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Behnke–Stein Theorem On Stein Manifolds
In mathematics, especially several complex variables, the Behnke–Stein theorem states that a connected, non-compact (open) Riemann surface is a Stein manifold. In other words, it states that there is a nonconstant single-valued holomorphic function (univalent function) on such a Riemann surface. It is a generalization of the Runge approximation theorem and was proved by Heinrich Behnke and Karl Stein in 1948. Method of proof The study of Riemann surfaces typically belongs to the field of one-variable complex analysis, but the proof method uses the approximation by the polyhedron domain used in the proof of the Behnke–Stein theorem on domains of holomorphy and 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 .... References Several complex variables ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Function Of Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. As in complex analysis of functions of one variable, which is the case , the functions studied are ''holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subset \mathbb C), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Karl Stein (mathematician)
Karl Stein (1 January 1913 in Hamm, Westphalia – 19 October 2000) was a German mathematician. He is well known for complex analysis and cryptography. Stein manifolds and Stein factorization are named after him. Career Karl Stein received his doctorate with his dissertation on the topic ''Zur Theorie der Funktionen mehrerer komplexer Veränderlichen; Die Regularitätshüllen niederdimensionaler Mannigfaltigkeiten'' at the University of Münster under the supervision of Heinrich Behnke in 1937. Karl Stein was conscripted into the Wehrmacht sometime before 1942, and trained as a cryptographer to work at OKW/Chi, the Cipher Department of the High Command of the Wehrmacht. He was assigned to manage the OKW/Chi IV, Subsection a, which was a unit responsible for security of own processes, cipher devices testing, and invention of new cipher devices. He managed a staff of 11 In 1955, he became professor at the Ludwig Maximilian University of Munich and emeritated in 1981. In 1990 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. As in complex analysis of functions of one variable, which is the case , the functions studied are '' holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subset \mathbb C), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 states that if ''X'' is a Stein space and ''K'' is a compact \mathcal(X)-convex subset of ''X'', then every holomorphic function in an open neighborhood of ''K'' can be approximated uniformly on ''K'' by holomorphic functions on \mathcal(X) (in particular, by polynomials). Applications Since Runge's theorem may not hold for several complex variables, the Oka–Weil theorem is often used as an approximation theorem for several complex variables. The Behnke–Stein theorem was originally proved using the Oka–Weil theorem. See also * Oka coherence theorem In mathematics, the Oka coherence theorem, proved by , states that the sheaf \mathcal_ of holomorphic functions on \mathbb^n (and subsequently the sheaf \mathcal_ of holomorphic functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Analytic Polyhedron
In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space of the form :P = \ where is a bounded connected open subset of , f_j are holomorphic on and is assumed to be relatively compact in .See and . If f_j above are polynomials, then the set is called a polynomial polyhedron. Every analytic polyhedron is a domain of holomorphy and it is thus pseudo-convex. The boundary of an analytic polyhedron is contained in the union of the set of hypersurfaces : \sigma_j = \, \; 1 \le j \le N. An analytic polyhedron is a ''Weil polyhedron'', or Weil domain if the intersection of any of the above hypersurfaces has dimension no greater than .. See also * Behnke–Stein theorem * Bergman–Weil formula * Oka–Weil theorem Notes References *. * (also available as ). *. *. *. *. *. Notes from a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica The Istituto Nazionale di Alta Matematica Francesco Seve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. The journal is devoted to shorter research articles. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Heinrich Behnke
Heinrich Adolph Louis Behnke (9 October 1898 in Horn – 10 October 1979 in Münster) was a German mathematician and rector at the University of Münster. Life and career He was born into a Lutheran family in Horn, a suburb of Hamburg. He attended the University of Göttingen and submitted his doctoral thesis to the University of Hamburg. He was noted for work on complex analysis with Henri Cartan and Peter Thullen. His first wife, Aenne Albersheim, was Jewish, but she died soon after the birth of their son. He was concerned about his son's ethnicity during the Nazi period. In 1936 he was elected a member of the ''Deutsche Akademie der Naturforscher Leopoldina''. Selected publications *with Peter Thullen: Theorie der Funktionen mehrerer komplexer Veränderlicher, Springer Verlag, Ergebnisse der Mathematik und ihrer Grenzgebiete, 1934, 2nd edn. with collaboration by Reinhold Remmert 1970 *with Friedrich Sommer: Theorie der Funktionen einer komplexen Veränderlichen, Springe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Runge Approximation Theorem
Runge may refer to: Locations * Runge, Texas, United States * Runge (crater), lunar crater in Mare Smythii * Runge (surname) See also * Runge Newspapers, newspaper chain in Ontario, Canada *Inspector Heinrich Runge (although it is more often spelled "Lunge"), a character in the ''Monster'' series {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Univalent Function
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective. Examples The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, as f(z) = f(w) implies that f(z) - f(w) = (z-w)(z+w+2) = 0. As the second factor is non-zero in the open unit disc, z = w so f is injective. Basic properties One can prove that if G and \Omega are two open connected sets in the complex plane, and :f: G \to \Omega is a univalent function such that f(G) = \Omega (that is, f is surjective), then the derivative of f is never zero, f is invertible, and its inverse f^ is also holomorphic. More, one has by the chain rule :(f^)'(f(z)) = \frac for all z in G. Comparison with real functions For real analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
