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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
• Parametric complexes
  So far, homology has been used to describe the topology of abstract geometric objects. The nature of these objects, then, dictated that all homology clas ...their edges ($1$-cells) and vertices ($0$-cells). The result is a cubical complex $K$ for each $r$.
  45 KB (7,255 words) - 03:59, 29 November 2015
• Products
  More complex outcomes result from attaching to every point of $X$ a copy of $Y$: Simplicial complexes have proven to be the easiest to deal with, until now. The proble
  44 KB (7,951 words) - 02:21, 30 November 2015
• The topology of data
  ...three points are “close”, we add a face, etc. The result is a [[simplicial complex]] that approximates the [[manifold]] M behind the point cloud. More: [[Topo ...puter program that determines the topological features of multidimensional geometric figures that represent data via [[point cloud]]s. Working with JPlex, we ha
  9 KB (1,431 words) - 16:57, 20 February 2011
• Products of complexes
  More complex outcomes result from attaching to every point of $X$ a copy of $Y$: Simplicial complexes have proven to be the easiest to deal with, until now. The proble
  16 KB (2,892 words) - 22:39, 18 February 2016
• Discrete forms and cochains
  *$K$ is an oriented simplicial complex, and '''Proposition.''' The $k$-cochains on complex $K$ form a vector space denoted by $C^k=C^k(K)$.
  34 KB (5,619 words) - 16:00, 30 November 2015
• Robustness of topology: persistence
  The result is a [[simplicial complex]] $K$ for each $r$. ...rm a sequence called [[filtration]] as a sequence of "nested" [[simplicial complex]]es:
  12 KB (2,000 words) - 22:54, 5 April 2014
• Students research projects
  ...three points are “close”, we add a face, etc. The result is a [[simplicial complex]] that approximates the [[manifold]] M behind the point cloud. More: [[Topo #John Stonestreet, Charlie Lowe, ''Geometric Study of the Pattern of Microscopic Hair Development on the Wings of the Mu
  11 KB (1,674 words) - 23:20, 25 October 2011
• Computational Homology by Kaczynski, Mischaikow, Mrozek
  ...ology theory is developed for cubical complexes instead of the traditional simplicial complexes as necessary for studying digital images. I liked this approach ( **2.1 [[cubical complex|Cubical Sets]]
  5 KB (616 words) - 14:03, 6 October 2016
• Approaches to image analysis
  *CGAL, Computational Geometry Algorithms Library [8] and Simplicial Homology for GAP [26] are collections of C++ code. ...used to represent its tunnels. However, this representation fails in more complex settings such as porous material. The representation of tunnels in 3D will
  13 KB (2,018 words) - 13:55, 12 May 2011
• Computational topology
  *''Digital Geometry: Geometric Methods for Digital Image Analysis'' by Klette and Rosenfeld is quite advan *[[Cell complex]]es and [[simplicial complex]]es
  5 KB (748 words) - 19:24, 31 January 2015
• Simplex
  A ''geometric $n$-simplex'' in ${\bf R}^{n+1}$ is defined as the [[convex hull]] (the set If we treat the simplex as a [[cell complex]], its topology is very simple:
  3 KB (397 words) - 22:24, 3 September 2011
• Topology for Computing by Zomorodian
  ...[[homology]], of geometric objects. The methods apply only to [[simplicial complex]]es. This is sufficient to cover [[point cloud]]s and [[topological data an **The [[Morse-Smale Complex]] Algorithm
  971 bytes (121 words) - 04:00, 2 May 2011
• Homology
  ...rithm]] is extended to process nD [[cell complex]]es (simplicial, cubical, geometric, or CW-). In general, cell complexes are made of cells that can have an arb Examples of such A and B are: two vertices in the same component of any cell complex ($k = 0$); two meridians of a torus, but not a meridian and the equator ($k
  8 KB (1,388 words) - 14:03, 1 June 2014
• Hodge duality of multivectors
  But does this isomorphism have any geometric meaning? '''Theorem.''' In a tangent space of a simplicial or a cubical complex $K$, every (non-zero) chain is represented by a single (non-zero) multivect
  3 KB (488 words) - 12:34, 14 August 2015