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Crank Of A Partition
In number theory, the crank of an integer partition is a certain number associated with the partition. It was first introduced without a definition by Freeman Dyson, who hypothesised its existence in a 1944 paper. Dyson gave a list of properties this yet-to-be-defined quantity should have. In 1988, George E. Andrews and Frank Garvan discovered a definition for the crank satisfying the properties hypothesized for it by Dyson. Dyson's crank Let ''n'' be a non-negative integer and let ''p''(''n'') denote the number of partitions of ''n'' (''p''(0) is defined to be 1). Srinivasa Ramanujan in a paper published in 1918 stated and proved the following congruences for the partition function ''p''(''n''), since known as Ramanujan congruences. * ''p''(5''n'' + 4) ≡ 0 (mod 5) * ''p''(7''n'' + 5) ≡ 0 (mod 7) * ''p''(11''n'' + 6) ≡ 0 (mod 11) These congruences imply that partitions of numbers of the form 5''n'' + 4 (respectively, of the forms 7''n'' + 5 and 11''n'' + 6 ) can be d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, mathematical formulation of quantum mechanics, condensed matter physics, nuclear physics, and nuclear engineering, engineering. He was professor emeritus in the Institute for Advanced Study in Princeton, New Jersey, Princeton and a member of the board of sponsors of the ''Bulletin of the Atomic Scientists''. Dyson originated several concepts that bear his name, such as Dyson's transform, a fundamental technique in additive number theory, which he developed as part of his proof of Mann's theorem; the Dyson tree, a hypothetical genetic engineering, genetically engineered plant capable of growing in a comet; the Dyson series, a Perturbation theory (quantum mechanics), perturbative series where each term is represented by Feynman diagrams; the Dys ... [...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] |
Integer Partition
In number theory and combinatorics, a partition of a non-negative integer , also called an integer partition, is a way of writing as a summation, sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition (combinatorics), composition.) For example, can be partitioned in five distinct ways: : : : : : The only partition of zero is the empty sum, having no parts. The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . An individual summand in a partition is called a part. The number of partitions of is given by the Partition function (number theory), partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George E
George may refer to: Names * George (given name) * George (surname) People * George (singer), American-Canadian singer George Nozuka, known by the mononym George * George Papagheorghe, also known as Jorge / GEØRGE * George, stage name of Giorgio Moroder * George, son of Andrew I of Hungary Places South Africa * George, South Africa, a city ** George Airport United States * George, Iowa, a city * George, Missouri, a ghost town * George, Washington, a city * George County, Mississippi * George Air Force Base, a former U.S. Air Force base located in California Computing * George (algebraic compiler) also known as 'Laning and Zierler system', an algebraic compiler by Laning and Zierler in 1952 * GEORGE (computer), early computer built by Argonne National Laboratory in 1957 * GEORGE (operating system), a range of operating systems (George 1–4) for the ICT 1900 range of computers in the 1960s * GEORGE (programming language), an autocode system invented by Charles Le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Garvan
Francis G. Garvan (born March 9, 1955) is an Australian-born mathematician who specializes in number theory and combinatorics. He holds the position Professor of Mathematics at the University of Florida. He received his Ph.D. from Pennsylvania State University (January, 1986) with George E. Andrews as his thesis advisor. Garvan's thesis, ''Generalizations of Dyson's rank'', concerned the rank of a partition and formed the groundwork for several of his later papers. Garvan is well-known for his work in the fields of ''q''-series and integer partitions. Most famously, in 1988, Garvan and Andrews discovered a definition of the crank of a partition. The crank of a partition is an elusive combinatorial statistic similar to the rank of a partition which provides a key to the study of Ramanujan congruences in partition theory. It was first described by Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Srinivasa Ramanujan
Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems then considered unsolvable. Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition Function (number Theory)
In number theory, the partition function represents the number of possible partitions of a non-negative integer . For instance, because the integer 4 has the five partitions , , , , and . No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of ends in the digit 4 or 9, the number of partitions of will be divisible by 5. Definition and examples For a positive integer , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Congruences
In mathematics, Ramanujan's congruences are the congruences for the partition function ''p''(''n'') discovered by Srinivasa Ramanujan: : \begin p(5k+4) & \equiv 0 \pmod 5, \\ p(7k+5) & \equiv 0 \pmod 7, \\ p(11k+6) & \equiv 0 \pmod . \end In plain words, the first congruence means that if a number is 4 more than a multiple of 5, i.e. it is in the sequence : 4, 9, 14, 19, 24, 29, . . . then the number of its partitions is a multiple of 5. Later, other congruences of this type were discovered, for numbers and for Tau-functions. Background In his 1919 paper, he proved the first two congruences using the following identities (using q-Pochhammer symbol notation): : \begin & \sum_^\infty p(5k+4)q^k=5\frac, \\ pt& \sum_^\infty p(7k+5)q^k=7\frac+49q\frac. \end He then stated that "It appears there are no equally simple properties for any moduli involving primes other than these". After Ramanujan died in 1920, G. H. Hardy extracted proofs of all three congruences from an unpubli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank Of A Partition
In number theory and combinatorics, the rank of an integer partition is a certain number associated with the partition. In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ... in a paper published in the journal Eureka. It was presented in the context of a study of certain congruence properties of the partition function discovered by the Indian mathematical genius Srinivasa Ramanujan. A different concept, sharing the same name, is used in combinatoric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vulcan (hypothetical Planet)
Vulcan () was a proposed planet that some pre-20th century astronomers thought existed in an orbit between Mercury (planet), Mercury and the Sun. Speculation about, and even purported observations of, intermercurial bodies or planets date back to the beginning of the 17th century. The case for their probable existence was bolstered by the support of the French mathematician Urbain Le Verrier, who had predicted the existence of Neptune using disturbances in the orbit of Uranus. By 1859, he had confirmed unexplained peculiarities in Mercury's orbit and predicted that they had to be the result of the gravitational influence of another unknown nearby planet or series of asteroids. A French amateur astronomer's report that he had observed an object passing in front of the Sun that same year led Le Verrier to announce that the long sought after planet, which he gave the name Vulcan, had been discovered at last. Many searches were conducted for Vulcan over the following decades but, de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bruce C
The English language name Bruce arrived in Scotland with the Normans, from the place name Brix, Manche in Normandy, France, meaning "the willowlands". Initially promulgated via the descendants of king Robert the Bruce (1274−1329), it has been a Scottish surname since medieval times; it is now a common male given name. The variant ''Lebrix'' and ''Le Brix'' are French variations of the surname. Note: A few people are notable in more than one field, and therefore appear in more than one section. Arts and entertainment Film and television * Bruce Altman (born 1955), American actor * Bruce Baillie (1931–2020), American filmmaker * Bruce Bennett (1906–2007), American actor and athlete * Bruce Berman (born 1952), American film producer * Bruce Boa (1930–2004), Canadian actor * Bruce Boxleitner (born 1950), American actor * Bruce Campbell (born 1958), American actor, director, writer, producer and author * Bruce Conner (1933–2008), American artist and filmmaker * Br ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Partitions
In number theory and combinatorics, a partition of a non-negative integer , also called an integer partition, is a way of writing as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition.) For example, can be partitioned in five distinct ways: : : : : : The only partition of zero is the empty sum, having no parts. The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . An individual summand in a partition is called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
