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Domain Of Holomorphy
In mathematics, in the theory of functions of Function of several complex variables, several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be analytic continuation, extended to a bigger domain. Formally, an open set \Omega in the ''n''-dimensional complex space ^n is called a ''domain of holomorphy'' if there do not exist non-empty open sets U \subset \Omega and V \subset ^n where V is connected space, connected, V \not\subset \Omega and U \subset \Omega \cap V such that for every holomorphic function f on \Omega there exists a holomorphic function g on V with f = g on U In the n=1 case, every open set is a domain of holomorphy: we can define a holomorphic function with zeros accumulation point, accumulating everywhere on the boundary (topology), boundary of the domain, which must then be a analytic continuation#Natural boundary, natural boundary for a domain of definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of Holomorphy Illustration
Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Domain (mathematical analysis), an open connected set *Domain of discourse, the set of entities over which logic variables may range *Algebraic structure, Domain of an algebraic structure, the set on which the algebraic structure is defined *Domain theory, the study of certain subsets of continuous lattices that provided the first denotational semantics of the lambda calculus *Domain (ring theory), a nontrivial ring without left or right zero divisors **Integral domain, a non-trivial commutative ring without zero divisors ***Atomic domain, an integral domain in which every non-zero non-unit is a finite product of irreducible elements ***Bézout domain, an integral domain in which the sum of two principal ideals is again a principal ideal ***Eu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Problem
In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of affine varieties or affine schemes in algebraic geometry. Definition Suppose X is a complex manifold of complex dimension n and let \mathcal O(X) denote the ring of holomorphic functions on X. We call X a Stein manifold if the following conditions hold: * X is holomorphically convex, i.e. for every compact subset K \subset X, the so-called ''holomorphically convex hull'', ::\bar K = \left \, :is also a ''compact'' subset of X. * X is holomorphically separable, i.e. if x \neq y are two points in X, then there exists f \in \mathcal O(X) such that f(x) \neq f(y). Non-compact Riemann surfaces are Stein manifolds Let ''X'' be a connected, non-compact R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solution Of The Levi Problem
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Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Solution, in problem solving * Solution, in solution selling Other uses * V-STOL Solution, an ultralight aircraft * Solution (band), a Dutch rock band ** ''Solution'' (Solution album), 1971 * Solution A.D., an American rock band * ''Solution'' (Cui Jian album), 1991 * ''Solutions'' (album), a 2019 album by K.Flay See also * The Solution (other) The Solution may refer to: * ''The Solution'' (Mannafest album), an album by indie rock band Mannafest * ''The Solution'' (Beanie Sigel album), an album by rapper Beanie Sigel * ''The Solution'' (Buckshot and 9th Wonder album), a collaboration albu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Pseudoconvex
In mathematics, more precisely in the theory of functions of several complex variables, a pseudoconvex set is a special type of open set in the ''n''-dimensional complex space C''n''. Pseudoconvex sets are important, as they allow for classification of domains of holomorphy. Let :G\subset ^n be a domain, that is, an open connected subset. One says that G is ''pseudoconvex'' (or '' Hartogs pseudoconvex'') if there exists a continuous plurisubharmonic function \varphi on G such that the set :\ is a relatively compact subset of G for all real numbers x. In other words, a domain is pseudoconvex if G has a continuous plurisubharmonic exhaustion function. Every (geometrically) convex set is pseudoconvex. However, there are pseudoconvex domains which are not geometrically convex. When G has a C^2 (twice continuously differentiable) boundary, this notion is the same as Levi pseudoconvexity, which is easier to work with. More specifically, with a C^2 boundary, it can be shown t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Behnke–Stein Theorem
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a union of an increasing sequence G_k \subset \mathbb^n (i.e., G_k \subset G_) of domains of holomorphy is again a domain of holomorphy. It was proved by Heinrich Behnke and Karl Stein in 1938. This is related to the fact that an increasing union of pseudoconvex domains is pseudoconvex and so it can be proven using that fact and the solution of the Levi problem. Though historically this theorem was in fact used to solve the Levi problem, and the theorem itself was proved using the Oka–Weil theorem In mathematics, especially the theory of several complex variables, the Oka–Weil theorem is a result about the uniform convergence of holomorphic functions on Stein spaces due to Kiyoshi Oka and André Weil. Statement The Oka–Weil theorem .... This theorem again holds for Stein manifolds, but it is not known if it holds for Stein space. References * * Several complex variable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cousin Problems
In mathematics, the Cousin problems are two questions in several complex variables, concerning the existence of meromorphic functions that are specified in terms of local data. They were introduced in special cases by Pierre Cousin in 1895. They are now posed, and solved, for any complex manifold ''M'', in terms of conditions on ''M''. For both problems, an open cover of ''M'' by sets ''Ui'' is given, along with a meromorphic function ''fi'' on each ''Ui''. First Cousin problem The first Cousin problem or additive Cousin problem assumes that each difference :f_i-f_j is a holomorphic function, where it is defined. It asks for a meromorphic function ''f'' on ''M'' such that :f-f_i is ''holomorphic'' on ''Ui''; in other words, that ''f'' shares the singular behaviour of the given local function. The given condition on the f_i-f_j is evidently ''necessary'' for this; so the problem amounts to asking if it is sufficient. The case of one variable is the Mittag-Leffler theorem on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lars Hörmander
Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Medal in 1962 and the Wolf Prize in 1988. In 2006 he was awarded the Steele Prize for Mathematical Exposition for his four-volume textbook ''Analysis of Linear Partial Differential Operators'', which is considered a foundational work on the subject. Hörmander completed his Ph.D. in 1955 at Lund University. Hörmander then worked at Stockholm University, at Stanford University, and at the Institute for Advanced Study in Princeton, New Jersey. He returned to Lund University as a professor from 1968 until 1996, when he retired with the title of professor emeritus. Biography Education Hörmander was born in Mjällby, a village in Blekinge in southern Sweden where his father was a teacher. Like his older brothers and sisters before him, he att ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kiyoshi Oka
was a Japanese mathematician who did fundamental work in the theory of several complex variables. Biography Oka was born in Osaka. He went to Kyoto Imperial University in 1919, turning to mathematics in 1923 and graduating in 1924. He was in Paris for three years from 1929, returning to Hiroshima University. He published solutions to the first and second Cousin problems, and work on domains of holomorphy, in the period 1936–1940. He received his Doctor of Science degree from Kyoto Imperial University in 1940. These were later taken up by Henri Cartan and his school, playing a basic role in the development of sheaf theory. The Oka–Weil theorem is due to a work of André Weil in 1935 and Oka's work in 1937. Oka continued to work in the field, and proved Oka's coherence theorem in 1950. Oka's lemma is also named after him. He was a professor at Nara Women's University from 1949 to retirement at 1964. He received many honours in Japan. Honors * 1951 Japan Academy Pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eugenio Elia Levi
Eugenio Elia Levi (18 October 1883 – 28 October 1917) was an Italian mathematician, known for his fundamental contributions in group theory, in the theory of partial differential operators and in the theory of functions of several complex variables. He was a younger brother of Beppo Levi and was killed in action during First World War. Work Research activity He wrote 33 papers, classified by his colleague and friend Mauro Picone according to the scheme reproduced in this section. Differential geometry Group theory He wrote only three papers in group theory: in the first one, discovered what is now called Levi decomposition, which was conjectured by Wilhelm Killing and proved by Élie Cartan in a special case. Function theory In the theory of functions of several complex variables he introduced the concept of pseudoconvexity during his investigations on the domain of existence of such functions: it turned out to be one of the key concepts of the theory. Cauchy and Gour ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oka's Lemma
In mathematics, Oka's lemma, proved by Kiyoshi Oka, states that in a domain of holomorphy In mathematics, in the theory of functions of several complex variables, a domain of holomorphy is a domain which is maximal in the sense that there exists a holomorphic function on this domain which cannot be extended to a bigger domain. For ... in \Complex^n, the function -\log d(z) is plurisubharmonic, where d is the distance to the boundary. This property shows that the domain is pseudoconvex. Historically, this lemma was first shown in the Hartogs domain in the case of two variables, also Oka's lemma is the inverse of the Levi's problem (unramified Riemann domain over \Complex^n). So maybe that's why Oka called Levi's problem as "problème inverse de Hartogs", and the Levi's problem is occasionally called Hartogs' Inverse Problem. References * * * * * Further reading * PDF [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |