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Simon B. Kochen
Simon Bernhard Kochen (; born 14 August 1934) is a Canadian mathematician, working in the fields of model theory, number theory and quantum mechanics. Education and career Kochen was born in Antwerp, Belgium, and escaped the Nazis with his family, thanks to a courageous Norwegian ship captain. Raised in England, he attended grammar school before moving to Canada. Kochen attended McGill University and obtained his bachelor’s and master’s degrees there. He moved to the US afterwards and received his Ph.D. (''Ultrafiltered Products and Arithmetical Extensions'') from Princeton University in 1958 under the direction of Alonzo Church. Since 1967 he has been a member of Princeton's Department of Mathematics. He chaired the department from 1989 to 1992 and became the Henry Burchard Fine Professor in mathematics in 1994. During 1966–1967 and 1978–1979, Kochen was at the Institute for Advanced Study. In 1967 he was awarded, together with James Ax, the seventh Cole Prize, Frank Nels ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antwerp
Antwerp (; ; ) is a City status in Belgium, city and a Municipalities of Belgium, municipality in the Flemish Region of Belgium. It is the capital and largest city of Antwerp Province, and the third-largest city in Belgium by area at , after Tournai and Couvin. With a population of 565,039, it is the List of most populous municipalities in Belgium, most populous municipality in Belgium, and with a metropolitan population of over 1.2 million people, the country's Metropolitan areas in Belgium, second-largest metropolitan area after Brussels. Definitions of metropolitan areas in Belgium. Flowing through Antwerp is the river Scheldt. Antwerp is linked to the North Sea by the river's Western Scheldt, Westerschelde estuary. It is about north of Brussels, and about south of the Netherlands, Dutch border. The Port of Antwerp is one of the biggest in the world, ranking second in Europe after Rotterdam and List of world's busiest container ports, within the top 20 globally. The city ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Particles
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a consequence of Flavour (particle physics), flavor and Quantum chromodynamics, color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively. Among the 61 elementary particles embraced by the Standard Model number: electrons and other leptons, quarks, and the fundamental bosons. Subatomic particles such as protons or neutrons, which Quark, contain two or more elementary particles, are known as composite particles. Ordinary matter is composed of atoms, themselves once thought to be indivisible elementary particles. The name ''atom'' comes from the Ancient Greek word ''ἄτομος'' (wikt:átomo#:~:text=Learned borrowing from Latin atomus,, "to cut")., atomos) which means ''indivisible'' or ''u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Will
Free will is generally understood as the capacity or ability of people to (a) choice, choose between different possible courses of Action (philosophy), action, (b) exercise control over their actions in a way that is necessary for moral responsibility, or (c) be the ultimate source or originator of their actions. There are different theories as to its nature, and these aspects are often emphasized differently depending on philosophical tradition, with debates focusing on whether and how such freedom can coexist with determinism, divine foreknowledge, and other constraints. Free will is closely linked to the concepts of moral responsibility, praise, culpability, and other judgements which apply only to actions that are freely chosen. It is also connected with the concepts of Advice (opinion), advice, persuasion, deliberation, and Prohibitionism, prohibition. Traditionally, only actions that are freely Will (philosophy), willed are seen as deserving credit or blame. Whether free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Will Theorem
The free will theorem of John H. Conway and Simon B. Kochen states that if we have a free will in the sense that our choices are not a function of the past, then, under specific assumptions drawn from quantum mechanics and relativity, so must some elementary particles. That is, if human experimenters possess a form of free will—defined as the ability to make choices not entirely determined by prior events—then certain elementary particles must also exhibit a corresponding form of indeterminacy. The theorem argues that stochastic processes do not satisfy this definition of "freedom," because random values can, in principle, be pre-determined or embedded in the past (for example, sampled from a pre-existing table). Consequently, the theorem implies that no physical theory relying solely on a combination of deterministic laws and pre-existing randomness can fully account for the observed outcomes of quantum measurements. Conway and Kochen's paper was published in ''Foundations o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Horton Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches of recreational mathematics, most notably the invention of the cellular automaton called the Game of Life. Born and raised in Liverpool, Conway spent the first half of his career at the University of Cambridge before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the rest of his career. On 11 April 2020, at age 82, he died of complications from COVID-19. Early life and education Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. By the time he was 11, his ambition was to become a mathematician. After leaving sixth form, he studied mathematics at Gonville and Caius Coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Contextuality
Quantum contextuality is a feature of the phenomenology of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so in a realistic hidden-variable theory leads to values that are dependent upon the choice of the other (compatible) observables which are simultaneously measured (the measurement context). More formally, the measurement result (assumed pre-existing) of a quantum observable is dependent upon which other commuting observables are within the same measurement set. Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem. The study of contextuality has developed into a major topic of interest in quantum foundations as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Specker
Ernst Paul Specker (11 February 1920, Zürich – 10 December 2011, Zürich) was a Swiss mathematician. Much of his most influential work was on Quine's New Foundations, a set theory with a universal set, but he is most famous for the Kochen–Specker theorem in quantum mechanics, showing that certain types of hidden-variable theories are impossible. He also proved the ordinal partition relation thereby solving a problem of Erdős. Specker received his Ph.D. in 1949 from ETH Zurich ETH Zurich (; ) is a public university in Zurich, Switzerland. Founded in 1854 with the stated mission to educate engineers and scientists, the university focuses primarily on science, technology, engineering, and mathematics. ETH Zurich ran ...,. where he remained throughout his professional career. See also * Specker sequence * Baer–Specker group References External links Biography at the University of St. Andrews Ernst Specker (1920-2011) Martin Fürer, January 25, 2012. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Numbers
In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right. For example, comparing the expansion of the rational number \tfrac15 in base vs. the -adic expansion, \begin \tfrac15 &= 0.01210121\ldots \ (\text 3) &&= 0\cdot 3^0 + 0\cdot 3^ + 1\cdot 3^ + 2\cdot 3^ + \cdots \\ mu\tfrac15 &= \dots 121012102 \ \ (\text) &&= \cdots + 2\cdot 3^3 + 1 \cdot 3^2 + 0\cdot3^1 + 2 \cdot 3^0. \end Formally, given a prime number , a -adic number can be defined as a series s=\sum_^\infty a_i p^i = a_k p^k + a_ p^ + a_ p^ + \cdots where is an integer (possibly negative), and each a_i is an integer such that 0\le a_i < p. A -adic integer is a -adic number such that < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Equation
''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated ... * Diophantine equation * Diophantine quintuple * Diophantine set {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Ax
James Burton Ax (10 January 1937 – 11 June 2006) was an American mathematician who made groundbreaking contributions in algebra and number theory using model theory. He shared, with Simon B. Kochen, the seventh Frank Nelson Cole Prize in Number Theory, which was awarded for a series of three joint papers on Diophantine problems. Education and career Ax was born in New York City and graduated from Stuyvesant High School in 1954. He then joined the Brooklyn Polytechnic University. He earned his Ph.D. from the University of California, Berkeley in 1961 under the direction of Gerhard Hochschild, with a dissertation on ''The Intersection of Norm Groups''. After a year at Stanford University, he joined the mathematics faculty at Cornell University. He spent the academic year 1965–1966 at Harvard University on a Guggenheim Fellowship. In 1969, he was recruited by his Berkeley classmate Jim Simons to move from Cornell to the mathematics department at Stony Brook Unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
