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Kemnitz's Conjecture
In additive number theory, Kemnitz's conjecture states that every set of lattice points in the plane has a large subset whose centroid is also a lattice point. It was proved independently in the autumn of 2003 by Christian Reiher Christian Reiher (born 19 April 1984 in Starnberg) is a German mathematician. He is the fifth most successful participant in the history of the International Mathematical Olympiad, having won four gold medals in the years 2000 to 2003 and a br ..., then an undergraduate student, and Carlos di Fiore, then a high school student. The exact formulation of this conjecture is as follows: :Let n be a natural number and S a set of 4n-3 lattice points in plane. Then there exists a subset S_1 \subseteq S with n points such that the centroid of all points from S_1 is also a lattice point. Kemnitz's conjecture was formulated in 1983 by Arnfried Kemnitz as a generalization of the Erdős–Ginzburg–Ziv theorem, an analogous one-dimensional result stating that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Number Theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigroups with an operation of addition. Additive number theory has close ties to combinatorial number theory and the geometry of numbers. Two principal objects of study are the sumset of two subsets ''A'' and ''B'' of elements from an abelian group ''G'', :A + B = \, and the h-fold sumset of ''A'', :hA = \underset\,. Additive number theory The field is principally devoted to consideration of ''direct problems'' over (typically) the integers, that is, determining the structure of ''hA'' from the structure of ''A'': for example, determining which elements can be represented as a sum from ''hA'', where ''A'' is a fixed subset.Nathanson (1996) II:1 Two classical problems of this type are the Goldbach conjecture (which is the conjecture that 2'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Point
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and every element of ''A'' is also an element of ''B'', then: :*''A'' is a subset of ''B'', denoted by A \subseteq B, or equivalently, :* ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centroid
In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any object in ''n''-dimensional Euclidean space. In geometry, one often assumes uniform mass density, in which case the '' barycenter'' or '' center of mass'' coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin. In physics, if variations in gravity are considered, then a '' center of gravity'' can be defined as the weighted mean of all points weighted by their specific weight. In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center. History The term "centroid" is of recent coinage (1814). It is used as a substitute for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christian Reiher
Christian Reiher (born 19 April 1984 in Starnberg) is a German mathematician. He is the fifth most successful participant in the history of the International Mathematical Olympiad, having won four gold medals in the years 2000 to 2003 and a bronze medal in 1999. Just after finishing his ''Abitur'', he proved Kemnitz's conjecture, an important problem in the theory of zero-sums. He went on to earn his Diplom in mathematics from the Ludwig Maximilian University of Munich. Reiher received his Dr. rer. nat. ''Doctor rerum naturalium'' ( for, , Latin, doctor of natural sciences, lit. 'doctor of the things of nature'), abbreviated Dr. rer. nat., is a doctoral academic degree awarded by universities in some European countries (e.g. Germany, Austria and C ... from the University of Rostock under supervision of in February 2010 (Thesis: ''A proof of the theorem according to which every prime number possesses property B'') and works now at the University of Hamburg. Selected publ ... [...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 disco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chevalley–Warning Theorem
In number theory, the Chevalley–Warning theorem implies that certain polynomial equations in sufficiently many variables over a finite field have solutions. It was proved by and a slightly weaker form of the theorem, known as Chevalley's theorem, was proved by . Chevalley's theorem implied Artin's and Dickson's conjecture that finite fields are quasi-algebraically closed fields . Statement of the theorems Let \mathbb be a finite field and \_^r\subseteq\mathbb _1,\ldots,X_n/math> be a set of polynomials such that the number of variables satisfies :n>\sum_^r d_j where d_j is the total degree of f_j. The theorems are statements about the solutions of the following system of polynomial equations :f_j(x_1,\dots,x_n)=0\quad\text\, j=1,\ldots, r. * The ''Chevalley–Warning theorem'' states that the number of common solutions (a_1,\dots,a_n) \in \mathbb^n is divisible by the characteristic p of \mathbb. Or in other words, the cardinality of the vanishing set of \_^r is 0 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ars Combinatoria (journal)
''Ars Combinatoria, a Canadian Journal of Combinatorics'' is an English language research journal in combinatorics, published by the Charles Babbage Research Centre, Winnipeg, Manitoba, Canada. From 1976 to 1988 it published two volumes per year, and subsequently it published as many as six volumes per year. The journal is indexed in ''MathSciNet'' and '' Zentralblatt''. As of 2019, SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestige of scholarly journals that accounts for both the number of citations received by a journal and the prestige of the journals where the citations come from. Rationale Cita ... listed it in the bottom quartile of miscellaneous mathematics journals. As of December 15, 2021, the editorial board of the journal resigned, asking that inquiries be directed to the publisher. References 1976 establishments in Canada Publications established in 1976 Academic journals published in Canada English-language j ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorica
''Combinatorica'' is an international journal of mathematics, publishing papers in the fields of combinatorics and computer science. It started in 1981, with László Babai and László Lovász as the editors-in-chief with Paul Erdős as honorary editor-in-chief. The current editors-in-chief are Imre Bárány and József Solymosi. The advisory board consists of Ronald Graham, Gyula O. H. Katona, Miklós Simonovits, Vera Sós, and Endre Szemerédi. It is published by the János Bolyai Mathematical Society and Springer Verlag. The following members of the ''Hungarian School of Combinatorics'' have strongly contributed to the journal as authors, or have served as editors: Miklós Ajtai, László Babai, József Beck, András Frank, Péter Frankl, Zoltán Füredi, András Hajnal, Gyula Katona, László Lovász, László Pyber, Alexander Schrijver, Miklós Simonovits, Vera Sós, Endre Szemerédi, Tamás Szőnyi, Éva Tardos, Gábor Tardos.{{cite web, url=https://www.spr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The very first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.87. Notable publications * The 1972 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applications of combinatorics. ''Series B'' is concerned primarily with graph and matroid theory. The two series are two of the leading journals in the field and are widely known as ''JCTA'' and ''JCTB''. The journal was founded in 1966 by Frank Harary and Gian-Carlo Rota.They are acknowledged on the journals' title pages and Web sites. SeEditorial board of JCTA Originally there was only one journal, which was split into two parts in 1971 as the field grew rapidly. An electronic, |
Theorems In Discrete Mathematics
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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