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• Cohomology
  '''Definition.''' Let the ''dual''<!--\index{dual space}--> of $V$ be defined by Following the idea of this terminology, we add “co” to a word to indicate its dual. Such is the relation between chains and cochains (forms). In that sense, $
  45 KB (6,860 words) - 16:46, 30 November 2015
• Duality: forms as cochains
  '''Definition.''' We let the ''dual module''<!--\index{dual module}--> of $V$ to be ...spirit of this terminology, we might add “co” to any word to indicate its dual. Such is the relation between chains and cochains (forms). In that sense, $
  29 KB (4,540 words) - 13:42, 14 March 2016
• Metric tensor
  ...''dot product'' defined on $V={\bf R}^n$ or any other module with a fixed basis. For ...al. In other words, every inner product is a weighted dot product, in some basis.
  42 KB (7,131 words) - 17:31, 30 November 2015
• Metric complexes
  ...''dot product'' defined on $V={\bf R}^n$ or any other module with a fixed basis. For ...al. In other words, every inner product is a weighted dot product, in some basis.
  35 KB (5,871 words) - 22:43, 7 April 2016
• Dual space
  ...ear]] "functionals", also called ''covectors'', on $V$. It is called the ''dual'' of $V$. Note. If $V$ is a [[module]] over a ring $R$, the dual space is still the set of all linear functionals on $V$:
  9 KB (1,390 words) - 16:14, 16 June 2014
• Topology Illustrated -- Index
  *[[basis of neighborhoods|basis of neighborhoods]] *[[basis of topology|basis of topology]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Multilinear algebra
  ...ients $R$? Then we can choose $V_A$ to be the $R$-module generated by some basis of ${\bf R}^n$, for each $A$. Suppose now that $\{v_1,...,v_n \}$ is a basis of $V$. Then any vector in $V$ can be represented as a linear combination o
  49 KB (8,852 words) - 00:30, 29 May 2015
• Introduction to discrete calculus
  These are two vector spaces with their basis served by $N$ and $E$ respectively. [[image:geometry of R and dual.png| center]]
  40 KB (6,983 words) - 19:24, 23 July 2016
• Multilinear forms
  Another way to look at this space is the [[dual space]] of the space of multivectors: ...s of this space $\Lambda^k$ with specific formulas for the elements of the basis.
  18 KB (3,325 words) - 13:32, 26 August 2013
• Linear operators: part 4
  *$U={\bf R}^N$ with basis being the set of all vectors and *$W={\bf R}^n$ with basis being the set of all edges.
  13 KB (2,067 words) - 01:11, 12 September 2011
• Social choice
  It appears that ratings alone aren't versatile enough to be a basis of a fair electoral system. We will consider higher order votes next. ...ucture to the whole cochain group $C^*(K)$, by specifying an ''orthonormal basis''.
  47 KB (8,030 words) - 18:48, 30 November 2015
• Discrete calculus article
  ...o-one correspondence with the set of ''k''-simplices in ''S''. To define a basis explicitly, one has to choose an orientation of each simplex. One standard .../sub>,...,''v''_''k'') be an oriented ''k''-simplex, viewed as a basis element of ''C_k''. The '''boundary operator'''
  27 KB (4,329 words) - 16:02, 1 September 2019
• Cochain complexes and cohomology
  ...ells is the number of degrees of freedom as given by the [[dual space|dual basis]]. ...e cohomology is also the same, but only in the sense of vector spaces! The basis elements in dimension $1$ behave differently under wedge product. In the sp
  17 KB (2,592 words) - 14:38, 14 April 2013
• Discrete forms and cochains
  Below we start to develop cohomology theory as the dual of that of homology, for the case when: ...$c$ in $K$, there is a corresponding “elementary” cochain. We define the ''dual'' cochain $c^*:C_k\to R$ of $c$ by
  34 KB (5,619 words) - 16:00, 30 November 2015
• Integration of differential forms of degree 2
  Look at the basis only, then expand by [[linearity]]: This is a [[vector space]], $C_k(R)$, with $k$-cells serving as a [[basis]].
  15 KB (2,341 words) - 20:53, 13 March 2013
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni
  ...imal $p$-cells is the transpose of the boundary operator on [[dual complex|dual]] $(n-p+1)$-cells". That's [[Poincare duality]]. ...duction is [[Principal Component Analysis]] (PCA)... finds suitable set of basis vectors and project the high-dimensional data onto these vectors."
  11 KB (1,663 words) - 16:03, 26 November 2012
• Hodge duality of multivectors
  Even when the the meaning of the dual cell is clear, the question remains about its orientation. We discuss the H An orientation of this space can be seen as a choice of a basis $\{p_1,...,p_n\}$ of $V$ and an $n$-vector
  3 KB (488 words) - 12:34, 14 August 2015
• Calculus on chains
  With the help of [[dual space|duality]] we have a bird's-eye view of (a large part of) calculus, as ...e we just look where the [[basis]] elements goes under $d$. The "standard" basis of $C^k(R)$ consists of forms $\alpha _a$ with a single non-zero entry -- f
  4 KB (556 words) - 14:03, 30 March 2013
• Lefschetz numbers in control theory
  ...or $H_{k}(M)$ and $\{x_{1}^{k},...,x_{m_{k}}^{k}\}$ the corresponding dual basis for $H^{k}(M).$ The ''Lefschetz class'' of $g$ corresponding to $v\in H_{\a ...} _{M}\frown g_{\ast}(O_{M}\otimes v)), \] where $\overline{O}_{M}$ is the dual of $O_{M}.$ The first term vanishes unless $|v|=0$ or $n.$ The second term
  17 KB (3,052 words) - 22:12, 15 July 2014
• Forms in Euclidean spaces
  ...tangent spaces have compatible bases: if $AB$ is the $i$th element of the basis of $T_A(K)$, then $-BA$ is the $i$th element of $T_B(K)$. ...near combinations are ''functions'' and we can't think of these forms as a basis of $\Omega ^k(X)$ -- as an $R$-module. However, it is often beneficial to l
  44 KB (7,778 words) - 23:32, 24 April 2015

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