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Abelian Group Theory
Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group where the commutator subgroup is abelian * Abelianisation Topology and number theory * Abelian variety, a complex torus that can be embedded into projective space * Abelian surface, a two-dimensional abelian variety * Abelian function, a meromorphic function on an abelian variety * Abelian integral, a function related to the indefinite integral of a differential of the first kind Other mathematics * Abelian category, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel * Abelian and Tauberian theorems, in real analysis, used in the summation of divergent series * Abelian extension, in Galois theory, a field extension for which the associated Galois group is abelian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
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 the Norwegian 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. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable group, solvable, i.e., if the group can be decomposed into a series of normal Group extension, extensions of an abelian group. Every finite extension of a finite field is a cyclic extension. Description Class field theory provides detailed information about the abelian extensions of number fields, function field of an algebraic variety, function fields of algebraic curves over finite fields, and local fields. There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for over 250 years. He was also an innovator in the field of elliptic functions and the discoverer of Abelian functions. He made his discoveries while living in poverty and died at the age of 26 from tuberculosis. Most of his work was done in six or seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "What a head the young Norwegian has!" Life Early life Niels Henrik Abel was born prematurely in Nedstrand, Norway, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pre-abelian Category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear); # C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts; # given any morphism ''f'': ''A'' → ''B'' in C, the equaliser of ''f'' and the zero morphism from ''A'' to ''B'' exists (this is by definition the kernel of ''f''), as does the coequaliser (this is by definition the cokernel of ''f''). Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(''A'',''B''), which is an abelian group by item 1; or as the unique mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Movses Abelian
Movses Abelian is an Armenian diplomat who has served as Under-Secretary-General for General Assembly and Conference Management and Coordinator for Multilingualism in the United Nations Secretariat since 2019.United Nations, Movses Abelian, Under-Secretary-General for General Assembly and Conference Management Retrieved 13 October 2024. Education and early career Abelian was educated in Armenia, Russia and the United States of America. He was an associate professor at from 1989 to 1992 before joining the[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hovhannes Abelian
Hovhannes Harutyuni Abelian (, October 23, 1865, Shamakhi, Baku Governorate, Russian Empire - July 1, 1936, Yerevan, Soviet Armenia) was an Armenian actor, People's Artist of the Armenian SSR (1925). Biography From 1882, he worked in Armenian and Russian theatres of Baku and Tiflis Tbilisi ( ; ka, თბილისი, ), in some languages still known by its pre-1936 name Tiflis ( ), ( ka, ტფილისი, tr ) is the Capital city, capital and List of cities and towns in Georgia (country), largest city of Georgia ( .... In 1908, he became the founder of " Abelian-Armenian Theatral Group", and realised artistic tours in different countries (Russia, Iran, Germany, France, USA). In 1925, Abelian entered to the Armenian State Theatre, and played in the cinema ("Namus", 1925). A realistic-style actor, he played more than 300 roles. Sources * Armenian Concise Encyclopedia, Ed. by acad. K. Khudaverdian, Yerevan, 1990, p. 11 * External links * * {{DEFAULTSOR ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Theory
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, the Lagrangian is invariant under these transformations. The term "gauge" refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the '' symmetry group'' or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Von Neumann Algebra
In functional analysis, a branch of mathematics, an abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commutative, commute. The prototypical example of an abelian von Neumann algebra is the algebra ''L''∞(''X'', μ) for μ a measure (mathematics), measure on ''X'' realized as an algebra of operators on the Hilbert space ''L''2(''X'', μ) as follows: Each ''f'' ∈ ''L''∞(''X'', μ) is identified with the multiplication operator :\psi \mapsto f \psi. Of particular importance are the abelian von Neumann algebras on separable space, separable Hilbert spaces, particularly since they are completely classifiable by simple invariants. Though there is a theory for von Neumann algebras on non-separable Hilbert spaces (and indeed much of the general theory still holds in that case) the theory is considerably simpler for algebras on separable spaces and most applications to other areas of mathematics or physics only use separab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian And Tauberian Theorems
In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing that if a series converges to some limit then its Abel sum is the same limit, and Tauber's theorem showing that if the Abel sum of a series exists and the coefficients are sufficiently small (o(1/''n'')) then the series converges to the Abel sum. More general Abelian and Tauberian theorems give similar results for more general summation methods. There is not yet a clear distinction between Abelian and Tauberian theorems, and no generally accepted definition of what these terms mean. Often, a theorem is called "Abelian" if it shows that some summation method gives the usual sum for convergent series, and is called "Tauberian" if it gives conditions for a series summable by some method that allows it to be summable in the usual sense. In th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Of Abelian Groups
In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of Ab is the trivial group which consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. Ab is a full subcategory of Grp, the category of ''all'' groups. The main difference between Ab and Grp is that the sum of two homomorphisms ''f'' and ''g'' between abelian groups is again a group homomorphism: :(''f''+''g'')(''x''+''y'') = ''f''(''x''+''y'') + ''g''(''x''+''y'') = ''f''(''x'') + ''f''(''y'') + ''g''(''x'') + ''g''(''y'') : = ''f''(''x'') + ''g''(''x'') + ''f''(''y'') + ''g''(''y'') = (''f''+''g'')(''x'') + (''f''+''g'')(''y' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Category
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, . 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. Mac Lane says Alexander Grothendieck defined the abelian category, but there is a reference that says Eilenberg's disciple, Buchsbaum, proposed the concept in his PhD thesis, and Groth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Integral
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form :\int_^z R(x,w) \, dx, where R(x,w) is an arbitrary rational function of the two variables x and w, which are related by the equation :F(x,w)=0, where F(x,w) is an irreducible polynomial in w, :F(x,w)\equiv\varphi_n(x)w^n+\cdots+\varphi_1(x)w +\varphi_0\left(x\right), whose coefficients \varphi_j(x), j=0,1,\ldots,n are rational functions of 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. Abelian integrals are natural generalizations of elliptic integrals, which arise when :F(x,w)=w^2-P(x), \, where 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), in the formula above, is a polynomial of degree greater than 4. Hist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
