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Dyson's Transform
Dyson's transform is a fundamental technique in additive number theory.Additive Number Theory: Inverse Problems and the Geometry of Sumsets By Melvyn Bernard Nathanson, Springer, Aug 22, 1996, , https://books.google.com/books?id=PqlQjNhjkKUC&dq=%22e-transform%22&source=gbs_navlinks_s, p. 42 It was developed by Freeman Dyson as part of his proof of Mann's theorem, is used to prove such fundamental results of Additive Number Theory as the Cauchy-Davenport theorem, and was used by Olivier Ramaré in his work on the Goldbach conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold ... that proved that every even integer is the sum of at most 6 primes. The term ''Dyson's transform'' for this technique is used by Ramaré. Halberstam and Roth call it the τ-transformation. This formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freeman Dyson
Freeman John Dyson (15 December 1923 – 28 February 2020) was an English-American theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrices, mathematical formulation of quantum mechanics, condensed matter physics, nuclear physics, and engineering. He was Professor Emeritus in the Institute for Advanced Study in 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 genetically engineered plant capable of growing in a comet; the Dyson series, a perturbative series where each term is represented by Feynman diagrams; the Dyson sphere, a thought experiment that attempts to explain how a spacefaring, space-faring civilization would meet its energy requirements with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Two principal objects of study are the sumset of two subsets ''A'' and ''B'' of elements from an abelian group ''G'', :A + B = \, and the h-fold sumset of ''A'', :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of ''hA'' from the structure of ''A'': for example, determining which elements can be represented as a sum from ''hA'', where ''A'' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mann's Theorem
In additive number theory, the Schnirelmann density of a sequence of numbers is a way to measure how "dense" the sequence is. It is named after Russian mathematician Lev Schnirelmann, who was the first to study it.Schnirelmann, L.G. (1930).On the additive properties of numbers, first published in "Proceedings of the Don Polytechnic Institute in Novocherkassk" (in Russian), vol XIV (1930), pp. 3-27, and reprinted in "Uspekhi Matematicheskikh Nauk" (in Russian), 1939, no. 6, 9–25.Schnirelmann, L.G. (1933). First published asÜber additive Eigenschaften von Zahlen in "Mathematische Annalen" (in German), vol 107 (1933), 649-690, and reprinted asOn the additive properties of numbers in "Uspekhin. Matematicheskikh Nauk" (in Russian), 1940, no. 7, 7–46. Definition The Schnirelmann density of a set of natural numbers ''A'' is defined as :\sigma A = \inf_n \frac, where ''A''(''n'') denotes the number of elements of ''A'' not exceeding ''n'' and inf is infimum.Nathanson (1996) pp.191–19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy-Davenport Theorem
In additive number theory and combinatorics, a restricted sumset has the form :S=\, where A_1,\ldots,A_n are finite nonempty subsets of a field ''F'' and P(x_1,\ldots,x_n) is a polynomial over ''F''. If P is a constant non-zero function, for example P(x_1,\ldots,x_n)=1 for any x_1,\ldots,x_n, then S is the usual sumset A_1+\cdots+A_n which is denoted by nA if A_1=\cdots=A_n=A. When :P(x_1,\ldots,x_n) = \prod_ (x_j-x_i), ''S'' is written as A_1\dotplus\cdots\dotplus A_n which is denoted by n^ A if A_1=\cdots=A_n=A. Note that , ''S'', > 0 if and only if there exist a_1\in A_1,\ldots,a_n\in A_n with P(a_1,\ldots,a_n)\not=0. Cauchy–Davenport theorem The Cauchy–Davenport theorem, named after Augustin Louis Cauchy and Harold Davenport, asserts that for any prime ''p'' and nonempty subsets ''A'' and ''B'' of the prime order cyclic group \mathbb/p\mathbb we have the inequalityGeroldinger & Ruzsa (2009) pp.141–142 :, A+B, \ge \min\ where A+B := \, i.e. we're usin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olivier Ramaré
Olivier Ramaré is a French mathematician who works as Senior researcher for the CNRS. He is currently attached to Aix-Marseille Université. Ramaré earned a doctorate in 1991 from the University of Bordeaux with a dissertation ''Contribution au problème de Goldbach : tout entier >1 est somme d'au plus treize nombres premiers'' supervised by Jean-Marc Deshouillers. In 1995, he sharpened earlier work on Schnirelmann's theorem by proving that every even number is a sum of at most six primes. This result may be compared with Goldbach's conjecture, which states that every even number except 2 is the sum of two primes. The truth of Ramaré's result for all sufficiently large even numbers is a consequence of Vinogradov's theorem, whereas the full result follows from Goldbach's weak conjecture. In turn, Ramaré's result was strengthened by Terence Tao who proved in 2014 that every odd number is the sum of at most five primes, and by Harald Helfgott's claimed proof of Goldbach's wea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goldbach Conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than 4 × 1018, but remains unproven despite considerable effort. History On 7 June 1742, the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII), in which he proposed the following conjecture: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would indeed be a sum of primes. He then proposed a second conjecture in the margin of his letter, which implies the first: Euler replied in a letter dated 30 June 1742 and reminded Goldbach of an earlier conversation they had had (), in which Goldbach had remarked that the first of those two conjectures would follow from the statement This is in fact equivalent to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sumset
In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets A and B of an abelian group G (written additively) is defined to be the set of all sums of an element from A with an element from B. That is, :A + B = \. The n-fold iterated sumset of A is :nA = A + \cdots + A, where there are n summands. Many of the questions and results of additive combinatorics and additive number theory can be phrased in terms of sumsets. For example, Lagrange's four-square theorem can be written succinctly in the form :4\Box = \mathbb, where \Box is the set of square numbers. A subject that has received a fair amount of study is that of sets with ''small doubling'', where the size of the set A+A is small (compared to the size of A); see for example Freiman's theorem. See also *Restricted sumset * Sidon set *Sum-free set *Schnirelmann density *Shapley–Folkman lemma *X + Y sorting References * * * *Terence Tao and Van Vu, ''Additive Combinatorics'', Cambridge Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
