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Schwarz Alternating Method
In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869–1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved. This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907. It furnished a scheme for uniformizing the union of two regions knowing how to uniformize each of them separately, provided their intersection was topologically a disk or an annulus. From 1870 onwards Carl Neumann also contributed to this theory. In the 1950s Schwarz's method was generalized in the theory of partial differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Hermann Amand Schwarz (1843-1921) By Louis Zipfel
Hermann or Herrmann may refer to: * Hermann (name), list of people with this name * Arminius, chieftain of the Germanic Cherusci tribe in the 1st century, known as Hermann in the German language * Éditions Hermann, French publisher * Hermann, Missouri, a town on the Missouri River in the United States ** Hermann AVA, Missouri wine region * The German SC1000 bomb of World War II was nicknamed the "Hermann" by the British, in reference to Hermann Göring * Herrmann Hall, the former Hotel Del Monte, at the Naval Postgraduate School, Monterey, California * Memorial Hermann Healthcare System, a large health system in Southeast Texas * The Herrmann Brain Dominance Instrument (HBDI), a system to measure and describe thinking preferences in people * Hermann station (other), stations of the name * Hermann (crater), a small lunar impact crater in the western Oceanus Procellarum * Hermann Huppen, a Belgian comic book artist * Hermann 19, an American sailboat design built by Ted Herma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Additive Schwarz Method
In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. Overview Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down. :(Model problem) The heat distribution in a square metal plate such that the left edge is kept at 1 degree, and the other edges are kept at 0 degree, after letting it sit for a long period of time satisfies the following boundary value problem: ::''f''''xx''(''x'',''y'') + ''f''''yy''(''x'',''y'') = 0 ::''f''(0,''y'') = 1; ''f''(''x'',0) = ''f''(''x'',1) = ''f''(1,''y'') = 0 :where ''f'' is the unknown functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Conformal Mappings
Conformal may refer to: * Conformal (software), in ASIC Software * Conformal coating in electronics * Conformal cooling channel, in injection or blow moulding * Conformal field theory in physics, such as: ** Boundary conformal field theory ** Coset conformal field theory ** Logarithmic conformal field theory ** Rational conformal field theory * Conformal fuel tanks on military aircraft * Conformal hypergraph, in mathematics * Conformal geometry, in mathematics * Conformal group, in mathematics * Conformal map, in mathematics * Conformal map projection In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere or an ellipsoid) is preserved in the image of the projection; that is, the projection is a conformal map in the mat ..., in cartography * Conformal prediction, in computer science {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Doklady Akademii Nauk SSSR
The ''Proceedings of the USSR Academy of Sciences'' (, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), ) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. The journal is indexed in Russian Science Citation Index. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dating from 1726. ''Doklady Akademii Nauk SSSR-Comptes Rendus de l'Académie des Sciences de l'URSS, Seriya ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Additive Schwarz Method
In mathematics, the additive Schwarz method, named after Hermann Schwarz, solves a boundary value problem for a partial differential equation approximately by splitting it into boundary value problems on smaller domains and adding the results. Overview Partial differential equations (PDEs) are used in all sciences to model phenomena. For the purpose of exposition, we give an example physical problem and the accompanying boundary value problem (BVP). Even if the reader is unfamiliar with the notation, the purpose is merely to show what a BVP looks like when written down. :(Model problem) The heat distribution in a square metal plate such that the left edge is kept at 1 degree, and the other edges are kept at 0 degree, after letting it sit for a long period of time satisfies the following boundary value problem: ::''f''''xx''(''x'',''y'') + ''f''''yy''(''x'',''y'') = 0 ::''f''(0,''y'') = 1; ''f''(''x'',0) = ''f''(''x'',1) = ''f''(1,''y'') = 0 :where ''f'' is the unknown functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Schwarz Reflection Principle
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane. In notation, if F(z) is a function that satisfies the above requirements, then its extension to the rest of the complex plane is given by the formula, F(\bar) = \overline. That is, we make the definition that agrees along the real axis. The result proved by Hermann Schwarz is as follows. Suppose that ''F'' is a continuous function on the closed upper half plane \left\ , holomorphic on the upper half plane \left\ , which takes real values on the real axis. Then the extension formula given above is an analytic continuation to the whole complex plane. In practice it would be better to have a theorem t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Schwarz Triangle Map
In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a Schwarz triangle, although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a Möbius triangle, the inverse of the Schwarz triangle function is a single-valued automorphic function for that triangle's triangle group. More specifically, it is a modular function. Formula Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle in radians. Each of ''α'', ''β'', and ''γ'' may take values between 0 and 1 inclusive. Following Nehari, these angles are in clockwise order, with the vertex having angle ''πα'' at the origin and the vertex having angle ''πγ'' lying on the real line. The Schwarz triangle function can be given in terms of hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Schwarzian Derivative
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces. It is named after the German mathematician Hermann Schwarz. Definition The Schwarzian derivative of a holomorphic function of one complex variable is defined by (Sf)(z) = \left( \frac\right)' - \frac\left(\frac\right)^2 = \frac-\frac\left(\frac\right)^2. The same formula also defines the Schwarzian derivative of a Smoothness, function of one Function of a real variable, real variable. The alternative notation \ = (Sf)(z) is frequently used. Properties The Schwarzian derivative of any Möbius transformation g(z) = \frac is zero. Conversely, the Möbius transformations are the only functions wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Dirichlet Conditions
Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory of Fourier series and was one of the first to give the modern formal definition of a function. In mathematical physics, he studied potential theory, boundary-value problems, and heat diffusion, and hydrodynamics. Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in particular for results named after him. Biography Early life (1805–1822) Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of Vienna in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Laplace's Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Solomon Mikhlin
Solomon Grigor'evich Mikhlin (, real name Zalman Girshevich Mikhlin) (the family name is also transliterated as Mihlin or Michlin) (23 April 1908 – 29 August 1990) was a Soviet mathematician of who worked in the fields of linear elasticity, singular integrals and numerical analysis: he is best known for the introduction of the symbol of a singular integral operator, which eventually led to the foundation and development of the theory of pseudodifferential operators.According to and the references cited therein: see also . For more information on this subject, see the entries on singular integral operators and on pseudodifferential operators. Biography He was born in , Rechytsa District, Minsk Governorate (in present-day Belarus) on 23 April 1908; himself states in his resume that his father was a merchant, but this assertion could be untrue since, in that period, people sometimes lied on the profession of parents in order to overcome political limitations in the access ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Elliptic Partial Differential Equation
In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which generally model phenomena that change in time. The canonical examples of elliptic PDEs are Laplace's Equation and Poisson's Equation. Elliptic PDEs are also important in pure mathematics, where they are fundamental to various fields of research such as differential geometry and optimal transport. Definition Elliptic differential equations appear in many different contexts and levels of generality. First consider a second-order linear PDE for an unknown function of two variables u = u(x,y), written in the form Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0, where , , , , , , and are functions of (x,y), using subscript notation for the partial derivatives. The PDE is called elliptic if B^2-AC 0 are hyperbolic. For a general linear second-order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |