|
Karl Menger
Karl Menger (; January 13, 1902 – October 5, 1985) was an Austrian-born American mathematician, the son of the economist Carl Menger. In mathematics, Menger studied the theory of algebra over a field, algebras and the dimension theory of low-regularity (smoothness), regularity ("rough") curves and regions; as well as topology. 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 (mathematician), Hans Hahn and received his PhD from the University of Vienna in 1924. Luitzen Egbertus Jan Brouwer, 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 University, Rice Institute. From 1937 to 1946 he was a professor at the University of Notre Dame. From 1946 to 1971, he w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vienna
Vienna ( ; ; ) is the capital city, capital, List of largest cities in Austria, most populous city, and one of Federal states of Austria, nine federal states of Austria. It is Austria's primate city, with just over two million inhabitants. Its larger metropolitan area has a population of nearly 2.9 million, representing nearly one-third of the country's population. Vienna is the Culture of Austria, cultural, Economy of Austria, economic, and Politics of Austria, political center of the country, the List of cities in the European Union by population within city limits, fifth-largest city by population in the European Union, and the most-populous of the List of cities and towns on the river Danube, cities on the river Danube. The city lies on the eastern edge of the Vienna Woods (''Wienerwald''), the northeasternmost foothills of the Alps, that separate Vienna from the more western parts of Austria, at the transition to the Pannonian Basin. It sits on the Danube, and is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inductive Dimension
In the mathematical field of topology, the inductive dimension of a topological space ''X'' is either of two values, the small inductive dimension ind(''X'') or the large inductive dimension Ind(''X''). These are based on the observation that, in ''n''-dimensional Euclidean space ''R''''n'', (''n'' − 1)-dimensional spheres (that is, the boundaries of ''n''-dimensional balls) have dimension ''n'' − 1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets. The small and large inductive dimensions are two of the three most usual ways of capturing the notion of "dimension" for a topological space, in a way that depends only on the topology (and not, say, on the properties of a metric space). The other is the Lebesgue covering dimension. The term "topological dimension" is ordinarily understood to refer to the Lebesgue covering dimension. For "sufficiently nice" s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chicago
Chicago is the List of municipalities in Illinois, most populous city in the U.S. state of Illinois and in the Midwestern United States. With a population of 2,746,388, as of the 2020 United States census, 2020 census, it is the List of United States cities by population, third-most populous city in the United States after New York City and Los Angeles. As the county seat, seat of Cook County, Illinois, Cook County, the List of the most populous counties in the United States, second-most populous county in the U.S., Chicago is the center of the Chicago metropolitan area, often colloquially called "Chicagoland" and home to 9.6 million residents. Located on the shore of Lake Michigan, Chicago was incorporated as a city in 1837 near a Chicago Portage, portage between the Great Lakes and the Mississippi River, Mississippi River watershed. It grew rapidly in the mid-19th century. In 1871, the Great Chicago Fire destroyed several square miles and left more than 100,000 homeless, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rice University
William Marsh Rice University, commonly referred to as Rice University, is a Private university, private research university in Houston, Houston, Texas, United States. Established in 1912, the university spans 300 acres. Rice University comprises eight undergraduate, graduate and professional schools, including Rice University School of Humanities, School of Humanities, Rice University School of Social Sciences, School of Social Sciences, Jesse H. Jones Graduate School of Business, George R. Brown School of Engineering, Wiess School of Natural Sciences, Susanne M. Glasscock School of Continuing Studies, Rice University School of Architecture, Rice School of Architecture, and Shepherd School of Music. Established as William M. Rice Institute for the Advancement of Literature, Science and Art after the murder of its namesake William Marsh Rice, Rice has been a member of the Association of American Universities since 1985 and is Carnegie Classification of Institutions of Higher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harvard University
Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyman John Harvard (clergyman), John Harvard, it is the oldest institution of higher learning in the United States. Its influence, wealth, and rankings have made it one of the most prestigious universities in the world. Harvard was founded and authorized by the Massachusetts General Court, the governing legislature of Colonial history of the United States, colonial-era Massachusetts Bay Colony. While never formally affiliated with any Religious denomination, denomination, Harvard trained Congregationalism in the United States, Congregational clergy until its curriculum and student body were gradually secularized in the 18th century. By the 19th century, Harvard emerged as the most prominent academic and cultural institution among the Boston B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Amsterdam
The University of Amsterdam (abbreviated as UvA, ) is a public university, public research university located in Amsterdam, Netherlands. Established in 1632 by municipal authorities, it is the fourth-oldest academic institution in the Netherlands still in operation. The UvA is one of two large, publicly funded research universities in the city, the other being the Vrije Universiteit Amsterdam (VU). It is also part 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 List of universities in the Netherlands, largest university in the Netherlands by enrollment. The main campus is located in Amsterdam-Centrum, central Amsterdam, with a few faculties located in adjacent Government of Amsterdam, boroughs. The university is organised into seven faculties: Humanities, Social science, Social and Psychology, Behavioural Sciences, Economics and Business, Science, Law, Medicine, Dentistry. Clo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan "Bertus" Brouwer (27 February 1881 – 2 December 1966) 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 one of the founders 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. In addition to his mathematical work, Brouwer also published the short philosophical tract ''Life, Art, and Mysticism'' (1905). Biography B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Game Theory
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed two-person zero-sum games, in which a participant's gains or losses are exactly balanced by the losses and gains of the other participant. In the 1950s, it was extended to the study of non zero-sum games, and was eventually applied to a wide range of Human behavior, behavioral relations. It is now an umbrella term for the science of rational Decision-making, decision making in humans, animals, and computers. Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum games 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 mathematical economics. His paper was f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''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 (mathematics), set of vertices (also called nodes or points); * ... [...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 (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. 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 ope ... [...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 foun ... [...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 map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set together with operations of multiplication and addition and scalar multiplication by elements of a field (mathematics), 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 where associativity of multiplication is assumed, and non-associative algebras, where associativity is not assumed (but not excluded, either). Given an integer ''n'', the ring (mathematics), ring of real matrix, real square matrix, 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-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
