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A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is a weighted sum or weighted average .
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain . Real-valued functions of a real variable (commonly called real functions ) and real-valued functions of several real variables are the main object of study of calculus and ...
A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ([x, y]) = 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra [g,g] and hence descends to a weight on the abelian Lie algebra g/[g,g]. Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple ...
For example, the equations = = form a parametric representation of the unit circle, where t is the parameter: A point (x, y) is on the unit circle if and only if there is a value of t such that these two equations generate that point.
Sometimes an extension of the domain of the edge weight function to is considered (with the resulting function still being called the edge weight function) by setting (,) = whenever (,). In applications each graph vertex x ∈ V {\displaystyle x\in V} usually represents a single entity in the given data, e.g., elements of a finite data set ...
In functional analysis, a weighted space is a space of functions under a weighted norm, which is a finite norm (or semi-norm) that involves multiplication by a particular function referred to as the weight. Weights can be used to expand or reduce a space of considered functions.
For example, if : [,] is the Dirichlet function that is on irrational numbers and on rational numbers, and [,] is equipped with Lebesgue measure, then the support of is the entire interval [,], but the essential support of is empty, since is equal almost everywhere to the zero function.
The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values. If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean. [1]
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