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In mathematics, an Eisenstein sum is a finite sum depending on a finite field and related to a Gauss sum. Eisenstein sums were introduced by Eisenstein in 1848, named "Eisenstein sums" by Stickelberger in 1890, and rediscovered by Yamamoto in 1985, who called them relative Gauss sums.
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https://en.wikipedia.org/wiki/Eisenstein_sum
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In mathematics, an Engel subalgebra of a Lie algebra with respect to some element x is the subalgebra of elements annihilated by some power of ad x. Engel subalgebras are named after Friedrich Engel. For finite-dimensional Lie algebras over infinite fields the minimal Engel subalgebras are the Cartan subalgebras.
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https://en.wikipedia.org/wiki/Engel_subalgebra
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In mathematics, an Enoki surface is compact complex surface with positive second Betti number that has a global spherical shell and a non-trivial divisor D with H0(O(D)) ≠ 0 and (D, D) = 0. Enoki (1980) constructed some examples. They are surfaces of class VII, so are non-Kähler and have Kodaira dimension −∞.
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https://en.wikipedia.org/wiki/Enoki_surface
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In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958). A cardinal κ is called α-Erdős if for every function f: κ< ω → {0, 1}, there is a set of order type α that is homogeneous for f . In the notation of the partition calculus, κ is α-Erdős if κ(α) → (α)< ω.The existence of zero sharp implies that the constructible universe L satisfies "for every countable ordinal α, there is an α-Erdős cardinal". In fact, for every indiscernible κ, Lκ satisfies "for every ordinal α, there is an α-Erdős cardinal in Coll(ω, α)" (the Levy collapse to make α countable).
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https://en.wikipedia.org/wiki/Erdös_cardinal
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However, the existence of an ω1-Erdős cardinal implies existence of zero sharp. If f is the satisfaction relation for L (using ordinal parameters), then the existence of zero sharp is equivalent to there being an ω1-Erdős ordinal with respect to f . Thus, the existence of zero sharp implies that the axiom of constructibility is false. If κ is α-Erdős, then it is α-Erdős in every transitive model satisfying "α is countable."
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https://en.wikipedia.org/wiki/Erdös_cardinal
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In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime. A perfect Euler brick is one whose space diagonal is also an integer, but such a brick has not yet been found.
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https://en.wikipedia.org/wiki/Euler_brick
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In mathematics, an Euler system is a collection of compatible elements of Galois cohomology groups indexed by fields. They were introduced by Kolyvagin (1990) in his work on Heegner points on modular elliptic curves, which was motivated by his earlier paper Kolyvagin (1988) and the work of Thaine (1988). Euler systems are named after Leonhard Euler because the factors relating different elements of an Euler system resemble the Euler factors of an Euler product. Euler systems can be used to construct annihilators of ideal class groups or Selmer groups, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some Tate-Shafarevich groups. This led to Karl Rubin's new proof of the main conjecture of Iwasawa theory, considered simpler than the original proof due to Barry Mazur and Andrew Wiles.
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https://en.wikipedia.org/wiki/Euler_systems
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In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an equidimensional equation. Because of its particularly simple equidimensional structure, the differential equation can be solved explicitly.
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https://en.wikipedia.org/wiki/Cauchy–Euler_equation
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In mathematics, an Fσ set (said F-sigma set) is a countable union of closed sets. The notation originated in French with F for fermé (French: closed) and σ for somme (French: sum, union).The complement of an Fσ set is a Gδ set.Fσ is the same as Σ 2 0 {\displaystyle \mathbf {\Sigma } _{2}^{0}} in the Borel hierarchy.
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https://en.wikipedia.org/wiki/F-sigma_set
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In mathematics, an H-matrix is a matrix whose comparison matrix is an M-matrix. It is useful in iterative methods. Definition: Let A = (aij) be a n × n complex matrix. Then comparison matrix M(A) of complex matrix A is defined as M(A) = αij where αij = −|Aij| for all i ≠ j, 1 ≤ i,j ≤ n and αij = |Aij| for all i = j, 1 ≤ i,j ≤ n. If M(A) is a M-matrix, A is a H-matrix. Invertible H-matrix guarantees convergence of Gauss–Seidel iterative methods.
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https://en.wikipedia.org/wiki/H-matrix_(iterative_method)
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In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed.
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https://en.wikipedia.org/wiki/H-space
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In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. An I-bundle is said to be twisted if it is not trivial. Two simple examples of I-bundles are the annulus and the Möbius band, the only two possible I-bundles over the circle S 1 {\displaystyle S^{1}} .
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https://en.wikipedia.org/wiki/I-bundle
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The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product S 1 × I {\displaystyle S^{1}\times I} , and the Möbius band is a non-trivial or twisted bundle. Both bundles are 2-manifolds, but the annulus is an orientable manifold while the Möbius band is a non-orientable manifold. Curiously, there are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle K {\displaystyle K} .
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https://en.wikipedia.org/wiki/I-bundle
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That surface has three I-bundles: the trivial bundle K × I {\displaystyle K\times I} and two twisted bundles. Together with the Seifert fiber spaces, I-bundles are fundamental elementary building blocks for the description of three-dimensional spaces.
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https://en.wikipedia.org/wiki/I-bundle
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These observations are simple well known facts on elementary 3-manifolds. Line bundles are both I-bundles and vector bundles of rank one. When considering I-bundles, one is interested mostly in their topological properties and not their possible vector properties, as one might be for line bundles.
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https://en.wikipedia.org/wiki/I-bundle
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In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set D of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of D. The set of all finite sums over D is often denoted as FS(D). Slightly more generally, for a sequence of natural numbers (ni), one can consider the set of finite sums FS((ni)), consisting of the sums of all finite length subsequences of (ni). A set A of natural numbers is an IP set if there exists an infinite set D such that FS(D) is a subset of A. Equivalently, one may require that A contains all finite sums FS((ni)) of a sequence (ni).
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https://en.wikipedia.org/wiki/IP_set
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Some authors give a slightly different definition of IP sets: They require that FS(D) equal A instead of just being a subset. The term IP set was coined by Hillel Furstenberg and Benjamin Weiss to abbreviate "infinite-dimensional parallelepiped". Serendipitously, the abbreviation IP can also be expanded to "idempotent" (a set is an IP if and only if it is a member of an idempotent ultrafilter).
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https://en.wikipedia.org/wiki/IP_set
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In mathematics, an Igusa curve is (roughly) a coarse moduli space of elliptic curves in characteristic p with a level p Igusa structure, where an Igusa structure on an elliptic curve E is roughly a point of order p of E(p) generating the kernel of V:E(p) → E. An Igusa variety is a higher-dimensional analogue of an Igusa curve. Igusa curves were studied by Igusa (1968) and Igusa varieties were introduced by Harris & Taylor (2001).
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https://en.wikipedia.org/wiki/Igusa_variety
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In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on.
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https://en.wikipedia.org/wiki/Igusa_zeta-function
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In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. The Riemann zeta function is an example of an L-function, and one important conjecture involving L-functions is the Riemann hypothesis and its generalization.
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https://en.wikipedia.org/wiki/L_function
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The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the Euler product formula there is a deep connection between L-functions and the theory of prime numbers. The mathematical field that studies L-functions is sometimes called analytic theory of L-functions.
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https://en.wikipedia.org/wiki/L_function
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In mathematics, an LB-space, also written (LB)-space, is a topological vector space X {\displaystyle X} that is a locally convex inductive limit of a countable inductive system ( X n , i n m ) {\displaystyle (X_{n},i_{nm})} of Banach spaces. This means that X {\displaystyle X} is a direct limit of a direct system ( X n , i n m ) {\displaystyle \left(X_{n},i_{nm}\right)} in the category of locally convex topological vector spaces and each X n {\displaystyle X_{n}} is a Banach space. If each of the bonding maps i n m {\displaystyle i_{nm}} is an embedding of TVSs then the LB-space is called a strict LB-space.
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https://en.wikipedia.org/wiki/LB-space
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This means that the topology induced on X n {\displaystyle X_{n}} by X n + 1 {\displaystyle X_{n+1}} is identical to the original topology on X n . {\displaystyle X_{n}.} Some authors (e.g. Schaefer) define the term "LB-space" to mean "strict LB-space," so when reading mathematical literature, its recommended to always check how LB-space is defined.
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https://en.wikipedia.org/wiki/LB-space
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In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system ( X n , i n m ) {\displaystyle (X_{n},i_{nm})} of Fréchet spaces. This means that X is a direct limit of a direct system ( X n , i n m ) {\displaystyle (X_{n},i_{nm})} in the category of locally convex topological vector spaces and each X n {\displaystyle X_{n}} is a Fréchet space. The name LF stands for Limit of Fréchet spaces.
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https://en.wikipedia.org/wiki/LF-space
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If each of the bonding maps i n m {\displaystyle i_{nm}} is an embedding of TVSs then the LF-space is called a strict LF-space. This means that the subspace topology induced on Xn by Xn+1 is identical to the original topology on Xn. Some authors (e.g. Schaefer) define the term "LF-space" to mean "strict LF-space," so when reading mathematical literature, it is recommended to always check how LF-space is defined.
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https://en.wikipedia.org/wiki/LF-space
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In mathematics, an LLT polynomial is one of a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon (1997) as q-analogues of products of Schur functions. J. Haglund, M. Haiman, N. Loehr (2005) showed how to expand Macdonald polynomials in terms of LLT polynomials. Ian Grojnowski and Mark Haiman (2007, preprint) proved a positivity conjecture for LLT polynomials that combined with the previous result implies the Macdonald positivity conjecture for Macdonald polynomials, and extended the definition of LLT polynomials to arbitrary finite root systems.
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https://en.wikipedia.org/wiki/LLT_polynomial
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In mathematics, an N-topological space is a set equipped with N arbitrary topologies. If τ1, τ2, ..., τN are N topologies defined on a nonempty set X, then the N-topological space is denoted by (X,τ1,τ2,...,τN). For N = 1, the structure is simply a topological space. For N = 2, the structure becomes a bitopological space introduced by J. C. Kelly.
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https://en.wikipedia.org/wiki/N-topological_space
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In mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were described by Borchers (1962) and Uhlmann (1962), who studied some examples of O*-algebras, called Borchers algebras, arising from the Wightman axioms of quantum field theory. Powers (1971) and Lassner (1972) began the systematic study of algebras of unbounded operators.
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https://en.wikipedia.org/wiki/O*-algebra
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In mathematics, an Ockham algebra is a bounded distributive lattice with a dual endomorphism, that is, an operation ~ satisfying ~(x ∧ y) = ~x ∨ ~y, ~(x ∨ y) = ~x ∧ ~y, ~0 = 1, ~1 = 0. They were introduced by Berman (1977), and were named after William of Ockham by Urquhart (1979). Ockham algebras form a variety. Examples of Ockham algebras include Boolean algebras, De Morgan algebras, Kleene algebras, and Stone algebras.
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https://en.wikipedia.org/wiki/Ockham_algebras
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In mathematics, an Oper is a principal connection, or in more elementary terms a type of differential operator. They were first defined and used by Vladimir Drinfeld and Vladimir Sokolov to study how the KdV equation and related integrable PDEs correspond to algebraic structures known as Kac–Moody algebras. Their modern formulation is due to Drinfeld and Alexander Beilinson.
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https://en.wikipedia.org/wiki/Oper_(mathematics)
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In mathematics, an Orlicz sequence space is any of certain class of linear spaces of scalar-valued sequences, endowed with a special norm, specified below, under which it forms a Banach space. Orlicz sequence spaces generalize the ℓ p {\displaystyle \ell _{p}} spaces, and as such play an important role in functional analysis.
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https://en.wikipedia.org/wiki/Orlicz_sequence_space
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In mathematics, an R-function, or Rvachev function, is a real-valued function whose sign does not change if none of the signs of its arguments change; that is, its sign is determined solely by the signs of its arguments.Interpreting positive values as true and negative values as false, an R-function is transformed into a "companion" Boolean function (the two functions are called friends). For instance, the R-function ƒ(x, y) = min(x, y) is one possible friend of the logical conjunction (AND). R-functions are used in computer graphics and geometric modeling in the context of implicit surfaces and the function representation. They also appear in certain boundary-value problems, and are also popular in certain artificial intelligence applications, where they are used in pattern recognition. R-functions were first proposed by Vladimir Logvinovich Rvachev (Russian: Влади́мир Логвинович Рвачёв) in 1963, though the name, "R-functions", was given later on by Ekaterina L. Rvacheva-Yushchenko, in memory of their father, Logvin Fedorovich Rvachev (Russian: Логвин Фёдорович Рвачёв).
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https://en.wikipedia.org/wiki/Rvachev_function
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In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative attempt to unify several cohomology theories by Alexander Grothendieck. Abelian categories are very stable categories, for example they are regular and they satisfy the snake lemma.
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https://en.wikipedia.org/wiki/Homological_algebra
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The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory.
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https://en.wikipedia.org/wiki/Homological_algebra
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Abelian categories are named after Niels Henrik Abel. More concretely, a category is abelian if it has a zero object, it has all binary products and binary coproducts, and it has all kernels and cokernels. all monomorphisms and epimorphisms are normal.
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https://en.wikipedia.org/wiki/Homological_algebra
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In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum.
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https://en.wikipedia.org/wiki/Abelian_category
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Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory.
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https://en.wikipedia.org/wiki/Abelian_category
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In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified.
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https://en.wikipedia.org/wiki/Fundamental_theorem_of_finite_abelian_groups
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In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form ∫ z 0 z R ( x , w ) d x , {\displaystyle \int _{z_{0}}^{z}R(x,w)\,dx,} where R ( x , w ) {\displaystyle R(x,w)} is an arbitrary rational function of the two variables x {\displaystyle x} and w {\displaystyle w} , which are related by the equation F ( x , w ) = 0 , {\displaystyle F(x,w)=0,} where F ( x , w ) {\displaystyle F(x,w)} is an irreducible polynomial in w {\displaystyle w} , F ( x , w ) ≡ φ n ( x ) w n + ⋯ + φ 1 ( x ) w + φ 0 ( x ) , {\displaystyle F(x,w)\equiv \varphi _{n}(x)w^{n}+\cdots +\varphi _{1}(x)w+\varphi _{0}\left(x\right),} whose coefficients φ j ( x ) {\displaystyle \varphi _{j}(x)} , j = 0 , 1 , … , n {\displaystyle j=0,1,\ldots ,n} are rational functions of x {\displaystyle x} . The value of an abelian integral depends not only on the integration limits, but also on the path along which the integral is taken; it is thus a multivalued function of z {\displaystyle z} . Abelian integrals are natural generalizations of elliptic integrals, which arise when F ( x , w ) = w 2 − P ( x ) , {\displaystyle F(x,w)=w^{2}-P(x),\,} where P ( x ) {\displaystyle P\left(x\right)} is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where P ( x ) {\displaystyle P(x)} , in the formula above, is a polynomial of degree greater than 4.
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https://en.wikipedia.org/wiki/Abelian_integral
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In mathematics, an abelian surface is a 2-dimensional abelian variety. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bilinear relations. Essentially, these are conditions on the parameter space of period matrices for complex tori which define an algebraic subvariety. This subvariety contains all of the points whose period matrices correspond to a period matrix of an abelian variety.
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https://en.wikipedia.org/wiki/Abelian_surface
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The algebraic ones are called abelian surfaces and are exactly the 2-dimensional abelian varieties. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Finding criteria for a complex torus of dimension 2 to be a product of two elliptic curves (up to isogeny) was a popular subject of study in the nineteenth century.
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https://en.wikipedia.org/wiki/Abelian_surface
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Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1×S1×S1×S1 so the fundamental group is Z4. Hodge diamond: Examples: A product of two elliptic curves. The Jacobian variety of a genus 2 curve.
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https://en.wikipedia.org/wiki/Abelian_surface
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In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A). The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century. One of the major achievements in algebraic number theory and algebraic geometry of the twentieth century was to find the correct formulations of the corresponding theory for abelian varieties of dimension d > 1. The problem is at a deeper level of abstraction, because it is much harder to manipulate analytic functions of several complex variables.
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https://en.wikipedia.org/wiki/Abelian_variety_of_CM-type
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The formal definition is that End Q ( A ) {\displaystyle \operatorname {End} _{\mathbb {Q} }(A)} the tensor product of End(A) with the rational number field Q, should contain a commutative subring of dimension 2d over Q. When d = 1 this can only be a quadratic field, and one recovers the cases where End(A) is an order in an imaginary quadratic field. For d > 1 there are comparable cases for CM-fields, the complex quadratic extensions of totally real fields. There are other cases that reflect that A may not be a simple abelian variety (it might be a cartesian product of elliptic curves, for example).
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https://en.wikipedia.org/wiki/Abelian_variety_of_CM-type
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Another name for abelian varieties of CM-type is abelian varieties with sufficiently many complex multiplications. It is known that if K is the complex numbers, then any such A has a field of definition which is in fact a number field. The possible types of endomorphism ring have been classified, as rings with involution (the Rosati involution), leading to a classification of CM-type abelian varieties.
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https://en.wikipedia.org/wiki/Abelian_variety_of_CM-type
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To construct such varieties in the same style as for elliptic curves, starting with a lattice Λ in Cd, one must take into account the Riemann relations of abelian variety theory. The CM-type is a description of the action of a (maximal) commutative subring L of EndQ(A) on the holomorphic tangent space of A at the identity element. Spectral theory of a simple kind applies, to show that L acts via a basis of eigenvectors; in other words L has an action that is via diagonal matrices on the holomorphic vector fields on A. In the simple case, where L is itself a number field rather than a product of some number of fields, the CM-type is then a list of complex embeddings of L. There are 2d of those, occurring in complex conjugate pairs; the CM-type is a choice of one out of each pair.
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https://en.wikipedia.org/wiki/Abelian_variety_of_CM-type
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It is known that all such possible CM-types can be realised. Basic results of Goro Shimura and Yutaka Taniyama compute the Hasse–Weil L-function of A, in terms of the CM-type and a Hecke L-function with Hecke character, having infinity-type derived from it. These generalise the results of Max Deuring for the elliptic curve case.
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https://en.wikipedia.org/wiki/Abelian_variety_of_CM-type
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In mathematics, an absolute presentation is one method of defining a group.Recall that to define a group G {\displaystyle G} by means of a presentation, one specifies a set S {\displaystyle S} of generators so that every element of the group can be written as a product of some of these generators, and a set R {\displaystyle R} of relations among those generators. In symbols: G ≃ ⟨ S ∣ R ⟩ . {\displaystyle G\simeq \langle S\mid R\rangle .}
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https://en.wikipedia.org/wiki/Absolute_presentation_of_a_group
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Informally G {\displaystyle G} is the group generated by the set S {\displaystyle S} such that r = 1 {\displaystyle r=1} for all r ∈ R {\displaystyle r\in R} . But here there is a tacit assumption that G {\displaystyle G} is the "freest" such group as clearly the relations are satisfied in any homomorphic image of G {\displaystyle G} . One way of being able to eliminate this tacit assumption is by specifying that certain words in S {\displaystyle S} should not be equal to 1.
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https://en.wikipedia.org/wiki/Absolute_presentation_of_a_group
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{\displaystyle 1.} That is we specify a set I {\displaystyle I} , called the set of irrelations, such that i ≠ 1 {\displaystyle i\neq 1} for all i ∈ I . {\displaystyle i\in I.}
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https://en.wikipedia.org/wiki/Absolute_presentation_of_a_group
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In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite. For a real-valued function, since where both ∫ f + ( x ) d x {\textstyle \int f^{+}(x)\,dx} and ∫ f − ( x ) d x {\textstyle \int f^{-}(x)\,dx} must be finite. In Lebesgue integration, this is exactly the requirement for any measurable function f to be considered integrable, with the integral then equaling ∫ f + ( x ) d x − ∫ f − ( x ) d x {\textstyle \int f^{+}(x)\,dx-\int f^{-}(x)\,dx} , so that in fact "absolutely integrable" means the same thing as "Lebesgue integrable" for measurable functions. The same thing goes for a complex-valued function.
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https://en.wikipedia.org/wiki/Absolutely_integrable
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Let us define where ℜ f ( x ) {\displaystyle \Re f(x)} and ℑ f ( x ) {\displaystyle \Im f(x)} are the real and imaginary parts of f ( x ) {\displaystyle f(x)} . Then so This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function is Lebesgue integrable only if all the four integrals are finite. So having a finite integral of the absolute value is equivalent to the conditions for the function to be "Lebesgue integrable".
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https://en.wikipedia.org/wiki/Absolutely_integrable
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In mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element itself. In semigroup theory, the absorbing element is called a zero element because there is no risk of confusion with other notions of zero, with the notable exception: under additive notation zero may, quite naturally, denote the neutral element of a monoid. In this article "zero element" and "absorbing element" are synonymous.
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https://en.wikipedia.org/wiki/Annihilating_element
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In mathematics, an absorbing set for a random dynamical system is a subset of the phase space. A dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. The absorbing set eventually contains the image of any bounded set under the cocycle ("flow") of the random dynamical system. As with many concepts related to random dynamical systems, it is defined in the pullback sense.
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https://en.wikipedia.org/wiki/Absorbing_set_(random_dynamical_systems)
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In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point. The complex is called “abstract” since its points, which are called “cells”, are not subsets of a Hausdorff space as is the case in Euclidean and CW complexes. Abstract cell complexes play an important role in image analysis and computer graphics.
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https://en.wikipedia.org/wiki/Abstract_cell_complexes
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In mathematics, an abstract differential equation is a differential equation in which the unknown function and its derivatives take values in some generic abstract space (a Hilbert space, a Banach space, etc.). Equations of this kind arise e.g. in the study of partial differential equations: if to one of the variables is given a privileged position (e.g. time, in heat or wave equations) and all the others are put together, an ordinary "differential" equation with respect to the variable which was put in evidence is obtained. Adding boundary conditions can often be translated in terms of considering solutions in some convenient function spaces. The classical abstract differential equation which is most frequently encountered is the equation d u d t = A u + f {\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=Au+f} where the unknown function u = u ( t ) {\displaystyle u=u(t)} belongs to some function space X {\displaystyle X} , 0 ≤ t ≤ T ≤ ∞ {\displaystyle 0\leq t\leq T\leq \infty } and A: X → X {\displaystyle A:X\to X} is an operator (usually a linear operator) acting on this space.
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https://en.wikipedia.org/wiki/Abstract_differential_equation
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An exhaustive treatment of the homogeneous ( f = 0 {\displaystyle f=0} ) case with a constant operator is given by the theory of C0-semigroups. Very often, the study of other abstract differential equations amounts (by e.g. reduction to a set of equations of the first order) to the study of this equation. The theory of abstract differential equations has been founded by Einar Hille in several papers and in his book Functional Analysis and Semi-Groups. Other main contributors were Kōsaku Yosida, Ralph Phillips, Isao Miyadera, and Selim Grigorievich Krein.
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https://en.wikipedia.org/wiki/Abstract_differential_equation
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In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines. A geometric polytope is said to be a realization of an abstract polytope in some real N-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.
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https://en.wikipedia.org/wiki/Projective_polytope
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In mathematics, an acceptable ring is a generalization of an excellent ring, with the conditions about regular rings in the definition of an excellent ring replaced by conditions about Gorenstein rings. Acceptable rings were introduced by Sharp (1977). All finite-dimensional Gorenstein rings are acceptable, as are all finitely generated algebras over acceptable rings and all localizations of acceptable rings.
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https://en.wikipedia.org/wiki/Acceptable_ring
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In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space. If X is locally compact, then an equivalent condition is that there is a compact subset K of X such that the image of K under the action of G covers X. It is sometimes referred to as mpact, a tongue-in-cheek reference to dual notions where prefixing with "co-" twice would "cancel out".
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https://en.wikipedia.org/wiki/Cocompact_group_action
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In mathematics, an acyclic space is a nonempty topological space X in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of X are isomorphic to the corresponding homology groups of a point. In other words, using the idea of reduced homology, H ~ i ( X ) = 0 , ∀ i ≥ − 1.
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https://en.wikipedia.org/wiki/Acyclic_space
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{\displaystyle {\tilde {H}}_{i}(X)=0,\quad \forall i\geq -1.} It is common to consider such a space as a nonempty space without "holes"; for example, a circle or a sphere is not acyclic but a disc or a ball is acyclic. This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any closed loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface."
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https://en.wikipedia.org/wiki/Acyclic_space
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The condition of acyclicity on a space X implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of X to the circle or to the higher spheres is null-homotopic. If a space X is contractible, then it is also acyclic, by the homotopy invariance of homology. The converse is not true, in general. Nevertheless, if X is an acyclic CW complex, and if the fundamental group of X is trivial, then X is a contractible space, as follows from the Whitehead theorem and the Hurewicz theorem.
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https://en.wikipedia.org/wiki/Acyclic_space
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In mathematics, an addition chain for computing a positive integer n can be given by a sequence of natural numbers starting with 1 and ending with n, such that each number in the sequence is the sum of two previous numbers. The length of an addition chain is the number of sums needed to express all its numbers, which is one less than the cardinality of the sequence of numbers.
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https://en.wikipedia.org/wiki/Addition_chain
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In mathematics, an addition theorem is a formula such as that for the exponential function: ex + y = ex · ey,that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y). Slightly more generally, as is the case with the trigonometric functions sin and cos, several functions may be involved; this is more apparent than real, in that case, since there cos is an algebraic function of sin (in other words, we usually take their functions both as defined on the unit circle). The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for elliptic functions. To "classify" addition theorems it is necessary to put some restriction on the type of function G admitted, such that F(x + y) = G(F(x), F(y)).In this identity one can assume that F and G are vector-valued (have several components).
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https://en.wikipedia.org/wiki/Addition_theorem
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An algebraic addition theorem is one in which G can be taken to be a vector of polynomials, in some set of variables. The conclusion of the mathematicians of the time was that the theory of abelian functions essentially exhausted the interesting possibilities: considered as a functional equation to be solved with polynomials, or indeed rational functions or algebraic functions, there were no further types of solution. In more contemporary language this appears as part of the theory of algebraic groups, dealing with commutative groups.
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https://en.wikipedia.org/wiki/Addition_theorem
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The connected, projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an abelian variety by rather weak conditions on its group law. The so-called quasi-abelian functions are all known to come from extensions of abelian varieties by commutative affine group varieties. Therefore, the old conclusions about the scope of global algebraic addition theorems can be said to hold. A more modern aspect is the theory of formal groups.
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https://en.wikipedia.org/wiki/Addition_theorem
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In mathematics, an additive set function is a function mapping sets to numbers, with the property that its value on a union of two disjoint sets equals the sum of its values on these sets, namely, μ ( A ∪ B ) = μ ( A ) + μ ( B ) . {\textstyle \mu (A\cup B)=\mu (A)+\mu (B).} If this additivity property holds for any two sets, then it also holds for any finite number of sets, namely, the function value on the union of k disjoint sets (where k is a finite number) equals the sum of its values on the sets. Therefore, an additive set function is also called a finitely additive set function (the terms are equivalent).
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https://en.wikipedia.org/wiki/Sigma_additive
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However, a finitely additive set function might not have the additivity property for a union of an infinite number of sets. A σ-additive set function is a function that has the additivity property even for countably infinite many sets, that is, μ ( ⋃ n = 1 ∞ A n ) = ∑ n = 1 ∞ μ ( A n ) . {\textstyle \mu \left(\bigcup _{n=1}^{\infty }A_{n}\right)=\sum _{n=1}^{\infty }\mu (A_{n}).}
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https://en.wikipedia.org/wiki/Sigma_additive
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Additivity and sigma-additivity are particularly important properties of measures. They are abstractions of how intuitive properties of size (length, area, volume) of a set sum when considering multiple objects. Additivity is a weaker condition than σ-additivity; that is, σ-additivity implies additivity. The term modular set function is equivalent to additive set function; see modularity below.
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https://en.wikipedia.org/wiki/Sigma_additive
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In mathematics, an adherent point (also closure point or point of closure or contact point) of a subset A {\displaystyle A} of a topological space X , {\displaystyle X,} is a point x {\displaystyle x} in X {\displaystyle X} such that every neighbourhood of x {\displaystyle x} (or equivalently, every open neighborhood of x {\displaystyle x} ) contains at least one point of A . {\displaystyle A.} A point x ∈ X {\displaystyle x\in X} is an adherent point for A {\displaystyle A} if and only if x {\displaystyle x} is in the closure of A , {\displaystyle A,} thus x ∈ Cl X A {\displaystyle x\in \operatorname {Cl} _{X}A} if and only if for all open subsets U ⊆ X , {\displaystyle U\subseteq X,} if x ∈ U then U ∩ A ≠ ∅ . {\displaystyle x\in U{\text{ then }}U\cap A\neq \varnothing .}
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https://en.wikipedia.org/wiki/Adherent_point
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This definition differs from that of a limit point of a set, in that for a limit point it is required that every neighborhood of x {\displaystyle x} contains at least one point of A {\displaystyle A} different from x . {\displaystyle x.} Thus every limit point is an adherent point, but the converse is not true.
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https://en.wikipedia.org/wiki/Adherent_point
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An adherent point of A {\displaystyle A} is either a limit point of A {\displaystyle A} or an element of A {\displaystyle A} (or both). An adherent point which is not a limit point is an isolated point. Intuitively, having an open set A {\displaystyle A} defined as the area within (but not including) some boundary, the adherent points of A {\displaystyle A} are those of A {\displaystyle A} including the boundary.
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https://en.wikipedia.org/wiki/Adherent_point
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In mathematics, an adhesive category is a category where pushouts of monomorphisms exist and work more or less as they do in the category of sets. An example of an adhesive category is the category of directed multigraphs, or quivers, and the theory of adhesive categories is important in the theory of graph rewriting. More precisely, an adhesive category is one where any of the following equivalent conditions hold: C has all pullbacks, it has pushouts along monomorphisms, and pushout squares of monomorphisms are also pullback squares and are stable under pullback. C has all pullbacks, it has pushouts along monomorphisms, and the latter are also (bicategorical) pushouts in the bicategory of spans in C.If C is small, we may equivalently say that C has all pullbacks, has pushouts along monomorphisms, and admits a full embedding into a Grothendieck topos preserving pullbacks and preserving pushouts of monomorphisms.
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https://en.wikipedia.org/wiki/Adhesive_category
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In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles have important applications in the theory of connections as well as in gauge theory.
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https://en.wikipedia.org/wiki/Adjoint_bundle
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In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f: A → X be a continuous map (called the attaching map). One forms the adjunction space X ∪f Y (sometimes also written as X +f Y) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. Formally, X ∪ f Y = ( X ⊔ Y ) / ∼ {\displaystyle X\cup _{f}Y=(X\sqcup Y)/\sim } where the equivalence relation ~ is generated by a ~ f(a) for all a in A, and the quotient is given the quotient topology. As a set, X ∪f Y consists of the disjoint union of X and (Y − A). The topology, however, is specified by the quotient construction. Intuitively, one may think of Y as being glued onto X via the map f.
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https://en.wikipedia.org/wiki/Adjunction_space
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In mathematics, an admissible algebra is a (possibly non-associative) commutative algebra whose enveloping Lie algebra of derivations splits into the sum of an even and an odd part. Admissible algebras were introduced by Koecher (1967).
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https://en.wikipedia.org/wiki/Admissible_algebra
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In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials.
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https://en.wikipedia.org/wiki/Affine_Hecke_algebra
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In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras.
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https://en.wikipedia.org/wiki/Affine_Kac–Moody_algebra
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As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra g {\displaystyle {\mathfrak {g}}} , one considers the loop algebra, L g {\displaystyle L{\mathfrak {g}}} , formed by the g {\displaystyle {\mathfrak {g}}} -valued functions on a circle (interpreted as the closed string) with pointwise commutator. The affine Lie algebra g ^ {\displaystyle {\hat {\mathfrak {g}}}} is obtained by adding one extra dimension to the loop algebra and modifying a commutator in a non-trivial way, which physicists call a quantum anomaly (in this case, the anomaly of the WZW model) and mathematicians a central extension.
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https://en.wikipedia.org/wiki/Affine_Kac–Moody_algebra
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More generally, if σ is an automorphism of the simple Lie algebra g {\displaystyle {\mathfrak {g}}} associated to an automorphism of its Dynkin diagram, the twisted loop algebra L σ g {\displaystyle L_{\sigma }{\mathfrak {g}}} consists of g {\displaystyle {\mathfrak {g}}} -valued functions f on the real line which satisfy the twisted periodicity condition f(x + 2π) = σ f(x). Their central extensions are precisely the twisted affine Lie algebras. The point of view of string theory helps to understand many deep properties of affine Lie algebras, such as the fact that the characters of their representations transform amongst themselves under the modular group.
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https://en.wikipedia.org/wiki/Affine_Kac–Moody_algebra
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In mathematics, an affine algebraic group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable group defined over a field k is said to split over k or k-split if the isomorphism is defined over k. This coincides with the usual notion of split for an algebraic group. Every diagonalizable group splits over the separable closure ks of k. Any closed subgroup and image of diagonalizable groups are diagonalizable. The torsion subgroup of a diagonalizable group is dense.
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https://en.wikipedia.org/wiki/Diagonalizable_group
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The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian groups with Gal(ks/k)-equivariant morphisms without p-torsion, if k is of characteristic p. This is an analog of Poincaré duality and motivated the terminology. A diagonalizable k-group is said to be anisotropic if it has no nontrivial k-valued character.
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https://en.wikipedia.org/wiki/Diagonalizable_group
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The so-called "rigidity" states that the identity component of the centralizer of a diagonalizable group coincides with the identity component of the normalizer of the group. The fact plays a crucial role in the structure theory of solvable groups. A connected diagonalizable group is called an algebraic torus (which is not necessarily compact, in contrast to a complex torus). A k-torus is a torus defined over k. The centralizer of a maximal torus is called a Cartan subgroup.
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https://en.wikipedia.org/wiki/Diagonalizable_group
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In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation h(x, y, t) = 0 can be restricted to the affine algebraic plane curve of equation h(x, y, 1) = 0.
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https://en.wikipedia.org/wiki/Algebraic_curves
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These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve.
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https://en.wikipedia.org/wiki/Algebraic_curves
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If the curve is contained in an affine space or a projective space, one can take a projection for such a birational equivalence. These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must be studied on non-plane curves.
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https://en.wikipedia.org/wiki/Algebraic_curves
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This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula). A non-plane curve is often called a space curve or a skew curve.
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https://en.wikipedia.org/wiki/Algebraic_curves
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In mathematics, an affine braid group is a braid group associated to an affine Coxeter system. Their group rings have quotients called affine Hecke algebras. They are subgroups of double affine braid groups.
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https://en.wikipedia.org/wiki/Affine_braid_group
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In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.
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https://en.wikipedia.org/wiki/Affine_bundle
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In mathematics, an affine combination of x1, ..., xn is a linear combination ∑ i = 1 n α i ⋅ x i = α 1 x 1 + α 2 x 2 + ⋯ + α n x n , {\displaystyle \sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n},} such that ∑ i = 1 n α i = 1. {\displaystyle \sum _{i=1}^{n}{\alpha _{i}}=1.} Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients α i {\displaystyle \alpha _{i}} are elements of K. The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the α i {\displaystyle \alpha _{i}} are elements of K (or R {\displaystyle \mathbb {R} } for a Euclidean space), and the affine combination is also a point. See Affine space § Affine combinations and barycenter for the definition in this case.
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https://en.wikipedia.org/wiki/Affine_combination
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This concept is fundamental in Euclidean geometry and affine geometry, because the set of all affine combinations of a set of points forms the smallest subspace containing the points, exactly as the linear combinations of a set of vectors form their linear span. The affine combinations commute with any affine transformation T in the sense that T ∑ i = 1 n α i ⋅ x i = ∑ i = 1 n α i ⋅ T x i . {\displaystyle T\sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\sum _{i=1}^{n}{\alpha _{i}\cdot Tx_{i}}.} In particular, any affine combination of the fixed points of a given affine transformation T {\displaystyle T} is also a fixed point of T {\displaystyle T} , so the set of fixed points of T {\displaystyle T} forms an affine subspace (in 3D: a line or a plane, and the trivial cases, a point or the whole space). When a stochastic matrix, A, acts on a column vector, b→, the result is a column vector whose entries are affine combinations of b→ with coefficients from the rows in A.
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https://en.wikipedia.org/wiki/Affine_combination
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In mathematics, an affine representation of a topological Lie group G on an affine space A is a continuous (smooth) group homomorphism from G to the automorphism group of A, the affine group Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of A. An example is the action of the Euclidean group E(n) on the Euclidean space En. Since the affine group in dimension n is a matrix group in dimension n + 1, an affine representation may be thought of as a particular kind of linear representation. We may ask whether a given affine representation has a fixed point in the given affine space A. If it does, we may take that as origin and regard A as a vector space; in that case, we actually have a linear representation in dimension n. This reduction depends on a group cohomology question, in general.
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https://en.wikipedia.org/wiki/Affine_representation
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In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space. They are used in the classification of affine Lie algebras and superalgebras, and semisimple p-adic algebraic groups, and correspond to families of Macdonald polynomials. The reduced affine root systems were used by Kac and Moody in their work on Kac–Moody algebras. Possibly non-reduced affine root systems were introduced and classified by Macdonald (1972) and Bruhat & Tits (1972) (except that both these papers accidentally omitted the Dynkin diagram ).
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https://en.wikipedia.org/wiki/Affine_root_system
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In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space.
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https://en.wikipedia.org/wiki/Affine_line
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Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine space; this amounts to forgetting the special role played by the zero vector.
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https://en.wikipedia.org/wiki/Affine_line
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In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine subspace.
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https://en.wikipedia.org/wiki/Affine_line
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One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace.
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https://en.wikipedia.org/wiki/Affine_line
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Linear subspaces, in contrast, always contain the origin of the vector space. The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension 2 is an affine plane. An affine subspace of dimension n – 1 in an affine space or a vector space of dimension n is an affine hyperplane.
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https://en.wikipedia.org/wiki/Affine_line
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In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that the transition functions are algebra isomorphisms. Since algebras are also vector spaces, every algebra bundle is a vector bundle. Examples include the tensor-algebra bundle, exterior bundle, and symmetric bundle associated to a given vector bundle, as well as the Clifford bundle associated to any Riemannian vector bundle.
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https://en.wikipedia.org/wiki/Algebra_bundle
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