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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. 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, V \not\subset \Omega and U \subset \Omega \cap V such that for every holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ... 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 that is not identically zero, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain Of Holomorphy Illustration
A domain is a geographic area controlled by a single person or organization. Domain may also refer to: Law and human geography * Demesne, in English common law and other Medieval European contexts, lands directly managed by their holder rather than being delegated to subordinate managers * Domaine, a large parcel of land under single ownership, which would historically generate income for its owner. * Eminent domain, the right of a government to appropriate another person's property for public use * Hotzaah#Domains, Private domain / Public domain, places defined under Jewish law where it is either permitted or forbidden to move objects on the Sabbath day * Public domain, creative work to which no exclusive intellectual property rights apply * Territory (subdivision), a non-sovereign geographic area which has come under the authority of another government Science * Domain (biology), a taxonomic subdivision larger than a kingdom * Domain of discourse, the collection of entities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Problem
Levi ( ; ) was, according to the Book of Genesis, the third of the six sons of Jacob and Leah (Jacob's third son), and the founder of the Israelite Tribe of Levi (the Levites, including the Kohanim) and the great-grandfather of Aaron, Moses and Miriam. Certain religious and political functions were reserved for the Levites. Most scholars view the Torah as projecting the origins of the Levites into the past to explain their role as landless cultic functionaries. Origins The Torah suggests that the name ''Levi'' refers to Leah's hope for Jacob to ''join'' with her, implying a derivation from Hebrew ''yillaweh'', meaning ''he will join'', but scholars suspect that it may simply mean "priest", either as a loanword or by referring to those people who were ''joined'' to the Ark of the Covenant. Another possibility is that the Levites were a tribe of Judah not from the clan of Moses or Aaron and that the name "Levites" indicates their ''joining'' - either with the Israelites in gener ... [...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 * A business solution is a method of organizing people and resources that can be sold as a product ** 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 * Nature-based solutions * The Solution (other) The Solution may refer to: *The Solution (Mannafest album), ''The Solution'' (Mannafest album), an album by indie rock band Mannafest *The Solution (Beanie Sigel album), ''The Solution'' (Beanie Sigel album), an album by rapper Beanie Sigel *The So ... [...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 ... [...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 sta .... This theorem again holds for Stein manifolds, but it is not known if it holds for Stein space. References * * Several complex variab ... [...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 pre ... [...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 attended ... [...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 domain of holomorphy, 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 (mathematics), 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 i ... [...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 Goursat ... [...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. Forma ... 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. Furthermore, Oka's lemma is the inverse of Levi's problem (unramified Riemann domain over \Complex^n). Perhaps, this is why Oka referred to Levi's problem as "problème inverse de Hartogs", and could explain why Levi's problem is occasionally referred to as Hartogs' Inverse Problem. References * * * * * Further reading * PDF [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
