Finite valued feedback laws and piecewise classical solutions

Often, in engineering literature, we find control systems in which the open loop inputs are piecewise constant and take values in a finite set. Such open loop inputs cause the system to have fairly regular solutions. On the other hand, when acting in closed loop, feedback laws taking values in a finite set may not be reinterpreted as open loop inputs of the considered type. In fact, pathological behaviours such as the accumulation of discontinuities may appear (Zeno phenomenon). We give some conditions which can be used as tools for building finite valued feedback laws not causing such pathological behaviours.     

Discrete valued feedback laws and the Zeno phenomenon

We consider closed loop control systems in which feedback laws take values in a discrete set U$U$ and we study the Zeno phenomenon, i.e. the accumulation of discontinuities with respect to time. The results generalize those obtained in [F. Ceragioli, Finite valued feedback laws and piecewise classical solutions, Nonlinear Analysis 65 (2006) 984–998] for the case where U$U$ is finite to the case where U$U$ may be infinite. An application to a quantized control problem is also shown.     

Two valued measure and summability of double sequences

In this paper, following the methods of Connor [2], we extend the idea of statistical convergence of a double sequence (studied by Muresaleen and Edely [12]) to μ-statistical convergence and convergence in μ-density using a two valued measure μ. We also apply the same methods to extend the ideas of divergence and Cauchy criteria for double sequences. We then introduce a property of the measure μ called the (APO₂) condition, inspired by the (APO) condition of Connor [3]. We mainly investigate the interrelationships between the two types of convergence, divergence and Cauchy criteria and ultimately show that they become equivalent if and only if the measure μ has the condition (APO₂).     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

Two valued measure and some new double sequence spaces in 2-normed spaces

The purpose of this paper is to introduce some new generalized double difference sequence spaces using summability with respect to a two valued measure and an Orlicz function in 2-normed spaces which have unique non-linear structure and to examine some of their properties. This approach has not been used in any context before.     

Scalar conservation laws with fractional stochastic forcing: Existence,uniqueness and invariant measure

We study a fractional stochastic perturbation of a first-order hyperbolic equation of nonlinear type. The existence and uniqueness of the solution are investigated via a Lax–Ole?nik formula. To construct the invariant measure we use two main ingredients. The first one is the notion of a generalized characteristic in the sense of Dafermos. The second one is the fact that the oscillations of the fractional Brownian motion are arbitrarily small for an infinite number of intervals of arbitrary length.     

Nearby variables with nearby conditional laws and a strong approximation theorem for Hilbert space valued martingales

Summary In this paper we focus on sequences of random vectors which do not admit a strong approximation of their partial sums by sums of independent random vectors. In the first part we prove conditional versions of the Strassen-Dudley theorem. We apply these in the second part of the paper to obtain strong invariance principles for vector-valued martingales which, when properly normalized, converge in law to a mixture of Gaussian distributions.Research partially supported by the Air Force Office of Scientific Research Contract NO. F49260 85 C 0144Part of this work was done while the second author was visiting the Department of Statistics and the Center for Stochastic Processes, University of North Carolina, Chapel Hill. He thanks the members of the Department of Statistics for their hospitality     

Wave scattering in hyperbolic balance laws and the zero relaxation limit

Ricerche di Matematica - In the context of a simple hyperbolic system of balance laws that relaxes to a scalar conservation law, the paper investigates the process by which the synergy of waves...     

Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem

The Generalized Riemann Problem (GRP) for a nonlinear hyperbolic system of m balance laws (or alternatively “quasi-conservative” laws) in one space dimension is now well-known and can be formulated as follows: Given initial-data which are analytic on two sides of a discontinuity, determine the time evolution of the solution at the discontinuity. In particular, the GRP numerical scheme (second-order high resolution) is based on an analytical evaluation of the first time derivative. It turns out that this derivative depends only on the first-order spatial derivatives, hence the initial data can be taken as piecewise linear. The analytical solution is readily obtained for a single equation (m = 1) and, more generally, if the system is endowed with a complete (coordinate) set of Riemann invariants. In this case it can be “diagonalized” and reduced to the scalar case. However, most systems with m > 2 do not admit such a set of Riemann invariants. This paper introduces a generalization of this concept: weakly coupled systems (WCS). Such systems have only “partial set” of Riemann invariants, but these sets are weakly coupled in a way which enables a “diagonalized” treatment of the GRP. An important example of a WCS is the Euler system of compressible, nonisentropic fluid flow (m = 3). The solution of the GRP discussed here is based on a careful analysis of rarefaction waves. A “propagation of singularities” argument is applied to appropriate Riemann invariants across the rarefaction fan. It serves to “rotate” initial spatial slopes into “time derivative”. In particular, the case of a “sonic point” is incorporated easily into the general treatment. A GRP scheme based on this solution is derived, and several numerical examples are presented. Special attention is given to the “acoustic approximation” of the analytical solution. It can be viewed as a proper linearization (different from the approach of Roe) of the nonlinear system. The resulting numerical scheme is the simplest (second-order, high-resolution) generalization of the Godunov scheme.     

Leibnizian,Robinsonian, and Boolean valued monads

This is an overview of the present-day versions of monadology with some applications to vector lattices and linear inequalities. Two approaches to combining nonstandard set-theoretic models are sketched and illustrated by order convergence, principal projection, and polyhedrality.     

On the hellinger square integral with respect to an operator valued measure and stationary processes

A construction of the Hellinger square integral with respect to a semispectral measure in a Banach space B is given. It is proved that the space of values of a B-valued stationary stochastic process is unitarily isomorphic to the space of all B¹-valued measures that are Hellinger square integrable with respect to the spectral measure of the process. Some applications of the above theorem in the prediction theory (especially to interpolation problem) are also considered.     

Admissibility and uniqueness of weak solutions to hyperbolic systems of balance laws

A system of quasi-linear first-order equations written in the divergence form and constrained by the unilateral differential inequality (the second law of thermodynamics) with a strictly concave entropy function is analysed. In the class BV, i.e. a subset of regular distributions represented by functions of bounded variation in the sense of Tonelli-Cesari, a weak solution to the system is defined. The parabolized version of the system is also discussed in order to define an admissible weak solution as a limit of a sequence of Lipschitz continuous solutions to the parabolic problem. It is proved that an admissible weak solution of the Cauchy problem is unique in the class BV.     

Extinction probability,regularity and asymptotic growth of Markovian populations

"The distribution of the maximum and the extinction probability for a Markovian population is derived. Asymptotic growth is described, using the sequence of sojourn times. A regularity criterion for the processes under consideration exists under certain assumptions. For a class of processes with specific population-dependent transition rates the asymptotic behaviour is given explicitly."     

A new family of convex weakly compact valued random variables in Banach space and applications to laws of large numbers

We consider a new family of convex weakly compact valued integrable random sets which is called an adapted array of convex weakly compact valued integrable random variables of type p (1?p?2). By this concept, more general laws of large numbers will be established. Some illustrative examples are provided.     

Ultrametric Cantor sets and growth of measure

A class of ultrametric Cantor sets (C, d _u ) introduced recently (S. Raut and D. P. Datta, Fractals 17, 45–52 (2009)) is shown to enjoy some novel properties. The ultrametric d _u is defined using the concept of relative infinitesimals and an inversion rule. The associated (infinitesimal) valuation which turns out to be both scale and reparametrization invariant, is identified with the Cantor function associated with a Cantor set $ \tilde C $ \tilde C , where the relative infinitesimals are supposed to live in. These ultrametrics are both metrically as well as topologically inequivalent compared to the topology induced by the usual metric. Every point of the original Cantor set C is identified with the closure of the set of gaps of $ \tilde C $ \tilde C . The increments on such an ultrametric space is accomplished by following the inversion rule. As a consequence, Cantor functions are reinterpreted as locally constant functions on these extended ultrametric spaces. An interesting phenomenon, called growth of measure, is studied on such an ultrametric space. Using the reparametrization invariance of the valuation it is shown how the scale factors of a Lebesgue measure zero Cantor set might get deformed leading to a deformed Cantor set with a positive measure. The definition of a new valuated exponent is introduced which is shown to yield the fatness exponent in the case of a positive measure (fat) Cantor set. However, the valuated exponent can also be used to distinguish Cantor sets with identical Hausdorff dimension and thickness. A class of Cantor sets with Hausdorff dimension log₃ 2 and thickness 1 are constructed explicitly.     

Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra

During the past 10 years multifractal analysis has received an enormous interest. For a sequence n(φn) of functions φn:X→M on a metric space X, multifractal analysis refers to the study of the Hausdorff dimension of the level sets

     

Multivariate Polynomials,Duality, and Structured Matrices

We first review the basic properties of the well known classes of Toeplitz, Hankel, Vandermonde, and other related structured matrices and reexamine their correlation to operations with univariate polynomials. Then we define some natural extensions of such classes of matrices based on their correlation to multivariate polynomials. We describe the correlation in terms of the associated operators of multiplication in the polynomial ring and its dual space, which allows us to generalize these structures to the multivariate case. Multivariate Toeplitz, Hankel, and Vandermonde matrices, Bezoutians, algebraic residues, and relations between them are studied. Finally, we show some applications of this study to rootfinding problems for a system of multivariate polynomial equations, where the dual space, algebraic residues, Bezoutians, and other structured matrices play an important role. The developed techniques enable us to obtain a better insight into the major problems of multivariate polynomial computations and to improve substantially the known techniques of the study of these problems. In particular, we simplify and/or generalize the known reduction of the multivariate polynomial systems to the matrix eigenproblem, the derivation of the Bézout and Bernshtein bounds on the number of the roots, and the construction of multiplication tables. From the algorithmic and computational complexity point, we yield acceleration by one order of magnitude of the known methods for some fundamental problems of solving multivariate polynomial systems of equations.