applications of the complex interpolation method to a von neumann algebra non commutative lp spaces non commutative lp spaces 1 p associated with a von neumann algebra are considered the paper consists of two parts in part i by making use of the complex interpolation method non commutative lp spaces are defined as interpolation spaces between the von neumann algebra in question and its predual also all expected properties such as duality and uniform convexity are proved in the frame of interpolaton theory and relative modular theory in part ii these lp spaces are compared with haagerup s lp spaces based on this comparison a non commutative analogue of the classical stein weiss interpolation theorem is obtained

minimal symmetric factorizations of symmetric real and complex rational matrix functions abstract real and complex rational matrix functions w such that w w t where 1 and their minimal factorizations w l t d l are studied such factorizations are described geometrically in terms of invariant subspaces with certain isotropic properties with respect to a quadratic form using results partly known and partly proved in the paper concerning the dimensions of such subspaces upper bounds are given for the degrees of the rational matrix functions l in the above factorizations

hermitian polynomial matrices isometries and inertial signs nonsingular hermitian polynomial matrices are described by two inner product spaces and selfadjoint operators the associated hermitian linear pencils are used to characterize isometries as an application a new proof of a factorization result on polynomial matrices with constant signature is given

existence of minimal nonsquare j symmetric factorizations for self adjoint rational matrix functions in this paper we show that any rational matrix function having hermitian values on the imaginary axis and with constant signature and constant pole signature admits a minimal symmetric factorization with possibly nonsquare factors our proof is based on a construction which shows that any such function can be extended preserving its mcmillan degree to a function that admits j symmetric factorization with square factors also we consider other properties of the factors in j symmetric factorizations particular attention is given to the study of the common invariant zero structure of these factors

inertia and eigenvalue relations between symmetrized and symmetrizing matrices for the real and the general field case abstract starting from a theorem of frobenius that every n n matrix is the product of two symmetric ones we study relations between the similarity invariants of a square matrix and the congruence invariants of its symmetric factors section 1 treats the real case sec 2 the arbitrary field case and sec 3 the indefinite inner product case for krein spaces the proofs are obtained from the real canonical pair form in secs 1 and 3 and from the recently found rational canonical pair form in sec 2 each time via combinatorial type arguments on weighted partitions of n the resulting theorems typically give bounds for the elementary divisor structure of a in terms of the index or signature of one or both of its symmetric factors or vice versa our results greatly extend and generalize the classic results of klein loewy taussky et al and in sec 2 put new light on waterhouse s recent characterization of hereditarily euclidean fields a short survey on the history of the subject from the early 1800s on completes the paper

remarks on orthogonal polynomials and balanced realizations given a proper antistable rational transfer function g a balanced realization of g is constructed as a matrix representation of the abstract shift realization introduced by fuhrmann 1976 the required basis is constructed as a union of sets of polynomials orthogonal with respect to weights given by the squares of the absolute values of minimal degree schmidt vectors of the corresponding hankel operators this extends results of fuhrmann 1991 obtained in the generic case

a duality theory for robust stabilization and model reduction fuhrmann 1991 1993 developed a duality theory in the context of hanel norm approximation and nehari complementation problems the class of functions involved were the scalar antistable transfer functions this work was extended using normalized coprime factorizations by fuhrmann and ober 1993a to the class of all minimal transfer functions in this paper we extend the scope of the duality theory significantly the paper presents a unified approach to problems of hankel norm approximation model reduction and robust control of rational multivariable transfer functions the unification is achieved by considering two classes of transfer functions and corresponding normalized coprime factorization using the youla kucera parametrization of all stabilizing controllers we single out a unique controller by imposing a mcmillan degree minimization restriction on the doubly coprime factorizations with this controller we construct an associated stable transfer function which we call the characteristic function many problems on the original system can be reduced to the study of the characteristic function

quadratic optimal controller to stabilise symmetrical systems it is considered a controller design in order to stabilize a control system the technique that is used for designing the controller includes a linear regulator and an asymptotical estimator which form the controller the linear regulator is designed by state estimators feedback for symmetrical systems minimizing a quadratic performance index computing the gain matrix of optimal feedback is solved by riccati s equation whilst the gain estimator matrix is computed by making use of symmetrical systems properties the symmetrical systems properties allow us to find the solution for riccati s dual equation without needing to solve two equations one for the feedback and other for the estimator using the controllability and observability matrices can be found a symmetrical and invertible matrix which can be used for computing the riccati s dual solution having the regulator and the estimator gain matrices we proceed to compute the transfer function of the controller based on estimator then we use this m

generalized inversion of finite rank hankel and toeplitz operators with rational matrix symbols abstract the goal of the paper is a generalized inversion of finite rank hankel operators and hankel or toeplitz operators with block matrices having finitely many rows to attain it a left coprime fractional factorization of a strictly proper rational matrix function and the bezout equation are used generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization

on pade type model order reduction of j hermitian linear dynamical systems a simple yet powerful approach to model order reduction of large scale linear dynamical systems is to employ projection onto block krylov subspaces the transfer functions of the resulting reduced order models of such projection methods can be characterized as pade type approximants of the transfer function of the original large scale system if the original system exhibits certain symmetries then the reduced order models are considerably more accurate than the theory for general systems predicts in this paper the framework of j hermitian linear dynamical systems is used to establish a general result about this higher accuracy in particular it is shown that in the case of j hermitian linear dynamical systems the reduced order transfer functions match twice as many taylor coefficients of the original transfer function as in the general case an application to the sprim algorithm for order reduction of general rcl electrical networks is discussed

on the partial realization problem abstract in this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra most of which were originally developed in other contexts we approach the partial realization problem from several different angles and explore the connections to such topics as factoriza tion of hankel matrices block tridiagonalization generalizations of the lanczos process for biorthogonalization the euclidean algorithm and the principal part con tinued fractions of arne magnus the pade table and the berlekamp masseyalgo rithm in this way we are able to clarify some previous results by rissanen kalman and others and place them in a broader context this leads to several results and concepts which we think are new our analysis is restricted to the scalar case but some definitions and formulations have been rigged to facilitate an extension to the matrix case

un renforcement de la propriete t nous proposons un renforcement de la propriete t de kazhdan en remplacant les representations unitaires par les representations dans des espaces de hilbert qui ne sont pas necessairement unitaires mais a croissance exponentielle suffisamment petite on sait qu un groupe localement compact a la propriete t si et seulement si il existe un idempotent p dans c max g tel que pour toute representation unitaire continue h de g p est le projecteur orthogonal sur le sous espace ferme de h forme des vecteurs ginvariants pour la propriete t renforcee nous posons une definition dans ce style on appelle longueur sur un groupe localement compact g une fonction continue g r verifiant g 1 g et g1g2 g1 g2 pour g g1 g2 g dans la suite pour tout groupe localement compact g on supposera choisie une mesure de haar a gauche qui munit cc g d une structure d algebre par convolution

on the bezoutian for polynomial matrices abstract the bezoutian b of two polynomial matrices can be described as a solution of a linear matrix equation this fact yields a new proof of the barnett factorization of b

on invertibility of nonsquare generalized bezoutians abstract conditions for inverting h bezoutians of nonsquare matrix polynomials are studied necessary and sufficient conditions for invertibility of the anderson jury bezoutian are obtained an application to the inversion of block hankel matrices is given

approximation properties for noncommutative lp spaces associated with lattices in lie groups abstract in 2010 lafforgue and de la salle gave examples of noncommutative l p spaces without the operator space approximation property oap and hence without the completely bounded approximation property cbap to this purpose they introduced the property of completely bounded approximation by schur multipliers on s p denoted ap p cb schur and proved that for p 1 4 3 4 the groups sl n z with n 3 do not have the ap p cb schur since for p 1 the ap p cb schur is weaker than the approximation property of haagerup and kraus ap these groups were also the first examples of exact groups without the ap recently haagerup and the author proved that also the group sp 2 r does not have the ap without using the ap p cb schur in this paper we prove that sp 2 r does not have the ap p cb schur for p 1 12 11 12 it follows that a large class of noncommutative l p spaces does not have the oap or cbap

a polynomial characterization of mathcal a mathcal b invariant and reachability subspaces based on the state space model of p fuhrmann a link is laid between the geometric approach to linear system theory as developed by w m wonham and a s morse and the approach based on polynomial matrices in particular polynomial characterizations of a b invariant and reachability subspaces are given these characterizations are used to prove the equivalence of the disturbance decoupling problem and the exact model matching problem and also to connect the polynomial matrix and the geometric approach to the construction of observers finally constructive procedures and conditions are given for computing the supremal a b invariant subspace and reachability space and for checking the solvability of the exact model matching problem

indefinite inner product spaces associated with hermitian polynomial matrices abstract nonsingular hermitian polynomial matrices are completely characterized by bases and shift operators of suitably defined inner product spaces

approximation properties for group algebras and group von neumann algebras let g be a locally compact group let c g resp vn g be the c algebra resp the von neumann algebra associated with the left regular representation i of g let a g be the fourier algebra of g and let moa g be the set of completely bounded multipliers of a g with the completely bounded norm moa g is a dual space and we say that g has the approximation property ap if there is a net ua of functions in a g with compact support such that ua a 1 in the associated weak topology in particular g has the ap if g is weakly amenable a g has an approximate identity that is bounded in the completely bounded norm for a discrete group f we show that f has the ap x c f has the slice map property for subspaces of any c algebra x vn jt has the slice map property for a weakly closed subspaces of any von neumann algebra property se the semidirect product of weakly amenable groups need not be weakly amenable we show that the larger class of groups with the ap is stable with respect to semidirect products and more generally this class is stable with respect to group extensions we also obtain some results concerning crossed products for example we show that the crossed product m a g of a von neumann algebra m with property s by a group g with the ap also has property s

canonical realizations by factorization of constant matrices in van der schaft and rapisarda 2011 3 we showed that a state variable for a lti system can be computed factorizing a two variable polynomial matrix computed directly from the system equations different factorizations of this matrix yield different state maps and consequently different state equations based on this fact we present a unifying point of view on some classical canonical forms for linear systems

autonomous subbehaviours and output nulling subspaces the aim of the present paper is to point out some connections between geometric control theory and the theory of behaviours specifically given a behaviour we analyse the set of all autonomous subbehaviours and relate those to some polynomial matrix completion problems we use this to analyse the different module structures on some vectorial polynomial spaces subsequently we apply this analysis to the study and the polynomial characterization of output nulling subspaces controlled invariant subspaces and the natural feedback induced module structures in these spaces toeplitz operators and wiener hopf factorizations play an important role

a fast and faithful collocation method with efficient matrix assembly for a two dimensional nonlocal diffusion model abstract the peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities which cannot be described properly by the classical theory of solid mechanics however the operator in the peridynamic theory is nonlocal so the resulting numerical methods generate dense or full coefficient matrices which require o n 2 of memory where n is the number of unknowns in the discretized system gaussian types of direct solvers which were traditionally used to solve these problems require o n 3 of operations furthermore due to the singularity of the kernel in the peridynamic model the evaluation and assembly of the coefficient matrix can be very expensive numerous numerical experiments have shown that in many practical simulations the evaluation and assembly of the coefficient matrix often constitute the main computational cost the significantly increased computational work and memory requirement of the peridynamic model over those for the classical partial differential equation models severely limit their applications especially in multiple space dimensions we develop a fast and faithful collocation method for a two dimensional nonlocal diffusion model which can be viewed as a scalar valued version of a peridynamic model without using any lossy compression but rather by exploiting the structure of the coefficient matrix the new method reduces the evaluation and assembly of the coefficient matrix by o n reduces the computational work from o n 3 required by the traditional methods to o n log 2 n and the memory requirement from o n 2 to o n numerical results are presented to show the utility of the fast method

functional models in linear algebra abstract this paper examines the role of functional models in linear algebra and some related areas of application notably linear systems theory this approach adopts an intermidiate level of abstraction somewhere between abstract module theory on the one hand and matrix theory on the other it is a powerful method which simplifies many proofs provides insights and illuminates the less obvious connections between linear algebra and operator theory approximation theory moment problems interpolation and control theory

the kernel of the bezoutian of polynomial matrices abstract a definition of anderson and jury is extended to define a bezoutian b for a quadruple f g u d of polynomial matrices which have the property that the strictly proper rational parts of f 1 g and ud 1 are equal the right kernel of b is described by means of block hankel matrices associated to d and to a greatest common right divisor of d and u

a global parametrization of symptotically stable linear systems abstract in this paper we construct a homeomorphism from the set of p m transfer functions of mcmillan degree n onto the open subspace of asymptotically stable linear systems this homeomorphism yields a one to one correspondence between 1 i canonical forms for state space equivalence of minimal systems and 2 ii canonical forms for state space equivalence of asymptotically stable minimal systems implications for the topology of various sets of asymptotically stable systems are given

complex static skew symmetric output feedback control we study the problem of feedback control for skew symmetric and skew hamiltonian transfer functions using skew symmetric controllers this extends work of helmke et al who studied static symmetric feedback control of symmetric and hamiltonian linear systems we identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a grassmannian give necessary and sufficient conditions for pole placement by static skew symmetric complex feedback and use schubert calculus for the orthogonal grassmannian to count the number of complex feedback laws when there are finitely many of them finally we also construct a real skew symmetric linear system with only real feedback for any set of real poles

a fast galerkin method with efficient matrix assembly and storage for a peridynamic model peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities which cannot be described properly by classical theory of solid mechanics however the operators in the peridynamic models are nonlocal so the resulting numerical methods generate dense or full stiffness matrices gaussian types of direct solvers were traditionally used to solve these problems which requires o n 3 of operations and o n 2 of memory where n is the number of spatial nodes this imposes significant computational and memory challenge for a peridynamic model especially for problems in multiple space dimensions a simplified model which assumes that the horizon of the material d o n 1 was proposed to reduce the computational cost and memory requirement to o n however the drawback is that the corresponding error estimate becomes one order suboptimal furthermore the assumption of d o n 1 does not seem to be physically reasonable since the horizon d represents a physical property of the material that should not depend on computational mesh size we develop a fast galerkin method for the non simplified peridynamic model by exploiting the structure of the stiffness matrix the new method reduces the computational work from o n 3 required by the traditional methods to o nlog 2n and the memory requirement from o n 2 to o n without using any lossy compression the significant computational and memory reduction of the fast method is better reflected in numerical experiments when solving a one dimensional peridynamic model with 2 1 4 16 384 unknowns the traditional method consumed cpu time of 6days and 11h while the fast method used only 3 3s in addition on the same computer with 128gb memory the traditional method with a gaussian elimination or conjugate gradient method ran out of memory when solving the problem with 2 1 6 131 072 unknowns in contrast the fast method was able to solve the problem with 2 2 8 268 435 456 unknowns using 3days and 11h of cpu time this shows the benefit of the significantly reduced memory requirement of the fast method

a unifying approach to some old and new theorems on distribution and clustering abstract a unifying approach is proposed to studying the distributions of eigenvalues and singular values of toeplitz matrices associated with a fourier series and multilevel toeplitz matrices associated with a multidimensional fourier series obtained are the extensions of the szego and avram parter theorems where the generating function is now required to belong to l 2 and not necessarily to l analogous extensions are given for multilevel toeplitz matrices in particular it is proved that if f x 1 x p l 2 then the p level complex toeplitz matrices allied with f have their singular values distributed as f x 1 x p the distribution results for the cesaro optimal circulants hold even if f l 1 also suggested are new theorems on clustering that have to do with the preconditioning of multilevel toeplitz matrices by multilevel circulants

on hamiltonian rational transfer functions abstract a sequel to a previous paper fuhrmann 1983 this paper contains a detailed study of rational transfer functions having hamiltonian symmetry a detailed realization procedure is described that reflects this symmetry as a corollary the williamson canonical form for hamiltonian maps in symplectic spaces is derived

output feedback pole assignment for transfer functions with symmetries this paper studies the problem of pole assignment for symmetric and hamiltonian transfer functions a necessary and sufficient condition for pole assignment by complex symmetric output feedback transformations is given moreover in the case where the mcmillan degree coincides with the number of parameters appearing in the symmetric feedback transformations we derive an explicit combinatorial formula for the number of pole assigning symmetric feedback gains the proof uses intersection theory in projective space as well as a formula for the degree of the complex lagrangian grassmann manifold

circulant preconditioned iterative methods for peridynamic model simulation the galerkin finite element formulation is employed to discretize the peridynamic model for microelastic materials on a finite bar the coefficient matrix of the resulting system possesses the symmetric toeplitz plus tridiagonal structure as the symmetric toeplitz plus tridiagonal matrix vector multiplication can be efficiently implemented via using the fast fourier transforms ffts which can reduce the computational cost of the matrix vector product in this paper we consider the conjugate gradient cg method with a circulant preconditioner to solve the discretized linear systems moreover the spectrum of the preconditioned matrix is proven to be mostly clustered around 1 thus the accelerating benefits of circulant preconditioners are theoretically guaranteed numerical experiments are reported to demonstrate the effectiveness and robustness of circulant preconditioners in terms of the number of iterations and cpu time finally the comprehensive performances evaluation of different circulant preconditioners are also given

unitary solutions of a class of algebraic riccati equations and factorization abstract we consider the algebraic riccati equation xd 1 x xd 2 d 3 x d 1 0 where d 2 and d 3 are self adjoint we are interested in contractive and unitary solutions x of this equation stability properties of such solutions under perturbations of the coefficients are studied further the connection between unitary solutions of such equations and symmetric factorizations of nonnegative rational matrix functions is investigated

on a theorem of hermite and hurwitz abstract the hermite hurwitz theorem computes the degree over r of a real rational function in terms of the signature of an associated quadratic form known today as the hankel matrix of this formula which hermite was led to by his work on the problem of representing integers as sums of squares gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations in this paper this theorem and various generalizations to the matrix valued case are discussed and described in terms of signature formulae these include its relation to stability theory and the matrix hermite hurwitz theorem of bitmead anderson as applied to questions of circuit synthesis this also includes a global form of hormander s signature formula for the maslov index of a rational loop in a lagrangian grassmannian due to byrnes and duncan and applications to the topology of spaces of rational matrix valued functions following the work of brockett byrnes and duncan this includes in particular a topological proof of the matrix hermite hurwitz theorem

multipliers of the fourier algebras of some simple lie groups and their discrete subgroups for any amenable locally compact group g the space of multipliers ma g of the fourier algebra a g coincides with the space b g of functions on g that are linear combinations of continuous positive definite functions we prove that ma g b g 0 for many non amenable connected groups more specifically we prove that moa g b g 0 for the classical complex lie groups and the general lorentz groups soo n 1 n 2 moa g is a certain subspace of ma g which we call the space of completely bounded multipliers of a g unlike ma g the space moa g has nice stability properties with respect to direct products of groups it is known that the fourier algebra of the free group on n generators n 2 2 admits an unbounded approximate unit pn which is bounded in the multiplier norm we extend this result to any closed subgroup of the general lorentz group soo n 1 moreover we show that for these groups pn can be chosen to be bounded with respect to the moa g norm by a duality argument we obtain that the reduced c algebra of every discrete subgroup of soo n 1 has the completely bounded approximation property in particular this property holds for c f2 the reduced c algebra of the free group on two generators we also prove

on tangential matrix interpolation abstract the paper presents an algebraic approach using polynomial and rational models over an arbitrary field to tangential interpolation problems both by polynomial as well as rational matrix functions appropriate extensions of scalar problems associated with the names of lagrange first order hermite high order and newton recursive are derived the relation of interpolation problems to the matrix chinese remainder theorem are clarified some two sided interpolation problems are discussed using the theory of tensored functional models

bezoutians of polynomial matrices and their generalized inverses abstract the bezoutian b of a quadruple f g u d of polynomial matrices is studied it is shown that under certain coprimeness conditions there is a 1 or a 1 2 inverse of b which is a block hankel matrix

polynomial interpolation and gaussian quadrature for matrix valued functions abstract the techniques for polynomial interpolation and gaussian quadrature are generalized to matrix valued functions it is shown how the zeros and rootvectors of matrix orthonormal polynomials can be used to get a quadrature formula with the highest degree of precision

stable invariant lagrangian subspaces factorization of symmetric rational matrix functions and other applications several applications of earlier results by the authors concerning various notions of stability of invariant lagrangian subspaces are studied the applications include stability of symmetric minimal factorizations of real symmetric rational matrix functions stability of factorizations of certain classes of symmetric matrix polynomials and stably well posed matricial boundary value problems with symmetries an interpretation of many stability results in terms of equilibria of lie group actions on subspaces is given as well

remarks on the inversion of hankel matrices the paper continues the investigation into the links between algebraic system theory more specifically the partial realization problem and the problem of hankel matrix inversion the representation of the inverse of a hankel matrix as a bezoutiant and the bezout equation play a principal role

reduction to generalized hessenberg form and inverse spectral problems abstract lie algebraic generalizations of hessenberg matrices are considered we address the questions under what conditions an element of the classical lie algebra can be reduced by a similarity transformation to the generalized hessenberg form we completely solve an inverse spectral problem for symmetric generalized hessenberg matrices related to classical lie algebras

on structured singular values of reciprocal matrices computing the structured singular value is a bottleneck of the robustness analysis and synthesis of control systems in this paper it is shown that the structured singular value is identical to the maximum singular value for an important class of matrices called reciprocal which reflects the intrinsic symmetry of physical world

brief necessary and sufficient conditions for existence of j spectral factorization for para hermitian rational matrix functions for a para hermitian rational matrix function g l j c li p a 1b where j j is invertible and which has no poles and zeros on the imaginary line we give necessary and sufficient conditions in terms of a b c and j for the existence of a j spectral factorization as well as an algorithm to obtain the j spectral factor in case it exists

bezout and hankel matrices associated with row reduced matrix polynomials barnett type formulas abstract based on the approach introduced by b d o anderson and e i jury in 1976 the definition of finite hankel and bezout matrices corresponding to matrix polynomials is extended to the case where the denominator of the corresponding rational matrix function is not necessarily monic but is row reduced the matrices introduced keep most of the well known properties that hold in the monic case in particular we derive extensions of formulas giving a connection with polynomials in the companion matrix usually called barnett formulas of the inversion theorem and of formulas concerning alternating products of hankel and bezout matrices

generalized bezoutian and matrix equations abstract a natural generalization of the classical bezout matrix of two polynomials is introduced for a family of several matrix polynomials the main aim of the paper is to show that this generalized bezoutian serves as an adequate connecting link between the class of equations in matrix polynomials m y z l r and the class of linear matrix equations ax xb c each equation in one of these classes is coupled with a certain equation in the other class so that for each couple the generalized bezoutian corresponding to a solution y z of the equation in matrix polynomials is a solution of the matrix equation and conversely any solution x of the matrix equation is a generalized bezoutian corresponding to a certain solution of the equation in matrix polynomials in particular either both equations are solvable or both have no solutions explicit formulas connecting the solutions of the two equations are given also various representation formulas for the generalized bezoutian are derived and its relation to the resultant matrix and the greatest common divisor of several matrix polynomials is discussed

principal component analysis of flexible systems open loop case a generic class of flexible systems characterized by finitely many lightly damped harmonic oscillators is analyzed by means of the open loop principal component analysis that is singular value analysis and gramian balancing as the main result it is shown that as the damping ratio goes to zero the balanced state coordinates are decoupled and coincide with the modal coordinates further simple formulas expressing the asymptotic singular values as functions of the modal parameters are derived

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

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

tangent spaces of rational matrix functions we consider the task of determining tangent spaces for classes of rational matrix valued functions our analysis is based on methods from control theory and in particular the theory of polynomial models explicit descriptions of tangent spaces of rational transfer functions stable rational transfer functions rational inner functions and symmetric rational transfer functions are obtained moreover a new proof of delchamps s decomposition formula for the tangent bundle of rational transfer functions is given a riemannian metric as well as a symplectic structure is defined 0 1998 elsevier science inc

duality in polynomial models with some applications to geometric control theory duality is studied in the context of polynomial models for linear systems the output injection group the dual of the feedback group is studied and a polynomial characterization of c a invariant subspaces as well as of the maximal reachability subspaces contained in ker c is given

continuous and discontinuous finite element methods for a peridynamics model of mechanics in contrast to classical partial differential equation models the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro differential equation that does not involve spatial derivatives of the displacement field as a result the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems the peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions based on a variational formulation continuous and discontinuous galerkin finite element methods are developed for the peridynamic model discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges through a series of simple one dimensional computational experiments we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates one issue addressed is the effect of the relative sizes of the horizon and the grid for problems with smooth solutions we find that continuous and discontinuous piecewise linear approximations result in the same accuracy as that obtained by continuous piecewise linear approximations for classical models piecewise constant approximations are less robust and require the grid size to be small with respect to the horizon we then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions in case a grid point is placed at the locations of the jump discontinuities such results are directly obtained in the general case we show that such results can be obtained through a simple automated abrupt local refinement of elements containing the discontinuity in order to reduce the number of degrees of freedom while preserving accuracy we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities

on the parametrization of conditioned invariant subspaces and observer theory the present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair c a we do this analysis in two different ways one based on polynomial models starting with a characterization obtained in p a fuhrmann linear operators and systems in hilbert space 1981 ieee trans automat control ac 26 1981 284 the other being a state space approach toeplitz operators projections in polynomial and rational models wiener hopf factorizations and factorization indices all appear and are tools in the characterizations we single out an important subclass of conditioned invariant subspaces namely the tight ones which already made an appearance in p a fuhrmann u helmke systems control lett 30 1997 217 a precursor of the present paper of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair c a is the structural map that associates with any reachable pair with only the input dimension constrained a uniquely determined conditioned invariant subspace the construction of this map uses polynomial models and the shift realization new objects the partial observability and reachability matrices are introduced which are needed for the state space characterizations kernel and image representations for conditioned invariant subspaces are derived uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness we pass on to an analysis and derivation of the kronecker hermite canonical form for full column rank rectangular polynomial matrices this extends the work of a e eckberg a characterization of linear systems via polynomial matrices and module theory ph d thesis mit cambridge ma 1974 g d forney siam j control optim 13 1973 493 and d hinrichsen h f munzner and d pratzel wolters systems

the spectrum of a family of circulant preconditioned toeplitz systems the solutions of symmetric positive definite toeplitz systems ax b are studied by the preconditioned conjugate gradient method the preconditioner is the circulant matrix c that minimizes the frobenius norm c a f t chan an optimal circulant preconditioner for toeplitz systems ucla department of mathematics cam report 87 06 june 1987 the convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of c 1 a for toeplitz matrix a with entries which are fourier coefficients of a positive function in the wiener class this paper establishes the invertibility of c finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix c 1 a as the dimension increases and proves that they are clustered around 1

minimal factorizations of rational matrix functions in terms of polynomial models the problem of minimal factorization of rational bicausal matrices is considered using polynomial models the factors are given explicitly in terms of polynomial matrices

the algebraic riccati equation a polynomial approach abstract polynomial models are used to give a unified approach to the problem of classifying the set of all real symmetric solutions of the algebraic riccati equation

a polynomial approach to hankel norm and balanced approximations abstract a new unified approach to problems of hankel norm and balanced approximations is presented which is based on a combination of polynomial algebra and the geometry of invariant subspaces contrary to state space methods where contact with external properties of systems is indirect the approach presented yields new insights into basic properties of hankel norm approximation and balanced realizations several approximation results are interpreted geometrically in terms of projections also duality results in this area are printed out

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

transfer functions and operator theory abstract the notion of a transfer function from systems theory serves as an important tool in dealing with certain problems in operator and matrix theory this is illustrated with a variety of material taken from existing literature a striking feature of the results is their high degree of explicitness

optimal trigonometric preconditioners for nonsymmetric toeplitz systems abstract this paper is concerned with the solution of systems of linear equations tnxn bn where tn n n denotes a sequence of nonsingular nonsymmetric toeplitz matrices arising from a generating function of the wiener class we present a technique for the fast construction of optimal trigonometric preconditioners mn mn t ntn of the corresponding normal equation which can be extended to toeplitz least squares problems in a straightforward way moreover we prove that the spectrum of the preconditioned matrix mn1t ntn is clustered at 1 such that the pcg method applied to the normal equation converges superlinearly numerical tests confirm the theoretical expectations

a new realization theory a realization theory that is based on the kalman system theory and motivated by the fuhrmann realization theory is presented in the fuhrmann realization theory the realization state space is defined by a natural polynomial module k sub q which is related to a nonsingular polynomial matrix q the module k sub q is formed of all f where q sup 1 f is strictly proper in the realization theory proposed the realization state space is defined by a different module m sub q which is formed of properly truncated strictly proper formal power series of q sup 1 f the realization theory can be used to prove the dual realization scheme induced by the fuhrmann realization theory