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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
• Navier–Stokes equations
  $$\frac{\partial v}{\partial t} = − \star (v ∧ \star dv) + \frac{1}{2}d||v||^2 − dp + \mu d^∗dv. $$ $$\frac{\partial v}{\partial t} + \star (v ∧ \star dv) - \frac{1}{2}d||v||^2 = − dp + \mu d^∗dv. $$
  5 KB (742 words) - 03:32, 30 August 2012
• Calculus I, the discrete version
  *an edge $a=[x,x+h]$ corresponds to the node $a^\star=x+h/2$; and *a node $x$ corresponds to the edge $x^\star=[x-h/2,x+h/2]$. $\\$
  64 KB (11,521 words) - 19:48, 22 June 2017
• Rating votes
  ...r\quad \star\star\star\quad \star\star\star\star\quad \star\star\star\star\star\ .$$
  9 KB (1,553 words) - 06:12, 22 June 2016
• Triangulations
  ...n a simplicial complex $K$ and a vertex $A$ in $K$, the ''star''<!--\index{star}--> of $A$ in $K$ is the collection of all simplices in $K$ that contain $A The ''open star''<!--\index{open star}--> is the union of the insides of all these cells:
  30 KB (5,172 words) - 21:52, 26 November 2015
• Wave equation from Maxwell equations
  In terms of the [[Hodge duality|Hodge star operator]], the ''constitutive relations'' are: $$D=\varepsilon_0 \star E,$$
  4 KB (655 words) - 14:51, 13 July 2012
• Higher order votes
  ...r\quad \star\star\star\quad \star\star\star\star\quad \star\star\star\star\star\ .$$
  41 KB (6,942 words) - 05:04, 22 June 2016
• Discrete forms and cochains
  ...uad \star\star\star\quad \star\star\star\star\quad \star\star\star\star\star\ .$$
  34 KB (5,619 words) - 16:00, 30 November 2015
• Forms
  ...of $K$ is the set of $1$-chains over $R$ generated by the $1$-dimensional star of the vertex $A$: ...the edges adjacent to $A$, we can also think of all $1$-''chains'' in the star of $A$ as directions at $A$. They are subject to algebraic operations on ch
  13 KB (2,459 words) - 03:27, 25 June 2015
• Hodge duality of multivectors
  $$\star: \Lambda _k(V) \to \Lambda _{n-k}(V),$$ $$\star (p_{s(1)} \wedge p_{s(2)}\wedge ... \wedge p_{s(k)}):= (-1)^{\pi(s)}p_{s(k+
  3 KB (488 words) - 12:34, 14 August 2015
• Modeling with discrete exterior calculus
  ...{n}^{(i,j)})=-\frac{k}{4} \Big[ d_{x}(\star d_{x}(T_{n}^{(i-1,j)}))+d_{y}(\star d_{y}(T_{n}^{(i,j-1)})) \Big]\end{equation} where $d_{x}$ denotes an exterior derivative with respect to $x$ and $\star$ denotes the use of the Hodge Duality.
  31 KB (5,254 words) - 17:57, 21 July 2012
• Maxwell equations
  In terms of the [[Hodge duality|Hodge star operator]], the ''constitutive relations'' are: $$D=\varepsilon_0 \star E,$$
  6 KB (922 words) - 00:30, 9 April 2016
• ODEs
  ...K={\mathbb R}$ with the standard geometry: $|a|=1$ for all $a\in K,a\in K^\star$. for a given $a$. Here $r' '=\star d \star d r$, where $\star$ is the Hodge star operator of $K$.
  47 KB (8,415 words) - 15:46, 1 December 2015
• Topology Illustrated -- Index
  *[[Hodge star operator|Hodge star operator]] *[[open star|open star]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Maps of polyhedra
  ..., we realize that we are talking about $a$ and $f(a)$ located within the ''star'' of the corresponding vertex! Recall that given a complex $K$ and a vertex $A$ in $K$, the star<!--\index{star}--> of $A$ in $K$ is the collection of all cells in $K$ that contain $A$:
  51 KB (9,162 words) - 15:33, 1 December 2015
• Hodge dual
  ...- k)$-cochain is defined by its value on the [[Hodge duality|dual cell]] $\star a$ by $$\frac{1}{|\star a|}<\star \phi, \star a> = \frac{1}{|a|}<\phi, a>$$
  1,003 bytes (159 words) - 14:01, 27 July 2012
• Hodge duality of cubical forms
  Of course, the star operator is now extended to [[chains]] by [[linearity]]. With this arrangement we also ensure that we have $\star \star =Id$.
  7 KB (1,114 words) - 18:10, 27 August 2015
• Axioms of calculus
  ...we start with the set of all cells that contain $A$ called the (open) ''[[star]] of vertex'' $A$, Then the ''Hodge star operators'' are linear operators on the chain complexes:
  6 KB (1,124 words) - 14:17, 4 August 2013
• Convergence of the discrete to the continuous
  where $A_n$ is in the star of $A$ and $B_n$ is in the star of $B$ in $K_{nm}$, and where $A_n$ is in the star of $A$ and $B_n$ is in the star of $B$ in $K_{nm}$, and
  21 KB (3,664 words) - 02:02, 18 July 2018
• Manifolds
  ...erse may be ''curved''. For example, the observation that the light from a star passing the sun deviates from a straight line may be considered as evidence ...definition. First, by the above theorem, $K$ has to be a graph. Since the star of a vertex with more than one adjacent edge isn't homeomorphic to the open
  51 KB (8,919 words) - 01:58, 30 November 2015
• Codiffferential
  $$\delta = (-1)^{nk + n + 1}s\, {\star d\star} = (-1)^k\,{\star^{-1}d\star},$$
  382 bytes (57 words) - 03:15, 5 October 2012
• Tangent bundles
  ...K$ is the set of $1$-chains over ring $R$ generated by the $1$-dimensional star of the vertex $A$: ...the edges adjacent to $A$, we can also think of all $1$-''chains'' in the star of $A$ as directions at $A$. They are subject to algebraic operations on ch
  16 KB (2,753 words) - 13:55, 16 March 2016
• Nerve of cover
  Given a simplicial complex $K$ and a vertex $A$ in $K$, the ''star'' of $A$ in $K$ is the collection of all simplices in $K$ that contain $A$: ...easy to prove that this is a subcomplex of $K$. We will also use the word "star" for the union of ${A}$ and the interiors of all the simplices that contain
  8 KB (1,389 words) - 13:35, 12 August 2015
• Combinatorial cell complexes
  ...tion. First, by the above theorem, $K$ has to be a graph. Furthermore, the star of a vertex with more than one adjacent edge isn't homeomorphic to the open Of course, we recognize this collection of simplices as the star of vertex $A$.
  34 KB (5,710 words) - 22:27, 18 February 2016
• Advanced Topology: exercises
  #A set $Y \subset {\bf R}^n$ is called star-shaped if there is $a\in Y$ such that for any $x \in Y$ the segment from $x #Define the Hodge $\star$ operator for discrete forms on the plane. Give examples.
  9 KB (1,487 words) - 18:18, 9 May 2013
• Discrete forms
  ...of $K$ is the set of $1$-chains over $R$ generated by the $1$-dimensional star of the vertex $A$: ...the edges adjacent to $A$, we can also think of all $1$-''chains'' in the star of $A$ as directions at $A$. They are subject to algebraic operations on ch
  35 KB (6,055 words) - 13:23, 24 August 2015
• Advanced Topology -- Spring 2013 -- final exam
  #A set $Y \subset {\bf R}^n$ is called star-shaped if there is $a\in Y$ such that for any $x \in Y$ the segment from $x #A set $Y \subset {\bf R}^n$ is called star-shaped if there is $a\in Y$ such that for any $x \in Y$ the segment from $x
  3 KB (532 words) - 15:09, 8 May 2013
• Calculus on cubical complexes
  ...of $K$ is the set of $1$-chains over $R$ generated by the $1$-dimensional star of the vertex $A$: ...the edges adjacent to $A$, we can also think of all $1$-''chains'' in the star of $A$ as directions at $A$. They are subject to algebraic operations on ch
  36 KB (6,218 words) - 16:26, 30 November 2015
• Topology II -- Spring 2014
  ...e chapters below within a week. The most current material is marked with $\star$. #[[Homology theory]] $\star$
  3 KB (445 words) - 16:04, 20 May 2014
• Topology I -- Fall 2013
  ...e chapters below within a week. The most current material is marked with $\star$. $\star$
  7 KB (881 words) - 19:04, 10 December 2013
• Forms in Euclidean spaces
  ...ertex $A$ consists, as before, of the edges adjacent to $A$, i.e., the $1$-star $St(A)$. However, this time the algebra of $T_A(K)$ doesn't come from the $ supplies the star with algebra of edges.
  44 KB (7,778 words) - 23:32, 24 April 2015

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