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As the parenthetical ordering indicates, the mode-m column vectors are arranged by sweeping all the other mode indices through their ranges, with smaller mode indexes varying more rapidly than larger ones; thus [1]
An n-dimensional multi-index is an -tuple = (,, …,) of non-negative integers (i.e. an element of the -dimensional set of natural numbers, denoted ).. For multi ...
The flattening transformation is an algorithm that transforms nested data parallelism into flat data parallelism. It was pioneered by Guy Blelloch as part of the NESL programming language. [ 1 ] The flattening transformation is also sometimes called vectorization , but is completely unrelated to automatic vectorization .
In Python NumPy arrays implement the flatten method, [note 1] while in R the desired effect can be achieved via the c() or as.vector() functions. In R , function vec() of package 'ks' allows vectorization and function vech() implemented in both packages 'ks' and 'sn' allows half-vectorization.
There are three variants: the flattening , [1] sometimes called the first flattening, [2] as well as two other "flattenings" ′ and , each sometimes called the second flattening, [3] sometimes only given a symbol, [4] or sometimes called the second flattening and third flattening, respectively.
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 ...
To use column-major order in a row-major environment, or vice versa, for whatever reason, one workaround is to assign non-conventional roles to the indexes (using the first index for the column and the second index for the row), and another is to bypass language syntax by explicitly computing positions in a one-dimensional array.
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.