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Random Measures in Infinite-Dimensional Dynamics
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Palle E.T. Jorgensen, Feng Tian
Theorem 1.5.1. Let X be a locally compact Hausdorff space with Borel σ-algebra B, and let λ be a fixed positive regular measure on (X, B). Then a positive definite (p.d.) system is a pair (H, λ) with a Hilbert spaceH⊂Lloc2(λ), and such that, for allφ∈Cc(X), there is aKφ∈H with ∫φfdλ=〈f,Kφ〉H, ∀f∈H; the reproducing property.
Production Scheduling
Published in Katsundo Hitomi, Manufacturing Systems Engineering, 2017
In solving flow-shop scheduling problems, wherein all jobs are processed through an identical sequence of multiple machines arranged in series, two properties given below are dominant. Here a regular measure of performance is an amount to be minimised that is expressed as a function of job completion times and increases only if at least one of the completion times increases. Scheduling criteria introduced so far are mostly regular measures.
Boundedness of paraproducts on spaces of homogeneous type I
Published in Applicable Analysis, 2022
Der-Chen Chang, Xing Fu, Dachun Yang
It has been proven that many classical results on Euclidean spaces from harmonic analysis still hold true on spaces of homogeneous type, in the sense of Coifman and Weiss [24,25], which are generalizations of many important underlying spaces such as non-isotropic Euclidean spaces, quasi-metric spaces equipped with the Ahlfors n-regular measure, compact Riemannian manifolds equipped with the Euclidean distance and the Lebesgue measure, the boundary of a Lipschitz domain in equipped with the Euclidean distance and the harmonic measure, etc.; see [25, pp. 588-590] for more examples. Recall that a quasi-metric space is a non-empty set equipped with a quasi-metricd such that, for any , ; if and only if x = y;d satisfies the quasi-triangle inequality, where is called the quasi-triangle constant which is independent of x, y, and z.
Controllability of multi-term time-fractional differential systems
Published in Journal of Control and Decision, 2020
Vikram Singh, Dwijendra N. Pandey
Further, from (14) and (11), we have Thus, we conclude that From (12), we obtain In view of the definition of ν, have From , we obtain this implies that is relatively compact for almost all in . Further, using the fact that , by and (A2), we conclude that is uniformly integrable for a.e. . So, by the Definition 2.10, is semicompact. Moreover, by Lemma 2.11, we have is relatively compact in . Then, by (13) is also relatively compact in . Since ϒ is a nonsingular, monotone and regular measure of noncompactness, then by Mönch condition, we have which shows that is relatively compact in . This completes the proof.
Vibrations of a beam between two rigid stops: strong solutions in the framework of vector-valued measures
Published in Applicable Analysis, 2018
In particular, it follows that the reaction shear stress generated by the rigid stops (that prevents any penetration) at the contacting end of the beam can be decomposed, see (4.2) in Proposition 4.4, into a regular measure and a singular measure. The singular component may be interpreted as instantaneous impulses when the beam end recoils immediately upon impact, while the regular part allows for a more extended contact in time. Thus, our results provide a much more detailed description of the contact process.