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 \( a_n \) be a sequence of positive numbers such that \( 0 \leq \frac{a_{n+1}}{a_n} \leq 1 + \frac{1}{n^2} \) for all \( n \). Given that the series \( \sum_{k=1}^{\infty} \frac{1}{k^2} \) converges, does this imply that the sequence \( a_n \) converges to a finite limit? If so, what is the limit? | automath | [
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0.012600241228938103,
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-... | 0.823974 | 0.802118 | 0.021857 |
In the context of category theory, can you verify that the canonical map $$\kappa : \text{colim}_J \text{lim}_IF(i,j) \longrightarrow \text{lim}_I\text{colim}_JF(i,j)$$ is a bijection when $J$ is a small filtered category and $I$ is a finite index set? If so, demonstrate the necessary steps to establish this bijection,... | automath | [
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0.001224352396093309,
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0.000... | 0.812335 | 0.792988 | 0.019347 |
In a problem where applicants have specific apartment size demands and there are apartments of various sizes available, suppose we have \( n \) applicants with demands \( d_1, d_2, \ldots, d_n \) sorted in increasing order and \( m \) apartments with sizes \( s_1, s_2, \ldots, s_m \) also sorted in increasing order. If... | automath | [
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0.032399509102106094,
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-0.0... | 0.813384 | 0.791027 | 0.022357 |
Given a rectangular table with \( n \) rows and \( m \) columns, where \( 2 \leq n, m \leq 300 \) and \( 2 \leq 2k \leq n \cdot m \), Valera wants to place exactly \( k \) non-overlapping tubes on the table. Each tube is a sequence of connected cells that fulfill the conditions stated above.
1. What is the maximum po... | automath | [
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0.04057451710104942,
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-0.01174... | 0.825426 | 0.804118 | 0.021308 |
Given a unit disk \( D \) and a square \( R = \{(x,y) : -\frac{1}{\sqrt{2}} \leq x, y \leq \frac{1}{\sqrt{2}}\} \) inscribed within it, we generate \( n \) independent and identically distributed observations \( Z_1, Z_2, \ldots, Z_n \) uniformly from \( D \). Let \( T \) be the number of observations that fall within ... | automath | [
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0.... | 0.837987 | 0.808112 | 0.029875 |
Find all real numbers \( \lambda \) and the corresponding continuous functions \( f \) on the interval \([0,1]\) such that the equation
\[
\int_0^1 \min(x,y) f(y) \,dy = \lambda f(x)
\]
has a non-zero solution \( f \). | automath | [
0.014409903436899185,
0.019329575821757317,
-0.040424492210149765,
-0.036667097359895706,
0.05450957641005516,
0.012182317674160004,
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0.05768134444952011,
0.03400404006242752,
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0.020573170855641365,
0.027278080582618713,
-0.019034922122955322,
-0... | 0.827573 | 0.811073 | 0.0165 |
Given the recurrence relation \( c[n] = \sqrt{n-1} \cdot c[n-1] + \sqrt{n} \cdot c[n+1] \), with the constraint \( \sum_{n=0}^{\infty} |c_n|^2 = 1 \), find the value of \( \bar{n} \) for which there exists a sequence \( \{c_n\} \) satisfying the additional condition \( \sum_{n=0}^{\infty} n |c_n|^2 = \bar{n} \) and det... | automath | [
0.0157830398529768,
0.05656253546476364,
-0.040010057389736176,
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0.038676917552948,
0.01192563772201538,
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0.08754613250494003,
0.021145660430192947,
-0.004163719248026609,
-0.005174841266125441,
0.009783917106688023,
-0.03717629611492157,
-0.0149... | 0.842328 | 0.815587 | 0.026741 |
Is it possible to derive the Recursion Theorem, specifically $$\forall e \exists k [\Phi_e \text{ is total } \implies \Phi_{\Phi_e(k)} \cong \Phi_k],$$ from Lawvere's Fixed Point Theorem, which states that if $\mathcal{C}$ is Cartesian closed and $f: A \rightarrow B^A$ is an epimorphism, then every $g: B \rightarrow B$... | automath | [
0.012703184969723225,
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0.01528481300920248,
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0.00... | 0.808434 | 0.789718 | 0.018716 |
Let \( R \) be a ring, and let \( f_1, \ldots, f_n \in R \) generate the unit ideal. If the ring \( A = \prod_{i} R_{f_i} \) is a faithfully flat \( R \)-algebra, how many elements \( f_1, \ldots, f_n \) are required such that the Čech cohomology of quasi-coherent sheaves on affine schemes is trivial? Provide a specifi... | automath | [
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0.024925120174884796,
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-0.... | 0.82014 | 0.797497 | 0.022643 |
Given a set of \( n \) points in a plane, determine the maximum number of nested polygons that can be formed such that:
1. Each polygon contains only one child polygon (no siblings).
2. Each polygon has only one immediate parent and does not lie in the intersection of two or more parents.
Assuming that the outermost ... | automath | [
0.033802203834056854,
0.02475854381918907,
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0.05835395306348801,
0.03482130169868469,
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0.004369131289422512,
0.043321315199136734,
-0.03167436644434929,
-0.0... | 0.838111 | 0.815859 | 0.022252 |
Consider two closed subspaces \(\mathcal{H}_1\) and \(\mathcal{H}_2\) of an infinite-dimensional Hilbert space \(\mathcal{H}\) such that \(\mathcal{H} = \mathcal{H}_1 \oplus \mathcal{H}_2\). If \(\mathcal{H}_1\) and \(\mathcal{H}_2\) are orthogonal, what is the dimension of the external direct sum \(\mathcal{H}_1 \oplu... | automath | [
0.0011172863887622952,
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0.027406444773077965,
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0.024556227028369904,
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-0... | 0.824853 | 0.806253 | 0.0186 |
Given the theorem \( \text{sub_of_eql} \) which states that if \( EQL(a, b) \) is true and \( a \neq b \), then \( BND(a - b) \) must also be true, find a pair of specific real numbers \( a \) and \( b \) such that \( |a - b| < 1 \) and both \( a \) and \( b \) are distinct. What is the interval in which \( a - b \) li... | automath | [
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0.03915151208639145,
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-0.0... | 0.837612 | 0.814928 | 0.022684 |
Let $\mathcal{C}$ be a category with 5 objects and 7 morphisms. If $\mathcal{S}$ is fibred in groupoids over $\mathcal{C}$, how many unique lifts $\chi : z \to y$ can you construct for a given pair of morphisms $\phi : y \to x$ and $\psi : z \to x$, assuming that there exists at least one morphism $f : p(z) \to p(y)$ s... | automath | [
0.017534110695123672,
0.03150441125035286,
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0.026594128459692,
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0.058601029217243195,
0.0448014922440052,
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0.03637658804655075,
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-0.00... | 0.821599 | 0.803292 | 0.018307 |
Let \( R \) be a commutative ring. Determine the structure of the set of units in the polynomial ring \( R[x] \), and provide an explicit characterization of these units in terms of the elements of \( R \). Specifically, describe the conditions on the coefficients of a polynomial \( a_0 + a_1x + a_2x^2 + \ldots + a_nx^... | automath | [
0.04934742674231529,
0.02830888144671917,
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0.019621143117547035,
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0.06346738338470459,
0.042684778571128845,
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0.007062159478664398,
0.01849476993083954,
-0.036181431263685226,
-0.01... | 0.815198 | 0.790971 | 0.024227 |
Let \( G \) be a connected graph with \( n \) edges and \( m \) vertices. Consider the equivalence relation \( \sim \) defined on the edges of the graph such that for edges \( e_1 \) and \( e_2 \), \( e_1 \sim e_2 \) if they lie on a common cycle or are identical. Determine the maximum number of equivalence classes of ... | automath | [
0.015507864765822887,
0.007754790596663952,
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0.0355198010802269,
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0.05440884828567505,
0.012470073066651821,
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0.016123030334711075,
0.020772013813257217,
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0.0... | 0.823664 | 0.80132 | 0.022344 |
Consider the optimization problem defined by the following components:
1. A vector of variables \(\mathbf{x} = (x_1, x_2, \ldots, x_N)\).
2. The objective function given by:
\[
\sum_{n=1}^N \log\left(1+\frac{x_n}{\alpha+\sum_{m=1}^M \beta_m^{(n)} x_m}\right)
\]
3. The constraint:
\[
\mathbf{A}\mathb... | automath | [
0.030823925510048866,
0.02552047185599804,
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0.03071035072207451,
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0.0658487007021904,
0.026560407131910324,
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0.0029353229328989983,
-0.03607490658760071,
0.001... | 0.822068 | 0.800054 | 0.022014 |
Consider the following linear programming problem:
$$
\begin{array}{ll}
\text{maximize} & 0 x_1 + 0 x_2 + 0 x_3\\
\text{subject to} & x_1 + 3x_2 + 2x_3 = 3\\
& 2x_1 + 7x_2 + x_3 = 4\\
& 3x_1 + x_2 + 2x_3 = 5\\
& x_1, x_2, x_3 \geq 0
\end{array}
$$
What is the unique nonnegative feasible solution \( (x_1, x_2, x_3) \)... | automath | [
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0.02029494382441044,
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0.08395695686340332,
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0.0017747849924489856,
0.010281865485012531,
-0.022594135254621506,
-0.0... | 0.827416 | 0.806972 | 0.020443 |
Let \( n \) be a positive integer and \( p \) be the probability of success on a single trial in a binomial distribution. If \( X \) represents the number of successes in \( n \) trials, determine the probability \( P(X < r) \) in terms of \( n \), \( r \), and \( p \). Then, confirm that this probability is equal to \... | automath | [
-0.0014487769221886992,
0.008198878727853298,
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0.02352151833474636,
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0.060380153357982635,
0.03380771726369858,
0.008333656005561352,
-0.0059649827890098095,
0.02937537617981434,
-0.027849482372403145,
... | 0.825029 | 0.801268 | 0.023761 |
Consider the non-commutative algebra \( A = M_n(\mathbb{C}) \) for some integer \( n > 1 \).
1. What can be said about the maximal ideals of \( A \)?
2. Specifically, how many maximal ideals does \( A \) have, and what does this imply about the structure of the quotient \( A/M \) for a maximal ideal \( M \)?
Answe... | automath | [
0.0615534633398056,
0.022518767043948174,
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0.03888769820332527,
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0.04998333752155304,
0.028938306495547295,
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0.0058565703220665455,
-0.011591976508498192,
-0.028037915006279945,
-0... | 0.806501 | 0.782737 | 0.023763 |
Let \( a_n = \frac{(-1)^n}{n} \) for \( n \in \mathbb{N} \). Determine whether the infinite product \( \prod_{n=1}^\infty \left(1 + a_n\right) \) is absolutely convergent. Provide reasoning for your answer. | automath | [
0.024137528613209724,
0.04193206503987312,
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0.03529568389058113,
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0.07799532264471054,
0.025155778974294662,
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0.010971141047775745,
-0.035800401121377945,
... | 0.831704 | 0.814579 | 0.017125 |
Let \( A \subset \mathbb{R}^n \) be a measurable set such that \( m(A) < +\infty \). Suppose there exists a sequence of continuous functions \( \{ f_k \} \) such that \( f_k \to f \) almost everywhere on \( A \) and \( f_k \in L^1(A) \) for all \( k \). If \( L^1(A) \) is the space of integrable functions, what conditi... | automath | [
0.008402216248214245,
0.03958645090460777,
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0.04034345597028732,
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0.004007645882666111,
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0.02250542864203453,
-0.0281345397233963,
-0.01... | 0.808115 | 0.776643 | 0.031472 |
Let \( G \) be the group presented by \( \langle g_1, g_2 \mid r(g_1, g_2) = 1 \rangle \), where \( r(g_1, g_2) \) is some relator. If \( G \) has order \( n \) and every element can be represented as a product of the generators \( g_1 \) and \( g_2 \), what can be said about the order of the group presented by \( \lan... | automath | [
0.039132460951805115,
0.03654397279024124,
-0.031847018748521805,
-0.029209380969405174,
0.020061539486050606,
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0.059598732739686966,
0.03330647572875023,
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0.008915388956665993,
0.019139107316732407,
-0.030731050297617912,
-... | 0.811945 | 0.788467 | 0.023478 |
Given a vector space \( V \) of dimension \( n \) and \( m \) linearly independent vectors \( \vec{a}_1, \vec{a}_2, \ldots, \vec{a}_m \) where \( m < n \), is it true that there exists a vector \( \vec{v} \in V \) that is orthogonal to all of the \( m \) vectors? If so, can you describe a method to find such a vector? | automath | [
0.017909368500113487,
0.041107695549726486,
-0.03553592041134834,
-0.02913578599691391,
0.0428956113755703,
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0.003769171889871359,
0.06922945380210876,
0.026068201288580894,
-0.024166131392121315,
-0.005977727472782135,
0.016296619549393654,
-0.0269907359033823,
-0.02... | 0.816746 | 0.799282 | 0.017463 |
Let \( f: [-1, 1] \to [0, 1] \) be a continuous function such that \( x^2 + (f(x))^2 \le 1 \) for all \( x \in [-1, 1] \) and \( \int_0^1 f(x) \, dx = \frac{\pi}{4} \). If we define \( f(x) = \sqrt{1-x^2} \) for all \( x \in [0, 1] \), what is the value of the integral
\[
\int_{-1/2}^{1/\sqrt{2}} \frac{f(x)}{1-x^2} \... | automath | [
0.007250762078911066,
0.050577905029058456,
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0.05688382312655449,
0.0028629926964640617,
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0.05928657203912735,
0.03379911929368973,
0.0030540807638317347,
0.006378438789397478,
0.024222178384661674,
-0.012505692429840565,
-0.0... | 0.820457 | 0.803732 | 0.016725 |
Let \( I = [a, b] \) be a compact interval in \( \mathbb{R} \), and let \( \{ F_n \} \) be a sequence of increasing functions \( F_n : I \to \mathbb{R} \) that converge pointwise to a continuous function \( F : I \to \mathbb{R} \). If \( F \) is continuous and bounded, under what conditions can we conclude that \( F_n ... | automath | [
0.0073816534131765366,
0.03280074894428253,
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0.024937618523836136,
0.0034320885315537453,
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0.06518694013357162,
0.027730461210012436,
0.006603497080504894,
0.018917595967650414,
0.011340770870447159,
-0.040453702211380005,
0... | 0.813685 | 0.78078 | 0.032906 |
Suppose \( S \) is a Noetherian ring and \( f: S \to R \) is a morphism such that \( R \) is a finite \( S \)-algebra. If we denote \( T \) as a \( S \)-algebra that is locally of finite presentation and can be factored as \( T = \varinjlim T_i \) with \( T_i \) finite and of finite presentation over some \( S \), what... | automath | [
0.01986568234860897,
0.0436750128865242,
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0.03589731454849243,
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0.058601826429367065,
0.0292274858802557,
-0.003908208571374416,
0.018635721877217293,
0.04871174693107605,
-0.026804618537425995,
0.001... | 0.818982 | 0.802522 | 0.01646 |
Consider a grid of width \( n \) where elements are inserted from the top left going left-to-right and top-to-bottom. Each insertion causes the elements to shift right circularly and the contents of the first column to move down by one position, with the new element occupying the top-left corner.
Given \( N \) consec... | automath | [
0.023163652047514915,
0.027204392477869987,
-0.03568859025835991,
-0.034517209976911545,
0.02283322811126709,
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0.0656413808465004,
0.026002461090683937,
-0.001120934379287064,
0.02522972971200943,
0.020721616223454475,
-0.04196782782673836,
-0.00... | 0.819396 | 0.797313 | 0.022083 |
Let \( f: [0,1]^2 \to \mathbb{R} \) be defined by \( f(x,y) = 1 \) for all \( (x,y) \in [0,1]^2 \). Determine the Riemann integral of \( f \) over the compact interval \( [0,1]^2 \). What is the value of \( \int_{[0,1]^2} f \, d\mu \)? | automath | [
0.016125645488500595,
0.051511555910110474,
-0.03986189886927605,
-0.04359319806098938,
0.03435421735048294,
-0.00003496195495245047,
-0.014934615232050419,
0.051096249371767044,
0.04222666472196579,
0.006318465806543827,
0.0027614054270088673,
0.026072831824421883,
-0.033477816730737686,
... | 0.820964 | 0.802326 | 0.018638 |
A gambler starts with $1,000,000 and plays a game where he wins $1 with probability \( \frac{1}{3} \) and loses $1 with probability \( \frac{2}{3} \). He quits as soon as he is ahead by $2. What is the probability that he will ever reach $1,000,002 before going broke, and demonstrate that this probability is less than ... | automath | [
-0.0002716279122978449,
0.029290543869137764,
-0.03583580628037453,
-0.03253928944468498,
0.028320174664258957,
-0.002586289308965206,
-0.021692389622330666,
0.06335607916116714,
0.03337462991476059,
-0.010977181605994701,
0.003582093631848693,
0.0208855252712965,
-0.025107508525252342,
-0... | 0.838984 | 0.822646 | 0.016338 |
Consider the construction of a Riemannian manifold \( M \) that is flat and simply connected, such as the universal cover of the punctured Euclidean plane. If you were to choose a point \( p \in M \) and a radius \( r > 0 \), what can be concluded about the volume of the closed ball \( B_r(p) \)? Specifically, can you ... | automath | [
0.018184473738074303,
0.045642539858818054,
-0.018855132162570953,
-0.042378537356853485,
0.022150348871946335,
-0.016430089250206947,
-0.013002422638237476,
0.046859368681907654,
0.029448440298438072,
-0.008664276450872421,
0.0094828512519598,
0.04173393175005913,
-0.026574110612273216,
-... | 0.80828 | 0.791197 | 0.017083 |
Let \( p \) and \( q \) be distinct Fermat's primes. Calculate the number of sides of the regular polygon that can be constructed by compass and straightedge using the theorem which states that a regular \( n \)-gon can be constructed if and only if \( n \) is the product of distinct Fermat's primes. | automath | [
0.03664775937795639,
0.029566152021288872,
-0.024678561836481094,
-0.030970871448516846,
0.02883833833038807,
-0.020505573600530624,
-0.020183740183711052,
0.06724073737859726,
0.03724372014403343,
-0.000790972902905196,
0.0010478174081072211,
0.03141731768846512,
-0.030985325574874878,
-0... | 0.832335 | 0.80831 | 0.024025 |
Let \( V \) be a finite-dimensional Euclidean vector space, and let \( A, A' \) be affine subspaces of \( V \) such that \( \dim(A) = \dim(A') \). Suppose there exist points \( p \in A \) and \( p' \in A' \) such that the Euclidean distances from these points to their respective affine subspaces are equal, i.e., \( d(A... | automath | [
0.01760142669081688,
0.026121709495782852,
-0.02875303104519844,
-0.03606540337204933,
0.03398086875677109,
-0.021428434178233147,
-0.0019677728414535522,
0.06408939510583878,
0.034786079078912735,
-0.006840063259005547,
0.008100146427750587,
0.015041735954582691,
-0.0325922966003418,
-0.0... | 0.833668 | 0.797521 | 0.036148 |
Consider the integral equation of the form
\[
f(x) - \lambda \int_{a}^{b} K(x,y)f(y) \, dy = g(x)
\]
where \(K(x,y)\) is a known kernel function, \(\lambda\) is a scalar parameter, and \(g(x)\) is a given function defined on the interval \([a, b]\). Given \(K(x,y) = e^{-x} + x \sin(y) - x \cos(y^2)\) and \(g(x) = \s... | automath | [
-0.005864572245627642,
0.03430255502462387,
-0.04313446581363678,
-0.045628659427165985,
0.050853975117206573,
0.015284213237464428,
-0.005480974446982145,
0.08065897971391678,
0.03631884977221489,
0.0038239252753555775,
0.021271826699376106,
0.023562146350741386,
-0.030753713101148605,
-0... | 0.822092 | 0.797665 | 0.024428 |
Consider the following mappings:
1. \( T: \mathbb{R}^3 \to \mathbb{R} \) defined by \( T(x) = x_1 x_2 x_3 \).
2. \( T: P \to P \) defined by \( T(p) = p + 2p' + 3p'' \) where \( P \) is the vector space of all polynomials.
3. \( \text{det}: M_{22} \to \mathbb{R} \) defined by \( \text{det} \left( \begin{array}{cc} a ... | automath | [
0.02113124169409275,
0.0421418771147728,
-0.035572122782468796,
-0.0589074082672596,
0.04812344163656235,
-0.029210496693849564,
0.004351799376308918,
0.08665258437395096,
0.006992674432694912,
-0.009893356822431087,
0.006783101707696915,
-0.001874621957540512,
-0.027826597914099693,
-0.03... | 0.808696 | 0.787815 | 0.020881 |
What is the smallest positive real constant \( \varepsilon \) such that a metric space \( X \) satisfying the inequality
\[
\sum_{i=1}^n d(x_i, x'_i)^2 \leq (1+\varepsilon)^2 \sum_{j=1}^m d(y_j, y'_j)^2
\]
for any choice of points \( x_1, x_1', \ldots, x_n, y_1, y_1', \ldots, y_m \) admits a bi-Lipschitz embedding ... | automath | [
-0.016997963190078735,
0.010018398053944111,
-0.04676179215312004,
-0.037468038499355316,
0.02966916561126709,
0.01642763428390026,
-0.02428603544831276,
0.0684284120798111,
0.053655192255973816,
-0.0037959017790853977,
-0.014459151774644852,
0.021193455904722214,
-0.01500784046947956,
-0.... | 0.825861 | 0.808162 | 0.017699 |
Given a sequence of nonnegative measurable functions \( f_n \) such that \( \int_X f_n d\mu = 1 \) for all \( n \) and a probability measure \( \mu \) such that \( \int_A f_n d\mu \to \int_A fd\mu \) for all measurable sets \( A \), can we conclude whether \( f_n \to f \) pointwise almost everywhere, in measure, or in ... | automath | [
0.017783259972929955,
0.016521548852324486,
-0.0403103344142437,
-0.021000957116484642,
0.017696773633360863,
-0.0024817325174808502,
-0.025487145408988,
0.0654541403055191,
0.029706260189414024,
-0.0023332745768129826,
0.005546295549720526,
0.023541944101452827,
-0.02713521383702755,
0.00... | 0.812605 | 0.779668 | 0.032937 |
Given a sequence of independent random variables \(X_1, X_2, \ldots, X_n\) with a common probability measure \(P\) and let \(C \in \mathcal{C}\) be a set. Define the empirical measure \(P_n(C) = \frac{1}{n} \#\{1 \leq i \leq n : X_i \in C\}\). What is the asymptotic distribution of \(\sqrt{n}(P_n(C) - P(C))\) as \(n \t... | automath | [
-0.000550928758457303,
0.02168680541217327,
-0.022239264100790024,
-0.027391692623496056,
0.025316117331385612,
-0.021879348903894424,
-0.023460302501916885,
0.057234253734350204,
0.027925729751586914,
-0.01650085859000683,
0.0038218859117478132,
0.02677936665713787,
-0.029476428404450417,
... | 0.825608 | 0.803189 | 0.022418 |
Consider the function \( f(x, y) = x^y \) defined for all \( (x, y) \) in \( \mathbb{R}^2 \) except at \( (0, 0) \). If we define \( 0^0 = 1 \), what is the limit of \( f(x,y) \) as \( (x,y) \) approaches \( (0,0) \) along the line \( y = x \)? Calculate the limit \( \lim_{(x,y) \to (0,0)} f(x,y) \) based on this defin... | automath | [
0.02117147669196129,
0.03733259066939354,
-0.027473084628582,
-0.025062810629606247,
0.030576881021261215,
-0.013890440575778484,
-0.02153833582997322,
0.07662233710289001,
0.039623770862817764,
-0.016342012211680412,
-0.002843983005732298,
0.017138967290520668,
-0.028560610488057137,
-0.0... | 0.821005 | 0.803789 | 0.017217 |
Is the sequence of functions \( f_n(x) = \frac{x^2 + nx}{n} \) uniformly convergent on \(\mathbb{R}\)? If not, demonstrate why it does not converge uniformly by calculating \( \lim_{n \to \infty} \sup_{x \in \mathbb{R}} |f_n(x) - f(x)| \), where \( f(x) = x \). | automath | [
-0.0007689393241889775,
0.042259830981492996,
-0.04317966103553772,
-0.041856445372104645,
0.026924170553684235,
-0.007836375385522842,
-0.0386931337416172,
0.07712678611278534,
0.013854996301233768,
-0.0035624406300485134,
0.004052965901792049,
0.017446544021368027,
-0.028745517134666443,
... | 0.818436 | 0.795728 | 0.022708 |
Consider the dihedral group \( D_8 \) of order 8. Let \( N_1 \) and \( N_2 \) be the two normal Klein four-subgroups of \( D_8 \). If there exists an automorphism \( \sigma \) of \( D_8 \) that interchanges \( N_1 \) and \( N_2 \), determine the order of the automorphism group \( \text{Aut}(D_8) \). Is it equal to the ... | automath | [
0.018381930887699127,
0.02737477980554104,
-0.017064789310097694,
-0.03425165265798569,
0.01463937759399414,
-0.0008781475480645895,
-0.0173132736235857,
0.05314859002828598,
0.04403943195939064,
0.007515766192227602,
0.008487655781209469,
0.011181698180735111,
-0.05874546989798546,
-0.021... | 0.816827 | 0.798422 | 0.018405 |
Given a set of up to 100,000 points in \( \mathbb{Z}^3 \) with coordinates bounded by \([-10^{18}, 10^{18}]\), determine the point \( M(x, y, z) \) in \( \mathbb{Z}^3 \) that minimizes the maximum Manhattan distance to all points in the set. What is the minimum possible value of this maximum Manhattan distance? | automath | [
0.036998022347688675,
0.02109634317457676,
-0.03674519434571266,
-0.029509112238883972,
0.0397123284637928,
-0.0216441098600626,
-0.028968805447220802,
0.024485142901539803,
0.05367250740528107,
-0.015067238360643387,
0.004332154057919979,
0.031423285603523254,
-0.03487684577703476,
-0.017... | 0.838825 | 0.822144 | 0.01668 |
Let \( G = (\mathbb{R}, +) \) and consider the subgroup \( H \) defined as \( H = \{ u + v\alpha : u, v \in \mathbb{Z} \} \), where \( \alpha \) is an irrational number. Given that \( H \) is the internal direct product of the subgroups \( K_1 = \mathbb{Z} \) and \( K_2 = \alpha \mathbb{Z} \), is it true that the closu... | automath | [
0.029034078121185303,
0.049695730209350586,
-0.03584052249789238,
-0.027249203994870186,
0.03825617581605911,
0.00386077375151217,
-0.0006833893712610006,
0.06412765383720398,
0.03539394959807396,
-0.0043646967969834805,
-0.0023778213653713465,
0.019565271213650703,
-0.02723916806280613,
-... | 0.81459 | 0.789944 | 0.024646 |
Given $k$ independent samples drawn from normal distributions $X_{ij} \sim N(\mu_i, \sigma^2)$, with sample sizes $n_i$, let $F = \frac{S_2/(k-1)}{S_1/(N-k)}$ be the ANOVA F-statistic, where $S_1$ and $S_2$ are defined as the within samples and between samples sum of squares, respectively.
If the null hypothesis stat... | automath | [
-0.007145105395466089,
0.015173299238085747,
-0.06276270002126694,
-0.033824335783720016,
0.04666027054190636,
-0.015716513618826866,
-0.015978585928678513,
0.06245056539773941,
0.021514805033802986,
-0.016708778217434883,
0.019459310919046402,
0.03292684629559517,
-0.048937588930130005,
0... | 0.822934 | 0.802299 | 0.020636 |
Let \( J \) be an arc on the unit circle \( \mathbb{S}^1 \) with measure \( |J| \), and let \( \alpha \) be an angle such that \( \frac{\alpha}{\pi} \) is irrational. The indicator sequence \( s(J) = \{s_n\} \) is defined such that \( s_n = 1 \) if \( z_n = e^{in\alpha} \in J \) and \( s_n = 0 \) otherwise. If we know ... | automath | [
0.013275409117341042,
0.034674786031246185,
-0.03719967231154442,
-0.020712068304419518,
0.020595451816916466,
-0.007935944944620132,
-0.02100764587521553,
0.0508197657763958,
0.02832261100411415,
0.002915089949965477,
0.020964624360203743,
0.007209291215986013,
-0.02427012287080288,
-0.00... | 0.829613 | 0.8 | 0.029613 |
Consider the set \( X = \{0, 1, 2, 3, 4, 5\} \) with the operation defined by the following table:
\[
\begin{array}{c|cccccc}
\circ & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline
0 & 0 & 2 & 1 & 3 & 4 & 5 \\
1 & 1 & 0 & 2 & 4 & 5 & 3 \\
2 & 2 & 1 & 0 & 5 & 3 & 4 \\
3 & 3 & 4 & 5 & 0 & 2 & 1 \\
4 & 4 & 5 & 3 & 1 & 0 & 2 \\
5 & 5 & ... | automath | [
0.048122502863407135,
0.02646557055413723,
-0.05997451767325401,
-0.012057137675583363,
0.035918716341257095,
-0.04555937275290489,
-0.00758056016638875,
0.09395920485258102,
0.04648229479789734,
-0.003789474256336689,
-0.0033532097004354,
0.021525658667087555,
-0.043152064085006714,
-0.01... | 0.826173 | 0.80253 | 0.023643 |
Construct two finite sets \( A \) and \( B \) such that \( |A \cap B| < |A| \) and \( |A| \leq |B| \). Provide specific examples of sets \( A \) and \( B \) to demonstrate this counterexample. | automath | [
0.036848362535238266,
0.016174515709280968,
-0.02826584503054619,
-0.024592332541942596,
0.016787823289632797,
-0.025238847360014915,
-0.027724020183086395,
0.06451237201690674,
0.04561033844947815,
-0.01514388807117939,
0.007737805601209402,
0.028326280415058136,
-0.04316030442714691,
-0.... | 0.82931 | 0.806445 | 0.022865 |
Let \( J \) be an \( n \times n \) real matrix that is not necessarily symmetric or invertible. For a matrix \( B \in \mathbb{R}^{n \times n} \), define the Cayley transform of \( B \) as
\[
A = (I - \alpha B)^{-1} (I + \alpha B)
\]
for some non-zero real constant \( \alpha \). Under what conditions on \( B \) does ... | automath | [
0.03067011944949627,
0.0485491007566452,
-0.04575842246413231,
-0.031028905883431435,
0.033785879611968994,
-0.0019170996965840459,
0.0009173686266876757,
0.05949689447879791,
0.020545674487948418,
0.00010934095189440995,
-0.0033181123435497284,
-0.004518568050116301,
-0.041645489633083344,
... | 0.813903 | 0.791289 | 0.022615 |
Consider a hill climbing optimization algorithm that aims to find the maximum value of a function \( f(x) \). Suppose the function \( f(x) \) is defined over the discrete states \( x_1, x_2, ..., x_n \) where \( f \) produces the following values:
\[
f(x_1) = 3, \quad f(x_2) = 2, \quad f(x_3) = 5, \quad f(x_4) = 7, \... | automath | [
0.006550685036927462,
0.023579251021146774,
-0.036371197551488876,
-0.036328598856925964,
0.030091963708400726,
0.0031990569550544024,
-0.0331452302634716,
0.08774899691343307,
0.016415974125266075,
-0.012469585053622723,
-0.004637413192540407,
0.023872297257184982,
-0.03210636228322983,
-... | 0.819058 | 0.799823 | 0.019235 |
Let \( P \) and \( Q \) be complex polynomials such that \( \deg(P) < \deg(Q) \) and \( Q(x) = c \prod_{i=1}^n (x - \alpha_i)^{m_i} \) where \( \deg(Q) = \sum_{i=1}^n m_i \). If \( \deg(Q) = d > 1 \) and there exist unique coefficients \( a_{i,j} \) such that
\[
\frac{P(x)}{Q(x)} = \sum_{i=1}^n \sum_{j=1}^{m_i} \frac... | automath | [
0.026372745633125305,
0.03342686966061592,
-0.03576304391026497,
-0.04000070318579674,
0.02875886671245098,
-0.0022182390093803406,
-0.016529155895113945,
0.07087873667478561,
0.03703347221016884,
0.000031972056603990495,
-0.0030461852438747883,
0.030535131692886353,
-0.03901941329240799,
... | 0.829577 | 0.797682 | 0.031895 |
Given a coprime pair of integers \( x \) and \( N \), let \( r \) denote the order such that \( x^r \equiv 1 \mod N \). If we sample \( k_1/r \) and \( k_2/r \) from integers such that \( 0 \leq k_1, k_2 < r \), how many such pairs \( (k_1, k_2) \) exist that allow us to uniquely determine the value of \( r \) from the... | automath | [
0.017216894775629044,
0.035022150725126266,
-0.05259653553366661,
-0.020779242739081383,
0.01897752843797207,
-0.0015834784135222435,
-0.004384142812341452,
0.05591975152492523,
0.02486640401184559,
-0.0063727181404829025,
-0.0008995095849968493,
0.006318686995655298,
-0.024001101031899452,
... | 0.835132 | 0.814418 | 0.020714 |
Given the tetrahedron defined by the points \( P_1 = (1, 0, 2/3) \), \( P_2 = (4, 1, 0) \), \( P_3 = (1, 2, 4) \), and \( P_4 = (0, 3, 1) \) in 3D space, how can we derive the area of the triangular cross-section formed by the intersection of the tetrahedron with a plane perpendicular to the \( x \)-axis at \( y = 0.8 ... | automath | [
0.022588716819882393,
0.04368061572313309,
-0.035737138241529465,
-0.04002360999584198,
0.039397288113832474,
-0.017923317849636078,
-0.03373568132519722,
0.07049555331468582,
0.033746540546417236,
0.024227356538176537,
0.0026547296438366175,
0.029865430667996407,
-0.023938501253724098,
-0... | 0.811316 | 0.791874 | 0.019442 |
Given a square matrix \( A \) of size \( n \times n \) such that the null space of \( A \) contains only the zero vector, what can we conclude about the solutions to the system of linear equations \( \linearsystem{A}{\vect{b}} \) for any vector \( \vect{b} \) in \( \mathbb{R}^n \)? Specifically, how many solutions will... | automath | [
0.007846560329198837,
0.02647806517779827,
-0.06654105335474014,
-0.040011465549468994,
0.04528370872139931,
-0.002662291284650564,
-0.015264138579368591,
0.06922802329063416,
0.041340045630931854,
-0.024103006348013878,
-0.010204342193901539,
0.02530813403427601,
-0.025492776185274124,
-0... | 0.811962 | 0.795157 | 0.016805 |
Given a continuous function \( f(x) \) defined on the closed interval \([a, b]\) that is differentiable on the open interval \((a, b)\), and satisfies \( f(a) \neq f(b) \), can we conclude that there exists a point \( c \in (a, b) \) such that \( f'(c) = 0 \) if \( f(c) \) lies strictly between \( f(a) \) and \( f(b) \... | automath | [
0.019250892102718353,
0.0280190110206604,
-0.04295995458960533,
-0.02745366469025612,
0.04218173772096634,
-0.012537047266960144,
-0.023499537259340286,
0.08228786289691925,
0.032854288816452026,
-0.01076591107994318,
0.00793603528290987,
0.022863034158945084,
-0.01733117364346981,
-0.0094... | 0.821618 | 0.802215 | 0.019404 |
Let \( A \) be a set with at least two distinct elements. Is the operation of symmetric difference, defined as \( g_1 \triangle g_2 = (g_1 \cup g_2) - (g_1 \cap g_2) \), a group operation on the power set \( \mathcal{P}(A) \)? If so, determine whether this group is finite or infinite, and justify whether it is abelian. | automath | [
0.049639396369457245,
0.028422817587852478,
-0.02790263295173645,
-0.027217285707592964,
0.017172623425722122,
-0.05303658917546272,
-0.03182711824774742,
0.07368414103984833,
0.04432728886604309,
-0.011298980563879013,
0.011588686145842075,
0.02198614552617073,
-0.024438800290226936,
-0.0... | 0.832659 | 0.814623 | 0.018037 |
Suppose you have a spherical object of radius \( R \) and mass \( M \) that is claimed to have a uniform density. If you measure its center of mass and find it at the geometric center, and then you calculate its moment of inertia around any axis and find it to match the theoretical value \( I = \frac{2}{5} M R^2 \), ca... | automath | [
0.016405468806624413,
0.01567061059176922,
-0.03631481155753136,
-0.04382497817277908,
0.01892172358930111,
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-0.03704576566815376,
0.0543937087059021,
0.02572529762983322,
-0.016623809933662415,
0.015936415642499924,
0.011700280010700226,
-0.041914425790309906,
-0.022... | 0.81069 | 0.792093 | 0.018597 |
Determine all compact Riemannian manifolds \( A \) and \( B \) for which the isometry group of the product manifold \( A \times B \) is isomorphic to the direct product of the isometry groups \( \text{Isom}(A) \times \text{Isom}(B) \). Are there any conditions on \( A \) and \( B \) that guarantee this isomorphism hold... | automath | [
0.022314060479402542,
0.03772275522351265,
-0.031207438558340073,
-0.02323736436665058,
0.03312200680375099,
-0.013435699045658112,
-0.0048800199292600155,
0.06994187831878662,
0.03437070548534393,
-0.005202235653996468,
-0.007770442869514227,
0.0283620934933424,
-0.025467675179243088,
-0.... | 0.812036 | 0.785764 | 0.026272 |
Given a finite set \( U \) and a predicate \( p \) on subsets of \( U \) such that \( (p(A) \land (A \subseteq B)) \Longrightarrow p(B) \), if \( p(U) \) holds, how many minimal satisfying sets exist if \( |U| = n \) and \( p(A) \) is true for all subsets \( A \) containing at least \( \lceil n/2 \rceil \) elements? | automath | [
0.027977431192994118,
0.022746287286281586,
-0.040608763694763184,
-0.012596961110830307,
0.03598739206790924,
-0.03231174871325493,
-0.020122384652495384,
0.045337606221437454,
0.028884662315249443,
-0.014017377980053425,
0.0156345646828413,
0.033597949892282486,
-0.006846001837402582,
-0... | 0.830309 | 0.810607 | 0.019702 |
Given two arcs in 3D Cartesian space, the first arc has a radius \( r_1 \) and subtends an angle \( \alpha_1 \) in a plane, while the second arc, which is extruded from the first and lies on a surface perpendicular to the plane of the first arc, has a radius \( r_2 \) and subtends an angle \( \alpha_2 \). If the two ar... | automath | [
0.017101019620895386,
0.03338796645402908,
-0.051938194781541824,
-0.05538387596607208,
0.021050341427326202,
-0.001728213974274695,
-0.037419047206640244,
0.06123242527246475,
0.02951172925531864,
-0.0005211863317526877,
0.01788024790585041,
0.010156100615859032,
-0.03818320855498314,
-0.... | 0.820868 | 0.804442 | 0.016427 |
Let \( f: X \to Y \) be a continuous function between topological spaces \( X \) and \( Y \). Define a specific continuous function \( f: [0,1] \to [0,1] \) by \( f(x) = x^2 \).
1. Is \( f \) continuous on the interval \([0,1]\)?
2. What is the image of \( f \)?
3. If \( V = [0, 0.25] \), what is the preimage \( f^{... | automath | [
0.020322535187005997,
0.03525438904762268,
-0.026838364079594612,
-0.028179313987493515,
0.03813625127077103,
-0.019701940938830376,
-0.01703420653939247,
0.07646709680557251,
0.035117264837026596,
-0.006031803786754608,
0.017310848459601402,
0.019735420122742653,
-0.016957441344857216,
-0... | 0.813855 | 0.795228 | 0.018627 |
Given a spherical triangle \(A_1 A_2 A_3\) on the unit sphere with solid angle \(\Omega\), and a point \(P\) inside this triangle defined by the proportions \(t_1, t_2, t_3\) of the areas, express the location of the point \(P\) as a linear combination of the vertices \(A_1, A_2, A_3\). Specifically, derive an analytic... | automath | [
0.011739764362573624,
0.0271309707313776,
-0.04205349087715149,
-0.05023139342665672,
0.01819276250898838,
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0.06101854518055916,
0.044561371207237244,
0.015143396332859993,
0.018832456320524216,
0.0465727299451828,
-0.03467784449458122,
-0.024696... | 0.820551 | 0.79683 | 0.023721 |
Consider a distribution on the manifold \(\mathbb{R}^3\) defined by the vector fields \(X = \frac{\partial}{\partial x} + y \frac{\partial}{\partial z}\) and \(Y = \frac{\partial}{\partial y}\). Determine whether this distribution is integrable, involutive, and completely integrable, and justify your answers based on t... | automath | [
0.01755572110414505,
0.018188737332820892,
-0.030592698603868484,
-0.03799421712756157,
0.04116697236895561,
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0.05758729204535484,
0.04316968098282814,
-0.010478351265192032,
0.013320298865437508,
0.024106435477733612,
-0.04727395996451378,
-0.... | 0.811457 | 0.794254 | 0.017203 |
Let \( S^1 \) denote the unit circle and \( x_0 \) be a fixed point on \( S^1 \). Determine whether \( S^1 \times \{ x_0 \} \) is a retract of \( S^1 \times S^1 \) and explain why \( S^1 \times \{ x_0 \} \) is not a deformation retract of \( S^1 \times S^1 \). Additionally, find the fundamental groups of both \( S^1 \t... | automath | [
0.01637873612344265,
0.04444907233119011,
-0.028907466679811478,
-0.05217056721448898,
0.0364612378180027,
-0.026904145255684853,
0.004106780514121056,
0.07239724695682526,
0.037881407886743546,
0.01769721694290638,
0.004491067957133055,
0.028360646218061447,
-0.02661491185426712,
-0.02617... | 0.824849 | 0.802733 | 0.022115 |
Given a sequence of \(n\) independent and identically distributed Bernoulli random variables, where \(m\) of them result in ones, the maximum likelihood estimator (MLE) for the probability \(\theta\) of obtaining a one is \(\hat{\theta} = \frac{m}{n}\). However, suppose we know that \(\theta = \frac{1}{2} + \frac{1}{2}... | automath | [
0.002019495004788041,
0.03778548166155815,
-0.04559679329395294,
-0.05205734446644783,
0.04715551435947418,
0.0065664565190672874,
-0.03755677118897438,
0.061855997890233994,
0.027597080916166306,
-0.006776429247111082,
0.004355129785835743,
0.03498496115207672,
-0.020555773749947548,
0.00... | 0.810469 | 0.787182 | 0.023286 |
Given a symmetric matrix \( A \) with polynomial expression \( x^T A x = 5x_3^2 + 4x_2 x_3 + 2x_1 x_3 + 2x_1 x_2 + 4x_1^2 \), demonstrate that \( A \) is not positive definite by finding specific values for \( x_1, x_2, \) and \( x_3 \) such that \( x^T A x < 0 \). What specific choice of \( x_1, x_2, x_3 \) will satis... | automath | [
0.030655577778816223,
0.02137432061135769,
-0.057741016149520874,
-0.04043806716799736,
0.030180828645825386,
-0.03952184319496155,
-0.011978351511061192,
0.09120684862136841,
0.04086477681994438,
-0.013634049333631992,
-0.00017851946176961064,
0.015423733741044998,
-0.031159689649939537,
... | 0.825263 | 0.808288 | 0.016975 |
Given the recurrence relation
$$ a_n = 1 + \left(\dfrac{n+1}{2n}\right) a_{n-1}, $$
with the initial condition \( a_1 = \sum_{r=0}^{1} \dfrac{1}{\binom{1}{r}} \) and the closed form
$$ a_n = \sum_{r=0}^{n} \dfrac{1}{\binom{n}{r}}, $$
can you provide a finite algorithmic method that derives \( a_n \) from the re... | automath | [
0.010565260425209999,
0.04133094847202301,
-0.025531120598316193,
-0.023115426301956177,
0.02752322517335415,
-0.011528692208230495,
-0.0030633530113846064,
0.08159853518009186,
0.02363637089729309,
-0.017226003110408783,
0.008938392624258995,
0.01204147282987833,
-0.013999106362462044,
-0... | 0.830018 | 0.810355 | 0.019663 |
Let \( x \), \( y \), and \( z \) be three consecutive integers such that \( x < y < z \). Define the average of these three integers. If it is given that \( x = 11 \), what is the average of \( x \), \( y \), and \( z \)?
Additionally, if it is given that the average of \( y \) and \( z \) is \( 12.5 \), what is the... | automath | [
0.03022943064570427,
0.02865685150027275,
-0.041767023503780365,
-0.04746581241488457,
0.04832812398672104,
-0.008960281498730183,
-0.03125838562846184,
0.03579175844788551,
0.019748985767364502,
-0.011909129098057747,
0.0051561384461820126,
0.010013699531555176,
-0.05291101336479187,
-0.0... | 0.835824 | 0.81543 | 0.020394 |
Let \( S \) be a scheme, and consider two sheaves of sets \( \mathcal{F} \) and \( \mathcal{G} \) over \( S \). If the map \( a: \mathcal{F} \to \mathcal{G} \) is such that the induced map on stalks \( a_{\overline{s}} : \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}} \) is surjective for all geometric points... | automath | [
0.03105931542813778,
0.015006913803517818,
-0.024019086733460426,
-0.0247137863188982,
0.03905392810702324,
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-0.02840004488825798,
0.07368267327547073,
0.03738880529999733,
-0.004988333210349083,
0.007851838134229183,
0.045889340341091156,
-0.0381002314388752,
-0.0035... | 0.822685 | 0.790739 | 0.031946 |
Given an integer \( n \) (where \( 2 \leq n \leq 50 \)), construct a square matrix of size \( n \times n \) using integers from \( 1 \) to \( n^2 \) such that each integer occurs exactly once. Your objective is to maximize the "beauty" of the matrix, defined as the number of different absolute differences \( |x - y| \)... | automath | [
0.03479049727320671,
0.03136049211025238,
-0.04812641069293022,
-0.02752271667122841,
0.03506947681307793,
-0.01175240520387888,
-0.040991462767124176,
0.06347005814313889,
0.04330681264400482,
-0.01295034121721983,
0.013316510245203972,
0.04010483995079994,
-0.03956061229109764,
-0.006443... | 0.834174 | 0.809565 | 0.024609 |
Let \( f(x) \) be a continuous function defined on the interval \([a, b]\), where \( a < b \). If \( c \) is a point in the interval \((a, b)\), express the integral \( \int_a^b f(x) \, dx \) in terms of the integrals with respect to point \( c \). Furthermore, if we know that \( \int_a^c f(x) \, dx = 5 \) and \( \int_... | automath | [
0.04325253888964653,
0.04893627017736435,
-0.050048548728227615,
-0.026463190093636513,
0.031935930252075195,
-0.002692026551812887,
-0.015906386077404022,
0.05696114897727966,
0.043988753110170364,
0.00031007977668195963,
0.008501917123794556,
0.03055064007639885,
-0.03146756812930107,
-0... | 0.822754 | 0.802816 | 0.019938 |
Given two bases \(\mathcal{B} = \{v_1, v_2, \ldots, v_n\}\) and \(\mathcal{C} = \{u_1, u_2, \ldots, u_n\}\) for an \(n\)-dimensional vector space \(V\) over a field \(\mathbb{F}\), if the change of basis map \(P: \mathbb{F}^n \rightarrow \mathbb{F}^n\) is defined such that \(P(v_i) = u_i\) for all \(i = 1, \ldots, n\),... | automath | [
0.013283882290124893,
0.023801205679774284,
-0.038410767912864685,
-0.030478112399578094,
0.036125872284173965,
-0.03610909357666969,
-0.012962037697434425,
0.0624769888818264,
0.028181783854961395,
-0.013268345035612583,
0.0024732532911002636,
0.016291502863168716,
-0.022960303351283073,
... | 0.827087 | 0.809256 | 0.017831 |
Given an \( n \times n \) real symmetric matrix \( A \) with eigenvalues \( \lambda_1, \lambda_2, \ldots, \lambda_n \), describe the algorithm to find the nearest positive definite matrix to \( A \) in terms of the Frobenius norm, and state the conditions under which the original matrix \( A \) can be modified to ensur... | automath | [
0.01344889122992754,
0.021382683888077736,
-0.039769403636455536,
-0.014128046110272408,
0.023500313982367516,
-0.016249755397439003,
-0.023522162809967995,
0.06268229335546494,
0.02616734430193901,
-0.012256569229066372,
-0.00963650830090046,
0.013587933033704758,
-0.03162265568971634,
-0... | 0.82134 | 0.800363 | 0.020977 |
Let \( M \) be a smooth manifold defined by the equations:
\[
w = (L \cos(c) + a) \sin(t) \cos(p)
\]
\[
x = (L \cos(c) + a) \sin(t) \sin(p)
\]
\[
y = (L \cos(c) + a) \cos(t)
\]
\[
z = \sqrt{4(L + a)^2 - (L \cos(c) + a)^2}
\]
where \( a > L > 0 \) are constants, and \( c, t, p \) are free parameters. Determine whether... | automath | [
0.02056019753217697,
0.03032110445201397,
-0.03179168328642845,
-0.03542374446988106,
0.050861310213804245,
-0.012261186726391315,
-0.033820267766714096,
0.060219284147024155,
0.03625068441033363,
0.0052191573195159435,
-0.00979792419821024,
0.03227077051997185,
-0.029569942504167557,
-0.0... | 0.821306 | 0.80116 | 0.020146 |
Given a domain \( \Omega \subset \mathbb{R}^n \) and an open subset \( \omega_1 \subset \omega_2 \subset \Omega \), if \( \theta \in H^2(\omega_1) \) is a function in the Sobolev space, under what conditions does there exist a function \( \tilde{\theta} \in H^2(\Omega) \) such that \( \tilde{\theta} = \theta \) on \( \... | automath | [
-0.009830635040998459,
0.03400034457445145,
-0.047707699239254,
-0.05448395758867264,
0.04218992590904236,
0.0017472871113568544,
-0.019585907459259033,
0.0713496282696724,
0.03747415170073509,
-0.0045781927183270454,
-0.007567322347313166,
0.017266927286982536,
-0.017271507531404495,
-0.0... | 0.808048 | 0.791754 | 0.016293 |
Let \( S \) be a family of finite sets satisfying Hall's marriage condition, and let \( T \) be a set constructed as a transversal of \( S \). If \( |S| = n \) and the largest set in \( S \) has size \( k \), what is the minimum number of elements in \( T \) required to ensure that \( T \) can serve as a transversal fo... | automath | [
0.02918298915028572,
0.03850580379366875,
-0.04981619492173195,
-0.027303513139486313,
0.054244622588157654,
-0.006249456200748682,
-0.022701751440763474,
0.0626453384757042,
0.01583150029182434,
0.0016914474545046687,
0.023185107856988907,
0.037738677114248276,
-0.029936548322439194,
-0.0... | 0.82287 | 0.800557 | 0.022313 |
Given the functions \( f(x, y, z) = z \) and \( g(x, y, z) = x^2 - y^2 + z \), find the values of \( (x, y, z) \) that satisfy \( \nabla f(x, y, z) = \delta \nabla g(x, y, z) \) for some constant \( \delta \), and verify whether these points correspond to extremal points of \( f \) under the constraint \( g(x, y, z) = ... | automath | [
0.019105112180113792,
0.03174145147204399,
-0.03494458645582199,
-0.04337889328598976,
0.0350070595741272,
-0.008483123034238815,
-0.014527952298521996,
0.06744012981653214,
-0.0002376746851950884,
0.0013348052743822336,
-0.00489792600274086,
0.02995908632874489,
-0.028133971616625786,
-0.... | 0.8231 | 0.799957 | 0.023143 |
Let \( A \) be a normal operator on a Hilbert space \( H \) over the complex field, with spectrum \( \sigma \) and a spectral resolution of the identity \( E \). If we select a unit vector \( x \) in \( H \), define the measure \( d\mu_{x}(\lambda) = d\|E(\lambda)x\|^{2} \) as a Borel probability measure on \( \sigma(A... | automath | [
-0.012269892729818821,
0.036060839891433716,
-0.037285346537828445,
-0.02124565839767456,
0.040393244475126266,
-0.0024746579583734274,
-0.019599853083491325,
0.053285785019397736,
0.035797182470560074,
-0.004748399835079908,
0.006616873200982809,
0.035598598420619965,
-0.020046120509505272,... | 0.789969 | 0.765763 | 0.024206 |
Using the Gram-Schmidt process, determine the first three polynomials that are orthogonal with respect to the inner product defined by
\[
\langle f, g \rangle = \int_{-1}^{1} f(x)g(x) \, dx.
\]
Specifically, find the explicit forms of the polynomials \( p_1(x) \), \( p_2(x) \), and \( p_3(x) \) that result from the ... | automath | [
0.03579059988260269,
0.039480775594711304,
-0.028138786554336548,
-0.04300345852971077,
0.03548060357570648,
-0.03123384527862072,
-0.0037962242495268583,
0.06431598961353302,
0.02953234501183033,
0.000671481539029628,
0.009655343368649483,
0.007714043837040663,
-0.03027721866965294,
-0.03... | 0.817162 | 0.798905 | 0.018257 |
Consider the homogeneous linear system given by:
\[
\begin{cases}
2x_1 + 4x_2 - 6x_3 = 0 \\
3x_1 + 6x_2 - 9x_3 = 0 \\
x_1 + 2x_2 - 3x_3 = 0
\end{cases}
\]
Determine whether this system has nontrivial solutions and, if so, find the values of \( k \) for which the system has infinitely many solutions. | automath | [
0.020217668265104294,
0.021343043074011803,
-0.043463047593832016,
-0.05258588492870331,
0.05275272950530052,
-0.033332709223032,
-0.005592485889792442,
0.07539226859807968,
0.029499290511012077,
-0.017501626163721085,
0.00719572464004159,
0.02332909032702446,
-0.02893971838057041,
-0.0376... | 0.831686 | 0.814171 | 0.017515 |
Given a smooth manifold \( M \) of dimension \( n \) with a smooth nonempty boundary \( \partial M \), suppose there is a function \( f: M \setminus \partial M \to \mathbb{R} \) such that \( f(p) \to -\infty \) as \( p \) approaches \( \partial M \), and for any \( A \leq \alpha \leq \beta \), the sublevel sets \( \{f ... | automath | [
0.021376870572566986,
0.03431490808725357,
-0.0394781231880188,
-0.036100324243307114,
0.03059161640703678,
0.01415789034217596,
-0.013573846779763699,
0.08432885259389877,
0.03408496454358101,
0.006803613621741533,
-0.004240244626998901,
0.04046378284692764,
-0.026350466534495354,
-0.0140... | 0.809295 | 0.787231 | 0.022064 |
Given the space \( X \) obtained from the torus \( S^1 \times S^1 \) by attaching a Möbius band via a homeomorphism from the boundary circle of the Möbius band to the circle \( S^1 \times \{x_0\} \) in the torus, what is the presentation of the fundamental group \( \pi_1(X) \) and how does the universal cover \( \widet... | automath | [
-0.00016979713109321892,
0.037935901433229446,
-0.030314497649669647,
-0.03908934071660042,
0.02400163561105728,
-0.025286942720413208,
0.0025806725025177,
0.057737790048122406,
0.03981301188468933,
0.011267902329564095,
0.00675151078030467,
0.03033582679927349,
-0.028583690524101257,
-0.0... | 0.800072 | 0.780661 | 0.019412 |
Let \( K: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R} \) be a kernel function that corresponds to a feature mapping \( \phi: \mathbb{R}^n \to \mathbb{R}^m \) in an infinite-dimensional space. Consider the Radial Basis Function (RBF) kernel defined as:
\[
K(x, x') = e^{-\gamma \|x - x'\|^2}
\]
for \( x, x' \in \ma... | automath | [
0.0031668436713516712,
0.04847493767738342,
-0.04433464631438255,
-0.03604639321565628,
0.033109910786151886,
-0.00790494866669178,
-0.01381275337189436,
0.0662979781627655,
0.04558492451906204,
-0.0012300248490646482,
0.008823366835713387,
0.01146688312292099,
-0.03166739270091057,
0.0049... | 0.803996 | 0.784958 | 0.019038 |
Given the function \( f(x) = \sin\left(\frac{1}{x}\right) \) on the interval \( (0, 1) \), determine whether \( f \) is uniformly continuous on that interval. If it is not uniformly continuous, provide a sequence \( (x_n) \) approaching zero for which \( |f(x_n) - f(y_n)| \) does not approach zero as \( n \) approaches... | automath | [
-0.00337059679441154,
0.04303229600191116,
-0.04275186359882355,
-0.04336364567279816,
0.019951332360506058,
0.004283736925572157,
-0.027519334107637405,
0.09209366887807846,
0.027149982750415802,
-0.0008010847377590835,
0.037503935396671295,
0.01126039493829012,
-0.009090103209018707,
0.0... | 0.803169 | 0.783424 | 0.019745 |
Let \( R = k[x_1, \ldots, x_n] \) and consider an ideal \( I \) in \( R \) such that \( I = \cap_{j} I_j \) is a primary decomposition where each \( I_j \) is a primary ideal. If \( I \) is not a radical ideal, what additional geometric information does the primary decomposition provide compared to the decomposition in... | automath | [
0.04769538715481758,
0.047746650874614716,
-0.01949046365916729,
-0.03086945042014122,
0.02186042256653309,
-0.009996959939599037,
-0.003704219125211239,
0.0714883878827095,
0.03693429380655289,
-0.005152471829205751,
0.019815806299448013,
-0.01511468831449747,
-0.04185032472014427,
-0.023... | 0.809737 | 0.782377 | 0.027361 |
Given a space curve defined by the parametric equations \( x(t) = a \cos(t) \), \( y(t) = a \sin(t) \), and \( z(t) = bt \), where \( a \) and \( b \) are positive constants, determine the values of \( a \) and \( b \) such that the curvature \( \kappa \) and the torsion \( \omega \) satisfy the condition \( \frac{\kap... | automath | [
0.03784897178411484,
0.027390630915760994,
-0.03615637496113777,
-0.032481543719768524,
0.039120614528656006,
-0.018119243904948235,
-0.019850915297865868,
0.068973109126091,
0.026315443217754364,
-0.008827208541333675,
0.002980536548420787,
0.034688737243413925,
-0.026441842317581177,
0.0... | 0.824362 | 0.804214 | 0.020148 |
Let \( X \) be a locally compact Hausdorff space. Define the supremum norm on \( C_0(X) \) as \( \|f\|_{\infty} = \sup_{x \in X} |f(x)| \) for \( f \in C_0(X) \).
1. What is the appropriate topology for \( C_c(X) \) such that every positive linear functional on \( C_c(X) \) extends uniquely to a continuous linear fun... | automath | [
0.0019455993315204978,
0.039872366935014725,
-0.0553935244679451,
-0.03041754849255085,
0.023792296648025513,
0.0068559832870960236,
-0.026149559766054153,
0.06006940081715584,
0.04039052128791809,
-0.010518846102058887,
-0.00002550443787185941,
0.018978163599967957,
-0.034287091344594955,
... | 0.812078 | 0.784428 | 0.027649 |
Given a set \( X \) and a Kuratowski closure operator \( \mathbf{c} : \wp(X) \to \wp(X) \) satisfying axioms K1 to K4, is it possible for \( \mathbf{c}(\{x\}) \) to contain more than one element if \( x \in X \) and all singleton sets in \( X \) are required to be stable under closure (i.e., \( \mathbf{c}(\{x\}) = \{x\... | automath | [
0.02261456288397312,
0.04651586711406708,
-0.0456429123878479,
-0.021968171000480652,
0.042991191148757935,
-0.02128497324883938,
-0.007046391721814871,
0.07296976447105408,
0.038245536386966705,
-0.01953061856329441,
0.007935850881040096,
0.02470068447291851,
-0.03895971179008484,
-0.0027... | 0.817185 | 0.788532 | 0.028654 |
Given two topological spaces \((Y, \tau)\) and \((X_i, \tau_i)\) for \(i = 1, 2, \ldots, n\), and continuous mappings \(f_i: (Y, \tau) \to (X_i, \tau_i)\), what can be said about the continuity of the mapping \(f: (Y, \tau) \to \prod_{i=1}^n (X_i, \tau_i)\) defined by \(f(y) = (f_1(y), f_2(y), \ldots, f_n(y))\)? Specif... | automath | [
0.020670419558882713,
0.03889726474881172,
-0.03586099296808243,
-0.018804557621479034,
0.03849666938185692,
0.007065825164318085,
-0.026386313140392303,
0.09441833943128586,
0.030105115845799446,
-0.0061419932171702385,
-0.004760523326694965,
0.01481730304658413,
-0.00491779251024127,
-0.... | 0.800906 | 0.7824 | 0.018507 |
Let \( f(s) \) be a polynomial of degree \( d \) and define \( I(s) = e^s \sum_{j=0}^d f^{(j)}(0) - \sum_{j=0}^d f^{(j)}(s) \). If \( |I(s)| \leq |s| e^{|s|} F(|s|) \), where \( F(s) \) is the polynomial whose coefficients are the absolute values of those of \( f(s) \), what is the maximum degree \( d \) such that this... | automath | [
0.02247626520693302,
0.026248503476381302,
-0.03780684992671013,
-0.034579746425151825,
0.021402966231107712,
-0.014260821975767612,
-0.023589687421917915,
0.05533882975578308,
0.04088617488741875,
-0.003487809095531702,
-0.0015989759704098105,
0.04401803016662598,
-0.050930365920066833,
-... | 0.825043 | 0.802537 | 0.022507 |
Given a finite group \( G \) and subgroups \( H \) and \( K \), where there exists a normal series \( G = G_0 \triangleright G_1 \triangleright \ldots \triangleright G_r = H \), demonstrate that \( (H : H \cap K) \) divides \( (G : K) \) and that \( |H K| \) divides \( |G| \). If \( |G| = 240 \), \( |H| = 20 \), and \(... | automath | [
0.023233400657773018,
0.043660011142492294,
-0.031013986095786095,
-0.05225975811481476,
0.029616182669997215,
-0.04128521308302879,
-0.02101755142211914,
0.0490269809961319,
0.03743734955787659,
-0.01208542287349701,
0.02102651260793209,
0.017622938379645348,
-0.03692634031176567,
0.00216... | 0.83598 | 0.817596 | 0.018384 |
Given the sequence defined by \( a_1=1 \) and \( a_{n+1}=a_n+\sqrt{1+a_n^2} \) for all \( n \in \mathbb{N} \), compute the limit
\[
\lim_{n \to \infty} \frac{a_n}{2^n}.
\]
What is the value of this limit? | automath | [
0.0099399508908391,
0.05070791393518448,
-0.019244089722633362,
-0.023617392405867577,
0.03454477712512016,
-0.0117226243019104,
-0.02937275357544422,
0.07838357985019684,
0.0405845083296299,
-0.031931523233652115,
0.002304362365975976,
0.020491361618041992,
-0.026744693517684937,
-0.00648... | 0.828337 | 0.811458 | 0.016879 |
Consider the curve defined by \( y = f(x) \) such that the distance from the origin to the tangent line at any point \( (x_0, f(x_0)) \) on the curve equals \( x_0 \). Find the equation of the curve \( f(x) \). Additionally, determine the shape of the curve and any special properties it satisfies. | automath | [
0.012255254201591015,
0.028557054698467255,
-0.05333573743700981,
-0.04855678230524063,
0.03303403779864311,
-0.04241834208369255,
-0.0058384547010064125,
0.07589872926473618,
0.028832364827394485,
-0.0029949406161904335,
-0.003151585115119815,
0.015243448317050934,
-0.04009488224983215,
0... | 0.814414 | 0.796354 | 0.018061 |
Let \( A \) be a non-empty set and let \( \mathcal{P}(A) \) represent the power set of \( A \). Define a choice function \( f: \mathcal{P}(A) - \{\emptyset\} \rightarrow A \) such that for every non-empty subset \( B \) of \( A \), \( f(B) \in B \). If \( |A| = n \), how many distinct functions \( f \) can be defined u... | automath | [
0.04068213701248169,
0.025334395468235016,
-0.04169059917330742,
-0.0034796169493347406,
0.02840832807123661,
-0.025823580101132393,
-0.02267332375049591,
0.04778684303164482,
0.03687015175819397,
-0.007217532955110073,
0.012434358708560467,
0.020143257454037666,
-0.027167104184627533,
-0.... | 0.829576 | 0.800188 | 0.029388 |
Let \( S \) be a subset of \( \mathbb{R}^2 \) defined as the closed disk centered at the origin with radius 1, i.e., \( S = \{ (x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1 \} \).
1. Is the set \( S \) convex?
2. Provide an example of two points in \( S \) whose sum is not in \( S \).
Consider the implications of thes... | automath | [
0.011782130226492882,
0.04398616403341293,
-0.0471462719142437,
-0.026658760383725166,
0.03467812016606331,
0.0006367516471073031,
-0.021421188488602638,
0.06412602961063385,
0.04507629945874214,
-0.0007428642711602151,
0.03159906715154648,
0.034141961485147476,
-0.04322429746389389,
-0.02... | 0.824675 | 0.794961 | 0.029714 |
Let \( f(x, y) = x^2 + y^2 \) be a scalar function, and let \( \mathbf{r}(t) = (t, t^2) \) for \( t \) ranging from \( 0 \) to \( 1 \). Calculate the line integral \( \int_{\mathbf{r}} \nabla f \cdot d\mathbf{r} \) using the gradient theorem, and determine the exact value of this integral. | automath | [
0.012284335680305958,
0.036916375160217285,
-0.04215100780129433,
-0.03081856481730938,
0.04598505049943924,
0.0025245225988328457,
-0.02372988685965538,
0.07342517375946045,
0.03788578137755394,
-0.020046569406986237,
-0.0003960210015065968,
0.02839173749089241,
-0.031076189130544662,
-0.... | 0.811324 | 0.79479 | 0.016534 |
Let \( M \) be a simple matroid with rank \( n \) and let \( q \) be a prime power. If \( M \) has more than \( \frac{q^{n}-1}{q-1} \) elements, what is the minimum possible rank of a minor of \( M \) that is isomorphic to \( U_{2,q+2} \)? | automath | [
0.027455389499664307,
0.02242003194987774,
-0.023165296763181686,
-0.04330693930387497,
0.03734612464904785,
-0.027725132182240486,
0.0024627589154988527,
0.060259222984313965,
0.0238791573792696,
-0.015799157321453094,
0.018005620688199997,
0.025836234912276268,
-0.01976351998746395,
-0.0... | 0.828719 | 0.808689 | 0.02003 |
Let \( X = \{1, 2, 3, 4\} \) and consider the topology \( \mathcal{T} = \{ \varnothing, \{1, 2\}, \{2, 3\}, \{2\}, \{1, 2, 3\}, \{1, 2, 3, 4\} \} \). If we define a subbasis \( \mathcal{S} \) to be the collection \( \{ \{1, 2\}, \{2, 3\} \} \), what is the basis generated from \( \mathcal{S} \) and does it include the ... | automath | [
0.04280520975589752,
0.022445527836680412,
-0.05461733043193817,
-0.004708381835371256,
0.040462102741003036,
-0.01655479148030281,
-0.0038653111550956964,
0.053586915135383606,
0.039439234882593155,
-0.001959686866030097,
0.017585529014468193,
0.03576923906803131,
-0.038913507014513016,
-... | 0.822135 | 0.798983 | 0.023152 |
Consider the statement: "For every point \( x \) in a set \( U \), there exists a radius \( r > 0 \) such that all points \( y \) within \( r \) of \( x \) are also in \( U \)." What is the correct negation of this statement? Provide a clear representation of the negation using quantifiers and logical symbols. | automath | [
0.011342757381498814,
0.022649487480521202,
-0.043700505048036575,
-0.023641522973775864,
0.03186307102441788,
-0.04354485124349594,
-0.010853770188987255,
0.056054580956697464,
0.05958154797554016,
-0.002118489006534219,
0.010413541458547115,
0.025539131835103035,
-0.02673823945224285,
-0... | 0.827236 | 0.802306 | 0.02493 |
Let \( S = \{a, b, c\} \) be a semigroup under a binary operation \( \circ \). Define the operation \( \circ_\mathcal{P} \) on the power set \( \mathcal{P}(S) \).
If \( A = \{a\} \), \( B = \{b\} \), and \( C = \{c\} \), how many distinct subsets of \( S \) can be formed by evaluating \( A \circ_\mathcal{P} (B \circ_... | automath | [
0.030030852183699608,
0.03215114772319794,
-0.04150845855474472,
-0.003336760215461254,
0.03321997821331024,
-0.027733774855732918,
-0.02220141515135765,
0.07272429019212723,
0.04044153541326523,
-0.00014036288484930992,
-0.0003509051166474819,
0.016402484849095345,
-0.02551187016069889,
-... | 0.838273 | 0.819436 | 0.018838 |
What are all positive integers \( n \) such that there exists a surjective homomorphism from the symmetry group \( \mathbb{D}_3 \) onto \( (\mathbb{Z}_n, +) \)? Additionally, what is the only positive integer \( n \) for which a homomorphism exists from \( \mathbb{R} \) onto \( \mathbb{Z}_n \)? | automath | [
0.03472375497221947,
0.01630406081676483,
-0.031941741704940796,
-0.037081003189086914,
0.03310171142220497,
-0.014431297779083252,
0.0011552160140126944,
0.04138001799583435,
0.039617810398340225,
-0.032377246767282486,
0.0022376582492142916,
0.0230301171541214,
-0.04555841535329819,
-0.0... | 0.830451 | 0.813758 | 0.016693 |
Let \( A \) and \( B \) be finite sets with \( |A| = m \) and \( |B| = n \). Suppose \( f: A \rightarrow B \) is a function such that \( f \) is surjective and \( g: B \rightarrow A \) is a function such that \( f \circ g \) is the identity function on \( B \). If it is also given that the composition \( h \circ f \) i... | automath | [
0.039276041090488434,
0.04001958295702934,
-0.03954111784696579,
-0.004033188335597515,
0.035171542316675186,
-0.02066369540989399,
-0.02509108930826187,
0.053091101348400116,
0.03371192887425423,
-0.0005489145405590534,
0.014426848851144314,
0.023737017065286636,
-0.02138332836329937,
-0.... | 0.823187 | 0.799266 | 0.023921 |
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