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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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Bernoulli_polynomials |
In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical systems. Many important dynamical systems (such as Axiom A systems) exhibit a... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Bernoulli_scheme |
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://en.wikipedia.org/wiki/Bernstein–Sato_polynomial |
In mathematics, the Bernstein–Zelevinsky classification, introduced by Bernstein and Zelevinsky (1977) and Zelevinsky (1980), classifies the irreducible complex smooth representations of a general linear group over a local field in terms of cuspidal representations. | https://en.wikipedia.org/wiki/Bernstein–Zelevinsky_classification |
In mathematics, the Berry–Robbins problem asks whether there is a continuous map from configurations of n points in R3 to the flag manifold U(n)/Tn that is compatible with the action of the symmetric group on n points. It was posed by Berry and Robbins (1997) and solved positively by Atiyah (2000). | https://en.wikipedia.org/wiki/Berry–Robbins_problem |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Besicovitch_inequality |
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://en.wikipedia.org/wiki/Besov_space |
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://en.wikipedia.org/wiki/Bessel_polynomials |
k ! ( x 2 ) k . {\displaystyle y_{n}(x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k! | https://en.wikipedia.org/wiki/Bessel_polynomials |
}}\,\left({\frac {x}{2}}\right)^{k}.} Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials: 8: 15 θ n ( x ) = x n y n ( 1 / x ) = ∑ k = 0 n ( n + k ) ! ( n − k ) ! | https://en.wikipedia.org/wiki/Bessel_polynomials |
k ! x n − k 2 k . {\displaystyle \theta _{n}(x)=x^{n}\,y_{n}(1/x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k! | https://en.wikipedia.org/wiki/Bessel_polynomials |
}}\,{\frac {x^{n-k}}{2^{k}}}.} The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is y 3 ( x ) = 15 x 3 + 15 x 2 + 6 x + 1 {\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1} while the third-degree reverse Bessel polynomial is θ 3 ( x ) = ... | https://en.wikipedia.org/wiki/Bessel_polynomials |
In mathematics, the Bessel potential is a potential (named after Friedrich Wilhelm Bessel) similar to the Riesz potential but with better decay properties at infinity. If s is a complex number with positive real part then the Bessel potential of order s is the operator ( I − Δ ) − s / 2 {\displaystyle (I-\Delta )^{-s/2... | https://en.wikipedia.org/wiki/Bessel_potential |
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://en.wikipedia.org/wiki/Bessel–Maitland_function |
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://en.wikipedia.org/wiki/Beurling–Lax_theorem |
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://en.wikipedia.org/wiki/Bhatia–Davis_inequality |
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://en.wikipedia.org/wiki/Bianchi_classification |
In mathematics, the Bing–Borsuk conjecture states that every n {\displaystyle n} -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and 2, and it is known that the 3-dimensional version of the conjecture implies the Poincaré conjecture... | https://en.wikipedia.org/wiki/Bing–Borsuk_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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture |
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://en.wikipedia.org/wiki/Birkhoff–Grothendieck_theorem |
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://en.wikipedia.org/wiki/Birman–Wenzl_algebra |
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://en.wikipedia.org/wiki/Bishop-Gromov_inequality |
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://en.wikipedia.org/wiki/Bishop–Phelps_theorem |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Bismut_connection |
Denote the anti-symmetrization as T ( Z , X , Y ) + cyc~in~ X , Y , Z = T ( Z , X , Y ) + S ( Z , X , Y ) {\displaystyle T(Z,X,Y)+{\textrm {cyc~in~}}X,Y,Z=T(Z,X,Y)+S(Z,X,Y)} , with S {\displaystyle S} given explicitly as S ( Z , X , Y ) = − 1 2 ⟨ X , J ( ∇ Y J ) Z ⟩ − 1 2 ⟨ Y , J ( ∇ Z J ) X ⟩ . {\displaystyle S(Z,X,Y)... | https://en.wikipedia.org/wiki/Bismut_connection |
S ( Z , X , J Y ) + S ( J Z , X , Y ) = − 1 2 ⟨ J X , ( − ( ∇ J Y J ) Z − ( J ∇ Z J ) Y + ( J ∇ Y J ) Z + ( ∇ J Z J ) Y ) ⟩ = − 1 2 ⟨ J X , R e ( ( 1 − i J ) ) ⟩ . {\displaystyle {\begin{aligned}S(Z,X,JY)+S(JZ,X,Y)&=-{\frac {1}{2}}\langle JX,{\big (}-(\nabla _{JY}J)Z-(J\nabla _{Z}J)Y+(J\nabla _{Y}J)Z+(\nabla _{JZ}J)Y{... | https://en.wikipedia.org/wiki/Bismut_connection |
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://en.wikipedia.org/wiki/Bloch_group |
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://en.wikipedia.org/wiki/Blumberg_theorem |
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://en.wikipedia.org/wiki/Bochner_integral |
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://en.wikipedia.org/wiki/Bochner–Kodaira–Nakano_identity |
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://en.wikipedia.org/wiki/Bochner–Martinelli_formula |
In mathematics, the Bockstein spectral sequence is a spectral sequence relating the homology with mod p coefficients and the homology reduced mod p. It is named after Meyer Bockstein. | https://en.wikipedia.org/wiki/Bockstein_spectral_sequence |
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://en.wikipedia.org/wiki/Bogomolov_conjecture |
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://en.wikipedia.org/wiki/Bogomolov–Miyaoka–Yau_inequality |
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://en.wikipedia.org/wiki/Bohr_compactification |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Bolza_surface |
In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } (there is also a version for non homogeneous univariate polynomials). This norm has many remarkable properties, the most important bei... | https://en.wikipedia.org/wiki/Bombieri_norm |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Bombieri–Vinogradov_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://en.wikipedia.org/wiki/Bony–Brezis_theorem |
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://en.wikipedia.org/wiki/Boole_polynomials |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Boolean_prime_ideal_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://en.wikipedia.org/wiki/Borel_fixed-point_theorem |
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://en.wikipedia.org/wiki/Borell–Brascamp–Lieb_inequality |
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://en.wikipedia.org/wiki/Borel–Carathéodory_theorem |
In mathematics, the Borromean rings are three simple closed curves in three-dimensional space that are topologically linked and cannot be separated from each other, but that break apart into two unknotted and unlinked loops when any one of the three is cut or removed. Most commonly, these rings are drawn as three circl... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Borromean_rings |
It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In knot theory, the Borromean rings can be proved to be linked by counting their Fox n-colorings. As links, they are Brunnian, alte... | https://en.wikipedia.org/wiki/Borromean_rings |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Borsuk–Ulam_theorem |
In mathematics, the Bott cannibalistic class, introduced by Raoul Bott (1962), is an element θ k ( V ) {\displaystyle \theta _{k}(V)} of the representation ring of a compact Lie group that describes the action of the Adams operation ψ k {\displaystyle \psi ^{k}} on the Thom class λ V {\displaystyle \lambda _{V}} of a c... | https://en.wikipedia.org/wiki/Bott_cannibalistic_class |
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://en.wikipedia.org/wiki/Bott_periodicity_theorem |
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://en.wikipedia.org/wiki/Bott_residue_formula |
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://en.wikipedia.org/wiki/Bourbaki–Witt_theorem |
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://en.wikipedia.org/wiki/Brascamp–Lieb_inequality |
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://en.wikipedia.org/wiki/Brauer_group |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Brauer–Nesbitt_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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Brauer–Siegel_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://en.wikipedia.org/wiki/Brauer–Suzuki_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://en.wikipedia.org/wiki/Brauer–Suzuki–Wall_theorem |
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://en.wikipedia.org/wiki/Clifford_invariant |
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://en.wikipedia.org/wiki/Bretherton_equation |
The original equation studied by Bretherton has quadratic nonlinearity, p = 2. {\displaystyle p=2.} | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Bretherton_equation |
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://en.wikipedia.org/wiki/Browder–Minty_theorem |
In mathematics, the Brown measure of an operator in a finite factor is a probability measure on the complex plane which may be viewed as an analog of the spectral counting measure (based on algebraic multiplicity) of matrices. It is named after Lawrence G. Brown. | https://en.wikipedia.org/wiki/Brown_measure |
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://en.wikipedia.org/wiki/Bruhat_decomposition |
In mathematics, the Bruhat order (also called strong order or strong Bruhat order or Chevalley order or Bruhat–Chevalley order or Chevalley–Bruhat order) is a partial order on the elements of a Coxeter group, that corresponds to the inclusion order on Schubert varieties. | https://en.wikipedia.org/wiki/Weak_Bruhat_order |
In mathematics, the Brumer bound is a bound for the rank of an elliptic curve, proved by Brumer (1992). | https://en.wikipedia.org/wiki/Brumer_bound |
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://en.wikipedia.org/wiki/Brunn–Minkowski_theorem |
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://en.wikipedia.org/wiki/Buckmaster_equation |
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://en.wikipedia.org/wiki/Burkill_integral |
In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic ring structure is a more recent development, due to Solomon (1967). | https://en.wikipedia.org/wiki/Burnside_ring |
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://en.wikipedia.org/wiki/Burr–Erdős_conjecture |
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://en.wikipedia.org/wiki/Bussgang_theorem |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Butcher_group |
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://en.wikipedia.org/wiki/Calderón–Zygmund_theory |
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://en.wikipedia.org/wiki/Calkin_correspondence |
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://en.wikipedia.org/wiki/Calogero–Degasperis–Fokas_equation |
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://en.wikipedia.org/wiki/Cameron–Martin_formula |
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://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Cantor_function |
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883.Through consideration of this set, Cantor and others helped lay the foundations ... | https://en.wikipedia.org/wiki/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://en.wikipedia.org/wiki/Cantor_dust |
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://en.wikipedia.org/wiki/Carathéodory_kernel_theorem |
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://en.wikipedia.org/wiki/Carathéodory_metric |
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://en.wikipedia.org/wiki/Caristi_fixed-point_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://en.wikipedia.org/wiki/Carleson–Jacobs_theorem |
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