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Dushnik-Miller Dimension
In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see . Formal definition The dimension of a poset ''P'' is the least integer ''t'' for which there exists a family :\mathcal R=(<_1,\dots,<_t) of s of ''P'' so that, for every ''x'' and ''y'' in ''P'', ''x'' precedes ''y'' in ''P'' if and only if it precedes ''y'' in all of the linear extensions. That is, : An alternative definition of order dimension is the minimal number of [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Dimension
In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. first studied order dimension; for a more detailed treatment of this subject than provided here, see . Formal definition The dimension of a poset ''P'' is the least integer ''t'' for which there exists a family :\mathcal R=(<_1,\dots,<_t) of s of ''P'' so that, for every ''x'' and ''y'' in ''P'', ''x'' precedes ''y'' in ''P'' if and only if it precedes ''y'' in all of the linear extensions. That is, : An alternative definition of order dimension is the minimal number of |
Polynomial Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Computing
The ''SIAM Journal on Computing'' is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM). Although its official ISO abbreviation is ''SIAM J. Comput.'', its publisher and contributors frequently use the shorter abbreviation ''SICOMP''. SICOMP typically hosts the special issues of the IEEE Annual Symposium on Foundations of Computer Science (FOCS) and the Annual ACM Symposium on Theory of Computing (STOC), where about 15% of papers published in FOCS and STOC each year are invited to these special issues. For example, Volume 48 contains 11 out of 85 papers published in FOCS 2016. References * External linksSIAM Journal on Computing on |
Order (journal)
''Order'' (subtitled ''A Journal on the Theory of Ordered Sets and its Applications'') is a quarterly peer-reviewed academic journal on order theory and its applications, published by Springer Science+Business Media. It was established in 1984 by Ivan Rival (University of Calgary). From 2010 to 2018, its editor-in-chief was Dwight Duffus (Emory University). He was succeeded in 2019 by Ryan R. Martin (Iowa State University). Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2017 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 0.353. References External links * Order theory Mathematics journals Springer Science+Business Media academic journals Publications established in 198 ... [...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] |
American Journal Of Mathematics
The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United States, established in 1878 at the Johns Hopkins University by James Joseph Sylvester, an English-born mathematician who also served as the journal's editor-in-chief from its inception through early 1884. Initially W. E. Story was associate editor in charge; he was replaced by Thomas Craig in 1880. For volume 7 Simon Newcomb became chief editor with Craig managing until 1894. Then with volume 16 it was "Edited by Thomas Craig with the Co-operation of Simon Newcomb" until 1898. Other notable mathematicians who have served as editors or editorial associates of the journal include Frank Morley, Oscar Zariski, Lars Ahlfors, Hermann Weyl, Wei-Liang Chow, S. S. Chern, André Weil, Harish-Chandra, Jean Dieudonné, Henri Cartan, Stephen S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Dimension
In mathematics, especially order theory, the interval order for a collection of intervals on the real line is the partial order corresponding to their left-to-right precedence relation—one interval, ''I''1, being considered less than another, ''I''2, if ''I''1 is completely to the left of ''I''2. More formally, a countable poset P = (X, \leq) is an interval order if and only if there exists a bijection from X to a set of real intervals, so x_i \mapsto (\ell_i, r_i) , such that for any x_i, x_j \in X we have x_i , a left nesting is an i \in n/math> such that i < i+1 < f(i+1) < f(i) and a right nesting is an |
Inclusion Order
In the mathematical field of order theory, an inclusion order is the partial order that arises as the subset-inclusion relation on some collection of objects. In a simple way, every poset ''P'' = (''X'',≤) is (isomorphic to) an inclusion order (just as every group is isomorphic to a permutation group – see Cayley's theorem). To see this, associate to each element ''x'' of ''X'' the set : X_ = \ ; then the transitivity of ≤ ensures that for all ''a'' and ''b'' in ''X'', we have : X_ \subseteq X_ \text a \leq b . There can be sets S of cardinality less than , X, such that ''P'' is isomorphic to the inclusion order on ''S''. The size of the smallest possible ''S'' is called the 2-dimension of ''P''. Several important classes of poset arise as inclusion orders for some natural collections, like the Boolean lattice ''Q''''n'', which is the collection of all 2''n'' subsets of an ''n''-element set, the interval-containment orders, which are precisely the orders of order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain (order Theory)
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a (strongly connected, formerly called total). Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but refers generally to some sort of totally ordered subsets of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order. Strict and non-strict total orders A on a set X is a strict partial ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schnyder's Theorem
In graph theory, Schnyder's theorem is a characterization of planar graphs in terms of the order dimension of their incidence posets. It is named after Walter Schnyder, who published its proof in 1989. The incidence poset of an undirected graph with vertex set and edge set is the partially ordered set of height 2 that has as its elements. In this partial order, there is an order relation when is a vertex, is an edge, and is one of the two endpoints of . The order dimension of a partial order is the smallest number of total orderings whose intersection is the given partial order; such a set of orderings is called a ''realizer'' of the partial order. Schnyder's theorem states that a graph is planar if and only if the order dimension of is at most three. Extensions This theorem has been generalized by to a tight bound on the dimension of the height-three partially ordered sets formed analogously from the vertices, edges and faces of a convex polyhedron, or more genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Graph
In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See, for example, Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). As Dynkin diagrams In algebra, path graphs appear as the Dynkin diagrams of type A. As such, they classify the root system of type A and the Weyl group of ty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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