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wiki_5500_chunk_0 | Beltrami differential equation | Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5501_chunk_0 | Beltrami differential equation | The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5502_chunk_0 | Bendixson–Dulac theorem | In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a C 1 {\displaystyle C^{1}} function φ ( x , y ) {\displaystyle \varphi (x,y)} (called the Dulac function) such that the expression ∂ ( φ f ) ∂ x + ∂ ( φ g ) ∂ y {\displaystyle {\frac {\partial (\varphi f)}{\partial x}}+{\frac ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5503_chunk_0 | Benjamin–Ono equation | In mathematics, the Benjamin–Ono equation is a nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water. It was introduced by Benjamin (1967) and Ono (1975). The Benjamin–Ono equation is u t + u u x + H u x x = 0 {\displaystyle u_{t}+uu_{x}+Hu_{xx}=0} where H is the Hi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5504_chunk_0 | Bergman–Weil formula | In mathematics, the Bergman–Weil formula is an integral representation for holomorphic functions of several variables generalizing the Cauchy integral formula. It was introduced by Bergmann (1936) and Weil (1935). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5505_chunk_0 | Seidel triangle | In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5506_chunk_0 | Seidel triangle | The Bernoulli numbers are special values of the Bernoulli polynomials B n ( x ) {\displaystyle B_{n}(x)} , with B n − = B n ( 0 ) {\displaystyle B_{n}^{-{}}=B_{n}(0)} and B n + = B n ( 1 ) {\displaystyle B_{n}^{+}=B_{n}(1)} .The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Ber... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5507_chunk_0 | Bernoulli polynomials | In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series expansion of functions, and with the Euler–MacLaurin formula. These polynomials occur in the study of many special functions and, in particular, the Riemann zeta func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5508_chunk_0 | Bernoulli polynomials | For the Bernoulli polynomials, the number of crossings of the x-axis in the unit interval does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the sine and cosine functions. A similar set of polynomials, based on a generating function, is the family of Euler polynomial... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5509_chunk_0 | Bernoulli scheme | This is essentially the Markov partition. The term shift is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5510_chunk_0 | Bernstein–Sato polynomial | In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by Joseph Bernstein (1971) and Mikio Sato and Takuro Shintani (1972, 1974), Sato (1990). It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5511_chunk_0 | Besicovitch inequality | In mathematics, the Besicovitch inequality is a geometric inequality relating volume of a set and distances between certain subsets of its boundary. The inequality was first formulated by Abram Besicovitch.Consider the n-dimensional cube n {\displaystyle ^{n}} with a Riemannian metric g {\displaystyle g} . Let denote ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5512_chunk_0 | Besicovitch inequality | The Besicovitch inequality asserts that The inequality can be generalized in the following way. Given an n-dimensional Riemannian manifold M with connected boundary and a smooth map f: M → n {\displaystyle f:M\rightarrow ^{n}} , such that the restriction of f to the boundary of M is a degree 1 map onto ∂ n {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5513_chunk_0 | Besov space | In mathematics, the Besov space (named after Oleg Vladimirovich Besov) B p , q s ( R ) {\displaystyle B_{p,q}^{s}(\mathbf {R} )} is a complete quasinormed space which is a Banach space when 1 ≤ p, q ≤ ∞. These spaces, as well as the similarly defined Triebel–Lizorkin spaces, serve to generalize more elementary function... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5514_chunk_0 | Bessel polynomials | In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series: 101 y n ( x ) = ∑ k = 0 n ( n + k ) ! ( n − k ) ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5515_chunk_0 | Bessel–Maitland function | In mathematics, the Bessel–Maitland function, or Wright generalized Bessel function, is a generalization of the Bessel function, introduced by Edward Maitland Wright (1934). The word "Maitland" in the name of the function seems to be the result of confusing Edward Maitland Wright's middle and last names. It is given by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5516_chunk_0 | Beurling–Lax theorem | In mathematics, the Beurling–Lax theorem is a theorem due to Beurling (1949) and Lax (1959) which characterizes the shift-invariant subspaces of the Hardy space H 2 ( D , C ) {\displaystyle H^{2}(\mathbb {D} ,\mathbb {C} )} . It states that each such space is of the form θ H 2 ( D , C ) , {\displaystyle \theta H^{2}(\m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5517_chunk_0 | Bhatia–Davis inequality | In mathematics, the Bhatia–Davis inequality, named after Rajendra Bhatia and Chandler Davis, is an upper bound on the variance σ2 of any bounded probability distribution on the real line. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5518_chunk_0 | Bianchi classification | In mathematics, the Bianchi classification provides a list of all real 3-dimensional Lie algebras (up to isomorphism). The classification contains 11 classes, 9 of which contain a single Lie algebra and two of which contain a continuum-sized family of Lie algebras. (Sometimes two of the groups are included in the infin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5519_chunk_0 | Birch and Swinnerton-Dyer conjecture | In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5520_chunk_0 | Birch and Swinnerton-Dyer conjecture | The modern formulation of the conjecture relates arithmetic data associated with an elliptic curve E over a number field K to the behaviour of the Hasse–Weil L-function L(E, s) of E at s = 1. More specifically, it is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5521_chunk_0 | Birkhoff–Grothendieck theorem | In mathematics, the Birkhoff–Grothendieck theorem classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over C P 1 {\displaystyle \mathbb {CP} ^{1}} is a direct sum of holomorphic line bundles. The theorem was proved by Alexander Grothendieck (1957, Theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5522_chunk_0 | Birman–Wenzl algebra | In mathematics, the Birman–Murakami–Wenzl (BMW) algebra, introduced by Joan Birman and Hans Wenzl (1989) and Jun Murakami (1987), is a two-parameter family of algebras C n ( ℓ , m ) {\displaystyle \mathrm {C} _{n}(\ell ,m)} of dimension 1 ⋅ 3 ⋅ 5 ⋯ ( 2 n − 1 ) {\displaystyle 1\cdot 3\cdot 5\cdots (2n-1)} having the Hec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5523_chunk_0 | Bishop-Gromov inequality | In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5524_chunk_0 | Bishop–Phelps theorem | In mathematics, the Bishop–Phelps theorem is a theorem about the topological properties of Banach spaces named after Errett Bishop and Robert Phelps, who published its proof in 1961. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5525_chunk_0 | Bismut connection | In mathematics, the Bismut connection ∇ {\displaystyle \nabla } is the unique connection on a complex Hermitian manifold that satisfies the following conditions, It preserves the metric ∇ g = 0 {\displaystyle \nabla g=0} It preserves the complex structure ∇ J = 0 {\displaystyle \nabla J=0} The torsion T ( X , Y ) {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5526_chunk_0 | Bismut connection | Further let ∇ {\displaystyle \nabla } be the Levi-Civita connection. Define first a tensor T {\displaystyle T} such that T ( Z , X , Y ) = − 1 2 ⟨ Z , J ( ∇ X J ) Y ⟩ {\displaystyle T(Z,X,Y)=-{\frac {1}{2}}\langle Z,J(\nabla _{X}J)Y\rangle } . This tensor is anti-symmetric in the first and last entry, i.e. the new conn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5527_chunk_0 | Bismut connection | In concrete terms, the new connection is given by Γ β γ α − 1 2 J δ α ∇ β J γ δ {\displaystyle \Gamma _{\beta \gamma }^{\alpha }-{\frac {1}{2}}J_{~\delta }^{\alpha }\nabla _{\beta }J_{~\gamma }^{\delta }} with Γ β γ α {\displaystyle \Gamma _{\beta \gamma }^{\alpha }} being the Levi-Civita connection. The new connection... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5528_chunk_0 | Bloch group | In mathematics, the Bloch group is a cohomology group of the Bloch–Suslin complex, named after Spencer Bloch and Andrei Suslin. It is closely related to polylogarithm, hyperbolic geometry and algebraic K-theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5529_chunk_0 | Blumberg theorem | In mathematics, the Blumberg theorem states that for any real function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } there is a dense subset D {\displaystyle D} of R {\displaystyle \mathbb {R} } such that the restriction of f {\displaystyle f} to D {\displaystyle D} is continuous. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5530_chunk_0 | Bochner integral | In mathematics, the Bochner integral, named for Salomon Bochner, extends the definition of Lebesgue integral to functions that take values in a Banach space, as the limit of integrals of simple functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5531_chunk_0 | Bochner–Kodaira–Nakano identity | In mathematics, the Bochner–Kodaira–Nakano identity is an analogue of the Weitzenböck identity for hermitian manifolds, giving an expression for the antiholomorphic Laplacian of a vector bundle over a hermitian manifold in terms of its complex conjugate and the curvature of the bundle and the torsion of the metric of t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5532_chunk_0 | Bochner–Martinelli formula | In mathematics, the Bochner–Martinelli formula is a generalization of the Cauchy integral formula to functions of several complex variables, introduced by Enzo Martinelli (1938) and Salomon Bochner (1943). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5533_chunk_0 | Bogomolov conjecture | In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometry. The conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang in 1998. A further generalization to general abelian va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5534_chunk_0 | Bogomolov–Miyaoka–Yau inequality | In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality c 1 2 ≤ 3 c 2 {\displaystyle c_{1}^{2}\leq 3c_{2}} between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topological types of the underlying real 4-manifold. It was proved independ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5535_chunk_0 | Bohr compactification | In mathematics, the Bohr compactification of a topological group G is a compact Hausdorff topological group H that may be canonically associated to G. Its importance lies in the reduction of the theory of uniformly almost periodic functions on G to the theory of continuous functions on H. The concept is named after Har... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5536_chunk_0 | Bolza surface | In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), is a compact Riemann surface of genus 2 {\displaystyle 2} with the highest possible order of the conformal automorphism group in this genus, namely G L 2 ( 3 ) {\displaystyle GL_{2}(3)} of order 48 (the g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5537_chunk_0 | Bolza surface | The Bolza surface is the smooth completion of the affine curve. Of all genus 2 {\displaystyle 2} hyperbolic surfaces, the Bolza surface maximizes the length of the systole (Schmutz 1993). As a hyperelliptic Riemann surface, it arises as the ramified double cover of the Riemann sphere, with ramification locus at the six... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5538_chunk_0 | Bombieri–Vinogradov theorem | In mathematics, the Bombieri–Vinogradov theorem (sometimes simply called Bombieri's theorem) is a major result of analytic number theory, obtained in the mid-1960s, concerning the distribution of primes in arithmetic progressions, averaged over a range of moduli. The first result of this kind was obtained by Mark Barba... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5539_chunk_0 | Bombieri–Vinogradov theorem | This result is a major application of the large sieve method, which developed rapidly in the early 1960s, from its beginnings in work of Yuri Linnik two decades earlier. Besides Bombieri, Klaus Roth was working in this area. In the late 1960s and early 1970s, many of the key ingredients and estimates were simplified by... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5540_chunk_0 | Bony–Brezis theorem | In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant under the flow defined by a vector field, namely at each point of the closed set the vector field must have non-positive... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5541_chunk_0 | Boole polynomials | In mathematics, the Boole polynomials sn(x) are polynomials given by the generating function ∑ s n ( x ) t n / n ! = ( 1 + t ) x 1 + ( 1 + t ) λ {\displaystyle \displaystyle \sum s_{n}(x)t^{n}/n!={\frac {(1+t)^{x}}{1+(1+t)^{\lambda }}}} (Roman 1984, 4.5), (Jordan 1939, sections 113–117). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5542_chunk_0 | Boolean prime ideal theorem | In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5543_chunk_0 | Boolean prime ideal theorem | Although the various prime ideal theorems may appear simple and intuitive, they cannot be deduced in general from the axioms of Zermelo–Fraenkel set theory without the axiom of choice (abbreviated ZF). Instead, some of the statements turn out to be equivalent to the axiom of choice (AC), while others—the Boolean prime ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5544_chunk_0 | Borel fixed-point theorem | In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by Armand Borel (1956). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5545_chunk_0 | Borell–Brascamp–Lieb inequality | In mathematics, the Borell–Brascamp–Lieb inequality is an integral inequality due to many different mathematicians but named after Christer Borell, Herm Jan Brascamp and Elliott Lieb. The result was proved for p > 0 by Henstock and Macbeath in 1953. The case p = 0 is known as the Prékopa–Leindler inequality and was re-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5546_chunk_0 | Borel–Carathéodory theorem | In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5547_chunk_0 | Borromean rings | They have been used in Christian symbolism as a sign of the Trinity, and in modern commerce as the logo of Ballantine beer, giving them the alternative name Ballantine rings. Physical instances of the Borromean rings have been made from linked DNA or other molecules, and they have analogues in the Efimov state and Borr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5548_chunk_0 | Borsuk–Ulam theorem | In mathematics, the Borsuk–Ulam theorem states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point. Here, two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center. Formally: if f: S n → R n {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5549_chunk_0 | Borsuk–Ulam theorem | The same is true for any circle. This assumes the temperature varies continuously in space, which is, however, not always the case.The case n = 2 {\displaystyle n=2} is often illustrated by saying that at any moment, there is always a pair of antipodal points on the Earth's surface with equal temperatures and equal bar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5550_chunk_0 | Bott periodicity theorem | In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959), which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5551_chunk_0 | Bott residue formula | In mathematics, the Bott residue formula, introduced by Bott (1967), describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5552_chunk_0 | Bourbaki–Witt theorem | In mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed point theorem for partially ordered sets. It states that if X is a non-empty chain complete poset, and f: X → X {\displaystyle f:X\to X} such that f ( x ) ≥ x {\displaystyle f(x)\geq x} for all x , {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5553_chunk_0 | Brascamp–Lieb inequality | In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . It generalizes the Loomis–Whitney inequality and Hölder's inequality. The second is a result of probabilit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5554_chunk_0 | Brauer group | In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer. The Brauer group arose out of attempts to classify division algebras ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5555_chunk_0 | Brauer–Nesbitt theorem | In mathematics, the Brauer–Nesbitt theorem can refer to several different theorems proved by Richard Brauer and Cecil J. Nesbitt in the representation theory of finite groups. In modular representation theory, the Brauer–Nesbitt theorem on blocks of defect zero states that a character whose order is divisible by the hi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5556_chunk_0 | Brauer–Nesbitt theorem | Let G {\displaystyle G} be a group and E {\displaystyle E} be some field. If ρ i: G → G L n ( E ) , i = 1 , 2 {\displaystyle \rho _{i}:G\to GL_{n}(E),i=1,2} are two finite-dimensional semisimple representations such that the characteristic polynomials of ρ 1 ( g ) {\displaystyle \rho _{1}(g)} and ρ 2 ( g ) {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5557_chunk_0 | Brauer–Siegel theorem | In mathematics, the Brauer–Siegel theorem, named after Richard Brauer and Carl Ludwig Siegel, is an asymptotic result on the behaviour of algebraic number fields, obtained by Richard Brauer and Carl Ludwig Siegel. It attempts to generalise the results known on the class numbers of imaginary quadratic fields, to a more ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5558_chunk_0 | Brauer–Siegel theorem | {\displaystyle {\frac {}{\log |D_{i}|}}\to 0{\text{ as }}i\to \infty .} Assuming that, and the algebraic hypothesis that Ki is a Galois extension of Q, the conclusion is that log ( h i R i ) log | D i | → 1 as i → ∞ {\displaystyle {\frac {\log(h_{i}R_{i})}{\log {\sqrt {|D_{i}|}}}}\to 1{\text{ as }}i\to \infty } whe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5559_chunk_0 | Brauer–Siegel theorem | If one assumes that all the degrees {\displaystyle } are bounded above by a uniform constant N, then one may drop the assumption of normality - this is what is actually proved in Brauer's paper. This result is ineffective, as indeed was the result on quadratic fields on which it built. Effective results in the same di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5560_chunk_0 | Brauer–Suzuki theorem | In mathematics, the Brauer–Suzuki theorem, proved by Brauer & Suzuki (1959), Suzuki (1962), Brauer (1964), states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a center of order 2. In particular, such a group cannot be simple. A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5561_chunk_0 | Brauer–Suzuki–Wall theorem | In mathematics, the Brauer–Suzuki–Wall theorem, proved by Brauer, Suzuki & Wall (1958), characterizes the one-dimensional unimodular projective groups over finite fields. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5562_chunk_0 | Clifford invariant | In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field F is a group BW(F) classifying finite-dimensional graded central division algebras over the field. It was first defined by Terry Wall (1964) as a generalization of the Brauer group. The Brauer group of a field F is the set of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5563_chunk_0 | Bretherton equation | In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964: u t t + u x x + u x x x x + u = u p , {\displaystyle u_{tt}+u_{xx}+u_{xxxx}+u=u^{p},} with p {\displaystyle p} integer and p ≥ 2. {\displaystyle p\geq 2.} While u t , u x {\displaystyle u_{t},u... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5564_chunk_0 | Bretherton equation | The original equation studied by Bretherton has quadratic nonlinearity, p = 2. {\displaystyle p=2.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5565_chunk_0 | Bretherton equation | Nayfeh treats the case p = 3 {\displaystyle p=3} with two different methods: Whitham's averaged Lagrangian method and the method of multiple scales.The Bretherton equation is a model equation for studying weakly-nonlinear wave dispersion. It has been used to study the interaction of harmonics by nonlinear resonance. Br... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5566_chunk_0 | Browder–Minty theorem | In mathematics, the Browder–Minty theorem (sometimes called the Minty–Browder theorem) states that a bounded, continuous, coercive and monotone function T from a real, separable reflexive Banach space X into its continuous dual space X∗ is automatically surjective. That is, for each continuous linear functional g ∈ X∗,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5567_chunk_0 | Bruhat decomposition | In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) G = BWB of certain algebraic groups G into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5568_chunk_0 | Brunn–Minkowski theorem | In mathematics, the Brunn–Minkowski theorem (or Brunn–Minkowski inequality) is an inequality relating the volumes (or more generally Lebesgue measures) of compact subsets of Euclidean space. The original version of the Brunn–Minkowski theorem (Hermann Brunn 1887; Hermann Minkowski 1896) applied to convex sets; the gene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5569_chunk_0 | Buckmaster equation | In mathematics, the Buckmaster equation is a second-order nonlinear partial differential equation, named after John D. Buckmaster, who derived the equation in 1977. The equation models the surface of a thin sheet of viscous liquid. The equation was derived earlier by S. H. Smith and by P Smith, but these earlier deriva... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5570_chunk_0 | Burkill integral | In mathematics, the Burkill integral is an integral introduced by Burkill (1924a, 1924b) for calculating areas. It is a special case of the Kolmogorov integral. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5571_chunk_0 | Burr–Erdős conjecture | In mathematics, the Burr–Erdős conjecture was a problem concerning the Ramsey number of sparse graphs. The conjecture is named after Stefan Burr and Paul Erdős, and is one of many conjectures named after Erdős; it states that the Ramsey number of graphs in any sparse family of graphs should grow linearly in the number ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5572_chunk_0 | Bussgang theorem | In mathematics, the Bussgang theorem is a theorem of stochastic analysis. The theorem states that the cross-correlation of a Gaussian signal before and after it has passed through a nonlinear operation are equal up to a constant. It was first published by Julian J. Bussgang in 1952 while he was at the Massachusetts Ins... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5573_chunk_0 | Butcher group | In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by Hairer & Wanner (1974), is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic forma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5574_chunk_0 | Butcher group | Connes & Kreimer (1999) pointed out that the Butcher group is the group of characters of the Hopf algebra of rooted trees that had arisen independently in their own work on renormalization in quantum field theory and Connes' work with Moscovici on local index theorems. This Hopf algebra, often called the Connes–Kreimer... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5575_chunk_0 | Calderón–Zygmund theory | In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function f: Rd → C, where Rd denotes Euclidean space and C denotes the complex numbers, the lemma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5576_chunk_0 | Calkin correspondence | In mathematics, the Calkin correspondence, named after mathematician John Williams Calkin, is a bijective correspondence between two-sided ideals of bounded linear operators of a separable infinite-dimensional Hilbert space and Calkin sequence spaces (also called rearrangement invariant sequence spaces). The correspond... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5577_chunk_0 | Calogero–Degasperis–Fokas equation | In mathematics, the Calogero–Degasperis–Fokas equation is the nonlinear partial differential equation u t = u x x x − 1 8 u x 3 + u x ( A e u + B e − u ) . {\displaystyle \displaystyle u_{t}=u_{xxx}-{\frac {1}{8}}u_{x}^{3}+u_{x}\left(Ae^{u}+Be^{-u}\right).} This equation was named after F. Calogero, A. Degasperis, and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5578_chunk_0 | Cameron–Martin formula | In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5579_chunk_0 | Cantor function | In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure. Though it is continuous everywhere and has zero derivative almost everywhere,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5580_chunk_0 | Cantor function | Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow. It is also called the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5581_chunk_0 | Cantor dust | Cantor mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). By a theorem of L. E. J. Brouwer,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5582_chunk_0 | Carathéodory kernel theorem | In mathematics, the Carathéodory kernel theorem is a result in complex analysis and geometric function theory established by the Greek mathematician Constantin Carathéodory in 1912. The uniform convergence on compact sets of a sequence of holomorphic univalent functions, defined on the unit disk in the complex plane an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5583_chunk_0 | Carathéodory metric | In mathematics, the Carathéodory metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry. It is named after the Greek mathematician Constantin Carathéodory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5584_chunk_0 | Caristi fixed-point theorem | In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ε {\displaystyle \varepsilon } -variational principle of Ekeland (1974, 1979). The c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5585_chunk_0 | Carleson–Jacobs theorem | In mathematics, the Carleson–Jacobs theorem, introduced by Carleson and Jacobs (1972), describes the best approximation to a continuous function on the unit circle by a function in a Hardy space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5586_chunk_0 | Carlitz exponential | In mathematics, the Carlitz exponential is a characteristic p analogue to the usual exponential function studied in real and complex analysis. It is used in the definition of the Carlitz module – an example of a Drinfeld module. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5587_chunk_0 | Cartan involution | In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5588_chunk_0 | Cartan model | In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5589_chunk_0 | Cartan–Dieudonné theorem | In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described as the composition of at most n reflections. The notion of a symmetric bilinear space is a generalization of Euclidean ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5590_chunk_0 | Cartan–Dieudonné theorem | For example, in the two-dimensional Euclidean plane, every orthogonal transformation is either a reflection across a line through the origin or a rotation about the origin (which can be written as the composition of two reflections). Any arbitrary composition of such rotations and reflections can be rewritten as a comp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5591_chunk_0 | Cartan–Hadamard conjecture | In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after F... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5592_chunk_0 | Cartan–Kähler theorem | In mathematics, the Cartan–Kähler theorem is a major result on the integrability conditions for differential systems, in the case of analytic functions, for differential ideals I {\displaystyle I} . It is named for Élie Cartan and Erich Kähler. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5593_chunk_0 | Castelnuovo–de Franchis theorem | In mathematics, the Castelnuovo–de Franchis theorem is a classical result on complex algebraic surfaces. Let X be such a surface, projective and non-singular, and let ω1 and ω2be two differentials of the first kind on X which are linearly independent but with wedge product 0. Then this data can be represented as a pull... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5594_chunk_0 | Cauchy condensation test | In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence f ( n ) {\displaystyle f(n)} of non-negative real numbers, the series ∑ n = 1 ∞ f ( n ) {\textstyle \sum \limits _{n=1}^{\infty }f(n)} converges if and only ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5595_chunk_0 | Cauchy integral | In mathematics, the Cauchy integral theorem (also known as the Cauchy–Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if f ( z ) {\displaystyle f(z)} is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5596_chunk_0 | Cauchy principal part | In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the singularity (so the singularity is n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5597_chunk_0 | Cauchy–Hadamard theorem | In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard rediscovered it. Hadama... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5598_chunk_0 | Cauchy–Kowalevski theorem | In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by Augustin Cauchy (1842), and the full result by Sofya ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_5599_chunk_0 | Cayley–Bacharach theorem | In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P2. The original form states: Assume that two cubics C1 and C2 in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then eve... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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