on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

sample function properties of multi parameter stable processes the almost sure behaviour of the sample functions of the vectorvalued n parameter wiener process and its stable analogues is investigated especially their small fluctuations in particular laws of the simple and of the iterated logarithm are established for the supremum of the local time increments or sojourn times these results give precise information about the minimum oscillation of the sample functions in addition the hausdorff measure problem for the graph and the range of these processes is solved

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

semi cancellable fractions in system theory in this paper we propose to define the transfer function of a system in a new way namely as a semi cancellable fraction scf i e a fraction in which unstable pole zero cancellations are not allowed within this new framework it is possible to be rigorous and coherent in dealing with the stabilizability of a system from an input output viewpoint not focusing on the stabilizability of infinite dimensional input output systems

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

non solvability for a class of left invariant second order differential operators on the heisenberg group we study the question of local solvability for second order left invariant differential operators on the heisenberg group h n of the form formula math where a ij is a complex n x n matrix such operators never satisfy a cone condition in the sense of sjostrand and hormander we may assume that p a cannot be viewed as a differential operator on a lower dimensional heisenberg group under the mild condition that re im and their commutator are linearly independent we show that p is not locally solvable even in the presence of lower order terms provided that n 7 in the case n 3 we show that there are some operators of the form described above that are locally solvable this result extends to the heisenberg group h 3 a phenomenon first observed by karadzhov and muller in the case of h 2 it is interesting to notice that the analysis of the exceptional operators for the case n 3 turns out to be more elementary than in the case n 2 when 3 n 6 the analysis of these operators seems to become quite complex from a technical point of view and it remains open at this time

analysis of second order differential operators on heisenberg groups i given a matrixs sp n one finds a second order bi invariant differential operator s on the one dimensional extensiong s of the heisenberg grouph n induced bys we construct solutions to certain cauchy problems for s and fundamental solutions for these operators this analysis is connected with the discussion of the oscillator semigroup by r howe

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

discussing some examples of linear system interconnections the interconnection of linear systems may exhibit disturbing phenomena like hidden modes or a lack of control variables these features which cannot be taken into account by the transfer function approach are here explained by utilizing the language of modules which is strongly related to willems behavorial approach several examples are carefully discussed

the analysis of partial differential operators i hormander springer 1985 vol 1 391 pp vol 2 390 pp vol 3 525 pp vol 4 351 pp

periodic polynomial interpretation for structural properties of linear periodic discrete time systems the structural properties of linear periodic discrete time systems are analyzed in the periodic polynomial representation it is shown that the classical polynomial approach for linear time invariant systems can be extended to periodic systems new definitions and properties are given in terms of skew polynomial rings and periodic difference algebra necessary and sufficient conditions for the characterization of reachability controllability observability and reconstructibility are given in this framework

algebraic geometric aspects of feedback stabilization this paper develops a theory of feedback stabilization for siso transfer functions over a general integral domain which extends the well known coprime factorization approach to stabilization necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained these conditions are then refined in the special cases of unique factorization domains ufds noetherian rings and rings of fractions it is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring this interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant the set of transfer functions is topologized so as to restrict to the graph topology of vidyasagar ieee trans automatic control ac 29 1984 pp 403 418 when the ring is a bezout domain it is shown that stability of a feedback system is robust in this topology when the ring is a ufd this theory is then applied to the problem of sta

solvability for a class of doubly characteristic differential operators on 2 step nilpotent groups since h lewy s le discovery of a smooth linear partial differential operator without solution the problem of solvability of linear differential operators has been investigated intensively by various authors this has meanwhile led most notably through the work of hdrmander maslov egorov nirenberg erves and beals fefferman but with many important contributions also by others to a very satisfactory theory in the case of principal type operators see h6 much less is known about operators whose principal symbol vanishes of second order on part of its characteristic variety i e those which admit double characteristics even in the case of real principal symbols the main objective of this article is to present a model class of such operators for which we are able to discuss solvability in a complete way our class being composed of rather special operators will certainly not capture all phenomena which might arise in the study of general doubly characteristic operators with real symbol as the results in st already indicate but it is still rich enough to exhibit several new features if one compares with operators of principal type the operators that we shall consider will be left invariant differential operators on connected simply connected 2 step nilpotent lie groups g of the form m 0 1 le ajkvjvk u j k l

transition density of one dimensional diffusion with discontinuous drift the transition density of a one dimensional diffusion process with a discontinuous drift coefficient is studied a probabilistic representation of the transition density is given illustrating the close connections between discontinuities of the drift and brownian local times in addition some explicit results are obtained based upon the trivariate density of brownian motion its occupation and local times

measure of the multiple self intersection set of a markov process let x t t 0 be a markov process in rm with homogeneous transition density p t x y for a closed bounded set b rm x is said to have a self intersection of order r 2 in b if there are distinct points t1 tr such that x t1 b and x tj x t1 for j 2 r the focus of this work is the hausdorff measure suitably defined of the set of such r tuples the main result is that under general conditions on p t x y as well as the specific condition there is a measure function m t defined in terms of the integral above such that the corresponding hausdorff measure of self intersection set is positive with positive probability the results are applied to levy and diffusion processes and are shown to extend recent results in this area

approximation du temps local des surfaces gaussiennes letx be a centered stationary gaussian stochastic process with ad dimensional parameter d 2 f its spectral measure int limits r d x 2 f dx infty x denotes the euclidean norm ofx we consider regularizations of the trajectories ofx by means of convolutions of the formxe t e x t where e stands for an approximation of unity as e tends to zero satisfying certain regularity conditions

the modulator of the local time let x t 0 t 1 be a real measurable function having a local time x t which is a continuous function of t for almost all x it is also assumed that for some m 2 and some real interval b m x 1 is integrable over b the modulator is a function mm t b t 0 denned in terms of it is shown that the modulator serves as a measure of the smoothness of the lm b valued function t with respect to t then it is shown that the modulator plays a central role in precisely describing certain irregularity properties of x t the results are applied to the case where x t is the sample function of a real stochastic process in this way new results are obtained for large classes of gaussian and markov processes

a remarkable class of second order differential operators on the heisenberg group mathbb h _2 let l be a differential operator on mathbb r 5 whose principal part is of the form sum j k 1 2 alpha jk x j y k where x j partial x j and y j partial y j x j partial u j 1 2 are the usual vector fields generating the lie algebra of the heisenberg group mathbb h 2 we study the problem of local solvability of these doubly characteristic operators the whole class of operators splits into three subclasses depending on the sign of a respective determinant the operators in the first subclass when the determinant is negative are generically non solvable the operators in the second subclass when the determinant is positive are solvable for arbitrary left invariant lower order terms provided that the coefficient matrix alpha j k is non degenerate this fact seems remarkable since many of these operators have the property that the values taken by their principal symbol are not contained in any proper subcone of the complex plane under suitable conditions solvability even holds in the presence of non invariant lower order terms

on the solvability of homogeneous left invariant differential operators on the heisenberg group abstract we discuss the problem of solvability for the class of homogenoeus left invariant operators s on the heisenberg group h n introduced by f de mari m m peloso and f ricci 1995 j reine angew math 464 67 96 where s sp c n has a degenerated semi definite real part and n 2 l l 0 1 we completely characterize the solvable operators in this class as a consequence we obtain that certain operators s are solvable while their transposes t s are not this is the case for instance for the operator yz on h 1 where z x iy is the unsolvable lewy operator

fundamental solutions of a class of ultra hyperbolic operators on pseudo h type groups abstract pseudo h type lie groups g r s of signature r s are defined via a module action of the clifford algebra c l r s on a vector space v r 2 n they form a subclass of all 2 step nilpotent lie groups based on their algebraic structure they can be equipped with a left invariant pseudo riemannian metric let n r s denote the lie algebra corresponding to g r s in the case s 0 a choice of left invariant vector fields x 1 x 2 n which generate a complement of the center of n r s gives rise to a second order differential operator r s x 1 2 x n 2 x n 1 2 x 2 n 2 which we call ultra hyperbolic we prove that r s is locally solvable if and only if r 0 in particular it follows that r s does not admit a fundamental solution in the space d g r s of schwartz distributions whenever r 0 in terms of classical special functions we present families of fundamental solutions of 0 s in the class of tempered distributions s g 0 s and study their properties

a remark on willems trajectory characterization of linear controllability abstract willems trajectory characterization of linear controllability is related to some module theoretic property

analysis of the kohn laplacian on quadratic cr manifolds abstract we study the kohn laplacian b q acting on 0 q forms on quadratic cr manifolds we characterize the operators b q that are locally solvable and hypoelliptic respectively in terms of the signatures of the scalar components of the levi form

joint continuity of the local times of markov processes consider a markov process on the real line with a specified transition density function certain conditions on the latter are shown to be sufficient for the almost sure existence of a local time of the sample function which is jointly continuous in the state and time variables