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 |
|---|---|---|---|---|---|
Let \( X \) be a random variable that follows a Normal(0,1) distribution. Define the probability measure \( P \) associated with \( X \) and the Lebesgue measure \( \mu \) on \( \mathbb{R} \).
What is the integral expression for the probability \( P\{X \leq t\} \), and how would you verify that the density \( \frac{1... | automath | [
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0.034680113196372986,
-0.046686284244060516,
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0.03433142229914665,
0.0017309320392087102,
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0.0387653112411499,
0.04042983800172806,
0.01027876976877451,
-0.004603024572134018,
0.020676106214523315,
-0.004253941588103771,
-0.01... | 0.81526 | 0.794516 | 0.020744 |
Let $X \to B$ be a quasi-separated morphism of finite type algebraic spaces. If $W$ is an affine scheme and we have a diagram
\[
\vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ B \ar[r] & B_i } }
\]
for $i' \geq i$, let $X_{i'}$ be the scheme theoretic image of $X \to B_{i'} \times_{B_i} W$. Under what condition... | automath | [
0.008648714981973171,
0.024755187332630157,
-0.04813874885439873,
-0.025815874338150024,
0.02024923823773861,
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0.08190461248159409,
0.03740036115050316,
-0.023976698517799377,
0.011378091759979725,
0.03331851214170456,
-0.024016741663217545,
0.01... | 0.807808 | 0.787991 | 0.019816 |
Let $(X_n)_{n=1}^{\infty}$ be a sequence of random variables converging to a random variable $X$ in probability. Given the condition that $\Pr\{\sup_{k \ge n} |X_k - X| > \varepsilon\} \to 0$ for every $\varepsilon > 0$, what can we conclude about the almost sure convergence of $X_n$ to $X$? Specifically, what is the p... | automath | [
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0.026959232985973358,
-0.04196992516517639,
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0.02206149697303772,
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0.04867462441325188,
0.02076738141477108,
-0.026868492364883423,
-0.011324187740683556,
0.019139613956212997,
-0.011640055105090141,
... | 0.810872 | 0.794641 | 0.01623 |
In a connected undirected weighted graph \( G \) with distinct edge weights, consider the following statements about its minimum spanning trees (MST):
1. The edge with the second smallest weight is always part of any minimum spanning tree of \( G \).
2. At least one of the edges with the third smallest and the fourth ... | automath | [
0.019066911190748215,
0.032198064029216766,
-0.025962086394429207,
-0.045502349734306335,
0.06563596427440643,
-0.036504875868558884,
-0.01280196662992239,
0.060983095318078995,
0.03019036166369915,
0.007225211709737778,
0.035994309931993484,
0.008972314186394215,
-0.020658988505601883,
-0... | 0.808934 | 0.788254 | 0.020679 |
Let \( A = \{1, 2, 3\} \). Define two equivalence relations \( R \) and \( S \) on \( A \) as follows:
- \( R = \{(1,1), (2,2), (1,2), (2,1), (3,3)\} \)
- \( S = \{(1,1), (2,2), (2,3), (3,2), (3,3)\} \)
Determine whether \( R \cup S \) and \( R \cap S \) are equivalence relations. Provide a justification for your ans... | automath | [
0.0532246008515358,
0.014031222090125084,
-0.03473104536533356,
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0.023149237036705017,
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0.07245735079050064,
0.028098251670598984,
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0.01569814793765545,
0.020329490303993225,
-0.027441730722784996,
-0.... | 0.819386 | 0.797788 | 0.021597 |
Consider a continuous function \( f: \mathbb{R} \to \mathbb{R} \) such that its second derivative \( f''(x) \) is always positive on the interval \( [a, b] \). If you approximate \( f \) using a function \( g \) that is a quadratic polynomial which is tangent to \( f \) at a point \( c \in (a, b) \) and matches \( f \)... | automath | [
0.022493578493595123,
0.04253903403878212,
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-0.03487296402454376,
0.028785577043890953,
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0.056129924952983856,
0.013527111150324345,
-0.004683922976255417,
0.011898401193320751,
0.02753247693181038,
-0.027482470497488976,
0.... | 0.828499 | 0.802475 | 0.026023 |
Let \( K \) be a finite category with \( n \) objects. Given a functor \( F: \mathcal{C} \rightarrow \mathcal{D} \) and a full subcategory \( \mathcal{C}^0 \subseteq \mathcal{C} \), under what conditions will \( F \) be left Kan extended from \( \mathcal{C}^0 \) if it is established that every object \( C \) within \( ... | automath | [
0.033499691635370255,
0.010977870784699917,
-0.03750508651137352,
-0.023753952234983444,
0.03515072539448738,
-0.0020597646944224834,
-0.02030756138265133,
0.05504875257611275,
0.049570661038160324,
-0.0011825147084891796,
-0.0006168376421555877,
0.0281191635876894,
-0.03559127449989319,
0... | 0.80984 | 0.788981 | 0.020859 |
Let \( n \geq 2 \) and let \( \mathcal{S} \) be a family of subsets of \( \{1, 2, \ldots, n\} \) such that the union of any four subsets \( A, B, C, D \in \mathcal{S} \) has size at most \( n-2 \). What is the maximum possible size of the family \( \mathcal{S} \)? | automath | [
0.04287850484251976,
0.024783199653029442,
-0.03826135769486427,
-0.027753956615924835,
0.03083859384059906,
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0.04538479819893837,
0.03757301717996597,
-0.0004764371842611581,
0.024888837710022926,
0.04099717363715172,
-0.025750501081347466,
-0.03... | 0.830102 | 0.804908 | 0.025194 |
Given the functions \( y_1 = x^3 \) and \( y_2 = |x|^3 \), find a real number \( c \) such that the linear combination \( c_1 y_1 + c_2 y_2 = 0 \) holds for all \( x \in (-\infty, \infty) \). Determine if \( c_1 \) and \( c_2 \) must both be zero, showing whether these functions are linearly independent or dependent. A... | automath | [
-0.0004954961477778852,
0.04366646707057953,
-0.0648370161652565,
-0.046096544712781906,
0.03914983198046684,
0.0017355595482513309,
0.00012661810615099967,
0.08259885013103485,
0.029966646805405617,
-0.013698937371373177,
-0.004966943059116602,
0.02616047114133835,
-0.018612032756209373,
... | 0.821386 | 0.797826 | 0.02356 |
Consider the random variables \(X\) and \(Y\) that have a joint probability density function (pdf) given by \(\frac{1}{x^2 y^2}\) for \(x, y \geq 1\). Define new random variables \(U = XY\) and \(V = \frac{X}{Y}\).
Determine the joint support of the random variables \(U\) and \(V\), and explain why the set of points ... | automath | [
-0.008903704583644867,
0.03454429283738136,
-0.05546100065112114,
-0.029957124963402748,
0.03637069836258888,
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0.060919586569070816,
0.035907261073589325,
0.015943627804517746,
-0.003342418698593974,
0.0311566311866045,
0.0072604455053806305,
-0... | 0.816315 | 0.78589 | 0.030425 |
Let \( R \) and \( S \) be Noetherian rings, and let \( \varphi: R \to S \) be a flat ring map. Suppose the fiber rings \( S \otimes_R \kappa(\mathfrak{p}) \) for any prime \( \mathfrak{p} \) of \( R \) have depth at least \( k \), and \( R \) satisfies property \( (S_k) \). Under these conditions, what can we conclude... | automath | [
0.027747202664613724,
0.036167293787002563,
-0.03845469653606415,
-0.032392002642154694,
0.011570567265152931,
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0.06499572098255157,
0.016872411593794823,
-0.003481069812551141,
0.011702397838234901,
0.03292236477136612,
-0.0312171783298254,
-0... | 0.810935 | 0.788849 | 0.022086 |
Consider two six-sided dice thrown simultaneously. Define the events \( A \), \( B \), and \( C \) as follows:
- Event \( A \): The first die shows an even number.
- Event \( C \): The sum of the numbers on the two dice is less than or equal to 5.
Are the events \( A \) and \( C \) mutually exclusive? Provide a true... | automath | [
0.021329473704099655,
0.003093960229307413,
-0.02877696231007576,
-0.017753325402736664,
0.0368066243827343,
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0.07850929349660873,
0.04531191661953926,
0.0051546297036111355,
0.02188979648053646,
0.016974246129393578,
-0.020560821518301964,
-0.02... | 0.825829 | 0.808416 | 0.017413 |
Let \( p \) be a prime and consider the finite field \( \mathbb{F}_q \) where \( q = p^a \) for some positive integer \( a \). Determine the number of distinct elements in the image of the map \( f: \mathbb{F}_q \to \mathbb{F}_q \) defined by \( f(x) = x^p \). | automath | [
0.0411154180765152,
0.021841492503881454,
-0.027615269646048546,
-0.034876495599746704,
0.035890620201826096,
-0.04652968421578407,
0.0007241145358420908,
0.059899356216192245,
0.050412584096193314,
-0.003132003592327237,
0.006406331434845924,
0.0269643384963274,
-0.04232072830200195,
-0.0... | 0.82992 | 0.812715 | 0.017206 |
Consider a scenario where a fair six-sided die is rolled repeatedly until a "1" appears. Let \( X \) be the random variable representing the number of rolls required to get the first "1".
1. What is the probability that \( X \) is greater than 10, and what is the probability that \( X \) is greater than 13 given that... | automath | [
0.005814410280436277,
0.029669541865587234,
-0.019593514502048492,
-0.020090457051992416,
0.0317232646048069,
-0.004715959075838327,
-0.018760783597826958,
0.07704637944698334,
0.022293880581855774,
0.010781790129840374,
0.020602216944098473,
0.02485637739300728,
-0.03038369119167328,
-0.0... | 0.825974 | 0.807922 | 0.018052 |
Let \( S \) be a scheme, and let \( f: X \to Y \) be a morphism of algebraic spaces over \( S \). Assume that \( f \) is proper and that \( Y \) is locally Noetherian. If \( y \in |Y| \) is a point of codimension \( \leq 1 \) and the transcendence degree of every point \( x \in X^0 \) (the set of points of codimension ... | automath | [
0.015262614004313946,
0.032552625983953476,
-0.03800695389509201,
-0.03135710209608078,
0.022006550803780556,
-0.015846621245145798,
-0.011208796873688698,
0.05775051936507225,
0.04470854625105858,
-0.003684535389766097,
-0.0034412273671478033,
0.03960150107741356,
-0.026967475190758705,
0... | 0.807723 | 0.781936 | 0.025787 |
Consider a spherical surface centered at the origin that divides into two halves, \( L \) and \( R \). Given a current density \( \boldsymbol{J} \) flowing into the sphere from the left \( L \) and halting symmetrically at the origin, express and verify the relationship between the fluxes of \( \boldsymbol{J} \) across... | automath | [
0.000708326930180192,
0.04042042791843414,
-0.052359335124492645,
-0.021196655929088593,
0.006712810602039099,
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0.06035984307527542,
0.03287357836961746,
0.012947897426784039,
0.006480334792286158,
0.019031263887882233,
-0.022578740492463112,
-0.... | 0.808017 | 0.790666 | 0.017352 |
Let \( H \) be a finite group with a subgroup \( H_1 \). If \( H_1 \) is generated by \( k \) cyclic groups, how many distinct elements can be formed by taking products of these \( k \) cyclic groups taken \( n \) at a time, where \( n \leq k \)? Specifically, if each cyclic group has size \( m_i \) for \( i = 1, \ldot... | automath | [
0.039495401084423065,
0.03633446991443634,
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0.030890120193362236,
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0.05499488487839699,
0.03431668505072594,
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0.0062495931051671505,
0.03703095763921738,
-0.021839315071702003,
-0... | 0.818776 | 0.801751 | 0.017024 |
Given a ring \( A \) and a finitely generated ideal \( I \subset A \), consider the functor \( R\Gamma_I: D(A) \to D(I^\infty \text{-torsion}) \). If \( K \) is an object in \( D(A) \) such that the cohomology modules \( H^q(R\Gamma_I(K)) \) are nonzero for some \( q \), what can be said about \( K \) in relation to \(... | automath | [
0.043869227170944214,
0.02870584838092327,
-0.016462909057736397,
-0.017780110239982605,
0.0030162848997861147,
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0.013741787523031235,
0.059168491512537,
0.03682120889425278,
-0.018256261944770813,
0.010766244493424892,
0.017903033643960953,
-0.047520387917757034,
0.00... | 0.799505 | 0.77919 | 0.020316 |
Can you construct an explicit example of a point finite open cover of \(\mathbb{R}\) that is everywhere non-locally finite? Additionally, please explain why your example satisfies these properties. | automath | [
0.01580786146223545,
0.03568649664521217,
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0.003741665743291378,
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0.041120655834674835,
0.03289768844842911,
-0.017435388639569283,
0.032544609159231186,
0.02999136410653591,
-0.04461339861154556,
-0.011... | 0.816806 | 0.795649 | 0.021158 |
Let \(X\) be a Hausdorff space with a dense subset \(A \subseteq X\) such that \(|X|^{|A|} = |X|\). If \(|\text{Cont}(X, X)| = |X|\), is it possible for \(X\) to be zero-dimensional? Justify your answer. | automath | [
0.00992383249104023,
0.03751237690448761,
-0.02537781372666359,
-0.031111106276512146,
0.038091544061899185,
0.001368899131193757,
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0.07415585964918137,
0.042929813265800476,
-0.019938252866268158,
0.013883466832339764,
0.017489036545157433,
-0.041179753839969635,
-0.... | 0.822738 | 0.804504 | 0.018234 |
Let \( X \) be a topological space, and let \( A \subseteq X \) be a co-dense set, while \( B \subseteq X \) is a nowhere dense set. Verify that the union \( A \cup B \) is a co-dense set. Provide an example of two co-dense sets whose union is not co-dense, and explain why this happens in terms of their topological pro... | automath | [
0.033740561455488205,
0.027388744056224823,
-0.0362701378762722,
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0.006043994799256325,
0.007443082984536886,
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0.07361019402742386,
0.05376144126057625,
-0.003345690667629242,
0.02483660914003849,
0.010876412503421307,
-0.01651165261864662,
-0.00... | 0.803522 | 0.781757 | 0.021765 |
Consider the differential equation \( \frac{dx}{dt} = f(t, x) \) where \( f(t, x) \) is continuous. Let \( P(t_0, x_0) \) be a point in the plane defined by the integral curves of this differential equation. Assume that the integral funnel defined by point \( P \) contains all points \( (t, x) \) such that \( a \leq t ... | automath | [
0.018621429800987244,
0.030294345691800117,
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0.02816854976117611,
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0.08735957741737366,
0.03888405114412308,
0.010970499366521835,
0.002728995168581605,
0.03078443370759487,
-0.025597767904400826,
-0.0... | 0.822132 | 0.798539 | 0.023592 |
Given a graph \( G \) with \( n \) vertices and without loops, determine if there exists a labeling of the vertices with integers such that the degree of vertex \( i \) can be expressed as
\[
\deg(i) = \sum_{j=1}^n \mathbf{1}_L(\ell_i - \ell_j),
\]
for some set \( L \), where \( \ell_i \) is the label assigned to ve... | automath | [
0.013162435963749886,
0.02193639613687992,
-0.03473770245909691,
-0.03209513798356056,
0.014305000193417072,
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0.04866219684481621,
0.008740553632378578,
-0.015345567837357521,
0.021437929943203926,
0.019961079582571983,
-0.019141467288136482,
0.01... | 0.828284 | 0.803111 | 0.025174 |
Given two binary integers represented as $A = \langle a_1, a_2, \ldots, a_n \rangle$ and $B = \langle b_1, b_2, \ldots, b_n \rangle$, encoded as arrays with the least significant bit first, derive the pseudocode to compute their sum, storing the result in array $C = \langle c_1, c_2, \ldots, c_{n+1} \rangle$. Additiona... | automath | [
-0.00674845278263092,
0.04062524810433388,
-0.03155209496617317,
-0.011271453462541103,
0.038887981325387955,
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0.07012908160686493,
0.037545282393693924,
-0.02018159255385399,
0.007907739840447903,
0.035065166652202606,
-0.03111245669424534,
-0.0... | 0.81622 | 0.799006 | 0.017214 |
Given a sequence of real numbers \( \{a_k\}_{k \geq 1} \), what condition must hold for the sequence to be the derivatives \( f^{(k)}(0) \) of a smooth function \( f \) defined on some interval \( (-\epsilon, \epsilon) \) for some \( \epsilon > 0 \)? Specifically, what must be true about the limit supremum \( \limsup_{... | automath | [
0.003824241692200303,
0.04544907808303833,
-0.03501344844698906,
-0.03212834149599075,
0.03651314601302147,
0.015714284032583237,
-0.04061705619096756,
0.06771864742040634,
0.018802402541041374,
-0.01554846204817295,
-0.006011124234646559,
0.013103139586746693,
-0.03379804641008377,
-0.003... | 0.820476 | 0.800606 | 0.01987 |
Prove, using the Gram-Schmidt process, that if \(\{x_1, x_2, \ldots, x_n\}\) is a linearly independent set in an inner product space, then the orthonormal vectors \(\{e_1, e_2, \ldots, e_n\}\) defined by
\[
e_k = \frac{z_k}{||z_k||}, \quad z_k = x_k - \sum_{j=1}^{k-1} \langle x_k,e_j \rangle e_j
\]
satisfy \(\langle... | automath | [
0.0141809256747365,
0.03014925681054592,
-0.03152322769165039,
-0.04046780616044998,
0.04402376338839531,
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-0.020043030381202698,
0.07054108381271362,
0.0316849909722805,
-0.017046788707375526,
-0.0024806770961731672,
0.025052376091480255,
-0.03258465230464935,
-0.01061... | 0.823747 | 0.794329 | 0.029418 |
Given two topological spaces \( (A, \tau) \) and \( (C, \zeta) \), and a function \( f: E \to C \) with \( E \subseteq A \), under what conditions can the limit of \( f \) as \( x \) approaches an accumulation point \( x_0 \) in \( E \) exist, if the limit \( l \) is required to lie in a subset \( D \subseteq C \)? Spe... | automath | [
0.027059141546487808,
0.03655657172203064,
-0.02418198622763157,
-0.013414097018539906,
0.021098168566823006,
-0.007126247510313988,
-0.02690858207643032,
0.05889904126524925,
0.023569976910948753,
-0.012439259327948093,
0.007747163530439138,
0.034236058592796326,
-0.029215821996331215,
-0... | 0.813267 | 0.792566 | 0.020701 |
Consider the family of sets \( S = \{ A_n \}_{n \in \mathbb{N}} \), where each \( A_n \) is defined as \( A_n = \{ 1, 2, \ldots, n \} \). Determine the cardinality of the disjoint union \( \biguplus_{n=1}^\infty A_n \), and explain whether this disjoint union results in a set or a proper class within Zermelo-Fraenkel s... | automath | [
0.020391004160046577,
0.0249777939170599,
-0.03553369641304016,
-0.0152964536100626,
0.03227796405553818,
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-0.024059945717453957,
0.06109379976987839,
0.036322638392448425,
0.0012981604086235166,
0.025215772911906242,
0.013489952310919762,
-0.027248360216617584,
-0.0... | 0.825184 | 0.795428 | 0.029756 |
Let \( T = (S, \tau) \) be a topological space where \( S = \mathbb{R} \) and the topology \( \tau \) is the standard topology on the real numbers. If \( A = \{ \frac{1}{n} : n \in \mathbb{N} \} \cup \{ 0 \} \), how many distinct \( \omega \)-accumulation points does the set \( A \) have? | automath | [
0.04046071693301201,
0.013126994483172894,
-0.059811897575855255,
-0.024489358067512512,
0.025500714778900146,
-0.017048200592398643,
-0.023454397916793823,
0.06429269164800644,
0.038330379873514175,
-0.004344439599663019,
-0.019625898450613022,
0.023759989067912102,
-0.03452204912900925,
... | 0.831325 | 0.80695 | 0.024375 |
Given a 2-dimensional continuous non-degenerate martingale of the form \( Y(t) = \int_0^t \sigma(s) \, dW(s) \), with \( \sigma(t, \omega) \) satisfying the uniform non-degenerate condition, can we determine the probability \( \mathbb{P}\{ Y(t) = (1,1) \text{ for some } t \in (0, 1) \} \)? Specifically, under what cond... | automath | [
-0.009855068288743496,
0.01986098848283291,
-0.03572995960712433,
-0.010973346419632435,
0.03903498500585556,
0.004125318024307489,
-0.018856212496757507,
0.08092320710420609,
0.02636166848242283,
-0.004923785105347633,
0.012975283898413181,
0.011799492873251438,
-0.013410547748208046,
-0.... | 0.804689 | 0.787237 | 0.017453 |
Given a convex polygon with an area of \( A \) and a perimeter of \( P \), if the volume-to-boundary ratio corresponding to such a polygon's Cheeger set is given by a function of its angles, can you provide a formula or method to compute this ratio based solely on \( A \), \( P \), and the polygon's angle measures? Wha... | automath | [
0.032427236437797546,
0.024218710139393806,
-0.02828865498304367,
-0.019666774198412895,
0.006315284874290228,
-0.02251037210226059,
-0.02728232368826866,
0.047487422823905945,
0.027699248865246773,
-0.00661923922598362,
0.01870225928723812,
0.03670589253306389,
-0.024133509024977684,
-0.0... | 0.821554 | 0.802138 | 0.019416 |
Let \( X \) and \( Y \) be two continuous random variables defined on \( \mathbb{R}^m \) and \( \mathbb{R}^n \), respectively. Suppose \( I(X; Y) \) is finite. Given smooth embeddings \( f: \mathbb{R}^m \to \mathbb{R}^{m+k} \) and \( g: \mathbb{R}^n \to \mathbb{R}^{n+l} \), is it true that \( I(f(X); g(Y)) \) is also f... | automath | [
0.00511824619024992,
0.040862057358026505,
-0.045693714171648026,
-0.03511057049036026,
0.034601230174303055,
-0.003563697449862957,
-0.018668312579393387,
0.04345390573143959,
0.024921121075749397,
-0.02994735725224018,
0.01633807085454464,
0.013980861753225327,
-0.018478866666555405,
0.0... | 0.810292 | 0.78629 | 0.024002 |
Let \( \{a_n\} \) be a sequence of positive real numbers such that \( a_n = \frac{1}{n} \) for \( n \geq 1 \). Determine the convergence of the power series
\[
f(z) = \sum_{n=1}^\infty a_n z^n,
\]
when \( z \) is on the unit circle \( |z| = 1 \). Specify the behavior of \( f(z) \) at \( z = 1 \). | automath | [
0.014261631295084953,
0.060551341623067856,
-0.027230987325310707,
-0.02895418554544449,
0.03111780621111393,
-0.0021858084946870804,
-0.009363229386508465,
0.04191399738192558,
0.025224119424819946,
0.0019318016711622477,
0.014397142454981804,
-0.003415868617594242,
-0.02447965182363987,
... | 0.825234 | 0.798405 | 0.026829 |
Let \( f(x) = \begin{cases} \sin \frac{1}{x}, & x \ne 0 \\ 0, & x = 0 \end{cases} \). For the limit \( L = \lim_{x \to 0} f(x) \), determine whether the following assertion is true or false: "If we take any \( \delta > 0 \), it is possible to find an \( \epsilon > 0 \) such that \( |f(x) - L| < \epsilon \) whenever \( ... | automath | [
-0.0126970075070858,
0.038917798548936844,
-0.025384949520230293,
-0.032660745084285736,
0.034747201949357986,
-0.015599584206938744,
-0.019897006452083588,
0.0969366654753685,
0.02537541650235653,
-0.0004068361013196409,
0.006458186078816652,
0.014524975791573524,
-0.02488658018410206,
-0... | 0.811196 | 0.789681 | 0.021515 |
Let \( X_1, X_2, \ldots \) be a sequence of independent random variables with variances \( \sigma_k^2 \) and let \( s_n^2 = \sum_{k=1}^n \sigma_k^2 \). If it is known that
\[
\max_{k=1,\ldots,n} \frac{\sigma_k^2}{s_n^2} \to 0 \quad \text{as } n \to \infty,
\]
what can be concluded about the validity of the Central L... | automath | [
-0.007887343876063824,
0.047829002141952515,
-0.056720659136772156,
-0.023880260065197945,
0.02792460471391678,
0.015704533085227013,
-0.02818453125655651,
0.060883764177560806,
0.012990163639187813,
-0.009002242237329483,
-0.012608091346919537,
0.009738205932080746,
-0.03984776884317398,
... | 0.8151 | 0.783271 | 0.031829 |
Given the functional equations \( f(f(x)) = kx \) and \( f(x^2) = xf(x) \) for a fixed non-zero constant \( k \in \mathbb{R} \), determine the possible forms of the function \( f \) that satisfy these equations. Specifically, what can you say about the structure of the sets \( K^+ \) and \( K^- \) where \( f(x) = \sqrt... | automath | [
0.03316289186477661,
0.045443344861269,
-0.0371798574924469,
-0.03479662537574768,
0.042657654732465744,
-0.010068297386169434,
0.003284190082922578,
0.07683002203702927,
0.04235219955444336,
0.013668970204889774,
0.023958224803209305,
0.006380231585353613,
-0.03278493136167526,
-0.0105581... | 0.815446 | 0.796524 | 0.018922 |
Let \( f : [0, 1] \to \mathbb{R} \) be defined as follows:
\[
f(x) =
\begin{cases}
1 & \text{if } x \in [0, 0.5) \\
0 & \text{if } x \in [0.5, 1]
\end{cases}
\]
Determine whether the Riemann and Darboux integrals of \( f \) exist over the interval \( [0, 1] \), and if they do, what the values of both integrals are. | automath | [
0.018992861732840538,
0.04164203256368637,
-0.014139113016426563,
-0.034974757581949234,
0.03346772491931915,
0.013101392425596714,
-0.012659057974815369,
0.0772315040230751,
0.033818457275629044,
0.009946498088538647,
0.01369899045675993,
0.021707192063331604,
-0.0371541902422905,
-0.0427... | 0.812898 | 0.792885 | 0.020012 |
What is the relationship between the fundamental group, the second homotopy group, and the third homotopy group of a compact simply connected 3-manifold as it evolves under Ricci flow with surgery, and how does this relate to the existence of embedded copies of $\mathbb{RP}^2$ with trivial normal bundle? Specifically, ... | automath | [
0.014457834884524345,
0.028514979407191277,
-0.017770491540431976,
-0.03769415616989136,
0.028637221083045006,
-0.00043677777284756303,
-0.02666105329990387,
0.052069392055273056,
0.03524218499660492,
-0.005942902993410826,
0.008924526162445545,
0.006951161194592714,
-0.03643530234694481,
... | 0.792494 | 0.775595 | 0.016899 |
For the spaces defined in the text, verify whether the following functions qualify as norms and justify your answers with appropriate reasoning. Specifically, determine which of the following spaces can be classified as normed spaces:
1. \( V_a = \mathbb{R}^2 \) with \( ||(x,y)||_a = |x+y| \)
2. \( V_b = \mathbb{R}^2 ... | automath | [
0.00926993414759636,
0.0338536873459816,
-0.020311132073402405,
-0.03681487217545509,
0.038674093782901764,
-0.011288337409496307,
-0.03042357787489891,
0.0721074789762497,
0.025425268337130547,
0.011038676835596561,
0.008480092510581017,
0.03642304241657257,
-0.02381233312189579,
-0.01749... | 0.802198 | 0.782705 | 0.019493 |
Is there a joint distribution for two random variables \( X \) and \( Y \) such that both are uniformly distributed on \([0, 1]\), and the conditional expectation \( \mathbb{E}[Y | X = x] \) is linear in \( x \), specifically of the form \( \mathbb{E}[Y | X = x] = a + b x \) with \( a, b \) being constants, while ensur... | automath | [
-0.01211332157254219,
0.007894364185631275,
-0.05127657577395439,
-0.05099903419613838,
0.035947345197200775,
0.014495247043669224,
-0.01867016777396202,
0.06531082838773727,
0.039242349565029144,
-0.032481078058481216,
0.014571493491530418,
0.014214732684195042,
-0.02565968595445156,
-0.0... | 0.81703 | 0.793666 | 0.023364 |
Given an \( m \times n \) matrix \( A \) and a vector \( b \in \mathbb{R}^m \) that lies in the column space of \( A \), determine under what conditions the least squares solution \( \hat{x} \) to \( Ax = b \) is unique. Provide a detailed explanation and, if applicable, a counterexample to illustrate your findings. Sp... | automath | [
0.01722460426390171,
0.03737349435687065,
-0.0329892598092556,
-0.05039731785655022,
0.05081304907798767,
-0.007529717870056629,
-0.006433562841266394,
0.08361013978719711,
0.032039862126111984,
-0.015545768663287163,
0.006831945851445198,
-0.002864855108782649,
-0.033272091299295425,
-0.0... | 0.816425 | 0.790985 | 0.025441 |
Given the propositional variables \( p \) and \( q \), where \( p \) represents the statement "(\( \sqrt{2} \))^\(\sqrt{2}\) is rational" and \( q \) represents the statement "(\( \sqrt{2} \))^\(\sqrt{2}\) is irrational", determine how many distinct truth assignments exist for the well-formed formula \( (p \land \neg q... | automath | [
0.01391825545579195,
0.02669672667980194,
-0.04533405229449272,
-0.03632627800107002,
0.016458582133054733,
-0.029145769774913788,
-0.01845225691795349,
0.058084070682525635,
0.044074349105358124,
-0.002896255115047097,
-0.0038392336573451757,
0.01786002330482006,
-0.025792201980948448,
-0... | 0.828533 | 0.811455 | 0.017078 |
Let \( V \) be a finite-dimensional inner product space with an orthonormal basis \( \{ e_1, e_2, \ldots, e_n \} \). If for a vector \( v \in V \), the inner products \( <v, e_i> \) for \( i = 1, 2, \ldots, n \) are given as \( <v, e_1> = 3 \), \( <v, e_2> = 4 \), and \( <v, e_3> = 5 \), compute the norm \( \| v \|^2 \... | automath | [
0.025344407185912132,
0.031068352982401848,
-0.041640833020210266,
-0.024673810228705406,
0.040259215980768204,
-0.005884387064725161,
-0.027268582955002785,
0.07093819975852966,
0.024218685925006866,
0.008299450390040874,
-0.00004858314787270501,
0.039484959095716476,
-0.03628489002585411,
... | 0.821949 | 0.801348 | 0.020601 |
Given an aquarium with a fixed number \( n \) of critters, each seen with probability \( p \), if the maximum count observed from \( k \) independent counts is \( M \), what is the relationship between \( n \), \( p \), and the likelihood of observing \( M \)? Under what conditions can we determine that \( n \) must eq... | automath | [
0.008243455551564693,
0.00960501004010439,
-0.049481697380542755,
-0.020821180194616318,
0.026693692430853844,
-0.012029154226183891,
-0.047945477068424225,
0.030932877212762833,
0.019664844498038292,
0.0005769873969256878,
-0.0022301978897303343,
0.035358626395463943,
-0.027060452848672867,... | 0.816957 | 0.798551 | 0.018406 |
Given a fixed unitary matrix \( U \) of size \( n \times n \), can you provide an example of a unitary matrix that has exactly \( n \) distinct eigenvalues, all of which lie on the unit circle in \(\mathbb{C}\)? Additionally, can you compute these eigenvalues explicitly for your example? | automath | [
0.0026127018500119448,
0.040002837777137756,
-0.02706357277929783,
-0.038654960691928864,
0.03863715007901192,
-0.039862874895334244,
-0.004344021435827017,
0.044527336955070496,
0.03652028366923332,
-0.010509280487895012,
0.018055368214845657,
0.017798535525798798,
-0.02363704890012741,
-... | 0.823623 | 0.806417 | 0.017206 |
Can we establish a general condition under which the statement \( X_n = \mu + O_p(n^{-1}) \) implies \( E[X_n] = \mu + O(n^{-1}) \) for a uniformly integrable sequence of random variables \( X_n \)? If so, provide an example or counterexample to support your conclusion. | automath | [
-0.004371377639472485,
0.010903771035373211,
-0.047760073095560074,
-0.04035976901650429,
0.03602472320199013,
-0.01655861735343933,
-0.024521630257368088,
0.07020623236894608,
0.03242030367255211,
-0.003826727159321308,
0.010509154759347439,
0.01952582411468029,
-0.046901047229766846,
-0.... | 0.810847 | 0.79444 | 0.016408 |
If we have two full rank matrices \( A \) and \( B \) and a set of identity matrices \( Y_i \) for \( i = 1, 2, \dots, N \), what is the value of \( N \) such that the equation
\[
\sum_{i=1}^N Y_i Y_i^\top \Delta_i Y_i Y_i^\top = 0_n
\]
implies that \( Y_i^\top \Delta_i Y_i = 0_{m} \) for all \( i \), given that \( ... | automath | [
0.04053513705730438,
0.013890375383198261,
-0.05665648728609085,
-0.02699328027665615,
0.02290886454284191,
-0.028393786400556564,
-0.003952641971409321,
0.06457097828388214,
0.04610820487141609,
-0.005427030846476555,
-0.013648622669279575,
0.010677673853933811,
-0.030671296641230583,
-0.... | 0.826239 | 0.802039 | 0.0242 |
Let \( X_n \) be independent identically distributed random variables from a unit exponential distribution, meaning that \( P(X_n \leq x) = 1 - e^{-x} \) for \( x \geq 0 \). Find a sequence \( u_n \) such that \( \lim_{n \to \infty} u_n = \infty \) and \( P(M_n \geq u_n \text{ infinitely often}) = 0 \) where \( M_n = \... | automath | [
-0.017402401193976402,
0.042041487991809845,
-0.03295858949422836,
-0.029124729335308075,
0.019684894010424614,
-0.007874419912695885,
-0.022265955805778503,
0.06720705330371857,
0.03197149932384491,
-0.006592797115445137,
-0.0008261252078227699,
0.026377366855740547,
-0.023152505978941917,
... | 0.821626 | 0.788685 | 0.03294 |
Given two vectors \( \mathbf{A} \) and \( \mathbf{B} \) originating from the same point, and forming an angle \( \Theta \) between them, can you provide a geometric interpretation of the expression \( |\mathbf{A}| |\mathbf{B}| \cos(\Theta) \), specifically in terms of its relationship to areas or projections? Additiona... | automath | [
0.01944040134549141,
0.03528246656060219,
-0.05760178714990616,
-0.04936979338526726,
0.01720818690955639,
0.002062559826299548,
-0.019922735169529915,
0.07810639590024948,
0.04601424187421799,
-0.003635002998635173,
0.016226930543780327,
0.012624777853488922,
-0.021080387756228447,
-0.007... | 0.805664 | 0.780089 | 0.025575 |
Given a compact orientable surface of genus \( g \geq 2 \), consider the fundamental group \( G = \pi(X,x) \) with presentation \( G = \langle x_1,y_1,\ldots,x_g,y_g \ | \ [x_1,y_1] \cdots [x_g,y_g] = 1 \rangle \).
How can one demonstrate that the relation \( r = [x_1,y_1] \cdots [x_g,y_g] \) is not an element of the... | automath | [
0.013454495929181576,
0.04270721971988678,
-0.03345327451825142,
-0.03530487045645714,
0.030430536717176437,
-0.019520578905940056,
-0.010633544065058231,
0.050566576421260834,
0.037777725607156754,
-0.007887265644967556,
0.012835631147027016,
0.01757780648767948,
-0.017847944051027298,
-0... | 0.813098 | 0.790794 | 0.022305 |
Is it true that the inequality \( A(x) \geq \frac{M(x)}{x} \) holds for all integers \( x \), where \( A(x) = \sum_{n \leq x} \frac{\mu(n)}{n} \) and \( M(x) = \sum_{n \leq x} \mu(n) \)? If false, can you provide a specific counterexample? | automath | [
0.022993694990873337,
0.03343251347541809,
-0.036307260394096375,
-0.03180622681975365,
0.025242295116186142,
-0.019736019894480705,
-0.008909910917282104,
0.07783018797636032,
0.03512182831764221,
-0.030568938702344894,
-0.005398536566644907,
0.022271182388067245,
-0.042226869612932205,
0... | 0.816061 | 0.794969 | 0.021092 |
Given the differential equation
$$y' = \frac{y}{\sqrt{1-x^2}},$$
where \( y \) and \( x \) are real-valued functions and \( -1 < x < 1 \), find the general solution of this equation, expressing \( y \) in terms of \( x \). Additionally, determine the conditions under which the constant \( C \) can be chosen such th... | automath | [
-0.0004912652657367289,
0.04340820014476776,
-0.03610200062394142,
-0.041917357593774796,
0.03343559429049492,
-0.020916540175676346,
-0.012168203480541706,
0.07522282749414444,
0.040867701172828674,
-0.006910877302289009,
0.013896061107516289,
0.012452438473701477,
-0.026107031852006912,
... | 0.811024 | 0.79231 | 0.018715 |
Consider two undirected graphs \( G_1 = (V, E_1) \) and \( G_2 = (V, E_2) \) defined over the same set of vertices \( V \). Let \( w: E_1 \cup E_2 \rightarrow \mathbb{R} \) be a weight function assigned to the edges.
A path \( P = (v_0, e_1, v_1, e_2, \ldots, e_n, v_n) \) in \( G_1 \cup G_2 \) is said to be alternati... | automath | [
0.003739949781447649,
0.03196261450648308,
-0.018780792132019997,
-0.014855493791401386,
0.016558706760406494,
-0.0049848598428070545,
-0.027340667322278023,
0.05891558900475502,
0.02386930212378502,
-0.0027482295408844948,
0.03126313537359238,
0.015035144984722137,
-0.008000604808330536,
... | 0.813979 | 0.793086 | 0.020893 |
Given a bounded domain \(\Omega \subset \mathbb{R}^2\) and a divergence-free vector field \(u: \Omega \to \mathbb{R}^2\) with a 2D vorticity \(\omega = \text{curl } u\), does there exist a smooth function \(A^{x_0}(x)\) such that for all \(x\) in a ball \(B(x_0, r) \subset \Omega\), the following local version of the B... | automath | [
-0.016725411638617516,
0.04354693368077278,
-0.03894639015197754,
-0.028454244136810303,
0.02190203219652176,
-0.0016738154226914048,
-0.04342971369624138,
0.07143249362707138,
0.03465095907449722,
-0.003363034687936306,
0.003364246804267168,
0.02962525375187397,
-0.03154269978404045,
-0.0... | 0.811697 | 0.79197 | 0.019727 |
Given a cyclically ordered group \( (G, +) \) and a regular cut \( < \) on \( G \), define the conditions under which the set of all regular cuts \( C(G) \) can be considered a completion of \( (G, +) \). Specifically, what are the necessary and sufficient conditions for a cyclically ordered group \( (G, +) \) to be eq... | automath | [
0.022041048854589462,
0.014557715505361557,
-0.04663672298192978,
-0.036904796957969666,
0.026523947715759277,
-0.025741269811987877,
-0.012937557883560658,
0.05866633728146553,
0.013313007541000843,
-0.006254915613681078,
0.02191014029085636,
0.011404692195355892,
-0.03566114604473114,
0.... | 0.814653 | 0.797312 | 0.017341 |
Let \( D = \{ z \in \mathbb{C} : |z| < 1 \} \). For which of the following pairs \( (a,b,c,d) \) does there exist a holomorphic function \( f: D \to D \) such that \( f(a) = b \) and \( f'(c) = d \)?
1. \( a = 0, b = 0, c = 0, d = 2 \)
2. \( a = \frac{3}{4}, b = \frac{3}{4}, c = \frac{2}{3}, d = \frac{3}{4} \)
3. \( a... | automath | [
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0.02384369634091854,
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For a given rational number \( c \), does the equation \((x^2 - 1)(y^2 - 1) = cz^4\) have a non-trivial rational solution \((x, y, z) \in \mathbb{Q}^3\) with \( z \neq 0\)? If so, can you provide a specific example of such a solution? | automath | [
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0.026061207056045532,
0.012178587727248669,
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-0... | 0.834035 | 0.816383 | 0.017652 |
Let \( f : \{-1,1\}^n \to \mathbb{R} \) be a function such that \( k \in \mathbb{N} \) and the \( k \)-th component satisfies \( f^{=k} \). According to the Bonami Lemma, what is the relationship between \( \|\mathrm{T}_{1/\sqrt{3}} f^{=k}\|_4 \) and \( \|f^{=k}\|_2 \) provided that \( \|f^{=k}\|_4 \) is capped by \( \... | automath | [
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0.019566431641578674,
0.026318538933992386,
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-0.00320... | 0.837505 | 0.821268 | 0.016237 |
Given two isosceles triangles with equal side lengths \( L \) and base width \( w \), initially touching at a point and forming an apex angle \( \theta_i \), if both triangles are tilted towards each other by an angle \( \alpha \) about the angle bisector of \( \theta_i \) located at their common point, what is the rel... | automath | [
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0.03967522829771042,
0.025631481781601906,
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Given the ordinary differential equation
$$x' = 2t(1+x^2), \quad x(0) = 0,$$
which can be solved using separation of variables, determine the interval in which the solution
$$\lambda(t) = \tan(t^2)$$
is defined. What condition does the uniqueness of this solution satisfy, and for what values of \( t \) does \( \... | automath | [
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0.021097077056765556,
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0.015790237113833427,
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0.0168... | 0.825592 | 0.807119 | 0.018473 |
Consider the vector space \( \mathbb{R}^3 \) and the following subsets:
1. \( U_1 = \{(x, y, z) \in \mathbb R^3 : xyz > 0\} \)
2. \( U_2 = \{(x, y, z) \in \mathbb R^3 : y - 2 = 0\} \)
3. \( U_3 = \{s(1, 0, 0) + t(0, 0, 1) : s, t \in \mathbb R\} \)
4. \( U_4 = \{(0, 0, 0)\} \)
For each of these subsets, determine if ... | automath | [
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0.028296610340476036,
0.032479427754879,
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-0.032... | 0.817831 | 0.798756 | 0.019075 |
In a cubical box defined by the coordinates \(0 \leq x, y, z \leq 1\), with the potential on all faces except the top face at \(z=1\) set to zero, derive the expression for the coefficients \(a_{nm}\) in the series solution for the potential \(\phi(x, y, z)\). Given the boundary condition on the top face is \(\phi(x', ... | automath | [
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0.012923605740070343,
0.016661306843161583,
0.022970248013734818,
-0.03497540205717087,
-0.01361... | 0.829887 | 0.80637 | 0.023517 |
Let \( p \) be a prime number and consider a coherent sequence \( \alpha_n \) of \( p \)-adic numbers. If the first five terms of this sequence are given as follows:
- \( \alpha_0 = 1 \)
- \( \alpha_1 = 1 + p \)
- \( \alpha_2 = 1 + p + p^2 \)
- \( \alpha_3 = 1 + p + p^2 + p^3 \)
- \( \alpha_4 = 1 + p + p^2 + p^3 + p^... | automath | [
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0.021391848102211952,
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-0.0... | 0.845682 | 0.824139 | 0.021543 |
Given a locally compact abelian group \( G \), let \( \mu \) be a Haar measure on \( G \). Consider the regular representation \( \tau: G \rightarrow U(L^2(G, d\mu)) \) defined by \( \tau(g) f(x) = f(g^{-1} x) \) for \( f \in L^2(G, d\mu) \). If \( \psi: G \rightarrow \mathbb{C} \) is a continuous, compactly supported ... | automath | [
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0.02403290383517742,
-0.023256488144397736,
-0.0102... | 0.817711 | 0.791186 | 0.026524 |
What is the maximum size \( m \) of a subset \( S \) of the set \( \{ \frac{1}{\sqrt{n}}, -\frac{1}{\sqrt{n}} \}^n \) such that for any distinct vectors \( s_i, s_j \in S \), it holds that \( |\langle s_i, s_j \rangle| \leq \epsilon \), given that \( n \) is the dimension and \( \epsilon \leq \frac{1}{\sqrt{n}} \)? Pro... | automath | [
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0.029539907351136208,
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-0.014... | 0.826417 | 0.806822 | 0.019595 |
Let \( G \) be a locally compact Hausdorff group with a Haar measure \( \mu \), and let \( H \) be a closed normal subgroup of \( G \). If \( E \) is a union of \( H \)-cosets in \( G \) that is \( \mu \)-measurable, does it follow that the image \( q(E) \) under the quotient map \( q: G \to G/H \) is \( \nu \)-measura... | automath | [
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0.029247580096125603,
0.006149911321699619,
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-0.... | 0.805221 | 0.78328 | 0.021941 |
Consider the $3 \times 3$ matrix \( A = \begin{pmatrix} 3 & 1 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 1 \end{pmatrix} \). Determine whether the matrix \( A \) is diagonalizable. If it is, provide the matrices \( S \) and \( \Lambda \) such that \( A = S\Lambda S^{-1} \), where \( S \) is the matrix of eigenvectors and \( \Lambda \... | automath | [
0.021908065304160118,
0.03956013545393944,
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0.024241825565695763,
0.014634897001087666,
-0.05750182643532753,
-0.... | 0.808425 | 0.788765 | 0.01966 |
Given the Gronwall inequality where \( u(t) \leq \alpha(t) + \int_a^t \beta(s) u(s) \, ds \) for continuous functions \( \alpha \) and \( \beta \), define specific continuous functions \( \alpha(t) = t \) and \( \beta(t) = 1 \) for \( t \in [0, 1] \). What is the upper bound for \( u(1) \) according to the Gronwall ine... | automath | [
0.01771366596221924,
0.03821420669555664,
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0.029724562540650368,
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0.00904043484479189,
0.03710595518350601,
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0.00254... | 0.80486 | 0.785812 | 0.019048 |
Let \( R \) be a ring and consider the short exact sequence of complexes \( 0 \to K_1^\bullet \to K_2^\bullet \to K_3^\bullet \to 0 \). If \( K_3^n \) is flat for all \( n \in \mathbb{Z} \), and \( K_1^\bullet \) and \( K_2^\bullet \) are K-flat, what can we conclude about the K-flatness of \( K_3^\bullet \)? In how ma... | automath | [
0.023618122562766075,
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0.0008942582062445581,
0.029588129371404648,
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0.0... | 0.820185 | 0.797432 | 0.022753 |
Given a safe prime \( p \) of 2048 bits, let \( q = \frac{p - 1}{2} \) be a prime, and let \( (x_1, y_1) \) and \( (x_2, y_2) \) be two pairs of elements in \( \mathbb{Z}_p^* \) such that \( y_1 = x_1^{r_1} \mod p \) and \( y_2 = x_2^{r_2} \mod p \) for some unknown integers \( r_1 \) and \( r_2 \).
What is the compu... | automath | [
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0.05317645147442818,
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0.010610830038785934,
0.0026066340506076813,
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-0.0... | 0.821611 | 0.803873 | 0.017738 |
Given a positive integer \( p \) and a nonzero rational number \( r = \frac{a}{b} \), where \( a \) and \( b \) are integers, determine the \( p \)-adic valuation \( v_p(r) \) and explain how it relates to the largest power of \( p \) that divides \( r \). Additionally, if \( r = \frac{2^5 \cdot 3}{5^2} \), compute \( ... | automath | [
0.039948780089616776,
0.04828456789255142,
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0.03615489974617958,
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0.026037978008389473,
0.01566532999277115,
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-0.02... | 0.829648 | 0.812429 | 0.017219 |
In the context of Stokes' Theorem, if we consider a smooth surface \( S \) bounded by a positively oriented curve \( C \) and a vector field \( \vec{F} \), does the normal vector \( \vec{n} \) in the surface integral
\[
\iint_S \text{curl} \, \vec{F} \cdot \vec{n} \, dS
\]
necessarily have to be a unit vector, and i... | automath | [
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0.008789447136223316,
0.028527092188596725,
-0.023568376898765564,
-0.019... | 0.795176 | 0.77807 | 0.017106 |
Let \( f \) and \( g \) be non-negative integrable functions on the interval \( [0, 1] \). If \( \int_{0}^{1} f^2 \, dx = 4 \) and \( \int_{0}^{1} g^2 \, dx = 9 \), calculate the maximum possible value of \( \left| \int_{0}^{1} fg \, dx \right| \) using the Cauchy-Schwarz inequality. | automath | [
0.030852092429995537,
0.016482774168252945,
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0.04944552108645439,
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0.01570993848145008,
0.029653988778591156,
0.03087339550256729,
-0.032248515635728836,
-0.0263... | 0.823637 | 0.804236 | 0.019401 |
Consider the following statements about entire functions \( f : \mathbb{C} \rightarrow \mathbb{C} \):
1. There exists an entire function \( f \) that is real-valued and satisfies \( f(0) = 0 \) and \( f(1) = 1 \).
2. There exists an entire function \( f \) such that \( f(n + \frac{1}{n}) = 0 \) for all positive intege... | automath | [
0.03688502311706543,
0.058157142251729965,
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0.0414525642991066,
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-0.005187415983527899,
0.021460475400090218,
-0.04072114825248718,
-0.018... | 0.833186 | 0.806098 | 0.027089 |
Given two bivectors \( A \) and \( B \) in 3D space, express them in terms of a common vector \( u \) such that \( A = au \) and \( B = bu \) for some vectors \( a \) and \( b \). Specifically, what conditions on \( A \) and \( B \) guarantee the existence of such a vector \( u \), and how would you derive \( u \) in t... | automath | [
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0.03038705699145794,
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0.04397507384419441,
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0.010773580521345139,
0.0331684909760952,
-0.019633833318948746,
-0.021346... | 0.816545 | 0.799594 | 0.016951 |
In the context of convex optimization using Stochastic Gradient Descent (SGD), suppose we have a convex cost function \( f: \mathbb{R}^n \to \mathbb{R} \) with a unique global minimum at point \( x^* \).
If we begin at an arbitrary point \( x^{(0)} \) in the parameter space and compute the gradient \( \nabla f(x^{(0)... | automath | [
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0.013994383625686169,
0.031671635806560516,
-0.02569703757762909,
-0... | 0.81738 | 0.797882 | 0.019498 |
Let \( A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \) and \( B = \begin{pmatrix} 2 & 0 \\ 0 & -2 \end{pmatrix} \). Determine whether matrices \( A \) and \( B \) are similar, and justify your answer based on their characteristic polynomials and eigenvalues. Then consider the matrices \( C = \begin{pmatrix} 3 & 1 ... | automath | [
0.026295816525816917,
0.03086785040795803,
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0.04624555632472038,
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0.002013209741562605,
0.015682829543948174,
-0.05490105226635933,
-0.020... | 0.813506 | 0.791091 | 0.022415 |
Let \( X \) be a locally compact Hausdorff space, and let \( \mu \) be a Borel measure on \( X \) such that \( \mu(K) < \infty \) for all compact subsets \( K \subset X \). If \( K_1, K_2, \ldots, K_n \) are disjoint compact subsets of \( X \), express \( \mu\left(\bigcup_{i=1}^n K_i\right) \) in terms of the measures ... | automath | [
0.014542502351105213,
0.06560362875461578,
-0.0419924221932888,
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0.04755902290344238,
0.0056479014456272125,
-0.01844209060072899,
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0.0064528039656579494,
0.02925337851047516,
0.020070992410182953,
-0.047721415758132935,
0.00958... | 0.811925 | 0.78679 | 0.025135 |
Consider two independent random variables \(X_1\) and \(X_2\) that are identically distributed as \(N(0,1)\), and let \(Y\) be a random variable such that \(P(Y \leq 0) = P(Y > 0) = \frac{1}{2}\). Define a random variable \(Z\) by
\[
Z = \begin{cases}
1 & \text{if } Y \leq 0 \\
0 & \text{if } Y > 0
\end{cases}
\]
... | automath | [
-0.006327753886580467,
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0.018682336434721947,
-0.03927210345864296,
... | 0.81315 | 0.79662 | 0.016531 |
Let \( F \) be a distribution function for nonnegative weights assigned to the edges of a two-dimensional square lattice, and let \( p \) represent the proportion of edge weights that equal zero, i.e., \( F(0) \). For a critical case where \( F(0) = \frac{1}{2} \), what are the necessary and sufficient conditions under... | automath | [
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0.035602640360593796,
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0.028774362057447433,
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0.000010275186468788888,
0.017694130539894104,
0.008903315290808678,
-0.02159099094569683,
-... | 0.825915 | 0.805231 | 0.020684 |
Given a sequence of groups \( (G_n) \) such that \( G_0 = 1 \) and the derived subgroup \( G_n' \) is isomorphic to \( G_{n-1} \) for all \( n \ge 1 \), can you construct a group \( G = \bigoplus_{n \ge 1} G_{2n} \) that has a cyclical derived series with a period \( p \) (where \( p \ge 1 \)) such that \( G \) is not ... | automath | [
0.008870826102793217,
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0.015433155000209808,
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0.014266245998442173,
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0.01154647208750248,
0.021385585889220238,
-0.0455361045897007,
0.0011... | 0.827327 | 0.805866 | 0.021462 |
Let \( f(z) = e^{z} \) be an analytic function defined on the closed disk \( D = \{ z \in \mathbb{C} : |z| \leq 1 \} \). Calculate the value of the integral
\[
\oint_{C} f(z) \, dz
\]
where \( C \) is the circle of radius 1 centered at the origin. What is the result of this integral? | automath | [
0.03125102445483208,
0.042529232800006866,
-0.041228488087654114,
-0.028158927336335182,
0.02259339578449726,
0.00968980323523283,
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0.03863342106342316,
0.0565163679420948,
0.020038576796650887,
0.026432469487190247,
0.01061882171779871,
-0.03129464015364647,
-0.01003... | 0.813411 | 0.796952 | 0.016459 |
Construct a complete graph \(K_4\) (four nodes fully connected) alongside a graph \(G\) with five nodes, consisting of a central hub connected to all four of its satellites, each of which is connected to two satellites and to the hub. What is the maximum ratio \( \frac{|E(S)|}{|S|} \) achievable by an optimal subset \(... | automath | [
0.022367341443896294,
0.027299145236611366,
-0.03005465678870678,
-0.03703375160694122,
0.011927865445613861,
-0.020496457815170288,
-0.02593843825161457,
0.053512319922447205,
0.029400648549199104,
-0.007719316519796848,
0.0191725492477417,
0.02757694385945797,
-0.014744731597602367,
0.00... | 0.8268 | 0.801541 | 0.025259 |
Let \( h(x) = \sqrt{2x - 3} \). Determine the limit \( \lim_{x \to 2} h(x) \) and verify if \( h \) is continuous at \( c = 2 \) by finding an appropriate \( \delta \) such that for every \( \epsilon > 0 \), the condition \( |h(x) - 1| < \epsilon \) holds whenever \( 0 < |x - 2| < \delta \). What is the value of \( \de... | automath | [
0.003554487833753228,
0.04273972287774086,
-0.028975702822208405,
-0.026158656924962997,
0.02785925567150116,
-0.023298684507608414,
-0.03286231309175491,
0.09492267668247223,
0.02605181373655796,
-0.028719956055283546,
-0.006434012670069933,
0.041593633592128754,
-0.017436234280467033,
-0... | 0.826653 | 0.808543 | 0.01811 |
Given the domain \( D = \{(x,y) \in \mathbb{R}^2 \mid xy > 0\} \), which is disconnected because it consists of points in the first and third quadrants (excluding the axes), determine whether \( D \) can be classified as simply connected. Furthermore, if we modify \( D \) to include the origin, forming \( D' = \{(x,y) ... | automath | [
0.003693560604006052,
0.03507890924811363,
-0.012547189369797707,
-0.05052567645907402,
0.029450101777911186,
-0.0019015561556443572,
-0.013278038240969181,
0.06187408044934273,
0.03805195912718773,
0.01745450310409069,
0.014293656684458256,
0.022922934964299202,
-0.018662303686141968,
-0.... | 0.810442 | 0.787281 | 0.023161 |
Given three sets \(A_1\), \(A_2\), and \(A_3\) with \(N(A_1) = 100\), \(N(A_2) = 1000\), and \(N(A_3) = 10,000\), find the number of elements in \(A_1 \cup A_2 \cup A_3\) under the following scenarios:
1. If \(A_1 \subseteq A_2\) and \(A_2 \subseteq A_3\).
2. If \(A_1\), \(A_2\), and \(A_3\) are pairwise disjoint sets... | automath | [
0.03278765827417374,
0.013136197812855244,
-0.027024488896131516,
-0.005794824101030827,
0.027873415499925613,
-0.007207571994513273,
-0.04306947439908981,
0.056145042181015015,
0.025977566838264465,
0.016893727704882622,
0.017991770058870316,
0.020880471915006638,
-0.03939976170659065,
-0... | 0.823784 | 0.804542 | 0.019242 |
Given the set $\{1, \sqrt{2}, \sqrt{3}, \sqrt{6}\}$, determine whether it forms a basis for the vector space over $\mathbb{Q}$ spanned by these vectors. Additionally, calculate the dimension of this vector space. | automath | [
0.028183400630950928,
0.04875953122973442,
-0.032996855676174164,
-0.04113302752375603,
0.053160786628723145,
-0.01732366345822811,
-0.004426676779985428,
0.050961531698703766,
0.03209521993994713,
-0.02259420044720173,
0.01913379691541195,
0.02767149545252323,
-0.04125504940748215,
-0.040... | 0.834688 | 0.817415 | 0.017273 |
If \( u \) belongs to the Sobolev space \( H^s(\mathbb{R}^n) \) for some \( s > \frac{n}{2} \), what is the least upper bound \( C \) such that
\[
\|u\|_{L^\infty(\mathbb{R}^n)} \leq C \|u\|_{H^s(\mathbb{R}^n)}
\]
can occur, and how does \( C \) depend on \( n \) and \( s \)? | automath | [
-0.010862124152481556,
0.03802862763404846,
-0.03707360476255417,
-0.02443576231598854,
0.039071615785360336,
0.007925797253847122,
-0.03082684427499771,
0.05754980072379112,
0.0296047180891037,
-0.01088276319205761,
-0.007232132833451033,
0.02891622669994831,
-0.013692078180611134,
-0.034... | 0.806991 | 0.786902 | 0.020089 |
Given the Erdős-Ginzburg-Ziv theorem, which asserts that for any integers \( a_1, a_2, \ldots, a_{2n-1} \) there exist distinct indices \( i_1, i_2, \ldots, i_n \) such that the sum \( a_{i_1} + a_{i_2} + \cdots + a_{i_n} \equiv 0 \pmod{n} \), determine the minimum value of \( n \) such that the theorem holds true. Spe... | automath | [
0.02942582406103611,
0.026807749643921852,
-0.022768495604395866,
-0.0301771592348814,
0.047390248626470566,
-0.023942284286022186,
-0.02044416032731533,
0.042997684329748154,
0.015167242847383022,
-0.009877252392470837,
0.001914719701744616,
0.0043463693000376225,
-0.029346497729420662,
-... | 0.823626 | 0.798189 | 0.025437 |
Suppose \( q = \langle 1, 2 \rangle \) is a nonsingular diagonal quadratic space over \( \mathbb{R} \). Determine whether the direct sum \( q \oplus q' \) is isomorphic to a totally hyperbolic space, where \( q' = \langle -1, -2 \rangle \). If it is, provide the isomorphism as an explicit quadratic form. | automath | [
0.0014298780588433146,
0.04174476116895676,
-0.03665853291749954,
-0.03813566267490387,
0.020667584612965584,
0.0016811517998576164,
0.007307469844818115,
0.0674813836812973,
0.023488102480769157,
-0.011536667123436928,
-0.002922946587204933,
0.013074887916445732,
-0.043296653777360916,
-0... | 0.829712 | 0.802885 | 0.026828 |
Consider the following function defined on the real numbers:
\[
f(x) = \begin{cases}
1 & \text{if } x \text{ is rational} \\
0 & \text{if } x \text{ is irrational}
\end{cases}
\]
Is the limit
\[
\lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x)
\]
equal to 1 for any rational \( x \)? If so, for what v... | automath | [
0.011029074899852276,
0.05524564906954765,
-0.037034451961517334,
-0.028245586901903152,
0.039946239441633224,
-0.002534745493903756,
-0.00027157622389495373,
0.05564656853675842,
0.041292447596788406,
-0.01597294956445694,
-0.00335469632409513,
0.011814004741609097,
-0.027694428339600563,
... | 0.825283 | 0.80413 | 0.021153 |
Consider an ordered set \( (X, \geq) \) and let \( U \subseteq X \) be an up-set. If \( |U| = n \), find the maximum number of elements in any down-set \( D \) such that \( U \cap D = \emptyset \) under the assumption that \( |X| = m \). Express your answer in terms of \( n \) and \( m \). | automath | [
0.03257175534963608,
0.002233185339719057,
-0.046556439250707626,
-0.023511534556746483,
0.02444770187139511,
-0.010745354928076267,
-0.007524280808866024,
0.059185028076171875,
0.031899210065603256,
-0.011091492138803005,
0.017131857573986053,
0.03296341747045517,
-0.035070668905973434,
-... | 0.83239 | 0.809575 | 0.022815 |
Given a set of \( n \) points in two-dimensional space, determine the maximum number of points that can be placed such that it is impossible to find a straight line (hyperplane) that separates them into two distinct classes with at least one point from each class on either side. What is the theoretical maximum value of... | automath | [
0.017664168030023575,
0.021695652976632118,
-0.03813507780432701,
-0.0303080715239048,
0.02358219213783741,
-0.021324487403035164,
-0.026220588013529778,
0.04892163351178169,
0.04241297021508217,
-0.012238726019859314,
0.010121646337211132,
0.04534131661057472,
-0.02201857790350914,
-0.017... | 0.839739 | 0.820629 | 0.01911 |
Consider the second order partial differential equation given by
$$\frac{{\partial^2 u}}{{\partial t^2}} = k \frac{{\partial^2 u}}{{\partial x^2}} + c$$
with boundary conditions
$$u(0,t) = 0$$
$$u(1,t) = 2c_0$$
and the initial condition
$$u(x,0) = c_0(1 - \cos(\pi x)).$$
Assuming \(c\) is a constant, classi... | automath | [
0.0023868379648774862,
0.032972726970911026,
-0.037480004131793976,
-0.03839560225605965,
0.023836255073547363,
-0.010460560210049152,
-0.005321463569998741,
0.10238375514745712,
0.05304964259266853,
0.012243059463799,
0.009548407979309559,
0.026958603411912918,
-0.019339274615049362,
-0.0... | 0.812664 | 0.792113 | 0.020551 |
Let \( \gamma : D \rightarrow D \) be a bijection such that for every bijection \( \delta : D \rightarrow D \), we have \( \gamma \circ \delta = \delta \circ \gamma \). If \( \gamma \neq \text{id}_D \), what can you conclude about the cardinality \( |D| \)? Specifically, determine if \( |D| = 2 \) or \( |D| > 2 \), and... | automath | [
0.0023347334936261177,
0.014039736241102219,
-0.03480922430753708,
-0.023014526814222336,
0.018103381618857384,
-0.03519647940993309,
-0.022775091230869293,
0.07575340569019318,
0.039123598486185074,
-0.012107906863093376,
-0.006625842303037643,
0.02083432674407959,
-0.030809739604592323,
... | 0.828763 | 0.80721 | 0.021553 |
For a positive integer \( x \), define the function \( f(x) \) as follows:
\[
f(x) =
\begin{cases}
1 & \text{if } x = 1 \\
\frac{x}{10} & \text{if } x \text{ is divisible by } 10 \\
x + 1 & \text{otherwise}
\end{cases}
\]
A sequence is defined as \( x_1 = x \) and \( x_{n+1} = f(x_n) \) for all positive integer... | automath | [
0.023769576102495193,
0.04240962862968445,
-0.04434109106659889,
-0.02863272838294506,
0.02836517058312893,
-0.014769314788281918,
-0.024925077334046364,
0.06405705958604813,
0.02971695363521576,
0.006417660973966122,
-0.005232884082943201,
0.0191044919192791,
-0.04399450123310089,
-0.0120... | 0.834297 | 0.816825 | 0.017472 |
Let \( P_3(\mathbb{R}) \) be the space of real polynomials of degree at most 3, equipped with the inner product defined by
\[
\langle f(x), g(x) \rangle = \int_0^1 f(x) g(x) \, dx.
\]
Consider the subspace \( U \) spanned by the polynomials
\[
u_1 = 1 + 2x - 3x^2 \quad \text{and} \quad u_2 = x - x^2 + 2x^3.
\]
De... | automath | [
0.013816461898386478,
0.04639481380581856,
-0.041838523000478745,
-0.04338482394814491,
0.041746970266103745,
-0.009699254296720028,
0.0007885437225922942,
0.06811349838972092,
0.032434143126010895,
0.004582209978252649,
0.004511374980211258,
0.033892881125211716,
-0.0353364571928978,
-0.0... | 0.823522 | 0.796685 | 0.026837 |
Let \( A \) be a weakly Artinian integral domain, and let \( K \) be its field of fractions. If \( L \) is a finite extension field of \( K \) and \( B \) is a subring of \( L \) containing \( A \), how many non-zero prime ideals does \( B \) contain, and are they all maximal if the statement of the Krull-Akizuki theor... | automath | [
0.021227218210697174,
0.014881307259202003,
-0.02164708450436592,
-0.030283795669674873,
0.016925286501646042,
-0.00965254195034504,
0.005006195046007633,
0.033856142312288284,
0.02895781211555004,
-0.015683546662330627,
0.013923545368015766,
0.014369456097483635,
-0.02671726979315281,
0.0... | 0.825615 | 0.804048 | 0.021567 |
Given two non-negative sequences \( x = (x_1, x_2, \ldots, x_n) \) and \( y = (y_1, y_2, \ldots, y_n) \) such that \( x_1 \ge x_2 \ge \cdots \ge x_n \) and \( y_1 \ge y_2 \ge \cdots \ge y_n \), and the condition that for all positive integers \( p \),
\[
\sum_{i=1}^n x_i^p \geq \sum_{i=1}^n y_i^p,
\]
is it possible ... | automath | [
0.012516462244093418,
0.042296748608350754,
-0.03665352240204811,
-0.012539150193333626,
0.03623780980706215,
-0.011735194362699986,
-0.00493277283385396,
0.08803801238536835,
0.02774464152753353,
-0.01721459999680519,
0.0002080319682136178,
0.019795669242739677,
-0.026594405993819237,
-0.... | 0.835595 | 0.813427 | 0.022167 |
Given a function \( F : \{0, 1, \ldots, n-1\} \to \{0, 1, \ldots, m-1\} \) that satisfies the property \( F((x + y) \mod n) = (F(x) + F(y)) \mod m \), and a lookup table with \( \frac{1}{5} \) of its entries altered by an adversary, describe a randomized algorithm that outputs \( F(z) \) for any input \( z \) with a pr... | automath | [
0.03959399834275246,
0.026587454602122307,
-0.026155846193432808,
-0.02174040488898754,
0.04307200387120247,
-0.02092083729803562,
-0.023823197931051254,
0.050964999943971634,
0.020738162100315094,
-0.006859682034701109,
0.00564576406031847,
0.04054143279790878,
-0.03765726089477539,
-0.02... | 0.832399 | 0.816134 | 0.016265 |
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