test fields on compact space times in this paper some basic aspects of lorentzian field theory on compact lorentz manifolds are studied all compact space times are acausal i e possess closed timelike curves this makes them a useful testbed in analyzing some new notions of causality that will be introduced for more general acausal space times in addition studying compact space times in their own right raises a wide range of fascinating mathematical problems some of which will be explored it will be shown that it is reasonable to expect lorentzian field theory on a compact space time to provide information on the topology of the underlying manifold if this is true then this information is likely to be orthogonal or complementary to the information obtained through the study of euclidean field theory

a polynomial approach to nonlinear system controllability uses a polynomial approach to present a necessary and sufficient condition for local controllability of single input single output siso nonlinear systems the condition is presented in terms of common factors of a noncommutative polynomial expression this result exposes controllability properties of a nonlinear system in the input output framework and gives a computable procedure for examining nonlinear system controllability using computer algebra

forward and backward shifts of vector fields towards the dual algebraic framework an algebraic framework for discrete time nonlinear control systems is introduced based on the forward and backward shifts of the vector fields it appears to be the dual of the framework based on the differential 1 forms as applications the characterization of accessibility and the solution to the static state feedback linearization problem are stated in terms of the given vector fields

minimal realization of nonlinear systems described by input output difference equations we consider a general class of nonlinear discrete time systems described by their input output i o relationship we first present the necessary and sufficient conditions for existence of a local observable state space realization of the system and an explicit algorithm for computing it in cases where the i o map and its resulting realization are nonminimal we formulate an algorithm for extracting its minimal realization whenever possible the developed theory and algorithms are illustrated by means of several examples

state maps for linear systems modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential algebraic equations as such state space models are not a natural starting point for modeling while they have utmost importance in the simulation and control phase the paper addresses the problem of computing state variables for systems of linear differential algebraic equations of various forms the point of view from which the problem is considered is the behavioral one as put forward in j c willems automatica j ifac 22 1986 pp 561 580 dynamics reported 2 1989 pp 171 269 ieee trans automat control 36 1991 pp 259 294

brief transfer equivalence and realization of nonlinear higher order input output difference equations two fundamental modelling problems in nonlinear discrete time control systems are studied using the language of differential forms the discrete time nonlinear single input single output systems to be studied are described by input output i o difference equations i e a high order difference equation relating the input the output and a finite number of their time shifts a new definition of equivalence is introduced which generalizes the notion of transfer equivalence well known for the linear case our definition is based upon the notion of an irreducible differential form of the system and was inspired by the analogous definition for continuous time systems the second problem to be addressed is the realization problem the i o difference equation is assumed to be in the irreducible form so that one can obtain an accessible and observable realization necessary and sufficient conditions are given for the existence of a local state space realization of the irreducible i o difference equation these conditions are formulated in terms of the integrability of certain subspaces of one forms classified according to their relative degree the sufficiency part of the proof gives a constructive procedure up to finding the integrating factors and integration of the set of one forms for obtaining a locally observable and accessible state space system if the system is not in the irreducible form one has first to apply the reduction procedure to transform the system into the irreducible form

proof of the ghahramani lau conjecture abstract the ghahramani lau conjecture is established in other words the measure algebra of every locally compact group is strongly arens irregular to this end we introduce and study certain new classes of measures called approximately invariant respectively strongly singular which are of interest in their own right moreover we show that the same result holds for the measure algebra of any not necessarily locally compact polish group

linear input output equivalence and row reducedness of discrete time nonlinear systems the problem of linear input output i o equivalence of mero morphic nonlinear control systems described by implicit higher order difference equations is studied it is proved that any system is linearly i o equivalent to a row reduced form the constructive algorithm is given for finding the required transformation the latter amounts to 1 multiply the set of i o equations 0 from left by a unimodular matrix a whose entries are non commutative polynomials in the forward shift operator and 2 define certain multiplicative subset of the difference ring of analytic functions which introduces some inequations that should be satisfied

relaxing realizability conditions for discrete time nonlinear systems this paper addresses two related topics first a nonrealizable set of input output equations can be made realizable in the classical state space form by adding a postcompensator whereas this is impossible in the continuous time case construction of such postcompensator of minimal order is presented second a general subclass of discrete time systems is suggested for which the observable subspace is non integrable the construction is based on the nonrealizable system which is realized by adding a postcompensator and then combined with the output of the original system

linearization of discrete time systems the algebraic formalism developed in this paper unifies the study of the accessibility problem and various notions of feedback linearizability for discrete time nonlinear systems the accessibility problem for nonlinear discrete time systems is shown to be easy to tackle by means of standard linear algebraic tools whereas this is not the case for nonlinear continuous time systems in which case the most suitable approach is provided by differential geometry the feedback linearization problem for discrete time systems is recasted through the language of differential forms in the event that a system is not feedback linearizable the largest feedback linearizable subsystem is characterized within the same formalism using the notion of derived flag of a pfaffian system a discrete time system may be linearizable by dynamic state feedback though it is not linearizable by static state feedback necessary and sufficient conditions are given for the existence of a so called linearizing output which in turn is a sufficient condition for dynamic state feedback linearizability

controllability of linear stochastic systems abstract we discuss several concepts of controllability for stochastic systems and derive sufficient and in most cases also necessary algebraic conditions for the controllability of the state and the output of white and colored noise linear systems these criteria reduce to the well known deterministic ones in the absence of noise

multipliers and ideals in second conjugate algebras related to locally compact groups abstract let g be a locally compact group in this paper we study compact or weakly compact multipliers on the second conjugate algebras l 1 g and m g we prove among other things that g is amenable if and only if there is a non zero compact or weakly compact right multiplier on l 1 g or m g we show that if g is compact non compact then l 1 g cannot have any non zero weakly compact left multipliers t with t n 1 0 for some n l 1 g we also study maximal regular ideals of l 1 g w when g is a locally compact abelian group and w is a weight function on g and a p g 1 p g

degree of dieudonne determinant defines the order of nonlinear system abstractthe paper proves that the order of the minimal state space realisation of a set of irreducible implicit input output i o difference equations is equal to the degree of the dieudonne dete

orders of input output differential equations and state space dimensions this paper deals with the orders of input output equations satisfied by nonlinear systems such equations represent differential or difference in the discrete time case relations between high order derivatives or shifts respectively of input and output signals it is shown that under analyticity assumptions there cannot exist equations of order less than the minimal dimension of any observable realization this generalizes the known situation in the classical linear case the results depend on new facts themselves of considerable interest in control theory regarding universal inputs for observability in the discrete case and observation spaces in both the discrete and continuous cases included in the paper is also a new and simple self contained proof of sussmann s universal input theorem for continuous time analytic systems

convolution operators on group algebras which are tauberian or cotauberian abstract we study the convolution operators t acting on the group algebras l 1 g and m g where g is a locally compact abelian group and is a complex borel measure on g we show that a cotauberian convolution operator t acting on l 1 g is fredholm of index zero and that t is tauberian if and only if so is the corresponding convolution operator acting on the algebra of measures m g and we give some applications of these results

detectability of linear systems over a principal ideal domain with unknown inputs abstract observability and detectability conditions are obtained for a sort of linear time invariant systems affected by unknown inputs it is assumed that the system can be described using matrices whose elements are within a principal ideal domain pid it is shown that under suitable hypothesis the obtained conditions are necessary and sufficient the analysis is carried out by making use of the smith normal form of matrices over a pid the obtained conditions are a generalization of known conditions for particular sorts of systems which are included in those considered here

module maps on duals of banach algebras and topological centre problems we study various spaces of module maps on the dual of a banach algebra a and relate them to topological centres we introduce an auxiliary topological centre zt a a for the left quotient banach algebra a a of a our results indicate that zt a a is indispensable for investigating properties of module maps over a and for understanding some asymmetry phenomena in topological centre problems as well as the interrelationships between different arens irregularity properties for the class of banach algebras of type m introduced recently by the authors we show that strong arens irregularity can be expressed both in terms of automatic normality of a module maps on a and through certain commutation relations this in particular generalizes the earlier work on group algebras by ghahramani and mcclure 1992 13 and by ghahramani and lau 1997 12 we link a module map property over a to the space wap a of weakly almost periodic functionals on a generalizing a result by lau and ulger 1996 34 for banach algebras with a bounded approximate identity we also show that for a locally compact quantum group g the quotient strong arens irregularity of l1 g can be obtained from that of m g and can be characterized via the canonical c0 g module structure on luc g

realization of discrete time nonlinear input output equations polynomial approach the aim of this paper is to solve reduction and realization problems for discrete time multi input multi output nonlinear control systems by applying the theory of non commutative polynomials first the necessary and sufficient reducibility condition is presented in terms of the greatest common left divisor of two polynomial matrices associated with the set of the higher order input output i o difference equations of the system the condition also provides a method for system reduction i e for finding the irreducible representation of the set of the i o equations being transfer equivalent to the original system representation second to solve the realization problem a formula is presented for computing the differentials of the state coordinates directly from the polynomial description of the nonlinear system the polynomial approach addressed in this paper is more direct and requires noticeably less computations than earlier methods represented in terms of subspaces of differential one forms

on a conjecture by ghahramani lau and related problems concerning topological centres abstract let a be a banach algebra and consider a equipped with the first arens product we establish a general criterion which ensures that a is left strongly arens irregular i e the first topological centre of a is reduced to a itself using this criterion we prove that for a very large class of locally compact groups ghahramani lau s conjecture cf lau 94 and gha lau 95 stating the left strong arens irregularity of the measure algebra m g holds true our methods obviously yield as well the right strong arens irregularity in the situation considered furthermore the same condition used above implies that every linear left a module homomorphism on a is automatically bounded and w continuous we finally show that our criterion also yields a partial answer to a question raised by lau ulger trans amer math soc 348 3 1996 1191 on the topological centre of the algebra a a for a having a right approximate identity bounded by 1

duality and conformal structure in four dimensions two metrics that are conformally related define the same hodge dual operator on the space of two forms the converse namely that two metrics that have the same hodge dual are conformally related is established this is true for metrics of arbitrary nondegenerate signature for euclidean signature a stronger result namely that the conformal class of the metric is completely determined by choosing a dual operator on two forms satisfying certain conditions is proved

metrics and dual operators for a four dimensional vector space it is possible to define independently of a metric the notion of a dual operator acting on bivectors it is shown that the map which takes a conformal class of metrics together with an orientation to the induced dual operator is an isomorphism for each metric signature type this map is equivalent to an isomorphism of certain homogeneous spaces

geometry of c algebras and the bidual of their projective tensor product abstract given c algebras a and b consider the banach algebra a b where denotes the projective banach space tensor product if a and b are commutative this is the varopoulos algebra v a b we write v a for v a a it has been an open problem for almost 40 years to determine precisely when a b is arens regular see e g 33 48 49 we solve this classical question for arbitrary c algebras indeed we show that a b is arens regular if and only if a or b has the phillips property note that a has the latter property if and only if it is scattered and has the dunford pettis property a further equivalent condition is that a has the schur property or again equivalently the enveloping von neumann algebra a is finite atomic i e a direct sum of matrix algebras hence arens regularity of a b is entirely encoded in the geometry of the c algebras in case a and b are von neumann algebra we conclude that a b is arens regular if and only if a or b is finite dimensional we also show that this characterization does not generalize to the class of non selfadjoint dual even commutative operator algebras specializing to commutative c algebras a and b we obtain that v a b is arens regular if and only if a or b is scattered we further describe the centre z v a showing that it is banach algebra isomorphic to a e h a where e h denotes the extended haagerup tensor product we deduce that v a is strongly arens irregular if and only if a is finite dimensional hence v a is neither arens regular nor strongly arens irregular if and only if a is non scattered this is the case e g for a l

comparison of lpv and nonlinear system theory a realization problem abstract the paper explores the utilization of the nonlinear realization theory to address the problem of transforming linear parameter varying input output lpv io equations into a state space form with static dependence on the so called scheduling parameter the necessary and sufficient solvability conditions are given and three additional subclasses of lpv io equations are suggested that are guaranteed to have a realization of the considered type

a polynomial approach to discrete time non linear system controllability this paper uses a polynomial approach to present a necessary and sufficient condition for local controllability of single input single output siso discrete time non linear systems the condition is presented in terms of common factors of a non commutative polynomial expression this extends the result on controllability of linear systems and the recent result on controllability of continuous time non linear systems to expose controllability of discrete time non linear systems in the input output framework the controllability condition yields a finite step computing procedure for examining controllability of discrete time input output non linear systems

adjoint polynomial formulas for nonlinear state space realization this paper focuses on computational aspects of the realization of nonlinear multi input multi output systems instead of the algorithmic solutions provided in earlier works the explicit formulas are presented which enable to compute the differentials of the state coordinates directly from the polynomial description of the nonlinear system the solution is based on the concept of adjoint polynomials and requires a minimal amount of computations the formulas are implemented in computer algebra system mathematica and made available online via web mathematica tools

an algebraic framework generalizing the concept of transfer functions to nonlinear systems an algebraic point of view in nonlinear control systems built up by introducing a differential field of meromorphic functions and a vector space of differential forms is in this paper extended by introducing skew polynomials such polynomials act as operators over a vector space of differential forms the left skew polynomial ring defined over the field of meromorphic functions is embedded to its quotient field to provide a basis for a symbolic computation of nonlinear systems members of such a quotient field are suggested as the transfer functions of nonlinear systems whereby the concept of the transfer functions is generalized to nonlinear systems the theory is applied to nonlinear control systems and argues the invariance of introduced transfer functions of nonlinear systems to static state transformations the existence of input output descriptions for state space representations availability of transfer function algebra for nonlinear systems etc