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• Fundamental group
  Here, if one thinks of the circle as a subset of the complex plane, the projection is given by $\pi (x)=e^{2\pi ix}$. In particular, the standard $n$-times wrapping loop $\alpha _n$ of the circle is lifted to $\gamma_n$ given by $\gamma_n(s)=ns$:
  10 KB (1,673 words) - 18:23, 2 December 2012
• ODEs
  Given a complex $K$, this is the most elementary ODE with respect to a $0$-form $f$: ...model ''motion''. Our domain is then the standard $1$-dimensional cubical complex $K={\mathbb R}$ and we are to study differential forms over ring $R={\bf R}
  47 KB (8,415 words) - 15:46, 1 December 2015
• Applications of ODEs
  *Case #3: complex conjugate roots. $$x = C e^{\alpha t} \cos(\beta t) + K e^{\alpha t} \sin(\beta t),$$
  50 KB (8,692 words) - 14:29, 24 November 2018
• Forms in Euclidean spaces
  The idea is as follows. Suppose cell complex $K$ is realized in ${\bf R}^n$. Then the tangent space $T_A(K)$ at vertex $ Let's review. The complex ${\mathbb R}^n$ comes with a standard orientation of all edges -- along the
  44 KB (7,778 words) - 23:32, 24 April 2015
• Computing homology
  Recall that given a [[cell complex]] $K$, a $k$-[[the algebra of chains|chain]] is a "formal" [[linear combina ...hains of different dimensions in order to capture the topology of the cell complex. This relation is given by the ''[[boundary operator]]''.
  26 KB (4,370 words) - 21:55, 10 January 2014
• The algebra of cells
  <!--150-->[[image:Cubical complex in 3d.png|center]] Now, what about boundaries of more complex objects?
  34 KB (5,644 words) - 13:35, 1 December 2015
• Diffusion
  ...one cell to the next. So, $F=F(p,t)$ is a $(n,1)$-form, but over the dual complex. ...cs (the generalized Hodge star) and the simulations for progressively more complex situations.
  39 KB (6,850 words) - 15:29, 17 July 2015
• The algebra of cubical cells
  <!--150-->[[image:Cubical complex in 3d.png| center]] Now, what about boundaries of more complex objects?
  46 KB (7,844 words) - 12:50, 30 March 2016
• The algebra of oriented cells
  ...nation of directions and our evaluation of the topology of a given cubical complex should remain the same. ...a cubical complex $K$ is a “formal linear combination of $k$-cells” in the complex:
  36 KB (6,395 words) - 14:09, 1 December 2015
• Discrete forms
  ...se here to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple pieces. We deno ...they “look” identical. Frequently, one just assigns numbers to cells in a complex as we did above.
  35 KB (6,055 words) - 13:23, 24 August 2015
• Calculus on cubical complexes
  ...for now to concentrate on the ''cubical grid'', i.e., the infinite cubical complex acquired by dividing the Euclidean space into small, simple pieces (cubes). ...they “look” identical. Frequently, one just assigns numbers to cells in a complex as we did above.
  36 KB (6,218 words) - 16:26, 30 November 2015
• Differential calculus
  ...but let's review the tools at our disposal that allow us to deal with more complex functions. Examining the graph reveals that the maximum value lies somewher '''Example (quadratic polynomials).''' Things become much more complex if we need to analyze a quadratic function,
  84 KB (14,321 words) - 00:49, 7 December 2018
• Hodge duality of cubical forms
  [[image:dual complex dim 1.png|center]] ...x]] $K$ then the set of all of the duals of the cells of $K$ is the ''dual complex'' $K^*$.
  7 KB (1,114 words) - 18:10, 27 August 2015
• Elementary PDEs
  *the ''topology'' of the cell complex $L$ of the objects and springs, *the ''geometry'' given to that complex such as the lengths of the springs, and
  16 KB (2,843 words) - 21:41, 23 March 2016
• Manifolds
  ....''' Prove that for any ''countable'' ordinal $\alpha$, pasting together $\alpha$ copies of $[0,1)$ gives a space still homeomorphic to $[0,1)$. $$\alpha = \{U_y:y \in F\},\ \beta = \{V_y:y \in F\},$$
  51 KB (8,919 words) - 01:58, 30 November 2015
• Modeling with discrete exterior calculus
  ...the heat spreads though a grid or lattice of cells. These form a cellular complex composed of 0-cells, 1-cells, and 2-cells. In discretizing the heat equatio ...his combination of "rooms" and "walls" (and "columns") is called [[cubical complex]]. This approach is different from the numerical approach to the heat equat
  31 KB (5,254 words) - 17:57, 21 July 2012
• The algebra of oriented cubes
  ...nation of directions and our evaluation of the topology of a given cubical complex should remain the same. ...a cubical complex $K$ is a “formal linear combination of $k$-cells” in the complex:
  32 KB (5,480 words) - 02:23, 26 March 2016
• Advanced Topology: exercises
  ..., \gamma \in \Omega ^1({\bf R}^3)$ are linearly independent. Assuming $dd(\alpha)=dd(\beta)=dd(\gamma)=0$, prove that $dd(\psi ^1)=0$. ...example of a graph that cannot be represented by a one-dimensional cubical complex.
  9 KB (1,487 words) - 18:18, 9 May 2013
• Classes of functions
  '''Example.''' Constant functions are convenient building blocks for more complex functions. This is a familiar example of how we build from three constant f '''Example (quadratic polynomials).''' Things become much more complex if we need to analyze a quadratic function,
  143 KB (24,052 words) - 13:11, 23 February 2019
• Homology software
  ...sted below compute [[homology groups]] of [[cell complex]]es, [[simplicial complex]]es etc in a variety of applied scenarios, including [[persistence]]. ...tation, no support, etc. Proceed at your own risk. The commercial ones are Alpha Shapes by GeoMagic and Iris by [[Ayasdi]].
  4 KB (648 words) - 03:16, 30 March 2011

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