Nonlinear Sampling and Lebesgue's Integral Sums

We consider nonlinear, or "event-dependent", sampling, i.e. such that the sampling instances {tk} depend on the function being sampled. The use of such sampling in the construction of Lebesgue's integral sums is noted and discussed as regards physical measurement and also possible nonlinearity of singular systems. Though the limit of the sums, i.e. Lebesgue's integral, is linear with regard to the function being integrated, these sums are nonlinear in the sense of the sampling. A relevant method of frequency detection not using any clock, and using the nonlinear sampling, is considered. The mathematics and the realization arguments essentially complete each other.Comment: This is a continuation of my research of the classification of singular systems as linear and nonlinear (see IEEE CAS MAG, III, 2009 for switched systems, and here in the ArXiv) to sampling systems. The noted nonlinearity of Lebesgue's approximating sums, and an application of the "psy-transform", introduced by me earlier, to signal analysis are the examples. 5 pages, 4 figure

On two generalisations of the final value theorem : scientific relevance, first applications, and physical foundations

The present work considers two published generalisations of the Laplace-transform final value theorem (FVT) and some recently appeared applications of one of these generalisations to the fields of physical stochastic processes and Internet queueing. Physical sense of the irrational time functions, involved in the other generalisation, is one of the points of concern. The work strongly extends the conceptual frame of the references and outlines some new research directions for applications of the generalised theorem