If we define the genus of \( X \) as the maximum \( g \) such that a compact Riemann surface of genus \( g \) minus a point embeds into \( X \), can we conclude that \( X \) can be locally biholomorphic to an open subset of a compact Riemann surface if and only if \( X \) h... | automath | [
-0.006733396090567112,
0.01959608681499958,
-0.04048202186822891,
-0.04622548818588257,
0.028033005073666573,
-0.01539718545973301,
-0.017826832830905914,
0.051759637892246246,
0.03516740724444389,
0.00137803191319108,
0.0228372011333704,
0.04152596369385719,
-0.041517846286296844,
-0.0152... | 0.805909 | 0.780944 | 0.024965 |
Let \( \mathcal{H} \) be a separable Hilbert space and \( p \geq 2 \). Consider a bounded linear operator \( T \colon \mathcal{H} \to L^p(\mathbb{R}^d) \). Define the set
\[
M = \{ f \in \mathcal{H} : \|Tf\|_{L^p} = \|T\| \|f\|_{\mathcal{H}} \}.
\]
Is the set \( M \) weakly closed in \( \mathcal{H} \) if \( T \) sat... | automath | [
0.01901339739561081,
0.06044068560004234,
-0.03240472450852394,
-0.039816342294216156,
0.04811977222561836,
-0.01227042730897665,
-0.0008982890285551548,
0.08319734781980515,
0.04412141814827919,
-0.002938529010862112,
0.0010357586434111,
0.016717836260795593,
-0.02240917459130287,
-0.0145... | 0.804179 | 0.77925 | 0.024929 |
Let \( a_n \) and \( b_n \) be positive sequences that converge to limits \( a \) and \( b \) respectively, with \( b > 0 \). Define the sequence \( c_n = \frac{a_n}{b_n} \).
1. What is the value of \( \lim_{n \to \infty} c_n \)?
2. Under what conditions does \( c_n \) converge to this limit? | automath | [
0.011744584888219833,
0.04886065796017647,
-0.04204246401786804,
-0.0011915008071810007,
0.02072197012603283,
-0.005600959528237581,
-0.018403947353363037,
0.057485464960336685,
0.040590908378362656,
-0.007684989832341671,
-0.009803688153624535,
0.014779905788600445,
-0.03681495413184166,
... | 0.830404 | 0.809047 | 0.021358 |
Consider the function \( h: \mathbb{N}_0 \times \mathbb{N} \to \mathbb{N} \) defined by \( h(m, n) = m + n \). Determine two different right-inverse functions \( h': \mathbb{N} \to \mathbb{N}_0 \times \mathbb{N} \) such that for any \( k \in \mathbb{N} \), \( h(h'(k)) = k \). What are these right-inverse functions, and... | automath | [
0.042546529322862625,
0.048441991209983826,
-0.017680838704109192,
-0.026282327249646187,
0.03153308108448982,
-0.02052067406475544,
-0.014844144694507122,
0.06896762549877167,
0.041298989206552505,
0.017104532569646835,
-0.0049160257913172245,
0.015562119893729687,
-0.03781845420598984,
-... | 0.82194 | 0.801639 | 0.020301 |
Given the function \( f(x_1, x_2, \ldots, x_n) \) that is to be maximized subject to the constraint \( x_1^2 + x_2^2 + \ldots + x_n^2 = a \), where \( a \) is a positive constant, what can be concluded about the values of \( x_2, \ldots, x_n \) if it is found that the gradient \( \frac{df}{dx_1} \) at the point \( (0, ... | automath | [
-0.005788788665086031,
0.030927622690796852,
-0.05234348028898239,
-0.02606307901442051,
0.026649532839655876,
-0.0019377429271116853,
-0.02005898952484131,
0.0760967880487442,
0.03288954496383667,
-0.014765397645533085,
-0.006975426804274321,
0.007138907443732023,
-0.031608086079359055,
-... | 0.822919 | 0.799421 | 0.023498 |
Given two arbitrary positive integers \( p \) and \( q \), can you provide a number \( n \) such that \( \gcd(n, p) = 1 \) and \( \gcd(n, q) = 1 \) that can be computed in linear time with respect to the maximum of \( p \) and \( q \)? If possible, provide an example of such an \( n \) and explain how your solution is ... | automath | [
0.032464828342199326,
0.030377596616744995,
-0.03338012844324112,
-0.03711327910423279,
0.035982128232717514,
-0.015349241904914379,
-0.015995046123862267,
0.05324834957718849,
0.025292132049798965,
-0.01585320755839348,
0.024706028401851654,
0.026591740548610687,
-0.02704373374581337,
-0.... | 0.842902 | 0.822277 | 0.020624 |
Let \( f: \mathbb{R}^n \to \mathbb{R} \) be a differentiable function that is defined on a closed convex set \( K \). Suppose there exists a point \( x \in K \) such that \( f(x) = \min_{y \in K} f(y) \), and let \( F(x) \) denote the gradient of \( f \) at point \( x \). If \( F(x) = (1, 2, 3) \) and \( y = (4, 5, 6) ... | automath | [
0.009745855815708637,
0.04973798617720604,
-0.039382513612508774,
-0.029512502253055573,
0.05346764624118805,
-0.00691488292068243,
-0.020145582035183907,
0.0798143744468689,
0.03136930242180824,
-0.0002556817198637873,
0.003228698391467333,
0.04133312776684761,
-0.05191270262002945,
-0.00... | 0.82829 | 0.809019 | 0.019271 |
Given two families of sets \( (A_i)_{i \in I} \) and \( (B_i)_{i \in I} \), can you determine the necessary and sufficient conditions for the union of the mappings \( \bigcup_{i \in I} f_i \) to be a mapping from \( \bigcup_{i \in I} A_i \) to \( \bigcup_{i \in I} B_i \)? Specifically, under what conditions on the doma... | automath | [
0.0382845364511013,
0.02616424486041069,
-0.02228032797574997,
-0.030012056231498718,
0.032511334866285324,
-0.009645525366067886,
-0.013992968946695328,
0.03966750204563141,
0.031270623207092285,
0.003954305313527584,
0.011061444878578186,
0.03105822391808033,
-0.011598324403166771,
-0.01... | 0.816212 | 0.795791 | 0.020421 |
Consider the functions \( f(x) = \frac{x^2 - x}{x - 1} \) and \( g(x) = x \). Determine the values of \( x \) for which \( f(x) \) and \( g(x) \) are equal, and state whether \( f \) and \( g \) are the same function. Justify your answer by discussing their domains and the points where they may be defined. | automath | [
0.027340183034539223,
0.019447457045316696,
-0.047146521508693695,
-0.042161982506513596,
0.04855682700872421,
-0.037338737398386,
0.01333885733038187,
0.06913787126541138,
0.03723938763141632,
-0.014933086931705475,
0.007842465303838253,
0.018925676122307777,
-0.0336446687579155,
-0.01977... | 0.817627 | 0.801143 | 0.016484 |
Given a triangle \( ABC \) with cevians intersecting opposite sides at points \( D_+, E_+, F_+, D_-, E_-, F_- \), let \( K_A, K_B, K_C \) be the points defined by the barycentric coordinates derived from these cevian points. If the cevians \( AD, BE, CF \) meet at a point \( M \), what are the barycentric coordinates o... | automath | [
0.0479828380048275,
0.023367807269096375,
-0.049904823303222656,
-0.019955793395638466,
0.02870107628405094,
-0.010418491438031197,
0.0012916394043713808,
0.040640223771333694,
0.04152543842792511,
0.0014916150830686092,
0.006435918156057596,
0.028042200952768326,
-0.040819983929395676,
-0... | 0.825553 | 0.804362 | 0.021191 |
In triangle \( \triangle ABC \), let the lengths of sides \( a = 10 \), \( b = 6 \), and the angle \( C = 30^\circ \) be given. Using the law of cosines, calculate the length of side \( c \). Then, using the law of sines, find the possible values for angle \( A \). Determine the valid configuration of angles in the tri... | automath | [
0.023403752595186234,
0.024781029671430588,
-0.04049723595380783,
-0.015655256807804108,
0.03912145271897316,
-0.023073624819517136,
-0.012657987885177135,
0.0458773635327816,
0.023391978815197945,
-0.003395647741854191,
0.007835928350687027,
0.026406867429614067,
-0.03821897506713867,
-0.... | 0.813076 | 0.796666 | 0.01641 |
Given two data sets of approximately 150 positive integer samples that exhibit a density estimate resembling a Cauchy distribution, derive a condition under which the distribution of these samples can lead to such a Cauchy-like fit. Specifically, what type of underlying distributions or sampling mechanisms could cause ... | automath | [
0.020647911354899406,
0.007923929952085018,
-0.039233382791280746,
-0.046066898852586746,
0.025459086522459984,
-0.0010971432784572244,
-0.022234119474887848,
0.06275609135627747,
0.012425199151039124,
-0.009372059255838394,
0.006610190495848656,
0.019549278542399406,
-0.020448969677090645,
... | 0.82596 | 0.80939 | 0.016571 |
Given a monoid \( M \) acting on a set \( \alpha \), let \( s \) be a subset of \( \alpha \). Define \( \text{Fix}(s) \) as the set of all points in \( s \) that are fixed by every element of \( M \), i.e.,
\[
\text{Fix}(s) = \{ x \in s \mid m \cdot x = x \text{ for all } m \in M \}
\]
What is the relationship betwe... | automath | [
0.03608657792210579,
0.03115999326109886,
-0.03581537306308746,
-0.0028062888886779547,
0.034328315407037735,
-0.04959329962730408,
-0.009262721054255962,
0.04705169051885605,
0.0522819347679615,
0.0043518440797924995,
0.009089355356991291,
0.03817976266145706,
-0.02065163105726242,
-0.010... | 0.810203 | 0.790875 | 0.019328 |
Consider a bijective function \( f: [0, +\infty) \to [0, +\infty) \) satisfying the functional equation:
\[
f^{-1}(f(x)f(y))=\frac{f(x+y)-f(x)-f(y)}{2}
\]
for all non-negative real numbers \( x \) and \( y \). Given that \( f \) is continuous and satisfies the properties derived in the analysis, can you verify whethe... | automath | [
0.01912103407084942,
0.030223164707422256,
-0.03367563337087631,
-0.03575119748711586,
0.03067518211901188,
-0.02912277728319168,
-0.0018287329003214836,
0.08523277193307877,
0.03456825017929077,
-0.005832752678543329,
-0.0013413781998679042,
0.011813909746706486,
-0.017187708988785744,
-0... | 0.827008 | 0.795101 | 0.031907 |
Given the equation $$X = F_1 X F_1^T + F_2 X F_2^T + \ldots + F_p X F_p^T + C$$ where $C$ is a known positive semidefinite matrix, and assuming that $F_i$ are matrices corresponding to a certain structured form, what conditions must be satisfied for the unique solution $X$ to be positive semidefinite? Additionally, und... | automath | [
0.003276632633060217,
0.01649521477520466,
-0.0476379431784153,
-0.03670754283666611,
0.04297706112265587,
-0.024949073791503906,
-0.010604599490761757,
0.08416320383548737,
0.03309248760342598,
-0.012825398705899715,
0.012021568603813648,
0.006429408211261034,
-0.03097471036016941,
-0.020... | 0.82947 | 0.81032 | 0.01915 |
What is the exact nature of the isomorphism between \(\mathbb{R}^2\) and \(\mathbb{C}\) in terms of topological spaces, and does the Heine-Borel theorem hold true for compact sets in \(\mathbb{C}\) based on this identification? Additionally, how many different types of isomorphisms can be established between \(\mathbb{... | automath | [
0.027880802750587463,
0.04683351889252663,
-0.03802427276968956,
-0.004916633479297161,
0.03299751505255699,
0.002154361689463258,
-0.007973347790539265,
0.05270452797412872,
0.04553239420056343,
-0.032449427992105484,
0.007472471334040165,
0.001155111356638372,
-0.02081013098359108,
-0.02... | 0.797099 | 0.775575 | 0.021524 |
Consider a three-man jury consisting of two jurors who independently make the correct decision with probability \( p \), and a third juror who randomly makes a decision by flipping a coin (so the probability of making a correct decision for the third juror is \( \frac{1}{2} \)).
What is the probability that the three... | automath | [
0.017787832766771317,
0.02858828566968441,
-0.05119406059384346,
-0.017805902287364006,
0.04187610745429993,
-0.020613649860024452,
-0.012608564458787441,
0.05055002123117447,
0.0193560179322958,
-0.010800744406878948,
0.000007296644980669953,
0.025238988921046257,
-0.03258737921714783,
-0... | 0.817367 | 0.797486 | 0.019881 |
Given a directed graph \( G = (V, E) \) and a set of pairs \( (s_i, t_i) \) representing starting and ending points with associated time windows for each object traveling along the graph, if the time windows for the objects are structured such that an object \( A \) occupies node \( v \) from time \( t_{A1} \) to \( t_... | automath | [
0.013747740536928177,
0.010008201003074646,
-0.02878938429057598,
-0.005302140023559332,
0.03271931782364845,
-0.02000056952238083,
-0.034651413559913635,
0.07299666106700897,
0.023191917687654495,
-0.008564367890357971,
0.035312723368406296,
0.024211660027503967,
-0.0028129981365054846,
-... | 0.811769 | 0.784379 | 0.027391 |
Given the inhomogeneous recurrence relation defined by
\[
a_n = 1 + p a_{n-1+k} + (1-p) a_{n-1}, \quad a_0 = 0,
\]
where \(0 < p < 1\) and \(n, k\) are positive integers, determine the general form of the solution \(a_n\).
Specifically, for \(k = 2\), find the specific solution \(a_n\) and determine if it is uniqu... | automath | [
0.012974667362868786,
0.04166856035590172,
-0.027756836265325546,
-0.024627089500427246,
0.02948627434670925,
0.0009266424458473921,
-0.0021068742498755455,
0.1116541251540184,
0.026939887553453445,
-0.004359889309853315,
0.017337476834654808,
0.004841615911573172,
-0.02750353142619133,
-0... | 0.819251 | 0.79635 | 0.022901 |
Given the separable differential equation
\[
\frac{dy}{dx} = 2x(1 - y^2)^{\frac{1}{2}},
\]
determine whether the solution to the initial value problem
\[
\begin{cases}
\frac{dy}{dx} = 2x(1 - y^2)^{\frac{1}{2}}, & x > 0 \\
y(0) = y_0 \in (-1, 1]
\end{cases}
\]
is unique for all \(x > 0\). If not, provide condi... | automath | [
0.012263096868991852,
0.03832891955971718,
-0.025322087109088898,
-0.030245011672377586,
0.041486047208309174,
-0.004767908714711666,
-0.010042695328593254,
0.10845663398504257,
0.04330899566411972,
-0.002447582548484206,
0.004962431266903877,
0.012543956749141216,
-0.027326064184308052,
-... | 0.817713 | 0.795744 | 0.021968 |
Given the function \( f(\mathbf{x}) = \frac{1}{2} \|\mathbf{A}(\mathbf{y} - \mathbf{x})\|_2^2 \), where \( \mathbf{A} \) is an \( m \times n \) matrix, find the expression for the Legendre transform \( f^{\star}(\mathbf{s}) \) and identify the conditions under which the transform is maximized. Specifically, if \( \math... | automath | [
0.014921193942427635,
0.05055883526802063,
-0.051429539918899536,
-0.04091501981019974,
0.033211611211299896,
-0.001358881127089262,
-0.03136233612895012,
0.07382245361804962,
0.03669833391904831,
-0.007194219157099724,
0.016469230875372887,
0.015316728502511978,
-0.0294608473777771,
-0.00... | 0.80986 | 0.793483 | 0.016377 |
Let \( p \) be a prime number, and \( V \) be a vector space over \( \mathbb{Q} \) with dimension \( p \). Suppose \( \{ v_0, v_1, \ldots, v_{p-1} \} \) is a basis for \( V \), and let \( u_0 = v_{i_1} + v_{i_2} + \cdots + v_{i_k} \) where the indices \( i_1, i_2, \ldots, i_k \) are distinct and \( k < p \). Define \( ... | automath | [
0.04046797752380371,
0.04636877402663231,
-0.018924089148640633,
-0.02091197296977043,
0.04159080982208252,
-0.028811722993850708,
-0.0025847628712654114,
0.06486053019762039,
0.030804812908172607,
-0.007534422446042299,
0.005764358211308718,
0.025179166346788406,
-0.02758500725030899,
-0.... | 0.831394 | 0.804577 | 0.026817 |
In a circle with center \( O \), let \( A \) and \( C \) be points on the circle such that \( AC \) is a chord, and let \( AB \) be a tangent to the circle at point \( A \). If the measure of the central angle \( \angle COA \) is \( 60^\circ \), what is the measure of the angle \( \angle CAB \) formed by the tangent \(... | automath | [
0.027820680290460587,
0.019616149365901947,
-0.04943259805440903,
-0.03233415260910988,
0.018826324492692947,
-0.02702265977859497,
-0.012761689722537994,
0.05146254971623421,
0.01645227149128914,
0.007070716004818678,
0.0052912235260009766,
0.0013950965367257595,
-0.03675795719027519,
-0.... | 0.818466 | 0.800521 | 0.017945 |
Consider the operator \( T: \mathbb{C}^2 \to \mathcal{S}_2 \) defined by
\[
T(x) = \begin{bmatrix} x_1 & x_2 \\ \bar{x}_2 & x_1 \end{bmatrix}
\]
for \( x = (x_1, x_2) \in \mathbb{C}^2 \). Let \( A = \begin{bmatrix} a_1 & a_2 \\ \bar{a}_2 & a_1 \end{bmatrix} \in \mathcal{S}_2 \).
1. Compute the inner product \( \la... | automath | [
0.023410698398947716,
0.05461815744638443,
-0.031200889497995377,
-0.02834479510784149,
0.04655615612864494,
-0.011853777803480625,
-0.02491634711623192,
0.08879171311855316,
0.03830655291676521,
0.018634123727679253,
0.003299189265817404,
0.028265781700611115,
-0.040698014199733734,
-0.01... | 0.8145 | 0.797285 | 0.017215 |
Consider two PDE models of a chemical reaction, where the solutions of one model are denoted as \( u(x,t) \) and the solutions of the other as \( v(x,t) \). If it is known that as \( t \to \infty \), \( u(x,t) \) converges to a function \( L(x) \) and \( v(x,t) \) converges to the same function \( L(x) \), how would on... | automath | [
-0.012517855502665043,
0.038752246648073196,
-0.03285662457346916,
-0.024925479665398598,
0.028034044429659843,
-0.008317047730088234,
-0.03849918022751808,
0.06654191017150879,
0.029544156044721603,
-0.0011331605492159724,
-0.0018654725281521678,
0.016218284144997597,
-0.002113974653184414,... | 0.811999 | 0.795373 | 0.016626 |
Find a subset \( A \) of \( \mathbb{R}^2 \) such that the sets of accumulation points \( A' \), \( A'' \), and \( A''' \) of \( A \) are all distinct from each other and from \( A \). What is one possible construction for such a set \( A \)? | automath | [
0.04146861657500267,
0.02944420464336872,
-0.025183318182826042,
-0.025447236374020576,
0.030002744868397713,
-0.02926717884838581,
-0.019865062087774277,
0.0674465000629425,
0.05601809546351433,
-0.015538675710558891,
0.0033198585733771324,
0.014215272851288319,
-0.042457349598407745,
-0.... | 0.839894 | 0.812334 | 0.02756 |
Given a finite-dimensional complex vector space \( V \) with a Hermitian form \( h \) represented by a Hermitian matrix \( H \), if the quadratic form associated with \( h \) is defined by \( Q(z) = h(z,z) \), what condition must hold for \( Q(z) \) to be real for all \( z \in V \)? Also, if the same condition is satis... | automath | [
0.015773415565490723,
0.03630722686648369,
-0.03847423195838928,
-0.05158085748553276,
0.035405632108449936,
-0.009453858248889446,
0.004397023934870958,
0.059768740087747574,
0.029694948345422745,
-0.019656646996736526,
0.0030902486760169268,
0.027972450479865074,
-0.027475476264953613,
-... | 0.806126 | 0.785218 | 0.020909 |
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