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	Simplicial complex - Wikipedia, the free encyclopedia
		In mathematics, a simplicial complex is a topological space of a particular kind, built up of points, line segments, triangles, and their n-dimensional counterparts.
		Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory.
		The general finite simplicial complex is a set of instructions for joining a number of simplices of varying dimensions together, as a topological space in the abstract (not assumed to be a subset of Euclidean space).
	en.wikipedia.org /wiki/Simplicial_complex (580 words)

	Encyclopedia :: encyclopedia : Algebraic topology (Site not responding. Last check: 2007-11-01)
		The free rank of the n-th homology group of a simplicial complex is equal to the n-th Betti number, so one can use the homology groups of a simplicial complex to calculate its Euler-Poincaré characteristic.
		As another example, the top-dimensional integral cohomology group of a closed manifold detects orientability: this group is isomorphic to either the integers or 0, according as the manifold is orientable or not.
		Beyond simplicial homology, which is defined only for simplicial complexes, one can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.
	www.hallencyclopedia.com /Algebraic_topology (613 words)

	PlanetMath: simplicial complex (Site not responding. Last check: 2007-11-01)
		We do so not because the homology of a simplicial complex is so intrinsically interesting in and of itself, but because the resulting homology theory is identical to the singular homology of the associated topological space
		The proof of this theorem is considerably more difficult than what we have done to this point, requiring the techniques of barycentric subdivision and simplicial approximation, and we refer the interested reader to [1].
		This is version 6 of simplicial complex, born on 2002-04-11, modified 2006-07-31.
	planetmath.org /encyclopedia/SimplicialComplex.html (464 words)

	Abstracts of Papers (Site not responding. Last check: 2007-11-01)
		Applications of the acyclic graph complexes to outer automorphisms of free groups are outlined.
		Abstract: We combinatorially interpret the spectra of discrete Laplace operators from the boundary maps in the simplicial complex of independent sets of a matroid.
		Abstract: A new recursion for the unsigned reduced Euler characteristic of the independent complex of a matroid
	www.math.uri.edu /~andrewk/abstracts (340 words)

	[No title]
		Although homology can be defined in terms of huge (infinitely-generated) chain complexes, one may in fact show that the this complex is chain-homotopic to the finite chain complex freelyl generated by the cells of the simplicial complex.
		In the case of the cube, for example, we simply observe that the space is homeomorphic to the CW complex formed by adjoining a 2-cell (a closed disk in R^2) to a 0-cell (a point) by sending the whole boundary of the disk to the point.
		So if you wanted to know whether the (abstract) homology groups of a simplicial complex could be computed in finite time, the answer is certainly yes.
	www.math.niu.edu /~rusin/known-math/96/homology_calc (1030 words)

	Topology - Wikipedia, the free encyclopedia
		Topology has introduced a new geometric language (simplicial complexes, homotopy, cohomology, Poincaré duality, fibrations, vector bundles, sheaves, characteristic classes, Morse functions, homological algebra, spectral sequences).
		It has had a major impact on the fields of differential geometry, algebraic geometry, dynamical systems and partial differential equations in the large, and several complex variables.
		A toroid in three dimensions; A coffee cup and a donut are both topologically indistinguishable from this toroid.
	en.wikipedia.org /wiki/Topology (1755 words)

	[No title]
		It is of interest to which extend* * algebraic properties of the face ring are reflected by combinatorial or geometric propert* ies of the simplicial* complex.
		Let K be an abstract simplicial complex with m vertices given by the set V = * {1,..., m} That is, K = {oe1,..., oer} consists of a finite set of faces oei V, which i *s closed with respect to formation of subsets.
		We call a simplicial complex reduced, if for every vertex i, the in* *clusion st({i}) K is proper.
	hopf.math.purdue.edu /Notbohm/cmcomplex.txt (4560 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		A hypergraph $C$ is called a "simplicial complex" if it is closed downwards, namely if $A \in C$, and $B \subset A$, then $B \in C$.
		A simplicial complex $M$ is called a "matroid" if it has a property fulfilled by sets of independent sets of vectors, namely: if $A,B \in M$, and $B > A$, then $A\cup \{x\} \in M$ for some element $x$ of $B \setminus A$.
		We prove a generalization, in which one of the matroids is replaced by a general simplicial complex, and the "rank" of the restricted matroid is replaced by the topological connectivity of the restriction of the complex.
	www.math.technion.ac.il /~techm/20041228143020041228aha (248 words)

	Geometry of the Complex of Curves Abstract (Site not responding. Last check: 2007-11-01)
		The Complex of Curves on a Surface is a simplicial complex whose vertices are homotopy classes of simple closed curves, and whose simplices are sets of homotopy classes which can be realized disjointly.
		In this paper we prove that, endowed with a natural metric, the complex is hyperbolic in the sense of Gromov.
		The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family.
	www.math.yale.edu /~yhm3/research/complexI.html (233 words)

	Polytopal and Simplicial Complexes
		As an example of a polytopal complex, consider the set of all faces of a polytope.
		In the case of simplicial complexes things are easier because of the following simple lemma.
		Since any subset of vertices of a simplex is the vertex set of a face, it is more convenient to describe a simplicial complex in terms of the associated abstract simplicial complex in the set of its vertices.
	www.uni-bayreuth.de /departments/wirtschaftsmathematik/rambau/Diss/diss_MASTER/node36.html (252 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		****************************************************************************) (************************************************************************** An abstract simplicial complex consists of a set of vertices together with a downward-closed set of simplices on those vertices under the sub-simplex ordering.
		The dimension of a complex K is ** the maximum dimension of any of its simplices.
		We restrict attention to "pure" simplicial complexes, in which every simplex of lower dimension arises as a sub-simplex of a simplex of maximum ** dimension.
	www.cs.cmu.edu /afs/cs.cmu.edu/project/pscico/pscico/src/simpcomp/SIMPLICIAL_COMPLEX_.sml (170 words)

	AProPo -- 7 Beyond Polytopes (Site not responding. Last check: 2007-11-01)
		This section is concerned with problems on finite abstract simplicial complexes.
		A pure d-dimensional finite abstract simplicial complex whose dual graph (defined on the facets, where two facets are adjacent if they share a common (d-1)-face) is connected is a pseudo-manifold if every (d-1)-dimensional simplex is contained in at most two facets.
		The boundary of a simplicial (d+1)-dimensional polytope induces a d-dimensional pseudo-manifold.
	www.zib.de /pfetsch/apropo/HTML3/Beyond_polytopes.html (190 words)

	Talk 4939 data/Spring_2004/0322 (Site not responding. Last check: 2007-11-01)
		The number of faces of the abstract simplicial complex formed by independent sets of a matroid is a combinatorial invariant of theoretical and practical interest.
		Though it is a numerical invariant associated with an abstract and purely combinatorial object, we will show that it can be studied in a very geometric/topological manner.
		In particular, we will highlight the very close analogy between results for the number of faces of matroid complexes and the corresponding results for number of faces of simplicial convex polytopes.
	www.math.duke.edu /mcal?abstract-4939 (150 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		# # To specify a simplicial complex to "homology", we use the same # definition of an abstract simplicial complex that most people are # familiar with, namely a set of sets satisfying certain properties.
		Then there are four list # operations I like to use: "card" is the cardinality of a list or set, # appendb appends an element to the back of a list, "replace(l, i, x)" # replaces the ith element of l with x, and "delete(l, i)" deletes the # ith element of a list l.
		The basic # data structure used for computation in this worksheet is a simplicial # complex, but a complex is determined by the simplices that are # maximal with respect to inclusion, so it the user may define a complex # that way, and homology will use "complx" to fill it out.
	www.cis.upenn.edu /~rah/MOISE.txt (1200 words)

	Abstract (Site not responding. Last check: 2007-11-01)
		Abstract: This talk will discuss a new combinatorial approach to studying the minimal free resolution of square-free monomial ideals.
		I will concentrate on the class of quadratic square-free monomial ideals, which can be viewed as edge ideals of graphs.
		The method also extends quite successfully to an arbitrary square-free monomial ideal viewed as the facet ideal of a simplicial complex.
	www.nd.edu /~magic05/abstracts/ha.html (60 words)

	Math Seminars: Eli Berger (Site not responding. Last check: 2007-11-01)
		In this talk we generalize this theorem, replacing one of the matroids by a general simplicial complex.
		One application is a solution of the case $r=3$ of a matroidal version of Ryser's conjecture.
		We describe the case in which the complex is the complex of independent sets of a graph, and prove generalizations of known results on ``independent systems of representatives" (which are the special case in which the matroid is a partition matroid).
	www.math.ias.edu /abstract.php?event=2719 (183 words)

	Algebraic Combinatorics Seminar (Site not responding. Last check: 2007-11-01)
		We say a graph is t-colorable if one can assign one of t colors to each vertex of the graph such that no adjacent vertices have the same color.
		Since any subgraph of a t-colorable graph is t-colorable, the set of all t-colorable graphs on a fixed vertex set can be considered an abstract simplicial complex with the t-colorable graphs as faces.
		In their 2003 paper "Complexes of t-colorable graphs", Svante Linusson and John Shareshian apply the techniques of discrete Morse theory to the complex of t-colorable graphs on the fixed vertex set [n] to determine the homotopy type for t leq 2 and leq n-3.
	www.ms.uky.edu /~readdy/Seminar/wells.html (150 words)

	AProPo -- 33 f-Vector of Simplicial Complexes
		This problem is only known to be in NP for partitionable (see Problem 19) simplicial complexes (see Kleinschmidt and Onn [39]).
		Let V be the vertex set of a simplicial complex Δ that is defined by the minimal non-faces e ∈ E.
		(the dual complex), which is given by its facets.
	www.zib.de /pfetsch/apropo/f_vector_of_simplicial_complexes.html (483 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		A monomial ideal of a Stanley-Reisner ring associated to a subcomplex of an (abstract) simplicial complex.
		I will give a combinatorial topological formula for the multigraded Hilbert series, and in the case where the ambient complex is Gorenstein, compare this with a second formula that generalizes results of Mustata and Terai.
		The agreement between these two formulae is seen to be a disguised form of Alexander duality.
	www-math.mit.edu /~combin/abstracts/march02 (192 words)

	Complex and Shape
		These pieces of overlap are instrumental in the construction of a set system closed under the subset operation.
		In topology, such a system is referred to as an abstract simplicial complex.
		Figure: The decomposition of the union of disks and its dual complex (figures 2 and 3 overlapped).
	www.geom.uiuc.edu /~mucke/GeomDir/shapes95/node3.html (380 words)

	Math Forum Discussions (Site not responding. Last check: 2007-11-01)
		between an abstract simplicial complex and its geometric realization.
		Given complexes A and B, such that A is a proper subset of B, isn't it
		Let A, B, and C be complexes such that A < B, and f:B is a
	mathforum.org /kb/thread.jspa?threadID=1120466&messageID=3675350 (235 words)

	Concurrency Abstracts (Site not responding. Last check: 2007-11-01)
		Furthermore it is argued that in abstract interleaving semantics (at least for finite processes) branching bisimulation congruence is the finest reasonable congruence possible.
		The motivating application is a uniform theory of abstract or parametrized time in which to any given notion of time there corresponds an algebra of concurrent behaviors and their operations, always the same operations but interpreted automatically and appropriately for that notion of time.
		As expected we depend on the now standard technique of refinement of atomic events to complex events; what is not expected is that their complexity need be only that of nondeterminism, in that we refine one atomic event to a set of alternative atomic events, not to a set of sequences.
	boole.stanford.edu /abstracts.html (9716 words)

	Abstract for 2003/3/28: Klivans (Site not responding. Last check: 2007-11-01)
		Abstract for the Combinatorics and Number Theory Seminar
		Shifted complexes are a type of abstract simplicial complex.
		I will also establish connections between shifted complexes and totally symmetric plane partitions, standard Young tableaux, linear extensions of posets, and independence complexes of matroids.
	www.math.binghamton.edu /dept/ComboSem/abstract.200303kli.html (62 words)

	Dear
		Abstract: Computational topology is an emerging discipline which blends pure and applied mathematics.
		(1) The classical notion of a simplicial complex is relevant to surface intersection algorithms.
		Topological applications to aeronautical engineering will be presented.
	www.math.uri.edu /~eaton/AbstractOct2502.htm (124 words)

	conabs
		We then use these methods to solve interpolation problems on unions of non-commensurate lattices, which are created via specific number-theoretic guidelines.
		In the 1980's, M. Perles asked if the abstract simplicial complex of dissections of a convex (n+2)-gon could be viewed as the boundary complex of some convex polytope.
		C. Lee answered this question in the affirmative by constructing the desired polytope, called the associahedron; he also developed other properties of the associahedron.
	www.american.edu /academic.depts/cas/mathstat/MAA/spr00/conabs.html (1239 words)

	Homogeneous Multivariate Polynomials with the Half-Plane Property -- from Mathematica Information Center
		A polynomial P in n complex variables is said to have the "half-plane property" (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane.
		Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics.
		graph, matroid, jump system, abstract simplicial complex, spanning tree, basis, generating polynomial, reliability polynomial, Brown-Colbourn conjecture, half-plane property, Hurwitz polynomial, positive rational function, Lee-Yang theorem, Heilmann-Lieb theorem, Grace-Walsh-Szegö coincidence theorem, matrix-tree theorem, electrical network, nonnegative matrix, determinant, permanent
	library.wolfram.com /infocenter/Articles/1034 (280 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Reisner respectively Stanley explained in which sense Cohen-Macaulay and Gorenstein properties of the face ring are reflected by geometric and/or combinatoric properties of the simplicial complex.
		We give a new proof for these result by homotopy theoretic methods and constructions.
		Our approach is based on ideas used very successfully in the analysis of the homotopy theory of classifying spaces.
	hopf.math.purdue.edu /Notbohm/cmcomplex.abstract (59 words)

	Computational Homology Project (Site not responding. Last check: 2007-11-01)
		of a chain complex, as well as the homomorphisms induced in homology by chain maps
		homsimpl - compute the homology of a finite abstract simplicial complex or relative homology of a pair of simplicial complexes
		All trademarks, registered trademarks or brand names are property of the respective holders and used in this document for descriptive purposes only.
	www.math.gatech.edu /~chomp/advanced/programs.php (1023 words)

	The Complex of Maximal Lattice Free Simplices
		The simplicial complex K(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form {x : Ax Download Info
		If you experience problems downloading a file, check if you have the proper application to view it first.
		Authors registered on the RePEc Author Service receive monthly emails with details about downloads and abstract views of their works.
	ideas.repec.org /p/cwl/cwldpp/1032.html (194 words)

	Definition of Nerve (disambiguation)
		Nerve may refer to more than one thing:
		In mathematics, a nerve refers to an abstract simplicial complex formed from a family of objects by taking intersections: the 0-simplices are the objects themselves, the 1-simplices join pairs of objects with non-empty intersections, the 2-simplices join triples of objects with non-empty intersections etc. See nerve of an open covering.
		Nerve can also refer to a cocky form of self-confidence.
	www.wordiq.com /definition/Nerve_(disambiguation) (236 words)

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