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• Topology of graphs
  Here we take our first step toward ''algebraic topology''. *verifying that it is path-connected<!--\index{path-connectedness}--> requires testing ''infinite
  25 KB (4,214 words) - 16:08, 28 November 2015
• Compact spaces
  ...nnectedness}--> (under a continuous map<!--\index{continuous function}-->) is a path-connected. The second main topological theorem from Calculus 1 is:
  19 KB (3,207 words) - 13:06, 29 November 2015
• Cubical complex
  Suppose the Euclidean space ${\bf R}^N$ is given and so is its cubical grid ${\bf Z}^N$. Suppose also that we have its decomposition $ ...of cubical cells $K\subset {\mathbb R}^N$ for which the boundary operator is well defined. This requires us to include all “faces” of the cells alre
  29 KB (4,800 words) - 13:41, 1 December 2015
• Computing homology
  ..., typically, integer coefficients. Then the set of all $k$-chains $C_k(K)$ is an [[abelian group]] with respect to chain addition generated by the $k$-ce ...tyle\sum_i s_i \sigma_i \colon s_i \in {\bf Z}, \sigma_i {\rm \hspace{3pt} is \hspace{3pt} a \hspace{3pt}} k{\rm -cell \hspace{3pt} in \hspace{3pt}} K \r
  26 KB (4,370 words) - 21:55, 10 January 2014
• Simplicial maps and chain maps
  A graph map<!--\index{graph map}--> $f:K\to L$ is a function between graphs $K,L$ that satisfies, for each edge $e$, either: *1. (cloning) $f(e)$ is an edge $g$ and $f$ takes the end-points of $e$ to the end-points of $g$; o
  34 KB (5,897 words) - 16:05, 26 October 2015
• Robustness of topology of digital images and point clouds by Saveliev
  ''Robustness of topology of digital images and point clouds'' by [[Peter Saveliev]] ...ements of the filtration without double count. The second step of analysis is to discard the features that lie outside the user's choice of the acceptabl
  27 KB (4,547 words) - 04:08, 6 November 2012
• Modeling with discrete exterior calculus
  ...erential forms. Our main conclusion is that the isotropy of heat on a grid is heavily dependant upon the geometry of the grid. For example, the square gr ...T}{\partial t}=-k\nabla^{2} T$ as the [[Heat transfer |heat equation]]. It is continuous and its solution relies upon both initial and boundary condition
  31 KB (5,254 words) - 17:57, 21 July 2012
• Simplicial maps
  A graph map<!--\index{graph map}--> $f:K\to L$ is a function between graphs $K,L$ that satisfies, for each edge $e$, either: *1. (cloning) $f(e)$ is an edge $g$ and $f$ takes the end-points of $e$ to the end-points of $g$; o
  34 KB (5,929 words) - 03:31, 29 November 2015
• Selection votes
  ...seen as mutually exclusive but ''all'' of them may come true. What happens is determined by the probabilities assigned to the primary events. These conve ...epresents the lottery when either is equally likely to appear while “hail” is impossible. The probabilities of the three events give the vector $(\tfrac{
  24 KB (3,989 words) - 01:56, 16 May 2016
• Applied Topology and Geometry
  #REDIRECT[[Topology Illustrated]] ''Applied Topology and Geometry'' by [[Peter Saveliev]]
  16 KB (2,088 words) - 16:37, 29 November 2014
• Test
  ''Applied Topology and Geometry'' by [[Peter Saveliev]] ...appreciate your comments. If you are a beginner, you might want to start [[Topology Illustrated|here]] instead.
  16 KB (2,139 words) - 23:01, 9 February 2015
• Lefschetz numbers in control theory
  The goal is to develop some applications of the Lefschetz fixed point theory techniques ...one, $x\in M$. The ''equilibrium'' set $C=\{x\in M:f(x)=x\}$ of the system is the set of fixed points of $f.$
  17 KB (3,052 words) - 22:12, 15 July 2014
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  This is an informal review... <center>'''[[Calculus is topology]].'''</center>
  11 KB (1,663 words) - 16:03, 26 November 2012
• Products of complexes
  The idea of the product may be traced to the image of a stack, which is a simple arrangement of multiple copies of $X$: We can think of it as if a copy of the $y$-axis is attached to every point on the $x$-axis. Or, we can think in terms of ''pro
  16 KB (2,892 words) - 22:39, 18 February 2016
• Calculus as a part of topology
  Topology, Algebra, and Geometry are disciplines within Mathematics. In calculus we u ...d Geometry. However, take a look at where this is all happening. The locus is the ''Euclidean space''. Such a space has three different types of structur
  13 KB (2,233 words) - 14:41, 20 February 2015
• More about manifolds
  Suppose the circle is centered at $0$ on the $xy$-plane. In other words, each point is identified with the one symmetric with respect to the $x$-axis. Then
  9 KB (1,542 words) - 19:58, 21 January 2014
• Nerve of cover
  ...ebraic invariants. But how do you find this representation if all you have is a topological space, i.e., a collection of open sets. To solve this problem, let's start with a study of the topology of a [[simplicial complex]].
  8 KB (1,389 words) - 13:35, 12 August 2015
• Constructions
  ...ical sense. The Reflexivity Axiom, $A \sim A$, is: every spot of the sheet is glued to itself. The Symmetry Axiom, $A \sim B \Longrightarrow B \sim A$, b Thinking of a ''zipper'' is also appropriate:
  13 KB (2,270 words) - 22:14, 18 February 2016
• Algebra of differential forms
  For example, we see below that $\phi = x^2 dx + xy dy$ is a [[linear function]] of $dx$ and $dy$, non-linear for $x,y$. This is called ''[[additivity]]''.
  11 KB (1,947 words) - 18:14, 22 August 2015
• Equilibria of dynamical systems
  ...em]] is simply a map $F:S \to S$. Therefore the meaning of its equilibrium is simple: $F(a)=a$. It's a [[fixed point]]! ...ith an example outside the realm of physics, for a change. The application is to the existence of equilibria supply and demand in a simple market economy
  7 KB (1,251 words) - 15:00, 4 April 2014

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