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Karl Menger
Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebras and the dimension theory of low- regularity ("rough") curves and regions; in graph theory, he is credited with Menger's theorem. Outside of mathematics, Menger has substantial contributions to game theory and social sciences. Biography Karl Menger was a student of Hans Hahn and received his PhD from the University of Vienna in 1924. L. E. J. Brouwer invited Menger in 1925 to teach at the University of Amsterdam. In 1927, he returned to Vienna to accept a professorship there. In 1930 and 1931 he was visiting lecturer at Harvard University and the Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame. From 1946 to 1971, he was a professor at Illinois Institute of Technology (IIT) in Chicago. In 1983, IIT awarded Menger a Doctor of Humane Letters and Sciences degree. C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vienna
en, Viennese , iso_code = AT-9 , registration_plate = W , postal_code_type = Postal code , postal_code = , timezone = CET , utc_offset = +1 , timezone_DST = CEST , utc_offset_DST = +2 , blank_name = Vehicle registration , blank_info = W , blank1_name = GDP , blank1_info = € 96.5 billion (2020) , blank2_name = GDP per capita , blank2_info = € 50,400 (2020) , blank_name_sec1 = HDI (2019) , blank_info_sec1 = 0.947 · 1st of 9 , blank3_name = Seats in the Federal Council , blank3_info = , blank_name_sec2 = GeoTLD , blank_info_sec2 = .wien , website = , footnotes = , image_blank_emblem = Wien logo.svg , blank_emblem_size = Vienna ( ; german: Wien ; ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Menger's Theorem
In the mathematical discipline of graph theory, Menger's theorem says that in a finite graph, the size of a minimum cut set is equal to the maximum number of disjoint paths that can be found between any pair of vertices. Proved by Karl Menger in 1927, it characterizes the connectivity of a graph. It is generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs. Edge connectivity The edge-connectivity version of Menger's theorem is as follows: :Let ''G'' be a finite undirected graph and ''x'' and ''y'' two distinct vertices. Then the size of the minimum edge cut for ''x'' and ''y'' (the minimum number of edges whose removal disconnects ''x'' and ''y'') is equal to the maximum number of pairwise edge-independent paths from ''x'' to ''y''. :Extended to all pairs: a graph is ''k''-edge-connected (it remains connected after removing fewer than ''k'' edges) if and only if e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rice University
William Marsh Rice University (Rice University) is a Private university, private research university in Houston, Houston, Texas. It is on a 300-acre campus near the Houston Museum District and adjacent to the Texas Medical Center. Rice is ranked among the top universities in the United States. Opened in 1912 as the Rice Institute after the murder of its namesake William Marsh Rice, Rice is a research university with an undergraduate focus. Its emphasis on undergraduate education is demonstrated by its 6:1 student-faculty ratio. The university has a Research I university, very high level of research activity, with $156 million in sponsored research funding in 2019. Rice is noted for its applied science programs in the fields of artificial heart research, structural chemical analysis, signal processing, space science, and nanotechnology. Rice has been a member of the Association of American Universities since 1985 and is Carnegie Classification of Institutions of Higher Education ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher learning in the United States and one of the most prestigious and highly ranked universities in the world. The university is composed of ten academic faculties plus Harvard Radcliffe Institute. The Faculty of Arts and Sciences offers study in a wide range of undergraduate and graduate academic disciplines, and other faculties offer only graduate degrees, including professional degrees. Harvard has three main campuses: the Cambridge campus centered on Harvard Yard; an adjoining campus immediately across Charles River in the Allston neighborhood of Boston; and the medical campus in Boston's Longwood Medical Area. Harvard's endowment is valued at $50.9 billion, making it the wealthiest academic institution in the world. Endowment inco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Amsterdam
The University of Amsterdam (abbreviated as UvA, nl, Universiteit van Amsterdam) is a public research university located in Amsterdam, Netherlands. The UvA is one of two large, publicly funded research universities in the city, the other being the Vrije Universiteit Amsterdam (VU). Established in 1632 by municipal authorities and later renamed for the city of Amsterdam, the University of Amsterdam is the third-oldest university in the Netherlands. It is one of the largest research universities in Europe with 31,186 students, 4,794 staff, 1,340 PhD students and an annual budget of €600 million. It is the largest university in the Netherlands by enrollment. The main campus is located in central Amsterdam, with a few faculties located in adjacent boroughs. The university is organised into seven faculties: Humanities, Social and Behavioural Sciences, Economics and Business, Science, Law, Medicine, Dentistry. The University of Amsterdam has produced six Nobel Laureates and fiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the 20th century, he is known as the founder of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension. Brouwer also became a major figure in the philosophy of intuitionism, a constructivist school of mathematics which argues that math is a cognitive construct rather than a type of objective truth. This position led to the Brouwer–Hilbert controversy, in which Brouwer sparred with his formalist colleague David Hilbert. Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl. Biography Brouwer was born to Dutch Protestant parents. Early in his career, Brou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by those of other participants. In the 21st century, game theory applies to a wide range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regularity (smoothness)
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of differentiability class ''C^k'' if the derivatives f',f'',\dots,f^ exist and are continuous on U. If f is k-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension Theory
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found nec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra Over A Field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer ''n'', the ring of real square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi identity i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Menger
Carl Menger von Wolfensgrün (; ; 28 February 1840 – 26 February 1921) was an Austrian economist and the founder of the Austrian School of economics. Menger contributed to the development of the theories of marginalism and marginal utility, which rejected cost-of-production theory of value, such as developed by the classical economists such as Adam Smith and David Ricardo. As a departure from such, he would go on to call his resultant perspective, the subjective theory of value. Biography Family and education Carl Menger von Wolfensgrün was born in the city of Neu-Sandez in Galicia, Austrian Empire, which is now Nowy Sącz in Poland. He was the son of a wealthy family of minor nobility; his father, Anton Menger, was a lawyer. His mother, Caroline Gerżabek, was the daughter of a wealthy Bohemian merchant. He had two brothers, Anton and Max, both prominent as lawyers. His son, Karl Menger, was a mathematician who taught for many years at Illinois Institute of Technolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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