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Quasiregular Map
In the mathematical field of analysis, quasiregular maps are a class of continuous maps between Euclidean spaces R''n'' of the same dimension or, more generally, between Riemannian manifolds of the same dimension, which share some of the basic properties with holomorphic functions of one complex variable. Motivation The theory of holomorphic (=analytic) functions of one complex variable is one of the most beautiful and most useful parts of the whole mathematics. One drawback of this theory is that it deals only with maps between two-dimensional spaces (Riemann surfaces). The theory of functions of several complex variables has a different character, mainly because analytic functions of several variables are not conformal. Conformal maps can be defined between Euclidean spaces of arbitrary dimension, but when the dimension is greater than 2, this class of maps is very small: it consists of Möbius transformations only. This is a theorem of Joseph Liouville; relaxing the smoothne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simion Stoilov
Simion Stoilow or Stoilov ( – 4 April 1961) was a Romanian mathematician, creator of the Romanian school of complex analysis, and author of over 100 publications. Biography He was born in Bucharest, and grew up in Craiova. His father, Colonel Simion Stoilow, fought at the in the Romanian War of Independence. After studying at the Obedeanu elementary school and the Carol I High School, Stoilow went in 1907 to the University of Paris, where he earned a B.S. degree in 1910 and a Ph.D. in Mathematics in 1916. His doctoral dissertation was written under the direction of Émile Picard. He returned to Romania in 1916 to fight in the Romanian Campaign of World War I, first in Dobrudja, then in Moldavia. After the war, he became professor of mathematics at the University of Iași (1919–1921) and the University of Cernăuți (1921–1939). He was an Invited Speaker of the International Congress of Mathematicians in 1920 at Strasbourg, in 1928 at Bologna, and in 1936 at Oslo. In 192 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member of a given Set (mathematics), collection of Subset, subsets of a given set, a collection that has the property of containing every union (set theory), union of its members, every finite intersection (set theory), intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology (structure), topology. These conditions are very loose, and allow enormous flexibility in the choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as \nabla^2 f = 0 or \Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subharmonic Function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows. If the graph of a convex function and a line intersect at two points, then the graph of the convex function is ''below'' the line between those points. In the same way, if the values of a subharmonic function are no larger than the values of a harmonic function on the ''boundary'' of a ball, then the values of the subharmonic function are no larger than the values of the harmonic function also ''inside'' the ball. ''Superharmonic'' functions can be defined by the same description, only replacing "no larger" with "no smaller". Alternatively, a superharmonic function is just the negative of a subharmonic function, and for this reason any property of subharmonic functions can be easily transferred to superharm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Drasin
David Drasin (born 3 November 1940, Philadelphia) is an American mathematician, specializing in function theory. Drasin received in 1962 his bachelor's degree from Temple University and in 1966 his doctorate from Cornell University supervised by Wolfgang Fuchs and Clifford John Earle, Jr. with thesis ''An integral Tauberian theorem and other topics''. After that he was an assistant professor, from 1969 an associate professor, and from 1974 a full professor at Purdue University. He was visiting professor in 2005 at the University of Kiel and in 2005/2006 at the University of Helsinki. In 1976, Drasin gave a complete solution to the inverse problem of Nevanlinna theory (value distribution theory), which was posed by Rolf Nevanlinna in 1929. In the 1930s, the problem was investigated by Nevanlinna and by, among others, Egon Ullrich( de) (1902–1957) with later investigations by Oswald Teichmüller (1913–1943), Hans Wittich, Le Van Thiem (1918–1991) and other mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Picard's Theorem
In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of a function, range of an analytic function. They are named after Émile Picard. The theorems Little Picard Theorem: If a function (mathematics), function f: \mathbb \to\mathbb is entire function, entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted by \lambda, and which performs, using modern terminology, the holomorphic universal covering of the twice punctured plane by the unit disc. This function is explicitly constructed in the theory of elliptic functions. If f omits two values, then the composition of f with the inverse of the modular function maps the plane into the unit disc which implies that f is constant by Liouville's theorem (complex analysis), Liouville's theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zorich's Theorem
In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. The result was conjectured by M. A. Lavrentev in 1938. Theorem Every locally homeomorphic quasiregular map In the mathematical field of analysis, quasiregular maps are a class of continuous maps between Euclidean spaces R''n'' of the same dimension or, more generally, between Riemannian manifolds of the same dimension, which share some of the basic prop ...ping f : R^ \rightarrow R^ for n \geq 3, is a homeomorphism of R^. The fact that there is no such result for n = 2 is easily shown using the exponential function. References General topology Theorems in topology Conjectures that have been proved {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiconformal
In mathematical complex analysis, a quasiconformal mapping is a (weakly differentiable) homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity. Quasiconformal mappings are a generalization of conformal mappings that permit the bounded distortion of angles locally. Quasiconformal mappings were introduced by and named by , Intuitively, let ''f'' : ''D'' → ''D''′ be an orientation-preserving homeomorphism between open sets in the plane. If ''f'' is continuously differentiable, it is ''K''-quasiconformal if, at every point, its derivative maps circles to ellipses with the ratio of the major to minor axis bounded by ''K''. Definition Suppose ''f'' : ''D'' → ''D''′ where ''D'' and ''D''′ are two domains in C. There are a variety of equivalent definitions, depending on the required smoothness of ''f''. If ''f'' is assumed to have continuous partial derivatives, then ''f'' is quasiconformal pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
