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Marshall Stone
Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras. Biography Stone was the son of Harlan Fiske Stone, who was the Chief Justice of the United States in 1941–1946. Marshall Stone's family expected him to become a lawyer like his father, but he became enamored of mathematics while he was an undergraduate at Harvard University, where he was a classmate of future judge Henry Friendly. He completed a PhD there in 1926, with a thesis on differential equations that was supervised by George David Birkhoff. Between 1925 and 1937, he taught at Harvard, Yale University, and Columbia University. Stone was promoted to a full professor at Harvard in 1937. During World War II, Stone did classified research as part of the "Office of Naval Operations" and the "Office of the Chief of Staff" of the United States Department of War. In 1946, he became the chairma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New York City
New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on one of the world's largest natural harbors. The city comprises five boroughs, each coextensive with a respective county. The city is the geographical and demographic center of both the Northeast megalopolis and the New York metropolitan area, the largest metropolitan area in the United States by both population and urban area. New York is a global center of finance and commerce, culture, technology, entertainment and media, academics, and scientific output, the arts and fashion, and, as home to the headquarters of the United Nations, international diplomacy. With an estimated population in 2024 of 8,478,072 distributed over , the city is the most densely populated major city in the United States. New York City has more than double the population of Los Angeles, the nation's second-most populous city. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernard Galler
Bernard A. Galler (October 3, 1928 – September 4, 2006) was an American mathematician and computer scientist at the University of Michigan who was involved in the development of large-scale operating systems and computer languages including the MAD programming language and the Michigan Terminal System operating system. Education and career Galler attended the University of Chicago where he earned a BSc in mathematics at the University of Chicago (1947), followed by a MSc from UCLA and a PhD from the University of Chicago (1955), advised by Paul Halmos and Marshall Stone. He joined the mathematics department at the University of Michigan (1955) where he taught the first Computer programming, programming course (1956) using an IBM 704. Galler helped to develop the computer language called the Michigan Algorithm Decoder (1959-) in use at several universities. He formed the Communication Sciences dept (1965), renamed Computer Sciences (CS), which became the Computer and Communicat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josiah Willard Gibbs Lectureship
The Josiah Willard Gibbs Lectureship (also called the Gibbs Lecture) of the American Mathematical Society is an annually awarded mathematical prize, named in honor of Josiah Willard Gibbs. The prize is intended not only for mathematicians, but also for physicists, chemists, biologists, physicians, and other scientists who have made important applications of mathematics. The purpose of the prize is to recognize outstanding achievement in applied mathematics and "to enable the public and the academic community to become aware of the contribution that mathematics is making to present-day thinking and to modern civilization." The prize winner gives a lecture, which is subsequently published in the Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-expert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
National Medal Of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral science, behavioral and social sciences, biology, chemistry, engineering, mathematics and physics. The twelve member presidential Committee on the National Medal of Science is responsible for selecting award recipients and is administered by the National Science Foundation (NSF). It is the highest science award in the United States. History The National Medal of Science was established on August 25, 1959, by an act of the Congress of the United States under . The medal was originally to honor scientists in the fields of the "physical, biological, mathematical, or engineering sciences". The Committee on the National Medal of Science was established on August 23, 1961, by Executive order (United States), executive order 10961 of President John F. Kennedy. O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach–Stone Theorem
In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff space ''X'' from the Banach space structure of the space ''C''(''X'') of continuous real- or complex-valued functions on ''X''. If one is allowed to invoke the algebra structure of ''C''(''X'') this is easy – we can identify ''X'' with the spectrum of ''C''(''X''), the set of algebra homomorphisms into the scalar field, equipped with the weak*-topology inherited from the dual space ''C''(''X'')*. The Banach-Stone theorem avoids reference to multiplicative structure by recovering ''X'' from the extreme points of the unit ball of ''C''(''X'')*. Statement For a compact Hausdorff space ''X'', let ''C''(''X'') denote the Banach space of continuous real- or complex-valued functions on ''X'', equipped with the supremum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Weierstrass Theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval (mathematics), interval can be uniform convergence, uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in #Historical works, 1885 using the Weierstrass transform. Marshall H. Stone considerably generalized the theorem and simplified the proof. His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval , an arbitrary compact space, compact Hausdorff space is considered, and instead of the Algebra over a field, algebra of polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Čech Compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a Universal property, universal map from a topological space ''X'' to a Compact space, compact Hausdorff space ''βX''. The Stone–Čech compactification ''βX'' of a topological space ''X'' is the largest, most general compact Hausdorff space "generated" by ''X'', in the sense that any continuous map from ''X'' to a compact Hausdorff space List of mathematical jargon#factor through, factors through ''βX'' (in a unique way). If ''X'' is a Tychonoff space then the map from ''X'' to its image (mathematics), image in ''βX'' is a homeomorphism, so ''X'' can be thought of as a (Dense (topology), dense) subspace of ''βX''; every other compact Hausdorff space that densely contains ''X'' is a Quotient space (topology), quotient of ''βX''. For general topological spaces ''X'', the map from ''X'' to ''βX'' need not be Injective functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–von Neumann Theorem
In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann. Representation issues of the commutation relations In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces. For a single particle moving on the real line \mathbb, there are two important observables: position and momentum. In the Schrödinger representation quantum description of such a particle, the position operator and momentum operator p are respectively given by \begin[] [x \psi](x_0) &= x_0 \psi(x_0) \\[] [p \psi](x_0) &= - i \hbar \frac(x_0) \end on the domain V of infinitely differentiable functions of compact support on \mathbb. Assume \hbar to be a fixed ''non-zero'' real number—in quantum theory \hbar is the reduced Planck ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone's Representation Theorem For Boolean Algebras
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space. Stone spaces Each Boolean algebra ''B'' has an associated topological space, denoted here ''S''(''B''), called its Stone space. The points in ''S''(''B'') are the ultrafilters on ''B'', or equivalently the homomorphisms from ''B'' to the two-element Boolean algebra. The topology on ''S''(''B'') is generated by a basis consisting of all sets of the form \, where ''b'' is an element of ''B''. These sets are also closed and so are clopen (both closed and open). This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. Fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone's Theorem On One-parameter Unitary Groups
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families :(U_)_ of unitary operators that are strongly continuous, i.e., :\forall t_0 \in \R, \psi \in \mathcal: \qquad \lim_ U_t(\psi) = U_(\psi), and are homomorphisms, i.e., :\forall s,t \in \R : \qquad U_ = U_t U_s. Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem was proved by , and showed that the requirement that (U_t)_ be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable. This is an impressive result, as it allows one to define the derivative of the mapping t \mapsto U_t, which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras. Formal statem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone Space
In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact Hausdorff totally disconnected space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras. Equivalent conditions The following conditions on the topological space X are equivalent: * X is a Stone space; * X is homeomorphic to the projective limit (in the category of topological spaces) of an inverse system of finite discrete spaces; * X is compact and totally separated; * X is compact, T0, and zero-dimensional (in the sense of the small inductive dimension); * X is coherent and Hausdorff. Examples Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space \Z_p of p-adic integers, where p is any prime number. Generalizing these examples, any p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
