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Corners Theorem
In arithmetic combinatorics, the corners theorem states that for every \varepsilon>0, for large enough N, any set of at least \varepsilon N^2 points in the N\times N grid \^2 contains a corner, i.e., a triple of points of the form \ with h\ne 0. It was first proved by Miklós Ajtai and Endre Szemerédi in 1974 using Szemerédi's theorem.. In 2003, József Solymosi gave a short proof using the triangle removal lemma. Statement Define a corner to be a subset of \mathbb^2 of the form \, where x,y,h\in \mathbb and h\ne 0. For every \varepsilon>0, there exists a positive integer N(\varepsilon) such that for any N\ge N(\varepsilon), any subset A\subseteq\^2 with size at least \varepsilon N^2 contains a corner. The condition h\ne 0 can be relaxed to h>0 by showing that if A is dense, then it has some dense subset that is centrally symmetric. Proof overview What follows is a sketch of Solymosi's argument. Suppose A\subset\^2 is corner-free. Construct an auxiliary tripartite graph G w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Combinatorics
In mathematics, arithmetic combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. Scope Arithmetic combinatorics is about combinatorial estimates associated with arithmetic operations (addition, subtraction, multiplication, and division). Additive combinatorics is the special case when only the operations of addition and subtraction are involved. Ben Green explains arithmetic combinatorics in his review of "Additive Combinatorics" by Tao and Vu. Important results Szemerédi's theorem Szemerédi's theorem is a result in arithmetic combinatorics concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured. that every set of integers ''A'' with positive natural density contains a ''k'' term arithmetic progression for every ''k''. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem. Green–Tao theorem and extensions The Gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miklós Ajtai
Miklós Ajtai (born 2 July 1946) is a computer scientist at the IBM Almaden Research Center, United States. In 2003, he received the Knuth Prize for his numerous contributions to the field, including a classic sorting network algorithm (developed jointly with J. Komlós and Endre Szemerédi), exponential lower bounds, superlinear time-space tradeoffs for branching programs, and other "unique and spectacular" results. He is a member of the U.S. National Academy of Sciences. Selected results One of Ajtai's results states that the length of proofs in propositional logic of the pigeonhole principle for ''n'' items grows faster than any polynomial in ''n''. He also proved that the statement "any two countable structures that are second-order equivalent are also isomorphic" is both consistent with and independent of ZFC. Ajtai and Szemerédi proved the corners theorem, an important step toward higher-dimensional generalizations of the Szemerédi theorem. With Komlós and Szemerà ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Endre Szemerédi
Endre Szemerédi (; born August 21, 1940) is a Hungarian-American mathematician and computer scientist, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science at Rutgers University since 1986. He also holds a professor emeritus status at the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences. Szemerédi has won prizes in mathematics and science, including the Abel Prize in 2012. He has made a number of discoveries in combinatorics and computer science, including Szemerédi's theorem, the Szemerédi regularity lemma, the Erdős–Szemerédi theorem, the Hajnal–Szemerédi theorem and the Szemerédi–Trotter theorem. Early life Szemerédi was born in Budapest. Since his parents wished him to become a doctor, Szemerédi enrolled at a college of medicine, but he dropped out after six months (in an interview he explained it: "I was not sure I could do work bearing such r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szemerédi's Theorem
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers ''A'' with positive natural density contains a ''k''-term arithmetic progression for every ''k''. Endre Szemerédi proved the conjecture in 1975. Statement A subset ''A'' of the natural numbers is said to have positive upper density if :\limsup_\frac > 0. Szemerédi's theorem asserts that a subset of the natural numbers with positive upper density contains infinitely many arithmetic progressions of length ''k'' for all positive integers ''k''. An often-used equivalent finitary version of the theorem states that for every positive integer ''k'' and real number \delta \in (0, 1], there exists a positive integer :N = N(k,\delta) such that every subset of of size at least δ''N'' contains an arithmetic progression of length ''k''. Another formulation uses the function ''r''''k''(''N''), the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
József Solymosi
József Solymosi is a Hungarian-Canadian mathematician and a professor of mathematics at the University of British Columbia. His main research interests are arithmetic combinatorics, discrete geometry, graph theory, and combinatorial number theory. Education and career Solymosi earned his master's degree in 1999 under the supervision of László Székely from the Eötvös Loránd University and his Ph.D. in 2001 at ETH Zürich under the supervision of Emo Welzl. His doctoral dissertation was ''Ramsey-Type Results on Planar Geometric Objects''. From 2001 to 2003 he was S. E. Warschawski Assistant Professor of Mathematics at the University of California, San Diego. He joined the faculty of the University of British Columbia in 2002. He was editor in chief of the ''Electronic Journal of Combinatorics'' from 2013 to 2015. Contributions Solymosi was the first online contributor to the first Polymath Project, set by Timothy Gowers to find improvements to the Hales–Jewett theorem. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Removal Lemma
In graph theory, the graph removal lemma states that when a graph contains few copies of a given Subgraph (graph theory), subgraph, then all of the copies can be eliminated by removing a small number of edges. The special case in which the subgraph is a triangle is known as the triangle removal lemma. The graph removal lemma can be used to prove Roth's theorem on 3-term arithmetic progressions, and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem. It also has applications to property testing. Formulation Let H be a graph with h vertices. The graph removal lemma states that for any \epsilon > 0, there exists a constant \delta = \delta(\epsilon, H) > 0 such that for any n-vertex graph G with fewer than \delta n^h Subgraph isomorphism problem, subgraphs isomorphic to H, it is possible to eliminate all copies of H by removing at most \epsilon n^2 edges from G. An alternative way to state this is to say that for any n-vertex graph G with o( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Analysis
''Discrete Analysis'' is a mathematics journal covering the applications of analysis to discrete structures. ''Discrete Analysis'' is an arXiv overlay journal, meaning the journal's content is hosted on the arXiv. History ''Discrete Analysis'' was created by Timothy Gowers to demonstrate that a high-quality mathematics journal could be inexpensively produced outside of the traditional academic publishing industry. The journal is open access, and submissions are free for authors. The journal's 2018 MCQ is 1.21.''Discrete Analysis'', MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ..., 2019. Accessed 2019-09-02. References * * External links *{{Official, https://discreteanalysisjournal.com/ Open access journals Mathematics journals Publications established in 2016 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypergraph Removal Lemma
In graph theory, the hypergraph removal lemma states that when a hypergraph contains few copies of a given sub-hypergraph, then all of the copies can be eliminated by removing a small number of hyperedges. It is a generalization of the graph removal lemma. The special case in which the graph is a tetrahedron is known as the tetrahedron removal lemma. It was first proved by Nagle, Rödl, Schacht and Skokan and, independently, by Gowers. The hypergraph removal lemma can be used to prove results such as Szemerédi's theorem and the multi-dimensional Szemerédi theorem. Statement The hypergraph removal lemma states that for any \varepsilon, r, m > 0, there exists \delta = \delta(\varepsilon, r, m) > 0 such that for any r-uniform hypergraph H with m vertices the following is true: if G is any n-vertex r-uniform hypergraph with at most \delta n^ subgraphs isomorphic to H, then it is possible to eliminate all copies of H from G by removing at most \varepsilon n^r hyperedges from G. An e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roth's Theorem
In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a century, the meaning of ''very good'' here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of , , , and . Statement Roth's theorem states that every irrational algebraic number \alpha has approximation exponent equal to 2. This means that, for every \varepsilon>0, the inequality :\left, \alpha - \frac\ \frac with C(\alpha,\varepsilon) a positive number depending only on \varepsilon>0 and \alpha. Discussion The first result in this direction is Liouville's theorem on approximation of algebraic numbers, which gives an approximation exponent of ''d'' for an algebraic number α of degree ''d'' ≥ 2. This is already enough to demonstrate the existence of transcen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1974 Introductions
Major events in 1974 include the aftermath of the 1973 oil crisis and the resignation of President of the United States, United States President Richard Nixon following the Watergate scandal. In the Middle East, the aftermath of the 1973 Yom Kippur War determined politics; following List of Prime Ministers of Israel, Israeli Prime Minister Golda Meir's resignation in response to high Israeli casualties, she was succeeded by Yitzhak Rabin. In Europe, the Turkish invasion of Cyprus, invasion and occupation of northern Cyprus by Turkey, Turkish troops initiated the Cyprus dispute, the Carnation Revolution took place in Portugal, and Chancellor of Germany, Chancellor of West Germany Willy Brandt resigned following an Guillaume affair, espionage scandal surrounding his secretary Günter Guillaume. In sports, the year was primarily dominated by the 1974 FIFA World Cup, FIFA World Cup in West Germany, in which the Germany national football team, German national team won the championshi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
