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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 with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 Let H be r-uniform (every edge connects exactly r vertices) hypergraph with h vertices. The hypergraph removal lemma states that for any \varepsilon >0 exists \delta = \delta(r, m, \varepsilon) > 0 such that for any r-uniform, n-vertices hypergraph G with fewer than \delta n^ subhypergraphs isomorphic to H it is possible to remove all copies of H by removing at most \varepsilon n^ edges. An equivalent fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Ramsey Theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?" Examples A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity. For example, consider a complete graph of order ''n''; that is, there are ''n'' vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour each edge either red or blue. How large must ''n'' be in order to ensure that there is either a blue triangle or a re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
2003 In Mathematics
3 (three) is a number, numeral and digit. It is the natural number following 2 and preceding 4, and is the smallest odd prime number and the only prime preceding a square number. It has religious and cultural significance in many societies. Evolution of the Arabic digit The use of three lines to denote the number 3 occurred in many writing systems, including some (like Roman and Chinese numerals) that are still in use. That was also the original representation of 3 in the Brahmic (Indian) numerical notation, its earliest forms aligned vertically. However, during the Gupta Empire the sign was modified by the addition of a curve on each line. The Nāgarī script rotated the lines clockwise, so they appeared horizontally, and ended each line with a short downward stroke on the right. In cursive script, the three strokes were eventually connected to form a glyph resembling a with an additional stroke at the bottom: ३. The Indian digits spread to the Caliphate in the 9th ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
1974 In Science
The year 1974 in science and technology involved some significant events, listed below. Astronomy and space exploration * February 8 – After 84 days in space, the last crew of the temporary American space station Skylab return to Earth. * February 13–15 – Sagittarius A*, thought to be the location of a supermassive black hole, is identified by Bruce Balick and Robert Brown using the baseline interferometer of the United States National Radio Astronomy Observatory. * November 16 – Arecibo message transmitted from Arecibo Observatory (Puerto Rico) to Messier 13. * Hawking radiation is predicted by Stephen Hawking. Computer Science * The Mark-8 microcomputer based on the Intel 8008 microprocessor is designed by Jonathan Titus. It is announced on the cover of the July 1974 issue of Radio-Electronics as "Your Personal Minicomputer". History of science * F. W. Winterbotham publishes ''The Ultra secret: the inside story of Operation Ultra, Bletchley Park and Enigma'', the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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, the Greek junta's collapse paves the way for the establishment of a Metapolitefsi, parliamentary republic 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
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''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 extension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 Mathematical analysis journals Academic journals est ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
