model reduction of systems with zeros interlacing the poles the problem of reducing a system with zeros interlacing the poles zip on the real axis is considered it is proved that many model reduction methods such as the balanced truncation balanced residualization suboptimal and optimal hankel approximations inherit the zip property properties of the hankel singular values of zip systems are also listed

behaviour of the frobenius map in a noncommutative world abstract we study various properties of the frobenius map x x p on a noncommutative algebra r over a field f of positive characteristic p we call r perfect whenever is surjective more generally we say that r is additively respectively multiplicatively p power closed if for every x y r there exists a positive integer n such that x p n y p n respectively x p n y p n is a p power in r if for example n were a ring homomorphism as when r is commutative then r would be additively and multiplicatively p power closed we show that a finite dimensional unital algebra r is lie nilpotent if and only if r is additively respectively multiplicatively p power closed and each of its simple homomorphic images has schur index 1 since not all finite dimensional perfect division algebras are lie nilpotent the schur index condition cannot be omitted we deduce that similar results hold for the class of all finitely generated pi algebras moreover for this same class we give a positive solution to the following problem reminiscent of the problems of kurosh and levitzki is every finitely generated perfect algebra finite dimensional

a note on the existence of positive realizations the problem of existence of positive realizations for a class of continuous and discrete time systems is addressed and solved it is shown that the existence of a minimal positive realization can be decided on the basis of two sub matrices of the infinite hankel matrix associated with the transfer function to be realized applications of this result to the problem of existence and construction of non minimal realizations are also discussed

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

positive sesquilinear form measures and generalized eigenvalue expansions positive operator measures with values in the space of bounded operators on a hilbert space and their generalizations mainly positive sesquilinear form measures are considered with the aim of providing a framework for their generalized eigenvalue type expansions though there are formal similarities with earlier approaches to special cases of the problem the paper differs e g from standard rigged hilbert space constructions and avoids the introduction of nuclear spaces the techniques are predominantly measure theoretic and hence the hilbert spaces involved are separable the results range from a naimark type dilation result to direct integral representations and a fairly concrete generalized eigenvalue expansion for unbounded normal operators

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

diagonalization and representation results for nonpositive sesquilinear form measures we study decompositions of operator measures and more general sesquilinear form measures e into linear combinations of positive parts and their diagonal vector expansions the underlying philosophy is to represent e as a trace class valued measure of bounded variation on a new hilbert space related to e the choice of the auxiliary hilbert space fixes a unique decomposition with certain properties but this choice itself is not canonical we present relations to naimark type dilations and direct integrals

characterizations of the canonical phase observable in this paper we investigate various properties of phase observables which could serve to determine the canonical phase observable among the family of all phase observables we also show that any contractive weighted shift operator defines a unique phase observable and we characterize phase observables that give the most accurate phase distribution in coherent states in the classical limit

technical communique dissipative analysis and control of state space symmetric systems the paper addresses the problem of analysis and static output feedback control synthesis for strict quadratic dissipativity of linear time invariant systems with state space symmetry as a particular case of dissipative systems we consider the mixed h and positive real performance criterion and we develop an explicit expression for calculating the h norm of these systems subsequently an explicit parametrization of the static output feedback control gains that solve the mixed h and positive real performance problem is obtained numerical examples demonstrate the use and computational advantages of the proposed explicit solutions

wigner quantisation of arrival time and oscillator phase by means of the wigner distribution function an expression is derived for the quasiprobability that a free particle in one dimension arrives at the origin during a given time interval and also for the quasiprobability that the phase of a one dimensional harmonic oscillator lies in a given segment of the circle it is shown that the best bounds on these quasiprobabilities p are 0 1559 or p or 1 0077 independent of the size of the interval it is shown that on disjoint unions of infinitely many intervals the quasiprobabilities may add up to any positive or negative real value it is briefly indicated how negative probabilities can be avoided altogether in an alternative approach to the quantisation of arrival time and oscillator phase

the nilpotency of the radical in a finitely generated p i ring abstract let r x 1 x k be a p i ring satisfying a monic polynomial identity one of its coefficients is 1 where is a central noetherian subring it is proven that n r the nil radical of r is nilpotent as a corollary by taking f a field we settle affirmatively the open problem posed in c procesi rings with polynomial identities p 186 marcel dekker new york 1973 we prove the jacobson radical of a finitely generated p i algebra is nilpotent

gauge covariant observables and phase operators we characterize those hilbert space operators which are first moments of the gauge covariant positive operator valued measures introduced by lahti and pellonpaa we consider the suitability of such measures for describing quantum phase observing that the gauge angle defining the measure is not always closely related to the spectrum of the first moment operator we take the position that the first moment operator represents the quantal property the measure is imperfectly describing this is in contrast to the viewpoint that positive operator valued measures are the fundamental quantum observables

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

phase observables phase operators and operator orderings we represent quantum phase observables as phase shift covariant normalized positive operator measures the phase operators are the first moment operators of the phase observables a phase operator determines the associated phase observable uniquely we show that the cahill glauber s ordered phase operators are determined by phase shift covariant generalized operator measures which are ordinary operator measures whenever re s 0 and phase observables when s 1 the wigner weyl quantized phase operator is not determined by any phase observable we investigate the classical limit of covariant generalized operator measures in coherent states

convergence of the iterative rational krylov algorithm abstract the iterative rational krylov algorithm irka of gugercin et al 2008 8 is an interpolatory model reduction approach to the optimal h 2 approximation problem even though the method has been illustrated to show rapid convergence in various examples a proof of convergence has not been provided yet in this note we show that in the case of state space symmetric systems irka is a locally convergent fixed point iteration to a local minimum of the underlying h 2 approximation problem

model reduction for state space symmetric systems in this paper the model reduction problem for state space symmetric systems is investigated first it is shown that several model reduction methods such as balanced truncation balanced truncation which preserves the dc gain optimal and suboptimal hankel norm approximations inherit the state space symmetric property furthermore for single input and single output siso state space symmetric systems we prove that the h norm of its transfer functions can be calculated via two simple formulas moreover the siso state space symmetric systems are equivalent to systems with zeros interlacing the poles zip under mild conditions

quantum tomography phase space observables and generalized markov kernels we construct a generalized markov kernel which transforms the observable associated with the homodyne tomography into a covariant phase space observable with a regular kernel state illustrative examples are given in the cases of a schrodinger cat kernel state and the cahill glauber s parametrized distributions also we consider an example of a kernel state when the generalized markov kernel cannot be constructed

decentralized control of symmetric systems abstract this paper is concerned with the stabilizability problem of symmetric systems by means of decentralized controllers it is shown that the set of decentralized fixed modes of a symmetric system is equal to the set of uncontrollable and unobservable modes of the system and implies that the stabilizability of a state space symmetric system by means of a decentralized controller is equivalent to the stabilizability for the system by means of a centralized controller

optimal symmetric h sub 2 controllers for systems with collocated sensors and actuators this note addresses the problem of designing the optimal symmetric h 2 controller for a plant with collocated sensors and actuators but with possibly asymmetric disturbance injection and performance specification for such a control problem the generalized plant has a symmetric block representing the transfer matrix from the control input to the measured output a complete solution to the optimal h 2 control problem with the symmetric structural constraint is given in terms of the optimal solution to a standard h 2 model matching problem without the constraint

pi algebras generated by nilpotent elements of bounded index abstract let r be an associative algebra over a field f of positive characteristic p we address the following problem if r is generated by nilpotent elements with bounded index under what conditions can we conclude that r itself is nil of bounded index we prove that whenever r satisfies the engel condition and p 0 or r is lie soluble and p 2 then the conclusion holds along the way we prove that if r satisfies the engel condition then there exists a positive integer n such that the map x x p n is additive the converse also holds when f is infinite or r is generated by nilpotent elements

stabilization and h control of symmetric systems an explicit solution abstract in this paper we address the h control analysis the output feedback stabilization and the output feedback h control synthesis problems for state space symmetric systems using a particular solution of the bounded real lemma for an open loop symmetric system we obtain an explicit expression to compute the h norm of the system for the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state space symmetric systems for the h control synthesis problem we derive an explicit expression for the optimally achievable closed loop h norm and the optimal control gains extension to robust and positive real control of such systems are also examined these results are obtained from the linear matrix inequality formulations of the stabilization and the h control synthesis problems using simple matrix algebraic tools

on the structure of covariant phase observables we study the mathematical structure of covariant phase observables such observables can alternatively be expressed as phase matrices as sequences of unit vectors as sequences of phase states or as equivalence classes of covariant trace preserving operations covariant generalized operator measures are defined by structure matrices which form a w algebra with phase matrices as its subset the properties of the radon nikod m derivatives of phase probability measures are studied

when is a power of the frobenius map on a noncommutative ring a homomorphism abstract let x x p be the frobenius map on an associative unital ring r with prime characteristic p it is well known that whenever r is commutative n is a ring homomorphism for all positive integers n the converse however is not true in general indeed we prove that if r is m engel for some positive integer m then there exists a positive integer n 0 depending only on m such that for all n n 0 n is a ring homomorphism with central image conversely if any one of the following conditions holds n respects addition n respects multiplication n respects lie multiplication or the image of n is commutative then r is m engel for some m depending only on n consequently if is surjective and any one of the aforementioned conditions holds then r must be commutative

interpolatory h h model reduction abstract we introduce an approach to h model reduction that is founded on ideas originating in realization theory interpolatory h 2 optimal model reduction and complex chebyshev approximation within this new framework we are able to formulate a method that remains effective in large scale settings with the main cost dominated by sparse linear solves by employing loewner data driven partial realizations within each optimization cycle computationally demanding h norm calculations can be completely avoided several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better h performance than those produced by balanced truncation these models often are as good as and occasionally better than those models produced by optimal hankel norm approximation in all cases reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal hankel norm approximation