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Distributed Parameter Nonlinear Transmission Lines II: Existence of Weak Solutions
Published in Frederick Bloom, Mathematical problems of classical nonlinear electromagnetic theory, 2020
Combining (5.2.6) and (5.2.7) with (5.2.5) we have ∫ΩΦ(x)ω(dx)=0 thus establishing the absolute continuity of the measure ω(dx) with respect to Lebesgue measure dx. Because ω(dx) is absolutely continuous with respect to dx, the Radon-Nikodym theorem guarantees the existence of a density ωf (x) such that ω(dx)=ωf(x)dx
Random Vectors and Random Functions
Published in Richard M. Golden, Statistical Machine Learning, 2020
If x~ is an absolutely continuous random vector, the support specification measure ν specifies the support of P such that the integral in (8.4) is interpretable as a Riemann integral: P(B)=∫x∈Bp(x)dx.
Theory of elasticity and numerical methods in engineering
Published in Indrajit Chowdhury, Shambhu P. Dasgupta, Dynamics of Structure and Foundation – A Unified Approach, 2008
Indrajit Chowdhury, Shambhu P. Dasgupta
Bodies have a basic property that they have mass. In classical mechanics, mass is assumed to be conserved, that is the mass of a material body is the same at all times. In continuum mechanics, the mass is an absolutely continuous function of volume. It is assumed that a positive quantity, ρ, called density, can be obtained at every point in the body as ρ(X¯)=limk→∞massofVkvolumeofVk where Vk is a suitably chosen infinite sequence of particle sets shrinking down upon the point X̄, (x1, x2, x3). At time t = 0, the density at the point ξ̄ ≡ (ξ1, ξ2, ξ3) is defined by ρ0(ξ).
ℋ2 state-feedback control for continuous semi-Markov jump linear systems with rational transition rates
Published in International Journal of Control, 2023
M. de Almeida, M. Souza, A. R. Fioravanti, O. L. V. Costa
At last, in terms of notation, for real matrices or vectors, indicates transpose. For a square matrix X, denotes its trace, and for partitioned symmetric matrices, the symbol ★ denotes its symmetric blocks. The set of natural numbers is denoted by . On a probability space , the symbol denotes mathematical expectation and, for , represents the indicator function in A, that is, if , 0 otherwise. For any stochastic signal defined in the continuous-time domain, the quantity is its squared norm. The class of all signals , , such that is finite is denoted by . We recall that for an absolutely continuous function there exists an integrable function such that Note that is not unique. For simplicity, we write . We say that an matrix is absolutely continuous in if each element of , , is an absolutely continuous function. We represent by the matrix such that (1) is satisfied replacing and by, respectively, and for each , and write .