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Introduction to Wavelets
Published in Nirdosh Bhatnagar, Introduction to Wavelet Transforms, 2020
A useful technique to represent a function consists of specifying it as a linear combination of some simple functions. Let the set of useful functions be {gω|ω∈Ω}, where the set Ω is countable. A possible representation of a function f is f=∑ω∈Ωcωgω
Some advances on constrained Markov decision processes in Borel spaces with random state-dependent discount factors
Published in Optimization, 2022
Héctor Jasso-Fuentes, Raquiel R. López-Martínez, J. Adolfo Minjárez-Sosa
It is not difficult to prove that for any nonnegative measurable function , the following relation holds true: For each , where is the occupation measure defined in (11). The argument only requires to compare the above relation for G as an indicator function, then for G as a simple function (regarded as a sum of indicator functions), and finally for G as a nonnegative function (regarded as a monotone limit of simple functions).Part (a) allows us to rewrite the functions in (6) and in (7) as the linear product and , , with as in (11).
⋄α-Measurability and combined measure theory on time scales
Published in Applicable Analysis, 2022
Chao Wang, Guangzhou Qin, Ravi P. Agarwal, Donal O'Regan
The function defined by is called the characteristic function of A and a linear combination of the characteristic function like is called a simple function, where are pairwise disjoint sets with .
Prepositioning emergency supplies under uncertainty: a parametric optimization method
Published in Engineering Optimization, 2018
Xuejie Bai, Jinwu Gao, Yankui Liu
In prepositioning emergency supplies model (3)–(10), the parameter can be rewritten as a simple function of , , where real number represents the distance between nodes i and j in the normal state, and fuzzy variable represents the transportation cost per unit distance for unit commodity k. Consider the fuzzy demand in the service level constraint. In general, fuzzy demand is influenced by several factors, such as the number of victims and unit demand for commodity k. Suppose represents the percentage of victims in all peoples at node i, and describe as a linear function of , , where is the number of people at node i and is the unit demand for commodity k. Similarly, suppose that and represent the kth commodity's relative quality level and arc 's capacity under normal conditions. As a consequence, and can be expressed as and , where and represent the damage level of location i and arcs, respectively.