Random Sampling of a Continuous-time

We consider a dynamical system where the state equation is given by a linear stochastic di#erential equation and noisy measurements occur at discrete times, in correspondence of the arrivals of a Poisson process. Such a system models a network of a large number of sensors that are not synchronized with one another, where the waiting time between two measurements is modelled by an exponential random variable

C*-Algebras Associated with Quadratic Dynamical System

this paper we consider enveloping C # bygenerA]Q3 and definingrningA3' of the following for A = C#X,X XX # = f(X # X)#, wher f is a HerCLWUA mapping. Some prA erAW3 of thesealgebrQ associated with simple dynamical systems (f,R) ar studied. s an examplequadrAWL dynamical systems arconsiderUC Intro ductio It is well known that ther is close connection between thereA'2C3 tation theor of C # and strL'0A] of dynamical systems (f(),X). In the case when f is one-to-one mapping, the -algebr associated with thetrWWW00A]'LB3 grW had been studied by manyauthorA for example by Glimm,E#rm and Hahn. ThegenerQ theor of crB03LA] ducts of C # was elaborbAC by Doplicher Kastler and Robinson. InrCB3 t pap er (see [8] andrWWUBBA]' giventher in) a special class of by gener]"0 andrdA3QC" was consider] and some of thereA'B' frA thetheor ofcrC2W pr duct C # wer trW22A]'3U into non-bijective settings, which may be impor2L t in studying of multi-dimensionalnon-linear defor3A]'0 (see [11, 10, 3]), such as Witten's firQ deforWA]'0 of su(2), Quesne and Becker non-linear defor'A]WC of su(2) etc. Examples wer studied in connection with di#erW t quantumdeforCWA]WU ofalgebr2A such as Quantum Unit Disc (Klimek and Lesnievski), one-dimensional q-CCR andtheir nonlinear tr'C0QA]WB'WWA ets see [7, 8]. Thus, for example, for one-parW'Q33 Quantum Unit Disc ther cor'B onds the dynamical system: f(#)= (q+)#- ,wher is a parBU03A ofdeforCA]WW3 for two-parA]WW3 Quantum Unit Discther cor3U onds f(#)= (q+)#+1-q- , for Witten's fir' deforWA]W0 of su(2)ther corrC onds two-dimensionalquadrens map f(x y)=(p -1 (1 + p -1x ),g(gy +(p p -1 )x )), wher g = depending on the chosen ren for and p is a parBB0'A ofdefor2A]3U2 In theprA33 t paper we will deal with a one-dimensional polynomial map..