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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] |
Combinatorial Number Theory
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 Gr ... [...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] |
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] |
Paul Erdős
Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered on discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle; ''Time'' magazine called him "The Oddball's Oddba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 introduced by Paul Erdős in the early 1930s. The following are examples of covering systems: # \, # \, # \. A covering system is called ''disjoint'' (or ''exact'') if no two members overlap. A covering system is called ''distinct'' (or ''incongruent'') if all the moduli n_i are different (and bigger than 1). Hough and Nielsen (2019) proved that any distinct covering system has a modulus that is divisible by either 2 or 3. A covering system is called ''irredundant'' (or ''minimal'') if all the residue classes are required to cover the integers. The first two examples are disjoint. The third example is distinct. A system (i.e., an unordered multi-set) :\ of finitely many residue classes is called an m-cover if it covers every integer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Restricted Sumset
In additive number theory and combinatorics, a restricted sumset has the form :S=\, where A_1,\ldots,A_n are finite nonempty subsets of a field ''F'' and P(x_1,\ldots,x_n) is a polynomial over ''F''. If P is a constant non-zero function, for example P(x_1,\ldots,x_n)=1 for any x_1,\ldots,x_n, then S is the usual sumset A_1+\cdots+A_n which is denoted by nA if A_1=\cdots=A_n=A. When :P(x_1,\ldots,x_n) = \prod_ (x_j-x_i), ''S'' is written as A_1\dotplus\cdots\dotplus A_n which is denoted by n^ A if A_1=\cdots=A_n=A. Note that , ''S'', > 0 if and only if there exist a_1\in A_1,\ldots,a_n\in A_n with P(a_1,\ldots,a_n)\not=0. Cauchy–Davenport theorem The Cauchy–Davenport theorem, named after Augustin Louis Cauchy and Harold Davenport, asserts that for any prime ''p'' and nonempty subsets ''A'' and ''B'' of the prime order cyclic group \mathbb/p\mathbb we have the inequalityGeroldinger & Ruzsa (2009) pp.141–142 :, A+B, \ge \min\ where A+B := \, i.e. we're using mod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Progression
An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a_1 and the common difference of successive members is d, then the n-th term of the sequence (a_n) is given by :a_n = a_1 + (n - 1)d. A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series. History According to an anecdote of uncertain reliability, in primary school Carl Friedrich Gauss reinvented the formula \tfrac for summing the integers from 1 through n, for the case n=100, by grouping t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, generated by a single element. That is, it is a set (mathematics), set of Inverse element, invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer Exponentiation, power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a ''Generating set of a group, generator'' of the group. Every infinite cyclic group is isomorphic to the additive group \Z, the integers. Every finite cyclic group of Order (group theory), order n is isomorphic to the additive group of Quotient group, Z/''n''Z, the in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics, Probability And Computing
''Combinatorics, Probability and Computing'' is a peer-reviewed scientific journal in mathematics published by Cambridge University Press. Its editor-in-chief is Béla Bollobás ( DPMMS and University of Memphis). History The journal was established by Bollobás in 1992. Fields Medalist Timothy Gowers calls it "a personal favourite" among combinatorics journals and writes that it "maintains a high standard". Content The journal covers combinatorics, probability theory, and theoretical computer science. Currently, it publishes six issues annually. As with other journals from the same publisher, it follows a hybrid green/gold open access policy, in which authors may either place copies of their papers in an institutional repository after a six-month embargo period, or pay an open access charge to make their papers free to read on the journal's website. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...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] |
