instruction_seed stringlengths 120 1.22k | source stringclasses 1
value | embeddings sequencelengths 1.02k 1.02k | mean_positive_score float64 0.78 0.86 | mean_negative_score float64 0.76 0.83 | difference_score float64 0.02 0.04 |
|---|---|---|---|---|---|
Consider the equation $$\varphi(\varphi(a))=\varphi(\varphi(b))$$ where $\varphi$ is the Euler totient function and $\varphi(a) \neq \varphi(b)$. If $a = 2^{k-1}3^2$ and $b = 2^{k+1}$ for $k \geq 2$, compute the explicit values of $\phi(\phi(a))$ and $\phi(\phi(b))$ for $k = 2$ and $k = 3$, and verify if they are equal... | automath | [
0.04600238427519798,
0.045935045927762985,
-0.04080057889223099,
-0.04755646735429764,
0.025734473019838333,
-0.013690875843167305,
-0.0049880025908350945,
0.07500509917736053,
0.02116716280579567,
-0.01257823035120964,
0.020063847303390503,
0.02592398412525654,
-0.02182025834918022,
-0.02... | 0.856571 | 0.826255 | 0.030316 |
Let \(X_n\) be a sequence of random variables defined by
\[
P(X_n = 0) = 1 - \left(\frac{1}{2}\right)^n, \quad P(X_n = 1) = \left(\frac{1}{2}\right)^n \text{ for } n = 1, 2, 3, \ldots
\]
What is the limiting behavior of \( E(X_n) \) as \( n \) approaches infinity, and does \( E(X_n) \) converge almost surely and in ... | automath | [
-0.010324639268219471,
0.041191574186086655,
-0.04050051420927048,
-0.020398421213030815,
0.005423265974968672,
-0.008236466906964779,
-0.01498547475785017,
0.06462205946445465,
0.02227318473160267,
-0.011651325970888138,
-0.0108101861551404,
0.02998296171426773,
-0.03633501008152962,
-0.0... | 0.818617 | 0.798521 | 0.020095 |
Let \( A \) be an \( n \times n \) real matrix such that for all vectors \( v \) satisfying \( 1'v = 0 \), the condition \( v'Av \geq 0 \) holds. What is the minimum dimension \( n \) such that it is possible for such a matrix \( A \) to have a rank of \( n-1 \) where its first row and column contain arbitrary real num... | automath | [
0.022409703582525253,
0.020671017467975616,
-0.03584643825888634,
-0.038664452731609344,
0.04170046001672745,
-0.004301147535443306,
-0.010718461126089096,
0.06366799026727676,
0.032515980303287506,
-0.015284513123333454,
-0.020330224186182022,
0.016067199409008026,
-0.02923295646905899,
-... | 0.82793 | 0.798664 | 0.029266 |
Let \( \mathfrak{g} \) be a Lie algebra over \( \mathbb{C} \) and \( \mathfrak{h} \) be an ideal of \( \mathfrak{g} \). Consider the canonical homomorphism \( \pi: \mathfrak{g} \to \mathfrak{g/h} \) defined by \( \pi(x) = x + \mathfrak{h} \) for all \( x \in \mathfrak{g} \).
Compute the image under \( \pi \) of the e... | automath | [
0.04395950213074684,
0.04787546396255493,
-0.02836165577173233,
-0.03773186355829239,
0.04793958365917206,
-0.03162115439772606,
-0.004084398038685322,
0.07180482894182205,
0.05665863677859306,
-0.00027099475846625865,
0.0037574509624391794,
0.015501110814511776,
-0.03465479984879494,
-0.0... | 0.813672 | 0.79073 | 0.022942 |
Given the Cauchy-Riemann equations:
1. \( \frac{\partial u}{\partial x}(x_0, y_0) = \frac{\partial v}{\partial y}(x_0, y_0) \)
2. \( \frac{\partial u}{\partial y}(x_0, y_0) = -\frac{\partial v}{\partial x}(x_0, y_0) \)
determine the conditions under which the functions \( u(x, y) \) and \( v(x, y) \) are differentiab... | automath | [
0.01538369245827198,
0.051970358937978745,
-0.029115652665495872,
-0.038507502526044846,
0.043380022048950195,
-0.000712998618837446,
-0.0005471101612783968,
0.06796492636203766,
0.03319954872131348,
-0.00013314954412635416,
0.004766198806464672,
0.02015502378344536,
-0.03674563765525818,
... | 0.807541 | 0.78829 | 0.019251 |
Given that the divergence of a vector field \(\mathbf{A}\) is zero if and only if \(\mathbf{A}\) can be expressed as the curl of another vector field \(\mathbf{B}\), and that the curl of \(\mathbf{A}\) is zero if and only if \(\mathbf{A}\) can be expressed as the gradient of a scalar function \(\varphi\), what are the ... | automath | [
-0.007882325910031796,
0.021911010146141052,
-0.020352540537714958,
-0.04700283333659172,
0.04100365936756134,
0.0020774155855178833,
-0.040151942521333694,
0.06428495794534683,
0.013308471068739891,
-0.00493610929697752,
-0.0081244595348835,
0.003116789273917675,
-0.027653077617287636,
-0... | 0.814834 | 0.797854 | 0.016979 |
Let \( k \) be a field and \( \mathcal{F} \) be a coherent sheaf on \( \mathbf{P}^n_k \) with a Hilbert polynomial \( P \in \mathbf{Q}[t] \). If \( n \) is fixed and \( P(d) = \dim_k H^0(\mathbf{P}^n_k, \mathcal{F}(d)) \) holds for all \( d \gg 0 \), what is the value of \( P(0) \) if \( P(t) \) is defined as \( P(t) =... | automath | [
0.023889444768428802,
0.03893152251839638,
-0.04631821811199188,
-0.027308322489261627,
0.030852042138576508,
-0.025829458609223366,
-0.0011154067469760776,
0.06795307993888855,
0.0420355387032032,
-0.003554154420271516,
-0.012260040268301964,
0.03248239681124687,
-0.017825359478592873,
0.... | 0.82203 | 0.803171 | 0.018859 |
Given the function \( f : \mathbb{N} \times \mathbb{N} \to \mathbb{N} \) defined by a bijection that maps pairs of natural numbers to natural numbers, determine the number of distinct elements in the image of \( f \) if \( f(a, b) = a + b \). What is the maximum value of \( f(a, b) \) for \( 1 \leq a, b \leq n \)? | automath | [
0.049145475029945374,
0.019371356815099716,
-0.04412835091352463,
-0.019130639731884003,
0.036005742847919464,
-0.018362177535891533,
-0.013645408675074577,
0.06294510513544083,
0.028252525255084038,
0.01654122583568096,
0.005801638122648001,
0.02922958880662918,
-0.03801942989230156,
-0.0... | 0.83178 | 0.814315 | 0.017465 |
"Given the function \\( f(x) = x \\) for \\( 0 \\leq x \\leq 2\\pi \\) with periodic extension, its (...TRUNCATED) | automath | [-0.001890435814857483,0.05568404495716095,-0.034146204590797424,-0.02959735319018364,0.018970211967(...TRUNCATED) | 0.823232 | 0.801432 | 0.0218 |
"Let vectors \\(\\vec{v}\\), \\(\\vec{w}\\), and \\(\\vec{u}\\) be in \\(\\mathbb{R}^n\\). If \\(\\v(...TRUNCATED) | automath | [0.025601213797926903,0.0565556138753891,-0.045053135603666306,-0.04198604077100754,0.05196850001811(...TRUNCATED) | 0.814686 | 0.796262 | 0.018423 |
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