violation of the fluctuation dissipation theorem in glassy systems basic notions and the numerical evidence this review reports on the research done during past years on violations of the fluctuation dissipation theorem fdt in glassy systems it is focused on the existence of a quasi fluctuation dissipation theorem qfdt in glassy systems and the current supporting knowledge gained from numerical simulation studies it covers a broad range of non stationary aging and stationary driven systems such as structural glasses spin glasses coarsening systems ferromagnetic models at criticality trap models models with entropy barriers kinetically constrained models sheared systems and granular media the review is divided into four main parts 1 an introductory section explaining basic notions related to the existence of the fdt in equilibrium and its possible extension to the glassy regime qfdt 2 a description of the basic analytical tools and results derived in the framework of some exactly solvable models 3 a detailed report of the current evidence in favour of the qfdt and 4 a brief digression on the experimental evidence in its favour this review is intended for inexpert readers who want to learn about the basic notions and concepts related to the existence of the qfdt as well as for the more expert readers who may be interested in more specific results

large deviations condensation and giant response in a statistical system we study the probability distribution p of the sum of a large number of non identically distributed random variables n m condensation of fluctuations the phenomenon whereby one of such variables provides a macroscopic contribution to the global probability is discussed and interpreted by analogy with phase transitions in statistical mechanics a general expression for p is derived and its sensitivity to the details of the distribution of a single n m is worked out these general results are verified by the analytical and numerical solution of some specific examples

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

on singular implicit linear dynamical systems we investigate properties of existence unicity representation of the causal solutions of implicit linear systems or generalized systems when the underlying matrix pencil is singular we relate the geometric and the algebraic approaches the main conclusion is that if the underlying matrix pencil is column singular i e has a nonempty set of column minimal indices the causal solutions when they exist can exactly be represented as the output of a classical two player dynamical system where the second player accounts for the nonunicity properties of the equivalent system are related to those of the singular matrix pencils made with the given matrices

slow relaxation in the large n model for phase ordering the basic features of the slow relaxation phenomenology arising in phase ordering processes are obtained analytically in the large n model through the exact separation of the order parameter into the sum of thermal and condensation components the aging contribution in the response function x a g t t w is found to ohey a pattern of behavior under variation of dimensionality qualitatively similar to the one observed in ising systems there exists a critical dimensionality d 4 above which x a g t t w is proportional to the defect density d t while for d 4 it vanishes more slowly than d t and at d 2 does not vanish as in the ising case this behavior can be understood in terms of the dependence on dimensionality of the interplay between the defect density and the effective response associated to a single defect

descriptor representations without direct feedthrough term abstract descriptor representations are considered that are given by e a b c d with d 0 minimality under external equivalence is characterized in terms of the matrices e a b and c also transformations are given by which minimal e a b c representations are related under external equivalence the transformations turn out to be more simple than in the d 0 case algorithms for rewriting an e a b c d representation in e a b c form are also given finally a realization procedure is presented for obtaining a minimal e a b c representation for a system that is given in polynomial matrix fractional form

algebraic characterizations of almost invariance this paper stresses the algebraic as opposed to analytic nature of the concept of almost invarianoe which has been introduced by willems 1980 as a means of studying asymptotic aspects of linear systems based on an interpretation in terms of difference equations it is shown that the basic properties and a number of characterizations of tho four fundamental classes of subspaces can be derived in a relatively simple and purely algebraic fashion among other things this paper also extends the results of hautus on the frequency domain interpretation of ordinary a b invariant subspaces re derives the pencil characterization of jaffe and karcanias 1981 and obtains a new hybrid type of characterization the last result also loads to a rank test for almost invariance

feedback and pole placement in descriptor variable systems the effects and uses of applying linear feedback to continuous time descriptor systems are studied structural changes resulting from feeding back the slow and fast parts of the trajectory separately are characterized it is shown that under certain conditions related to controllability the poles of the slow subsystem may be shifted arbitrarily and the impulsive behaviour of the fast subsystem may be eliminated

generalized state space realizations of non proper rational transfer functions in this paper we propose an algebraic realization theory for polynomial transfer functions based on classical module theory realizations are given by linear systems whose generalized state space has a natural module structure over the ring of formal power series k z 1 existence and uniqueness of minimal realizations are proved and their properties are described the dimension of the minimal realization is called generalized degree of the polynomial transfer function and it is proved to be equal to the rank of an associated generalized hankel matrix finally a correspondence between realizations and fractional representations over k z 1 of polynomial matrices is established and some properties of such representations are investigated

generalized linear systems geometric and structural approaches abstract this paper presents a new way of introducing invariant subspaces for generalized systems this comes from somewhat known geometric algorithms but in most cases with initial conditions different from those usually seen in this context these definitions are shown to be consistent with other ones directly deduced from classical approaches like matrix pencil tools or discrete algebraic formalism the key starting point for their introduction is the extension of structural descriptions for these generalized systems such as have been available for proper systems many results from the proper case can thus be extended for instance the geometric definition for controllability indices and something similar to the famous morse canonical decomposition of the strictly proper case some links with the inversion algorithm are also sketched

a general realization theory for higher order linear differential equations in this note we show that the geometric quotient under a natural group action of the generalized state space systems recently considered by schumacher kuijper and geerts is algebraically isomorphic to the space of a homogeneous autoregressive system this result essentially follows from work of stromme published earlier in the algebraic geometry literature in particular these generalized state space systems represent a realization of the space of homogeneous autoregressive systems

common minimal multiples and divisors for rational matrix functions abstract we extend to rational matrix functions the theory of common multiples and common divisors of regular matrix polynomials developed in 12 attention is focused mainly on problems of existence uniqueness and construction from local data

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

maple code for the gamma algorithm for global optimization of uncertain functions in economy and finance problems with uncertainties are ubiquitous in many areas of life and the economy due to a lack of information as regards the economy and in finance problems with uncertainties stock prices marketing problems inflation unemployment are usually formulated by giving bounds on maximum and minimum values of certain parameters i e box constraints in such situations it is necessary to make a choice of better parameters that produce finite intervals of possible values for a given uncertain function at each point of the parameter space the gamma algorithm presents a method for making that choice a variant of the gamma algorithm based on the cubic algorithm is considered for global optimization of uncertain functions with box constraints in r n the set monotonic algorithm contains a block for problems with equality constraints and operates within the unit cube 0 1 n for all problems on this basis a maple code of modular structure is developed for full global optimization of uncertain functions in n variables the code does not create ill conditioned situations graphics are included and the solution set can be visualized in plane projections and sections an example related to minsky s financial instability hypothesis is presented with a graph to illustrate the use of the code

fixed poles in transfer function equations in this paper we study the pole structure of the solutions h z if any of equations of the form t z h z g z or dually t z f z h z where t z and g z or f z are given matrices of rational functions multivariable transfer functions over a field k this study is motivated by various design problems in linear dynamical system theory such as model matching or factorization problems whose solutions are systems with an internal dynamics determined by the pole structure of h z the methods we use are algebraic and module theoretic methods and the main tools are represented by the modules of the poles and of the zeros associated with a transfer function the main result is a complete description of the essential pole structure which is common to all the solutions h z this is given by means of a module whose invariant factors are explicitly computed in terms of fractional representations of the data t z and g z or f z the essential pole structure is shown to cons

poles and zeros of matrices of rational functions abstract this expository paper considers the problem of defining poles and zeros with multiplicity including those at infinity for a matrix of rational functions over a field the underlying theme question for the paper is does the number of poles of a matrix equal the number of zeros and what structural meaning underlies such an assertion our approach is motivated by ideas from linear system theory and control engineering control theoretic ideas are not a prerequisite and we hope to encourage specialists in classical linear and commutative algebra to investigate this promising source of new algebra problems

null pole subspaces of nonregular rational matrix functions abstract given a not necessarily regular rational matrix function w and a subset of the extended complex plane we associate with it a free module l w over the ring of scalar rational functions which are analytic on called the null pole subspace of w over in the scalar case the information encoded in this module is equivalent to the knowledge of all the poles and zeros including their multiplicities at all points of in the general matrix case the null pole subspace encodes the more complicated zero pole structure of a rational matrix function which is the key tool for the understanding of many factorization problems in this paper we show how various other modules which have been introduced in the literature in connection with pole zero structure e g pole module zero module can be read off from the null pole subspace and how conversely the null pole subspace can be recovered from its various pieces pole module null module a null pole coupling operator and left annihilator we give an analytic coordinate dependent description of these modules which has connection with realization theory and which is useful for computations we also solve the converse problem of finding a rational matrix function which has a given admissible module as its null pole subspace

spectral factorization of rectangular rational matrix functions with application to discrete wiener hopf equations the properties of a discrete wiener hopf equation are closely related to the factorization of the symbol of the equation we give a necessary and sufficient condition for existence of a canonical wiener hopf factorization of a possibly nonregular rational matrix function w relative to a contour which is a positively oriented boundary of a region in the finite complex plane the condition involves decomposition of the state space in a minimal realization of w and if it is satisfied we give explicit formulas for the factors the results are generalized by means of centered realizations to arbitrary rational matrix functions the proposed approach can be used to solve discrete wiener hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle

poles zeros and sheaf cohomology abstract the fundamental pole zero exact sequence for a transfer function matrix first enunciated in 1989 supplies a structural meaning to the hope that the number of zeros of a transfer function is the same as the number of poles this principle requires the inclusion of generic zeros counted using wedderburn forney spaces which measure the kernel and cokernel of the transfer function in this brief note we identify the wedderburn forney space as a cohomology group and give a new approach to the fundamental pole zero exact sequence in terms of exact sequences of sheaf cohomology groups

generalized state space realizations from matrix fraction descriptions the problem of the realization of a matrix fraction description mfd of a nonproper transfer function is considered an algorithm for the realization is developed and it is shown that it is a natural generalization of wolovich s method 1973 for the case of proper systems the resulting generalized state space system is shown to be reachable and the conditions for its minimality in terms of the coprimeness and column reducedness of the mfd are derived the relation between the controllability indexes of the realization and the forney order 1975 of the mfd is established

on the zeros and poles of a transfer function abstract the poles and zeros of a linear transfer function can be studied by means of the pole module and the transmission zero module these algebraic constructions yield finite dimensional vector spaces whose dimensions are the number of poles and the number of multivariable zeros of the transfer function in addition these spaces carry the structure of a module over a ring of polynomials which gives them a dynamical or state space structure the analogous theory at infinity gives finite dimensional spaces which are modules over the valuation ring of proper rational functions following ideas of wedderburn and forney we introduce new finite dimensional vector spaces which measure generic zeros which arise when a transfer function fails to be injective or surjective a new exact sequence relates the global spaces of zeros the global spaces of poles and the new generic zero spaces this sequence gives a structural result which can be summarized as follows the number of zeros of any transfer function is equal to the number of poles when everything is counted appropriately the same result unifies and extends a number of results of geometric control theory by relating global poles and zeros of general possibly improper transfer functions to controlled invariant and controllability subspaces including such spaces at infinity

the fluctuation dissipation relation in phase ordering kinetics abstract the properties of the response function and the fluctuation dissipation relation in phase ordering systems are discussed these issues are shown to be highly nontrivial even in the relatively simple case of coarsening dynamics considered here it is discussed how the connection between statics and dynamics fails in particular cases and the origin of this phenomenon is considered

rosenbrock models and their homotopy equivalence as is known the notion of homotopy equivalence is a fundamental notion of mathematics and was introduced in order to formalize a relation that is weaker than isomorphism in this note we define a homotopy equivalence of rosenbrock systems and show that it coincides with the classical equivalences of rosenbrock and fuhrmann next we show that the homotopy equivalence does preserve the important properties of a system including the properties at infinity when these are properly understood finally we define in a simple manner the states and motions of a system and claim that they are homotopy invariants

coupling operators wedderburn forney spaces and generalized inverses abstract the pole zero structure of a transfer function has been analyzed by two distinct approaches module theoretic coordinate free and analytic basis dependent the former the wyman sain conte perdon school emphasizes pole modules zero modules exact sequences of maps and abstract realization theory applications developed include the model matching problem zero structure of one sided inverses and connections with geometric systems theory the latter the gohberg school works with pole chains zero chains matrix equations and concrete factorization theory minimal wiener hopf inner outer and interpolation zero pole lagrange sylvester nevanlinna pick in this paper we make explicit the connection between the two approaches in particular we describe the zero pole exact sequence of a transfer function without a pole or zero at infinity in the coordinate system provided by a global left null pole triple an important tool for us and an object of interest in its own right is a generalized inverse g of a transfer function g we relate the poles of g with the zeros of g we also show that if the pole module of g is isomorphic to the zero module of g then left right pole pairs of g are left right null pairs of g

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

on coprime rational function matrices abstract the concept of coprimeness of matrices with elements in a field of fractions is introduced we focus on the field of rational functions and define when two rational matrices are coprime with respect to different rings the definition of coprimeness at infinity of polynomial matrices is obtained as a particular case moreover the fact that coprimeness is a local property is proved and it is shown how coprimeness changes by performing mobius transformations

a far from equilibrium fluctuation dissipation relation for an ising glauber like model we derive an exact expression of the response function to an infinitesimal magnetic field for an ising glauber like model with arbitrary exchange couplings the result is expressed in terms of thermodynamic averages and does not depend on the initial conditions or on the dimension of the space the response function is related to time derivatives of a complicated correlation function and so the expression is a generalization of the equilibrium fluctuation dissipation theorem in the special case of this model correspondence with the ising glauber model is discussed a discrete time version of the relation is implemented in monte carlo simulations and then used to study the ageing regime of the ferromagnetic two dimensional ising glauber model quenched from the paramagnetic phase to the ferromagnetic one our approach has the originality to give direct access to the response function and the fluctuation dissipation ratio

infinite structure and exact model matching problem a geometric approach we first generalize to a b c d quadruples the geometric characterization of the infinite zero structure we then give a new geometric proof for a solvability condition of the exact model matching problem expressed in terms of this concept and previously established in the polynomial context by morse verghese and pernebo this emphasizes the approach independent character of the infinite zero structure

homomorphisms of modules associated with polynomial matrices with infinite elementary divisors abstract if the inverse of a nonsingular polynomial matrix l has a polynomial part then one can associate with l a module over the ring of proper rational functions which is related to the structure of l at infinity in this paper we characterize homomorphisms of such modules

a simple procedure for the stabilization of a class of uncontrollable generalized systems this paper gives a simple procedure for the stabilization with a prescribed rate of convergence of the free motions of a class of possibly single input uncontrollable generalized continuous time time invariant linear systems preserving the impulse free property if any

almost invariant subspaces transmission and infinite zeros a lattice interpretation in the first part of the paper it is shown that the most important almost controlled invariant and almost conditionally invariant subspaces introduced by j c willems 4 5 are elements of the free distributive lattice generated by the well known subspaces download download full size image and download download full size image in the second part it is shown how the transmission zeros and the infinite zero structure may be obtained by duality within a state feedback context or an output injection one

a study of behaviors abstract the paper presents the results of the study of behavior theory as developed by j c willems from the point of view of polynomial and rational models considering behaviors in the discrete time case to be generalizations of rational models a natural focal point becomes the concept of a behavior homomorphism we give a characterization of behavior homomorphisms and analyze their invertibility properties in terms of embeddings in unimodular polynomial matrices these results which are of intrinsic interest are then applied to the uniform derivation of a large number of results for equivalence in different classes of behavior representations to a certain extent these are generalizations of the strict system equivalence concept for the class of polynomial matrix description of systems in the style of rosenbrock a study of behavioral controllability is undertaken and gives some new insights into connections with geometric control theory

on the zeros of a minimal realization in an earlier work the authors have introduced a coordinate free module theoretic definition of zeros for the transfer function g s of a linear multivariable system a b c the first contribution of this paper is the construction of an explicit k z module isomorphism from that zero module z g to v r where v is the supremal a b invariant subspace contained in kerc and r is the supremal a b controllable subspace contained in kerc and where a b c constitutes a minimal realization of g s the isomorphism is developed from an exact commutative diagram of k vector spaces the second contribution is the introduction of a zero signal generator and the establishment of a relation between this generator and the classic notion of blocked signal transmissions

module theoretic zero structures for system matrices the foundation for a coordinate free theory of the poles of a linear dynamical system was laid in 1965 by the module theoretic work of kalman however although the theoretical and application importance of transfer function zeros for feedback system design was widely known for single input single output systems by 1955 it remained for rosenbrock in 1970 to propose the ideas of transmission zeros input decoupling zeros and output decoupling zeros for multi input multi output systems by means of matrix theoretic methods a coordinate free module theoretic treatment of transmission zeros for a multi input mufti output transfer function was given in 1981 by wyman and sain this paper extends these coordinate free module theoretic studies to include systems which need not be controllable or observable interpretation of the rosenbrock system matrix is given on three levels rational finitely generated free modular and torsion divisible on the second level an omega zero module z omega is de

dynamical structures on pencils poles and fundamental subspaces abstract this paper provides a bridge between singular system representations module theory and the geometric theory of subspaces by means of a theorem the little pole space associated with a singular system is shown to be isomorphic to the global system pole space a module theoretic concept incorporating both finite and infinite system pole information these results hold true for any regular matrix pencil ze a and may be interpreted as special cases in terms of causal and anticausal systems moreover fundamental subspaces associated with matrix pencils are isomorphic to k z and o modules representing finite and infinite system poles as a result dynamical structures on these limit spaces may be determined

generalized reachability subspaces for singular systems one of the most important concepts of the geometric theory for proper linear systems is that of the controllability subspaces of a b in an attempt to extend this concept to singular systems it became clear that the nonequivalence of reachability and controllability for singular systems makes it necessary to define both controllability and reachability subspaces of e a b as different though naturally related entities a subspace bf r respectively bf c is defined to be a generalized reachability subspace respectively generalized controllability subspace if there exist linear maps f and g such that bf r respectively bf c is the reachable respectively controllable subspace of a regular singular system e dot x a bf x bgv it is proved that 1 bf r is a generalized reachability subspace if and only if it is a generalized a e bf b invariant almost reachability subspace 2 every subspace bf k contains a unique supremal generalized reachabi

off equilibrium generalization of the fluctuation dissipation theorem for ising spins and measurement of the linear response function we derive for ising spins an off equilibrium generalization of the fluctuation dissipation theorem which is formally identical to the one previously obtained for soft spins with langevin dynamics l f cugliandolo j kurchan and g parisi j phys i 4 1641 1994 the result is quite general and holds both for dynamics with conserved and nonconserved order parameters on the basis of this fluctuation dissipation relation we construct an efficient numerical algorithm for the computation of the linear response function without imposing the perturbing field which is alternative to those of chatelain j phys a 36 10 739 2003 and ricci tersenghi phys rev e 68 065104 r 2003 as applications of the new algorithm we present very accurate data for the linear response function of the ising chain with conserved and nonconserved order parameter dynamics finding that in both cases the structure is the same with a very simple physical interpretation we also compute the integrated response function of the two dimensional ising model confirming that it obeys scaling ensuremath chi t t w ensuremath simeq t w ensuremath a f t t w with a 0 26 ifmmode pm else textpm fi 0 01 as previously found with a different method

global optimization over general compact sets by the beta algorithm a maple code a variant of the beta algorithm based on the cubic algorithm 1 2 is presented for global optimization of holder continuous functions over general compact sets the set monotonic algorithm contains a block for problems with equality constraints and operates within the unit cube 0 1 n universal for all problems on this basis a maple code of modular structure is developed for full global optimization of functions of n variables the code does not create ill conditioned situations graphics are included and the solution set can be visualized in plane projections and sections the code is ready for engineering applications results of numerical experiments are presented with graphs to illustrate the use of the code

fluctuation dissipation relations for markov processes the fluctuation dissipation relation is calculated for stochastic models obeying a master equation with continuous time in the general case of a nonstationary process there appears to be no simple relation between the response and the correlation also if one considers stationary processes the linear response cannot be expressed via time derivatives of the correlation function alone in this case an additional function which has rarely been discussed previously is required this so called asymmetry depends on the two times also relevant for the response and the correlation and it vanishes under equilibrium conditions the asymmetry can be expressed in terms of the propagators and the transition rates of the master equation but it is not related to any physical observable in an obvious way it is found that the behavior of the asymmetry strongly depends on the nature of the dynamical variable considered in the calculation of the correlation and the response if one is concerned with a variable which randomizes with any transition among the states of the system the asymmetry vanishes in most cases this is in contrast to the situation for other classes of variables in particular for trap models of glassy relaxation the fluctuation dissipation ratio strongly depends on the observable and the asymmetry plays a dominant role in the determination of this ratio also if only neutral variables are considered some implications of a nonvanishing asymmetry with regard to the definition of an effective temperature are discussed

comment on scaling of the linear response in simple aging systems without disorder we have repeated the simulations of henkel paessens and pleimling hpp phys rev e 69 056109 2004 for the field cooled susceptibility ensuremath chi mathit fc t ensuremath ensuremath chi 0 ensuremath sim t ensuremath a in the quench of ferromagnetic systems to and below t c we show that contrary to the statement made by hpp the exponent a coincides with the exponent a of the linear response function r t s ensuremath sim s ensuremath 1 a f r t s we point out what are the assumptions in the argument of hpp that lead them to the conclusion ala

geometric characterization of complete controllability indices for singular systems abstract we show in this short note that our recent geometric definitions for singular systems 12 provide a natural characterization of the complete controllability indices introduced by kucera and zagalak 11 13

some geometric considerations about the kronecker normal form abstract in the first part of this paper we develop general algorithms from the kronecker normal form of any matrix pencil and give some of their properties in the second part these results are applied to the so called restricted matrix pencil associated with a given c a b system and introduced by jaffe and karcanias 1981 this allows us to make a link with well known algorithms of the geometric approach particularly the a b invariant subspace algorithm and the almost a b controllability subspace algorithm finally relationships between morse s invariants morse 1973 and the invariants of the restricted matrix pencil are exhibited

kinetics of the long range spherical model the kinetic spherical model with long range interactions is studied after a quench to t tc or to t tc for the two time response and correlation functions of the order parameter as well as for composite fields such as the energy density the ageing exponents and the corresponding scaling functions are derived the results are compared to the predictions which follow from local scale invariance

on matrix polynomials with the same finite and infinite elementary divisors abstract a criterion is presented that characterizes when two matrix polynomials of any size rank and degree have the same finite and infinite elementary divisors this characterization inherits a coprimeness condition of the extended unimodular equivalence defined by pugh and shelton 17 in the set of real or complex matrix polynomials satisfying the constraint that the difference between the number of rows and columns is constant this extended unimodular equivalence is first generalized to matrices of any size with elements in any principal ideal domain

off equilibrium response function in coarsening systems abstract in the late stage dynamics of systems evolving via domain coarsening the linear response function splits into the sum of a stationary term due to thermal fluctuations within domains and of an aging term due to off equilibrium interface dynamics the latter contribution is analyzed in the pure ising model under variation of space dimensionality and in the random field ising model in one dimension the variety of behaviors observed can be accounted for in terms of the effective response associated to a single interface implications for the connection between static and dynamic properties are discussed

a unified pole zero module for linear transfer functions abstract a new module theoretic tool is presented for the study of linear time invariant input output maps transfer functions for suitable subrings 0 of the field k s of rational functions an o module called the structure module can be attached to each transfer function this structure module naturally contains the classical kalman pole module as a submodule which gives the wyman sain multivariable zero module as the corresponding factor module a posteriori the resulting exact sequence splits and the structure module is isomorphic to the direct sum of the pole module and the zero module calculations over local rings echo valuation theoretic results obtained by kung and kailath

realization and partial fractions abstract we discuss the relation between two intrinsically different proposals that have been made in the literature concerning the representation by constant matrices of rational matrices given in fractional form it turns out that the relation is most naturally studied in the framework of partial fraction decompositions we develop the realization theory for decompositions with respect to arbitrary complementary parts of the extended complex plane which may for instance correspond to stability and instability an isomorphism is obtained which connects the spaces used in the two methods and several identities relating to the mcmillan degree are derived in a direct way finally a new computational procedure is given to obtain the partial fraction decomposition of a rational matrix given in fractional form

correlation functions and fluctuation dissipation relation in driven phase ordering systems an exactly solvable model the dynamics of a phase ordering system with non conserved order parameter under a plain shear flow with rate is solved analytically in the large n limit a phase transition is observed at a critical temperature tc after a quench from a high temperature equilibrium state to a lower temperature t a non equilibrium stationary state is entered when t tc while aging dynamics characterizes the phases with t tc two time quantities are computed and the off equilibrium generalization of the fluctuation dissipation theorem is provided

global optimization in problems with uncertainties the gamma algorithm abstract problems with uncertainties can be viewed and formalized making use of multifunctions or general set valued functions a new concept of global optimality is proposed which allows us to solve global optimization problems with uncertainties in natural setting without imposing artificial constraints on uncertainties nor introducing a kind of partial ordering in order to apply conventional optimality concepts and optimization techniques nor considering solution in probability with the new concept deterministic optimization requires two optimization procedures a study of the subject is presented with many illustrative examples then a monotonic iterative algorithm is developed which renders approximate solutions with precision specified in advance a notion of piecewise continuous function of several variables is proposed and the method is then generalized for uncertain functions defined by a closed set valued function with piecewise continuous upper and lower boundaries the max2 f reduction precision and decomposition lemmas are proved to facilitate practical applications deferred deletions of sets with discontinuities are introduced and convergence theorem is proved for the modified algorithm

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

minimal mcmillan degree rational matrix functions with prescribed local zero pole structure abstract given a rational m n matrix function w z and a subset of the complex plane c we give an explicit algorithm using column operations to produce a rational matrix function h z of minimal possible mcmillan degree which has the same left zero pole structure on as w z i e for which h wq where q and q 1 are rational n n matrix functions with no poles in