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Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
A multi-index of size is an element in (given that is fixed, if the size of multi-indices is omitted then the size should be assumed to be ). The length of a multi-index α = ( α 1 , … , α n ) ∈ N n {\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})\in \mathbb {N} ^{n}} is defined as α 1 + ⋯ + α n {\displaystyle \alpha _{1 ...
A vector treated as an array of numbers by writing as a row vector or column vector (whichever is used depends on convenience or context): = (), = Index notation allows indication of the elements of the array by simply writing a i, where the index i is known to run from 1 to n, because of n-dimensions. [1]
If is a multi-index, and a is a positive real number, then | | (). Any smooth function f with compact support is in S(R n).This is clear since any derivative of f is continuous and supported in the support of f, so has a maximum in R n by the extreme value theorem.
Concretely, in the case where the vector space has an inner product, in matrix notation these can be thought of as row vectors, which give a number when applied to column vectors. We denote this by V ∗ := Hom ( V , K ) {\displaystyle V^{*}:={\text{Hom}}(V,K)} , so that α ∈ V ∗ {\displaystyle \alpha \in V^{*}} is a linear map α : V → K ...
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