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- ==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
- ...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 simplices34 KB (5,929 words) - 03:31, 29 November 2015
- ==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
- <!--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
- <!--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
- 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 represented51 KB (9,162 words) - 15:33, 1 December 2015
- ==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
- [[image:cubical complex distorted.png|center]] [[image:cubical complex bent.png|center]]42 KB (7,131 words) - 17:31, 30 November 2015
- *[[abstract simplicial complex|abstract simplicial complex]] *[[augmented chain complex|augmented chain complex]]16 KB (1,773 words) - 00:41, 17 February 2016
- <!--75-->[[image:cubical complex distorted.png| center]] <!--75-->[[image:cubical complex bent.png| center]]35 KB (5,871 words) - 22:43, 7 April 2016
- ...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
- [[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
- '''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
- 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
- '''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
- 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 either42 KB (7,005 words) - 03:10, 30 November 2015
- [[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 satisfie27 KB (4,329 words) - 16:02, 1 September 2019
- 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
- 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 either41 KB (6,926 words) - 02:14, 21 October 2015
- ...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 ed41 KB (6,854 words) - 15:05, 28 October 2011
- 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
- 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 proble44 KB (7,951 words) - 02:21, 30 November 2015
- ...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 ha9 KB (1,431 words) - 16:57, 20 February 2011
- 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 proble16 KB (2,892 words) - 22:39, 18 February 2016
- *$K$ 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
- 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
- ...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 Mu11 KB (1,674 words) - 23:20, 25 October 2011
- ...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
- *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 will13 KB (2,018 words) - 13:55, 12 May 2011
- *''Digital Geometry: Geometric Methods for Digital Image Analysis'' by Klette and Rosenfeld is quite advan *[[Cell complex]]es and [[simplicial complex]]es5 KB (748 words) - 19:24, 31 January 2015
- 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]], 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]] Algorithm971 bytes (121 words) - 04:00, 2 May 2011
- ...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 ($k8 KB (1,388 words) - 14:03, 1 June 2014
- 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) multivect3 KB (488 words) - 12:34, 14 August 2015