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Irving Kaplansky
Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andrews. http://www-history.mcs.st-andrews.ac.uk/Biographies/Kaplansky.html. Biography Kaplansky or "Kap" as his friends and colleagues called him was born in Toronto, Ontario, Canada, to Polish-Jewish immigrants; his father worked as a tailor, and his mother ran a grocery and, eventually, a chain of bakeries. He went to Harbord Collegiate Institute receiving the Prince of Wales Scholarship as a teenager. He attended the University of Toronto as an undergraduate and finished first in his class for three consecutive years. In his senior year, he competed in the first William Lowell Putnam Mathematical Competition, becoming one of the first five recipients of the Putnam Fellowship, which paid for graduate studies at Harvard University. Administe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toronto
Toronto ( ; or ) is the capital city of the Canadian province of Ontario. With a recorded population of 2,794,356 in 2021, it is the most populous city in Canada and the fourth most populous city in North America. The city is the anchor of the Golden Horseshoe, an urban agglomeration of 9,765,188 people (as of 2021) surrounding the western end of Lake Ontario, while the Greater Toronto Area proper had a 2021 population of 6,712,341. Toronto is an international centre of business, finance, arts, sports and culture, and is recognized as one of the most multicultural and cosmopolitan cities in the world. Indigenous peoples have travelled through and inhabited the Toronto area, located on a broad sloping plateau interspersed with rivers, deep ravines, and urban forest, for more than 10,000 years. After the broadly disputed Toronto Purchase, when the Mississauga surrendered the area to the British Crown, the British established the town of York in 1793 and later designat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Samuel Ornstein
Donald Samuel Ornstein (born July 30, 1934, New York) is an American mathematician working in the area of ergodic theory. He received a Ph.D. from the University of Chicago in 1957 under the guidance of Irving Kaplansky. During his career at Stanford University he supervised the Ph. D. thesis of twenty three students, including David H. Bailey, Bob Burton, Doug Lind, Ami Radunskaya, Dan Rudolph, and Jeff Steif. He is most famous for his work on the isomorphism of Bernoulli shifts for which he won the 1974 Bôcher Prize. He has been a member of the National Academy of Sciences since 1981. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Algebras
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of operator algebras are phrased in algebraic terms, while the techniques used are highly analytic.''Theory of Operator Algebras I'' By Masamichi Takesaki, Springer 2012, p vi Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory. Overview Operator algebras can be used to study arbitrary sets of operators with little algebraic relation ''simultaneously''. From this point of view, operator algebras can be regarded as a generalization of spectral theory of a single operator. In general operator algebras are non-commutative rings. An operator alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Theory
In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological algebra, homological properties and Polynomial identity ring, polynomial identities. Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of ''commutative algebra'', a major area of modern mathematics. Because these three fields (algebraic geometry, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaplansky's Theorem On Quadratic Forms
In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Irving Kaplansky Irving Kaplansky (March 22, 1917 – June 25, 2006) was a mathematician, college professor, author, and amateur musician.O'Connor, John J.; Robertson, Edmund F., "Irving Kaplansky", MacTutor History of Mathematics archive, University of St Andr .... Statement of the theorem Kaplansky's theorem states that a prime ''p'' congruent to 1 modulo 16 is representable by both or none of ''x''2 + 32''y''2 and ''x''2 + 64''y''2, whereas a prime ''p'' congruent to 9 modulo 16 is representable by exactly one of these quadratic forms. This is remarkable since the primes represented by each of these forms individually are ''not'' describable by congruence conditions. Proof Kaplansky's proof uses the facts that 2 is a 4th power modulo ''p'' if and only if ''p'' is representable by ''x''2 + 64 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaplansky's Conjecture
The mathematician Irving Kaplansky is notable for proposing numerous conjectures in several branches of mathematics, including a list of ten conjectures on Hopf algebras. They are usually known as Kaplansky's conjectures. Group rings Let be a field, and a torsion-free group. Kaplansky's ''zero divisor conjecture'' states: * The group ring does not contain nontrivial zero divisors, that is, it is a domain. Two related conjectures are known as, respectively, Kaplansky's ''idempotent conjecture'': * does not contain any non-trivial idempotents, i.e., if , then or . and Kaplansky's ''unit conjecture'' (which was originally made by Graham Higman and popularized by Kaplansky): * does not contain any non-trivial units, i.e., if in , then for some in and in . The zero-divisor conjecture implies the idempotent conjecture and is implied by the unit conjecture. As of 2021, the zero divisor and idempotent conjectures are open. The unit conjecture, however, was disproved for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaplansky's Game
Kaplansky's game or Kaplansky's ''n''-in-a-line is an abstract board game in which two players take turns in placing a stone of their color on an infinite lattice board, the winner being the player who first gets ''k'' stones of their own color on a line which does not have any stones of the opposite color on it. It is named after Irving Kaplansky. General results * ''k'' ≤ 3 is a first-player win. * 4 ≤ ''k'' ≤ 7 is believed to be draw, but this remains unproven. * ''k'' ≥ 8 is a draw: Every player can draw via a "pairing strategy" or other "draw strategy" of ''m'',''n'',''k''-game. See also * ''m'',''n'',''k''-game * Hex (board game) * Harary's generalized tictactoe Harary's generalized tic-tac-toe or animal tic-tac-toe is a generalization of the game tic-tac-toe, defining the game as a race to complete a particular polyomino on a square grid of varying size, rather than being limited to "in a row" construction ... References {{Tic-Tac-Toe Abstract strategy game ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaplansky Density Theorem
In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led Gert Pedersen to comment in one of his books that, :''The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.'' Formal statement Let ''K''− denote the Strong operator topology, strong-operator closure of a set ''K'' in ''B(H)'', the set of bounded operators on the Hilbert space ''H'', and let (''K'')1 denote the intersection of ''K'' with the unit ball of ''B(H)''. :Kaplansky density theorem.Theorem 5.3.5; Richard Kadison, ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. . If A is a self-adjoint algebra of operators in B(H), then each element a in the unit ball of the strong-operator closure of A is in the strong-operator closure of the unit ball of A. In other words, (A)_1^ = (A^)_1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold Widom
Harold Widom (September 23, 1932 – January 20, 2021) was an American mathematician best known for his contributions to operator theory and random matrices. He was appointed to the Department of Mathematics at the University of California, Santa Cruz in 1968 and became professor emeritus in 1994. Education and research Widom was born in Newark, New Jersey. He studied at Stuyvesant High School, graduating in 1949, and was a member of the school's math team along with his brother Benjamin Widom (1944, 1948). Widom attended City College of New York until 1951, during which he was one of the winners of the William Lowell Putnam Mathematical Competition (1951). At the University of Chicago he obtained an M.S. (1952) and Ph.D., the latter on a thesis ''Embedding of AW*-algebras'' advised by Irving Kaplansky (1955). He taught mathematics at Cornell University (1955–68) where he started his work on Toeplitz and Wiener-Hopf operators, partly inspired by Mark Kac. Widom was appoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
