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Magnus And Oberhettinger
Hans Heinrich Wilhelm Magnus known as Wilhelm Magnus (February 5, 1907 in Berlin, Germany – October 15, 1990 in New Rochelle, New York) was a German-American mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations. Biography In 1931, Magnus received his PhD from the University of Frankfurt, in Germany. His thesis, written under the direction of Max Dehn, was entitled ''Über unendlich diskontinuierliche Gruppen von einer definierenden Relation (der Freiheitssatz)''. Magnus was a faculty member in Frankfurt from 1933 until 1938. He refused to join the Nazi Party and, as a consequence, was not allowed to hold an academic post during World War II. In 1947 he became a professor at the University of Göttingen. In 1948 he emigrated to the United States to collaborate on the Bateman Manuscript Project as a co-editor, while a visiting professor at the California Institute of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constituent states, Berlin is surrounded by the State of Brandenburg and contiguous with Potsdam, Brandenburg's capital. Berlin's urban area, which has a population of around 4.5 million, is the second most populous urban area in Germany after the Ruhr. The Berlin-Brandenburg capital region has around 6.2 million inhabitants and is Germany's third-largest metropolitan region after the Rhine-Ruhr and Rhine-Main regions. Berlin straddles the banks of the Spree, which flows into the Havel (a tributary of the Elbe) in the western borough of Spandau. Among the city's main topographical features are the many lakes in the western and southeastern boroughs formed by the Spree, Havel and Dahme, the largest of which is Lake Müggelsee. Due to its l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bateman Manuscript Project
The Bateman Manuscript Project was a major effort at collation and encyclopedic compilation of the mathematical theory of special functions. It resulted in the eventual publication of five important reference volumes, under the editorship of Arthur Erdélyi. Overview The theory of special functions was a core activity of the field of applied mathematics, from the middle of the nineteenth century to the advent of high-speed electronic computing. The intricate properties of spherical harmonics, elliptic functions and other staples of problem-solving in mathematical physics, astronomy and right across the physical sciences, are not easy to document completely, absent a theory explaining the inter-relationships. Mathematical tables to perform actual calculations needed to mesh with an adequate theory of how functions could be transformed into those already tabulated. Harry Bateman, a distinguished applied mathematician, undertook the somewhat quixotic task of trying to collate the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Lie Algebra
In mathematics, a free Lie algebra over a field ''K'' is a Lie algebra generated by a set ''X'', without any imposed relations other than the defining relations of alternating ''K''-bilinearity and the Jacobi identity. Definition The definition of the free Lie algebra generated by a set ''X'' is as follows: : Let ''X'' be a set and i\colon X \to L a morphism of sets (function) from ''X'' into a Lie algebra ''L''. The Lie algebra ''L'' is called free on ''X'' if i is the universal morphism; that is, if for any Lie algebra ''A'' with a morphism of sets f\colon X \to A, there is a unique Lie algebra morphism g\colon L\to A such that f = g \circ i. Given a set ''X'', one can show that there exists a unique free Lie algebra L(X) generated by ''X''. In the language of category theory, the functor sending a set ''X'' to the Lie algebra generated by ''X'' is the free functor from the category of sets to the category of Lie algebras. That is, it is left adjoint to the forgetful functo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator Collecting Process
In group theory, a branch of mathematics, the commutator collecting process is a method for writing an element of a group as a product of generators and their higher commutators arranged in a certain order. The commutator collecting process was introduced by Philip Hall in 1934 and articulated by Wilhelm Magnus in 1937. W. Magnus (1937), "Über Beziehungen zwischen höheren Kommutatoren", ''J. Grelle'' 177, 105-115. The process is sometimes called a "collection process". The process can be generalized to define a totally ordered subset of a free non-associative algebra, that is, a free magma; this subset is called the Hall set. Members of the Hall set are binary trees; these can be placed in one-to-one correspondence with words, these being called the Hall words; the Lyndon words are a special case. Hall sets are used to construct a basis for a free Lie algebra, entirely analogously to the commutator collecting process. Hall words also provide a unique factorization of monoids ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-relator Group
In the mathematical subject of group theory, a one-relator group is a group given by a group presentation with a single defining relation. One-relator groups play an important role in geometric group theory by providing many explicit examples of finitely presented groups. Formal definition A one-relator group is a group ''G'' that admits a group presentation of the form where ''X'' is a set (in general possibly infinite), and where r\in F(X) is a freely and cyclically reduced word. If ''Y'' is the set of all letters x\in X that appear in ''r'' and X'=X\setminus Y then :G=\langle Y\mid r=1\, \rangle \ast F(X'). For that reason ''X'' in () is usually assumed to be finite where one-relator groups are discussed, in which case () can be rewritten more explicitly as where X=\ for some integer n\ge 1. Freiheitssatz Let ''G'' be a one-relator group given by presentation () above. Recall that ''r'' is a freely and cyclically reduced word in ''F''(''X''). Let y\in X be a lette ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnus Expansion
In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first-order homogeneous linear differential equation for a linear operator. In particular, it furnishes the fundamental matrix of a system of linear ordinary differential equations of order with varying coefficients. The exponent is aggregated as an infinite series, whose terms involve multiple integrals and nested commutators. The deterministic case Magnus approach and its interpretation Given the coefficient matrix , one wishes to solve the initial-value problem associated with the linear ordinary differential equation : Y'(t) = A(t) Y(t), \quad Y(t_0) = Y_0 for the unknown -dimensional vector function . When ''n'' = 1, the solution simply reads : Y(t) = \exp \left( \int_^t A(s)\,ds \right) Y_0. This is still valid for ''n'' > 1 if the matrix satisfies for any pair of values of ''t'', ''t''1 an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kathryn F
Kathryn is a feminine given name and comes from the Greek meaning for 'pure'. It is a variant of Katherine. It may refer to: In television and film: * Kathryn Beaumont (born 1938), English voice actress and school teacher best known for her Disney animation film works * Kathryn Bernardo (born 1996), Filipina actress and recording artist * Kathryn Bigelow (born 1951), American film director, first woman to win the Academy Award, BAFTA, and DGA award for Best Director * Kathryn Busby, American television and film executive * Kathryn Cressida also known as "Kat" Cressida (born 1968), American voice actress * Kathryn Crosby (born 1933), American actress and singer who performed her most memorable roles under her birth-name Kathryn Grant * Kathryn Drysdale (born 1981), English actress * Kathryn Eames (1908 – 2004), American screen, stage, and television actress * Kathryn Erbe (born 1966), American actress best known for her lead role as Detective Eames on ''Law & Order: Criminal In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seymour Lipschutz
Seymour Saul Lipschutz (born 1931 died March 2018) was an author of technical books on pure mathematics and probability, including a collection of Schaum's Outlines. Lipschutz received his Ph.D. in 1960 from New York University's Courant Institute . He received his BA and MA degrees in Mathematics at Brooklyn College. He was a mathematics professor at Temple University, and before that on the faculty at the Polytechnic Institute of Brooklyn The New York University Tandon School of Engineering (commonly referred to as Tandon) is the engineering and applied sciences school of New York University. Tandon is the second oldest private engineering and technology school in the United Sta .... Bibliography *Schaum's Outline of Discrete Mathematics *Schaum's Outline of Probability *Schaum's Outline of Finite Mathematics *Schaum's Outline of Linear Algebra *Schaum's Outline of Beginning Linear Algebra *Schaum's Outline of Set Theory *Schaum's Outline of General Topology *Schaum's Out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Keller
Herbert Bishop Keller (19 June 1925 in Paterson, New Jersey – 26 January 2008 in Pasadena, California) was an American applied mathematician and numerical analyst. He was professor of applied mathematics, emeritus, at the California Institute of Technology. Early life and education Keller graduated from the Georgia Institute of Technology with a bachelor's in electrical engineering in 1945; and from New York University, later known as the Courant Institute, with a Ph.D. in 1954. Career A Fellow of the American Academy of Arts and Sciences, Keller was known for his contributions to the field of applied mathematics and numerical analysis including bifurcation theory, computational fluid dynamics (CFD), and scientific computing. He is the co-author, with Eugene Isaacson, of the classic textbook ''Analysis of Numerical Methods'', which is still in use in leading graduate programs in applied mathematics. It is interesting to observe that the first letters of each sentence in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edna Grossman
Edna Grossman (born Edna Kalka) is an American mathematician. She was born in Germany, grew up in Brooklyn, New York, and graduated with a B.S. in mathematics from Brooklyn College. She earned her M.S. in mathematics from New York University's Courant Institute of Mathematical Sciences, where she also received her Ph.D. in mathematics in 1972; her thesis, supervised by Wilhelm Magnus, concerned the symmetries of free groups. Grossman worked for IBM, where she was part of the team that designed and analyzed the Data Encryption Standard. She is known for her development, along with Bryant Tuckerman, of the first slide attack in cryptanalysis Cryptanalysis (from the Greek ''kryptós'', "hidden", and ''analýein'', "to analyze") refers to the process of analyzing information systems in order to understand hidden aspects of the systems. Cryptanalysis is used to breach cryptographic sec ..... References 20th-century American mathematicians 21st-century American mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joan Birman
Joan Sylvia Lyttle Birman (born May 30, 1927, in New York CityLarry Riddle., ''Biographies of Women Mathematicians'', at Agnes Scott College) is an American mathematician, specializing in low-dimensional topology. She has made contributions to the study of knots, 3-manifolds, mapping class groups of surfaces, geometric group theory, contact structures and dynamical systems. Birman is research professor emerita at Barnard College, Columbia University, where she has been since 1973. Family Her parents were George and Lillian Lyttle, both Jewish immigrants. Her father was from Russia but grew up in Liverpool, England. Her mother was born in New York and her parents were Russian-Polish immigrants. At age 17, George emigrated to the US and became a successful dress manufacturer. He appreciated the opportunities from having a business but he wanted his daughters to focus on education. She has three children, Kenneth P. Birman, Deborah Birman Shlider, and Carl David Birman. Her late ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytechnic University Of New York
The New York University Tandon School of Engineering (commonly referred to as Tandon) is the engineering and applied sciences school of New York University. Tandon is the second oldest private engineering and technology school in the United States. The school dates back to 1854 when its predecessor institutions, the University of the City of New York School of Civil Engineering and Architecture and the Brooklyn Collegiate and Polytechnic Institute, were founded. The school was renamed in 2015 in honor of NYU Trustees Chandrika and Ranjan Tandon following their donation of $100 million to the school. The school's main campus is in Brooklyn's MetroTech Center, an urban academic-industrial research park. It is one of several engineering schools that were founded based on a European polytechnic university model in the 1800s, in response to the increasing industrialization of the United States. It has been a key center of research in the development of microwave, wireless, radar, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
