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Hilbert Metric
In mathematics, the Hilbert metric, also known as the Hilbert projective metric, is an explicitly defined distance function on a bounded convex subset of the ''n''-dimensional Euclidean space R''n''. It was introduced by as a generalization of Cayley's formula for the distance in the Cayley–Klein model of hyperbolic geometry, where the convex set is the ''n''-dimensional open unit ball. Hilbert's metric has been applied to Perron–Frobenius theory and to constructing Gromov hyperbolic spaces. Definition Let Ω be a convex open domain in a Euclidean space that does not contain a line. Given two distinct points ''A'' and ''B'' of Ω, let ''X'' and ''Y'' be the points at which the straight line ''AB'' intersects the boundary of Ω, where the order of the points is ''X'', ''A'', ''B'', ''Y''. Then the Hilbert distance ''d''(''A'', ''B'') is the logarithm of the cross-ratio of this quadruple of points: : d(A,B)=\log\left(\frac\frac\right). The function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projectivization
In mathematics, projectivization is a procedure which associates with a non-zero vector space ''V'' a projective space (V), whose elements are one-dimensional subspaces of ''V''. More generally, any subset ''S'' of ''V'' closed under scalar multiplication defines a subset of (V) formed by the lines contained in ''S'' and is called the projectivization of ''S''. Properties * Projectivization is a special case of the factorization by a group action: the projective space (V) is the quotient of the open set ''V''\ of nonzero vectors by the action of the multiplicative group of the base field by scalar transformations. The dimension of (V) in the sense of algebraic geometry is one less than the dimension of the vector space ''V''. * Projectivization is functorial with respect to injective linear maps: if :: f: V\to W : is a linear map with trivial kernel then ''f'' defines an algebraic map of the corresponding projective spaces, :: \mathbb(f): \mathbb(V)\to \mathbb(W). : I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Association Of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. The MAA was founded in 1915 and is headquartered at 1529 18th Street, Northwest in the Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the ''American Mathematical Monthly'' (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to records on JSTOR. Mission and Vision The mission of the MAA is to advance the understanding of mathematics and its impact on our world. We envision a society that values the power and beauty of mathematics and fully realizes its potential to promote human flourishin ... [...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] |
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 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 university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Spo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Macbeath Regions
In mathematics, a Macbeath region is an explicitly defined region in convex analysis on a bounded convex subset of ''d''-dimensional Euclidean space \R^d. The idea was introduced by and dubbed by G. Ewald, D. G. Larman and C. A. Rogers in 1970. Macbeath regions have been used to solve certain complex problems in the study of the boundaries of convex bodies. Recently they have been used in the study of convex approximations and other aspects of computational geometry. Definition Let ''K'' be a bounded convex set in a Euclidean space. Given a point ''x'' and a scaler λ the λ-scaled the Macbeath region around a point ''x'' is: : (x)=K \cap (2x - K) = x + ((K-x)\cap(x-K)) = \ The scaled Macbeath region at ''x'' is defined as: : M_K^(x)=x + \lambda ((K-x)\cap(x-K)) = \ This can be seen to be the intersection of ''K'' with the reflection of ''K'' around ''x'' scaled by λ. Example uses * Macbeath regions can be used to create \epsilon approximations, with respect to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Mount
David Mount is a professor at the University of Maryland, College Park department of computer science whose research is in computational geometry. Biography Mount received a B.S. in Computer Science at Purdue University in 1977 and received his Ph.D. in Computer Science at Purdue University in 1983 under the advisement of Christoph Hoffmann. He began teaching at the University of Maryland in 1984 and is Professor in the department of Computer Science there. As a teacher, he has won the University of Maryland, College of Computer Mathematical and Physical Sciences Dean's Award for Excellence in Teaching in 2005 and 1997 as well as other teaching awards including the Hong Kong Science and Technology, School of Engineering Award for Teaching Excellence Appreciation in 2001. Research Mounts's main area of research is computational geometry, which is the branch of algorithms devoted to solving problems of a geometric nature. This field includes problems from classic geometry, l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Operators
In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: :f(t) = \int_^ (Tf)(u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Contraction Principle
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922. Statement ''Definition.'' Let (X, d) be a complete metric space. Then a map T : X \to X is called a contraction mapping on ''X'' if there exists q \in fixed-point x^* in ''X'' (i.e. T(x^*) = x^*). Furthermore, x^* can be found as follows: start with an arbitrary element x_0 \in X and define a sequence (x_n)_ by x_n = T(x_) for n \geq 1. Then \lim_ x_n = x^*. ''Remark 1.'' The following inequalities are equivalent and describe the speed of convergence: : \begin d(x^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician George David Birkhoff, Garrett was born in Princeton, New Jersey. He began the Harvard University BA course in 1928 after less than seven years of prior formal education. Upon completing his Harvard BA in 1932, he went to Cambridge University to study mathematical physics but switched to studying abstract algebra under Philip Hall. While visiting the University of Munich, he met Carathéodory who pointed him towards two important texts, Van der Waerden on abstract algebra and Speiser on group theory. Birkhoff held no Ph.D., a qualification British higher education did not emphasize at that time, and did not even bother obtaining an M.A. Nevertheless, after being a member of Harvard's Society of Fellows, 1933–36, he spent the rest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of Euclid's Elements. The names used for modern classification a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
