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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 ...
which in modern notation becomes: 1 , a , b , a b , c , a c , b c , a b c {\displaystyle 1,a,b,ab,c,ac,bc,abc} in which it is evident that the exponents in the modern notation names are simply the subscripts of the former (note that anything raised to the zeroth power is 1 and anything raised to the first power is itself).
Note that, since x + y is a vector and α is a multi-index, the expression on the left is short for (x 1 + y 1) α 1 ⋯(x n + y n) α n. Leibniz formula For smooth functions f {\textstyle f} and g {\textstyle g} , ∂ α ( f g ) = ∑ ν ≤ α ( α ν ) ∂ ν f ∂ α − ν g . {\displaystyle \partial ^{\alpha }(fg)=\sum _{\nu \leq \alpha ...
The Marshall-Edgeworth index, credited to Marshall (1887) and Edgeworth (1925), [11] is a weighted relative of current period to base period sets of prices. This index uses the arithmetic average of the current and based period quantities for weighting. It is considered a pseudo-superlative formula and is symmetric. [12]
Abstract index notation (also referred to as slot-naming index notation) [1] is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. [2] The indices are mere placeholders, not related to any basis and, in particular, are non-numerical.
Index mapping (or direct addressing, or a trivial hash function) in computer science describes using an array, in which each position corresponds to a key in the universe of possible values. [1] The technique is most effective when the universe of keys is reasonably small, such that allocating an array with one position for every possible key ...
For example, if x is a quantity, then x c is the characteristic unit used to scale it. As an illustrative example, consider a first order differential equation with constant coefficients: + = (). In this equation the independent variable here is t, and the dependent variable is x.