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Flip Graph
In mathematics, a flip graph is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are special cases of geometric graphs. Among noticeable flip graphs, one finds the 1-skeleton of polytopes such as associahedra or cyclohedra. Examples A prototypical flip graph is that of a convex n-gon \pi. The vertices of this graph are the triangulations of \pi, and two triangulations are adjacent in it whenever they differ by a single interior edge. In this case, the flip operation consists in exchanging the diagonals of a convex quadrilateral. These diagonals are the interior edges by which two triangulations adjacent in the flip graph differ. The resulting flip graph is both the Hasse diagram of the Tamari lattice and the 1-skeleton of the (n-3)-dimensional associahedron. This basic construction can be generalized in a number of wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flip Graphs
Flip, FLIP, or flips may refer to: People * Flip (nickname), a list of people * Lil' Flip (born 1981), American rapper * Flip Simmons, Australian actor and musician * Flip Wilson, American comedian Arts and entertainment Fictional characters * Flip (''Little Nemo''), a cartoon character * Flip, the title character of '' Flip's Twisted World'', a video game * Flip the Frog, a cartoon character * Flip the grasshopper, a character in the children's book '' The Adventures of Maya the Bee'' Music * Flip Records (1950s), a rhythm and blues and doo-wop label based in Los Angeles * Flip Records (1994), a record label in California * Flips, a short name of The Flaming Lips, an American rock band formed in 1983 * ''Flip'' (album), a 1985 solo album by Nils Lofgren * ''The Flip'' (album), a 1969 album by jazz saxophonist Hank Mobley Business * Flip or Flipping, an American term for buying and reselling something quickly, particularly real estate * Flip Burger Boutique, a chain of g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associahedron
In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari. Examples The one-dimensional associahedron ''K''3 represents the two parenthesizations ((''xy'')''z'') and (''x''(''yz'')) of three symbols, or the two triangulations of a square. It is itself a line segment. The two-dimensional associahedron ''K''4 represents the five parenthesizations of four symbols, or t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polytope
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domino Tiling
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a perfect matching in the grid graph formed by placing a vertex at the center of each square of the region and connecting two vertices when they correspond to adjacent squares. Height functions For some classes of tilings on a regular grid in two dimensions, it is possible to define a height function associating an integer to the vertices of the grid. For instance, draw a chessboard, fix a node A_0 with height 0, then for any node there is a path from A_0 to it. On this path define the height of each node A_ (i.e. corners of the squares) to be the height of the previous node A_n plus one if the square on the right of the path from A_n to A_ is black, and minus one otherwise. More details can be found in . Thurston's height condition describes a test for determining whether a simply- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclohedron
In geometry, the cyclohedron is a d-dimensional polytope where d can be any non-negative integer. It was first introduced as a combinatorial object by Raoul Bott and Clifford Taubes and, for this reason, it is also sometimes called the Bott–Taubes polytope. It was later constructed as a polytope by Martin Markl and by Rodica Simion. Rodica Simion describes this polytope as an associahedron of type B. The cyclohedron is useful in studying knot invariants. Construction Cyclohedra belong to several larger families of polytopes, each providing a general construction. For instance, the cyclohedron belongs to the generalized associahedra that arise from cluster algebra, and to the graph-associahedra, a family of polytopes each corresponding to a graph. In the latter family, the graph corresponding to the d-dimensional cyclohedron is a cycle on d+1 vertices. In topological terms, the configuration space of d+1 distinct points on the circle S^1 is a (d+1)-dimensional manifold, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop (topology)
In mathematics, a loop in a topological space is a continuous function from the unit interval to such that In other words, it is a path whose initial point is equal to its terminal point.. A loop may also be seen as a continuous map from the pointed unit circle into , because may be regarded as a quotient of under the identification of 0 with 1. The set of all loops in forms a space called the loop space of . See also * Free loop *Loop group *Loop space *Loop algebra *Fundamental group *Quasigroup In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that " division" is always possible. Quasigroups differ from groups mainly in that they need not be associative and need not have ... References Topology es:Grupo fundamental#Lazo {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Edges
In graph theory, multiple edges (also called parallel edges or a multi-edge), are, in an undirected graph, two or more edges that are incident to the same two vertices, or in a directed graph, two or more edges with both the same tail vertex and the same head vertex. A simple graph has no multiple edges and no loops. Depending on the context, a graph may be defined so as to either allow or disallow the presence of multiple edges (often in concert with allowing or disallowing loops): *Where graphs are defined so as to ''allow'' multiple edges and loops, a graph without loops or multiple edges is often distinguished from other graphs by calling it a ''simple graph.'' *Where graphs are defined so as to ''disallow'' multiple edges and loops, a multigraph or a pseudograph is often defined to mean a "graph" which ''can'' have loops and multiple edges. Multiple edges are, for example, useful in the consideration of electrical networks, from a graph theoretical point of view. Ad ... [...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 pap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (topology)
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world. In general In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of sol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangulation (topology)
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling. Motivation On the one hand, it is sometimes useful to forget about superfluous information of topological spaces: The replacement of the original spaces with simplicial complexes may help to recognize crucial properties and to gain a better understanding of the considered object. On the other hand, simplicial complexes are objects of combinatorial character and therefore one can assign them quantities rising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows now to assign such quantities to topological spaces. Investigations concerning the exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon's Theorem
In geometry, Radon's theorem on convex sets, published by Johann Radon in 1921, states that any set of ''d'' + 2 points in R''d'' can be partitioned into two sets whose convex hulls intersect. A point in the intersection of these convex hulls is called a Radon point of the set. For example, in the case ''d'' = 2, any set of four points in the Euclidean plane can be partitioned in one of two ways. It may form a triple and a singleton, where the convex hull of the triple (a triangle) contains the singleton; alternatively, it may form two pairs of points that form the endpoints of two intersecting line segments. Proof and construction Consider any set X=\\subset \mathbf^d of ''d'' + 2 points in ''d''-dimensional space. Then there exists a set of multipliers ''a''1, ..., ''a''''d'' + 2, not all of which are zero, solving the system of linear equations : \sum_^ a_i x_i=0,\quad \sum_^ a_i=0, because there are ''d'' + 2 un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Link (geometry)
The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local structure of the complex at the vertex. Link of a vertex Given an abstract simplicial complex and v a vertex in V(X), its link \operatorname(v,X) is a set containing every face \tau \in X such that v\not\in \tau and \tau\cup \ is a face of . * In the special case in which is a 1-dimensional complex (that is: a graph), \operatorname(v,X) contains all vertices u\neq v such that \ is an edge in the graph; that is, \operatorname(v, X)=\mathcal(v)=the neighborhood of v in the graph. Given a geometric simplicial complex and v\in V(X), its link \operatorname(v,X) is a set containing every face \tau \in X such that v\not\in \tau and there is a simplex in X that has v as a vertex and \tau as a face. Equivalently, the join v \star \tau is a face in X. * As an example, suppose v is the top vertex of the tetrahedron at the lef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
