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Roth's Theorem On Arithmetic Progressions
Roth's theorem on arithmetic progressions is a result in additive combinatorics concerning the existence of arithmetic progressions in subsets of the natural numbers. It was first proven by Klaus Roth in 1953. Roth's theorem is a special case of Szemerédi's theorem for the case k = 3. Statement A subset ''A'' of the natural numbers is said to have positive upper density if :\limsup_\frac > 0. Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of = \. Let r_3( be the size of the largest subset of /math> which contains no arithmetic progression. Roth's theorem on arithmetic progressions (finitary version): r_3( = o(N). Improving upper and lower bounds on r_3( is still an open research problem. History The first result in this direc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Combinatorics
Additive combinatorics is an area of combinatorics in mathematics. One major area of study in additive combinatorics are ''inverse problems'': given the size of the sumset is small, what can we say about the structures of and ? In the case of the integers, the classical Freiman's theorem provides a partial answer to this question in terms of multi-dimensional arithmetic progressions. Another typical problem is to find a lower bound for in terms of and . This can be viewed as an inverse problem with the given information that is sufficiently small and the structural conclusion is then of the form that either or is the empty set; however, in literature, such problems are sometimes considered to be direct problems as well. Examples of this type include the Erdős–Heilbronn Conjecture (for a restricted sumset) and the Cauchy–Davenport Theorem. The methods used for tackling such questions often come from many different fields of mathematics, including combinatorics, ergod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szemerédi Regularity Lemma
In extremal graph theory, Szemerédi’s regularity lemma states that a graph can be partitioned into a bounded number of parts so that the edges between parts are regular (in the sense defined below). The lemma shows that certain properties of random graphs can be applied to dense graphs like counting the copies of a given subgraph within graphs. Endre Szemerédi proved the lemma over bipartite graphs for his theorem on arithmetic progressions in 1975 and for general graphs in 1978. Variants of the lemma use different notions of regularity and apply to other mathematical objects like hypergraphs. Statement To state Szemerédi's regularity lemma formally, we must formalize what the edge distribution between parts behaving 'almost randomly' really means. By 'almost random', we're referring to a notion called -regularity. To understand what this means, we first state some definitions. In what follows is a graph with vertex set . Definition 1. Let be disjoint subsets of . The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Conlon
David Conlon (born 1982) is an Irish mathematician who is a Professor of Mathematics at the California Institute of Technology. His research interests are in Hungarian-style combinatorics, particularly Ramsey theory, extremal graph theory, combinatorial number theory, and probabilistic methods in combinatorics. He proved the first superpolynomial improvement on the Erdős–Szekeres bound on diagonal Ramsey numbers. He won the European Prize in Combinatorics in 2011 for his work in Ramsey theory and for his progress on Sidorenko's conjecture, and the Whitehead Prize in 2019. Life Conlon represented Ireland in the International Mathematical Olympiad in 1998 and 1999. He was an undergraduate in Trinity College Dublin, where he was List of Scholars of Trinity College Dublin, elected a Scholar in 2001 and graduated in 2003. He earned a PhD from Cambridge University in 2009. In 2019 he moved to California Institute of Technology, having been a fellow of Wadham College, Oxford and Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudorandomness
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Pseudorandom number generators are often used in computer programming, as traditional sources of randomness available to humans (such as rolling dice) rely on physical processes not readily available to computer programs, although developments in hardware random number generator technology have challenged this. Background The generation of random numbers has many uses, such as for random sampling, Monte Carlo methods, board games, or gambling. In physics, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions are radioactive decay and quantum measurement, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green–Tao Theorem
In number theory, the Green–Tao theorem, proven by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k, there exist arithmetic progressions of primes with k terms. The proof is an extension of Szemerédi's theorem. The problem can be traced back to investigations of Lagrange and Waring from around 1770.. Statement Let \pi(N) denote the number of primes less than or equal to N. If A is a subset of the prime numbers such that : \limsup_ \frac>0, then for all positive integers k, the set A contains infinitely many arithmetic progressions of length k. In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions. In their later work on the generalized Hardy–Littlewood conjecture, Green and Tao stated and conditionally proved the asymptotic formula : (\mathfrak_k + o(1))\frac for the number of ''k'' tuples of primes p_1 < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician, Fields medalist, and professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins Chair in the College of Letters and Sciences. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory. Tao was born to Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014, and is a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers, and is widely regarded as one of the greatest living mathematicians. Life and career Family Tao's parents are first generation immigrants from Hong Kong to Australia.'' Wen Wei Po'', Page A4, 24 August ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ben J
New Boyz were an American hip hop duo consisting of rappers Earl "Ben J" Benjamin (born October 13, 1991) and Dominic "Legacy" Thomas (born October 12, 1991) They debuted in the spring of 2009 with their viral hit " You're a Jerk" taken from their 2009 debut studio album '' Skinny Jeanz and a Mic''. The song peaked in the top thirty of the ''Billboard'' Hot 100, and it was the first song to bring the jerkin' style to the national forefront. A second single, " Tie Me Down" featuring Ray J, was also successful and peaked in the top thirty in early 2010. In May 2011, their second and final studio album, ''Too Cool to Care'', was released. It includes the top 40 hits " Backseat", featuring The Cataracs and Dev, and " Better with the Lights Off" featuring Chris Brown. The New Boyz have also been featured on Hot Chelle Rae's song " I Like It Like That", which peaked at No. 28 on the Hot 100. History 2005–2008: Early life and formation Benjamin and Thomas met as freshmen at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * Mathematical Reviews * Zentralblatt MATH * Science Citation Index * ISI Alerting Services * CompuMath Citation Index * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vitaly Bergelson
Vitaly Bergelson (born 1950 in Kiev) is a mathematical researcher and professor at Ohio State University in Columbus, Ohio. His research focuses on ergodic theory and combinatorics. Bergelson received his Ph.D. in 1984 under Hillel Furstenberg at the Hebrew University of Jerusalem. He gave an invited address at the International Congress of Mathematicians in 2006 in Madrid. Among Bergelson's best known results is a polynomial generalization of Szemerédi's theorem. The latter provided a positive solution to the famous Erdős–Turán conjecture from 1936 stating that any set of integers of positive upper density contains arbitrarily long arithmetic progressions. In a 1996 paper Bergelson and Leibman obtained an analogous statement for "polynomial progressions". The Bergelson-Leibman theoremAlexander Soifer, Branko Grünbaum, and Cecil RousseauMathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators. Springer-Verlag, New York, 2008, ; p. 358 and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal D'Analyse Mathématique
The ''Journal d'Analyse Mathématique'' is a triannual peer-reviewed scientific journal published by Springer Science+Business Media on behalf of Magnes Press (Hebrew University of Jerusalem). It was established in 1951 by Binyamin Amirà. The journal covers research in mathematics, especially classical analysis and related areas such as complex function theory, ergodic theory, functional analysis, harmonic analysis, partial differential equations, and quasiconformal mapping. Abstracting and indexing The journal is abstracted and indexed in *MathSciNet *Science Citation Index Expanded *Scopus * ZbMATH Open According to the ''Journal Citation Reports'', the journal has a 2022 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 1.0. References External links *{ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yitzhak Katznelson
Yitzhak Katznelson (; born 1934) is an Israeli mathematician. Katznelson was born in Jerusalem. He received his doctoral degree from the University of Paris in 1956. He is a professor of mathematics at Stanford University. He is the author of ''An Introduction to Harmonic Analysis'', which won the Steele Prize for Mathematical Exposition in 2002. 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, .... retrieved 2013-01-27. References External links [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
