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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 function and the Hurwitz zeta function. They are an Appell sequence (i.e. a Sheffer sequence for the ordinary derivative operator).
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https://en.wikipedia.org/wiki/Bernoulli_polynomials
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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 polynomials.
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https://en.wikipedia.org/wiki/Bernoulli_polynomials
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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 repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift.
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https://en.wikipedia.org/wiki/Bernoulli_scheme
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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.
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https://en.wikipedia.org/wiki/Bernoulli_scheme
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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 to the Bernstein polynomials used in approximation theory. It has applications to singularity theory, monodromy theory, and quantum field theory. Severino Coutinho (1995) gives an elementary introduction, while Armand Borel (1987) and Masaki Kashiwara (2003) give more advanced accounts.
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https://en.wikipedia.org/wiki/Bernstein–Sato_polynomial
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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.
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https://en.wikipedia.org/wiki/Bernstein–Zelevinsky_classification
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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).
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https://en.wikipedia.org/wiki/Berry–Robbins_problem
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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 the distance between opposite faces of the cube.
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https://en.wikipedia.org/wiki/Besicovitch_inequality
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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 {\displaystyle \partial ^{n}} , define Then ∏ i d i ≥ V o l ( M ) {\displaystyle \prod _{i}d_{i}\geq Vol(M)} . The Besicovitch inequality was used to prove systolic inequalities on surfaces.
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https://en.wikipedia.org/wiki/Besicovitch_inequality
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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 spaces such as Sobolev spaces and are effective at measuring regularity properties of functions.
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https://en.wikipedia.org/wiki/Besov_space
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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 ) !
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https://en.wikipedia.org/wiki/Bessel_polynomials
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k ! ( x 2 ) k . {\displaystyle y_{n}(x)=\sum _{k=0}^{n}{\frac {(n+k)!}{(n-k)!k!
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https://en.wikipedia.org/wiki/Bessel_polynomials
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}}\,\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 ) !
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https://en.wikipedia.org/wiki/Bessel_polynomials
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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!
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https://en.wikipedia.org/wiki/Bessel_polynomials
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}}\,{\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 ) = x 3 + 6 x 2 + 15 x + 15. {\displaystyle \theta _{3}(x)=x^{3}+6x^{2}+15x+15.} The reverse Bessel polynomial is used in the design of Bessel electronic filters.
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https://en.wikipedia.org/wiki/Bessel_polynomials
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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}} where Δ is the Laplace operator and the fractional power is defined using Fourier transforms. Yukawa potentials are particular cases of Bessel potentials for s = 2 {\displaystyle s=2} in the 3-dimensional space.
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https://en.wikipedia.org/wiki/Bessel_potential
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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 J μ , ν ( z ) = ∑ k ≥ 0 ( − z ) k Γ ( k μ + ν + 1 ) k ! . {\displaystyle J^{\mu ,\nu }(z)=\sum _{k\geq 0}{\frac {(-z)^{k}}{\Gamma (k\mu +\nu +1)k!}}.}
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https://en.wikipedia.org/wiki/Bessel–Maitland_function
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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}(\mathbb {D} ,\mathbb {C} ),} for some inner function θ {\displaystyle \theta } .
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https://en.wikipedia.org/wiki/Beurling–Lax_theorem
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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.
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https://en.wikipedia.org/wiki/Bhatia–Davis_inequality
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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 infinite families, giving 9 instead of 11 classes.) The classification is important in geometry and physics, because the associated Lie groups serve as symmetry groups of 3-dimensional Riemannian manifolds. It is named for Luigi Bianchi, who worked it out in 1898. The term "Bianchi classification" is also used for similar classifications in other dimensions and for classifications of complex Lie algebras.
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https://en.wikipedia.org/wiki/Bianchi_classification
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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.
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https://en.wikipedia.org/wiki/Bing–Borsuk_conjecture
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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. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2023, only special cases of the conjecture have been proven.
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https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
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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) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K (Wiles 2006). The conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof.
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https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
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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, Theorem 2.1), and is more or less equivalent to Birkhoff factorization introduced by George David Birkhoff (1909).
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https://en.wikipedia.org/wiki/Birkhoff–Grothendieck_theorem
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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 Hecke algebra of the symmetric group as a quotient. It is related to the Kauffman polynomial of a link. It is a deformation of the Brauer algebra in much the same way that Hecke algebras are deformations of the group algebra of the symmetric group.
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https://en.wikipedia.org/wiki/Birman–Wenzl_algebra
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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.
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https://en.wikipedia.org/wiki/Bishop-Gromov_inequality
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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.
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https://en.wikipedia.org/wiki/Bishop–Phelps_theorem
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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 ) {\displaystyle T(X,Y)} contracted with the metric, i.e. T ( X , Y , Z ) = g ( T ( X , Y ) , Z ) {\displaystyle T(X,Y,Z)=g(T(X,Y),Z)} , is totally skew-symmetric.Bismut has used this connection when proving a local index formula for the Dolbeault operator on non-Kähler manifolds. Bismut connection has applications in type II and heterotic string theory. The explicit construction goes as follows. Let ⟨ − , − ⟩ {\displaystyle \langle -,-\rangle } denote the pairing of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. ⟨ X , J Y ⟩ = − ⟨ J X , Y ⟩ {\displaystyle \langle X,JY\rangle =-\langle JX,Y\rangle } .
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https://en.wikipedia.org/wiki/Bismut_connection
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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 connection ∇ + T {\displaystyle \nabla +T} still preserves the metric.
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https://en.wikipedia.org/wiki/Bismut_connection
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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 also preserves the complex structure. However, the tensor T {\displaystyle T} is not yet totally anti-symmetric; the anti-symmetrization will lead to the Nijenhuis tensor.
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https://en.wikipedia.org/wiki/Bismut_connection
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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)=-{\frac {1}{2}}\langle X,J(\nabla _{Y}J)Z\rangle -{\frac {1}{2}}\langle Y,J(\nabla _{Z}J)X\rangle .} S {\displaystyle S} still preserves the complex structure, i.e. S ( Z , X , J Y ) = − S ( J Z , X , Y ) {\displaystyle S(Z,X,JY)=-S(JZ,X,Y)} .
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https://en.wikipedia.org/wiki/Bismut_connection
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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{\big )}\rangle \\&=-{\frac {1}{2}}\langle JX,Re{\big (}(1-iJ){\big )}\rangle .\end{aligned}}} So if J {\displaystyle J} is integrable, then above term vanishes, and the connection Γ β γ α + T β γ α + S β γ α . {\displaystyle \Gamma _{\beta \gamma }^{\alpha }+T_{~\beta \gamma }^{\alpha }+S_{~\beta \gamma }^{\alpha }.} gives the Bismut connection.
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https://en.wikipedia.org/wiki/Bismut_connection
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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.
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https://en.wikipedia.org/wiki/Bloch_group
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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.
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https://en.wikipedia.org/wiki/Blumberg_theorem
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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.
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https://en.wikipedia.org/wiki/Bochner_integral
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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 the manifold. It is named after Salomon Bochner, Kunihiko Kodaira, and Shigeo Nakano.
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https://en.wikipedia.org/wiki/Bochner–Kodaira–Nakano_identity
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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).
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https://en.wikipedia.org/wiki/Bochner–Martinelli_formula
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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.
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https://en.wikipedia.org/wiki/Bockstein_spectral_sequence
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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 varieties was also proved by Zhang in 1998.
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https://en.wikipedia.org/wiki/Bogomolov_conjecture
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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 independently by Shing-Tung Yau (1977, 1978) and Yoichi Miyaoka (1977), after Antonius Van de Ven (1966) and Fedor Bogomolov (1978) proved weaker versions with the constant 3 replaced by 8 and 4. Armand Borel and Friedrich Hirzebruch showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: William E. Lang (1983) and Robert W. Easton (2008) gave examples of surfaces in characteristic p, such as generalized Raynaud surfaces, for which it fails.
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https://en.wikipedia.org/wiki/Bogomolov–Miyaoka–Yau_inequality
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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 Harald Bohr who pioneered the study of almost periodic functions, on the real line.
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https://en.wikipedia.org/wiki/Bohr_compactification
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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 general linear group of 2 × 2 {\displaystyle 2\times 2} matrices over the finite field F 3 {\displaystyle \mathbb {F} _{3}} ). The full automorphism group (including reflections) is the semi-direct product G L 2 ( 3 ) ⋊ Z 2 {\displaystyle GL_{2}(3)\rtimes \mathbb {Z} _{2}} of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation y 2 = x 5 − x {\displaystyle y^{2}=x^{5}-x} in C 2 {\displaystyle \mathbb {C} ^{2}} .
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https://en.wikipedia.org/wiki/Bolza_surface
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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 vertices of a regular octahedron inscribed in the sphere, as can be readily seen from the equation above. The Bolza surface has attracted the attention of physicists, as it provides a relatively simple model for quantum chaos; in this context, it is usually referred to as the Hadamard–Gutzwiller model. The spectral theory of the Laplace–Beltrami operator acting on functions on the Bolza surface is of interest to both mathematicians and physicists, since the surface is conjectured to maximize the first positive eigenvalue of the Laplacian among all compact, closed Riemann surfaces of genus 2 {\displaystyle 2} with constant negative curvature.
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https://en.wikipedia.org/wiki/Bolza_surface
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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 being listed in this article.
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https://en.wikipedia.org/wiki/Bombieri_norm
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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 Barban in 1961 and the Bombieri–Vinogradov theorem is a refinement of Barban's result. The Bombieri–Vinogradov theorem is named after Enrico Bombieri and A. I. Vinogradov, who published on a related topic, the density hypothesis, in 1965.
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https://en.wikipedia.org/wiki/Bombieri–Vinogradov_theorem
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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 Patrick X. Gallagher.
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https://en.wikipedia.org/wiki/Bombieri–Vinogradov_theorem
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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 inner product with any exterior normal vector to the set. A vector is an exterior normal at a point of the closed set if there is a real-valued continuously differentiable function maximized locally at the point with that vector as its derivative at the point. If the closed subset is a smooth submanifold with boundary, the condition states that the vector field should not point outside the subset at boundary points. The generalization to non-smooth subsets is important in the theory of partial differential equations. The theorem had in fact been previously discovered by Mitio Nagumo in 1942 and is also known as the Nagumo theorem.
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https://en.wikipedia.org/wiki/Bony–Brezis_theorem
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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).
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https://en.wikipedia.org/wiki/Boole_polynomials
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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, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article focuses on prime ideal theorems from order theory.
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https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem
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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 ideal theorem, for instance—represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF + AC (ZFC) that the Boolean prime ideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT (for Boolean algebras) are sometimes used to refer to this additional axiom.
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https://en.wikipedia.org/wiki/Boolean_prime_ideal_theorem
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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).
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https://en.wikipedia.org/wiki/Borel_fixed-point_theorem
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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-discovered by Brascamp and Lieb in 1976, when they proved the general version below; working independently, Borell had done the same in 1975. The nomenclature of "Borell–Brascamp–Lieb inequality" is due to Cordero-Erausquin, McCann and Schmuckenschläger, who in 2001 generalized the result to Riemannian manifolds such as the sphere and hyperbolic space.
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https://en.wikipedia.org/wiki/Borell–Brascamp–Lieb_inequality
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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.
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https://en.wikipedia.org/wiki/Borel–Carathéodory_theorem
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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 circles in the plane, in the pattern of a Venn diagram, alternatingly crossing over and under each other at the points where they cross. Other triples of curves are said to form the Borromean rings as long as they are topologically equivalent to the curves depicted in this drawing. The Borromean rings are named after the Italian House of Borromeo, who used the circular form of these rings as an element of their coat of arms, but designs based on the Borromean rings have been used in many cultures, including by the Norsemen and in Japan.
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https://en.wikipedia.org/wiki/Borromean_rings
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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 Borromean nuclei, both of which have three components bound to each other although no two of them are bound. Geometrically, the Borromean rings may be realized by linked ellipses, or (using the vertices of a regular icosahedron) by linked golden rectangles.
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https://en.wikipedia.org/wiki/Borromean_rings
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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, alternating, algebraic, and hyperbolic. In arithmetic topology, certain triples of prime numbers have analogous linking properties to the Borromean rings.
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https://en.wikipedia.org/wiki/Borromean_rings
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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 {\displaystyle f:S^{n}\to \mathbb {R} ^{n}} is continuous then there exists an x ∈ S n {\displaystyle x\in S^{n}} such that: f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} . The case n = 1 {\displaystyle n=1} can be illustrated by saying that there always exist a pair of opposite points on the Earth's equator with the same temperature.
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https://en.wikipedia.org/wiki/Borsuk–Ulam_theorem
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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 barometric pressures, assuming that both parameters vary continuously in space. Since temperature, pressure or other such physical variables do not necessarily vary continuously, the predictions of the theorem are unlikely to be true in some necessary sense (as following from a mathematical necessity).The Borsuk–Ulam theorem has several equivalent statements in terms of odd functions. Recall that S n {\displaystyle S^{n}} is the n-sphere and B n {\displaystyle B^{n}} is the n-ball: If g: S n → R n {\displaystyle g:S^{n}\to \mathbb {R} ^{n}} is a continuous odd function, then there exists an x ∈ S n {\displaystyle x\in S^{n}} such that: g ( x ) = 0 {\displaystyle g(x)=0} . If g: B n → R n {\displaystyle g:B^{n}\to \mathbb {R} ^{n}} is a continuous function which is odd on S n − 1 {\displaystyle S^{n-1}} (the boundary of B n {\displaystyle B^{n}} ), then there exists an x ∈ B n {\displaystyle x\in B^{n}} such that: g ( x ) = 0 {\displaystyle g(x)=0} .
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https://en.wikipedia.org/wiki/Borsuk–Ulam_theorem
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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 complex representation V {\displaystyle V} . The term "cannibalistic" for these classes was introduced by Frank Adams (1965, p.151).
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https://en.wikipedia.org/wiki/Bott_cannibalistic_class
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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 groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory. There are corresponding period-8 phenomena for the matching theories, (real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of spheres.
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https://en.wikipedia.org/wiki/Bott_periodicity_theorem
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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.
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https://en.wikipedia.org/wiki/Bott_residue_formula
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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 , {\displaystyle x,} then f has a fixed point. Such a function f is called inflationary or progressive.
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https://en.wikipedia.org/wiki/Bourbaki–Witt_theorem
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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 probability theory which gives a concentration inequality for log-concave probability distributions. Both are named after Herm Jan Brascamp and Elliott H. Lieb.
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https://en.wikipedia.org/wiki/Brascamp–Lieb_inequality
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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 over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles.
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https://en.wikipedia.org/wiki/Brauer_group
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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 highest power of a prime p dividing the order of a finite group remains irreducible when reduced mod p and vanishes on all elements whose order is divisible by p. Moreover, it belongs to a block of defect zero. A block of defect zero contains only one ordinary character and only one modular character. Another version states that if k is a field of characteristic zero, A is a k-algebra, V, W are semisimple A-modules which are finite dimensional over k, and TrV = TrW as elements of Homk(A,k), then V and W are isomorphic as A-modules.
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https://en.wikipedia.org/wiki/Brauer–Nesbitt_theorem
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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 ) {\displaystyle \rho _{2}(g)} coincide for all g ∈ G {\displaystyle g\in G} , then ρ 1 {\displaystyle \rho _{1}} and ρ 2 {\displaystyle \rho _{2}} are isomorphic representations. If c h a r ( E ) = 0 {\displaystyle char(E)=0} or c h a r ( E ) > n {\displaystyle char(E)>n} , then the condition on the characteristic polynomials can be changed to the condition that Tr ρ 1 ( g ) {\displaystyle \rho _{1}(g)} =Tr ρ 2 ( g ) {\displaystyle \rho _{2}(g)} for all g ∈ G {\displaystyle g\in G} . As a consequence, let ρ: G a l ( K s e p / K ) → G L n ( Q ¯ l ) {\displaystyle \rho :Gal(K^{\rm {sep}}/K)\to GL_{n}({\overline {\mathbb {Q} }}_{l})} be a semisimple (continuous) l {\displaystyle l} -adic representations of the absolute Galois group of some field K {\displaystyle K} , unramified outside some finite set of primes S ⊂ M K {\displaystyle S\subset M_{K}} . Then the representation is uniquely determined by the values of the traces of ρ ( F r o b p ) {\displaystyle \rho (Frob_{p})} for p ∈ M K 0 − S {\displaystyle p\in M_{K}^{0}-S} (also using the Chebotarev density theorem).
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https://en.wikipedia.org/wiki/Brauer–Nesbitt_theorem
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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 general sequence of number fields K 1 , K 2 , … . {\displaystyle K_{1},K_{2},\ldots .\ } In all cases other than the rational field Q and imaginary quadratic fields, the regulator Ri of Ki must be taken into account, because Ki then has units of infinite order by Dirichlet's unit theorem. The quantitative hypothesis of the standard Brauer–Siegel theorem is that if Di is the discriminant of Ki, then log | D i | → 0 as i → ∞ .
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https://en.wikipedia.org/wiki/Brauer–Siegel_theorem
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{\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 } where hi is the class number of Ki.
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https://en.wikipedia.org/wiki/Brauer–Siegel_theorem
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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 direction were initiated in work of Harold Stark from the early 1970s.
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https://en.wikipedia.org/wiki/Brauer–Siegel_theorem
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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 generalization of the Brauer–Suzuki theorem is given by Glauberman's Z* theorem.
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https://en.wikipedia.org/wiki/Brauer–Suzuki_theorem
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In mathematics, the Brauer–Suzuki–Wall theorem, proved by Brauer, Suzuki & Wall (1958), characterizes the one-dimensional unimodular projective groups over finite fields.
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https://en.wikipedia.org/wiki/Brauer–Suzuki–Wall_theorem
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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 the similarity classes of finite dimensional central simple algebras over F under the operation of tensor product, where two algebras are called similar if the commutants of their simple modules are isomorphic. Every similarity class contains a unique division algebra, so the elements of the Brauer group can also be identified with isomorphism classes of finite dimensional central division algebras. The analogous construction for Z/2Z-graded algebras defines the Brauer–Wall group BW(F).
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https://en.wikipedia.org/wiki/Clifford_invariant
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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_{x}} and u x x {\displaystyle u_{xx}} denote partial derivatives of the scalar field u ( x , t ) . {\displaystyle u(x,t).}
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https://en.wikipedia.org/wiki/Bretherton_equation
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The original equation studied by Bretherton has quadratic nonlinearity, p = 2. {\displaystyle p=2.}
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https://en.wikipedia.org/wiki/Bretherton_equation
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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. Bretherton obtained analytic solutions in terms of Jacobi elliptic functions.
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https://en.wikipedia.org/wiki/Bretherton_equation
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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∗, there exists a solution u ∈ X of the equation T(u) = g. (Note that T itself is not required to be a linear map.) The theorem is named in honor of Felix Browder and George J. Minty, who independently proved it.
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https://en.wikipedia.org/wiki/Browder–Minty_theorem
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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.
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https://en.wikipedia.org/wiki/Brown_measure
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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 upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this. More generally, any group with a (B, N) pair has a Bruhat decomposition.
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https://en.wikipedia.org/wiki/Bruhat_decomposition
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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.
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https://en.wikipedia.org/wiki/Weak_Bruhat_order
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In mathematics, the Brumer bound is a bound for the rank of an elliptic curve, proved by Brumer (1992).
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https://en.wikipedia.org/wiki/Brumer_bound
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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 generalization to compact nonconvex sets stated here is due to Lazar Lyusternik (1935).
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https://en.wikipedia.org/wiki/Brunn–Minkowski_theorem
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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 derivations focused on the steady version of the equation. The Buckmaster equation is u t = ( u 4 ) x x + λ ( u 3 ) x {\displaystyle u_{t}=(u^{4})_{xx}+\lambda (u^{3})_{x}} where λ {\displaystyle \lambda } is a known parameter. == References ==
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https://en.wikipedia.org/wiki/Buckmaster_equation
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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.
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https://en.wikipedia.org/wiki/Burkill_integral
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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).
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https://en.wikipedia.org/wiki/Burnside_ring
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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 of vertices of the graph. The conjecture was proven by Choongbum Lee. Thus it is now a theorem.
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https://en.wikipedia.org/wiki/Burr–Erdős_conjecture
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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 Institute of Technology.
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https://en.wikipedia.org/wiki/Bussgang_theorem
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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 formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was Cayley (1857), prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics.
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https://en.wikipedia.org/wiki/Butcher_group
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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 algebra, is essentially equivalent to the Butcher group, since its dual can be identified with the universal enveloping algebra of the Lie algebra of the Butcher group. As they commented: We regard Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results.
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https://en.wikipedia.org/wiki/Butcher_group
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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 gives a precise way of partitioning Rd into two sets: one where f is essentially small; the other a countable collection of cubes where f is essentially large, but where some control of the function is retained. This leads to the associated Calderón–Zygmund decomposition of f , wherein f is written as the sum of "good" and "bad" functions, using the above sets.
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https://en.wikipedia.org/wiki/Calderón–Zygmund_theory
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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 correspondence is implemented by mapping an operator to its singular value sequence. It originated from John von Neumann's study of symmetric norms on matrix algebras. It provides a fundamental classification and tool for the study of two-sided ideals of compact operators and their traces, by reducing problems about operator spaces to (more resolvable) problems on sequence spaces.
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https://en.wikipedia.org/wiki/Calkin_correspondence
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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 A. Fokas.
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https://en.wikipedia.org/wiki/Calogero–Degasperis–Fokas_equation
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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.
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https://en.wikipedia.org/wiki/Cameron–Martin_formula
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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, its value still goes from 0 to 1 as its argument reaches from 0 to 1.
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https://en.wikipedia.org/wiki/Cantor_function
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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, and the Cantor–Lebesgue function. Georg Cantor (1884) introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack. The Cantor function was discussed and popularized by Scheeffer (1884), Lebesgue (1904) and Vitali (1905).
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https://en.wikipedia.org/wiki/Cantor_function
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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 of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments.
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https://en.wikipedia.org/wiki/Cantor_dust
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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, this is equivalent to being perfect nonempty, compact metrizable and zero dimensional.
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https://en.wikipedia.org/wiki/Cantor_dust
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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 and fixing 0, can be formulated purely geometrically in terms of the limiting behaviour of the images of the functions. The kernel theorem has wide application in the theory of univalent functions and in particular provides the geometric basis for the Loewner differential equation.
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https://en.wikipedia.org/wiki/Carathéodory_kernel_theorem
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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.
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https://en.wikipedia.org/wiki/Carathéodory_metric
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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 conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians James Caristi and William Arthur Kirk.Caristi fixed-point theorem can be applied to derive other classical fixed-point results, and also to prove the existence of bounded solutions of a functional equation.
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https://en.wikipedia.org/wiki/Caristi_fixed-point_theorem
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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.
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https://en.wikipedia.org/wiki/Carleson–Jacobs_theorem
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