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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 an arithmetic progression 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 \delta N contains an arithmetic progression of length ''k''. Another formulation uses the function ''r''''k''(''N''), the size of ... [...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] |
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] |
Bull
A bull is an intact (i.e., not Castration, castrated) adult male of the species ''Bos taurus'' (cattle). More muscular and aggressive than the females of the same species (i.e. cows proper), bulls have long been an important symbol cattle in religion and mythology, in many religions, including for sacrifices. These animals play a significant role in beef ranching, dairy farming, and a variety of sporting and cultural activities, including bullfighting and bull riding. Due to their temperament, handling of bulls requires precautions. Nomenclature The female counterpart to a bull is a cow, while a male of the species that has been Castration, castrated is a ''steer'', ''Oxen, ox'', or ''bullock'', although in North America, this last term refers to a young bull. Use of these terms varies considerably with area and dialect. Colloquially, people unfamiliar with cattle may also refer to steers and heifers as "cows", and bovines of aggressive or long-horned breeds as "bulls" reg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Israel Journal Of Mathematics
'' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem ( Magnes Press). History Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section F), the journal publishes articles on all areas of mathematics. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.70, and its 2009 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 ... was 0.754. External links * Mathematics journals Academic journals established in 1963 Academic journals of Israel English-language journals Bimonthly journals Hebrew University of Jerusalem {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Robert Alexander Rankin
Robert Alexander Rankin FRSE FRSAMD (27 October 1915 – 27 January 2001) was a Scottish mathematician who worked in analytic number theory. Life Rankin was born in Garlieston in Wigtownshire the son of Rev Oliver Rankin (1885–1954), minister of Sorbie and his wife, Olivia Theresa Shaw. His father took the name Oliver Shaw Rankin on marriage and became Professor of Old Testament Language, Literature and Theology in the University of Edinburgh. Rankin was educated at Fettes College then studied mathematics at Clare College, Cambridge, graduating in 1937. At Cambridge he was particularly influenced by J.E. Littlewood and A.E. Ingham. Rankin was elected a Fellow of Clare College in 1939, but his career was interrupted by the Second World War, during which he worked first for the Ministry of Supply then on rocketry research at Fort Halstead. In 1945 he returned to Cambridge as an assistant lecturer, and then moved to the University of Birmingham in 1951 as Mason professor of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Proceedings Of The National Academy Of Sciences
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2022 impact factor of 9.4. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the past, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open-access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are available ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Felix Behrend
Felix Adalbert Behrend (23 April 1911 – 27 May 1962) was a German mathematician of Jewish descent who escaped Nazi Germany and settled in Australia. His research interests included combinatorics, number theory, and topology. Behrend's theorem and Behrend sequences are named after him. Life Behrend was born on 23 April 1911 in Charlottenburg, a suburb of Berlin. He was one of four children of Dr. Felix W. Behrend, a politically liberal mathematics and physics teacher. Although of Jewish descent, their family was Lutheran. Behrend followed his father in studying both mathematics and physics, both at Humboldt University of Berlin and the University of Hamburg, and completed a doctorate in 1933 at Humboldt University. His dissertation, ''Über numeri abundantes'' abundant_number.html" ;"title="'On abundant number">'On abundant numbers''was supervised by Erhard Schmidt. With Adolf Hitler's rise to power in 1933, Behrend's father lost his job, and Behrend himself moved to Cambrid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
European Mathematical Society
The European Mathematical Society (EMS) is a European organization dedicated to the development of mathematics in Europe. Its members are different mathematical societies in Europe, academic institutions and individual mathematicians. The current president is Jan Philip Solovej, professor at the Department of Mathematics at the University of Copenhagen. Goals The Society seeks to serve all kinds of mathematicians in universities, research institutes and other forms of higher education. Its aims are to #Promote mathematical research, both pure and applied, #Assist and advise on problems of mathematical education, #Concern itself with the broader relations of mathematics to society, #Foster interaction between mathematicians of different countries, #Establish a sense of identity amongst European mathematicians, #Represent the mathematical community in supra-national institutions. The EMS is itself an Affiliate Member of the International Mathematical Union and an Associate Member ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rosetta Stone
The Rosetta Stone is a stele of granodiorite inscribed with three versions of a Rosetta Stone decree, decree issued in 196 BC during the Ptolemaic dynasty of ancient Egypt, Egypt, on behalf of King Ptolemy V Epiphanes. The top and middle texts are in Egyptian language, Ancient Egyptian using Egyptian hieroglyphs, hieroglyphic and Demotic (Egyptian), Demotic scripts, respectively, while the bottom is in Ancient Greek. The decree has only minor differences across the three versions, making the Rosetta Stone key to decipherment of ancient Egyptian scripts, deciphering the Egyptian scripts. The stone was carved during the Hellenistic period and is believed to have originally been displayed within a temple, possibly at Sais, Egypt, Sais. It was probably moved in late antiquity or during the Mamluk Sultanate (Cairo), Mamluk period, and was eventually used as building material in the construction of Fort Julien near the town of Rashid (Rosetta) in the Nile Delta. It was found there in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Gowers Norm
In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure present, or conversely, the amount of randomness. They are used in the study of arithmetic progressions in the group. They are named after Timothy Gowers, who introduced it in his work on Szemerédi's theorem. Definition Let f be a complex-valued function on a finite abelian group G and let J denote complex conjugation. The Gowers d-norm is :\Vert f \Vert_^ = \sum_ \prod_ J^ f\left(\right) \ . Gowers norms are also defined for complex-valued functions ''f'' on a segment = , where ''N'' is a positive integer. In this context, the uniformity norm is given as \Vert f \Vert_ = \Vert \tilde \Vert_/\Vert 1_ \Vert_, where \tilde N is a large integer, 1_ denotes the indicator function of 'N'' and \tilde f(x) is equal to f(x) for x \in /math> and 0 for all other x. This definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
