|
Invariance Of Domain
Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space \R^n. It states: :If U is an open subset of \R^n and f : U \rarr \R^n is an injective continuous map, then V := f(U) is open in \R^n and f is a homeomorphism between U and V. The theorem and its proof are due to L. E. J. Brouwer, published in 1912. The proof uses tools of algebraic topology, notably the Brouwer fixed point theorem. Notes The conclusion of the theorem can equivalently be formulated as: "f is an open map". Normally, to check that f is a homeomorphism, one would have to verify that both f and its inverse function f^ are continuous; the theorem says that if the domain is an subset of \R^n and the image is also in \R^n, then continuity of f^ is automatic. Furthermore, the theorem says that if two subsets U and V of \R^n are homeomorphic, and U is open, then V must be open as well. (Note that V is open as a subset of \R^n, and not just in the subspace topology. Openness ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the Norm (mathematics)#p-norm, -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Nicolas Bourbaki, Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as Central tendency#Solutions to variational problems, solutions to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Continuous Functions
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be scientific, belong to a non-scientific discipline, or no discipline at all. Depending on the context, a theory's assertions might, for example, include generalized explanations of how nature works. The word has its roots in ancient Greek, but in modern use it has taken on several related meanings. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction ("falsify") of it. Scientific theories are the most reliable, rigorous, and compre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comment
Comment may refer to: * Comment (linguistics) or rheme, that which is said about the topic (theme) of a sentence * Bernard Comment (born 1960), Swiss writer and publisher Computing * Comment (computer programming), explanatory text or information embedded in the source code of a computer program * Comment programming, a software development technique based on the regular use of comment tags Law * Public comment, a term used by various U.S. government agencies, referring to comments invited regarding a report or proposal * Short scholarly papers written by members of a law review * Comments on proposed rules under the rulemaking process in United States administrative law Media and entertainment * ''Comment'' (TV series), a 1958 Australian television series * ''Comment'' (album), a 1970 album by Les McCann * "Comment", a 1969 song by Charles Wright & the Watts 103rd Street Rhythm Band * ''Comment'', a quarterly journal published by Cardus * ''Comment'', later ''aCOMMENT'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amer , a town in Sindh province of Pakistan
{{disambiguation, geo ...
Amer may refer to: Places * Amer (river), a river in the Dutch province of North Brabant * Amer, Girona, a municipality in the province of Girona in Catalonia, Spain * Amber, India (also known as Amer, India), former city of Rajasthan state ** Amber Fort (also Amer Fort), India * AMER, a country grouping that refers to America or the Americas People * Amer (name) * Beni-Amer people, a mixed ethnic group inhabiting Sudan and Eritrea Other uses * Amer International Group, a Chinese company * Amer Sports, a Finnish headquartered sporting goods company * ''Amer'' (film), a 2009 Belgian-French thriller See also * Umerkot Umerkot (formerly known as Amarkot) is a city in the Sindh province of Pakistan. The local language is Dhatki, which is one of the Rajasthani languages of the Indo-Aryan language family. It is most closely related to Marwari. Sindhi, Urdu and ... [...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, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Leray
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology. Life and career He was born in Chantenay-sur-Loire (today part of Nantes). He studied at École Normale Supérieure from 1926 to 1929. He received his Ph.D. in 1933. In 1934 Leray published an important paper that founded the study of weak solutions of the Navier–Stokes equations. In the same year, he and Juliusz Schauder discovered a topological invariant, now called the Leray–Schauder degree, which they applied to prove the existence of solutions for partial differential equations lacking uniqueness. From 1938 to 1939 he was professor at the University of Nancy. He did not join the Bourbaki group, although he was close with its founders. His main work in topology was carried out while he was in a prisoner of war camp in Edelbach, Austria from 1940 to 1945. He concealed his expertise on differential equations, fearing th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Homeomorphism
In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an étale space over Y. Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps. A topological space X is locally homeomorphic to Y if every point of X has a neighborhood that is homeomorphic to an open subset of Y. For example, a manifold of dimension n is locally homeomorphic to \R^n. If there is a local homeomorphism from X to Y, then X is locally homeomorphic to Y, but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane \R^2, but there is no local homeomorphism S^2 \to \R^2. Formal definition A function f : X \to Y between two topological spaces is called a if for every point x \in X there exists an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighborhood (topology)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set. Definitions Neighbourhood of a point If X is a topological space and p is a point in X, then a of p is a subset V of X that includes an open set U containing p, p \in U \subseteq V \subseteq X. This is also equivalent to the point p \in X belonging to the topological interior of V in X. The neighbourhood V need be an open subset X, but when V is open in X then it is called an . Some authors have been known to require neighbourhoods to be open, so it is important to note conventions. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
