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Topic: Simplicial complex

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	PlanetMath: simplicial complex
		, is the simplicial complex consisting of all nonempty subsets of
		simplicial homology, simplicial cohomology, triangulation, abstract simplicial complex, abstract
		This is version 8 of simplicial complex, born on 2002-04-11, modified 2007-05-09.
	planetmath.org /encyclopedia/SimplicialComplex.html (471 words)

	Simplicial Complexes
		The library allows the programmer to build a simplicial complex or a set, associate data with the contained simplices, operate on the contained simplices and data, and query the complex or the set with powerful library functions.
		A simplicial set is similar to a simplicial complex but more general.
		Simplicial sets provide the programmer with functions similar to the simplicial complexes to manipulate the data and the simplices in the set, as well as query the set.
	www.cs.cmu.edu /afs/cs/project/pscico/pscico/src/simpcomp/README.html (499 words)

	Computability of Fault-Tolerant Distributed Protocols
		For instance, Figure 8 shows the simplicial set, called protocol complex, after one round of communication on a synchronous message-passing machine, which broadcasts the local states to all processes at each step.
		If there is a simplicial map from it to some suitable set of global states, respecting the specification of the problem, then there exists a corresponding wait-free protocol.
		Notice that the simplexes of the protocol complex are really the schedules, and this should be related to the directed homotopy approach of Section 2.
	www.di.ens.fr /~goubault/link004.html (1221 words)

	Definition of Simplicial complex
		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.
		Informally, a simplicial complex is made of a set of simplices that intersect with each other only by their common faces.
		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).
	www.wordiq.com /definition/Simplicial_complex (561 words)

	Simplicial complex - Definition, explanation
		Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory.
		A simplicial complex is a combinatorial object telling one how to construct a topological space out of a number of simplices.
		See also the discussion at polytope of simplicial complexes as subspaces of Euclidean space, made up of subsets each of which is a simplex.
	www.calsky.com /lexikon/en/txt/s/si/simplicial_complex.php (609 words)

	Complexity International
		As it is discussed, complexity of structure expressed in numeric form is non-informative, but even at the initial stage of analysis it becomes reasonable to express complexity linguistically, to fill the results of the mathematical modeling stage with meaning.
		A concept of complex system (or, complexity in general) is many-sided and rich (Cornacchio 1977; Edmonds 1997), and because of that we distinguish only structural features which could bring a valuable contribution to systems studying.
		Preliminary conclusions on complexity of system are drawn on the basis of observation of its behavior, which depends upon a system’s organization (Klir 1969).
	journal-ci.csse.monash.edu.au /ci_louise/vol07/degtia01/html (4250 words)

	Simplicial complex (Site not responding. Last check: )
		Informally, a simplicial complex ismade of a set of simplices that intersect with each other only by their common faces.In algebraic topology these spaces are found to be the easiestto deal with, in terms of concrete calculations.
		The general finite simplicial complex is a set ofinstructions 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).
		For the definition of homology groups of a simplicial complex, onecan read the corresponding chain complex directly - provided consistentorientations are made of all simplices.
	www.therfcc.org /simplicial-complex-46766.html (485 words)

	MathBus: Geometry Terms
		The embedding dimension of the simplicial complex is dimension of the space of the points in the simplices of the simplicial complex.
		First, since vertices in a simplicial complex are generally incident to many simplices, this device of referring to vertices by an index saves storage.
		A triangulated b-rep is the combination of three elements: a b-rep, a simplicial complex, and a mapping from simplicial complex vertex indices to faces of the b-rep (presumably through their bound names).
	www.cs.cornell.edu /Info/Projects/NuPrl/mathbus/geometry.htm (1440 words)

	QMG project: Geometric Objects and Datatypes
		A simplicial complex is a collection of simplices of one specific dimension embedded in a space of the same or higher dimension.
		Simplicial complexes are the output of the mesh generator and are an input to the finite-element program and to the graphics programs.
		The third representation of a brep or simplicial complex is as an Ascii string.
	www.cs.cornell.edu /info/people/vavasis/qmg1.1/geom.html (4392 words)

	[No title] (Site not responding. Last check: )
		Under certain favorable circumstances, a simplicial complex admits an ``optimal'' decision tree such that the evasive sets are enumerated by the reduced Betti numbers of the complex; we may hence read off the homology directly from the tree.
		be the simplicial complex of subsets of Ω
		As both complexes are known to be homotopically Cohen-Macaulay of dimension n-1, this yields that the two complexes are homotopy equivalent, thereby settling a conjecture due to Anders Björner and Volkmar Welker.
	www-math.mit.edu /~jakob/combinmain.html (2102 words)

	Good Math, Bad Math : Simplices and Simplicial Complexes
		A simplicial complex is a topological space formed from a set of simplices.
		So you can have two tetrahedrons intersecting along an edge - they're still a simplicial complex, because the intersection is a 1-simplex, so you consider it using the 2-simplices in K - and the line segment is a face of all of the triangles that meet at that edge.
		A simplicial complex where the largest dimension of any simplex in the complex is N is called a simplicial N-complex.
	scienceblogs.com /goodmath/2007/03/simplices_and_simplicial_compl.php (2246 words)

	Simplicial Families of Drawings
		A simplicial family is a representation of the set of all legal interpolations of an initial set of example drawings.
		Samples drawn from the cartoon character simplicial family are on the right and samples drawn from the entire interpolation space are on the left.
		The corresponding point in the simplicial complex is shown at the upper left of each drawing.
	www.cs.wisc.edu /graphics/Gallery/kovar.vol/Simplicial (608 words)

	[No title] (Site not responding. Last check: )
		Simplicial sets used to be called complete semisimplicial complexes(css complexes); the word complete referred to the incorporation of degeneracy operations with the face operations, so a semisimplicial complex only had face operators (i.e, the hom sets in the domain category consisted of strict monomorphisms [n]--->[m]).
		2) The barycentric subdivision of a regular semisimplicial CW complex is a (geometrical) simplicial complex.
		If the simplicial set $X$ is the 2-simplex $\Delta2$ with its > boundary $\partial \Delta2$ collapsed to a point, then the second > Kan/normal/barycentric subdivision $Z = Sd2(X)$ still has non-degenerate > 2-simplices that are not embedded, i.e., $Z$ is not a simplicial complex > in the obvious way.
	www.lehigh.edu /~dmd1/bg13.txt (727 words)

	The Flow Complex: A Data Structure for Geometric Modeling
		The authors wish to structure a given point cloud into a simplicial complex like the Delaunay triangulation, but which may be more suitable for certain types of point clouds and is still efficient to compute.
		The authors found that the flow complex is well suited for reconstructing the surface from which a finite set of points was sampled.
		Also the authors use the flow complex to model pockets in proteins in a way that is similar to earlier work that modeled pockets using the Delaunay triangulation.
	www.stanford.edu /~ctj/liteseer/segparmf/giesen03flow/index.html (314 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)

	topaz: Constructions
		Let C be the given simplicial complex and A the subcomplex induced by the given vertices.
		Since it produces a simplicial complex (from a polytope), it is counted as a client for the topaz application.
		Since it produces a simplicial complex (from a surface), it is counted as a client for the topaz application.
	www.math.tu-berlin.de /polymake/apps/topaz/clients.html (3564 words)

	Playing Nim on a simplicial complex - Ehrenborg, Steingr'imsson (ResearchIndex) (Site not responding. Last check: )
		Abstract: We introduce a generalization of the classical game of Nim by placing the piles on the vertices of a simplicial complex and allowing a move to affect the piles on any set of vertices that forms a face of the complex.
		These conditions are satisfied, for instance, when the simplicial complex consists of the independent sets of a binary matroid.
		Branching complexes are shown to be constrictive Section 5 and an algorithm to calculate the...
	citeseer.ist.psu.edu /10877.html (470 words)

	3.14 Assorted topics
		Hartle [124, 125] has suggested computing the wave functional of the universe in a simplicial approximation, and evaluating the discrete path integral semiclassically near stationary points of the Regge action.
		Other authors have suggested associating gauge-theoretic instead of metric variables with the building blocks of a simplicial complex, for the case of the Poincaré group [71], the Lorentz group [140], and for Ashtekar gravity with gauge group
		A recent proposal for constructing a canonical quantum theory is due to Mäkelä [158], who constructed a simplicial version of the Wheeler-DeWitt equation, based on the use of area instead of length variables (which however are known to be overcomplete).
	relativity.livingreviews.org /Articles/lrr-1998-13/node34.html (266 words)

	QUANTUM FIELD THEORY,STRING THEORY, AND TOPOLOGY
		One way to proceed is to adopt a simplicial approach to quantum gravity, known as Regge Calculus.
		In this method, one replaces a continuous spacetime manifold by a discrete lattice type structure (simplicial complex).
		In this way, one tries to construct path integrals on a simplicial complex which are invariant under arbitrary subdivisions of the simplicial complex.
	www.ucd.ie /math-phy/tqft.html (287 words)

	Simplicial Complex
		A geometric (finite) simplicial complex K is a (finite) collection
		A simplicial complex K inherits the subspace topology and we denote this topological space by
		Let M be a polyhedron with a triangulation as the union of two simplicial complexes
	www.maths.manchester.ac.uk /~kd/knots/node7.html (179 words)

	B
		This 1-D simplicial complex has an objective geometric realization as a highly singular curve, in the sense that every vertex of the graph is a multiple point of the curve.
		is the simplicial complex of a simplicial decomposition of the sphere
		is the simplicial complex of a manifold with a boundary.
	www.usc.edu /dept/LAS/CAMS/EE/Edmond/mine2.htm (2976 words)

	3.1 Path integral for Regge calculus
		One is interested in the behaviour of expectation values of local observables as the simplicial complex becomes large, and the existence of critical points and long-range correlations, in a scaling limit and as the cutoffs are removed.
		They obtained a simplicial lattice geometry by subdividing each unit cell of a hypercubic lattice into simplices.
		One may define analogues of local conformal transformations on a simplicial complex by multiplication with a positive scale factor at each vertex, but the global group property is incompatible with the existence of the generalized triangle inequalities.
	relativity.livingreviews.org /Articles/lrr-1998-13/node21.html (438 words)

	Simplicial Complex
		Each 1-simplex (or edge) has two faces, which are 0 -simplexes called vertices.
		To form a complex, the simplexes must fit together in a nice way.
		Figure 6.15: To become a simplicial complex, the simplex faces must fit together nicely.
	planning.cs.uiuc.edu /node274.html (163 words)

	MERL – TR2005-094 – Texture Design Using a Simplicial Complex of Morphable Textures
		We capture the structure of the induced space by a simplicial complex where vertices of the simplices represent input textures.
		We propose a morphable interpolation for textures, which also defines a metric used to build the simplicial complex.
		We allow users to continuously navigate in the simplicial complex and design new textures using a simple and efficient user interface.
	www.merl.com /publications/TR2005-094 (168 words)

	SPAMS Email Archive - Peter Clifford - Nim on a Simplicial Complex (Site not responding. Last check: )
		SPAMs is still on at 5pm in 2-338 but we now have some instant spam out of the cupboard by the pinch spammer Peter Clifford Title: Playing Nim on a simplicial complex Abstract: We discuss the classical game of Nim, and its known winning strategy (Bouton, 1901).
		Then we introduce a generalisation by placing the piles on the vertices of a simplicial complex.
		We allow a move to affect the piles on any set of vertices that forms a face of the complex.
	www-math.mit.edu /spams/spring1999/email4.htm (130 words)

	[No title] (Site not responding. Last check: )
		If the simplicial set $X$ is the 2-simplex $\Delta2$ with its boundary $\partial \Delta2$ collapsed to a point, then the second Kan/normal/barycentric subdivision $Z = Sd2(X)$ still has non-degenerate 2-simplices that are not embedded, i.e., $Z$ is not a simplicial complex in the obvious way.
		This was asserted in a 1956 Princeton preprint of Barratt's, titled "Simplical and semisimplicial complexes", by an argument involving two subdivisions, and some more.
		It seems to me that after 1970 or so, the algebraic topological interest in simplicial sets shifted entirely to their homotopy theory, rather than their topology, i.e., people were mostly concerned with (weak) homotopy equivalences rather than homeomorphisms of the geometric realizations of simplicial sets.
	www.lehigh.edu /dmd1/public/www-data/jr1229.txt (451 words)

	[No title]
		the boundary of the 3-chain [1234]+2[1235] in some simplicial* complex is equal to the boundary of [1234] plus twice the boundary of [1235], ([234]-[134]+[124]-[123])+2*([235]-[135]+[125]-[123]) = [234]-[134]+[124]-3[123]+2[235]-2[135]+2[125].
		A collection of simplices in some R^n is a simplicial complex if every pair of simplices which intersect do so in a common face; a face is simply the convex hull of any subset of the vertices of the simplex.
		Defined the complex numbers in both coordinate form and polar form, and showed that the two are isomorphic under multiplication; two structures (S,) and (T,%) are isomorphic if there is a bijection f from S to T with f(ab) = f(a)%f(b).
	math.berkeley.edu /~develin/math113/synopses.html (3604 words)

	Stepping Into Alpha Shapes (Site not responding. Last check: )
		The alpha-shape is the geometric object defined as the union of the elements in the complex.
		The discrete nature of the complex has computational advantages exploited in computing surface area and volume of a space filling diagram and in localizing and measuring voids.
		The local dimensionality of the complex controls the complexity of the sound through the recursive generation of waves of lower dimensions.
	evlweb.eecs.uic.edu /EVL/VROOM/HTML/PROJECTS/06Fu.html (627 words)

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