test.jsonl

{"query_key":"brownian-motion/lem:Predict_Part_Increasing","query_node_id":705,"repo":"brownian-motion","query_text":"\\((A^n_t)_{t\\in \\mathcal{D}_n^T}\\) is an increasing process.","positives":["brownian-motion/lem:nondecreasing_predictablePart_of_submartingale","brownian-motion/def:A"],"positive_ids":[2646,878]}
{"query_key":"PrimeNumberTheoremAnd/bklnw-prop-3-sub-3","query_node_id":3023,"repo":"PrimeNumberTheoremAnd","query_text":"\\(f(2^n) = 1 + u_n\\).","positives":["PrimeNumberTheoremAnd/bklnw-eq-2-4"],"positive_ids":[1636]}
{"query_key":"seymour/Matrix.IsTotallyUnimodular.linearIndependent_iff_support_linearIndependent_of_finite_of_finite","query_node_id":59,"repo":"seymour","query_text":"Let \\(A\\) be a rational TU matrix with finite number of rows and finite number of columns. Its rows are linearly independent iff the rows of its support matrix are linearly independent.","positives":["seymour/Matrix.IsTotallyUnimodular","seymour/Matrix.isUnit_iff_isUnit_det","seymour/Matrix.support_submatrix"],"positive_ids":[1724,1954,1767]}
{"query_key":"LBPTester/theorem : neg_one_smul","query_node_id":2497,"repo":"LBPTester","query_text":"In any vector space V over K, multiplying a vector by -1 gives its additive inverse: \\((v : V) : (-1 : K) \\bullet v = -v\\)","positives":["LBPTester/definition : VectorSpace"],"positive_ids":[1380]}
{"query_key":"toric/0-hopf-quot","query_node_id":3588,"repo":"toric","query_text":"If \\(A\\) is a Hopf algebra over \\(R\\) and \\(I\\) is a Hopf ideal, the quotient \\(A / I\\) is equipped with a canonical Hopf algebra structure over \\(R\\).","positives":["toric/0-biquot","toric/0-hopf-ideal"],"positive_ids":[1079,756]}
{"query_key":"PrimeNumberTheoremAnd/fks-theorem-3-1","query_node_id":3037,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(x {\\gt} e^{50}\\) and \\(50 {\\lt} T {\\lt} x\\). Then \\(E_\\psi (x) \\leq \\sum _{|\\gamma | {\\lt} T} |x^{\\rho -1}/\\rho | + 2 \\log ^2 x / T\\).","positives":["PrimeNumberTheoremAnd/zeroes-of-riemann-zeta","PrimeNumberTheoremAnd/Epsi-def"],"positive_ids":[2427,3055]}
{"query_key":"FLT_ch5/QHat.rat_meet_zHat","query_node_id":2627,"repo":"FLT_ch5","query_text":"The intersection of \\(\\mathbb {Q}\\) and \\(\\widehat{\\mathbb {Z}}\\) in \\(\\widehat{\\mathbb {Q}}\\) is \\(\\mathbb {Z}\\).","positives":["FLT_ch5/QHat","FLT_ch5/QHat.lowestTerms"],"positive_ids":[3428,3786]}
{"query_key":"PrimeNumberTheoremAnd/costa-pereira-sublemma-1-7","query_node_id":1532,"repo":"PrimeNumberTheoremAnd","query_text":"For every \\(x {\\gt} 0\\) we have \\[ \\psi (x) - \\theta (x) \\leqslant \\psi (x^{1/2}) + \\psi (x^{1/3}) + \\sum _{k \\geqslant 1} \\theta (x^{1/5k} \\]","positives":["PrimeNumberTheoremAnd/costa-pereira-sublemma-1-6"],"positive_ids":[3098]}
{"query_key":"PrimeNumberTheoremAnd/fks-corollary-3-12","query_node_id":1292,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(5/8 {\\lt} \\sigma _2 \\leq 1\\), \\(t_0 = t_0(\\sigma _2, x) = \\max \\left(H_{\\sigma _2}, \\exp \\left(\\sqrt{\\frac{\\log x}{R}}\\right)\\right)\\), \\(T {\\gt} t_0\\). Let \\(K \\geq 2\\), \\(\\lambda = (T/t_0)^{1/K}\\), and consider \\((t_k)_{k=0}^K\\) the sequence given by \\(t_k = t_0 \\lambda ^k\\). Then \\[ \\Sigma ^1_{\\sigma _2} = 2 \\sum _{\\substack {0 {\\lt} \\gamma {\\lt} T \\\\ \\sigma _2 \\leq \\beta {\\lt} 1}} \\frac{x^{\\beta -1}}{\\gamma } \\leq \\varepsilon _4(x, \\sigma _2, K, T), \\]","positives":["PrimeNumberTheoremAnd/zero-density-bound","PrimeNumberTheoremAnd/Riemann-von-Mangoldt-estimate","PrimeNumberTheoremAnd/RH-up-to","PrimeNumberTheoremAnd/classical-zero-free-region"],"positive_ids":[1474,1239,2065,3243]}
{"query_key":"PrimeNumberTheoremAnd/fks2-eq30","query_node_id":459,"repo":"PrimeNumberTheoremAnd","query_text":"For any \\(x \\geq x_0 {\\gt} 0\\), \\[ |\\pi (x) - \\operatorname {Li}(x)| \\leq \\left| \\frac{\\theta (x) - x}{\\log (x)} \\right| + \\left| \\pi (x_0) - \\operatorname {Li}(x_0) - \\frac{\\theta (x_0) - x_0}{\\log (x_0)} \\right| + \\left| \\int _{x_0}^{x} \\frac{\\theta (t) - t}{t(\\log (t))^2} \\, dt \\right|. \\]","positives":["PrimeNumberTheoremAnd/Epi-def","PrimeNumberTheoremAnd/Etheta-def","PrimeNumberTheoremAnd/Epsi-def"],"positive_ids":[23,892,3055]}
{"query_key":"toric/1-1-1-subgroup-torus","query_node_id":544,"repo":"toric","query_text":"Let \\(R\\) be a commutative ring of characteristic zero. Let \\(T\\) be a split torus. If \\(H \\subseteq T\\) is a connected closed subgroup, then \\(H\\) is a split torus.","positives":["toric/0-full-faithful-diag","toric/0-subgroup-diag","toric/0-diag-tors"],"positive_ids":[513,3029,2719]}
{"query_key":"BrauerGroup_new/a0000000113","query_node_id":1120,"repo":"BrauerGroup_new","query_text":"\\(\\sim _{\\operatorname{Br}}\\) is transitive.","positives":["BrauerGroup_new/con:matrix-matrix"],"positive_ids":[3590]}
{"query_key":"pfr/multidist-ruzsa-III","query_node_id":2068,"repo":"pfr","query_text":"Let \\(m \\ge 2\\), and let \\(X_{[m]}\\) be a tuple of \\(G\\)-valued random variables. If the \\(X_i\\) all have the same distribution, then \\(D[X_{[m]}] \\leq m d[X_i;X_i]\\) for any \\(1 \\leq i \\leq m\\).","positives":["pfr/sumset-lower","pfr/ruz-indep","pfr/klm-1","pfr/multidist-copy","pfr/ruz-copy","pfr/multidist-indep"],"positive_ids":[461,3276,1171,2722,282,2290]}
{"query_key":"FLT_ch3/hardly_ramified_lifts","query_node_id":1158,"repo":"FLT_ch3","query_text":"If \\(\\ell \\geq 3\\) is prime and \\(\\overline{\\rho }:\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathbb {Z}/\\ell \\mathbb {Z})\\) is hardly ramified and irreducible, then there exists a finite extension \\(K\\) of \\(\\mathbb {Q}_\\ell \\) with integer ring \\(\\mathcal{O}\\) and maximal ideal \\(\\mathfrak {m}\\) and a hardly ramified representation \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathcal{O})\\) whose reduction modulo \\(\\mathfrak {m}\\) is isomorphic to \\(\\rho \\).","positives":["FLT_ch3/hardly_ramified"],"positive_ids":[450]}
{"query_key":"brownian-motion/lem:elemStochIntegralLinear_assoc","query_node_id":1101,"repo":"brownian-motion","query_text":"For simple process \\(W\\) with values in \\(L(E, F)\\), simple process \\(V\\) with values in \\(L(F, G)\\), and stochastic process \\(X\\) with values in \\(E\\), we have \\begin{align*} V \\bullet _L (W \\bullet _L X) & = (V \\circ W) \\bullet _L X \\: , \\end{align*}","positives":["brownian-motion/lem:elemStochIntegralBilin_assoc"],"positive_ids":[2933]}
{"query_key":"brownian-motion/lem:M_uniform_integrable","query_node_id":1210,"repo":"brownian-motion","query_text":"The sequence \\((M^n_T)_{n\\in \\mathbb {N}}\\) is uniformly integrable (bounded in \\(L^1\\) norm).","positives":["brownian-motion/lem:A_uniform_integrable","brownian-motion/def:classD"],"positive_ids":[838,3521]}
{"query_key":"pfr/klm-3","query_node_id":402,"repo":"pfr","query_text":"If \\(n \\geq 1\\) and \\(X, Y_1, \\dots , Y_n\\) are jointly independent \\(G\\)-valued random variables, then \\[ d\\left[X; \\sum _{i=1}^n Y_i\\right] \\leq d\\left[X; Y_1\\right] + \\frac{1}{2}\\left(\\mathbb {H}\\left[ \\sum _{i=1}^n Y_i\\right] - \\mathbb {H}[Y_1]\\right). \\]","positives":["pfr/kv","pfr/ruz-indep"],"positive_ids":[3327,3276]}
{"query_key":"PrimeNumberTheoremAnd/initial-factorization-def","query_node_id":1345,"repo":"PrimeNumberTheoremAnd","query_text":"We perform an initial factorization by taking the natural numbers between \\(n-n/M\\) (inclusive) and \\(n\\) (exclusive) repeated \\(M\\) times, deleting those elements that are not \\(n/L\\)-smooth (i.e., have a prime factor greater than or equal to \\(n/L\\)).","positives":["PrimeNumberTheoremAnd/params-set","PrimeNumberTheoremAnd/factorization-def"],"positive_ids":[10,1253]}
{"query_key":"ChebyshevCircles/lem:sum_cos_m_phi_vanishes","query_node_id":1124,"repo":"ChebyshevCircles","query_text":"For any integers \\(N \\geq 1\\), \\(m\\) with \\(0 {\\lt} m {\\lt} N\\), and any \\(\\theta \\in \\mathbb {R}\\), \\begin{equation} \\sum _{k=0}^{N-1} \\cos \\left(m\\left(\\theta + \\frac{2\\pi k}{N}\\right)\\right) = 0. \\end{equation}","positives":["ChebyshevCircles/lem:discrete_orthogonality"],"positive_ids":[3410]}
{"query_key":"PrimeNumberTheoremAnd/Dusart_prop_5_12","query_node_id":843,"repo":"PrimeNumberTheoremAnd","query_text":"We have for \\(k \\geq 781\\), \\[ \\vartheta (p_k) \\leq k \\log k + \\log \\log k - 1 + \\frac{\\log \\log k - 2}{\\log k} - \\frac{0.782}{\\log ^2 k}. \\]","positives":["PrimeNumberTheoremAnd/theta-def"],"positive_ids":[108]}
{"query_key":"BlueprintRep/YuInv","query_node_id":3044,"repo":"BlueprintRep","query_text":"\\(Y_{\\mu }^{-1}\\) est une l’inverse de \\(Y_{\\mu }\\)","positives":["BlueprintRep/bijYu"],"positive_ids":[1783]}
{"query_key":"ClassFieldTheory/lem:unramified units trivial","query_node_id":3500,"repo":"ClassFieldTheory","query_text":"If \\(l/k\\) is unramified then \\({\\mathcal O}_l^\\times \\) has trivial group cohomology.","positives":["ClassFieldTheory/lem:finite field trivial","ClassFieldTheory/lem:serre_approx"],"positive_ids":[1777,2196]}
{"query_key":"amatelus/def:impl-chapter","query_node_id":138,"repo":"amatelus","query_text":"This chapter covers implementation guidance aspects of AMATELUS.","positives":["amatelus/def:crypto-assumptions"],"positive_ids":[3484]}
{"query_key":"GibbsMeasure/def:markov-spec","query_node_id":3806,"repo":"GibbsMeasure","query_text":"A specification \\(\\gamma \\) is a Markov specification iff \\(\\gamma _\\Lambda \\) is a probability kernel for every \\(\\Lambda \\in \\operatorname{Finset}(S)\\).","positives":["GibbsMeasure/def:spec"],"positive_ids":[765]}
{"query_key":"Sphere-Packing-Lean/lemma:theta-bounded-im-infty","query_node_id":3225,"repo":"Sphere-Packing-Lean","query_text":"For all \\(\\gamma \\in \\Gamma _1\\), \\(H_{2}|_2 \\gamma \\), \\(H_{3}|_2 \\gamma \\), and \\(H_{4}|_2 \\gamma \\) are holomorphic at \\(i\\infty \\).","positives":["Sphere-Packing-Lean/lemma:theta-slash-invariant","Sphere-Packing-Lean/lemma:Gamma-1-generators"],"positive_ids":[3215,1387]}
{"query_key":"seymour/Matrix.IsGraphic","query_node_id":2334,"repo":"seymour","query_text":"Let \\(A \\in \\mathbb {Q}^{X \\times Y}\\) be a matrix. If for all \\(j \\in Y\\), one has that \\(a_{i,j} = 0\\) for all \\(i \\in X\\), or that there exists \\(i_1,i_2 \\in X\\) such that \\[ a_{i,j} = \\begin{cases} 1 & \\text{ if $i = i_1$} \\\\ -1 & \\text{ if $i = i_2$} \\\\ 0 & \\text{ otherwise}, \\end{cases} \\]","positives":["seymour/Matrix"],"positive_ids":[2693]}
{"query_key":"brownian-motion/def:ProgMeasurable","query_node_id":1045,"repo":"brownian-motion","query_text":"A stochastic process \\(X\\) is said to be progressively measurable with respect to a filtration \\(\\mathcal{F}\\) if at each point in time \\(i\\), \\(X\\) restricted to \\((-\\infty , i] \\times \\Omega \\) is measurable with respect to the product \\(\\sigma \\)-algebra where the \\(\\sigma \\)-algebra used for \\(\\Omega \\) is \\(\\mathcal{F}_i\\).","positives":["brownian-motion/def:filtration"],"positive_ids":[2474]}
{"query_key":"brownian-motion/thm:Ito_formula","query_node_id":771,"repo":"brownian-motion","query_text":"Let \\(X^1, \\ldots , X^d\\) be continuous semi-martingales and let \\(f : \\mathbb {R}^d \\to \\mathbb {R}\\) be a twice continuously differentiable function. Then, writing \\(X = (X^1, \\ldots , X^d)\\), the process \\(f(X)\\) is a semi-martingale and we have \\begin{align*} f(X_t) & = f(X_0) + \\sum _{i=1}^d \\int _0^t \\frac{\\partial f}{\\partial x_i}(X_s) \\: dX^i_s + \\frac{1}{2} \\sum _{i,j=1}^d \\int _0^t \\frac{\\partial ^2 f}{\\partial x_i \\partial x_j}(X_s) \\: d[X^i, X^j]_s \\: . \\end{align*}","positives":["brownian-motion/thm:integration_by_parts"],"positive_ids":[2905]}
{"query_key":"Sphere-Packing-Lean/lemma:automorphy-factor-chain-rule","query_node_id":1500,"repo":"Sphere-Packing-Lean","query_text":"The automorphy factor satisfies the chain rule \\[ j_k(z,\\gamma _1\\gamma _2)=j_k(z,\\gamma _1)\\, j_k(\\gamma _2z,\\gamma _1). \\]","positives":["Sphere-Packing-Lean/def:automorphy-factor"],"positive_ids":[548]}
{"query_key":"PrimeNumberTheoremAnd/lem:Lprime-def","query_node_id":285,"repo":"PrimeNumberTheoremAnd","query_text":"There exists a positive integer \\(L'\\) such that \\[ L_n = q_1 q_2 q_3 \\, L' \\]","positives":["PrimeNumberTheoremAnd/lcm-criterion","PrimeNumberTheoremAnd/Ln-def"],"positive_ids":[3744,1520]}
{"query_key":"carleson/forest-convex","query_node_id":893,"repo":"carleson","query_text":"For each \\({\\mathfrak u}\\in {\\mathfrak U}_3(k,n,j)\\), the set \\(\\mathfrak {T}_2({\\mathfrak u})\\) satisfies the convexity condition 2.0.33.","positives":["carleson/C5-convex"],"positive_ids":[3804]}
{"query_key":"pfr/phi-first-estimate","query_node_id":898,"repo":"pfr","query_text":"\\(I_1\\le 2\\eta d[X_1;X_2]\\)","positives":["pfr/phi-min-def","pfr/first-fibre"],"positive_ids":[3787,2473]}
{"query_key":"BlueprintRep/bu","query_node_id":314,"repo":"BlueprintRep","query_text":"\\[ b_{\\mu } := \\frac{1}{|Q_{\\mu }|}\\sum _{g \\in Q_{\\mu }}(-1)^{g}g \\]","positives":["BlueprintRep/QuCard"],"positive_ids":[1657]}
{"query_key":"infinity-cosmos/lem:equiv-htpy-equiv","query_node_id":95,"repo":"infinity-cosmos","query_text":"A map \\(f \\colon A \\to B\\) between \\(\\infty \\)-categories in an \\(\\infty \\)-cosmos \\({\\mathord {\\mathcal{K}}}\\) is an equivalence if and only if it extends to the data of a “homotopy equivalence” with the free-living isomorphism \\(I\\) serving as the interval: that is, if there exist maps \\(g \\colon B \\to A\\)","positives":["infinity-cosmos/defn:equivalence"],"positive_ids":[1700]}
{"query_key":"FormalBook/quadratic_reciprocity1","query_node_id":830,"repo":"FormalBook","query_text":"Let \\(p\\) and \\(q\\) be different odd primes. Then \\[ (\\frac{q}{p})(\\frac{p}{q}) = (-1)^{\\frac{p-1}{2} \\frac{q-1}{2}}. \\]","positives":["FormalBook/gauss_lemma"],"positive_ids":[2757]}
{"query_key":"PrimeNumberTheoremAnd/div-remainder","query_node_id":356,"repo":"PrimeNumberTheoremAnd","query_text":"There exist integers \\(m \\ge 0\\) and \\(r\\) satisfying \\(0 {\\lt} r {\\lt} 4 p_1 p_2 p_3\\) and \\[ q_1 q_2 q_3 = 4 p_1 p_2 p_3 m + r \\]","positives":["PrimeNumberTheoremAnd/lcm-criterion"],"positive_ids":[3744]}
{"query_key":"BlueprintRep/PuCard","query_node_id":2690,"repo":"BlueprintRep","query_text":"Le nombre d’élément de \\(P_{\\mu }\\) est fini.","positives":["BlueprintRep/Pu"],"positive_ids":[2975]}
{"query_key":"exchangeability/lem:kallenberg_L2_bound","query_node_id":3642,"repo":"exchangeability","query_text":"The key L\\(^2\\) bound that drives the Cesaro convergence.","positives":["exchangeability/lem:l2_bound"],"positive_ids":[673]}
{"query_key":"pfr/rho-cond-relabeled","query_node_id":1749,"repo":"pfr","query_text":"If \\(f\\) is injective, then \\(\\rho (X|f(Y))=\\rho (X|Y)\\).","positives":["pfr/rho-cond-def"],"positive_ids":[1304]}
{"query_key":"seymour/MatrixSum3.IsCanonicalSigning.toMatrixLikeSum3","query_node_id":2457,"repo":"seymour","query_text":"Let \\(B''\\) be from Definition 53. Then \\(B'' \\in \\mathcal{C} (X_{\\ell }', Y_{\\ell }', X_{r}', Y_{r}'; c_{0}'', c_{1}'')\\) with \\(X_{\\ell }' = X_{\\ell } \\setminus \\{ x_{1}, x_{0}\\} \\), \\(X_{r}' = X_{r} \\setminus \\{ x_{2}, x_{1}, x_{0}\\} \\), \\(Y_{\\ell }' = Y_{\\ell } \\setminus \\{ y_{2}\\} \\), \\(Y_{r}' = Y_{r} \\setminus \\{ y_{0}, y_{1}\\} \\), \\(x_{0}\\) and \\(x_{1}\\) are the same, \\(c_{0}'' = B'' (X_{r}', y_{0})\\), and \\(c_{1}'' = B'' (X_{r}', y_{1})\\).","positives":["seymour/MatrixLikeSum3","seymour/MatrixSum3.IsCanonicalSigning.Al_D_isTotallyUnimodular"],"positive_ids":[524,1386]}
{"query_key":"pfr/cond-distance-lower","query_node_id":2589,"repo":"pfr","query_text":"For any \\(G\\)-valued random variables \\(X'_1,X'_2\\) and random variables \\(Z,W\\), one has \\[ d[X'_1|Z;X'_2|W] \\geq k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] ). \\]","positives":["pfr/cond-dist-def","pfr/distance-lower"],"positive_ids":[398,2256]}
{"query_key":"seymour/Matrix.fromRows_zero_reindex_toMatroid","query_node_id":3534,"repo":"seymour","query_text":"Adding extra zero rows to a full representation matrix of a vector matroid does not change the matroid.","positives":["seymour/VectorMatroid"],"positive_ids":[2016]}
{"query_key":"PrimeNumberTheoremAnd/bklnw-eq_A_26","query_node_id":2947,"repo":"PrimeNumberTheoremAnd","query_text":"For \\(100 \\leq x \\leq 10^{19}\\), one has \\[ | (x - \\psi (x)) / \\sqrt{x} | \\leq 0.94. \\]","positives":["PrimeNumberTheoremAnd/Epsi-def"],"positive_ids":[3055]}
{"query_key":"equational_theories/eq1689","query_node_id":1563,"repo":"equational_theories","query_text":"Equation 1689 is the law \\(0 \\simeq (1 \\diamond 0) \\diamond ((0 \\diamond 2) \\diamond 2)\\) (or the equation \\(x = (y \\diamond x) \\diamond ((x \\diamond z) \\diamond z)\\)).","positives":["equational_theories/magma-def"],"positive_ids":[3408]}
{"query_key":"Sphere-Packing-Lean/lemma:phi0-transform","query_node_id":2158,"repo":"Sphere-Packing-Lean","query_text":"We have \\begin{align} \\phi _0(z + 1) & = \\phi _0(z) \\label{eqn:phi0-trans-T} \\\\ \\phi _0\\left(-\\frac{1}{z}\\right) & = \\phi _0(z)-\\frac{12i}{\\pi }\\, \\frac{1}{z}\\, \\phi _{-2}(z)-\\frac{36}{\\pi ^2}\\, \\frac{1}{z^2}\\, \\phi _{-4}(z).\\label{eqn:phi0-trans-S} \\end{align}","positives":["Sphere-Packing-Lean/lemma:E2-transform-S","Sphere-Packing-Lean/def:phi4-phi2-phi0","Sphere-Packing-Lean/lemma:disc-cuspform","Sphere-Packing-Lean/lemma:Ek-is-modular-form"],"positive_ids":[3318,1367,2530,110]}
{"query_key":"M2/prop:homotopy_k_sheaf_compute_chech","query_node_id":3823,"repo":"M2","query_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves that verify 1 and 2 then the canonical map \\(H^{\\bullet }\\mathcal{F}^{\\bullet }\\to \\check{H}^{\\bullet }(X;\\mathcal{F}^{\\bullet })\\) is an isomorphism.","positives":["M2/def:homotopy_k_sheaf","M2/def:chech_of_compelx_of_k_prsh"],"positive_ids":[3076,1631]}
{"query_key":"FLT_ch9/IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv","query_node_id":3113,"repo":"FLT_ch9","query_text":"The natural map \\(B\\otimes _A\\mathbb {A}_K^\\infty \\to \\mathbb {A}_L^\\infty \\) is a \\(B\\)-algebra isomorphism.","positives":["FLT_ch9/IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1","FLT_ch9/IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2"],"positive_ids":[2445,2911]}
{"query_key":"carleson/convergence-for-twice-contdiff","query_node_id":2730,"repo":"carleson","query_text":"Let \\(f:{\\mathbb {R}}\\to {\\mathbb {C}}\\) be \\(2\\pi \\)-periodic and twice continuously differentiable. Then \\begin{equation} \\sup _{x\\in [0,2\\pi ]} |f(x) - S_Nf(x)| \\rightarrow 0 \\end{equation}","positives":["carleson/fourier-coeff-derivative","carleson/convergence-of-coeffs-summable"],"positive_ids":[2825,3134]}
{"query_key":"pfr/dist-diff-bound","query_node_id":1655,"repo":"pfr","query_text":"We have \\begin{align*} & \\sum _{i=1}^2 \\sum _{A,B \\in \\{ U,V,W\\} : A \\neq B} d[X_i^0;A|B, S] - d[X_i^0;X_i]\\\\ & \\qquad \\leq 12 k + \\frac{4(2 \\eta k - I_1)}{1-\\eta }. \\end{align*}","positives":["pfr/gen-ineq","pfr/second-estimate-aux"],"positive_ids":[1103,1495]}
{"query_key":"equational_theories/eq1648","query_node_id":264,"repo":"equational_theories","query_text":"Equation 1648 is the law \\(0 \\simeq (0 \\diamond 1) \\diamond ((0 \\diamond 1) \\diamond 1)\\) (or the equation \\(x = (x \\diamond y) \\diamond ((x \\diamond y) \\diamond y)\\)).","positives":["equational_theories/magma-def"],"positive_ids":[3408]}
{"query_key":"PrimeNumberTheoremAnd/costa-pereira-theorem-1a","query_node_id":388,"repo":"PrimeNumberTheoremAnd","query_text":"For every \\(x {\\gt} 0\\) we have \\(\\psi (x) - \\theta (x) \\leqslant \\psi (x^{1/2}) + \\psi (x^{1/3}) + \\psi (x^{1/5})\\).","positives":["PrimeNumberTheoremAnd/costa-pereira-sublemma-1-7"],"positive_ids":[1532]}
{"query_key":"ClassFieldTheory/def:ind₁ to coind₁","query_node_id":1744,"repo":"ClassFieldTheory","query_text":"There is a morphism of representations \\(\\mathrm{ind}_1(G,A) \\to \\mathrm{coind}_1(G,A)\\), which takes a finitely supported function \\(f : G \\to _0 A\\) to the function \\(f\\). If the group \\(G\\) is finite then this map is an isomorphism. More precisely, this is an isomorphism of functors \\(\\mathrm{ind}_1(G,-) \\cong \\mathrm{coind}_1(G,-)\\).","positives":["ClassFieldTheory/def:induced"],"positive_ids":[3596]}
{"query_key":"brownian-motion/lem:IsGaussian.ext_iff","query_node_id":1317,"repo":"brownian-motion","query_text":"Two Gaussian measures \\(\\mu \\) and \\(\\nu \\) on a separable Hilbert space are equal if and only if they have same mean and same covariance.","positives":["brownian-motion/lem:IsGaussian.charFun_eq","brownian-motion/thm:ext_of_charFun"],"positive_ids":[1907,415]}
{"query_key":"ClassFieldTheory/lem:induced trivial Tate","query_node_id":3513,"repo":"ClassFieldTheory","query_text":"If the group \\(G\\) is finite then \\(\\mathrm{ind}_1(G,A)\\) and \\(\\mathrm{coind}_1(G,A)\\) have trivial Tate cohomology.","positives":["ClassFieldTheory/lem:ind₁ trivial homology","ClassFieldTheory/def:ind₁ to coind₁","ClassFieldTheory/lem:Tate cohomology is cohomology or homology","ClassFieldTheory/lem:coind₁ trivial cohomology"],"positive_ids":[1014,1744,3774,2412]}
{"query_key":"carleson/Hardy-Littlewood","query_node_id":1224,"repo":"carleson","query_text":"Let \\(\\mathcal{B}\\) be a finite collection of balls in \\(X\\). If for some \\(\\lambda {\\gt}0\\) and some measurable function \\(u:X\\to [0,\\infty )\\) we have \\begin{equation} \\label{eq-ball-assumption} \\int _{B} u(x)\\, d\\mu (x)\\ge \\lambda \\mu (B) \\end{equation}","positives":["carleson/layer-cake-representation","carleson/covering-separable-space"],"positive_ids":[3673,2235]}
{"query_key":"ABC-Exceptions/prop:GeometryofNumbers","query_node_id":197,"repo":"ABC-Exceptions","query_text":"Let \\(d\\geq 1\\) and \\(\\varepsilon {\\gt}0\\) be fixed, and let \\[ X_1,\\ldots , X_d, Y_1,\\ldots , Y_d,Z_1,\\ldots , Z_d\\geq 1. \\]","positives":["ABC-Exceptions/thm:HeathBrownGeometry"],"positive_ids":[204]}
{"query_key":"Sphere-Packing-Lean/cor:ineqBnew","query_node_id":769,"repo":"Sphere-Packing-Lean","query_text":"212 holds.","positives":["Sphere-Packing-Lean/lemma:F-G-pos","Sphere-Packing-Lean/prop:Qdec","Sphere-Packing-Lean/lemma:Qlim"],"positive_ids":[1586,1923,679]}
{"query_key":"equational_theories/eq168","query_node_id":1336,"repo":"equational_theories","query_text":"Equation 168 is the law \\(0 \\simeq (1 \\diamond 0) \\diamond (0 \\diamond 2)\\) (or the equation \\(x = (y \\diamond x) \\diamond (x \\diamond z)\\)).","positives":["equational_theories/magma-def"],"positive_ids":[3408]}
{"query_key":"HoTTLean/defn:ProductsExponentials","query_node_id":801,"repo":"HoTTLean","query_text":"In the natural model we can construct these by considering first the map \\[ (\\mathsf{fst}, \\mathsf{snd}) : \\mathsf{Ty}_i \\times \\mathsf{Ty}_j \\to P_{\\mathsf{tp}_i}{\\mathsf{Ty}_j} \\]","positives":["HoTTLean/defn:NaturalModelPi","HoTTLean/defn:NaturalModelSigma","HoTTLean/prop:UvPoly.equiv"],"positive_ids":[2866,2870,3552]}
{"query_key":"carleson/linearized-truncation","query_node_id":2342,"repo":"carleson","query_text":"Let \\(\\sigma _1,\\sigma _2\\colon X\\to \\mathbb {Z}\\) be measurable functions with finite range and \\(\\sigma _1\\leq \\sigma _2\\). Then we have \\begin{equation} \\label{Sqlin} \\int \\mathbf{1}_{G}(x) \\left| {T}_{2,\\sigma _1,\\sigma _2}f(x)\\right|\\, d\\mu (x) \\le \\frac{2^{442a^3+2}}{(q-1)^6} \\mu (G)^{1-\\frac{1}{q}} \\mu (F)^{\\frac{1}{q}}, \\end{equation}","positives":["carleson/finitary-Carleson"],"positive_ids":[1008]}
{"query_key":"ChebyshevCircles/lem:chebyshev_roots","query_node_id":619,"repo":"ChebyshevCircles","query_text":"Let \\(\\xi _k = \\cos ((2k + 1)\\pi /(2N))\\) for \\(k = 0, \\ldots , N-1\\) denote the roots of \\(T_N(x)\\). Then for any \\(1 \\leq j {\\lt} N\\), \\begin{equation} \\sum _{k=0}^{N-1} \\xi _k^j = \\begin{cases} \\frac{N}{2^j} \\binom {j}{j/2} & \\text{if } j \\text{ is even}, \\\\ 0 & \\text{if } j \\text{ is odd.} \\end{cases} \\end{equation}","positives":["ChebyshevCircles/lem:chebyshev_angle_sum_odd","ChebyshevCircles/lem:chebyshev_angle_sum_even","ChebyshevCircles/lem:chebyshev_roots_characterization"],"positive_ids":[987,1981,2779]}
{"query_key":"toric/0-spec-cocomm-bialg","query_node_id":2838,"repo":"toric","query_text":"If \\(A\\) is a cocommutative bialgebra over \\(R\\), then \\(\\operatorname{Spec}A\\) is a commutative monoid scheme.","positives":["toric/0-spec-hopf"],"positive_ids":[1084]}
{"query_key":"equational_theories/confluent-theory","query_node_id":2924,"repo":"equational_theories","query_text":"Let \\(\\Gamma \\) be a theory. A word \\(w\\) can be reduced to another \\(w'\\) if one can get from \\(w\\) to \\(w'\\) by a series of substitutions of laws in \\(\\Gamma \\), where no substitution increases the length of the word this is a working definition, might not be the best one to keep.. A theory \\(\\Gamma \\) is confluent if whenever a word \\(w\\) can be reduced to both \\(w'\\) and \\(w''\\), then both \\(w'\\) and \\(w''\\) can be reduced further to a common reduction \\(\\tilde w\\). As such, each word \\(w \\in M_X\\) should have a unique simplification to a reduced word \\(w_\\Gamma \\) in some normal form, for instance the shortest reduction that is minimal with respect to some suitable ordering such as lexicographical ordering.","positives":["equational_theories/free-magma-def"],"positive_ids":[780]}
{"query_key":"PrimeNumberTheoremAnd/lem:abadeulmac'","query_node_id":3193,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(b{\\gt}0\\), \\(b\\in \\mathbb {Z}\\). Then, for all \\(s\\in \\mathbb {C}\\setminus \\{ 1\\} \\) with \\(\\Re s {\\gt} 0\\), \\begin{equation} \\label{eq:abak1'} \\sum _{n \\leq b} \\frac{1}{n^s} = \\zeta (s) + \\frac{b^{1-s}}{1-s} + \\frac{b^{-s}}{2} + s \\int _b^\\infty \\left(\\{ y\\} -\\frac{1}{2}\\right) \\frac{dy}{y^{s+1}}. \\end{equation}","positives":["PrimeNumberTheoremAnd/Zeta0EqZeta"],"positive_ids":[750]}
{"query_key":"FLT_ch7/instCentreAction","query_node_id":1167,"repo":"FLT_ch7","query_text":"Thus the centre \\(\\mathbb {Z}_n\\) of this universal enveloping algebra also acts on the smooth complex functions.","positives":["FLT_ch7/instUniversalEnvelopingAlgebraAction"],"positive_ids":[3462]}
{"query_key":"polytopes-in-LEAN/lem:ambient_is_primspace","query_node_id":3168,"repo":"polytopes-in-LEAN","query_text":"","positives":["polytopes-in-LEAN/def:primspace"],"positive_ids":[195]}
{"query_key":"PrimeNumberTheoremAnd/bklnw-prop-3-sub-7","query_node_id":3309,"repo":"PrimeNumberTheoremAnd","query_text":"\\(f(x) \\leq f(x_0)\\) for \\(x \\in [x_0, 2^{\\lfloor \\frac{\\log x_0}{\\log 2} \\rfloor + 1})\\).","positives":["PrimeNumberTheoremAnd/bklnw-prop-3-sub-2"],"positive_ids":[2417]}
{"query_key":"PrimeNumberTheoremAnd/ResidueOfTendsTo","query_node_id":1280,"repo":"PrimeNumberTheoremAnd","query_text":"If a function \\(f\\) is holomorphic in a neighborhood of \\(p\\) and \\(\\lim _{s\\to p} (s-p)f(s) = A\\), then \\(f(s) = \\frac{A}{s-p} + O(1)\\) near \\(p\\).","positives":["PrimeNumberTheoremAnd/existsDifferentiableOn_of_bddAbove"],"positive_ids":[3650]}
{"query_key":"FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact","query_node_id":829,"repo":"FLT_ch11","query_text":"\\(C\\) is compact.","positives":["FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.C","FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite","FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact"],"positive_ids":[1338,351,283]}
{"query_key":"pfr/rho-cond-invariant","query_node_id":2118,"repo":"pfr","query_text":"For any \\(s\\in G\\), \\(\\rho (X+s|Y)=\\rho (X|Y)\\).","positives":["pfr/rho-invariant","pfr/rho-cond-def"],"positive_ids":[956,1304]}
{"query_key":"Sphere-Packing-Lean/cor:logder-disc-E2","query_node_id":3508,"repo":"Sphere-Packing-Lean","query_text":"\\begin{equation} \\label{eqn:logder-disc-E2} \\Delta ' = E_2 \\Delta . \\end{equation}","positives":["Sphere-Packing-Lean/thm:ramanujan-formula"],"positive_ids":[3453]}
{"query_key":"M2/def:stalk","query_node_id":11,"repo":"M2","query_text":"If \\(\\mathcal{F}\\) is a presheaf on \\(X\\) and \\(p\\in X\\) then the stalk of \\(\\mathcal{F}\\) at \\(p\\) is the abelian group \\(\\mathcal{F}_p:=\\varinjlim \\limits _{p\\in U\\text{ open}}\\mathcal{F}(U)\\).","positives":["M2/def:pre_sheaves"],"positive_ids":[1352]}
{"query_key":"radii-polynomial/def:ScaledReal","query_node_id":766,"repo":"radii-polynomial","query_text":"For \\(\\nu {\\gt} 0\\) and \\(n \\in \\mathbb {N}\\), the type \\(\\texttt{ScaledReal}\\ \\nu \\ n\\) is \\(\\mathbb {R}\\) equipped with the scaled norm \\(\\| x\\| _n := |x| \\cdot \\nu ^n\\).","positives":["radii-polynomial/def:posreal"],"positive_ids":[3159]}
{"query_key":"M2/lem:hocolim_preserves_homotopy_k_sheaves","query_node_id":1236,"repo":"M2","query_text":"","positives":["M2/lem:homotpy_k_sheaves_stable_by_extension","M2/def:homotopy_colimit"],"positive_ids":[2106,85]}
{"query_key":"ClassFieldTheory/lem:herbrand ses","query_node_id":1186,"repo":"ClassFieldTheory","query_text":"Suppose we have a short exact sequence of representations of a finite cyclic group \\(G\\): \\[ 0 \\to A \\to B \\to C \\to 0. \\]","positives":["ClassFieldTheory/cor:periodic cohomology","ClassFieldTheory/def:herbrand quotient"],"positive_ids":[2898,3512]}
{"query_key":"PrimeNumberTheoremAnd/isIntegrable","query_node_id":1832,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(x{\\gt}0\\) and \\(\\sigma \\in \\mathbb {R}\\). Then \\[ \\int _{\\mathbb {R}}\\frac{x^{\\sigma +it}}{(\\sigma +it)(1+\\sigma + it)}dt \\]","positives":["PrimeNumberTheoremAnd/isHolomorphicOn"],"positive_ids":[3773]}
{"query_key":"carleson/estimate-bad","query_node_id":1113,"repo":"carleson","query_text":"We have \\begin{equation*} \\mu \\left({\\{ x\\in X: |T_r b(x)|{\\gt}\\alpha /2\\} }\\right) \\le \\frac{\\frac{2^{7a}}{c} + 2^{a^3+11a+4}}{\\alpha } \\int |f(y)|\\, d\\mu (y) \\, . \\end{equation*}","positives":["carleson/estimate-F-set","carleson/estimate-bad-partial"],"positive_ids":[329,113]}
{"query_key":"PrimeNumberTheoremAnd/ShiftTwoBound","query_node_id":2182,"repo":"PrimeNumberTheoremAnd","query_text":"For all \\(\\delta \\in (0,1)\\) and \\(t\\in \\mathbb {R}\\) with \\(|t|\\geq 2\\) we have \\[ -\\Re \\left(\\frac{\\zeta '}{\\zeta }(1+\\delta +2it)\\right)\\ll \\log |t|. \\]","positives":["PrimeNumberTheoremAnd/SumBoundI"],"positive_ids":[2766]}
{"query_key":"toric/0-grp-like-lin-indep","query_node_id":1592,"repo":"toric","query_text":"The group-like elements in a bialgebra \\(A\\) over a domain are linearly independent.","positives":["toric/0-tensor-lin-indep","toric/0-grp-like"],"positive_ids":[1874,2697]}
{"query_key":"toric/0-grp-alg-grp-like-gc","query_node_id":1778,"repo":"toric","query_text":"Let \\(R\\) be a domain, \\(G\\) a commutative group and \\(A\\) a \\(R\\)-bialgebra. Then bialgebra homs \\(R[G] \\to A\\) are in bijection with group homs \\(G \\to \\operatorname{GrpLike}A\\).","positives":["toric/0-grp-like-grp-alg","toric/0-grp-like-map"],"positive_ids":[2102,774]}
{"query_key":"equational_theories/eq39","query_node_id":1257,"repo":"equational_theories","query_text":"Equation 39 is the law \\(0 \\diamond 0 \\simeq 1 \\diamond 0\\) (or the equation \\(x \\diamond x = y \\diamond x\\)).","positives":["equational_theories/magma-def"],"positive_ids":[3408]}
{"query_key":"BonnAnalysis/def:fourier-L2","query_node_id":3138,"repo":"BonnAnalysis","query_text":"Let \\(f\\in L^2(V,E)\\) and take a sequence \\((f_n)_n\\subset L^1(V,E)\\cap L^2(V,E)\\) with \\(f_n\\xrightarrow [L^2]{}f\\). Set \\[ \\mathcal Ff:=\\widehat f:=\\lim _{n\\to \\infty }\\widehat{f_n}, \\]","positives":["BonnAnalysis/lem:fourier12-cauchy"],"positive_ids":[812]}
{"query_key":"ClassFieldTheory/cor:cohomology sub Sylow","query_node_id":32,"repo":"ClassFieldTheory","query_text":"Let \\(M\\) be a representation of a finite group \\(G\\) and let \\(S_p\\) be a Sylow \\(p\\)-subgroup of \\(G\\) for some prime number \\(p\\). Then for any \\(n \\in {\\mathbb Z}\\), \\(H^n_{\\mathrm{Tate}}(G,M)[p^\\infty ]\\) is isomorphic to an \\(R\\)-submodule of \\(H^n_{\\mathrm{Tate}}(S_p,M)\\).","positives":["ClassFieldTheory/cor:Tate up down isos","ClassFieldTheory/lem:cor comp rest"],"positive_ids":[3586,1077]}
{"query_key":"ClassFieldTheory/lem:cohomology S-ideles decomp","query_node_id":416,"repo":"ClassFieldTheory","query_text":"There are isomorphisms for all \\(n {\\gt} 0\\) \\[ H^n(l/k, {\\mathbb A}_{S,l}^\\times ) \\cong \\prod _{v \\in S} H^n(l_{\\hat v} / k_v, l_{\\hat v}^\\times ). \\]","positives":["ClassFieldTheory/lem:unramified units trivial","ClassFieldTheory/lem:semi-local iso coind"],"positive_ids":[3500,3660]}
{"query_key":"brownian-motion/lem:logSizeBallSequence_eq_zero","query_node_id":406,"repo":"brownian-motion","query_text":"For any log-size ball sequence \\((V_i, t_i, r_i)_{i \\in \\mathbb {N}}\\) for \\((J, a, c, n)\\), for all \\(k \\ge \\vert J \\vert \\), \\(V_k = \\emptyset \\).","positives":["brownian-motion/lem:logSizeBallSequence_V_anti"],"positive_ids":[2847]}
{"query_key":"polytopes-in-LEAN/lem:ch_of_empty_is_empty","query_node_id":1057,"repo":"polytopes-in-LEAN","query_text":"","positives":["polytopes-in-LEAN/def:convex_hull","polytopes-in-LEAN/lem:empty_is_convex"],"positive_ids":[2525,1919]}
{"query_key":"exchangeability/lem:blockAvg_tendsto","query_node_id":2132,"repo":"exchangeability","query_text":"For a shift-invariant measure, block averages converge in \\(L^1\\) to the conditional expectation given the shift-invariant \\(\\sigma \\)-algebra: \\(\\int |A_{m,n,k}(f) - \\mathbb {E}[f \\circ \\pi _0 \\mid \\mathcal{I}]| \\, d\\mu \\to 0\\) as \\(n \\to \\infty \\).","positives":["exchangeability/thm:birkhoff_condexp","exchangeability/def:blockAvg"],"positive_ids":[1748,435]}
{"query_key":"pfr/rho-sums","query_node_id":2724,"repo":"pfr","query_text":"If \\(X,Y\\) are independent, one has \\[ \\rho ^-(X+Y) \\leq \\rho ^-(X) \\]","positives":["pfr/kl-sums"],"positive_ids":[919]}
{"query_key":"PrimeNumberTheoremAnd/bound-score-1","query_node_id":2441,"repo":"PrimeNumberTheoremAnd","query_text":"If \\(M\\) is sufficiently large depending on \\(\\varepsilon \\), then \\(n \\log (1-1/M)^{-1} \\leq \\varepsilon n\\).","positives":["PrimeNumberTheoremAnd/params-set"],"positive_ids":[10]}
{"query_key":"carleson/partial-Fourier-sum-bound","query_node_id":3346,"repo":"carleson","query_text":"Let \\(g:{\\mathbb {R}}\\to {\\mathbb {C}}\\) be a measurable \\(2\\pi \\)-periodic function such that for some \\(\\delta {\\gt}0\\) and every \\(x\\in {\\mathbb {R}}\\), \\begin{equation} |g(x)|\\le \\delta \\, . \\end{equation}","positives":["carleson/Dirichlet-Hilbert"],"positive_ids":[1462]}
{"query_key":"brownian-motion/lem:charFun_gaussianReal","query_node_id":2762,"repo":"brownian-motion","query_text":"The characteristic function of a real Gaussian measure with mean \\(\\mu \\) and variance \\(\\sigma ^2\\) is given by \\(x \\mapsto \\exp \\left(i \\mu x - \\frac{\\sigma ^2 x^2}{2}\\right)\\).","positives":["brownian-motion/def:gaussianReal","brownian-motion/def:charFun"],"positive_ids":[1801,1576]}
{"query_key":"pfr/submodularity","query_node_id":552,"repo":"pfr","query_text":"With three random variables \\(X,Y,Z\\), one has \\(\\mathbb {H}[X|Y,Z] \\leq \\mathbb {H}[X|Z]\\).","positives":["pfr/cond-reduce"],"positive_ids":[2666]}
{"query_key":"PrimeNumberTheoremAnd/lem:sigmaLn","query_node_id":1788,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(L'\\) be as in Lemma 10.1.2. Then \\begin{equation} \\label{eq:sigmaLn} \\frac{\\sigma (L_n)}{L_n} \\; =\\; \\frac{\\sigma (L')}{L'} \\prod _{i=1}^3 \\Bigl(1 + \\frac{1}{q_i}\\Bigr). \\end{equation}","positives":["PrimeNumberTheoremAnd/lem:Lprime-def","PrimeNumberTheoremAnd/sigma-def"],"positive_ids":[285,387]}
{"query_key":"brownian-motion/lem:Submartingale.integrable_stoppedValue","query_node_id":460,"repo":"brownian-motion","query_text":"Let \\(X\\) be a submartingale. Then for all bounded stopping times \\(\\tau \\), the stopped value \\(X_\\tau \\) is integrable.","positives":["brownian-motion/def:Submartingale","brownian-motion/def:stoppedProcess"],"positive_ids":[542,915]}
{"query_key":"brownian-motion/lem:doob_Lp_countable","query_node_id":331,"repo":"brownian-motion","query_text":"Let \\(X : I \\rightarrow \\Omega \\rightarrow \\mathbb {R}\\) be a non-negative sub-martingale. Let \\(I\\) be countable. For every \\(M\\in I,\\lambda {\\gt} 0\\) and \\(p{\\gt}1\\) we have \\begin{align*} \\mathbb {E}\\left[ \\sup _{i\\in I, i \\leq M}X_i^p \\right] \\leq \\left(\\frac{p}{p-1}\\right)^p\\mathbb {E}[X_M^p] \\: . \\end{align*}","positives":["brownian-motion/lem:doob_countable"],"positive_ids":[1108]}
{"query_key":"brownian-motion/lem:posSemidef_gramMatrix","query_node_id":1807,"repo":"brownian-motion","query_text":"A gram matrix is positive semidefinite.","positives":["brownian-motion/def:gramMatrix"],"positive_ids":[906]}
{"query_key":"brownian-motion/cor:doob_ineq_stop_exp_val","query_node_id":608,"repo":"brownian-motion","query_text":"Let \\(X:\\mathbb {R}\\times \\Omega \\rightarrow E\\) be a right-continuous martingale with values in a normed space \\(E\\). For every \\(\\lambda {\\gt}0\\) and \\(p{\\gt}1\\) and \\(\\tau \\) stopping time a.s. bounded by \\(T{\\gt}0\\), we have \\[ \\mathbb {E}\\left[ \\sup _{t\\in [0,\\tau ]}\\lVert X_t \\rVert ^p \\right]\\leq \\left(\\frac{p}{p-1}\\right)^p\\mathbb {E}[\\lVert X_\\tau \\rVert ^p]. \\]","positives":["brownian-motion/cor:Martingale.submartingale_norm","brownian-motion/lem:doob_ineq_stop_exp_val"],"positive_ids":[167,454]}
{"query_key":"BrauerGroup_new/thm:wed-artin-algebra","query_node_id":649,"repo":"BrauerGroup_new","query_text":"Let \\(K\\) be a field and \\(B\\) an finite dimensional simple algebra over \\(K\\). There exists a non-zero \\(n\\in \\mathbb {N}\\) and a division \\(K\\)-algebra \\(S\\) such that \\(B \\cong \\operatorname{Mat}_{n}(S)\\).","positives":["BrauerGroup_new/con:end-vec-iso-matrix","BrauerGroup_new/thm:wed-artin-ideal"],"positive_ids":[1094,2164]}
{"query_key":"ABC-Exceptions/lem:Subcase2.4","query_node_id":233,"repo":"ABC-Exceptions","query_text":"\\(\\nu \\leq 0.66\\) in the case \\(s.2 {\\lt} 0.3\\) and \\(4s_1+s_2{\\lt}0.4\\).","positives":["ABC-Exceptions/lem:Subcase2.2","ABC-Exceptions/lem:Case2Basic2"],"positive_ids":[1416,3789]}
{"query_key":"Sphere-Packing-Lean/prop:H2-fourier","query_node_id":485,"repo":"Sphere-Packing-Lean","query_text":"\\(H_2\\) admits a Fourier series of the form \\begin{equation} H_2(z) = \\sum _{n \\ge 1} c_{H_2}(n) e^{\\pi i n z} \\end{equation}","positives":["Sphere-Packing-Lean/def:H2-H3-H4"],"positive_ids":[3629]}
{"query_key":"FLT_ch10/MeasureTheory.addEquivAddHaarChar_eq","query_node_id":3578,"repo":"FLT_ch10","query_text":"\\(d_A(\\phi )\\) is independent of choice of regular Haar measure.","positives":["FLT_ch10/MeasureTheory.addEquivAddHaarChar"],"positive_ids":[818]}
{"query_key":"brownian-motion/lem:komlos_aux","query_node_id":180,"repo":"brownian-motion","query_text":"Let \\(H\\) be a Hilbert space and \\((f_n)_{n\\in \\mathbb {N}}\\) a bounded sequence in \\(H\\). Then there exist functions \\(g_n\\in convex(f_n,f_{n+1},\\cdots )\\) such that \\((g_n)_{n\\in \\mathbb {N}}\\) converges in \\(H\\).","positives":["brownian-motion/lem:komlos_convex"],"positive_ids":[922]}
{"query_key":"ABC-Exceptions/thm:BdExceptionalBound","query_node_id":2468,"repo":"ABC-Exceptions","query_text":"There exists \\(\\varepsilon {\\gt}0\\) such that for all \\(\\mathbf{c}\\in \\mathbb {Z}^3\\) and \\(\\mathbf{X},\\mathbf{Y},\\mathbf{Z}\\in \\mathbb {R}_{{\\gt}0}^{d}\\). we have \\begin{align} B_d({\\bf c, X,Y,Z}) \\ll X^{0.66-\\varepsilon }. \\end{align}","positives":["ABC-Exceptions/thm:ABCExponentNu"],"positive_ids":[1165]}
{"query_key":"radii-polynomial/def:sq_map","query_node_id":365,"repo":"radii-polynomial","query_text":"The squaring map \\(\\text{sq}: \\ell ^1_\\nu \\rightarrow \\ell ^1_\\nu \\) is \\(\\text{sq}(a) = a \\star a\\).","positives":["radii-polynomial/def:l1Weighted","radii-polynomial/def:CauchyProduct"],"positive_ids":[749,3226]}
{"query_key":"toric/0-diag-hom","query_node_id":72,"repo":"toric","query_text":"Let \\(R\\) be a domain. The functor \\(G \\rightsquigarrow \\operatorname{Spec}R[G]\\) from the category of groups to the category of group schemes over \\(\\operatorname{Spec}R\\) is a group isomorphism on hom sets.","positives":["toric/0-full-faithful-diag","toric/0-full-faithful-grp-hom"],"positive_ids":[513,1058]}
{"query_key":"ABC-Exceptions/thm:BombierSchmidtforThueEqs","query_node_id":3601,"repo":"ABC-Exceptions","query_text":"Given integers \\(n,a_1,a_2\\neq 0\\) and \\(p\\ge 3\\), there are at most \\(O(p^{1+\\omega (|n|)})\\) many solutions \\((x_1,x_2)\\in \\mathbb {Z}^2\\) such that \\(|x_i|\\le X_i\\) and \\begin{align*} n = a_1 x_1^p + a_2 x_2^p. \\end{align*}","positives":["ABC-Exceptions/def:omega"],"positive_ids":[265]}
{"query_key":"ClassFieldTheory/lem:herbrand local units","query_node_id":50,"repo":"ClassFieldTheory","query_text":"If \\(l/k\\) is a cyclic extension then \\(h(l/k, {\\mathcal O}_l^\\times ) = 1\\).","positives":["ClassFieldTheory/lem:herbrand ses","ClassFieldTheory/lem:serre_cor","ClassFieldTheory/lem:herbrand finite"],"positive_ids":[1186,2821,2394]}
{"query_key":"FormalBook/cayley_formala_proof4","query_node_id":391,"repo":"FormalBook","query_text":"There are \\(n^{n - 2}\\) different labeled trees on \\(n\\) nodes.","positives":["FormalBook/double_counting"],"positive_ids":[2828]}
{"query_key":"carleson/lower-oscillation-bound","query_node_id":3015,"repo":"carleson","query_text":"For all \\(J \\in \\mathcal{J}'\\), we have that \\[ d_{B(J)}({\\mathcal{Q}}({\\mathfrak u}_1), {\\mathcal{Q}}({\\mathfrak u}_2)) \\ge 2^{-201a^3} 2^{Zn/2}\\, . \\]","positives":["carleson/overlap-implies-distance"],"positive_ids":[578]}
{"query_key":"toric/1-2-6-face-face","query_node_id":2695,"repo":"toric","query_text":"A face of a face of a polyhedral cone \\(\\sigma \\) is again a face of \\(\\sigma \\).","positives":["toric/1-2-5-face","toric/1-2-1-polyhedral-cone"],"positive_ids":[1233,2279]}
{"query_key":"PrimeNumberTheoremAnd/fks2-lemma-20a","query_node_id":2252,"repo":"PrimeNumberTheoremAnd","query_text":"The function \\(\\operatorname {Li}(x) - \\frac{x}{\\log x}\\) is strictly increasing for \\(x {\\gt} 6.58\\).","positives":["PrimeNumberTheoremAnd/li-def"],"positive_ids":[3275]}
{"query_key":"LeanAPAP/flat-on-bohr","query_node_id":1944,"repo":"LeanAPAP","query_text":"There is a constant \\(c{\\gt}0\\) such that the following holds. Let \\({\\epsilon ,\\delta \\in (0,1)}\\) and \\(p,k\\geq 1\\) be integers such that \\((k,\\left\\lvert G\\right\\rvert )=1\\). For any \\(A\\subseteq G\\) with density \\(\\alpha \\) there is a regular Bohr set \\(B\\) with \\[ d=\\operatorname{rank}(B) =O_{\\epsilon }\\left(\\mathcal{L}{\\alpha }^5p^4\\right) \\quad \\text{and}\\quad \\left\\lvert B\\right\\rvert \\geq \\exp \\left(-O_{\\epsilon ,\\delta }(\\mathcal{L}{\\alpha }^6p^5\\mathcal{L}{\\alpha /p})\\right)\\left\\lvert G\\right\\rvert \\]","positives":["LeanAPAP/prop-it","LeanAPAP/bourgain-trick"],"positive_ids":[3324,3681]}
{"query_key":"equational_theories/non_imp_3994_3588_thm","query_node_id":1161,"repo":"equational_theories","query_text":"There exists a magma which satisfies E3994 and not E3588.","positives":["equational_theories/models-def","equational_theories/eq3994","equational_theories/eq3588"],"positive_ids":[1745,3165,3511]}
{"query_key":"brownian-motion/lem:martingale_integral_of_forall_eq_zero","query_node_id":1506,"repo":"brownian-motion","query_text":"Let \\(X \\in L^2(M)\\) such that \\(\\langle X, V\\rangle = 0\\) for any simple process \\(V\\). Let \\(A_t = \\int _0^t X_s \\: d\\langle M \\rangle _s\\). Then \\(A_t\\) is a martingale.","positives":["brownian-motion/def:L2M","brownian-motion/lem:martingale_iff_integral_elemStochIntegral_eq_zero"],"positive_ids":[300,3030]}
{"query_key":"polytopes-in-LEAN/def:convex_hull","query_node_id":2525,"repo":"polytopes-in-LEAN","query_text":"The convex hull of a set \\(A\\) is the intersection of all convex subsets containing \\(A\\).","positives":["polytopes-in-LEAN/def:is_convex"],"positive_ids":[3198]}
{"query_key":"ClassFieldTheory/lem:local image inertia","query_node_id":2859,"repo":"ClassFieldTheory","query_text":"Let \\(l/k\\) be a finite abelian extension of local fields and let \\(I \\subseteq \\mathrm{Gal}(l/k)\\) be the inertia subgroup. Then the image of \\(I\\) in \\(k^\\times / N(l^\\times )\\) is the subgroup \\({\\mathcal O}_k^\\times / N({\\mathcal O}_l^\\times )\\).","positives":["ClassFieldTheory/lem:subgroup compatibility","ClassFieldTheory/lem:local unramified norms"],"positive_ids":[3664,1429]}
{"query_key":"carleson/correlation-distant-tree-parts","query_node_id":3099,"repo":"carleson","query_text":"We have for all \\({\\mathfrak u}_1 \\ne {\\mathfrak u}_2 \\in {\\mathfrak U}\\) with \\({\\mathcal{I}}({\\mathfrak u}_1) \\subset {\\mathcal{I}}({\\mathfrak u}_2)\\) and all bounded \\(g_1, g_2\\) with bounded support \\begin{equation} \\label{eq-lhs-big-sep-tree} \\left| \\int _X \\sum _{{\\mathfrak p}_1 \\in {\\mathfrak T}({\\mathfrak u}_1)} \\sum _{{\\mathfrak p}_2 \\in {\\mathfrak T}({\\mathfrak u}_2) \\cap \\mathfrak {S}} T^*_{{\\mathfrak p}_1}g_1 \\overline{T^*_{{\\mathfrak p}_2}g_2 }\\, \\mathrm{d}\\mu \\right| \\end{equation}","positives":["carleson/Lipschitz-partition-unity","carleson/Holder-van-der-Corput","carleson/lower-oscillation-bound","carleson/Holder-correlation-tree"],"positive_ids":[522,943,3015,1978]}
{"query_key":"pfr/neg-ent","query_node_id":2537,"repo":"pfr","query_text":"If \\(X\\) is \\(G\\)-valued, then \\(\\mathbb {H}[-X]=\\mathbb {H}[X]\\).","positives":["pfr/relabeled-entropy"],"positive_ids":[3480]}
{"query_key":"PrimeNumberTheoremAnd/prop:applem","query_node_id":2796,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(s = \\sigma + i \\tau \\), \\(\\sigma \\geq 0\\), \\(\\tau \\in \\mathbb {R}\\). Let \\(a,b\\in \\mathbb {Z} + \\frac{1}{2}\\), \\(b{\\gt}a{\\gt}\\frac{|\\tau |}{2\\pi }\\). Write \\(\\vartheta = \\frac{\\tau }{2\\pi a}\\). Then, for any integer \\(n\\geq 1\\), \\[ \\begin{aligned} \\int _a^b t^{-s} \\cos 2\\pi n t\\, dt & = \\left. \\left(\\frac{(-1)^n t^{-s}}{2\\pi i} \\cdot \\frac{\\frac{\\tau }{2\\pi t}}{n^2 - \\left(\\frac{\\tau }{2\\pi t}\\right)^2}\\right)\\right|_a^b \\\\ & \\quad + \\frac{a^{-\\sigma -1}}{4\\pi ^2}\\, O^*\\left(\\frac{\\sigma }{(n-\\vartheta )^2} + \\frac{\\sigma }{(n+\\vartheta )^2} + \\frac{|\\vartheta |}{|n-\\vartheta |^3} + \\frac{|\\vartheta |}{|n+\\vartheta |^3}\\right).\\end{aligned} \\]","positives":["PrimeNumberTheoremAnd/lem:aachfour","PrimeNumberTheoremAnd/lem:aachcanc"],"positive_ids":[2880,3292]}
{"query_key":"PrimeNumberTheoremAnd/Dusart_prop_5_11a","query_node_id":2170,"repo":"PrimeNumberTheoremAnd","query_text":"We have for \\(p_k \\geq 10^{11}\\), \\[ \\vartheta (p_k) \\geq k \\log k + \\log \\log k - 1 + \\frac{\\log \\log k - 2.050735}{\\log k}. \\]","positives":["PrimeNumberTheoremAnd/theta-def"],"positive_ids":[108]}
{"query_key":"FormalBook/four_proofs_euler_series","query_node_id":3563,"repo":"FormalBook","query_text":"Collecting the proofs from the chapter.","positives":["FormalBook/euler_series_2","FormalBook/euler_series_4","FormalBook/euler_series","FormalBook/euler_series_3"],"positive_ids":[2444,2181,593,2282]}
{"query_key":"radii-polynomial/lem:norm_pow_le","query_node_id":2974,"repo":"radii-polynomial","query_text":"For \\(a \\in \\ell ^1_\\nu \\) and \\(n \\in \\mathbb {N}\\): \\begin{equation*} \\| a^n\\| _{1,\\nu } \\leq \\| a\\| _{1,\\nu }^n. \\end{equation*}","positives":["radii-polynomial/thm:l1Weighted_NormedRing"],"positive_ids":[364]}
{"query_key":"toric/0-spec-hopf","query_node_id":1084,"repo":"toric","query_text":"Spec is a contravariant functor from the category of \\(R\\)-Hopf algebras to the category of group schemes over \\(\\operatorname{Spec}_R\\).","positives":["toric/0-full-faithful-spec-alg"],"positive_ids":[2178]}
{"query_key":"PrimeNumberTheoremAnd/tendsto_rpow_atTop_nhds_zero_of_norm_gt_one","query_node_id":1249,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(x{\\gt}1\\). Then \\[ \\lim _{\\sigma \\to -\\infty }x^\\sigma =0. \\]","positives":["PrimeNumberTheoremAnd/tendsto_rpow_atTop_nhds_zero_of_norm_lt_one"],"positive_ids":[2656]}
{"query_key":"infinite-primes-via-log/thm:infinite_primes","query_node_id":2592,"repo":"infinite-primes-via-log","query_text":"The set \\(\\mathbb {P} = \\big\\{ p \\in \\mathbb {N} \\ \\mid \\ p \\ \\mathrm{prime} \\big\\} \\) is infinite.","positives":["infinite-primes-via-log/thm:primeCountingReal_unbounded"],"positive_ids":[3319]}
{"query_key":"brownian-motion/lem:elemStochIntegral_assoc","query_node_id":2585,"repo":"brownian-motion","query_text":"\\(V \\bullet _{\\mathbb {R}} (W \\bullet _{\\mathbb {R}} X) = (V \\times W) \\bullet _{\\mathbb {R}} X\\) for real-valued simple processes \\(V, W\\) and stochastic process \\(X\\).","positives":["brownian-motion/cor:elemStochIntegral_assoc_real_bilin"],"positive_ids":[3033]}
{"query_key":"pfr/rho-invariant","query_node_id":956,"repo":"pfr","query_text":"For any \\(s \\in G\\), \\(\\rho (X+s) = \\rho (X)\\).","positives":["pfr/kl-div-inj","pfr/rho-def"],"positive_ids":[492,102]}
{"query_key":"ClassFieldTheory/def:reciprocity iso","query_node_id":2531,"repo":"ClassFieldTheory","query_text":"The theorem implies that we have isomorphisms for all \\(n\\in {\\mathbb Z}\\) (which depend on \\(\\sigma \\)): \\[ H^{n}_{\\mathrm{Tate}}(G,\\mathrm{aug}(R,G)) \\cong H^{n+1}_{\\mathrm{Tate}}(G,M) . \\]","positives":["ClassFieldTheory/thm:trivial cohomology implies trivial Tate","ClassFieldTheory/lem:Tate -1 aug","ClassFieldTheory/thm:splitting module trivial"],"positive_ids":[2449,709,538]}
{"query_key":"equational_theories/eq3994","query_node_id":3165,"repo":"equational_theories","query_text":"Equation 3994 is the law \\(0 \\diamond 1 \\simeq (2 \\diamond (0 \\diamond 1)) \\diamond 2\\) (or the equation \\(x \\diamond y = (z \\diamond (x \\diamond y)) \\diamond z\\)).","positives":["equational_theories/magma-def"],"positive_ids":[3408]}
{"query_key":"carleson/dens2-antichain","query_node_id":1394,"repo":"carleson","query_text":"We have that \\begin{equation} \\label{eqttt9} \\left|\\int \\overline{g(x)} \\sum _{{\\mathfrak p}\\in \\mathfrak {A}} T_{{\\mathfrak p}} f(x)\\, d\\mu (x)\\right|\\le 2^{103a^3}({q}-1)^{-1} \\operatorname{\\operatorname {dens}}_2(\\mathfrak {A})^{\\frac1{\\tilde{q}}-\\frac12} \\| f\\| _2\\| g\\| _2\\, . \\end{equation}","positives":["carleson/Hardy-Littlewood","carleson/maximal-bound-antichain"],"positive_ids":[1224,1203]}
{"query_key":"toric/1-2-18-dim-aff-tor-var-rat-polyhedral-cone","query_node_id":240,"repo":"toric","query_text":"\\[ \\dim U_\\sigma = \\dim N \\iff \\text{ the torus of $U_\\sigma $ is } T_N = N \\otimes _[{\\mathbb Z}] {\\mathbb C}^* \\iff \\sigma \\text{ is salient}. \\]","positives":["toric/1-1-14-spec-aff-mon-alg","toric/1-2-17-gordan-lemma","toric/1-2-12-salient-cone-tfae","toric/1-2-18-aff-tor-var-rat-polyhedral-cone"],"positive_ids":[1199,2505,1574,372]}
{"query_key":"equational_theories/eq374794","query_node_id":3448,"repo":"equational_theories","query_text":"Equation 374794 is the law \\(0 \\simeq (((1 \\diamond 1) \\diamond 1) \\diamond 0) \\diamond ((1 \\diamond 1) \\diamond 2)\\) (or the equation \\(x = (((y \\diamond y) \\diamond y) \\diamond x) \\diamond ((y \\diamond y) \\diamond z)\\)).","positives":["equational_theories/magma-def"],"positive_ids":[3408]}
{"query_key":"equational_theories/claim-4","query_node_id":1916,"repo":"equational_theories","query_text":"If \\(a \\to b \\to c \\to d \\to e \\to a\\) is a 5-cycle in the directed graph, and \\(a \\to b \\to c\\) and \\(c \\to d \\to e\\) are good paths, then \\(b \\to c \\to d\\) is also good.","positives":["equational_theories/graph-dual"],"positive_ids":[3827]}
{"query_key":"PrimeNumberTheoremAnd/LogDerivZetaFinalBound","query_node_id":602,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(t\\in \\mathbb {R}\\) with \\(|t|\\geq 2\\) and \\(0 {\\lt} r' {\\lt} r {\\lt} R' {\\lt} R{\\lt}1\\). If \\(f(z)=\\zeta (z+3/2+it)\\), then for all \\(z\\in \\overline{\\mathbb {D}_R'}\\setminus \\mathcal{K}_f(R')\\) we have that \\[ \\left|\\frac{f'}{f}(z)-\\sum _{\\rho \\in \\mathcal{K}_f(R')}\\frac{m_f(\\rho )}{z-\\rho }\\right| \\ll \\left(\\frac{16r^2}{(r-r')^3}+\\frac{1}{(R^2/R'-R')\\, \\log (R/R')}\\right)\\log |t|. \\]","positives":["PrimeNumberTheoremAnd/FinalBound","PrimeNumberTheoremAnd/GlobalBound","PrimeNumberTheoremAnd/ZetaFixedLowerBound"],"positive_ids":[3443,3175,994]}
{"query_key":"Sphere-Packing-Lean/prop:ineqB","query_node_id":2218,"repo":"Sphere-Packing-Lean","query_text":"Consider the function \\(B:(0,\\infty )\\to \\mathbb {C}\\) defined as \\begin{equation} \\label{eqn:defB} B(t) := -t^2\\phi _0(i/t)+\\frac{36}{\\pi ^2}\\, \\psi _I(it) \\end{equation}","positives":["Sphere-Packing-Lean/lemma:ineqABnew-equiv","Sphere-Packing-Lean/cor:ineqBnew"],"positive_ids":[3036,769]}
{"query_key":"M2/lem:chech_long_exact_sequence","query_node_id":589,"repo":"M2","query_text":"Let \\(0\\to \\mathcal{F}^{\\bullet }\\to \\mathcal{G}^{\\bullet }\\to \\mathcal{H}^{\\bullet }\\to 0\\) be a short exact sequence of complex of \\(\\mathcal{K}\\)-presheaves. Then there is a long exact seuquence in čech cohomology: \\[ \\ldots \\to \\check{H}^k(X;\\mathcal{F}^{\\bullet })\\to \\check{H}^k(X;\\mathcal{G}^{\\bullet })\\to \\check{H}^k(X;\\mathcal{H}^{\\bullet })\\to \\ldots \\]","positives":["M2/def:chech_of_compelx_of_k_prsh"],"positive_ids":[1631]}
{"query_key":"exchangeability/lem:directing_measure_isProb_L2","query_node_id":2676,"repo":"exchangeability","query_text":"The Stieltjes measure constructed from the limiting CDF is a probability measure a.e.","positives":["exchangeability/lem:cdf_mono","exchangeability/lem:cdf_bounds"],"positive_ids":[3655,2831]}
{"query_key":"pfr/multidist-chain-rule-cond","query_node_id":2685,"repo":"pfr","query_text":"Let \\(\\pi \\colon G \\to H\\) be a homomorphism of abelian groups. Let \\(I\\) be a finite index set and let \\(X_{[m]}\\) be a tuple of \\(G\\)-valued random variables. Let \\(Y_{[m]}\\) be another tuple of random variables (not necessarily \\(G\\)-valued). Suppose that the pairs \\((X_i, Y_i)\\) are jointly independent of one another (but \\(X_i\\) need not be independent of \\(Y_i\\)). Then \\begin{align} \\nonumber D[ X_{[m]} | Y_{[m]} ] & = D[ X_{[m]} \\, |\\, \\pi (X_{[m]}), Y_{[m]}] + D[ \\pi (X_{[m]}) \\, |\\, Y_{[m]}] \\\\ & \\quad \\qquad + \\mathbb {I}[ \\sum _{i=1}^m X_i : \\pi (X_{[m]}) \\; \\big| \\; \\pi \\bigl(\\sum _{i=1}^m X_i \\bigr), Y_{[m]} ].\\label{chain-eq-cond} \\end{align}","positives":["pfr/multidist-chain-rule"],"positive_ids":[1335]}
{"query_key":"FLT_ch5/Hurwitz.norm_one","query_node_id":1609,"repo":"FLT_ch5","query_text":"The norm of \\(1\\) is \\(1\\).","positives":["FLT_ch5/Hurwitz.norm"],"positive_ids":[1708]}
{"query_key":"GibbsMeasure/lem:lebesgue-cond-exp-char","query_node_id":2374,"repo":"GibbsMeasure","query_text":"If \\(f : X \\to [0, \\infty ]\\) is a \\(\\mathcal X\\)-measurable function, then \\(\\mu [f | \\mathcal B]\\) is the \\(\\mu \\)-ae unique \\(\\mathcal B\\)-measurable function \\(X \\to [0, \\infty ]\\) such that \\[ \\int _B \\mu [f | \\mathcal B]\\ \\partial \\mu = \\int _B f\\ \\partial \\mu \\]","positives":["GibbsMeasure/def:lebesgue-cond-exp"],"positive_ids":[3470]}
{"query_key":"brownian-motion/def:Martingale","query_node_id":1156,"repo":"brownian-motion","query_text":"Let \\(\\mathcal{F}\\) be a filtration on a measurable space \\(\\Omega \\) with measure \\(P\\) indexed by \\(T\\). A family of functions \\(M : T \\to \\Omega \\to E\\) is a martingale with respect to a filtration \\(\\mathcal{F}\\) if \\(M\\) is adapted with respect to \\(\\mathcal{F}\\) and for all \\(i \\le j\\), \\(P[M_j \\mid \\mathcal{F}_i] = M_i\\) almost surely.","positives":["brownian-motion/def:adapted"],"positive_ids":[158]}
{"query_key":"equational_theories/3744_implies_3722_381","query_node_id":84,"repo":"equational_theories","query_text":"E3744 \\((x \\diamond y = (x \\diamond z) \\diamond (w \\diamond y))\\) implies E3722 \\((x \\diamond y = (x \\diamond y) \\diamond (x \\diamond y))\\) and E381 \\((x \\diamond y = (x \\diamond z) \\diamond y)\\).","positives":["equational_theories/eq3744","equational_theories/eq381","equational_theories/eq3722"],"positive_ids":[2338,1711,2753]}
{"query_key":"equational_theories/duality","query_node_id":718,"repo":"equational_theories","query_text":"The law \\(w \\simeq w'\\) implies \\(w''\\simeq w'''\\), if and only if \\(w^{\\mathrm{op}} \\simeq (w')^{\\mathrm{op}}\\) implies \\(w''^{\\mathrm{op}} \\simeq (w''')^{\\mathrm{op}}\\).","positives":["equational_theories/pre-order"],"positive_ids":[1076]}
{"query_key":"brownian-motion/lem:predictable_quadraticVariation","query_node_id":2396,"repo":"brownian-motion","query_text":"The quadratic variation \\(\\langle M \\rangle \\) of a local martingale \\(M\\) is a predictable process.","positives":["brownian-motion/def:quadraticVariation"],"positive_ids":[1883]}
{"query_key":"M2/def:pushforward_of_K-sheaf","query_node_id":29,"repo":"M2","query_text":"If \\(\\mathcal{F}\\) is a pre-\\(\\mathcal{K}\\)-sheaf over \\(X\\), then the rule \\(K\\mapsto \\mathcal{F}(f^{-1}(K))\\) defines a pre-\\(\\mathcal{K}\\)-sheaf over \\(Y\\). The functor obtained is denoted \\(f_*\\) and named the pushforward by \\(f\\). \\(f_*\\) send \\(\\mathcal{K}\\)-sheaves over \\(X\\) into \\(\\mathcal{K}\\)-sheaves over \\(Y\\).","positives":["M2/def:pushforward_of_sheaf","M2/def:K_sheaves","M2/lem:basis_of_open_of_fm1K"],"positive_ids":[2876,293,144]}
{"query_key":"ClassFieldTheory/lem:local H2 upper bound","query_node_id":3813,"repo":"ClassFieldTheory","query_text":"Let \\(l/k\\) be a Galois extension of local fields. Then \\(|H^2(l/k,l^\\times )| \\le [l:k]\\).","positives":["ClassFieldTheory/cor:cohomology sub Sylow","ClassFieldTheory/cor:cohomology G-torsion","ClassFieldTheory/lem:local H2 l*","ClassFieldTheory/thm:inflation restriction sequence"],"positive_ids":[32,1614,2556,1533]}
{"query_key":"radii-polynomial/cor:sqrt2_unique","query_node_id":2515,"repo":"radii-polynomial","query_text":"There exists a unique \\(\\tilde{x} \\in \\overline{B}_{3/20}(13/10)\\) with \\(\\tilde{x}^2 = 2\\).","positives":["radii-polynomial/ex:sqrt2"],"positive_ids":[2035]}
{"query_key":"equational_theories/477-confl","query_node_id":1344,"repo":"equational_theories","query_text":"E477 \\((x = y \\diamond (x \\diamond (y \\diamond (y \\diamond y))))\\) is confluent.","positives":["equational_theories/eq477","equational_theories/confluent-theory","equational_theories/477-lemma"],"positive_ids":[770,2924,3401]}
{"query_key":"radii-polynomial/thm:simple_radii_poly_ef","query_node_id":2634,"repo":"radii-polynomial","query_text":"Given \\(f: E \\rightarrow F\\) Fréchet differentiable and injective \\(A: F \\rightarrow E\\) satisfying \\(\\| A(f(\\bar{x}))\\| \\leq Y_0\\), \\(\\| I_E - A \\circ \\text{Df}(\\bar{x})\\| \\leq Z_0\\), and \\(\\| A \\circ [\\text{Df}(c) - \\text{Df}(\\bar{x})]\\| \\leq Z_2(r) \\cdot r\\) for \\(c \\in \\overline{B}_r(\\bar{x})\\). If \\(p(r_0) = Z_2(r_0)r_0^2 - (1-Z_0)r_0 + Y_0 {\\lt} 0\\), then there exists unique \\(\\tilde{x} \\in \\overline{B}_{r_0}(\\bar{x})\\) with \\(f(\\tilde{x}) = 0\\).","positives":["radii-polynomial/lem:radii_poly_alt_form","radii-polynomial/lem:newton_Y_bound","radii-polynomial/thm:general_fixed_point","radii-polynomial/prop:fixed_zero_ef","radii-polynomial/lem:newton_deriv_bound_simple"],"positive_ids":[2580,14,2512,1419,2681]}
{"query_key":"brownian-motion/lem:Submartingale.locally_classD","query_node_id":2092,"repo":"brownian-motion","query_text":"Every cadlag submartingale is locally of class D.","positives":["brownian-motion/lem:Submartingale.locallyIntegrableSup","brownian-motion/lem:locally_classD_iff_locallyIntegrableSup"],"positive_ids":[1695,3213]}
{"query_key":"brownian-motion/lem:uniformIntegrableNorm","query_node_id":3752,"repo":"brownian-motion","query_text":"If \\((X_t)_{t \\in T}\\) is a family of uniformly integrable random variables, then so is \\((\\| X_t\\| )_{t \\in T}\\).","positives":["brownian-motion/lem:uniformIntegrableDominated"],"positive_ids":[2786]}
{"query_key":"radii-polynomial/lem:explicit_inverse","query_node_id":3613,"repo":"radii-polynomial","query_text":"If \\(\\| I_E - B\\| {\\lt} 1\\) for \\(B: E \\rightarrow E\\), then there exists \\(B^{-1}: E \\rightarrow E\\) such that \\begin{equation*} B \\circ B^{-1} = I_E \\quad \\text{and} \\quad B^{-1} \\circ B = I_E. \\end{equation*}","positives":["radii-polynomial/thm:neumann_series"],"positive_ids":[807]}
{"query_key":"FLT_ch3/hardly_ramified_spreads_out","query_node_id":1437,"repo":"FLT_ch3","query_text":"If \\(\\ell \\geq 3\\) is prime, \\(K\\) is a finite extension of \\(\\mathbb {Q}_\\ell \\) with integers \\(\\mathcal{O}\\) and if \\(\\rho :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(\\mathcal{O})\\) is a hardly ramified representation whose reduction is irreducible, then there exists a number field \\(M\\) and, for each finite place \\(\\mu \\) of \\(M\\) of characteristic prime to 2, with completion \\(M_\\mu \\) having integer ring \\(R_\\mu \\), a hardly ramified semisimple representation \\(\\rho _\\mu :\\operatorname{Gal}(\\overline{\\mathbb {Q}}/\\mathbb {Q})\\to \\operatorname{GL}_2(R_\\mu )\\) (by which we mean the generic fibre is semisimple), with the following properties: There is some \\(\\lambda \\mid \\ell \\) of \\(M\\) such that \\(\\rho _\\lambda \\cong \\rho \\), the isomorphism happening over some appropriate local field containing a copy of \\(M_\\lambda \\) and a copy of \\(K\\); If \\(\\mu _1\\) and \\(\\mu _2\\) are two finite places of \\(M\\) with odd residue characteristics \\(m_1\\) and \\(m_2\\), and if \\(p\\nmid 2m_1m_2\\) is prime, then \\(\\rho _{\\mu _1}\\) and \\(\\rho _{\\mu _2}\\) are both unramified at \\(p\\) and the characteristic polynomials \\(\\rho _{\\mu _1}(\\operatorname{Frob}_p)\\) and \\(\\rho _{\\mu _2}(\\operatorname{Frob}_p)\\) lie in \\(M[X]\\) and are equal.","positives":["FLT_ch3/hardly_ramified"],"positive_ids":[450]}
{"query_key":"Sphere-Packing-Lean/lemma:disc-E4E6","query_node_id":3510,"repo":"Sphere-Packing-Lean","query_text":"We have \\begin{equation} \\Delta (z) = (E_4^3-E_6^2)/1728. \\end{equation}","positives":["Sphere-Packing-Lean/def:disc-definition"],"positive_ids":[597]}
{"query_key":"Sphere-Packing-Lean/lemma:Ek-is-modular-form","query_node_id":110,"repo":"Sphere-Packing-Lean","query_text":"For all \\(k\\), \\(E_k\\in M_k(\\Gamma _1)\\). Especially, we have \\begin{equation} \\label{eqn:Ek-trans-S} E_k \\left(-\\frac{1}{z}\\right) = z^k E_k(z). \\end{equation}","positives":["Sphere-Packing-Lean/def:Mk","Sphere-Packing-Lean/def:Ek"],"positive_ids":[1488,776]}
{"query_key":"radii-polynomial/thm:CauchyProduct_mul_smul","query_node_id":486,"repo":"radii-polynomial","query_text":"Scalars pull out on the right: \\(a \\star (c \\cdot b) = c \\cdot (a \\star b)\\). Requires commutativity of the coefficient ring.","positives":["radii-polynomial/thm:CauchyProduct_comm"],"positive_ids":[993]}
{"query_key":"seymour/MatrixSum3.IsCanonicalSigning.Al_D_isTotallyUnimodular","query_node_id":1386,"repo":"seymour","query_text":"Let \\(B''\\) be from Definition 53. Let \\(c_{0}'' = B'' (X_{r}, y_{0})\\), \\(c_{1}'' = B'' (X_{r}, y_{1})\\), and \\(c_{2}'' = c_{0}'' - c_{1}''\\). Similarly, let \\(d_{0}'' = B'' (x_{0}, Y_{\\ell })\\), \\(d_{1}'' = B'' (x_{1}, Y_{\\ell })\\), and \\(d_{2}'' = d_{0}'' - d_{1}''\\). Then the following statements hold. For every \\(i \\in X_{r}\\), \\(c_{2}'' (i) \\in \\{ 0, \\pm 1\\} \\). If \\(D_{0}'' = \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix}\\), then \\(D'' = c_{0}'' \\otimes d_{0}'' - c_{1}'' \\otimes d_{1}''\\). If \\(D_{0}'' = \\begin{bmatrix} 1 & 1 \\\\ 0 & 1 \\end{bmatrix}\\), then \\(D'' = c_{0}'' \\otimes d_{0}'' - c_{0}'' \\otimes d_{1}'' + c_{1}'' \\otimes d_{1}''\\). For every \\(j \\in Y_{\\ell }\\), \\(D'' (X_{r}, j) \\in \\{ 0, \\pm c_{0}'', \\pm c_{1}'', \\pm c_{2}''\\} \\). For every \\(i \\in X_{r}\\), \\(D'' (i, Y_{\\ell }) \\in \\{ 0, \\pm d_{0}'', \\pm d_{1}'', \\pm d_{2}''\\} \\). \\(\\begin{bmatrix} A_{\\ell }” \\\\ D” \\end{bmatrix}\\) is TU.","positives":["seymour/MatrixSum3.toCanonicalSigning","seymour/MatrixSum3.HasTuBr.dddAl_isTotallyUnimodular"],"positive_ids":[2359,440]}
{"query_key":"PrimeNumberTheoremAnd/I3Bound","query_node_id":505,"repo":"PrimeNumberTheoremAnd","query_text":"Assuming a bound of the form of Lemma 3.4.28 we have that \\[ \\left|I_{3}(\\nu , \\epsilon , X, T)\\right| \\ll \\frac{X}{\\epsilon }\\, X^{-\\frac{A}{(\\log T)^9}} . \\]","positives":["PrimeNumberTheoremAnd/log_pow_over_xsq_integral_bounded","PrimeNumberTheoremAnd/MellinOfSmooth1b","PrimeNumberTheoremAnd/I3"],"positive_ids":[3195,2129,748]}
{"query_key":"seymour/MatrixSum3.IsCanonicalSigning.isTotallyUnimodular","query_node_id":409,"repo":"seymour","query_text":"\\(B''\\) from Definition 53 is TU.","positives":["seymour/MatrixLikeSum3.isTotallyUnimodular","seymour/MatrixSum3.IsCanonicalSigning.toMatrixLikeSum3"],"positive_ids":[666,2457]}
{"query_key":"carleson/Hilbert-kernel-bound","query_node_id":358,"repo":"carleson","query_text":"For \\(x,y\\in {\\mathbb {R}}\\) with \\(x\\neq y\\) we have \\begin{equation} \\label{eqcarl30} |\\kappa (x-y)|\\le 2^2(2|x-y|)^{-1}\\, . \\end{equation}","positives":["carleson/lower-secant-bound"],"positive_ids":[744]}
{"query_key":"pfr/first-cond","query_node_id":1875,"repo":"pfr","query_text":"We have \\begin{align*} & d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] \\\\ & \\qquad \\quad \\geq k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) \\\\ & \\qquad \\qquad \\qquad \\qquad - \\eta (d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]). \\end{align*}","positives":["pfr/cond-distance-lower"],"positive_ids":[2589]}
{"query_key":"FormalBook/book.irrational.Theorem_2","query_node_id":1407,"repo":"FormalBook","query_text":"\\(\\pi ^2\\) is irrational.","positives":["FormalBook/lem_aux_ii","FormalBook/lem_aux_iii"],"positive_ids":[335,3263]}
{"query_key":"pfr/first-useful","query_node_id":758,"repo":"pfr","query_text":"Let \\(X, Y, Z\\) be random variables taking values in some abelian group of characteristic \\(2\\), and with \\(Y, Z\\) independent. Then we have \\begin{align} \\nonumber d[X ; Y + Z] -d[X ; Y] & \\leq \\tfrac {1}{2} (\\mathbb {H}[Y+ Z] - \\mathbb {H}[Y]) \\\\ & = \\tfrac {1}{2} d[Y; Z] + \\tfrac {1}{4} \\mathbb {H}[Z] - \\tfrac {1}{4} \\mathbb {H}[Y]. \\label{lem51-a} \\end{align}","positives":["pfr/kv","pfr/cond-dist-fact"],"positive_ids":[3327,3450]}
{"query_key":"exchangeability/lem:tail_factorization","query_node_id":3021,"repo":"exchangeability","query_text":"The tail \\(\\sigma \\)-algebra factorization follows from the finite-level factorization via reverse martingale convergence.","positives":["exchangeability/lem:finite_level_factorization","exchangeability/lem:tailSigma_le_futureFiltration"],"positive_ids":[1671,2562]}
{"query_key":"brownian-motion/lem:predictableSet_elementaryPredictableSet","query_node_id":1872,"repo":"brownian-motion","query_text":"An elementary predictable set is measurable with respect to the predictable \\(\\sigma \\)-algebra.","positives":["brownian-motion/lem:predictable_Ioc_prod","brownian-motion/def:elementaryPredictableSet"],"positive_ids":[1274,2692]}
{"query_key":"carleson/third-exception","query_node_id":1424,"repo":"carleson","query_text":"We have \\begin{equation} \\mu (G_3)\\le 2^{-4} \\mu (G)\\, . \\end{equation}","positives":["carleson/boundary-exception","carleson/top-tiles","carleson/tree-count"],"positive_ids":[683,1583,413]}
{"query_key":"PrimeNumberTheoremAnd/Ramanujan-Soldner-constant","query_node_id":3507,"repo":"PrimeNumberTheoremAnd","query_text":"\\(\\operatorname {li}(2) = 1.0451\\dots \\).","positives":["PrimeNumberTheoremAnd/li-def"],"positive_ids":[3275]}
{"query_key":"brownian-motion/lem:sq_norm_elemStochIntegral","query_node_id":612,"repo":"brownian-motion","query_text":"For \\(V \\in \\mathcal{E}\\) and \\(M \\in \\mathcal{M}^2\\), then \\(V \\bullet M \\in \\mathcal{M}^2\\) (by Lemma 13.23) and \\begin{align*} \\Vert V \\bullet M \\Vert _{\\mathcal{M}^2}^2 & = \\Vert V \\Vert _{L^2(M)}^2 \\: . \\end{align*}","positives":["brownian-motion/def:L2M","brownian-motion/lem:L2Predictable.sq_norm_simpleProcess","brownian-motion/lem:isSquareIntegrable_elemStochIntegralBilin"],"positive_ids":[300,2410,3647]}
{"query_key":"carleson/tile-uncertainty","query_node_id":2960,"repo":"carleson","query_text":"Let \\({\\mathfrak p}_1, {\\mathfrak p}_2\\in {\\mathfrak P}\\) with \\(B({\\mathrm{c}}({\\mathfrak p}_1),5D^{{\\mathrm{s}}({\\mathfrak p}_1)}) \\cap B({\\mathrm{c}}({\\mathfrak p}_2),5D^{{\\mathrm{s}}({\\mathfrak p}_2)}) \\ne \\emptyset \\) and \\({\\mathrm{s}}({{\\mathfrak p}_1})\\leq {\\mathrm{s}}({{\\mathfrak p}_2})\\). For each \\(x_1\\in E({\\mathfrak p}_1)\\) and \\(x_2\\in E({\\mathfrak p}_2)\\) we have \\begin{equation} \\label{tgeo} 1+d_{{\\mathfrak p}_1}({\\mathcal{Q}}({\\mathfrak p}_1), {\\mathcal{Q}}({\\mathfrak p}_2))\\le 2^{8a}(1 + d_{B(x_1, D^{{\\mathrm{s}}({\\mathfrak p}_1)})}({Q}(x_1),{Q}(x_2)))\\, . \\end{equation}","positives":["carleson/monotone-cube-metrics","carleson/tile-range-support"],"positive_ids":[1590,1497]}
{"query_key":"GibbsMeasure/lem:gibbs-meas-tfae","query_node_id":1107,"repo":"GibbsMeasure","query_text":"Let \\(\\gamma \\) be a proper specification with parameter set \\(S\\) and state space \\((E, \\mathcal{E})\\), and let \\(\\nu \\in \\mathfrak {P}(E^S, \\mathcal{E}^S)\\). TFAE: \\(\\nu \\in \\mathcal{G}(\\gamma )\\). \\(\\gamma _\\Lambda \\circ _m\\nu = \\nu \\) for all \\(\\Lambda \\in \\operatorname{Finset}(S)\\). \\(\\gamma _\\Lambda \\circ _m\\nu = \\nu \\) frequently as \\(\\Lambda \\to S\\).","positives":["GibbsMeasure/lem:cond-exp-proper-ker-lintegral","GibbsMeasure/def:proper-spec","GibbsMeasure/def:gibbs-meas"],"positive_ids":[1298,3545,1460]}
{"query_key":"PrimeNumberTheoremAnd/lem:aachcanc","query_node_id":3292,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(s = \\sigma + i \\tau \\), \\(\\sigma ,\\tau \\in \\mathbb {R}\\). Let \\(n\\in \\mathbb {Z}_{{\\gt}0}\\). Let \\(a,b\\in \\mathbb {Z} + \\frac{1}{2}\\), \\(b{\\gt}a{\\gt}\\frac{|\\tau |}{2\\pi n}\\). Write \\(\\varphi _\\nu (t) = \\nu t - \\frac{\\tau }{2\\pi } \\log t\\). Then \\[ \\frac{1}{2} \\sum _{\\nu = \\pm n} \\left. \\frac{t^{-\\sigma } e(\\varphi _\\nu (t))}{2\\pi i \\varphi _\\nu '(t)}\\right|_a^b = \\left. \\frac{(-1)^n t^{-s} \\cdot \\frac{\\tau }{2\\pi t}}{2\\pi i \\left(n^2 - \\left(\\frac{\\tau }{2\\pi t}\\right)^2\\right)}\\right|_a^b. \\]","positives":["PrimeNumberTheoremAnd/e-def"],"positive_ids":[2050]}
{"query_key":"PrimeNumberTheoremAnd/ZetaLowerBound3","query_node_id":2070,"repo":"PrimeNumberTheoremAnd","query_text":"There exists a \\(c{\\gt}0\\) such that for all \\(1 {\\lt} \\sigma {\\lt}= 2\\) and \\(3 {\\lt} |t|\\), \\[ c \\frac{(\\sigma -1)^{3/4}}{(\\log |t|)^{1/4}} \\le |\\zeta (\\sigma + tI)|. \\]","positives":["PrimeNumberTheoremAnd/ZetaUpperBnd","PrimeNumberTheoremAnd/ZetaNear1BndExact"],"positive_ids":[1691,3762]}
{"query_key":"LeanAPAP/mzi_complex","query_node_id":1537,"repo":"LeanAPAP","query_text":"Let \\(m\\geq 1\\). If \\(f:G\\to \\mathbb {C}\\) is such that \\(\\mathbb {E}_x f(x)=0\\) and \\(\\left\\lvert f(x)\\right\\rvert \\leq 2\\) for all \\(x\\) then \\[ \\mathbb {E}_{x_1,\\ldots ,x_n} \\left\\lvert \\sum _{i=1}^n f(x_i)\\right\\rvert ^{2m} \\leq (8mn)^{m}. \\]","positives":["LeanAPAP/mzi"],"positive_ids":[3572]}
{"query_key":"radii-polynomial/lem:summable_eval","query_node_id":2215,"repo":"radii-polynomial","query_text":"If \\(a \\in \\ell ^1_\\nu \\) and \\(|z| \\leq \\nu \\), then \\(\\sum _{n=0}^\\infty a_n z^n\\) converges absolutely.","positives":["radii-polynomial/def:eval"],"positive_ids":[2535]}
{"query_key":"brownian-motion/def:IsCadlag","query_node_id":1878,"repo":"brownian-motion","query_text":"A function \\(f : T \\to E\\) is cadlag if it is right continuous and the limit \\(\\lim _{s \\uparrow t} f(s)\\) exists for all \\(t \\in T\\).","positives":["brownian-motion/def:RightContinuous"],"positive_ids":[1591]}
{"query_key":"FormalBook/e_pow_4_irrational","query_node_id":585,"repo":"FormalBook","query_text":"\\(e^4\\) is irrational.","positives":["FormalBook/little_lemma","FormalBook/e_pow_2_irrational"],"positive_ids":[1083,1451]}
{"query_key":"PrimeNumberTheoremAnd/fks2-def-5","query_node_id":2189,"repo":"PrimeNumberTheoremAnd","query_text":"Let \\(x_0 {\\gt} 2\\). We say a (step) function \\(ε_{\\diamond ,num}(x_0)\\) gives an admissible numerical bound for \\(E_\\diamond (x)\\) if \\(E_\\diamond (x) \\leq ε_{\\diamond ,num}(x_0)\\) for all \\(x \\geq x_0\\).","positives":["PrimeNumberTheoremAnd/Epi-def"],"positive_ids":[23]}
{"query_key":"PrimeNumberTheoremAnd/Dusart_cor_5_2_e","query_node_id":3054,"repo":"PrimeNumberTheoremAnd","query_text":"For \\(x \\geq 88789\\), we have \\[ \\pi (x) \\geq \\frac{x}{\\log x} \\Bigl(1 + \\frac{1}{\\log x} + \\frac{2}{\\log ^2 x}\\Bigr). \\]","positives":["PrimeNumberTheoremAnd/pi-def"],"positive_ids":[2667]}
{"query_key":"ClassFieldTheory/thm:first inequality","query_node_id":3777,"repo":"ClassFieldTheory","query_text":"For any finite Galois extension \\(l/k\\) be have \\[ |H^0_{\\mathrm{Tate}}(l/k, \\mathrm{Cl}_{l}) | \\le [l : k]. \\]","positives":["ClassFieldTheory/lem:Dirichlet density split","ClassFieldTheory/lem:density bound"],"positive_ids":[896,2939]}
{"query_key":"LeanAPAP/main-int-count","query_node_id":3769,"repo":"LeanAPAP","query_text":"If \\(A\\subseteq \\{ 1,\\ldots ,N\\} \\) has size \\(\\left\\lvert A\\right\\rvert =\\alpha N\\), then \\(A\\) contains at least \\[ \\exp (-O(\\mathcal{L}{\\alpha }^{12}))N^2 \\]","positives":["LeanAPAP/th-int-gen","LeanAPAP/hoelder-lifting"],"positive_ids":[3675,924]}
{"query_key":"PrimeNumberTheoremAnd/MediumPNT","query_node_id":1135,"repo":"PrimeNumberTheoremAnd","query_text":"We have \\[ \\sum _{n \\leq x} \\Lambda (n) = x + O(x \\exp (-c(\\log x)^{1/10})). \\]","positives":["PrimeNumberTheoremAnd/SmoothedChebyshevPull2","PrimeNumberTheoremAnd/SmoothedChebyshevPull1","PrimeNumberTheoremAnd/I5Bound","PrimeNumberTheoremAnd/LogDerivZetaHolcLargeT","PrimeNumberTheoremAnd/I3Bound","PrimeNumberTheoremAnd/LogDerivZetaBndUnif","PrimeNumberTheoremAnd/I8Bound","PrimeNumberTheoremAnd/I1Bound","PrimeNumberTheoremAnd/LogDerivZetaHolcSmallT","PrimeNumberTheoremAnd/MellinOfSmooth1c","PrimeNumberTheoremAnd/SmoothedChebyshevClose","PrimeNumberTheoremAnd/I4Bound","PrimeNumberTheoremAnd/SmoothExistence"],"positive_ids":[3454,2533,2487,876,505,734,2114,477,2034,238,2243,3260,3174]}
{"query_key":"toric/0-char-cochar-pairing","query_node_id":3485,"repo":"toric","query_text":"Let \\(R\\) be a domain and \\(G\\) a group scheme over \\(\\operatorname{Spec}R\\). Then there is a \\({\\mathbb Z}\\)-valued perfect pairing between \\(X(G)\\) and \\(X^*(G)\\).","positives":["toric/0-cochar","toric/0-char"],"positive_ids":[2645,1173]}
{"query_key":"brownian-motion/lem:logSizeBallSequence_V_anti","query_node_id":2847,"repo":"brownian-motion","query_text":"The sets \\(V_i\\) of a log-size ball sequence \\((V_i, t_i, r_i)_{i \\in \\mathbb {N}}\\) are a decreasing sequence of sets. That is, \\(V_{i+1} \\subseteq V_i\\) for all \\(i \\in \\mathbb {N}\\).","positives":["brownian-motion/def:logSizeBallSequence"],"positive_ids":[2293]}
{"query_key":"equational_theories/finite_imp_3994_3588","query_node_id":836,"repo":"equational_theories","query_text":"All finite magmas which satisfy E3994 \\((x \\diamond y = (z \\diamond (x \\diamond y)) \\diamond z)\\) also satisfy E3588 \\((x \\diamond y = z \\diamond ((x \\diamond y) \\diamond z))\\).","positives":["equational_theories/models-def","equational_theories/eq3994","equational_theories/eq3588"],"positive_ids":[1745,3165,3511]}
{"query_key":"brownian-motion/def:preBrownian","query_node_id":1975,"repo":"brownian-motion","query_text":"Let \\(\\Omega = \\mathbb {R}^{\\mathbb {R}_+}\\) and consider the probability space \\((\\Omega , P_B)\\) (where \\(P_B\\) is the measure defined in Definition 4.10). The identity on that space is a function \\(\\Omega \\to \\mathbb {R}_+ \\to \\mathbb {R}\\). We reorder the arguments to define a stochastic process \\(X : \\mathbb {R}_+ \\to \\Omega \\to \\mathbb {R}\\), which we call the pre-Brownian process.","positives":["brownian-motion/def:gaussianLimit"],"positive_ids":[2864]}
{"query_key":"ChandraFurstLipton/thm:eval-rand-complexity","query_node_id":533,"repo":"ChandraFurstLipton","query_text":"\\[ R_\\epsilon (\\operatorname{eval}) \\le 2d\\lceil \\log _2\\epsilon ^{-1}\\rceil \\]","positives":["ChandraFurstLipton/lem:rand-eq-test-protocol-eval-valid","ChandraFurstLipton/def:rand-complex","ChandraFurstLipton/def:eval"],"positive_ids":[2134,449,3066]}
{"query_key":"brownian-motion/lem:LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn","query_node_id":2148,"repo":"brownian-motion","query_text":"If a real-valued function has bounded variation on a set, then it is a difference of monotone functions there.","positives":["brownian-motion/def:LocallyBoundedVariationOn"],"positive_ids":[2261]}
{"query_key":"radii-polynomial/thm:mean_value","query_node_id":2191,"repo":"radii-polynomial","query_text":"Suppose that \\(U \\subset \\mathbb {R}^n\\) is open and that \\(f: U \\rightarrow \\mathbb {R}^n\\) is \\(C^1\\). Consider \\(x, y \\in U\\) such that the line segment \\((1-t)x + ty\\), \\(t \\in [0,1]\\), is contained in \\(U\\). Then, \\begin{equation*} f(y) - f(x) = \\left(\\int _0^1 \\text{Df}((1-t)x + ty) \\, dt\\right)(y-x), \\end{equation*}","positives":["radii-polynomial/def:operator_norm"],"positive_ids":[2365]}
{"query_key":"PrimeNumberTheoremAnd/initial-factorization-medium-prime-ge","query_node_id":639,"repo":"PrimeNumberTheoremAnd","query_text":"A medium prime \\(\\sqrt{n} {\\lt} p ≤ n/L\\) can be in deficit by at most \\(M\\).","positives":["PrimeNumberTheoremAnd/initial-factorization-def","PrimeNumberTheoremAnd/balance-def"],"positive_ids":[1345,917]}
{"query_key":"exchangeability/thm:birkhoff_condexp","query_node_id":1748,"repo":"exchangeability","query_text":"Birkhoff averages converge in L\\(^2\\) to the conditional expectation given the shift-invariant \\(\\sigma \\)-algebra.","positives":["exchangeability/lem:koopman_condexp_comm"],"positive_ids":[1861]}
{"query_key":"brownian-motion/def:IsSquareIntegrable","query_node_id":2190,"repo":"brownian-motion","query_text":"Let \\(T\\) be a linear order with bottom element 0, on which we have a filtration \\(\\mathcal{F}\\) satisfying the usual conditions. We say that a martingale \\(M : T \\to \\Omega \\to E\\) is square integrable if it is càdlàg and \\(\\sup _{t \\in T} \\Vert M_t \\Vert _{L^2} {\\lt} \\infty \\) (Lean remark: use eLpNorm (M t) 2).","positives":["brownian-motion/def:Martingale"],"positive_ids":[1156]}
{"query_key":"brownian-motion/lem:predictablePart_add_one","query_node_id":1270,"repo":"brownian-motion","query_text":"For any integer \\(n \\ge 0\\), \\(A_{n+1} = A_n + \\mathbb {E}[X_{n+1} - X_n \\mid \\mathcal{F}_n]\\).","positives":["brownian-motion/def:predictablePart"],"positive_ids":[925]}
{"query_key":"brownian-motion/thm:finite_set_bound_of_dist_le_of_le_diam","query_node_id":2610,"repo":"brownian-motion","query_text":"Suppose that \\(T\\) is a finite set with bounded internal covering number with constant \\(c_1{\\gt}0\\) and exponent \\(d {\\gt} 0\\). Let \\(X : T \\to \\Omega \\to E\\) be a process that satisfies the Kolmogorov condition for exponents \\((p,q)\\) with constant \\(M\\), with \\(q {\\gt} d\\) and \\(p {\\gt} 0\\). For all \\(\\delta \\in (0, 4\\mathrm{diam}(T)]\\), \\begin{align*} & \\mathbb {E}\\left[ \\sup _{s, t \\in T; d_T(s, t) \\le \\delta } d_E(X_s, X_t)^p \\right] \\\\ & \\le 2^{2p+4q+1} M \\delta ^{q-d} \\left(\\delta ^d \\left(\\log _2 N^{int}_{\\delta /4}(T) \\right)^q N^{int}_{\\delta /4}(T) + c_1 C_p\\right) \\: . \\end{align*}","positives":["brownian-motion/lem:second_term_bound","brownian-motion/lem:integral_sup_rpow_dist_cover_rescale","brownian-motion/lem:lintegral_sup_cover_eq_of_lt_iInf_dist","brownian-motion/cor:scale_change_rpow","brownian-motion/lem:IsKolmogorovProcess.lintegral_sup_rpow_edist_eq_zero"],"positive_ids":[2246,3212,3438,871,1528]}
{"query_key":"ClassFieldTheory/lem:Tate -1 aug","query_node_id":709,"repo":"ClassFieldTheory","query_text":"Let \\(S\\) be a subgroup of a finite group \\(G\\). Then there is an isomorphism of \\(R\\)-modules \\[ S^{\\mathrm{ab}} \\otimes _{{\\mathbb Z}} R \\cong H^{-1}_{\\mathrm{Tate}}(S,\\mathrm{aug}(R,G)), \\]","positives":["ClassFieldTheory/lem:homology 1 trivial","ClassFieldTheory/lem:Tate cohomology is cohomology or homology","ClassFieldTheory/def:augmentation module"],"positive_ids":[3808,3774,2791]}
{"query_key":"exchangeability/lem:indicator_product_bridge_contractable","query_node_id":583,"repo":"exchangeability","query_text":"For contractable sequences, indicator products satisfy the bridge condition.","positives":["exchangeability/def:contractable"],"positive_ids":[606]}
{"query_key":"M2/lem:homotpy_k_sheaves_stable_by_extension","query_node_id":2106,"repo":"M2","query_text":"Let \\(\\mathcal{F}^{\\bullet }\\) be a complex of \\(\\mathcal{K}\\)-presheaves. If \\(\\mathcal{F}^{\\bullet }\\) has a finite filtration whose associated graded is a Homotopy-\\(\\mathcal{K}\\)-sheaf, then \\(\\mathcal{F}^{\\bullet }\\) is a Homotopy-\\(\\mathcal{K}\\)-sheaf.","positives":["M2/def:homotopy_k_sheaf"],"positive_ids":[3076]}
{"query_key":"PrimeNumberTheoremAnd/bklnw-eq_A_21","query_node_id":263,"repo":"PrimeNumberTheoremAnd","query_text":"We define \\[ B = 5/2 - \\sigma , \\]","positives":["PrimeNumberTheoremAnd/zero-density-bound","PrimeNumberTheoremAnd/RH-up-to","PrimeNumberTheoremAnd/classical-zero-free-region"],"positive_ids":[1474,2065,3243]}
{"query_key":"radii-polynomial/thm:CauchyProduct_mul_one","query_node_id":2288,"repo":"radii-polynomial","query_text":"\\(a \\star e = a\\).","positives":["radii-polynomial/thm:toPowerSeries_mul","radii-polynomial/def:CauchyProduct_one"],"positive_ids":[133,1153]}
{"query_key":"exchangeability/lem:product_blockAvg_L1","query_node_id":3639,"repo":"exchangeability","query_text":"Products of block averages converge in L\\(^1\\).","positives":["exchangeability/lem:blockAvg_tendsto","exchangeability/lem:integral_prod_blockAvg"],"positive_ids":[2132,2558]}
{"query_key":"pfr/relabeled-entropy-cond","query_node_id":591,"repo":"pfr","query_text":"If \\(X: \\Omega \\to S\\), \\(Y: \\Omega \\to T\\), and \\(Z: \\Omega \\to U\\) are random variables, and \\(Y = f(X,Z)\\) almost surely for some map \\(f: S \\times U \\to T\\) that is injective for each fixed \\(U\\), then \\(\\mathbb {H}[X|Z] = \\mathbb {H}[Y|Z]\\). Similarly, if \\(g: T \\to U\\) is injective, then \\(\\mathbb {H}[X|g(Y)] = \\mathbb {H}[X|Y]\\).","positives":["pfr/conditional-entropy-def","pfr/relabeled-entropy"],"positive_ids":[1044,3480]}
{"query_key":"brownian-motion/def:approxSeq","query_node_id":3693,"repo":"brownian-motion","query_text":"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 \\).","positives":["brownian-motion/def:IsStoppingTime"],"positive_ids":[3129]}
{"query_key":"brownian-motion/def:stdGaussian","query_node_id":642,"repo":"brownian-motion","query_text":"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\\).","positives":["brownian-motion/def:gaussianReal"],"positive_ids":[1801]}
{"query_key":"amatelus/def:multi-chapter","query_node_id":3003,"repo":"amatelus","query_text":"This chapter covers multi-device support aspects of AMATELUS.","positives":["amatelus/thm:did-local-generation"],"positive_ids":[104]}
{"query_key":"LBPTester/theorem : subspace_contains_zero","query_node_id":2179,"repo":"LBPTester","query_text":"This is a proof that any subspace contains the zero vector:\\((0 : V) \\in W\\)","positives":["LBPTester/theorem : zero_smul","LBPTester/definition : Subspace"],"positive_ids":[2272,2123]}
{"query_key":"PrimeNumberTheoremAnd/HolomorphicOn_riemannZeta0","query_node_id":225,"repo":"PrimeNumberTheoremAnd","query_text":"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\\} \\).","positives":["PrimeNumberTheoremAnd/riemannZeta0"],"positive_ids":[3461]}
{"query_key":"equational_theories/free-confluent","query_node_id":3724,"repo":"equational_theories","query_text":"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 }\\) is the identity embedding (note that every single-letter word is already in normal form).","positives":["equational_theories/confluent-theory"],"positive_ids":[2924]}
{"query_key":"PrimeNumberTheoremAnd/I8","query_node_id":2931,"repo":"PrimeNumberTheoremAnd","query_text":"\\[ 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 \\]","positives":["PrimeNumberTheoremAnd/Smooth1"],"positive_ids":[3771]}
{"query_key":"FLT_ch9/NumberField.InfiniteAdeleRing.piEquiv","query_node_id":693,"repo":"FLT_ch9","query_text":"There is a natural \\(K_{\\infty }\\)-linear homeomorphism \\(K_{\\infty }^{[L:K]} \\cong _{K_{\\infty }} L_{\\infty }\\).","positives":["FLT_ch9/NumberField.InfinitePlace.Completion.piEquiv"],"positive_ids":[2749]}
{"query_key":"brownian-motion/cor:doob_lp_norm","query_node_id":3363,"repo":"brownian-motion","query_text":"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 \\rVert ^p] \\: . \\end{align*}","positives":["brownian-motion/cor:Martingale.submartingale_norm","brownian-motion/thm:doob_lp"],"positive_ids":[167,1677]}
{"query_key":"Sphere-Packing-Lean/def:holomorphic-modular-form","query_node_id":1477,"repo":"Sphere-Packing-Lean","query_text":"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 holomorphic on \\(\\mathbb {H}\\). For all \\(\\gamma \\in \\mathrm{SL}_2(\\mathbb {Z})\\), there exist \\(A, B \\in \\mathbb {R}\\) such that for all \\(z \\in \\mathbb {H}\\), with \\( A \\le \\mathrm{Im}(z)\\), we have \\(|(f \\mid _k \\gamma ) (z) |\\le B\\). Here \\(| - |\\) denotes the standard complex absolute value.","positives":["Sphere-Packing-Lean/def:congruence-subgroup"],"positive_ids":[907]}
{"query_key":"ClassFieldTheory/lem:Dirichlet density split","query_node_id":896,"repo":"ClassFieldTheory","query_text":"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]}\\).","positives":["ClassFieldTheory/lem:Dirichlet density degree one"],"positive_ids":[1018]}
{"query_key":"pfr/cond-multidist-alt","query_node_id":652,"repo":"pfr","query_text":"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}","positives":["pfr/conditional-entropy-def","pfr/cond-multidist-def"],"positive_ids":[1044,2700]}
{"query_key":"brownian-motion/def:simpleProcess","query_node_id":65,"repo":"brownian-motion","query_text":"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}\\)-measurable for \\(k \\ge 1\\). Then the simple process for that sequence is the process \\(V : T \\to \\Omega \\to E\\) defined by \\begin{align*} V_t = \\eta _0 \\mathbb {1}_{\\{ 0\\} }(t) + \\sum _{k=1}^{n} \\eta _k \\mathbb {1}_{(s_k, t_k]}(t) \\: . \\end{align*}","positives":["brownian-motion/def:filtration"],"positive_ids":[2474]}
{"query_key":"FLT_ch8/fixed_of_fixed1_aux2","query_node_id":2961,"repo":"FLT_ch8","query_text":"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 Q \\neq Q\\), then \\(g \\cdot b \\equiv 0 \\pmod{Q}\\).","positives":["FLT_ch8/fixed_of_fixed1_aux1"],"positive_ids":[1886]}
{"query_key":"Sphere-Packing-Lean/E8-vector-norms","query_node_id":3654,"repo":"Sphere-Packing-Lean","query_text":"All vectors in \\(\\Lambda _8\\) have norm of the form \\(\\sqrt{2n}\\), where \\(n\\) is a nonnegative integer.","positives":["Sphere-Packing-Lean/E8-defs-equivalent"],"positive_ids":[529]}
{"query_key":"ABC-Exceptions/lem:Subcase2.6","query_node_id":2745,"repo":"ABC-Exceptions","query_text":"\\(\\nu \\leq 0.66\\) in the case \\(s_2{\\lt}0.3\\) and \\(2s_1-s_2{\\gt}0.025\\).","positives":["ABC-Exceptions/prop:FourierBound","ABC-Exceptions/lem:Subcase2.3","ABC-Exceptions/lem:Subcase2.5","ABC-Exceptions/lem:Subcase2.4"],"positive_ids":[3214,825,1212,233]}
{"query_key":"equational_theories/austin-two","query_node_id":3006,"repo":"equational_theories","query_text":"Any law with at most two variables has a non-trivial finite model.","positives":["equational_theories/models-def"],"positive_ids":[1745]}
{"query_key":"GibbsMeasure/def:gibbs-meas","query_node_id":1460,"repo":"GibbsMeasure","query_text":"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)\\).","positives":["GibbsMeasure/def:spec","GibbsMeasure/def:cond-exp-ker"],"positive_ids":[765,1310]}
{"query_key":"toric/0-diag-aff-hom","query_node_id":501,"repo":"toric","query_text":"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.","positives":["toric/0-diag"],"positive_ids":[858]}
{"query_key":"PrimeNumberTheoremAnd/BlaschkeNonZero","query_node_id":3069,"repo":"PrimeNumberTheoremAnd","query_text":"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}\\).","positives":["PrimeNumberTheoremAnd/BlaschkeB","PrimeNumberTheoremAnd/ZeroFactorization"],"positive_ids":[3062,3498]}
{"query_key":"radii-polynomial/thm:norm_mul_le","query_node_id":164,"repo":"radii-polynomial","query_text":"\\(\\| 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\\).","positives":["radii-polynomial/def:l1Weighted_mul"],"positive_ids":[3101]}
{"query_key":"radii-polynomial/def:param_power_series","query_node_id":1824,"repo":"radii-polynomial","query_text":"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*}","positives":["radii-polynomial/def:param_seq"],"positive_ids":[909]}
{"query_key":"FLT_ch14/automorphic_representation","query_node_id":1602,"repo":"FLT_ch14","query_text":"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.","positives":["FLT_ch14/automorphic_form_actions"],"positive_ids":[1400]}
{"query_key":"FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.antidiag_mem_C","query_node_id":129,"repo":"FLT_ch11","query_text":"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.\\)","positives":["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"],"positive_ids":[1338,438,1254]}
{"query_key":"brownian-motion/thm:doob_ineq","query_node_id":143,"repo":"brownian-motion","query_text":"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 } \\leq \\frac{\\mathbb {E}[X_T]}{\\lambda } \\: . \\end{align*}","positives":["brownian-motion/lem:doob_countable","brownian-motion/def:RightContinuous"],"positive_ids":[1108,1591]}
{"query_key":"brownian-motion/lem:isGaussian_add_const","query_node_id":775,"repo":"brownian-motion","query_text":"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\\).","positives":["brownian-motion/def:IsGaussian"],"positive_ids":[3158]}
{"query_key":"ClassFieldTheory/lem:herbrand local l*","query_node_id":2758,"repo":"ClassFieldTheory","query_text":"If \\(l/k\\) is a cyclic extension of local fields then \\(h(l/k, l^\\times )= [l:k]\\).","positives":["ClassFieldTheory/lem:herbrand local units"],"positive_ids":[50]}
{"query_key":"pfr/ruz-copy","query_node_id":282,"repo":"pfr","query_text":"If \\(X',Y'\\) are copies of \\(X,Y\\) respectively then \\(d[X';Y']=d[X ;Y]\\).","positives":["pfr/ruz-dist-def","pfr/copy-ent"],"positive_ids":[1818,2358]}
{"query_key":"ClassFieldTheory/lem:Dirichlet density union","query_node_id":3617,"repo":"ClassFieldTheory","query_text":"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). \\]","positives":["ClassFieldTheory/def:Dirichlet density"],"positive_ids":[1246]}
{"query_key":"ChebyshevCircles/thm:newton_identities","query_node_id":954,"repo":"ChebyshevCircles","query_text":"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}","positives":["ChebyshevCircles/lem:discrete_orthogonality"],"positive_ids":[3410]}
{"query_key":"ClassFieldTheory/thm:splitting module trivial","query_node_id":538,"repo":"ClassFieldTheory","query_text":"Let \\(\\sigma '\\) be a 2-cocycle representing a fundamental class in \\(H^2(G,M)\\). Then \\(\\mathrm{split}(\\sigma ')\\) has trivial cohomology.","positives":["ClassFieldTheory/lem:splits in splitting module","ClassFieldTheory/thm:triviality criterion","ClassFieldTheory/lem:restriction fundamental class generates","ClassFieldTheory/lem:linear injective of surjective"],"positive_ids":[3356,156,1539,2325]}
{"query_key":"brownian-motion/lem:martingale_iff_integral_elemStochIntegral_eq_zero","query_node_id":3030,"repo":"brownian-motion","query_text":"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\\).","positives":["brownian-motion/lem:tendsto_elemStochIntegral_limitProcess","brownian-motion/def:Martingale","brownian-motion/def:elemStochIntegral"],"positive_ids":[1499,1156,490]}
{"query_key":"toric/0-biquot","query_node_id":1079,"repo":"toric","query_text":"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.","positives":["toric/0-biideal","toric/0-coquot"],"positive_ids":[2039,2265]}
{"query_key":"carleson/Lipschitz-partition-unity","query_node_id":522,"repo":"carleson","query_text":"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}","positives":["carleson/dyadic-partition-1","carleson/moderate-scale-change"],"positive_ids":[946,3224]}
{"query_key":"ABC-Exceptions/prop:DeterminantBound","query_node_id":1073,"repo":"ABC-Exceptions","query_text":"\\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*}","positives":["ABC-Exceptions/def:nu","ABC-Exceptions/prop:DeterminantMethod","ABC-Exceptions/def:aibici"],"positive_ids":[686,767,3643]}
{"query_key":"FLT_ch9/IsDedekindDomain.FiniteAdeleRing.tensorProduct_algEquiv","query_node_id":1163,"repo":"FLT_ch9","query_text":"The natural map \\(B\\otimes _AK\\to L\\) is a \\(B\\)-algebra isomorphism.","positives":["FLT_ch9/IsDedekindDomain.dvd_norm"],"positive_ids":[1585]}