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• Hodge star operator
  #REDIRECT [[Discrete Hodge star operator]]
  42 bytes (5 words) - 03:46, 21 April 2013
• Discrete Hodge star operator
  Hodge star is a [[linear operator]] on the [[cochain complex]]: $$\star : C^k(K) \rightarrow C^{n-k}(K^*),$$
  5 KB (825 words) - 20:02, 25 April 2013

Page text matches

• Diffusion
  $$d_t U(a)=- F^\star(\partial a)$$ $$d_t U(a) = −(d_x F^\star)(a).$$
  39 KB (6,850 words) - 15:29, 17 July 2015
• PDEs
  $$d_t U(\alpha)=- F^\star(\partial \alpha)$$ $$d_t U(\alpha) = −(d_x F^\star)(\alpha).$$
  44 KB (7,469 words) - 18:12, 30 November 2015
• More metric complexes
  ...n extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$. ...n ''arbitrary'' dimension, the first step in the construction its dual $K^\star$ is to choose the ''dimension'' $n$. The rule remains the same:
  21 KB (3,445 words) - 13:53, 19 February 2016
• Metric tensor
  ...at $A$ of $K$ is a submodule of $C_1(K)$ generated by the $1$-dimensional star of the vertex $A$: ...ods. These rods (connected by a new set of hinges) form a new complex $K^{\star}$.
  42 KB (7,131 words) - 17:31, 30 November 2015
• Geometric complexes
  ...n extra set of rods (and hinges) that form a new, Hodge-dual, complex $K^{\star}$. ...K$ of arbitrary dimension, the first step in the construction its dual $K^\star$ is to choose the ''ambient dimension'' $n$. The rule remains the same:
  20 KB (3,354 words) - 17:37, 30 November 2015
• Metric complexes
  ...at $A$ of $K$ is a submodule of $C_1(K)$ generated by the $1$-dimensional star of $A$: ...connected by a new set of hinges) form a new complex '''denoted''' by $K^{\star}$.
  35 KB (5,871 words) - 22:43, 7 April 2016
• Geometry
  ...at $A$ of $K$ is a submodule of $C_1(K)$ generated by the $1$-dimensional star of the vertex $A$: ...ods. These rods (connected by a new set of hinges) form a new complex $K^{\star}$.
  41 KB (6,928 words) - 17:31, 26 October 2015
• Diffusion equation
  $$d_t U(\alpha)= -F^\star(\partial \alpha)$$ $$d_t U(\alpha) = -(d_x F^\star)(\alpha).$$
  35 KB (5,917 words) - 12:51, 30 June 2016
• Elementary PDEs
  *$p=A^\star,q=B^\star$ are the two pipes from $a$, left and right. $$d_t U(a)=-\big( F^\star(A)-F^\star(B) \big) = F^\star(A)-F^\star(B),$$
  16 KB (2,843 words) - 21:41, 23 March 2016
• Calculus of chain maps
  Recall also that the domain ${\mathbb R}^\star$ is a full copy of the domain ${\mathbb R}$ with the chains and the boundar ...{\mathbb R}_x^\star \text{ and } {\mathbb R}_y \text{ vs. } {\mathbb R}_y^\star.$$
  41 KB (7,344 words) - 12:52, 25 July 2016
• Introduction to discrete calculus
  ...}$ with (possibly different) lengths. It is '''denoted''' by ${\mathbb R}^\star$. ...edge $a$ in ${\mathbb R}$ corresponds to a node $a^\star$ in ${\mathbb R}^\star$; and
  40 KB (6,983 words) - 19:24, 23 July 2016
• Wave equation
  ...e between the centers of the springs has length $\Delta x^\star, \Delta y^\star$. We think of $u$ as a form of degree $0$ -- with respect to $x,y$. $$u' '=\star d_x\star d_xu = \Delta u.$$
  10 KB (1,775 words) - 02:40, 9 April 2016
• Diffusion with various geometry
  The [[Hodge star operator]] has been defined as a [[linear operator]] between the primal and $$\star = \star ^m:C^m(K)\rightarrow C^{n-m}(K^*).$$
  13 KB (2,121 words) - 16:33, 7 June 2013
• Geometric Hodge duality
  ...pological" Hodge duality. Consider this ''Hodge duality diagram'', where $\star$ stands for [[Hodge duality of differential forms]]: &\ua{\star} & \ne & \da{\star} & \\
  5 KB (867 words) - 13:24, 19 May 2013
• Codifferential
  Consider this ''Hodge duality diagram'', where $\star$ stands for [[Hodge duality of differential forms]]: ...& \da{\star} & \ne & \da{\star} & & & & \da{\star} \\
  4 KB (532 words) - 00:15, 26 April 2013
• Links
  Then the preservation of the material in cell $\sigma=A^\star$, where $A$ is the dual vertex, is given by $$d_t M(A,t) = −\int_{\partial A^\star} \star F(·,t) + S(A,t).$$
  6 KB (998 words) - 12:40, 31 August 2015
• Hodge duality of differential forms
  ...ed the ''Hodge duality operator'', or Hodge star operator, or simply the $\star$-operator: $$\star \colon \Omega^k \rightarrow \Omega^{2-k},$$
  8 KB (1,072 words) - 17:59, 24 April 2013
• Freshman's introduction to discrete calculus
  *an edge $a$ in the domain corresponds to a new node $a^\star$; and *a node $A$ in the domain corresponds to a new edge $A^\star$. $\\$
  42 KB (7,443 words) - 14:18, 1 August 2016
• Second derivative and the Laplacian
  ...ackrel{\star}{\longmapsto} f' \stackrel{d}{\longmapsto} f' ' dx \stackrel{\star}{\longmapsto} f' '.$$ What is $\star df$ in the discrete case?
  4 KB (608 words) - 13:13, 28 August 2015
• Discrete Hodge star operator
  Hodge star is a [[linear operator]] on the [[cochain complex]]: $$\star : C^k(K) \rightarrow C^{n-k}(K^*),$$
  5 KB (825 words) - 20:02, 25 April 2013

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