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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 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]
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 ...
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. [1]
The fundamental property of this norm is the Bombieri inequality: let , be two homogeneous polynomials respectively of degree () and () with variables, then, the following inequality holds:
Where denotes the partial derivative of order (see multi-index notation). When σ = 1 {\displaystyle \sigma =1} , G σ ( Ω ) {\displaystyle G^{\sigma }(\Omega )} coincides with the class of analytic functions C ω ( Ω ) {\displaystyle C^{\omega }(\Omega )} , but for σ > 1 {\displaystyle \sigma >1} there are compactly supported functions in ...