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Zero-sum Problem
In number theory, zero-sum problems are certain kinds of combinatorial problems about the structure of a finite abelian group. Concretely, given a finite abelian group ''G'' and a positive integer ''n'', one asks for the smallest value of ''k'' such that every sequence of elements of ''G'' of size ''k'' contains ''n'' terms that sum to 0. The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv. They proved that for the group \mathbb/n\mathbb of integers modulo ''n'', k = 2n - 1. Explicitly this says that any multiset of 2''n'' − 1 integers has a subset of size ''n'' the sum of whose elements is a multiple of ''n'', but that the same is not true of multisets of size 2''n'' − 2. (Indeed, the lower bound is easy to see: the multiset containing ''n'' − 1 copies of 0 and ''n'' − 1 copies of 1 contains no ''n''-subset summing to a multiple of ''n''.) This result is known as the Erdős–Ginzburg–Ziv theorem after its discover ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighted EGZ Theorem
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average. Weight functions occur frequently in statistics and analysis, and are closely related to the concept of a measure. Weight functions can be employed in both discrete and continuous settings. They can be used to construct systems of calculus called "weighted calculus" and "meta-calculus".Jane Grossma''Meta-Calculus: Differential and Integral'' , 1981. Discrete weights General definition In the discrete setting, a weight function w \colon A \to \R^+ is a positive function defined on a discrete set A, which is typically finite or countable. The weight function w(a) := 1 corresponds to the ''unweighted'' situation in which all elements have equal weight. One can then apply this weight to various conce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Arie Bialostocki
Arie Bialostocki () is an Israeli American mathematician with expertise and contributions in discrete mathematics and finite groups. Education and career Arie received his BSc, MSc, and PhD (1984) degrees from Tel-Aviv University in Israel. His dissertation was done under the supervision of Marcel Herzog. at the Mathematics Genealogy Project After a year of postdoc at University of Calgary, Canada, he took a faculty position at the University of Idaho, became a professor in 1992, and continued to work there until he retired at the end of 2011. At Idaho, Arie maintained correspondence and collaborations with researchers from around the world who would share similar interests in mathematics. His Erdős number is 1. He has supervised seven PhD students and numerous undergraduate students who enjoyed his colorful anecdotes and advice. He organized the Research Experience for Undergraduates (REU) program at the University of Idaho from 1999 to 2003 attracting many promising u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zhi-Wei Sun
Sun Zhiwei (, born October 16, 1965) is a Chinese mathematician, working primarily in number theory, combinatorics, and group theory. He is a professor at Nanjing University. Biography Sun Zhiwei was born in Huai'an, Jiangsu. Sun and his twin brother Sun Zhihong proved a theorem about what are now known as the Wall–Sun–Sun primes. Sun proved Sun's curious identity in 2002. In 2003, he presented a unified approach to three topics of Paul Erdős in combinatorial number theory: covering system In mathematics, a covering system (also called a complete residue system) is a collection :\ of finitely many residue classes : a_i\pmod = \, whose union contains every integer. Examples and definitions The notion of covering system was i ...s, restricted sumsets, and zero-sum problems or EGZ Theorem. With Stephen Redmond, he posed the Redmond–Sun conjecture in 2006. In 2013, he published a paper containing many conjectures on primes, one of which states that for any ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbach (1997, 2 ... [...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 theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero-sum Ramsey Theory
In mathematics, zero-sum Ramsey theory or zero-sum theory is a branch of combinatorics. It deals with problems of the following kind: given a combinatorial structure whose elements are assigned different weights (usually elements from an Abelian group A), one seeks for conditions that guarantee the existence of certain substructure whose weights of its elements sum up to zero (in A). It combines tools from number theory, algebra, linear algebra, graph theory, discrete analysis, and other branches of mathematics. The classic result in this area is the 1961 theorem of Paul Erdős, Abraham Ginzburg, and Abraham Ziv: for any 2m - 1 elements of \mathbb_m, there is a subset of size m that sums to zero. (This bound is tight, as a sequence of m - 1 zeroes and m - 1 ones cannot have any subset of size m summing to zero.) There are known proofs of this result using the Cauchy-Davenport theorem, Fermat's little theorem In number theory, Fermat's little theorem states that if is a prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset Sum Problem
The subset sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S of integers and a target-sum T, and the question is to decide whether any subset of the integers sum to precisely T''.'' The problem is known to be NP-complete. Moreover, some restricted variants of it are NP-complete too, for example: * The variant in which all inputs are positive. * The variant in which inputs may be positive or negative, and T=0. For example, given the set \, the answer is ''yes'' because the subset \ sums to zero. * The variant in which all inputs are positive, and the target sum is exactly half the sum of all inputs, i.e., T = \frac(a_1+\dots+a_n) . This special case of SSP is known as the partition problem. SSP can also be regarded as an optimization problem: find a subset whose sum is at most ''T'', and subject to that, as close as possible to ''T''. It is NP-hard, but there are several algorithms that can solve it reasonably q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Davenport Constant
In mathematics, the Davenport constant is an invariant of a group studied in additive combinatorics, quantifying the size of nonunique factorizations. Given a finite abelian group , is defined as the smallest number such that every sequence of elements of that length contains a non-empty subsequence adding up to 0. In symbols, this is :D(G) = \min\left\. Example * The Davenport constant for the cyclic group G = \mathbb Z/n\mathbb Z is . To see this, note that the sequence of a fixed generator, repeated times, contains no subsequence with sum . Thus . On the other hand, if \_^n is an arbitrary sequence, then two of the sums in the sequence \left\_^n are equal. The difference of these two sums also gives a subsequence with sum . Properties * Consider a finite abelian group , where the are invariant factors. Then ::D(G) \ge M(G) = 1-r+\sum_i. : The lower bound is proved by noting that the sequence " copies of , copies of , etc." contains no subsequence with sum . * fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycentric-sum Problem
Combinatorial number theory deals with number theoretic problems which involve combinatorial ideas in their formulations or solutions. Paul Erdős is the main founder of this branch of number theory. Typical topics include covering system, zero-sum problems, various restricted sumsets, and arithmetic progressions in a set of integers. Algebraic or analytic methods are powerful in this field. In combinatorial number theory, the barycentric-sum problems are questions that can be answered using combinatorial techniques. The context of barycentric-sum problems are the barycentric sequences. Example Let Z_n be the cyclic group of integers modulo ''n''. Let ''S'' be a sequence of elements of Z_n, where the repetition of elements is allowed. Let , S, be the length of ''S''. A sequence S \subseteq Z_n with , S, \geq 2 is barycentric or has a barycentric-sum if it contains one element a_j such that \sum\limits_ a_i=, S, a_j. Informally, if S contains one element a_j, which is the ”a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
