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Hilbert's Twenty-first Problem
The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, concerns the existence of a certain class of linear differential equations with specified singular points and monodromic group. Statement The original problem was stated as follows (English translation from 1902): :''Proof of the existence of linear differential equations having a prescribed monodromic group'' :''In the theory of linear differential equations with one independent variable z, I wish to indicate an important problem one which very likely Riemann himself may have had in mind. This problem is as follows: To show that there always exists a linear differential equation of the Fuchsian class, with given singular points and monodromic group. The problem requires the production of n functions of the variable z, regular throughout the complex z-plane except at the given singular points; at these points the functions may become infinite of only finite ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the ''Bulletin of the American Mathematical Society''. Earlier publications (in the original German) appeared in and Nature and influence of the problems Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer. For other problems ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Problems
Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 at the Sorbonne. The complete list of 23 problems was published later, in English translation in 1902 by Mary Frances Winston Newson in the ''Bulletin of the American Mathematical Society''. Earlier publications (in the original German) appeared in and Nature and influence of the problems Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer. For other problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomonodromic Deformation
In mathematics, the equations governing the isomonodromic deformation of meromorphic linear systems of ordinary differential equations are, in a fairly precise sense, the most fundamental exact nonlinear differential equations. As a result, their solutions and properties lie at the heart of the field of exact nonlinearity and integrable systems. Isomonodromic deformations were first studied by Richard Fuchs, with early pioneering contributions from Lazarus Fuchs, Paul Painlevé, René Garnier, and Ludwig Schlesinger. Inspired by results in statistical mechanics, a seminal contribution to the theory was made by Michio Jimbo, Tetsuji Miwa, and Kimio Ueno, who studied cases with arbitrary singularity structure. Fuchsian systems and Schlesinger's equations Consider the Fuchsian system of linear differential equations :\frac=Ay=\sum_^\fracy where the independent variable ''x'' takes values in the complex projective line P1(C), the solution ''y'' takes values in C''n'' and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Röhrl
Helmut Röhrl or Rohrl (born 22 March 1927 in Straubing, died 30 January 2014) was a German mathematician. Besides complex analysis (including among other subjects the Riemann–Hilbert problem), he worked on algebra and category theory and totally convex spaces. In 1964 he edited the new edition of the classic textbook on complex analysis by Adolf Hurwitz and Richard Courant. Röhrl received his doctorate in 1949 at the Ludwig-Maximilians-Universität München under Robert König (and Oskar Perron) with doctoral thesis ''Über Differentialsysteme, welche aus multiplikativen Klassen mit exponentiellen Singularitäten entspringen''published in 3 parts, ''Mathematische Annalen'', vol. 123, 1951, pp. 53–75; vol. 124, 1952, pp. 187–218; vol. 125, 1953, pp. 448–466 and his habilitation in 1953 with habilitation thesis ''Abelsche Integrale auf Riemannschen Flächen endlichen Geschlechts''. He was a docent in mathematics from 1949 to 1951 in Würzburg, from 1951 to 1953 in M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuliy S
Yuliy is a given name. Notable people with the name include: *Yuliy Aykhenval'd or Yuly Aykhenvald (1872–1928), Russian Jewish literary critic who developed a native brand of Aestheticism * Yuliy Daniel or Yuli Daniel (1925–1988), Soviet dissident writer, poet, translator, and political prisoner * Yuliy Dmitrievich Engel (1868–1927), music critic, composer and one of the leading figures in the Jewish art music movement *Yuliy Ganf (1898–1973), Soviet Russian graphic artist, People's Artist of the USSR *Yuliy Kim (born 1936), one of Russia's foremost bards and playwrights *Yuliy Meitus (1903–1997), distinguished Ukrainian composer, famous for his operas *Yuliy Sannikov Yuliy Sannikov (born November 3, 1978) is a Ukrainian economist known for his contributions to mathematical economics, game theory, and corporate finance. He is an economics professor at the Stanford Graduate School of Business, and won both ... (born 1978), Ukrainian economist * Yuliy Osipovich Tseder ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josip Plemelj
Josip Plemelj (December 11, 1873 – May 22, 1967) was a Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory. He was the first chancellor of the University of Ljubljana. Life Plemelj was born in the village of Bled near Bled Castle in Austria-Hungary (now Slovenia); he died in Ljubljana, Yugoslavia (now Slovenia). His father, Urban, a carpenter and crofter, died when Josip was only a year old. His mother Marija, née , found bringing up the family alone very hard, but she was able to send her son to school in Ljubljana, where Plemelj studied from 1886 to 1894. Due to a bench thrown into Tivoli Pond by him or his friends, he could not attend the school after he finished the fourth class and had to pass the final exam privately. After leaving and obtaining the necessary examination results he went to the University of Vienna in 1894 where he had applied to Faculty of Arts to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Hilbert Correspondence
In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generalizations of this. The original setting appearing in Hilbert's twenty-first problem was for the Riemann sphere, where it was about the existence of systems of linear regular differential equations with prescribed monodromy representations. First the Riemann sphere may be replaced by an arbitrary Riemann surface and then, in higher dimensions, Riemann surfaces are replaced by complex manifolds of dimension > 1. There is a correspondence between certain systems of partial differential equations (linear and having very special properties for their solutions) and possible monodromies of their solutions. Such a result was proved for algebraic connections with regular singularities by Pierre Deligne (1970, generalizing existing work in the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Hilbert Problem
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others (see the book by Clancey and Gohberg (1981)). The Riemann problem Suppose that \Sigma is a closed simple contour in the complex plane dividing the plane into two parts denoted by \Sigma_ (the inside) and \Sigma_ (the outside), determined by the index of the contour with respect to a point. The classical problem, considered in Riemann's PhD dissertation (see ), was that of finding a function :M_+(z) = u(z) + i v(z) analytic inside \Sigma_ such that the boundary values of ''M''+ along \Sigma satisfy the equation :a(z)u(z) - b(z)v(z) = c(z) for all z\in \Sigma, where ''a'', ''b'', and ''c'' are given real-valued functions . By the Riemann mapping theorem, it suffices to consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of its domain. It is not required that be unique; the function may map one or more elements of to the same element of . The term ''surjective'' and the related terms '' injective'' and '' bijective'' were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. The French word '' sur'' means ''over'' or ''above'', and relates to the fact that the image of the domain of a surjective function completely covers the function's codomain. Any function induces a surjection by restricting its codomain to the image of its domain. Every surjective function has a right inverse assuming the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
