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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
• 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
• 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
• 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
• Advanced Topology: midterm
  ..., \gamma \in \Omega ^1({\bf R}^3)$ are linearly independent. Assuming $dd(\alpha)=dd(\beta)=dd(\gamma)=0$, prove that $dd(\psi ^1)=0$. ...wedge \psi ^2$, where the latter is equal to $1$ on a single square, say $\alpha$, parallel to the $xy$-plane and equal to $0$ elsewhere.
  2 KB (282 words) - 16:11, 14 March 2013
• Algebraic operations with forms and cohomology
  We will assume the following: $R$ is any [[cubical complex]], i.e., a set of cells such that if cell $s \in R$, then all of its [[boun on $\alpha$.
  8 KB (1,289 words) - 15:11, 9 October 2012
• Transformations
  '''Example (rotation with stretch-shrink).''' Let's consider a more complex function: $$V_{\perp}=||V||\sin \alpha,$$
  46 KB (7,625 words) - 13:08, 26 February 2018
• Integration of differential forms of degree 0 and 1
  ...screte manifold = a [[combinatorial manifold]] = a manifold-like [[cubical complex]]. ...$p(t)=x$, $q(s)=x$, then we can assume $p'(t) = \alpha q'(s)$, for some $\alpha(x)$, for every $x$ on $C$. Then
  12 KB (1,906 words) - 17:44, 31 December 2012
• Vector fields
  $$F_{||}=||F||\cos \alpha,$$ where $\alpha$ is the angle of $F$ with $D$.
  91 KB (16,253 words) - 04:52, 9 January 2019
• Approaches to image analysis
  *Alpha Shapes [5] runs only in Linux/Unix/Sun and only computes Betti numbers of 3 ...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
• Acyclic models
  $Ch(\mathcal{Mod})$ be the category of [[chain complex]]es of $R$-modules. *$\mathcal{F}=S_k$ is the functor of the $k$-th term of the singular chain complex;
  8 KB (1,126 words) - 15:32, 14 July 2013
• Calculus on chains
  .... Combined with the [[boundary operator]] $\partial$ they form the [[chain complex]] $\{C_*,\partial\}$ of $K$: But this is simply the [[cochain complex]] $\{C^*,d\}$:
  4 KB (556 words) - 14:03, 30 March 2013
• The derivative
  What about the latter, “touching”, line? It is much more complex. ...with which the cannonball leaves the muzzle -- no matter what the angle, $\alpha$, is. That's where the ''initial'' horizontal and the vertical velocities c
  75 KB (13,000 words) - 15:12, 7 December 2018
• Calculus of chain maps
  Suppose we have two copies of the complex ${\mathbb R}$, ${\mathbb R}_x$ and ${\mathbb R}_y$, possibly representing t $$d(\alpha f+\beta g)=\alpha df+\beta dg,$$
  41 KB (7,344 words) - 12:52, 25 July 2016
• Tangent cone
  A $2$-manifold is a [[surface]] and as a cubical complex it will look like this, compared to a $1$-manifold: $$TC_x(A)={\bf R} \cdot TC_x(A)=\{\alpha v: v \in TC_x(A),\alpha \in {\bf R} \}.$$
  5 KB (882 words) - 02:14, 26 March 2013
• Inner product
  ...cations when some measurements intended to evaluate the difference between complex entities, such molecules, may be more important than others. $$\cos \alpha = \frac{< u, v >}{ \lVert u \rVert \cdot \lVert v \rVert }$$
  4 KB (749 words) - 20:12, 1 May 2013
• JPlex
  ...ata as a list of vertices. After defining an epsilon value (similar to the alpha-value in [[persistent homology]]), PLEX calculates the appropriate edges fo ...logy]] is computed for [[point cloud]]s or [[filtration]]s of [[simplicial complex]]es.
  2 KB (231 words) - 03:21, 30 March 2011
• Visual image search engines
  ...nd if you can't solve a simpler problem how can you expect to solve a more complex one? I tried this color image. These are the tags: <em>texture, red, food, FAQ: "will be launching their alpha version to a small group of users on February 28th, 2006".
  15 KB (2,424 words) - 19:20, 14 June 2011
• Image search
  ...nd if you can't solve a simpler problem how can you expect to solve a more complex one? I tried this color image. These are the tags: <em>texture, red, food, FAQ: "will be launching their alpha version to a small group of users on February 28th, 2006".
  17 KB (2,793 words) - 17:40, 14 September 2008

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