query_key stringlengths 9 102 | query_node_id int64 11 3.82k | repo stringclasses 35
values | query_text stringlengths 0 1.28k | positives listlengths 1 13 | positive_ids listlengths 1 13 |
|---|---|---|---|---|---|
brownian-motion/def:approxSeq | 3,693 | brownian-motion | Given a stopping time \(\tau : \Omega \to T \cup \{ \infty \} \), a sequence of stopping times \((\tau _n)_{n \in \mathbb {N}}\) is called an discrete approximation of \(\tau \) if \(\tau _n(\Omega )\) is countable for each \(n\) and \(\tau _n \downarrow \tau \) a.s. as \(n \to \infty \). | [
"brownian-motion/def:IsStoppingTime"
] | [
3129
] |
brownian-motion/def:stdGaussian | 642 | brownian-motion | Let \((e_1, \ldots , e_d)\) be an orthonormal basis of \(E\) and let \(\mu \) be the standard Gaussian measure on \(\mathbb {R}\). The standard Gaussian measure on \(E\) is the pushforward measure of the product measure \(\mu \times \ldots \times \mu \) by the map \(x \mapsto \sum _{i=1}^d x_i \cdot e_i\). | [
"brownian-motion/def:gaussianReal"
] | [
1801
] |
amatelus/def:multi-chapter | 3,003 | amatelus | This chapter covers multi-device support aspects of AMATELUS. | [
"amatelus/thm:did-local-generation"
] | [
104
] |
LBPTester/theorem : subspace_contains_zero | 2,179 | LBPTester | This is a proof that any subspace contains the zero vector:\((0 : V) \in W\) | [
"LBPTester/theorem : zero_smul",
"LBPTester/definition : Subspace"
] | [
2272,
2123
] |
PrimeNumberTheoremAnd/HolomorphicOn_riemannZeta0 | 225 | PrimeNumberTheoremAnd | For any \(N\ge 1\), the function \(\zeta _0(N,s)\) is holomorphic on \(\{ s\in \mathbb {C}\mid \Re (s){\gt}0 ∧ s \ne 1\} \). | [
"PrimeNumberTheoremAnd/riemannZeta0"
] | [
3461
] |
equational_theories/free-confluent | 3,724 | equational_theories | Let \(\Gamma \) be a confluent theory. Then the free magma \(M_{X,\Gamma }\) subject to this theory can be described as the space of reduced words in \(M_X\) in normal form, where the operation \(w \diamond _\Gamma w'\) on this magma is defined as the normal form reduction of \(w \diamond w'\), and \(\iota _{X,\Gamma }... | [
"equational_theories/confluent-theory"
] | [
2924
] |
PrimeNumberTheoremAnd/I8 | 2,931 | PrimeNumberTheoremAnd | \[ I_8(\nu , \epsilon , X, T, \sigma _1) := \frac{1}{2\pi i} \int _{\sigma _1}^{\sigma _0} \left( \frac{-\zeta '}\zeta (\sigma + T i) \right) \mathcal M(\widetilde1_\epsilon )(\sigma + T i) X^{\sigma + T i} \ d\sigma \] | [
"PrimeNumberTheoremAnd/Smooth1"
] | [
3771
] |
FLT_ch9/NumberField.InfiniteAdeleRing.piEquiv | 693 | FLT_ch9 | There is a natural \(K_{\infty }\)-linear homeomorphism \(K_{\infty }^{[L:K]} \cong _{K_{\infty }} L_{\infty }\). | [
"FLT_ch9/NumberField.InfinitePlace.Completion.piEquiv"
] | [
2749
] |
brownian-motion/cor:doob_lp_norm | 3,363 | brownian-motion | Let \(X : \mathbb {R} \rightarrow \Omega \rightarrow E\) be a right-continuous martingale with values in a normed space \(E\). For every \(T, \lambda {\gt}0\) and \(p{\gt}1\) we have \begin{align*} \mathbb {E}\left[ \sup _{t\in [0,T]} \lVert X_t \rVert ^p \right] \leq \left(\frac{p}{p-1}\right)^p\mathbb {E}[\lVert X_T ... | [
"brownian-motion/cor:Martingale.submartingale_norm",
"brownian-motion/thm:doob_lp"
] | [
167,
1677
] |
Sphere-Packing-Lean/def:holomorphic-modular-form | 1,477 | Sphere-Packing-Lean | Let \(\Gamma \) denote a subgroup of \(\mathrm{SL}_2(\mathbb {Z})\), then a modular form of level \(\Gamma \) and weight \(k \in \mathbb {Z}\) is a function \(f : \mathbb {H} \to \mathbb {C}\) such that: For all \(\gamma \in \Gamma \) we have \(f\mid _k \gamma = f\) (such functions are called slash invariant). \(f\) is... | [
"Sphere-Packing-Lean/def:congruence-subgroup"
] | [
907
] |
ClassFieldTheory/lem:Dirichlet density split | 896 | ClassFieldTheory | Let \(l/k\) be a finite Galois extension of number fields. Then the set of degree \(1\) primes of \(k\) which split completely in \(l\) has density \(\frac{1}{[l:k]}\). | [
"ClassFieldTheory/lem:Dirichlet density degree one"
] | [
1018
] |
pfr/cond-multidist-alt | 652 | pfr | If the \((X_i,Y_i)\) are independent, \begin{equation} \label{multi-def-cond} D[ X_{[m]} | Y_{[m]}] := \mathbb {H}[\sum _{i=1}^m X_i \big| (Y_j)_{1 \leq j \leq m} ] - \frac{1}{m} \sum _{i=1}^m \mathbb {H}[ X_i | Y_i]. \end{equation} | [
"pfr/conditional-entropy-def",
"pfr/cond-multidist-def"
] | [
1044,
2700
] |
brownian-motion/def:simpleProcess | 65 | brownian-motion | Let \((s_k {\lt} t_k)_{k \in \{ 1, ..., n\} }\) be points in a linear order \(T\) with a bottom element 0. Let \((\eta _k)_{0 \le k \le n}\) be bounded random variables with values in a normed real vector space \(E\) such that \(\eta _0\) is \(\mathcal{F}_0\)-measurable and \(\eta _k\) is \(\mathcal{F}_{s_k}\)-measurab... | [
"brownian-motion/def:filtration"
] | [
2474
] |
FLT_ch8/fixed_of_fixed1_aux2 | 2,961 | FLT_ch8 | Let \(b_0 \in B\). Suppose that the image of \(b_0\) in the quotient \(B/Q\) is fixed by the stabilizer subgroup \(D_Q\). Then there exist elements \(a,b \in B\), with \(a\notin Q\) and \(a\) in the image of \(A\) such that for all \(g\in G\), If \(g \cdot Q = Q\), then \(g \cdot b \equiv ab_0 \pmod{Q}\). If \(g \cdot ... | [
"FLT_ch8/fixed_of_fixed1_aux1"
] | [
1886
] |
Sphere-Packing-Lean/E8-vector-norms | 3,654 | Sphere-Packing-Lean | All vectors in \(\Lambda _8\) have norm of the form \(\sqrt{2n}\), where \(n\) is a nonnegative integer. | [
"Sphere-Packing-Lean/E8-defs-equivalent"
] | [
529
] |
ABC-Exceptions/lem:Subcase2.6 | 2,745 | ABC-Exceptions | \(\nu \leq 0.66\) in the case \(s_2{\lt}0.3\) and \(2s_1-s_2{\gt}0.025\). | [
"ABC-Exceptions/prop:FourierBound",
"ABC-Exceptions/lem:Subcase2.3",
"ABC-Exceptions/lem:Subcase2.5",
"ABC-Exceptions/lem:Subcase2.4"
] | [
3214,
825,
1212,
233
] |
equational_theories/austin-two | 3,006 | equational_theories | Any law with at most two variables has a non-trivial finite model. | [
"equational_theories/models-def"
] | [
1745
] |
GibbsMeasure/def:gibbs-meas | 1,460 | GibbsMeasure | Given a specification \(\gamma \), a Gibbs measures specified by \(\gamma \) is a measure \(\nu \in \mathfrak {M}(E^S, \mathcal{E}^S)\) such that \(\gamma _\Lambda (A|\cdot )\) is a conditional expectation kernel for \(\nu \) for all \(A \in \mathcal E^S\) and \(\Lambda \in \operatorname{Finset}(S)\). | [
"GibbsMeasure/def:spec",
"GibbsMeasure/def:cond-exp-ker"
] | [
765,
1310
] |
toric/0-diag-aff-hom | 501 | toric | Let \(S\) be a scheme. Let \(M, N\) be commutative monoids and \(f : M \to N\) a monoid hom. Then the map \(D_S(f) : D_S(N) \to D_S(M)\) is affine. | [
"toric/0-diag"
] | [
858
] |
PrimeNumberTheoremAnd/BlaschkeNonZero | 3,069 | PrimeNumberTheoremAnd | Let \(0 {\lt} r {\lt} R{\lt}1\) and \(f:\overline{\mathbb {D}_1}\to \mathbb {C}\) be analytic on neighborhoods of points in \(\overline{\mathbb {D}_1}\). Then \(B_f(z)\neq 0\) for all \(z\in \overline{\mathbb {D}_r}\). | [
"PrimeNumberTheoremAnd/BlaschkeB",
"PrimeNumberTheoremAnd/ZeroFactorization"
] | [
3062,
3498
] |
radii-polynomial/thm:norm_mul_le | 164 | radii-polynomial | \(\| a \star b\| _{1,\nu } \leq \| a\| _{1,\nu } \cdot \| b\| _{1,\nu }\). This is axiom (4) of Definition 7.4.1, the key analytic property. The proof uses Mertens’ theorem and weight factorization \(\nu ^n = \nu ^k \cdot \nu ^l\). | [
"radii-polynomial/def:l1Weighted_mul"
] | [
3101
] |
radii-polynomial/def:param_power_series | 1,824 | radii-polynomial | For \(\lambda _0 \in \mathbb {R}\), the parameter element \(c \in \ell ^1_\nu \) represents \(\lambda _0 + z\): \begin{equation*} c = (\lambda _0, 1, 0, 0, \ldots ) \in \ell ^1_\nu . \end{equation*} | [
"radii-polynomial/def:param_seq"
] | [
909
] |
FLT_ch14/automorphic_representation | 1,602 | FLT_ch14 | An automorphic representation is an irreducible \((G(\mathbb {A}_N^f)\times U_\infty ,\mathfrak {g})\)-module isomorphic to an irreducible subquotient of the space of automorphic forms. | [
"FLT_ch14/automorphic_form_actions"
] | [
1400
] |
FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C | 129 | FLT_ch11 | For every \(\beta \in D_{\mathbb {A}}^{(1)}\), there exists \(b\in D^\times \) and \(\nu \in D_{\mathbb {A}}^{(1)}\) such that \(\beta =b\nu \) and \((\nu ,\nu ^{-1})\in C.\) | [
"FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.C",
"FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel'",
"FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.X_meets_kernel"
] | [
1338,
438,
1254
] |
brownian-motion/thm:doob_ineq | 143 | brownian-motion | Let \(X: \mathbb {R}_+ \to \Omega \to \mathbb {R}\) be a right-continuous non-negative sub-martingale. For every \(T \in \mathbb {R}_+\) and \(\lambda {\gt}0\) we have \begin{align*} P\left( \sup _{t\in [0,T]}X_t \geq \lambda \right) \leq \frac{\mathbb {E}[X_T \mathbb {I}_{\sup _{i \le T}X_i \ge \lambda }]}{\lambda } \... | [
"brownian-motion/lem:doob_countable",
"brownian-motion/def:RightContinuous"
] | [
1108,
1591
] |
brownian-motion/lem:isGaussian_add_const | 775 | brownian-motion | Let \(\mu \) be a Gaussian measure on \(F\) and let \(c \in F\). Then the measure \(\mu \) translated by \(c\) (the map of \(\mu \) by \(x \mapsto x + c\)) is a Gaussian measure on \(F\). | [
"brownian-motion/def:IsGaussian"
] | [
3158
] |
ClassFieldTheory/lem:herbrand local l* | 2,758 | ClassFieldTheory | If \(l/k\) is a cyclic extension of local fields then \(h(l/k, l^\times )= [l:k]\). | [
"ClassFieldTheory/lem:herbrand local units"
] | [
50
] |
pfr/ruz-copy | 282 | pfr | If \(X',Y'\) are copies of \(X,Y\) respectively then \(d[X';Y']=d[X ;Y]\). | [
"pfr/ruz-dist-def",
"pfr/copy-ent"
] | [
1818,
2358
] |
ClassFieldTheory/lem:Dirichlet density union | 3,617 | ClassFieldTheory | Suppose \(M_1\) and \(M_2\) are disjoint sets of primes of \({\mathcal O}_l\). If two of the sets \(M_1, M_2, M_1 \cup M_2\) have a Dirichlet density, then so does the third and we have \[ \mathrm{density}(M_1 \cup M_2) = \mathrm{density}(M_1) + \mathrm{density}(M_2). \] | [
"ClassFieldTheory/def:Dirichlet density"
] | [
1246
] |
ChebyshevCircles/thm:newton_identities | 954 | ChebyshevCircles | The power sums and elementary symmetric polynomials are related by: \begin{equation} k \cdot e_k = \sum _{i=1}^k (-1)^{i-1} e_{k-i} p_i, \quad k = 1, 2, \ldots , N, \end{equation} | [
"ChebyshevCircles/lem:discrete_orthogonality"
] | [
3410
] |
ClassFieldTheory/thm:splitting module trivial | 538 | ClassFieldTheory | Let \(\sigma '\) be a 2-cocycle representing a fundamental class in \(H^2(G,M)\). Then \(\mathrm{split}(\sigma ')\) has trivial cohomology. | [
"ClassFieldTheory/lem:splits in splitting module",
"ClassFieldTheory/thm:triviality criterion",
"ClassFieldTheory/lem:restriction fundamental class generates",
"ClassFieldTheory/lem:linear injective of surjective"
] | [
3356,
156,
1539,
2325
] |
brownian-motion/lem:martingale_iff_integral_elemStochIntegral_eq_zero | 3,030 | brownian-motion | An adapted integrable process \(X\) is a martingale if and only if for every bounded simple process \(V\) with values in \(\mathbb {R}\), \(\mathbb {E}[(V \bullet _{\mathbb {R}} X)_\infty ] = 0\). | [
"brownian-motion/lem:tendsto_elemStochIntegral_limitProcess",
"brownian-motion/def:Martingale",
"brownian-motion/def:elemStochIntegral"
] | [
1499,
1156,
490
] |
toric/0-biquot | 1,079 | toric | If \(B\) is a bialgebra over \(R\) and \(I\) is a bialgebra ideal, the quotient \(B / I\) is equipped with a canonical \(R\)-bialgebra structure. | [
"toric/0-biideal",
"toric/0-coquot"
] | [
2039,
2265
] |
carleson/Lipschitz-partition-unity | 522 | carleson | There exists a family of functions \(\chi _J\), \(J \in \mathcal{J}'\) such that \begin{equation} \label{eq-pao-1} \mathbf{1}_{{\mathcal{I}}({\mathfrak u}_1)} = \sum _{J \in \mathcal{J}'} \chi _J\, , \end{equation} | [
"carleson/dyadic-partition-1",
"carleson/moderate-scale-change"
] | [
946,
3224
] |
ABC-Exceptions/prop:DeterminantBound | 1,073 | ABC-Exceptions | \begin{align*} \nu {\lt} \min _{p,q\ge 1} \left( 1+\delta - a_p - b_q +\min \left(\frac{a_p}{q}, \frac{b_q}{p}\right)\right). \end{align*} | [
"ABC-Exceptions/def:nu",
"ABC-Exceptions/prop:DeterminantMethod",
"ABC-Exceptions/def:aibici"
] | [
686,
767,
3643
] |
FLT_ch9/IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv | 1,163 | FLT_ch9 | The natural map \(B\otimes _AK\to L\) is a \(B\)-algebra isomorphism. | [
"FLT_ch9/IsDedekindDomain.dvd_norm"
] | [
1585
] |
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