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Loop Group
In mathematics, a loop group (not to be confused with a loop) is a group of loops in a topological group ''G'' with multiplication defined pointwise. Definition In its most general form a loop group is a group of continuous mappings from a manifold to a topological group . More specifically, let , the circle in the complex plane, and let denote the space of continuous maps , i.e. :LG = \, equipped with the compact-open topology. An element of is called a ''loop'' in . Pointwise multiplication of such loops gives the structure of a topological group. Parametrize with , :\gamma:\theta \in S^1 \mapsto \gamma(\theta) \in G, and define multiplication in by :(\gamma_1 \gamma_2)(\theta) \equiv \gamma_1(\theta)\gamma_2(\theta). Associativity follows from associativity in . The inverse is given by :\gamma^:\gamma^(\theta) \equiv \gamma(\theta)^, and the identity by :e:\theta \mapsto e \in G. The space is called the free loop group on . A loop group is any subgroup of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng^ \in N for all g \in G and n \in N. The usual notation for this relation is N \triangleleft G. Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G are precisely the kernels of group homomorphisms with domain G, which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. Definitions A subgroup N of a group G is called a normal subgroup of G if it is invariant under conjugation; that is, the conjugation of an element of N by an element of G is always in N. The usual notation fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Algebra
In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. Definition For a Lie algebra \mathfrak over a field K, if K ,t^/math> is the space of Laurent polynomials, then L\mathfrak := \mathfrak\otimes K ,t^ with the inherited bracket \otimes t^m, Y\otimes t^n= ,Yotimes t^. Geometric definition If \mathfrak is a Lie algebra, the tensor product of \mathfrak with , the algebra of (complex) smooth functions over the circle manifold (equivalently, smooth complex-valued periodic functions of a given period), \mathfrak\otimes C^\infty(S^1), is an infinite-dimensional Lie algebra with the Lie bracket given by _1\otimes f_1,g_2 \otimes f_2 _1,g_2otimes f_1 f_2. Here and are elements of \mathfrak and and are elements of . This isn't precisely what would correspond to the direct product of infinitely many copies of \mathfrak, one for each point in , because of the smoothness restriction. Instead, it can be thought of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an ''A''∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of Ω''X'', i.e. the set of based-homotopy equivalence classes of based loops in ''X'', is a group, the fundamental group ''π''1(''X''). The iterated loop spaces of ''X'' are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space ''X'' is the space of maps from the circle ''S''1 to ''X'' with the compact-open topology. The free loop space of ''X'' is often denoted by \mathcalX. As a functor, the free loop space construction is rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karen Uhlenbeck
Karen Keskulla Uhlenbeck ForMemRS (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richardson Foundation Regents Chair. She is currently a distinguished visiting professor at the Institute for Advanced Study and a visiting senior research scholar at Princeton University. Uhlenbeck was elected to the American Philosophical Society in 2007. She won the 2019 Abel Prize for "her pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics." She is the first, and so far only, woman to win the prize since its inception in 2003. She donated half of the prize money to organizations which promote more engagement of women in research mathematics. Life and career Uhlenbeck was born in Clevel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chuu-Lian Terng
Chuu-Lian Terng () is a Taiwanese-American mathematician. Her research areas are differential geometry and integrable systems, with particular interests in completely integrable Hamiltonian partial differential equations and their relations to differential geometry, the geometry and topology of submanifolds in symmetric spaces, and the geometry of isometric actions. Education and career She received her B.S. from National Taiwan University in 1971 and her Ph.D. from Brandeis University in 1976 under the supervision of Richard Palais, whom she later married. She was a postdoctoral researcher at the University of California, Berkeley from 1976-1978, an assistant professor at Princeton University from 1978-1982, and was faculty at Northeastern University from 1982-2004. She was the first female assistant professor in mathematics at Princeton University. She is currently a professor emerita at the University of California at Irvine, which she joined in 2004. She also spent two yea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a balanced cancellation of nonlinear and dispersive effects in the medium.Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency. Solitons were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the " Wave of Translation". The Korteweg–de Vries equation was later formulated to model such waves, and the term "soliton" was coined by Zabu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bäcklund Transform
In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions. They are an important tool in soliton theory and integrable systems. A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter. It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other. A Bäcklund transform which relates solutions of the ''same'' equation is called an invariant Bäcklund transform or auto-Bäcklund transform. If such a transform can be found, much can be deduced about the solutions of the equation especially if the Bäcklund transform contains a parameter. However, no systematic way of finding Bäcklund transforms is known. History Bäcklund transforms have t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of ''H'' as time then ''H'' describes a ''continu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-space
In mathematics, an H-space is a homotopy-theoretic version of a generalization of the notion of topological group, in which the axioms on associativity and inverses are removed. Definition An H-space consists of a topological space , together with an element of and a continuous map , such that and the maps and are both homotopic to the identity map through maps sending to . This may be thought of as a pointed topological space together with a continuous multiplication for which the basepoint is an identity element up to basepoint-preserving homotopy. One says that a topological space is an H-space if there exists and such that the triple is an H-space as in the above definition. Alternatively, an H-space may be defined without requiring homotopies to fix the basepoint , or by requiring to be an exact identity, without any consideration of homotopy. In the case of a CW complex, all three of these definitions are in fact equivalent. Examples and properties The standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an ''A''∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of Ω''X'', i.e. the set of based-homotopy equivalence classes of based loops in ''X'', is a group, the fundamental group ''π''1(''X''). The iterated loop spaces of ''X'' are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space ''X'' is the space of maps from the circle ''S''1 to ''X'' with the compact-open topology. The free loop space of ''X'' is often denoted by \mathcalX. As a functor, the free loop space construction is rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
