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• Diffusion with various geometry
  ...defined as a [[linear operator]] between the primal and the dual [[cochain complex]]es: ...the ''discrete (geometric) Hodge star'' of $\phi$ is a cochain on the dual complex and it is defined by its values on the dual cells: for a $m$-[[chain]] $a$
  13 KB (2,121 words) - 16:33, 7 June 2013
• Axioms of chain complexes
  '''2. [[Refinement]]''': Refinement/subdivision doesn't change the chain complex. That is, if $(X,A)$ is a pair and $\gamma$ is an open cover of $X$, then t ...t space $P$ is acyclic. That is, the boundary operator $\partial$ of chain complex $C(P)$ of $P$ satisfies
  4 KB (592 words) - 14:13, 4 August 2013
• Combinatorial cell maps
  ...all possible ways complex $K=\{A,a,\alpha\}$ can be mapped to another cell complex. '''Exercise.''' Choose a different cell complex to represent $Y$ above in such a way that the projection is then a cell map
  31 KB (5,330 words) - 22:14, 14 March 2016
• Boundary operator
  #redirect[[Chain complex]] [[Image:1dim complex.jpg|center|150px]]
  5 KB (837 words) - 16:24, 1 June 2014
• Integration in dimension n
  Note: we can compute the volume of a complex figure G by putting it in a box and setting the density equal to zero in th What about more complex, curved domains? What if the domain of integration $G$ is represented as a
  33 KB (5,415 words) - 05:58, 20 August 2011
• Cell maps
  ...all possible ways complex $K=\{A,a,\alpha\}$ can be mapped to another cell complex. '''Exercise.''' Choose a different cell complex to represent $Y$ above in such a way that the projection is then a cell map
  42 KB (7,005 words) - 03:10, 30 November 2015
• Cubical calculus
  ...for now to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into cubes, ${\mathbb R}^n$. ...they “look” identical. Frequently, one just assigns numbers to cells in a complex as we did above. The difference is that these numbers aren't the coefficien
  25 KB (4,238 words) - 02:30, 6 April 2016
• Precalculus
  '''Theorem (Slope).''' The angle $\alpha$ between the $x$-axis and the line from $O$ to $P=(x,y)\ne O$ is given by i $$\tan \alpha =\frac{y}{x}.$$
  100 KB (16,148 words) - 20:04, 18 January 2017
• The gradient
  ...ctor attached to that point. This is just a clever way to visualize such a complex -- in comparison to the ones we have seen so far -- function. It's a ''loca '''Example.''' Three-dimensional vector fields are more complex. The one below is similar to the first example above:
  74 KB (13,039 words) - 14:05, 24 November 2018
• Dual cells and dual forms
  ...$(2-k)$-cell (dual) $\alpha^*$ with $\alpha^*$ centered at the center of $\alpha$. $$\alpha^{**}=\alpha$$
  2 KB (266 words) - 18:11, 27 August 2015
• Cochain complexes and cohomology
  Generally, if $K$ is a [[cubical complex]], $\dim C^k(K) = $ number of $k$-cells in $K$. Consider the [[cochain complex]]:
  17 KB (2,592 words) - 14:38, 14 April 2013
• Geometric Hodge duality
  ==Dual complex== [[image:metric tensor vs dual complex.png|center]]
  5 KB (867 words) - 13:24, 19 May 2013
• Convergence of the discrete to the continuous
  Suppose we have a sequence of copies of the standard cubical complex ${\mathbb R}$, ...\subset {\bf R}$ and suppose $f$ is integrable on $[A,B]$. Then, for every complex $K$ representing $[A,B]$ we define $g$ as the $1$-form acquired by integrat
  21 KB (3,664 words) - 02:02, 18 July 2018
• Applications of Computational Topology by Christopher Johnson
  ...s to bypass the computationally intensive process of building a simplicial complex. Using persistent homology allows us to locate important features in the im
  2 KB (282 words) - 16:46, 20 February 2011
• What shape of sword is best for cutting?
  ...hrough but, because the sword is rigid, the rest is also slowed down. This complex motion-control problem is routinely solved by a skillful swordsman. However ...help of the rotation matrix (shown at the top) for an angle of rotation $\alpha$:
  14 KB (2,504 words) - 14:59, 17 September 2019
• Modelling with discrete vector fields and forms
  ...[u,v]$, let $f(\alpha u+(1-\alpha)v)=\alpha f(u)+(1-\alpha)f(v)$ for all $\alpha \in [0,1]$; Dynamical systems are known for exhibiting a complex behavior even with a simple choice of $F$:
  9 KB (1,561 words) - 16:06, 27 August 2015
• Hodge decomposition
  *''[[co-exact]]'' if $\omega=\delta\alpha$ for some form $\alpha$; $$\phi = d\alpha +\delta \beta + \gamma \,$$
  1 KB (241 words) - 20:52, 13 March 2013
• Products
  More complex outcomes result from attaching to every point of $X$ a copy of $Y$: $$\alpha := \{V_b:b\in Y\}.$$
  44 KB (7,951 words) - 02:21, 30 November 2015
• Calculus of discrete differential forms
  Our definition, and the theorem, is applicable to any [[cubical complex]] $R \subset {\bf R}$ because, by definition, it is a set of cells such tha We need to find the value of the $2$-form $d \varphi$ on the $2$-cells $\alpha$ and $\beta$. We subtract vertically and horizontally, as before, and then
  9 KB (1,503 words) - 18:30, 22 August 2015
• Cup product
  where $\sigma$ is a $(p+q)$-[[simplex]] in the [[simplicial complex]] $X$ and $\sigma _{0,1, ..., p}$ is the $p$-th "front face" and $\sigma_{p For a [[cubical complex]] in the $n$-dimensional space, cochains are defined on the [[cube]]s:
  3 KB (460 words) - 20:57, 13 March 2013

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