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John R. Isbell
John Rolfe Isbell (October 27, 1930 – August 6, 2005) was an American mathematician. For many years he was a professor of mathematics at the University at Buffalo (SUNY). Biography Isbell was born in Portland, Oregon, the son of an army officer from Isbell, a town in Franklin County, Alabama... He attended several undergraduate institutions, including the University of Chicago, where professor Saunders Mac Lane was a source of inspiration. He began his graduate studies in mathematics at Chicago, briefly studied at Oklahoma A&M University and the University of Kansas, and eventually completed a Ph.D. in game theory at Princeton University in 1954 under the supervision of Albert W. Tucker. After graduation, Isbell was drafted into the U.S. Army, and stationed at the Aberdeen Proving Ground. In the late 1950s he worked at the Institute for Advanced Study in Princeton, New Jersey, from which he then moved to the University of Washington and Case Western Reserve University. He joi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buffalo News
''The Buffalo News'' is the daily newspaper of the Buffalo–Niagara Falls metropolitan area, located in downtown Buffalo, New York. It was for decades the only paper fully owned by Warren Buffett's Berkshire Hathaway. On January 29, 2020, the paper reported that it was being sold to Lee Enterprises. History ''The Buffalo News'' was founded as a Sunday paper with the name ''The Buffalo Sunday Morning News'' in 1873 by Edward Hubert Butler, Sr.Frequently Asked Questions , www.buffalonews.com On October 11, 1880, it began publishing daily editions as well, and in 1914, it became an inversion of its original existence by publishing Monday to Saturday, with no publication on Sunday. During most of its life, the ''News'' was known as ''The Buffalo Evening News''. A gentleman's agreement between the ''Evenin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton, New Jersey
The Municipality of Princeton is a Borough (New Jersey), borough in Mercer County, New Jersey, United States. It was established on January 1, 2013, through the consolidation of the Borough of Princeton, New Jersey, Borough of Princeton and Princeton Township, New Jersey, Princeton Township, both of which are now defunct. As of the 2020 United States census, the borough's population was 30,681, an increase of 2,109 (+7.4%) from the 2010 United States census, 2010 census combined count of 28,572. In the 2000 United States census, 2000 census, the two communities had a total population of 30,230, with 14,203 residents in the borough and 16,027 in the township. Princeton was founded before the American Revolutionary War. The borough is the home of Princeton University, one of the world's most acclaimed research universities, which bears its name and moved to the community in 1756 from the educational institution's previous location in Newark, New Jersey, Newark. Although its associ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Heyting Algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras. Locales and frames form the foundation of pointless topology, which, instead of building on point-set topology, recasts the ideas of general topology in categorical terms, as statements on frames and locales. Definition Consider a partially ordered set (''P'', ≤) that is a complete lattice. Then ''P'' is a complete Heyting algebra or frame if any of the following equivalent conditions hold: * ''P'' is a Heyting algebra, i.e. the operation (x\land\cdot) has a right adjoint (also called the lower a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sober Space
In mathematics, a sober space is a topological space ''X'' such that every (nonempty) irreducible space, irreducible closed subset of ''X'' is the closure (topology), closure of exactly one point of ''X'': that is, every nonempty irreducible closed subset has a unique generic point. Definitions Sober spaces have a variety of cryptomorphic definitions, which are documented in this section. In each case below, replacing "unique" with "at most one" gives an equivalent formulation of the Kolmogorov space, T0 axiom. Replacing it with "at least one" is equivalent to the property that the T0 quotient space (topology), quotient of the space is sober, which is sometimes referred to as having "enough points" in the literature. With irreducible closed sets A closed set is Hyperconnected space, irreducible if it cannot be written as the union of two proper closed subsets. A space is sober if every nonempty irreducible closed subset is the closure of a unique point. In terms of morphisms o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone Duality
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation theorem for Boolean algebras. These concepts are named in honor of Marshall Stone. Stone-type dualities also provide the foundation for pointless topology and are exploited in theoretical computer science for the study of formal semantics. This article gives pointers to special cases of Stone duality and explains a very general instance thereof in detail. Overview of Stone-type dualities Probably the most general duality that is classically referred to as "Stone duality" is the duality between the category Sob of sober spaces with continuous functions and the category SFrm of spatial frames with appropriate frame homomorphisms. The dual category of SFrm is the category of spatial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality (mathematics)
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the dual of is . In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called ''primal''). Such involutions sometimes have fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Space
In the mathematical field of topology, a uniform space is a topological space, set with additional mathematical structure, structure that is used to define ''uniform property, uniform properties'', such as complete space, completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in mathematical analysis, analysis. In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "''x'' is closer to ''a'' than ''y'' is to ''b''" make sense in uniform spaces. By comparison, in a general topological space, given sets ''A,B'' it is meaningful to say that a point ''x'' is ''arbitrarily close'' to ''A'' (i.e., in the Closure (topology), closure of ''A''), or perhaps that ''A'' is a ''smaller neighborhood'' of ''x'' than ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. In particular, many constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics 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, Torsion (mechanics), 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 List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, 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 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Rainwater
The fictitious mathematician John Rainwater was created as a student prank but has become known as the author of important results in functional analysis. At the University of Washington in 1952, John Rainwater was invented and enrolled in a mathematics course by graduate students who were in possession of a duplicate student-registration form. Later, mathematicians published under the pseudonym of John Rainwater. Papers were published under the name Rainwater mainly in functional analysis, particularly in the geometric theory of Banach spaces and in convex functions. Rainwater's theorem is an important result in summability theory and functional analysis. The University of Washington's seminar in functional analysis is called the Rainwater seminar, and the associated Rainwater notes have influenced Banach-space theory and convex analysis. The concept of a fictional pseudonym used by multiple people creating valuable mathematics is not unique. Most notably, Nicolas Bourbaki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
