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 |
|---|---|---|---|---|---|
Given a graph \( G = (V, E) \) with \( n \) vertices and a positive parameter \( \epsilon \), what is the maximum number \( M \) of vertex partitions \( V_1, V_2, \ldots, V_M \) that can exist such that for all but at most \( \epsilon M^2 \) pairs \( (i, j) \) (where \( 1 \leq i < j \leq M \)), the pair \( (V_i, V_j) \... | automath | [
0.034559763967990875,
0.010342448018491268,
-0.030999399721622467,
-0.015348784625530243,
0.019069693982601166,
-0.01147942803800106,
-0.03939207270741463,
0.039764534682035446,
0.016812417656183243,
-0.023947138339281082,
0.0032459916546940804,
0.02951614372432232,
-0.013374965637922287,
... | 0.832791 | 0.804826 | 0.027965 |
Given that the function to maximize is \(E(B^TX^TR) - E(XB)^TE(R)\) subject to the constraint \(E(XB) = 1\), where \(X\) is a random matrix with dimensions \(n \times m\), \(B\) is an \(m \times 1\) vector of variables, and \(R\) is a random vector of size \(n \times 1\), under what conditions can the maximum of this f... | automath | [
0.02788751944899559,
0.03262903913855553,
-0.04598744213581085,
-0.020904764533042908,
0.03630378842353821,
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-0.026727084070444107,
0.0623956099152565,
0.025455690920352936,
0.00335996737703681,
0.003396637737751007,
0.02721424028277397,
-0.02810141257941723,
0.0167294... | 0.810888 | 0.791674 | 0.019214 |
Let \( A \subset \mathbb{Z}^3 \) be a finite set, and let \( A_{xy}, A_{yz}, A_{xz} \) be its projections onto the corresponding two-dimensional planes. If the cardinalities of these sets are given by \( \# A_{xy} = a \), \( \# A_{yz} = b \), and \( \# A_{zx} = c \), what is the minimum possible value of \( \# A \) in ... | automath | [
0.02304997108876705,
0.04619334265589714,
-0.036350999027490616,
-0.02338816039264202,
0.03090638853609562,
0.0031936459708958864,
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0.02812957763671875,
-0.021003104746341705,
0.006094519980251789,
0.03275798633694649,
-0.03183950111269951,
-0.0095... | 0.836532 | 0.809938 | 0.026593 |
Let \( \{x_n\}_{n \in \mathbb{N}} \) be a sequence of real numbers that is strictly increasing and diverges to infinity. For a real number \( x \), determine the largest value of \( N \in \mathbb{N} \) such that \( x < x_N \). What can we conclude about the relationship between \( x \) and the sequence \( \{x_n\} \) as... | automath | [
0.021190522238612175,
0.047419387847185135,
-0.0395088754594326,
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0.01258509699255228,
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0.002031995914876461,
0.01689113862812519,
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0.0009675018... | 0.830641 | 0.803189 | 0.027452 |
Consider the equivalence relation $\sim$ defined on sequences $a : \mathbb{Z}^+ \rightarrow A$ such that $a \sim b$ if there are only finitely many indices $i$ where $a_i \neq b_i$. If we take the set $A = \{0, 1\}$ and analyze the equivalence classes under this relation, how many distinct equivalence classes can be fo... | automath | [
0.024929814040660858,
0.00875717680901289,
-0.020211130380630493,
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0.03093152679502964,
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0.016467269510030746,
0.01931348443031311,
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-0.00... | 0.829675 | 0.80551 | 0.024165 |
Consider a finite matroid \( M \) with rank \( r \) and ground set \( E \). If the ground set \( E \) contains \( n \) elements, how many distinct bases can \( M \) have if the rank \( r \) of the matroid is known? Specifically, if \( r = k \), express the number of bases in terms of \( n \) and \( k \).
Additionally... | automath | [
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0.02655346319079399,
-0.05276741459965706,
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0.023601461201906204,
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0.008146962150931358,
0.03787568211555481,
-0.042639799416065216,
-0.02... | 0.819658 | 0.802652 | 0.017006 |
Given a polynomial ideal $I$ in $k[x_1, x_2, \ldots, x_n]$ that defines a subvariety $X \subseteq Y = \mathbb{A}^n_k$, where $k$ is a field, let $g_1, g_2, \ldots, g_m$ be a set of generators of $I$. If the rank of the Jacobian matrix $J_x$ associated with $g_1, \ldots, g_m$ at a point $x \in X$ is $r$, what is the dim... | automath | [
0.01588061824440956,
0.027385881170630455,
-0.048114169389009476,
-0.031383831053972244,
0.006900427863001823,
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0.06366514414548874,
0.03396886587142944,
0.002144614467397332,
-0.0025465269573032856,
0.02114391326904297,
-0.036602236330509186,
-... | 0.821608 | 0.798784 | 0.022824 |
Consider the logical statement: "If \( P \) is a multiple of 6, then \( Q \) is divisible by 2." Let \( P \) be the integer 18 and \( Q \) be the integer 10.
1. Is the statement true or false based on the implication \( P \Rightarrow Q \)?
2. Construct a truth table for this implication and explain under what condit... | automath | [
0.03527715429663658,
0.03395689278841019,
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0.04280051589012146,
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0.07899069041013718,
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0.006436550058424473,
0.02525091916322708,
-0.023026691749691963,
0.0000... | 0.820664 | 0.800973 | 0.019691 |
Consider the series defined by \( u(x) = \sum_{i=1}^{\infty} h_i \psi(2^i x) \), where \( \psi \) is a function supported on \( B_2 \), linear on \( B_1 \), and \( |\nabla \psi| < c \). Let \( h_i = \frac{2^{-i}}{i} \).
1. What is the behavior of the \( W^{2,n} \) norm of \( u \) as \( n \) approaches infinity?
2. In... | automath | [
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0.05354521423578262,
-0.034265436232089996,
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0.016243841499090195,
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0.004110474139451981,
0.0003044009499717504,
0.007726415526121855,
-0.014191736467182636,
... | 0.812723 | 0.795091 | 0.017631 |
Let points \( P_0, P_1, P_2 \) be defined in \( \mathbb{R}^3 \) with position vectors \( \mathbf{r}_0, \mathbf{r}_1, \mathbf{r}_2 \). If the point \( (x,y,z) \) lies on the plane determined by these three points, express the condition that the vector \( \mathbf{r} \) from the origin to \( (x,y,z) \) satisfies the plane... | automath | [
0.024748006835579872,
0.04530363157391548,
-0.05963427573442459,
-0.056231141090393066,
0.028298111632466316,
0.006251422688364983,
-0.019149739295244217,
0.06200047954916954,
0.033682502806186676,
-0.009076963178813457,
0.00283628236502409,
0.03857104480266571,
-0.03008677437901497,
-0.01... | 0.818169 | 0.799039 | 0.01913 |
Let \( S \) be the closed set defined by the unit ball in \( \mathbb{R}^2 \), that is, \( S = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1 \} \). Let \( T \) be the closed set represented by the circle of radius 2 centered at the origin, defined as \( T = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 = 4 \} \). What is the d... | automath | [
0.010018303990364075,
0.042120955884456635,
-0.0626271441578865,
-0.0306406170129776,
0.051187463104724884,
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0.03609270602464676,
0.04472405090928078,
0.005174514837563038,
0.02141009084880352,
0.03683798015117645,
-0.017205389216542244,
-0.00672... | 0.82679 | 0.804169 | 0.022621 |
For which even value of \( p \) such that \( p \leq 1000 \), does the equation \( a^2 + b^2 = c^2 \) yield the maximum number of integer right triangles \( (a, b, c) \) with the perimeter equal to \( p \)? | automath | [
0.04201586917042732,
0.024632945656776428,
-0.04779079928994179,
-0.008698093704879284,
0.017796337604522705,
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0.04865610599517822,
0.026183979585766792,
-0.01702762022614479,
0.008001015521585941,
0.06042921170592308,
-0.023378098383545876,
-0.0... | 0.843406 | 0.825189 | 0.018217 |
Let \( X \) and \( Y \) be two compact Hausdorff spaces, each of Cantor-Bendixson rank \( \alpha + 1 \) and degree 1. Describe a method to construct a homeomorphism between \( X \) and \( Y \). Specifically, how many clopen sets are required in the partitioning of \( X \setminus X^\alpha \) and \( Y \setminus Y^\alpha ... | automath | [
0.0156865157186985,
0.02654382586479187,
-0.04032142832875252,
-0.022530345246195793,
0.02930542081594467,
0.0086821299046278,
-0.017074519768357277,
0.06373615562915802,
0.05239458382129669,
-0.021469272673130035,
0.004526182077825069,
0.03846106678247452,
-0.03463560715317726,
-0.0170380... | 0.8173 | 0.796813 | 0.020486 |
Let \( A \) and \( B \) be nonempty finite sets with \( |A| = m \) and \( |B| = n \). Suppose there are functions \( f: A \rightarrow B \) and \( g: B \rightarrow A \) such that \( g \circ f = i_A \), the identity function on \( A \). If \( g \) is injective, what can you conclude about the relationship between \( m \)... | automath | [
0.04354432597756386,
0.03673545643687248,
-0.04243859648704529,
-0.005590829998254776,
0.030221063643693924,
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-0.025288671255111694,
0.03749284893274307,
0.03058636747300625,
-0.0034059633035212755,
0.013630171306431293,
0.028778722509741783,
-0.018175490200519562,
-0... | 0.825267 | 0.799398 | 0.02587 |
Let \( I = (0,1] \) and \( T = \{ (x,y) \in I^2 : x \geq y \} \). Given functions \( f: I \to \mathbb{R} \) and \( w: T \to \mathbb{R} \) that are both analytic, consider the function defined by the integral
\[
A_w f(y) = \int_y^1 \frac{f(x) w(x,y)}{\sqrt{x^2 - y^2}} \, dx.
\]
Is the function \( A_w f \) analytic in... | automath | [
0.018542764708399773,
0.03571400046348572,
-0.03610057011246681,
-0.021285608410835266,
0.03952232003211975,
0.007694442756474018,
-0.02714822068810463,
0.07628355175256729,
0.035341281443834305,
0.003950434271246195,
0.016646940261125565,
0.011520943604409695,
-0.027561133727431297,
-0.00... | 0.814201 | 0.787219 | 0.026982 |
Given two parameters \(\theta_0 < \theta_1\) associated with the likelihood functions of random samples \(X_1, \dots, X_n\), let the n-th order statistic be denoted as \(x_{(n)}\). If it is true that the ratio of likelihood functions is expressed as
\[
\frac{L(x; \theta_1)}{L(x; \theta_0)} = \frac{I(x_{(n)} < \theta_... | automath | [
-0.01158206444233656,
0.025192750617861748,
-0.05640239268541336,
-0.03586629778146744,
0.03993382677435875,
0.010458124801516533,
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0.06967023015022278,
0.021144291386008263,
-0.012615042738616467,
-0.009088193997740746,
0.016565145924687386,
-0.03648170828819275,
0.00... | 0.807705 | 0.790417 | 0.017289 |
Let \(\alpha = \sqrt{2} + \sqrt{3}\). For which positive integers \(n\) can we find a polynomial \(r(x) \in \mathbb{Q}[x]\) such that \(r(\alpha) = \sqrt{n}\)? Please provide the square-free part of \(n\) that meets this criterion. | automath | [
0.03555040434002876,
0.039209168404340744,
-0.04839872196316719,
-0.02874533087015152,
0.02145136520266533,
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0.04712240770459175,
0.032057516276836395,
-0.021648192778229713,
0.004960114136338234,
0.03413078188896179,
-0.04014046490192413,
-0.056... | 0.840263 | 0.817378 | 0.022885 |
Given stochastic differential equations of the form
$$dX_t = B(t,X_t)dt + \Sigma(t,X_t)dW_t$$
where the coefficients $B$ and $\Sigma$ satisfy specific properties, under what conditions can we guarantee either the uniqueness of the strong solution or the possibility of multiple solutions? Specifically, if we define ... | automath | [
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0.009223214350640774,
-0.040046535432338715,
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0.031836047768592834,
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0.08150152862071991,
0.034603625535964966,
-0.03177361190319061,
0.02051597647368908,
0.010516906157135963,
-0.030040318146348,
0.... | 0.794437 | 0.778073 | 0.016364 |
Given a convex set \( C \) and an affine set \( D \) represented by \( D = \{ Fu + g \mid u \in \mathbb{R}^m \} \), where \( F \in \mathbb{R}^{n \times m} \) and \( C \) and \( D \) are disjoint, if we know that there exists a vector \( a \neq 0 \) such that \( a^\top F = 0 \) and \( b \leq a^\top g \) for all \( x \in... | automath | [
0.013806398957967758,
0.045705147087574005,
-0.05010225996375084,
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0.03565290570259094,
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0.07355453073978424,
0.0546773225069046,
0.008449932560324669,
0.02013293094933033,
0.029812218621373177,
-0.04635320603847504,
-0.021... | 0.834809 | 0.809849 | 0.02496 |
Is it possible to pack an infinite family of disjoint discs with radii \( \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \) inside a unit disc such that each disc is completely contained within the unit disc and no two discs overlap? If so, provide a constructive method to demonstrate that such a packing exists. | automath | [
0.013829754665493965,
0.042703840881586075,
-0.023486468940973282,
-0.01583731733262539,
0.02383752539753914,
0.0014760988997295499,
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0.043567970395088196,
0.03483237326145172,
0.0034378129057586193,
0.015690160915255547,
0.014325111173093319,
-0.023616701364517212,
-... | 0.826561 | 0.796218 | 0.030343 |
Consider the Schrödinger semigroup \( e^{it\Delta} \) acting on the Sobolev space \( W^{m,2}(\mathbb{R}^n) \). If \( f \in W^{m,2}(\mathbb{R}^n) \), what is the relationship between the norms of \( e^{it\Delta}(f) \) in the weighted Bergman space \( \mathcal{H}{\mathcal{L}}^{2}(\mathbb{C}^n, u_t^{m}(z)dz) \) and the So... | automath | [
-0.007608288433402777,
0.03533519059419632,
-0.03953452408313751,
-0.0392887108027935,
0.03373929485678673,
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0.0680566281080246,
0.03303759545087814,
-0.005413927137851715,
0.00007562851533293724,
0.022438792511820793,
-0.02310802787542343,
-0.... | 0.806689 | 0.788671 | 0.018018 |
Consider a set of \( n \) distinct positive integers. Let \( S \) be the set of all possible subset sums that can be formed from these integers. Define \( S_{\text{sorted}} \) as the sorted sequence of the elements in \( S \).
1. What is the time complexity of generating the sorted sequence \( S_{\text{sorted}} \) fr... | automath | [
0.03026415966451168,
0.016564954072237015,
-0.04525395855307579,
-0.011408135294914246,
0.016223488375544548,
-0.013064409606158733,
-0.02995717152953148,
0.05369975417852402,
0.03685673698782921,
0.0037006502971053123,
0.026037855073809624,
0.016257300972938538,
-0.04684411734342575,
-0.0... | 0.825793 | 0.802316 | 0.023477 |
Given a point \( A \) outside a circle where \( AT \) is a tangent to the circle at point \( T \) and \( AC \) is a secant that intersects the circle at points \( B \) and \( C \). Let the lengths of the perpendiculars from points \( B \) and \( C \) to line \( AT \) be \( BD = b \) and \( CE = c \), respectively, with... | automath | [
0.019642334431409836,
0.03121238388121128,
-0.03912854939699173,
-0.021709460765123367,
0.030505666509270668,
-0.009029721841216087,
-0.0392325296998024,
0.05323529615998268,
0.044447749853134155,
0.003660793649032712,
0.005078624468296766,
0.027489088475704193,
-0.03715839982032776,
-0.00... | 0.828329 | 0.804518 | 0.023811 |
Let \( a(z) \) and \( b(z) \) be two non-zero complex polynomials. Under what conditions is the product \( a(z) \overline{b(z)} \) analytic? Provide a detailed explanation of which of the following statements must be true:
1. \( a(z) \) is constant.
2. \( a(z)b(z) \) is constant.
3. \( \overline{a(z)}b(z) \) is consta... | automath | [
0.03604529798030853,
0.03633378818631172,
-0.03660308197140694,
-0.03979632258415222,
0.03517930209636688,
0.006318805273622274,
0.0006239578942768276,
0.061328303068876266,
0.03716253861784935,
0.010105391032993793,
0.007786857895553112,
0.015195834450423717,
-0.029194584116339684,
-0.018... | 0.814152 | 0.795318 | 0.018835 |
Consider the quadratic equation \( x^2 - kx + (k^2 + k - 5) = 0 \) with real roots \( \alpha \) and \( \beta \). The expression for \( \alpha^2 + \beta^2 \) can be rewritten as \( -k^2 - 2k + 10 \). Determine the maximum and minimum values of \( \alpha^2 + \beta^2 \) as \( k \) varies over all real numbers, and compute... | automath | [
0.037618305534124374,
0.03739058971405029,
-0.051262736320495605,
-0.03290491923689842,
0.0416109599173069,
-0.03414028510451317,
0.003516711760312319,
0.08384472876787186,
0.035727858543395996,
-0.013024291023612022,
0.00923274364322424,
0.0249685849994421,
-0.034552235156297684,
-0.02301... | 0.82184 | 0.803706 | 0.018134 |
Is it true that for the sequences satisfying \( 0 \le a_n \le 1 \) and the conditions that \( \sum_{n=0}^{\infty} a_n s^n \) is convergent for all \( 0 < s < 1 \) while \( \sum_{n=0}^{\infty} a_n \) diverges, the equality
\[
\lim_{s\to 1^{-}} \sum_{n=0}^{\infty} (a_n-a_{n-1}) s^n = \lim_{m\to\infty} \frac{1}{m+1} \su... | automath | [
0.009177863597869873,
0.03995294123888016,
-0.027978044003248215,
-0.028570614755153656,
0.024101082235574722,
-0.012060335837304592,
-0.022199679166078568,
0.06159115210175514,
0.027393369004130363,
-0.011104181408882141,
-0.005410634446889162,
0.011738641187548637,
-0.042608845978975296,
... | 0.818828 | 0.79943 | 0.019398 |
Consider a Poisson problem defined on a unit square domain \(\Omega\) with the following boundary conditions:
- Dirichlet conditions \(u = 1\) on \(x=0\) (denoted \(\GD^0\)) and \(u = 3\) on \(x=1\) (denoted \(\GD^1\)).
- A Robin condition \( -\kappa \frac{\partial u}{\partial n} = r(u-s) \) on \(y=0\) (denoted \(\GR... | automath | [
0.005737501662224531,
0.0417206771671772,
-0.032678328454494476,
-0.0519462451338768,
0.040749162435531616,
-0.015458215028047562,
-0.013488813303411007,
0.06627646833658218,
0.0346248596906662,
0.022291945293545723,
0.01057520043104887,
0.03501346334815025,
-0.021204734221100807,
-0.02026... | 0.820346 | 0.801547 | 0.018799 |
Given the polygonal presentation of a compact surface represented by the word $\langle S | W_1, W_2, \ldots, W_k \rangle$, where $W_i$ are the distinguishing edge labels, how can I systematically determine the number of equivalence classes of vertices, the number of distinct edges, and the number of faces in order to c... | automath | [
0.006455175578594208,
0.020414628088474274,
-0.03348344564437866,
-0.03861450031399727,
0.02492307312786579,
-0.019491517916321754,
-0.022530246526002884,
0.05426687002182007,
0.026704173535108566,
-0.002344558946788311,
0.011658847332000732,
0.03640177100896835,
-0.023942768573760986,
-0.... | 0.820701 | 0.803805 | 0.016896 |
Evaluate the following limits:
1. \( L_1 = \lim\limits_{n\rightarrow\infty}\int_{0}^{1}\frac{1+n^2x^2}{(1+x)^n}dx \)
2. \( L_2 = \lim_{n\rightarrow\infty}\int_{0}^{\infty}ne^{-nx}\sin(1/x)dx \)
What are the values of \( L_1 \) and \( L_2 \)? | automath | [
0.007239079102873802,
0.05621224269270897,
-0.042004745453596115,
-0.021887870505452156,
0.03387913480401039,
-0.005605389829725027,
-0.019830625504255295,
0.06024497374892235,
0.0405416376888752,
0.004688648506999016,
0.008594930171966553,
0.02089785411953926,
-0.026722807437181473,
-0.02... | 0.817462 | 0.800869 | 0.016593 |
In the context of an infinite-dimensional complex vector space \( V \), let \( A : V \times V \to \mathbb{C} \) be a sesquilinear form represented by its matrix entries \( A_{ij} = A(e_i, e_j) \), where \( \{ e_i \} \) is a basis for \( V \). What are the necessary and sufficient conditions on the entries \( A_{ij} \) ... | automath | [
0.019738461822271347,
0.037602197378873825,
-0.03867289051413536,
-0.032539840787649155,
0.030624261125922203,
0.0003352594212628901,
-0.014577159658074379,
0.07265636324882507,
0.030986053869128227,
-0.010165205225348473,
-0.00724043371155858,
0.026991184800863266,
-0.015184606425464153,
... | 0.818994 | 0.790681 | 0.028314 |
Let \( x_1, x_2, \ldots, x_n \) be \( n \) positive real numbers and consider the function \( f(x) = \log(x) \). What is the value of the quasi-arithmetic mean \( M_f(x_1, x_2, \ldots, x_n) \) for these numbers, and how does it relate to the geometric mean of \( x_1, x_2, \ldots, x_n \)? | automath | [
0.015291151590645313,
0.04490165784955025,
-0.0740106925368309,
-0.04520437493920326,
0.031511325389146805,
0.000013748122910328675,
-0.029559258371591568,
0.037807971239089966,
0.024718446657061577,
-0.009041986428201199,
-0.00017728196689859033,
0.007751204073429108,
-0.03986318036913872,
... | 0.832951 | 0.815373 | 0.017577 |
Given the expenditure minimization problem defined by the objective function $$\min_{x \geq 0, y \geq 0} p_X x + p_Y y$$ subject to the constraint $$x + 3y \geq U$$ where $$p_X > 0$$, $$p_Y > 0$$, and $$U \geq 0$$, determine the conditions under which the optimal value of $$x$$, denoted as $$x^h(p_X, p_Y, U)$$, takes o... | automath | [
-0.0005843672552146018,
0.02507585473358631,
-0.04950330778956413,
-0.017089836299419403,
0.04812794178724289,
0.00661064125597477,
-0.023316288366913795,
0.06152528524398804,
0.05252017080783844,
-0.005863114260137081,
0.01755806989967823,
0.029270779341459274,
-0.014702760614454746,
-0.0... | 0.822707 | 0.804785 | 0.017922 |
Given a trapezoid \( ABCD \) where \( AD \parallel BC \) with lengths \( a = AD \) and \( b = BC \), let \( M \) be the midpoint of \( AB \) and \( K \) the midpoint of \( CD \). The diagonals \( AC \) and \( BD \) intersect the line \( KM \) at points \( P \) and \( Q \), respectively. Express the distance \( PQ \) in... | automath | [
0.04583169147372246,
0.008970088325440884,
-0.0403493233025074,
-0.05634472146630287,
0.020045999437570572,
-0.015679389238357544,
-0.029432279989123344,
0.046579111367464066,
0.043649449944496155,
0.009641562588512897,
0.02594926208257675,
0.026228591799736023,
-0.021804682910442352,
-0.0... | 0.828715 | 0.805588 | 0.023127 |
Let \( f : X \to S \) be a morphism of schemes satisfying the equivalent conditions for being étale as stated in Lemma 29.35.2. Suppose \( f \) is étale and \( X \) and \( S \) are both affine schemes with \( X = \text{Spec}(R) \) and \( S = \text{Spec}(A) \). If \( R \) is a finite \( A \)-algebra such that \( R \) is... | automath | [
0.03538239747285843,
0.03307104855775833,
-0.026734979823231697,
-0.037453606724739075,
0.023915767669677734,
-0.04123625531792641,
-0.013353445567190647,
0.053768567740917206,
0.020018797367811203,
-0.005008087493479252,
0.030684933066368103,
0.03286872059106827,
-0.043832022696733475,
0.... | 0.807671 | 0.790365 | 0.017306 |
Let \( g \) be the genus of a smooth projective curve \( X \) over an algebraically closed field \( k \). If \( n \geq 1 \) is an integer invertible in \( k \), determine the rank of the group \( \mathop{\mathrm{Pic}}\nolimits (X)[n] \). | automath | [
0.013235337100923061,
0.035868994891643524,
-0.055803727358579636,
-0.0402841717004776,
0.036867234855890274,
-0.0339399129152298,
-0.01679418422281742,
0.039823342114686966,
0.044797465205192566,
-0.008008765988051891,
0.013661346398293972,
0.03367135301232338,
-0.029262633994221687,
-0.0... | 0.821393 | 0.796223 | 0.02517 |
Given a rectangle ABCO, with vertices C and B on the x-axis and y-axis respectively, and another vertex A on the line defined by the equation \( 2x + y = 4 \), find the following:
1. Calculate the area \( R \) of rectangle ABCO in terms of the x-coordinate \( a \) of point A.
2. Determine the range of values for \( a ... | automath | [
0.03252561017870903,
0.046507276594638824,
-0.03456434607505798,
-0.026815140619874,
0.014545035548508167,
-0.023393329232931137,
-0.01589406281709671,
0.06422443687915802,
0.05201932415366173,
0.01863427832722664,
0.00393378920853138,
0.04122917354106903,
-0.014405556954443455,
-0.0052253... | 0.821689 | 0.799841 | 0.021849 |
Given two complex numbers \( z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \) and \( z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \), express the product \( z_1 z_2 \) in terms of its magnitude and angle. Specifically, calculate the magnitude of \( z_1 z_2 \) and find the argument of \( z_1 z_2 \) in terms of \( r_1 \), \... | automath | [
0.038876425474882126,
0.04557371884584427,
-0.03539252653717995,
-0.04999352619051933,
0.02804238349199295,
-0.0046361349523067474,
-0.02392527088522911,
0.052784498780965805,
0.02467815764248371,
-0.009824549779295921,
0.00574962142854929,
0.029602810740470886,
-0.03593887761235237,
-0.02... | 0.813332 | 0.79672 | 0.016612 |
Let $(R, \mathfrak m)$ be a local Noetherian ring, and let $M$ be a nonzero finite $R$-module. If $f_1, \ldots, f_c \in \mathfrak m$ form an $M$-quasi-regular sequence, what can be concluded about the number of elements in a minimal generating set for the ideal generated by $f_1, \ldots, f_c$ in $M$, given that the ele... | automath | [
0.024617193266749382,
0.03163404017686844,
-0.04119579866528511,
-0.03703445568680763,
0.029597289860248566,
-0.01497061550617218,
-0.021875297650694847,
0.060798391699790955,
0.025813184678554535,
-0.03113534487783909,
0.001916899811476469,
0.025562064722180367,
-0.024219980463385582,
-0.... | 0.814586 | 0.794428 | 0.020158 |
Consider the function defined by the recursion
\[
f_k(n) = \sum_{1 \leq i \leq \frac{n}{2}} \binom{n}{i} f_k(i) f_k(n - i)
\]
with the initial conditions \( f_k(0) = 1 \) and \( f_k(1) = k \).
1. Determine an explicit expression or an asymptotic bound for \( f_{2^{n+1}}(2^n) \).
2. Does the inequality \( f_{2^{n+... | automath | [
0.03125997632741928,
0.055565863847732544,
-0.039413075894117355,
-0.023136671632528305,
0.02475729025900364,
-0.009316756390035152,
-0.03612252697348595,
0.07875830680131912,
0.014335647225379944,
0.012161435559391975,
0.016059542074799538,
0.013539253734052181,
-0.025654809549450874,
0.0... | 0.824064 | 0.802505 | 0.021559 |
Given that \( A_n = \frac{q^n}{n!} \int_0^\pi (x(\pi - x))^n \sin x \, dx \) is shown to be an integer for all integers \( n \), and it is established that \( 0 < A_n < \pi \frac{(q \pi^2 / 4)^n}{n!} \), what can you conclude about the limit of \( A_n \) as \( n \) approaches infinity if \( \pi \) is assumed to be rati... | automath | [
0.017740963026881218,
0.04969462379813194,
-0.01627556048333645,
-0.02065042220056057,
0.013137546367943287,
-0.0021587726660072803,
-0.008942047134041786,
0.044828154146671295,
0.033041778951883316,
-0.02092100866138935,
0.017508434131741524,
0.01699863187968731,
-0.04274008423089981,
-0.... | 0.828296 | 0.809595 | 0.018701 |
Is there a known characterization of partial orders $\mathbb{P}$ for which the Boolean completion of $\mathbb{P}$ has a $\kappa$-closed dense subset, in terms of properties of their generic extensions? If such a characterization exists, can you provide it and discuss its implications? | automath | [
0.024844985455274582,
0.02669852040708065,
-0.0410592257976532,
-0.033112574368715286,
0.014776689000427723,
-0.029071301221847534,
0.0021725164260715246,
0.03661366179585457,
0.025146303698420525,
-0.021107839420437813,
0.03719387203454971,
0.015171250328421593,
-0.038534924387931824,
0.0... | 0.812314 | 0.795364 | 0.01695 |
Let \( N \) be a positive integer with the prime factorization \( N = p_1^{a_1} p_2^{a_2} \cdots p_m^{a_m} \), where \( p_1, p_2, \ldots, p_m \) are distinct primes and \( a_1, a_2, \ldots, a_m \) are their respective positive integer exponents. Define the function \( \omega(N) \) to be the number of distinct prime fac... | automath | [
0.0330086313188076,
0.0271598007529974,
-0.03584115579724312,
-0.010998763144016266,
0.023583322763442993,
-0.020520970225334167,
-0.020568732172250748,
0.046253375709056854,
0.03589113429188728,
-0.012244722805917263,
-0.005299828015267849,
0.029717225581407547,
-0.03550713509321213,
-0.0... | 0.825136 | 0.80882 | 0.016316 |
Consider the generalized affine transformation matrix in 2D given by
\[
\begin{pmatrix} a_{11} & a_{12} & t_{1} \\ a_{21} & a_{22} & t_{2} \\ z_{1} & z_{2} & j \end{pmatrix},
\]
where \( z_{1}, z_{2} \) are any real numbers and \( j \) is a nonzero constant. If this matrix is applied to the homogeneous coordinate ve... | automath | [
0.03161415085196495,
0.03769869729876518,
-0.025776926428079605,
-0.05080891028046608,
0.060310497879981995,
-0.02131502330303192,
-0.011470370925962925,
0.06866859644651413,
0.022309837862849236,
-0.0005762689979746938,
0.0027665274683386087,
0.023384589701890945,
-0.041725266724824905,
-... | 0.804644 | 0.786904 | 0.01774 |
Is there an explicit set of rational numbers \( X \) such that for any interval \( Y \) of rational numbers, the limit
\[
\lim_{n \to \infty} \frac{\text{card}(X[n] \cap Y)}{\text{card}(\mathbb{Q}[n] \cap Y)} = \frac{1}{2} ?
\]
If so, describe the construction of such a set \( X \). | automath | [
0.025115275755524635,
0.03356907516717911,
-0.03560298681259155,
-0.014090686105191708,
0.0167225431650877,
-0.009559346362948418,
0.0017047792207449675,
0.03255017474293709,
0.0532652921974659,
-0.032877154648303986,
0.014721032232046127,
0.02161233127117157,
-0.04486384987831116,
-0.0092... | 0.840534 | 0.815754 | 0.024779 |
A square with an area of 4 is inscribed in a square with an area of 5, with one vertex of the smaller square on each side of the larger square. If a vertex of the smaller square divides a side of the larger square into two segments of lengths \( a \) and \( b \), what is the product \( ab \)?
• \( \frac{1}{5} \)
• ... | automath | [
0.03921377286314964,
0.043172042816877365,
-0.056527648121118546,
-0.03795411437749863,
0.040309805423021317,
-0.006697016302496195,
-0.021379433572292328,
0.0744633600115776,
0.0331815741956234,
0.014917665161192417,
0.007652013096958399,
0.03158145025372505,
-0.024224095046520233,
-0.023... | 0.835012 | 0.818796 | 0.016216 |
Given a sequence of non-negative measurable functions \( f_n: \mathbb{N} \rightarrow \mathbb{R}^+ \) such that \( \sum_{k=1}^{\infty} f_n(k) < \infty \) for each \( n \), use Fatou's lemma to determine whether the inequality
\[
\sum_{k=1}^{\infty} \liminf_{n \to \infty} f_n(k) \leq \liminf_{n \to \infty} \sum_{k=1}^{... | automath | [
0.011963929049670696,
0.03874606639146805,
-0.02694104053080082,
-0.04002754017710686,
0.027787944301962852,
0.004782800562679768,
-0.0143183134496212,
0.06554172933101654,
0.02734154649078846,
-0.002391104819253087,
-0.008420199155807495,
0.02864215336740017,
-0.04299585893750191,
0.00574... | 0.823495 | 0.794921 | 0.028573 |
Given the system of linear equations in three variables:
\[
\begin{align*}
{{a}_{1}}x + {{b}_{1}}y + {{c}_{1}}z + {{d}_{1}} = 0 \\
{{a}_{2}}x + {{b}_{2}}y + {{c}_{2}}z + {{d}_{2}} = 0 \\
{{a}_{3}}x + {{b}_{3}}y + {{c}_{3}}z + {{d}_{3}} = 0
\end{align*}
\]
If it is known that \(\Delta = \left| \begin{matrix} {{a}_{1}}... | automath | [
0.023784514516592026,
0.02491850033402443,
-0.05556340888142586,
-0.045399460941553116,
0.04518737271428108,
-0.014292370527982712,
-0.013244575820863247,
0.07506957650184631,
0.03912346437573433,
-0.024477455765008926,
0.006734069902449846,
0.020624537020921707,
-0.027691630646586418,
-0.... | 0.827091 | 0.810098 | 0.016993 |
Given two rotated ellipses defined by their centers \((x_1, y_1)\) and \((x_2, y_2)\), semimajor axes \(a_1\) and \(a_2\), semiminor axes \(b_1\) and \(b_2\), and rotation angles \(\theta_1\) and \(\theta_2\), describe how to find the pairs of points \((x_1, y_1)\) on the first ellipse and \((x_2, y_2)\) on the second ... | automath | [
-0.005820841528475285,
0.03594309836626053,
-0.0371488481760025,
-0.036136694252491,
0.021812859922647476,
-0.004667568486183882,
-0.008300112560391426,
0.055084820836782455,
0.0424436554312706,
-0.012177330441772938,
0.0009037401177920401,
0.013868221081793308,
-0.027041105553507805,
-0.0... | 0.813483 | 0.786727 | 0.026755 |
Let \( A \) and \( B \) be rings such that the morphism \( f: \text{Spec}(A) \to \text{Spec}(B) \) is a birational morphism with finitely many irreducible components. If \( \mathfrak{q} \) is the unique minimal prime of \( B \) and \( \mathfrak{p} \) is the unique minimal prime of \( A \) lying over \( \mathfrak{q} \),... | automath | [
0.03745900094509125,
0.049914803355932236,
-0.0296043548732996,
-0.01957050710916519,
0.03186759725213051,
-0.031143885105848312,
0.004374686628580093,
0.06633056700229645,
0.05454163998365402,
-0.022392094135284424,
0.003098537214100361,
-0.0066966223530471325,
-0.011863269843161106,
-0.0... | 0.816441 | 0.796809 | 0.019631 |
Consider a bipartite graph \( G \) where one set of nodes contains 2 vertices and the other set contains 3 vertices. If a maximum matching can be achieved and the resulting minimum vertex cover has been determined, how many vertices are included in the minimum vertex cover of the graph? What is the maximum independent ... | automath | [
0.021099308505654335,
0.005231237504631281,
-0.04948610067367554,
-0.024374276399612427,
0.02054622955620289,
-0.02057192102074623,
-0.015392672270536423,
0.034567009657621384,
0.03166695311665535,
-0.021080760285258293,
0.03338955342769623,
0.012476595118641853,
-0.0185810886323452,
0.001... | 0.825601 | 0.80839 | 0.017211 |
Let \( p(x) \) and \( q(x) \) be non-zero polynomials with real coefficients, and suppose the ratio \( \frac{p(x)}{q(x)} \) is also a polynomial. Let \( \deg(p) = a \) and \( \deg(q) = b \).
What is the relationship between the degrees \( a \) and \( b \) of the polynomials \( p \) and \( q \), and what condition mus... | automath | [
0.02519592083990574,
0.02845931239426136,
-0.0437135249376297,
-0.03361892327666283,
0.0007167725125327706,
0.005142468959093094,
0.006282411050051451,
0.03425009176135063,
0.02946127951145172,
-0.004926295485347509,
-0.0003827504988294095,
0.03473344072699547,
-0.024095522239804268,
-0.02... | 0.816805 | 0.796979 | 0.019826 |
Consider a completely regular Hausdorff space \( X \) such that there exists an ideal \( I \subseteq C(X) \) with \( |Z(I)| = n \) for some positive integer \( n \). Given that \( Z(I) \) represents the collection of zero sets of all functions in \( I \), characterize the structure of the ideal \( I \) in terms of its ... | automath | [
0.032323602586984634,
0.04387044534087181,
-0.047552838921546936,
-0.01582259312272072,
0.0381711982190609,
0.001724654110148549,
-0.0039977701380848885,
0.054571766406297684,
0.03309981897473335,
-0.01648589037358761,
0.002036427380517125,
0.014395887032151222,
-0.02374143712222576,
-0.01... | 0.818043 | 0.785978 | 0.032065 |
In triangle \( OAB \), let \( OA = a \) and \( OB = b \). Given that point \( P \) divides segment \( AB \) in the ratio \( AP:PB = 2:1 \) and point \( Q \) is such that \( OP = 3 \cdot QP \), can you find the position of point \( R \) on \( OA \) such that line \( RQ \) is parallel to \( AB \), and demonstrate that \(... | automath | [
0.04216902703046799,
0.01921454258263111,
-0.04804660752415657,
-0.04789389669895172,
0.011505108326673508,
-0.005133835133165121,
-0.006635311990976334,
0.07173608243465424,
0.028234858065843582,
0.007863609120249748,
0.011570390313863754,
0.03147135674953461,
-0.010618259198963642,
-0.00... | 0.829247 | 0.810634 | 0.018613 |
Let \( R \) be a commutative ring with unity, and \( M \) be a finitely generated module over \( R \). If \( \phi: M \to M \) is an injective \( R \)-module homomorphism, under what conditions on \( R \) does this imply that the dual map \( \phi^*: M^* \to M^* \) is surjective? Further, if \( \phi^*: M^* \to M^* \) is ... | automath | [
0.035040829330682755,
0.02485586889088154,
-0.0032069231383502483,
-0.02850014716386795,
0.02274268865585327,
-0.027765780687332153,
-0.008273943327367306,
0.06139259785413742,
0.043048735707998276,
-0.016725413501262665,
0.01331284362822771,
0.01924039050936699,
-0.025220798328518867,
-0.... | 0.799539 | 0.777841 | 0.021698 |
Consider the problem where you flip two fair coins and observe their outcomes. Define the random variable \( X \) as the value of the first coin, where \( X \) can be 0 or 1; and let \( Y \) be the sum of the outcomes of the two coins, so \( Y \) can take values in \{0, 1, 2\}. If you are observing \( X \) and want to ... | automath | [
-0.0014585898024961352,
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0.03352560102939606,
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0.... | 0.817482 | 0.798741 | 0.018741 |
Let \(\mathbf{F} = (3y, -xz, yz^2)\) and the surface \(S\) be defined as \(S = \{(x,y,z): z = \frac{1}{2}(x^2+y^2), z \leq 2\}\). Calculate the value of the surface integral \(\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\).
Provide detailed calculations and confirm whether the result matches with the applicat... | automath | [
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0.022099973633885384,
0.038617052137851715,
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-0.028... | 0.804779 | 0.7864 | 0.018379 |
Let \( X = [0, 1] \) be a closed interval in the real numbers, and let \( D \) be the set of all rational numbers in \( [0, 1] \). Define \( E = \mathbb{Q} \) (the set of all rational numbers).
1. Is the set \( D \) dense in the interval \( X \)?
2. Is the closure of \( D \) in \( \mathbb{R} \) equal to \( X \)?
Pr... | automath | [
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-0.02... | 0.836541 | 0.813462 | 0.023079 |
Let \( S = \{ \lambda_1, \lambda_2, \ldots, \lambda_n \} \) be a finite set of \( n \) real numbers, not all equal. Determine which of the following statements is true:
1. There exists an \( n \times n \) matrix with complex entries that is not self-adjoint, such that the eigenvalues of this matrix are precisely given... | automath | [
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Consider the metric space \( \mathbb{R} \) and the set \( E = \{0\} \cup \{1/n \mid n = 1, 2, 3, \ldots\} \). Define \( E' \) to be the set of all limit points of \( E \). What are the limit points of \( E \), and are the limit points of \( E \) the same as the limit points of \( E' \)? Please provide a detailed explan... | automath | [
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Let \( A \) be the coefficient matrix given by
\[
A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\]
and consider the augmented matrix
\[
\left( A \mid b \right) = \begin{pmatrix} 0 & 1 & 0 & -7 \\ 0 & 0 & 1 & 10 \end{pmatrix}.
\]
Determine the geometric interpretation of the solution set for the system ... | automath | [
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0.03095223195850849,
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-0.02... | 0.819305 | 0.800485 | 0.01882 |
Let \( S \) be a graded ring such that \( \text{Proj}(S) \) is nonempty and integral, but not isomorphic to \( \mathbf{P}^n_A \) for any ring \( A \) and any integer \( n \geq 0 \). Determine the number of such examples \( S \) that can be constructed, providing at least one specific graded ring that meets these criter... | automath | [
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... | 0.827108 | 0.793244 | 0.033864 |
Given a bivariate normal distribution with means \(\mu_X\) and \(\mu_Y\), standard deviations \(\sigma_X\) and \(\sigma_Y\), and correlation coefficient \(\rho\), determine the conditions under which the expected values \(\Bbb E(Y\mid X=x)\) and \(\Bbb E(X\mid Y=y)\) yield the same linear representation in their respec... | automath | [
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... | 0.812677 | 0.794417 | 0.018261 |
Define a binary operation \( \circ \) on \( \mathbb{R} \) by \( a \circ b = a^3 b \). Determine which axioms of a field are satisfied by the set \( \mathbb{R} \) with the usual addition and this new operation as multiplication, and identify which specific axiom fails. Provide a detailed justification for your findings. | automath | [
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0.02984260767698288,
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0.008079033344984055,
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Given a large number \( N \) of independent and identically distributed random variables \( X_i \) that follow a standard normal distribution \( N(0, 1) \), let \( Q \) be defined such that
\[
P[-Q \le \min(X_1, \ldots, X_N) \le \max(X_1, \ldots, X_N) \le Q] = C.
\]
Using the derived expression for \( Q \):
\[
Q = ... | automath | [
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0.00755462097004056,
0.03515465557575226,
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0.00... | 0.82328 | 0.804761 | 0.018519 |
Consider two sets: \( A = \{ \text{Ace, King, Queen} \} \) representing the ranks of cards and \( B = \{ \heartsuit, \diamondsuit, \clubsuit, \spadesuit \} \) representing the suits.
1. What is the cardinality of the Cartesian product \( A \times B \)?
2. How many elements are in the Cartesian product \( B \times A ... | automath | [
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0.02773166261613369,
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-0... | 0.824019 | 0.80309 | 0.02093 |
Consider a smooth $n$-dimensional manifold $M$, where $n > 2$. Suppose we perform a handle slide operation involving two framed $(k-1)$-dimensional spheres, $K_p$ and $K_q$, in $M$. Specifically, take the case where $n = 5$ and $k = 3$. Can you describe the resulting new framed $(k-1)$-dimensional sphere $K_q'$ that re... | automath | [
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0.020567424595355988,
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-0.01268... | 0.819639 | 0.793614 | 0.026025 |
Let \( x \) and \( y \) be real numbers. Determine if the equations \( x^2 = xy \) and \( x^2 = y^2 \) can be used to conclusively determine if \( x = y \). What are the valid values of \( x \) and \( y \), and how many distinct pairs \((x, y)\) satisfy this condition? | automath | [
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0.023735126480460167,
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Given the system of equations:
1. \(\frac{1}{4 \sin{x}} - \frac{a \sin{x}}{(1 + a^2 + 2a \cos{x})^{3/2}} = 0\)
2. \(r^3 - \frac{3}{a^3} + \frac{2(a + \cos{x})}{(1 + a^2 + 2a \cos{x})^{3/2}} = 0\)
3. \(r^3 - 3 + \frac{1 + a \cos{x}}{(1 + a^2 + 2a \cos{x})^{3/2}} + \frac{1}{4 \sin{x}} = 0\)
with \(r > 0\), \(a > 0\), a... | automath | [
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0.001886342535726726,
0.012921039946377277,
-0.014085931703448296,
-0.03... | 0.834172 | 0.814953 | 0.01922 |
Let \( E \) be an elliptic curve. Define the following properties of the addition law on \( E \):
1. \( P + \mathcal{O} = \mathcal{O} + P = P \) for all \( P \in E \) (Identity)
2. \( P + (-P) = \mathcal{O} \) for all \( P \in E \) (Inverse)
3. \( (P + Q) + R = P + (Q + R) \) for all \( P, Q, R \in E \) (Associativity... | automath | [
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0.021010437980294228,
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-... | 0.83312 | 0.815531 | 0.017589 |
Consider the special linear group \( \text{SL}_n(\mathbb{C}) \) for \( n > 1 \). Can you provide an explicit characterization of the matrices in \( \text{SL}_n(\mathbb{C}) \) that do not possess a \( p \)-th root in \( \text{SL}_n(\mathbb{C}) \) for any prime \( p \)? In particular, is the set of such matrices a finite... | automath | [
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0.030705340206623077,
0.013076250441372395,
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-0.... | 0.814101 | 0.792614 | 0.021487 |
Given a hexagon \( ABCDEF \) inscribed in a conic section, if points \( A, B, C, D, E, F \) are positioned such that \( AD \), \( BE \), and \( CF \) are concurrent, what can be concluded about the relationships between the angles at points \( A, B, C, D, E, \) and \( F \) in terms of their cyclic nature, specifically ... | automath | [
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0.02066408470273018,
0.0166945680975914,
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-0.0... | 0.828053 | 0.809731 | 0.018322 |
Given the function \( f(x,y) = x+y \) and the constraint \( g(x,y) = \sqrt{x} + \sqrt{y} - 1 \), determine the maximum value of \( f \) on the constraint curve, considering both critical points found using Lagrange Multipliers and the boundary points of the constraint region. What are these maximum points and their cor... | automath | [
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0.025305595248937607,
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0.04831275716423988,
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... | 0.823444 | 0.807192 | 0.016252 |
Find all sets of positive integers \( (A, B, C, X, Y, Z) \) such that the equations \( A^2 + B^2 = X^2 \), \( A^2 + C^2 = Y^2 \), and \( B^2 + C^2 = Z^2 \) hold true, where \( A, B, C, X, Y, Z \) represent the sides of right triangles and the corresponding hypotenuses. List the first three sets of such integers. | automath | [
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0.026513831689953804,
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0.0367763489484787,
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-0.006541753187775612,
0.04023142158985138,
0.042379774153232574,
-0.0325222946703434,
0.009912317618727684,
0.029893677681684494,
-0.029111018404364586,
-0.0361... | 0.853863 | 0.820994 | 0.032869 |
Let \( X_1, X_2, \ldots, X_n \) be a sequence of independent and identically distributed random variables drawn from a probability distribution with cumulative distribution function (CDF) \( F_X(x) \). According to the Glivenko-Cantelli theorem, as \( n \to \infty \), the empirical cumulative distribution function \( \... | automath | [
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0.00743097485974431,
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0.03462289646267891,
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0.07138533890247345,
0.026083581149578094,
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0.007189577911049128,
0.018482157960534096,
-0.03276725485920906,
0.0... | 0.812334 | 0.787901 | 0.024433 |
Let \( M \) be a smooth manifold and consider two diffeomorphisms \( f, g : M \to M \). Given that \( f \) and \( g \) are diffeotopic, define a smooth diffeotopy \( h : M \times I \to M \) such that \( h_0 = f \) and \( h_1 = g \). If every diffeomorphism on \( M \) that is diffeotopic to the identity map forms a grou... | automath | [
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0.017121534794569016,
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-0.031... | 0.80995 | 0.791181 | 0.01877 |
Given the sequence of functions defined by \( f_n(x) = \frac{x^{2n}}{1+x^{2n}} \), determine the pointwise limit \( f(x) \) of this sequence as \( n \to \infty \). Specifically, find the value of \( f(0) \), \( f(1) \), and \( f(-1) \). Additionally, is the function \( f(x) \) continuous at these points? Provide a deta... | automath | [
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0.03817436099052429,
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0.025229478254914284,
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0.030912304297089577,
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0... | 0.807368 | 0.787286 | 0.020081 |
Given a simply connected space \( X \) with trivial rational homology in dimensions less than \( D \), and given that the rational homotopy group \( \pi_D(X) \) is generated by the images under the Hurewicz map of elements in \( \pi_D(X) \) which form a basis for \( \tilde{H}_D(X, \mathbb{Q}) \), what is the necessary ... | automath | [
0.01103202998638153,
0.017427947372198105,
-0.017586268484592438,
-0.04942028224468231,
0.029045768082141876,
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0.00514296256005764,
0.043168045580387115,
0.0522647500038147,
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0.0029999162070453167,
0.018645498901605606,
-0.013357381336390972,
-0.0... | 0.804847 | 0.783026 | 0.021821 |
In triangle \(ABC\), the internal angle at vertex \(A\) measures \(40^\circ\). The angles at vertices \(B\) and \(C\) must be greater than \(40^\circ\). If the internal bisectors of angles \(A\), \(B\), and \(C\) intersect the circumcircle at points \(A_1\), \(B_1\), and \(C_1\) respectively, and the internal bisectors... | automath | [
0.015392372384667397,
0.029011115431785583,
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0.03347349539399147,
0.0008678978192619979,
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0.03913338854908943,
0.02026531845331192,
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0.01611889712512493,
0.01932484097778797,
-0.022059764713048935,
-0.0244... | 0.837341 | 0.812684 | 0.024657 |
Let \( p \) and \( q \) be distinct prime numbers, and define \( n = p \cdot q \). If \( a \) is an integer such that \( \gcd(a, n) = 1 \), what is the least positive integer \( k \) such that \( a^k \equiv 1 \pmod{n} \)? | automath | [
0.0463050901889801,
0.03699330613017082,
-0.023824088275432587,
-0.04447320103645325,
0.0425528809428215,
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0.05086569860577583,
0.03323327377438545,
-0.00771332997828722,
-0.0007301968871615827,
0.022351497784256935,
-0.03921389579772949,
-0.02... | 0.837917 | 0.820313 | 0.017603 |
Let \( A = \{1, 2, 3, \ldots, n\} \) be a finite set and \( f: A \to A \) be a bijective function. If every element in \( A \) is mapped to itself after applying \( f \) a certain number of times, can you determine the smallest positive integer \( M > 1 \) such that \( f^M(i) = f(i) \) for all \( i \in A \)? What is th... | automath | [
0.051008131355047226,
0.030586646869778633,
-0.03017904795706272,
-0.02542440965771675,
0.04033973440527916,
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0.03791069984436035,
0.03911365568637848,
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0.005721022840589285,
0.013982871547341347,
-0.026384560391306877,
-0.01... | 0.84026 | 0.816915 | 0.023345 |
Consider two relatively prime integers \( a \) and \( b \). Does there exist a uniform bound \( g(a, b) \) such that any sequence of \( g(a, b) \) consecutive integers contains at least one integer congruent to \( b \) modulo \( a \) and not divisible by any integers in the range from \( x+1 \) to \( y \), where \( y \... | automath | [
0.02299598976969719,
0.02676878497004509,
-0.02276736870408058,
-0.03008001297712326,
0.022625019773840904,
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-0.02404267154633999,
0.052985623478889465,
0.025610456243157387,
-0.021943431347608566,
0.012377697974443436,
0.005550604313611984,
-0.0372513085603714,
-0.01... | 0.843452 | 0.820974 | 0.022479 |
Given a graded ring \( S \) and a graded \( S \)-module \( M \), let \( \mathfrak{p} \in \text{Proj}(S) \) and \( f \in S \) be a homogeneous element of positive degree such that \( f \not\in \mathfrak{p} \). If we define the map \( \psi : M_{(\mathfrak{p})} \to (M_{(f)})_{\mathfrak{p'}} \) as follows:
\[
\psi\left(\f... | automath | [
0.0509740449488163,
0.013329584151506424,
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0.03926847502589226,
-0.045058950781822205,
0.002488761441782117,
0.07258641719818115,
0.0466163270175457,
-0.003923931159079075,
0.02102941833436489,
0.021986324340105057,
-0.025438250973820686,
-0.0156... | 0.814892 | 0.79368 | 0.021212 |
Let \( D, E, F \) be points on the sides \( BC, CA, AB \) respectively of a triangle \( ABC \) such that \( BD = CE = AF = k \) for some positive real number \( k \), and suppose that \( \angle BDF = \angle CED = \angle AFE = \theta \). Calculate the value of \( k \) for which triangle \( ABC \) is equilateral using ge... | automath | [
0.04105036333203316,
0.014265897683799267,
-0.03882686793804169,
-0.04524226486682892,
0.01822495646774769,
-0.030602585524320602,
0.004962339997291565,
0.056375548243522644,
0.03408736735582352,
0.005890494212508202,
0.0049860612489283085,
0.04620988294482231,
-0.043782200664281845,
-0.02... | 0.82851 | 0.802618 | 0.025892 |
In the context of the Intermediate Value Theorem, let \( f \) be a continuous function defined on the interval \([a, b]\), and let \( S = \{ x \in [a, b] : f(x) \le u \} \) for some \( u \in \mathbb{R} \). Suppose \( c = \sup S \) and that \( f(c) < u \).
Given that continuity implies for every \( \epsilon > 0 \) the... | automath | [
0.01835683546960354,
0.02007937617599964,
-0.04914199560880661,
-0.025246260687708855,
0.018626471981406212,
0.00858326070010662,
-0.0287725068628788,
0.08043631911277771,
0.02659764140844345,
-0.01713053695857525,
0.01829947903752327,
0.033108144998550415,
-0.019397543743252754,
-0.015425... | 0.814242 | 0.790204 | 0.024037 |
Let \( X = \{ \pi, e, \sqrt{2} \} \) and \( Y = \{ e, \gamma, \pi \} \) be subsets of the universal set of irrational numbers (\( \mathbb{R} \setminus \mathbb{Q} \)). Compute the set \( X \cap (X \cup Y) \) and state whether the Absorbing Law \( A \cap (A \cup B) = A \) holds for these sets \( X \) and \( Y \). | automath | [
0.012529179453849792,
0.052209366112947464,
-0.017631610855460167,
-0.0339989960193634,
0.004856331739574671,
-0.017230328172445297,
-0.014312966726720333,
0.05999744310975075,
0.04047836363315582,
-0.01226001512259245,
0.015676027163863182,
0.009350783191621304,
-0.03384300693869591,
-0.0... | 0.826996 | 0.798672 | 0.028324 |
Given binary relations \( f \), \( g \), and \( b \) on a finite set \( \mho \), define the predicate \( F \) such that \( F(a) \Leftrightarrow (a \circ f^{-1}) \cap (g^{-1} \circ b) \neq \emptyset \) for every binary relation \( a \) on \( \mho \).
Is it always possible to define a binary relation \( z \) such that ... | automath | [
0.05228430777788162,
0.03676991164684296,
-0.0391944944858551,
-0.018133535981178284,
0.05073310434818268,
-0.018920959904789925,
-0.008600556291639805,
0.067532978951931,
0.036558810621500015,
-0.011519256979227066,
0.006257534492760897,
0.026914410293102264,
-0.008608323521912098,
-0.006... | 0.820014 | 0.793874 | 0.02614 |
Let \( p \) be a positive real number. Consider the sequence of functions \( f_n(x) \) defined on the interval \([0, 1]\) by
\[
f_n(x) =
\begin{cases}
n^{p+1} x & \text{if } 0 \le x < \frac{1}{n} \\
\frac{1}{x^p} & \text{if } \frac{1}{n} \le x \le 1
\end{cases}
\]
What can you conclude about the uniform convergen... | automath | [
0.02451227605342865,
0.04523569345474243,
-0.026962099596858025,
-0.031680189073085785,
0.030714988708496094,
-0.02214187942445278,
-0.029426217079162598,
0.0602070577442646,
0.04082965850830078,
0.008090157061815262,
0.013383573852479458,
0.023181704804301262,
-0.03151274845004082,
-0.016... | 0.817809 | 0.79125 | 0.026559 |
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function such that \( f(x+y) = f(x)f(y) \) for all \( x, y \in \mathbb{R} \), and suppose that \( f \) is continuous. Given that the only continuous solutions to this functional equation can be expressed in the form \( f(x) = a^x \) for some constant \( a > 0 \), determine th... | automath | [
0.030406145378947258,
0.05070323497056961,
-0.04335942491889,
-0.029767196625471115,
0.03395724669098854,
0.0049518258310854435,
-0.0097907530143857,
0.0989808663725853,
0.0411222018301487,
-0.006428493186831474,
-0.0036634067073464394,
0.03201949968934059,
-0.03755626082420349,
-0.0187725... | 0.826524 | 0.807814 | 0.01871 |
Consider a Cauchy sequence \((x_n)\) in a metric space and a convergent subsequence \((x_{n_k})\) that converges to a limit \(x\). How can we show, using the properties of Cauchy sequences and the convergence of subsequences, that for any \(\epsilon > 0\), there exists an integer \(N\) such that for all \(n > N\), \(d(... | automath | [
0.021162476390600204,
0.02835630252957344,
-0.025868335738778114,
-0.028000028803944588,
0.01602964662015438,
-0.0026531964540481567,
-0.0155824925750494,
0.06598062068223953,
0.02902795374393463,
0.003489617258310318,
0.0019737202674150467,
0.01824556104838848,
-0.03779250383377075,
-0.00... | 0.826756 | 0.795923 | 0.030833 |
Given a group \( G \) that fits into a right-split short exact sequence
\[
1 \longrightarrow \mathbb{Z}_4 \longrightarrow G \longrightarrow \mathbb{Z}_2 \longrightarrow 1,
\]
and given that \( \mathbb{Z}_2 \longrightarrow G \) is defined by a homomorphism such that the composition \( \gamma \circ \alpha = 1_{\mathbb... | automath | [
0.030692102387547493,
0.034352753311395645,
-0.010135200805962086,
-0.022631792351603508,
0.034434329718351364,
-0.04067641869187355,
-0.014965563081204891,
0.06370196491479874,
0.037436310201883316,
-0.005183792673051357,
0.003510931273922324,
0.006630311720073223,
-0.015611625276505947,
... | 0.817201 | 0.795503 | 0.021698 |
Let \( A \) be an \( n \times n \) real matrix for which there exist two distinct \( n \)-dimensional real column vectors \( v_1 \) and \( v_2 \) such that \( A v_1 = A v_2 \). What can we conclude about the properties of \( A \)? Choose one of the following statements that must be true:
1. All eigenvalues of \( A \) ... | automath | [
0.04090321809053421,
0.03026912733912468,
-0.03038058429956436,
-0.04492249712347984,
0.040624093264341354,
-0.019779477268457413,
-0.0013467242242768407,
0.0912313163280487,
0.03432614356279373,
-0.011417928151786327,
-0.004226663615554571,
0.020642824470996857,
-0.03661506995558739,
-0.0... | 0.817836 | 0.792615 | 0.025221 |
Given a concrete category \( (\mathbf{A}, \mathscr{U}) \) over \( \mathbf{Set} \) and a singleton set \( \{x\} \), under what conditions can we establish a natural isomorphism between \( \mathscr{U} \) and the functor \( \operatorname{Hom}_{\mathbf{A}}(A, -) \), and can you explicitly describe the bijection \( \zeta_B:... | automath | [
0.04171575978398323,
0.014780499041080475,
-0.024787748232483864,
-0.018579687923192978,
0.03992859274148941,
-0.01993834786117077,
-0.017373891547322273,
0.06979955732822418,
0.046140749007463455,
-0.02462666481733322,
0.0037586349062621593,
0.01847461797297001,
-0.02706119231879711,
-0.0... | 0.806882 | 0.78936 | 0.017522 |
Given \( d \) independent identically distributed random points \( X_1, \ldots, X_{d+1} \) in \( \mathbb{R}^d \) sampled from a continuous probability density function \( f \), and a point \( x_0 \in \mathbb{R}^d \) that is a Lebesgue point of \( f \) with \( f(x_0) > 0 \), what is the probability that \( x_0 \) lies i... | automath | [
-0.002278820378705859,
0.04210521653294563,
-0.04416584596037865,
-0.03377804532647133,
0.021325191482901573,
0.003256673226132989,
-0.022989168763160706,
0.05085976794362068,
0.036930788308382034,
-0.009016585536301136,
0.006734185852110386,
0.015723787248134613,
-0.028298720717430115,
-0... | 0.828288 | 0.808348 | 0.01994 |
Let \( (a_n)_{n=1}^{\infty}, (b_n)_{n=1}^{\infty}, (c_n)_{n=1}^{\infty} \) be sequences defined as follows:
\[
a_n = \frac{1}{n}, \quad b_n = \frac{1}{n} + \frac{(-1)^n}{n^2}, \quad c_n = \frac{1}{n} + \frac{1}{n^2}
\]
Determine the limit to which the sequence \( (b_n)_{n=1}^{\infty} \) converges. Provide justificati... | automath | [
0.005671844817698002,
0.05410657078027725,
-0.028414415195584297,
-0.0272532906383276,
0.03094780072569847,
-0.013313377276062965,
-0.020091945305466652,
0.05706769600510597,
0.04251817241311073,
-0.006834996398538351,
-0.0021009815391153097,
0.014846588484942913,
-0.03145197406411171,
0.0... | 0.824567 | 0.797561 | 0.027006 |
Given the processes defined as \( M_t = M_0 + \int_0^t \sigma_s dB_s \) where \( (B_t) \) is standard Brownian motion, and with the conditions that \( E[M_1 | M_0] = M_0 \) and \( \int_0^1 \sigma_s^2 ds < \infty \) almost surely, can we conclude that \( (M_t) \) is a martingale? If not, provide an example of a stochast... | automath | [
0.007924829609692097,
0.022359348833560944,
-0.035364557057619095,
-0.010001248680055141,
0.04386553540825844,
-0.036632928997278214,
-0.023472324013710022,
0.08134447783231735,
0.04064269736409187,
-0.012241271324455738,
0.02169206738471985,
0.01215413399040699,
-0.037193480879068375,
-0.... | 0.79777 | 0.779536 | 0.018234 |
What is the unique radius \( r \) at which the global maximum of the function
\[
f(x) = \int_{\mathbb{S}^{n-1}} e^{-\alpha \| x - y \|^2} \, dS(y)
\]
is attained for \( \alpha > \frac{n}{2} \), and how does this radius depend on \( \alpha \) and \( n \)? | automath | [
0.02751481905579567,
0.03316950798034668,
-0.05472217872738838,
-0.026831241324543953,
0.017657261341810226,
0.0009477907442487776,
-0.04096951708197594,
0.04371652752161026,
0.04167891666293144,
-0.007703911513090134,
0.019859913736581802,
0.04164863005280495,
-0.022953292354941368,
-0.01... | 0.818376 | 0.798155 | 0.020221 |
Consider the cylinder defined by the equation \(x^2 + y^2 = 1\) in \(\mathbb{R}^3\). Parameterize this cylinder using cylindrical coordinates \((r, \theta, z)\) where \(r = 1\).
1. Write down the parameterization \(S(\theta, z)\) for the cylinder.
2. Verify that the parameterization \(S(\theta, z)\) | automath | [
0.02311432734131813,
0.037724122405052185,
-0.037771131843328476,
-0.03382766246795654,
0.03909725695848465,
-0.02652468904852867,
-0.013829841278493404,
0.068634532392025,
0.02011461742222309,
-0.0058553824201226234,
0.00995903741568327,
0.04060656949877739,
-0.026638872921466827,
-0.0190... | 0.817372 | 0.795009 | 0.022364 |
Let $X$, $Y$, and $Z$ be schemes over a scheme $S$. Suppose there is a morphism of schemes $m: X \times_S Y \to Z \times_S Z$. If $X(T)$, $Y(T)$, and $Z(T)$ denote the $T$-valued points of the respective schemes for a test scheme $T$ over $S$, how many distinct maps can be derived from the morphism $m$ from $X(T) \time... | automath | [
0.023752503097057343,
0.027946708723902702,
-0.04585102200508118,
-0.02916976809501648,
0.03157303109765053,
-0.026957601308822632,
-0.018117567524313927,
0.05497999116778374,
0.027910271659493446,
-0.01341311912983656,
-0.0007606047438457608,
0.04253927245736122,
-0.019354762509465218,
-0... | 0.830993 | 0.811202 | 0.019791 |
Given a sequence of positive constants \( \{a_n\} \) that diverges to infinity, consider a point estimator \( T_n(X) \) for a parameter \( \theta \). If \( a_n \) consistency of \( T_n(X) \) implies weak consistency, can you provide a specific example of a point estimator \( T_n(X) \) for which the conditions are satis... | automath | [
0.017620358616113663,
0.03177056834101677,
-0.033746279776096344,
-0.03974650800228119,
0.021585483103990555,
-0.0169016532599926,
-0.014414331875741482,
0.06220236048102379,
0.03671722486615181,
-0.007657631766051054,
0.013907830230891705,
0.01917113922536373,
-0.032365888357162476,
0.006... | 0.818823 | 0.796663 | 0.02216 |
Let the sequence of real numbers \( a_0, a_1, a_2, \ldots \) satisfy \( 1 = a_0 \leq a_1 \leq a_2 \leq \ldots \). Define the sequence \( b_n = \sum_{k=1}^n \frac{1 - \frac{a_{k-1}}{a_k}}{\sqrt{a_k}} \). What is the largest possible value of \( b_n \) as \( n \) approaches infinity, given the conditions on \( a_k \)? | automath | [
0.026735959574580193,
0.0532793328166008,
-0.025096211582422256,
-0.034579891711473465,
0.02064182050526142,
-0.0013123264070600271,
-0.027601774781942368,
0.060934584587812424,
0.03465431556105614,
-0.016216320917010307,
-0.0018518816214054823,
0.03292528912425041,
-0.03867848590016365,
-... | 0.847444 | 0.821034 | 0.026409 |
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