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Partition Regularity
In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set X, a collection of subsets \mathbb \subset \mathcal(X) is called ''partition regular'' if every set ''A'' in the collection has the property that, no matter how ''A'' is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is, for any A \in \mathbb, and any finite partition A = C_1 \cup C_2 \cup \cdots \cup C_n, there exists an ''i'' ≤ ''n'', such that C_i belongs to \mathbb. Ramsey theory is sometimes characterized as the study of which collections \mathbb are partition regular. Examples * the collection of all infinite subsets of an infinite set ''X'' is a prototypical example. In this case partition regularity asserts that every finite partition of an infinite set has an infinite cell (i.e. the infinite pigeonhole principle.) * sets with positive upper density in \mathbb: the ''u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in 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 is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halpern–Läuchli Theorem
In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern–Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern–Läuchli–Laver–Pincus or HLLP (named after James D. Halpern, Hans Läuchli, Richard Laver, and David Pincus), following . Let ''d'',''r'' < ω, \langle T_i: i \in d \rangle be a sequence of finitely splitting trees of height ω. Let :\bigcup_ \left(\prod_T_i(n)\right) = C_1 \cup \cdots \cup C_r, then there exists a sequence of subtrees \langle S_i: i \in d \rangle strongly embedded in \langle T_i: i \in d \rangle such that :\bigcup_ \left(\prod_S_i(n)\right) \subset C_k\textk \le r. Alternatively, let : S^d_ = \bigcup_ \left(\prod_T_i(n)\right) and : \mathbb^d=\bigcup_ S^d_.. The HLLP the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Rado's Theorem (Ramsey Theory)
Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the German mathematician Richard Rado. It was proved in his thesis, ''Studien zur Kombinatorik''. Statement Let A \mathbf = \mathbf be a system of linear equations, where A is a matrix with integer entries. This system is said to be r''-regular'' if, for every r-coloring of the natural numbers 1, 2, 3, ..., the system has a monochromatic solution. A system is ''regular'' if it is ''r-regular'' for all ''r'' ≥ 1. Rado's theorem states that a system A \mathbf = \mathbf is regular if and only if the matrix ''A'' satisfies the ''columns condition''. Let ''ci'' denote the ''i''-th column of ''A''. The matrix ''A'' satisfies the columns condition provided that there exists a partition ''C''1, ''C''2, ..., ''C''''n'' of the column indices such that if s_i = \Sigma_c_j, then # ''s''1 = 0 # for all ''i'' ≥ 2, ''si'' can be written as a ration ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents. Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called ''Diophantine geometry''. The word ''Diophantine'' refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–Čech Compactification
In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone–Čech compactification ''βX'' of a topological space ''X'' is the largest, most general compact Hausdorff space "generated" by ''X'', in the sense that any continuous map from ''X'' to a compact Hausdorff space factors through ''βX'' (in a unique way). If ''X'' is a Tychonoff space then the map from ''X'' to its image in ''βX'' is a homeomorphism, so ''X'' can be thought of as a (dense) subspace of ''βX''; every other compact Hausdorff space that densely contains ''X'' is a quotient of ''βX''. For general topological spaces ''X'', the map from ''X'' to ''βX'' need not be injective. A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milliken–Taylor Theorem
In mathematics, the Milliken–Taylor theorem in combinatorics is a generalization of both Ramsey's theorem and Hindman's theorem. It is named after Keith Milliken and Alan D. Taylor. Let \mathcal_f(\mathbb) denote the set of finite subsets of \mathbb, and define a partial order on \mathcal_f(\mathbb) by α<β max α |
IP Set
In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set ''D'' of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonempty subset of ''D''. The set of all finite sums over ''D'' is often denoted as FS(''D''). Slightly more generally, for a sequence of natural numbers (''n''i), one can consider the set of finite sums FS((''n''i)), consisting of the sums of all finite length subsequences of (''n''i). A set ''A'' of natural numbers is an IP set if there exists an infinite set ''D'' such that FS(''D'') is a subset of ''A''. Equivalently, one may require that ''A'' contains all finite sums FS((''n''i)) of a sequence (''n''i). Some authors give a slightly different definition of IP sets: They require that FS(''D'') equal ''A'' instead of just being a subset. The term IP set was coined by Hillel Furstenberg and Benjamin Weiss to abbreviate "infinite-dimensional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Rado
Richard Rado FRS (28 April 1906 – 23 December 1989) was a German-born British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Nazi persecution. He earned two PhDs: in 1933 from the University of Berlin, and in 1935 from the University of Cambridge. He was interviewed in Berlin by Lord Cherwell for a scholarship given by the chemist Sir Robert Mond which provided financial support to study at Cambridge. After he was awarded the scholarship, Rado and his wife left for the UK in 1933. He was appointed Professor of Mathematics at the University of Reading in 1954 and remained there until he retired in 1971. Contributions Rado made contributions in combinatorics and graph theory including 18 papers with Paul Erdős. In graph theory, the Rado graph, a countably infinite graph containing all countably infinite graphs as induced subgraphs, is named after Rado. He rediscovered it in 1964 after previous works on the same g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jon Folkman
Jon Hal Folkman (December 8, 1938 – January 23, 1969) was an American mathematician, a student of John Milnor, and a researcher at the RAND Corporation. Schooling Folkman was a Putnam Fellow in 1960. He received his Ph.D. in 1964 from Princeton University, under the supervision of Milnor, with a thesis entitled ''Equivariant Maps of Spheres into the Classical Groups''. Research Jon Folkman contributed important theorems in many areas of combinatorics. In geometric combinatorics, Folkman is known for his pioneering and posthumously-published studies of oriented matroids; in particular, the Folkman–Lawrence topological representation theorem is "one of the cornerstones of the theory of oriented matroids". In lattice theory, Folkman solved an open problem on the foundations of combinatorics by proving a conjecture of Gian–Carlo Rota; in proving Rota's conjecture, Folkman characterized the structure of the homology groups of "geometric lattices" in terms of the free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Syndetic
In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers. A set S \sub \mathbb is called ''piecewise syndetic'' if there exists a finite subset ''G'' of \mathbb such that for every finite subset ''F'' of \mathbb there exists an x \in \mathbb such that :x+F \subset \bigcup_ (S-n) where S-n = \. Equivalently, ''S'' is piecewise syndetic if there is a constant ''b'' such that there are arbitrarily long intervals of \mathbb where the gaps in ''S'' are bounded by ''b''. Properties * A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set. * If ''S'' is piecewise syndetic then ''S'' contains arbitrarily long arithmetic progressions. * A set ''S'' is piecewise syndetic if and only if there exists some ultrafilter ''U'' which contains ''S'' and ''U'' is in the smallest two-sided ideal of \beta \mathbb, the Stone–Čech compactification of the natural numbers. * Partition regularity: if S is piecewis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crispin St
Saints Crispin and Crispinian are the Christian patron saints of cobblers, curriers, tanners, and leather workers. They were beheaded during the reign of Diocletian; the date of their execution is given as 25 October 285 or 286. History Born to a noble Roman family in the 3rd century AD, Crispin and Crispinian fled persecution for their faith, ending up at Soissons, where they preached Christianity to the Gauls while making shoes by night. It is stated that they were twin brothers. They earned enough by their trade to support themselves and also to aid the poor. Their success attracted the ire of Rictus Varus, governor of Belgic Gaul, who had them tortured and thrown into the river with millstones around their necks. Though they survived, they were beheaded by the Emperor 286. Veneration The feast day of Saints Crispin and Crispinian is 25 October. Although this feast was removed from the Roman Catholic Church's universal liturgical calendar following the Second Vatican Coun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
