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• 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
• 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
• 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
• 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

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