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• Simplicial maps and chain maps
  ==Simplicial maps== ...to see map $h$ as a “realization”<!--\index{realization}--> $h=|f|$ of a “simplicial” map $f:K\to L$ between these complexes:
  34 KB (5,897 words) - 16:05, 26 October 2015
• Simplicial maps
  ...to see map $h$ as a “realization”<!--\index{realization}--> $h=|f|$ of a “simplicial” map $f:K\to L$ between these complexes: ...s of a “collapse”<!--\index{collapse}-->. Since graphs are $1$-dimensional simplicial complexes, we can rewrite those definitions using the language of simplices
  34 KB (5,929 words) - 03:31, 29 November 2015
• Triangulations
  ==Simplicial vs cell complexes== *''simplicial complexes''<!--\index{simplicial complex}-->: cells are homeomorphic to points, segments, triangles, tetrahedra, ...
  30 KB (5,172 words) - 21:52, 26 November 2015
• Simplicial complexes
  <!--s-->[[Image:example graph and simplicial complex.png|center]] This data set is called a ''simplicial complex''<!--\index{simplicial complex}--> (or sometimes even a “multi-graph”). Its elements are called $0$-,
  30 KB (5,021 words) - 13:42, 1 December 2015
• Data made Euclidean
  <!--s-->[[Image:example graph and simplicial complex.png|center]] This data set is called a ''simplicial complex''<!--\index{simplicial complex}--> (or sometimes even a “multi-graph”). Its elements are called $0$-,
  31 KB (5,219 words) - 15:07, 2 April 2016
• Maps of polyhedra
  Previously, we proved that if complex $K^1$ is obtained from complex $K$ via a sequence of elementary collapses, then Suppose the circle is given by the simplest cell complex with just two cells $A,a$. Let's list ''all'' maps that can be represented
  51 KB (9,162 words) - 15:33, 1 December 2015
• Simplicial homology
  ==Simplicial complexes== Recall that a chain complex<!--\index{chain complex}--> is a sequence of vector spaces and linear operators:
  31 KB (5,170 words) - 13:44, 1 December 2015
• Metric tensor
  [[image:cubical complex distorted.png|center]] [[image:cubical complex bent.png|center]]
  42 KB (7,131 words) - 17:31, 30 November 2015
• Topology Illustrated -- Index
  *[[abstract simplicial complex|abstract simplicial complex]] *[[augmented chain complex|augmented chain complex]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Metric complexes
  <!--75-->[[image:cubical complex distorted.png| center]] <!--75-->[[image:cubical complex bent.png| center]]
  35 KB (5,871 words) - 22:43, 7 April 2016
• Geometric complexes
  ...recall the mechanical interpretation of a realization $|K|$ of a geometric complex $K$ of ambient dimension $n=1$: ...rods using an extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$.
  20 KB (3,354 words) - 17:37, 30 November 2015
• Multilinear algebra
  [[image:simplicial tangent spaces on graph.png|center]] '''Definition.''' For each vertex $A$ in a cell complex $K$, the (dimension $1$) ''tangent space'' at $A$ of $K$ is the set of $1$-
  49 KB (8,852 words) - 00:30, 29 May 2015
• Homotopy
  '''Theorem (Fundamental Theorem of Algebra).''' Every non-constant (complex) polynomial has a root. ...is question seems too challenging indicates that the ''domain space is too complex''!
  46 KB (7,846 words) - 02:47, 30 November 2015
• More metric complexes
  Let's recall the mechanical interpretation of a realization $|K|$ of a metric complex $K$ of dimension $n=1$: ...rods using an extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$.
  21 KB (3,445 words) - 13:53, 19 February 2016
• Homotopy and homotopy equivalence
  '''Theorem (Fundamental Theorem of Algebra).''' Every non-constant (complex) polynomial has a root. ...is question seems too challenging indicates that the ''domain space is too complex''!
  45 KB (7,738 words) - 15:18, 24 October 2015
• Cell maps
  Below, we will see how the theory of simplicial maps and their homology is extended to general cell complexes. We take the lead from simplicial maps. Under a simplicial map $f$, every $n$-cell $s$ is either
  42 KB (7,005 words) - 03:10, 30 November 2015
• Discrete calculus article
  [[File:Simplicial complex example.svg|thumb|200px|A simplicial 3-complex.]] A '''simplicial complex''' <math>\mathcal{K}</math> is a set of [[Simplex|simplices]] that satisfie
  27 KB (4,329 words) - 16:02, 1 September 2019
• Combinatorial cell maps
  Below, we will see how the theory of simplicial maps is extended to general cell complexes. We take the lead from simplicial maps: every $n$-cell $s$ is either cloned, $f(s) \approx s$, or collapsed,
  31 KB (5,330 words) - 22:14, 14 March 2016
• Homology of maps
  Below we will see how the theory of simplicial maps and their homology is extended to general cell complexes. We take the lead from simplicial maps. Under a simplicial map $f$, every $n$-cell $s$ is either
  41 KB (6,926 words) - 02:14, 21 October 2015
• A graph, non-tree representation of the topology of a gray scale image by Saveliev
  ...s approach has been extensively applied to digital image analysis [11] and geometric modeling [15]. This method of cell decomposition differs in technical detai The cell decomposition also makes certain geometric concepts more straightforward. First, an object and its background share ed
  41 KB (6,854 words) - 15:05, 28 October 2011

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