Search results

Create the page "Group of chains of nodes" on this wiki! See also the search results found.

• Homology groups of graphs
  ==The algebra of plumbing== ...pology of the graph. Next, we will pursue this analysis via a certain kind of ''algebra''. We introduce this algebra with the following metaphor:
  28 KB (4,685 words) - 17:25, 28 November 2015
• Algebra on graphs
  The examples of graphs shown come as ''networks'' of bridges, atoms in a molecule, related individuals, or computers. '''Definition.''' A ''graph''<!--\index{graph}--> $G =(N,E)$ consists of two finite sets:
  36 KB (6,177 words) - 02:47, 21 February 2016
• The algebra of oriented cells
  ==Are chains just “combinations” of cells?== The nature of the problem is topological because the ''rubber'' band is allowed to stretc
  36 KB (6,395 words) - 14:09, 1 December 2015
• The algebra of cells
  ...dy such topological concepts as path-connectedness. ''Are curves also made of tiles?'' Such a curve would look like this: ...e in the usual topological sense! After all, a curve is a continuous image of the $1$-dimensional interval $[a,b]$. In that sense, a curve is supposed to
  34 KB (5,644 words) - 13:35, 1 December 2015
• Introduction to discrete calculus
  We next examine the combined ''domain'' of these new functions. We are to make the usual domain of functions -- the reals ${\bf R}$ -- discrete. We divide this set into unit
  40 KB (6,983 words) - 19:24, 23 July 2016
• Simplicial homology
  Recall that a chain complex<!--\index{chain complex}--> is a sequence of vector spaces and linear operators: This property allows us to study the ''homology'' of this chain complex:
  31 KB (5,170 words) - 13:44, 1 December 2015
• Calculus on graphs
  ==The algebra of plumbing, continued== We think of a graph as a plumbing system that consists of a network of pipes and joints, and each joint and each pipe is equipped with an on-off s
  16 KB (2,578 words) - 00:14, 18 February 2016
• Cochains on graphs
  ==The algebra of plumbing, continued== We think of a graph as a plumbing system that consists of a network of pipes and joints, and each joint and each pipe is equipped with an on-off s
  15 KB (2,523 words) - 18:08, 28 November 2015
• Simplicial complexes and maps
  A graph<!--\index{graph}--> is pure data. It consists of two sets: *the nodes, say, $N=\{A, B, C, D\}$, representing some agents, and
  27 KB (4,625 words) - 12:52, 30 March 2016
• Topology Illustrated -- Index
  *[[0th homology group|0th homology group]] *[[1st homology group|1st homology group]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Chain maps
  Let's consider maps of the “cubical circle”<!--\index{circle}--> to itself ...d then find an appropriate representation $g:K\to L$ for each $f$ in terms of their cells. More precisely, we are after
  47 KB (8,115 words) - 16:19, 20 July 2016
• Maps of graphs
  ...ogy}--> would be incomplete without considering continuous transformations of objects, also known as maps<!--\index{maps}-->. ...aces doesn't reveal the complete picture. Instead, we want to ''track each of these features'' and record what map does to it.
  29 KB (5,042 words) - 17:57, 28 November 2015
• Discrete forms and cochains
  ...the definite integral over $I$ is often thought of as a function the input of which is any integrable ''function'' $f$ while the output is a real number. ...iemann integral is introduced in calculus as the limit of the Riemann sums of $f$. The student then discovers that this function is ''linear'':
  34 KB (5,619 words) - 16:00, 30 November 2015
• Simplicial maps and chain maps
  ...wo topological spaces. To fully understand this map, we want to track each of the topological features, i.e., the homology classes, in $X$ as they are tr ...e want 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
  ...wo topological spaces. To fully understand this map, we want to track each of the topological features, i.e., the homology classes, in $X$ as they are tr ...e want to see map $h$ as a “realization”<!--\index{realization}--> $h=|f|$ of a “simplicial” map $f:K\to L$ between these complexes:
  34 KB (5,929 words) - 03:31, 29 November 2015
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  ...In contrast to traditional goals of finding an accurate ''discretization'' of conventional multivariate calculus, discrete calculus establishes a separat ...complex]] in terms of $(k-1$-cells, you also know the exterior derivative of all discrete [[differential forms]] ([[co-chain]]s). So, you know calculus.
  11 KB (1,663 words) - 16:03, 26 November 2012