from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

transcendental and interpolation methods in simultaneous stabilization and simultaneous partial pole placement problems in this paper we investigate the existence of a compensator which simultaneously renders a given r tuple of multiinput multioutput p times m linear dynamical systems internally stable in particular we parametrize a set of simultaneously stabilizable r tuples of plants and show that provided r leqq max m p the above set is semialgebraic and dense in the space sigma of r tuples of plants furthermore we also consider an extension of the classical pole placement and stabilization problems and investigate the simultaneous partial pole placement problem consequently we consider a suitable topology in and obtain a necessary condition and a sufficient condition for the generic partial pole placement problem surprisingly enough the problem of simultaneously stabilizing a triplet of m times m plants chosen generically is shown equivalent to the problem of partially pole placing a single m times m plant by a stable minimum phase compensator on the other hand the problem of simultaneous

differential calculi and linear connections a method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra as an example the generalized quantum plane is studied it is found that there is a strong correlation but not a one to one correspondence between the module structure of the 1 forms and the metric torsion free connections on it in the commutative limit the connection remains as a shadow of the algebraic structure of the 1 forms

soliton equations and the zero curvature condition in noncommutative geometry familiar nonlinear and in particular soliton equations arise as zero curvature conditions for connections with noncommutative differential calculi the burgers equation is formulated in this way and the cole hopf transformation for it attains the interpretation of a transformation of the connection to a pure gauge in this mathematical framework the kdv modified kdv equation and the miura transformation are obtained jointly in a similar setting and a rather straightforward generalization leads to the kp and a modified kp equation furthermore a differential calculus associated with the boussinesq equation is derived from the kp calculus

dynamical evolution in non commutative discrete phase space and the derivation of classical kinetic equations by considering a lattice model of extended phase space and using techniques of non commutative differential geometry we are led to a the concept of vector fields as generators of motion and transition probability distributions on the lattice b the emergence of the time direction on the basis of the encoding of probabilities in the lattice structure c the general prescription for the evolution of the observables in analogy with classical dynamics we show that in the limit of a continuous description these results lead to the time evolution of observables in terms of the adjoint of generalized fokker planck equations having 1 a diffusion coefficient given by the limit of the correlation matrix of the lattice coordinates with respect to the probability distribution associated with the generator of motion 2 a drift term given by the microscopic average of the dynamical equations in the present context these results are applied to one and two dimensional problems specifically we derive i the equations of diffusion smoluchowski and fokker planck in velocity space thus indicating the way random walk models are incorporated in the present context ii kramers equation by further assuming that motion is deterministic in coordinate space

non commutative geometry and kinetic theory of open systems the basic mathematical assumptions for autonomous linear kinetic equations for a classical system are formulated leading to the conclusion that if they are differential equations on its phase space m they are of at most second order for open systems interacting with a bath at canonical equilibrium they have the particular form of an equation of a generalized fokker planck type we show that it is possible to obtain them as liouville equations of hamiltonian dynamics on m with a particular non commutative differential structure provided that certain conditions geometric in character are fulfilled to this end symplectic geometry on m is developed in this context and an outline of the required tensor analysis and differential geometry is given certain questions as regards the possible mathematical interpretation of this structure are also discussed

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

convergence of local charges and continuity properties of w inclusions the local generators of symmetry transformations which have recently been constructed from a quantum field theoretical version of noether s theorem are shown to converge to the global ones as the volume tends to the whole space the proof relies on the continuous volume dependence of the universal localizing maps which are associated to the local splitw inclusions

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

on the embedding of von neumann subalgebras for a von neumann algebra with a cyclic and separating vector it will be shown that the von neumann subalgebras with the same cyclic vector can uniquely be characterized by one parametric operator valued functions obeying a set of conditions since the properties contain no reference to the subalgebra these operator valued functions will be called characteristic functions on the set of characteristic functions there exists a natural topology under which this set is complete

markov dilations and quantum detailed balance we construct quantum stochastic processes whose multi time correlation functions with suitable time ordering can be obtained from a quantum dynamical semigroup we prove that such a process defines a stationary markov dilation of the associated semigroup if and only if up to technicalities the semigroup satisfies the quantum detailed balance condition with respect to its stationary state

bicovariant differential calculi on glp q 2 and quantum subgroups in a previous paper all bicovariant differential calculi on the two parameter quantum group glp q 2 were determined in this work the authors elaborate some of their properties and discuss relations with work by other authors furthermore they show that there are two different ways in which bicovariant differential calculi on glq 2 where p q induce corresponding calculi on the quantum subgroup slq 2 it is shown that on slq 2 there are only two different bicovariant differential calculi from these one obtains the 4d or calculi on suq 2 the classical limit of the two differential calculi on slq 2 is investigated revealing a relation with recent work on deformed differential calculi on commutative function algebras

an approach to simultaneous system design part 11 nonswitching gain and dynamic feedback compensation by algebraic geometric methods this paper studies structured uncertainty problems in feedback system design considers a compact parameterization of the space of linear dynamical systems and introduces base points and critical points as two algebraic geometric objects that have significance in sensitivity and robustness studies respectively using the nevanlinna pick interpolation theory the author obtains a necessary and sufficient condition for simultaneous stabilization of a structured one parameter family of plants a recent result due to kharitonov on the simultaneous stability of a parameterized family of polynomials leads to a sufficiency condition for simultaneous stabilization of a structured multiparameter family of plants furthermore the author considers simultaneous pole placement of an r tuple of plants as a means to arbitrarily tune the natural frequencies of a multimode linear dynamical system the concept of nondegenerate and twisted r tuples of plants is introduced as the pole placement problem is studie

diffusion processes quantum dynamical semigroups and the classical kms condition we study some properties of the generator of a diffusion process describing the reduced dynamics of a classical system weakly coupled to an external reservoir we show that such processes may be equivalently obtained as classical limits of quantum dynamical semigroups derived in the weak coupling limit we prove that the stationary state is independent of the coupling if and only if the reservoir is in a state of thermal equilibrium

kinetic theory in the weak coupling approximation ii a classical harmonic oscillator model abstract the kinetic theory of open systems presented in a previous paper is applied to a classical harmonic oscillator model it leads to a kinetic equation very different from the conventionally used linearized landau equation that shows explicitly the influence of the field on the interaction of the harmonic particle with its surroundings contrary to what is often assumed a self consistent approach based on our equation suggests a one dimensional generalization of kramers model equation the limit of our equation when the harmonic frequency vanishes contains a divergent term whose nature is briefly discussed

differential calculus and gauge theory on finite sets we develop differential calculus and gauge theory on a finite set g an elegant formulation is obtained when g is supplied with a group structure and in particular for a cyclic group connes two point model 1986 which is an essential ingredient of his reformulation of the standard model of elementary particle physics is recovered in our approach reductions of the universal differential calculus to lower dimensional differential calculi are considered the complete reduction leads to a differential calculus on a periodic lattice

causal independence and the energy level density of states in local quantum field theory within the general framework of local quantum field theory a physically motivated condition on the energy level density of well localized states is proposed and discussed it is shown that any model satisfying this condition obeys a strong form of the principle of causal statistical independence which manifests itself in a specific algebraic structure of the local algebras split property it is also shown that the proposed condition holds in a free field theory

noncommutative differential calculus and lattice gauge theory the authors study consistent deformations of the classical differential calculus on algebras of functions and more generally commutative algebras such that differentials and functions satisfy nontrivial commutation relations for a class of such calculi it is shown that the deformation parameters correspond to the spacings of a lattice these differential calculi generate a lattice on a space continuum the whole setting of a lattice theory can then be deduced from the continuum theory via deformation of the standard differential calculus in this framework one just has to express the lagrangian for the continuum theory in terms of differential forms this expression then also makes sense for the deformed differential calculus there is a natural integral associated with the latter integration of the lagrangian over a space continuum then produces the correct lattice action for a large class of theories this is explicitly shown for the scalar field action and the action for su m gauge theory

sufficient conditions for generic simultaneous pole assignment and stabilization of linear mimo dynamical systems it has been shown that a generic r tuple of m input p output dynamical systems is simultaneously stabilizable pole assignable if r m p and the mcmillian degrees of the systems are not too different this result improves the currently known upper bound r spl les max m p on the number of plants in the case r spl les max m p an upper bound on the degree of the simultaneous pole assigning compensator has also been derived which improves the bound obtained by ghosh and byrnes 1983

on minimal degree simultaneous pole assignment problems in this paper we show that a generic r tuple of m input p output linear systems is simultaneously pole assignable if r m p and the mcmillan degrees of the systems are not too different we also obtain upper bounds for the degrees of the compensators which simultaneously assign the characteristic polynomials of the r tuple of closed loop systems the upper bounds are obtained for each of the two cases r max m p and max m p r m p 2002 elsevier science inc all rights reserved

kinetic equation for a classical test particle model in the weak coupling approximation abstract a kinetic equation for a harmonically bound classical particle weakly coupled to a thermal reservoir is presented the derivation based on the formalism of master equations developed in brussels is outlined and compared to other approaches some comments are also made as to the consistency of the so called kramers equation in the presence of an external field

quantum mechanics as noncommutative symplectic geometry the authors construct consistent differential calculi on the algebra generated by operators satisfying the canonical commutation relation this leads to a mathematical framework in which quantum mechanics can be understood as noncommutative symplectic geometry they point out the possibility to describe interactions geometrically as deformations of the differential calculus

simultaneous stabilization of three or more plants conditions on the positive real axis do not suffice the problem of the simultaneous stabilizability of a finite family of single input single output time invariant systems by a time invariant controller is studied the link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants these necessary conditions are proved to be in general not sufficient this result also disproves a long standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller the main result is to show that unlike the case of two plants the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles

a mechanical model of brownian motion we consider a dynamical system consisting of one large massive particle and an infinite number of light point particles we prove that the motion of the massive particle is in a suitable limit described by the ornstein uhlenbeck process this extends to three dimensions previous results by holley in one dimension

synthesis of simultaneous stabilizing h controller controller in the chain scattering framework it has been shown in lee and soh 2004 that the problem of simultaneous stabilization of a finite collection of l linear time invariant plants such that each closed loop transfer function has an norm less than a prescribed performance level is equivalent to a simultaneous strongly stabilizing problem i e finding a single stable and norm bounded controller for associated l 1 unimodular plants such that each resulting closed loop system is stable and norm bounded this paper attempts to provide a solution to the simultaneous strongly stabilization problem a new sufficient condition in the form of the solution to only one algebraic riccati equation is provided for the existence of a strongly stabilizing controller for a given stable system obtained from the parameterization of stabilizing controllers with the parameterization of strongly stabilizing controllers provided in this paper a solution is given for the simultaneous strongly stabilization problem and hence

kinetic theory in the weak coupling approximation iii classical systems extension of the theory and further applications abstract the kinetic theory of open classical systems in the weak coupling approximation presented in a previous paper paper i is extended here to a wider class of systems so that it is applicable when the liouville operator of the open system has a continuous spectrum it is applied i to an anharmonic oscillator and ii to a free test particle model thus leading to a generalization of the conventionally used linearized landau equation problems related to the divergences that appear when the kinetic equations obtained by the formalism of paper i are considered in the limit of a continuous spectrum of the open system s liouville operator are also discussed

differential calculi on commutative algebras a differential calculus on an associative algebra a is an algebraic analogue of the calculus of differential forms on a smooth manifold it supplies a with a structure on which dynamics and field theory can to some extent be formulated in very much the same way we are used to in the geometrical arena underlying classical physical theories and models in previous work certain differential calculi on a commutative algebra exhibited relations with lattice structures stochastics and parametrized quantum theories this motivated the present systematic investigation of differential calculi on commutative and associative algebras various results about their structure are obtained in particular it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of i forms an example of such a product is provided by the ito calculus of stochastic differentials for the case where the algebra a is freely generated by coordinates xi i 1 n we study calculi for which the differentials dxi constitute a basis of the space of 1 forms as a left a module these may be regarded as deformations of the ordinary differential calculus on rn for n or 3 a classification of all orbits under the general linear group of such calculi with constant structure functions is presented we analyse whether these calculi are reducible i e a skew tensor product of lower dimensional calculi or whether they are the extension as defined in this paper of a one dimension lower calculus furthermore generalizations to arbitrary n are obtained for all these calculi

an approach to simultaneous system design i semialgebraic geometric methods this paper introduces semialgebraic parameterization as an approach to analyze simultaneous stabilization and pole placement problems rational families of plants of a given mcmillan degree that are simultaneously stabilizable by a fixed family of compensators are parameterized for a discrete family of plants the parameterization problem reduces to the simultaneous stabilization or the pole placement problem of a r tuple of multi input multi output plants by a nonswitching compensator it is shown that by removing a semialgebraic subset of a proper algebraic set the space of plants can be decomposed into components that are either simultaneously stabilizable or simultaneously unstabilizable under special cases explicit parameterization of the semialgebraic set is obtained finally a necessary condition for the simultaneous stabilization of single input or single output plants is obtained

the stationary state and the heat equation for a variant of davies model of heat conduction we study a variant of davies model of heat conduction consisting of a chain of classical or quantum harmonic oscillators whose ends are coupled to thermal reservoirs at different temperatures and where neighboring oscillators interact via intermediate reservoirs in the weak coupling limit we show that a unique stationary state exists and that a discretized heat equation holds we give an explicit expression of the stationary state in the case of two classical oscillators the heat equation is obtained in the hydrodynamic limit and it is proved that it completely describes the macroscopic behavior of the model

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

output feedback pole placement with dynamic compensators derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q by using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system the authors proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles