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Formal Reliability Analysis of Railway Systems Using Theorem Proving Technique
Published in Qamar Mahboob, Enrico Zio, Handbook of RAMS in Railway Systems, 2018
Waqar Ahmad, Osman Hasan, Sofiène Tahar
Mathematically, a measure space is defined as a triple (Ω, Σ, µ), where Ω is a set, called the sample space; Σ represents a σ algebra of subsets of Ω, where the subsets are usually referred to as measurable sets; and µ is a measure with domain Σ. A probability space is a measure space (Ω, Σ, Pr), such that the measure, referred to as the probability and denoted by Pr, of the sample space is 1. In the HOL formalization of probability theory [10], given a probability space p, the functions , , and return the corresponding Ω, Σ, and Pr, respectively. This formalization also includes the formal verification of some of the most widely used probability axioms, which play a pivotal role in formal reasoning about reliability properties.
Control systems described by a class of fractional semilinear evolution hemivariational inequalities and their relaxation property
Published in Optimization, 2022
Let be a σ-finite measure space, E be a Banach space and . If weakly in and for μ-a.e. and all , where for μ=a.e. , then where the set is the Kruatowski upper limit of the set (cf. Definition 3.14 of [29]).
Solvability and optimal control of fractional differential hemivariational inequalities
Published in Optimization, 2021
Guangming Xue, Funing Lin, Bin Qin
Let be a σ-finite measure space, X be a Banach space and . Given and , if and for each and for μ-a.e. , where for μ-a.e. , then where denotes the closed convex hull of a set.
Optimal quantization via dynamics
Published in Dynamical Systems, 2020
Joseph Rosenblatt, Mrinal Kanti Roychowdhury
Here is the full range of how one might consider distortion errors generated by some sequence of stochastic processes . We assume that the processes are defined on an underlying state space which might be a probability space, but generally would be a σ-finite measure space. For , the range values of are in another measure space which is also naturally a metric space with metric . Elements are the parameters and the values the quantizers associated with the parameter ω.