poles and zeros at infinity of linear time varying systems the notions of poles and zeros at infinity and their relations are extended to the case of linear continuous time varying systems this study is based on the notion of a newborn systems which is in a mathematical point of view a graded module extension over the noncommutative ring of differential operators it is proved to be a relevant generalization to the time varying case of the equivalence class for the so called restricted equivalence of rosenbrock s polynomial matrix descriptions the authors approach is intrinsic and unifies the definitions previously given in the literature in the time invariant case

a new indirect adaptive control scheme for time varying plants structures are developed for the control and estimation parts of the indirect adaptive scheme which are more appropriate for time varying plants in contrast to the usual pointwise schemes the new controller structure can meet the control objective exactly for a wide class of plants with smooth but otherwise unrestricted parameter variations on the other hand the possibly available knowledge of the form or frequency of variation of the fast parameters is included in the new estimator structure so that for successful estimation the overall plant is not restricted to varying slowly with time the new estimator and control law are combined using the certainty equivalence principle to develop an indirect adaptive control scheme meeting the control objective for plants with time varying parameters whose fast varying parts are of known form and unknown parts are slow in the mean

weak exponential stability of linear time varying differential behaviors abstract we develop a new approach to exponential stability of linear time varying ltv differential behaviors that is analogous to that in our paper on exponential stability of discrete ltv behaviors h bourles et al 2015 5 stability theory for differential state space systems with smooth coefficients is an important subject in the literature for differential ltv behaviors with arbitrary smooth coefficients there is no reasonable stability theory therefore we restrict the smooth varying coefficients to functions that are defined by means of locally convergent puiseux series all rational functions are of this type we introduce a new kind of behaviors and prove a module behavior duality for these we define a new notion of weak exponential stability w e s of a behavior b and its associated finitely generated f g module m and show that the w e s modules and behaviors are closed under isomorphisms subobjects factor objects and extensions the standard uniform exponential stability of state space equations is not preserved under behavior isomorphisms and unsuitable for a behavioral theory in the main result we assume a nonzero f g torsion module m and its associated autonomous behavior b such a module may be regular or irregular singular according to the galois theory of differential equations if it is nonzero and regular singular it is never w e s for irregular singular m we characterize w e s of most b algebraically and constructively

global well posedness of the dynamic φ4 model in the plane we show global well posedness of the dynamic 4 4 model in the plane the model is a nonlinear stochastic pde that can only be interpreted in a renormalised sense solutions take values in suitable weighted besov spaces of negative regularity

continuous varying linear systems abstract we discuss implicit systems of ordinary linear differential equations with time variable coefficients their solutions in the signal space of hyperfunctions according to sato and their solution spaces called time varying linear systems or behaviours from the system theoretic point of view the basic result inspired by an analogous one for multidimensional constant linear systems is a duality theorem which establishes a categorical one one correspondence between time varying linear systems or behaviours and finitely generated modules over a suitable skew polynomial ring of differential operators this theorem is false for the signal spaces of infinitely often differentiable functions or of meromorphic hyper functions or of distributions on r it is used to obtain various results on key notions of linear system theory several new algorithms for modules over rings of differential operators and in particular new grobner basis algorithms due to insa and pauer make the system theoretic results effective

some basic structural properties of generalized linear systems abstract this paper shows how to give a kalman state variable realization to generalized linear systems simple characterizations of controllability observability of their duality and of a minimal realization are proposed our techniques based on elementary module theory are applicable to time varying systems as well as constant systems the note ends with a brief discussion of asymptotic observers in the constant case

tempered random attractors for parabolic equations in weighted spaces this paper deals with the asymptotic behavior of solutions of parabolic equations on unbounded domains which contain stochastic terms as well as non autonomous deterministic terms we define a continuous random dynamical system for the equations in weighted sobolev spaces that allow functions to have certain polynomial growth rate at infinity and hence include all bounded solutions we prove pullback asymptotic compactness of solutions as well as the existence and uniqueness of tempered random attractors in the weighted spaces the structures of the tempered attractors are fully characterized by tempered complete solutions in the case where the non autonomous deterministic terms are periodic in time we show that the tempered random attractors are also periodic in time to overcome the difficulty of non compactness of sobolev embeddings on unbounded domains the idea of uniform pathwise estimates on the tails of solutions is employed to show asymptotic compactness

on the turing patterns in one dimensional gradient skew gradient dissipative systems in this article fundamental properties concerning the turing patterns are considered in one dimensional dissipative systems with gradient skew gradient structure introduced in m kuwamura and e yanagida phys d 175 2003 pp 185 195 it is a natural extension of free energy which covers reaction diffusion systems of activator inhibitor type the theory based on this concept provides a new perspective on a fundamental problem of what unique turing pattern is to be selected among many

relative risk trees for censored survival data a method is developed for obtaining tree structured relative risk estimates for censored survival data the first step of a full likelihood estimation procedure is used in a recursive partitioning algorithm that adopts most aspects of the widely used classification and regression tree cart algorithm of breiman et al 1984 classification and regression trees belmont california wadsworth the performance of the technique is investigated through stimulation and compared to the tree structured survival methods proposed by davis and anderson 1989 statistics in medicine 8 947 961 and therneau grambsch and fleming 1990 biometrika 77 147 160

transfer matrices realization and control of continuous time linear time varying systems via polynomial fractional representations abstract an algebraic theory of transfer matrices fractional representations and control for linear continuous time time varying systems based on the realization theory of input output maps is given it is shown for the first time that the realization of such systems specified by an abstract input output map as a module homomorphism over noncommutative polynomial rings can be established using an abstract kalman input output map defined over a ring of skew polynomials with time varying coefficients it is shown that in fact transfer matrices can be defined as formal power

survival trees by goodness of split abstract a tree based method for censored survival data is developed based on maximizing the difference in survival between groups of patients represented by nodes in a binary tree the method includes a pruning algorithm with optimal properties analogous to the classification and regression tree cart pruning algorithm uniform convergence of the estimates of the conditional cumulative hazard and survival functions is discussed and an example is given to show the utility of the algorithm for developing prognostic classifications for patients

phase dynamics in the complex ginzburg landau equation abstract for 1 stable time periodic solutions a x t aqeiqx i qt are the locally preferred planform for the complex ginzburg landau equation t a a 1 i x 2 a 1 i a a 2 in order to describe the spatial global behavior an evolution equation for the local wave number q can be derived formally the local wave number q satisfies approximately a conservation law q h q it is the purpose of this paper to explain the extent to which the conservation law is valid by proving estimates for this formal approximation

modulation equation and spdes on unbounded domains we consider the approximation via modulation equations for nonlinear spdes on unbounded domains with additive space time white noise close to a bifurcation an infinite band of eigenvalues changes stability and we study the impact of small space time white noise on the dynamics close to this bifurcation as a first example we study the stochastic swift hohenberg equation on the whole real line here due to the weak regularity of solutions the standard methods for modulation equations fail and we need to develop new tools to treat the approximation as an additional result we sketch the proof for local existence and uniqueness of solutions for the stochastic swift hohenberg and the complex ginzburg landau equations on the whole real line in weighted spaces that allow for unboundedness at infinity of solutions which is natural for translation invariant noise like space time white noise we use energy estimates to show that solutions of the ginzburg landau equation are holder continuous and have moments in those functions spaces this gives just enough regularity to proceed with the error estimates of the approximation result

an infinite dimensional stochastic differential equation with state spacec ℝ we consider a time evolution of unbounded continuous spins on the real line the evolution is described by an infinite dimensional stochastic differential equation with local interaction introducing a condition which controls the growth of paths at infinity we can construct a diffusion process taking values inc in view of quantum field theory this is a time dependent model ofp 1 field in parisi and wu s scheme

regression analysis of multivariate incomplete failure time data by modeling marginal distributions abstract many survival studies record the times to two or more distinct failures on each subject the failures may be events of different natures or may be repetitions of the same kind of event in this article we consider the regression analysis of such multivariate failure time observations each marginal distribution of the failure times is formulated by a cox proportional hazards model no specific structure of dependence among the distinct failure times on each subject is imposed the regression parameters in the cox models are estimated by maximizing the failure specific partial likelihoods the resulting estimators are shown to be asymptotically jointly normal with a covariance matrix that can be consistently estimated simultaneous inferential procedures are then proposed extensive monte carlo studies indicate that the normal approximation is adequate for practical use the new methods allow time dependent covariates missing observations and arbitrary patterns of censorship they are illustrat

analyticity of ginzburg landau modes ginzburg landau formalism applies for dissipative systems defined on cylindrical domains which are close to the threshold of instability and for which the unstable fourier modes belong to non zero wave numbers in these situations the real part of the curve of critical eigenvalues as function drawn over the wave numbers k is positive of height o 2 and of width o here it is shown that the set of solutions which can be described by the ginzburg landau formalism is attractive to do this we demonstrate that in fourier space peaks appear at integer multiples of the critical wave number kc these peaks called ginzburg landau modes concentrate in time like e k mkc t for 0 t o 1 2 and m z the inverse fourier transform of such a ginzburg landau mode is an analytic function in a strip of width t this result extends a former work of w eckhaus

adaptive control of linear time varying plants a new model reference controller structure the problem of developing a control law which can force the output of a linear time varying plant to track the output of a stable linear time invariant reference model is discussed it is shown that the standard model reference controller used for linear time invariant plants cannot guarantee zero tracking error in general when the plant is time varying a new model reference controller is proposed which guarantees stability and zero tracking error for a general class of linear time varying plants with known parameters when the time varying plant parameters are unknown but vary slowly with time it is shown that the new controller can be combined with a suitable adaptive law so that all the signals in the closed loop remain bounded for any bounded initial conditions and the tracking error is small in the mean the assumption of slow parameter variations in the adaptive case can be relaxed if some information about the frequency or the form of the fast varying parameters is available a priori such information can be incorporated in an appropriately designed adaptive law so that stability and improved tracking performance is guaranteed for a class of plants with fast varying parameters

model reference adaptive control of linear time varying plants the case of jump parameter variations in this paper we consider the model reference adaptive control problem of single input single output linear time varying plants we assume that the plant parameters are uniformly bounded and piecewise lipschitz continuous inside the intervals of continuity we assume that the plant satisfies certain controllability observability relative degree and zero dynamics assumptions but the speed of parameter variations is not restricted to be small under these general conditions we design a control law of the model reference type which guarantees the boundedness of the closed loop plant and small tracking error in the mean square sense provided that the number of discontinuities of the plant parameters is small on average furthermore in the case of partially unknown plant parameters we show that the adaptive controller obtained by combining this control law with a suitable estimation algorithm guarantees closed loop boundedness and small tracking error in the mean square sense provided that in additi

approximation and attractivity properties of the degenerated ginzburg landau equation abstract we are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so called degenerated ginzburg landau equation takes the role of the classical ginzburg landau equation as the amplitude equation of the system this is the case when the relevant nonlinear terms vanish at the bifurcation point here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated ginzburg landau equation in this paper we restrict ourselves to a swift hohenberg kuramoto shivashinsky equation as a model for such a pattern forming system

the scaling limit for a stochastic pde and the separation of phases we investigate the problem of singular perturbation for a reaction diffusion equation with additive noise or a stochastic partial differential equation of ginzburg landau type under the situation that the reaction term is determined by a potential with double wells of equal depth as the parameter e the temperature of the system tends to 0 the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit we derive a stochastic differential equation which describes the random movement of the phase separation point the proof consists of two main steps we show that the solution stays near a manifoldm e of minimal energy configurations based on a lyapunov type argument then the limit equation is identified by introducing a nice coordinate system in a neighborhood ofm e

une interpretation algebrique de la transformation de laplace et des matrices de transfert abstract the tensor product of the module of a linear system with the quotient field of the ring of linear differential operators is a vector space where even in the time varying case a formal laplace transform and the transfer matrix are most naturally defined several classic problems are examined in this algebraic setting the relationship between left right coprime matrix decomposition and controllability observability the state variable canonical realization the transfer algebra with respect to parallel and series connections the input output inversion and model matching

trees for correlated survival data by goodness of split with applications to tooth prognosis in this article the regression tree method is extended to correlated survival data and applied to the problem of developing objective prognostic classification rules in periodontal research the robust logrank statistic is used as the splitting statistic to measure the between node difference in survival while adjusting for correlation among failure times from the same patient the partition based survival function estimator is shown to converge to the true conditional survival function tooth loss data from 100 periodontal patients 2 509 teeth was analyzed using the proposed method the goal is to assign each tooth to one of the five prognosis categories good fair poor questionable or hopeless after the best sized tree was identified an amalgamation procedure was used to form five prognostic groups the prognostic rules established here may be used by periodontists general dentists and insurance companies in devising appropriate treatment plans for periodontal patients

propagating fronts and the center manifold theorem we prove the existence of propagating front solutions for the swift hohenberg equation sh using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations they describe stationary solutions and front solutions of the partial differential equation sh we identify the well known amplitude equation as the lowest order approximation to the equation of motion on the center manifold

restricted markov uniqueness for the stochastic quantization of p φ 2 and its applications abstract in this paper we obtain restricted markov uniqueness of the generator and uniqueness of martingale probabilistically weak solutions for the stochastic quantization problem in both the finite and infinite volume case by clarifying the precise relation between the solutions to the stochastic quantization problem obtained by the dirichlet form approach and those obtained in 10 and in 24 we prove that the solution x z where x is obtained by the dirichlet form approach in 4 and z is the corresponding o u process satisfies the corresponding shifted equation see 1 4 below moreover we obtain that the infinite volume p 2 quantum field is an invariant measure for the x y z where y is the unique solution to the shifted equation

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

duality in time varying linear systems a module theoretic approach abstract using elementary module theory an intrinsic definition of the dual or adjoint of a generalized time varying linear system is given with this the duality of controllability and observability is recovered from their intrinsic module theoretical definitions the duality of state feedback and output injection is discussed both in the static case and for its quasistatic generalization related brunovsky type canonical forms are derived in the quasistatic case the corresponding controllability indices and their duals the observability indices are defined intrinsically

stabilizability of systems with exponential dichotomy in this paper we introduce the concept of controllability into a closed subspace for time varying linear systems various characterizations are given and the dual relation is discussed this concept is used to present a necessary and sufficient condition for the stabilizability of systems with exponential dichotomy

bifurcations of poiseuille flow between parallel plates three dimensional solutions with large spanwise wavelength the spatial dynamics approach is applied to the analysis of bifurcations of the three dimensional poiseuille flow between parallel plates in contrast to the classical studies we impose time periodicity as well as spatial periodicity with period 2 in the streamwise direction however we make no assumptions on the behavior in the spanwise direction except the uniform closeness of the bifurcating solution to the basic flow in an abstract setting it is shown how the dimension of the critical eigenspace of the spatial dynamics analysis can be uniquely determined from the classical linear stability problem for the three dimensional poiseuille problem we are able to find all relevant coefficients from the analysis of the purely two dimensional problem moreover we are able to analyze precisely the influence of a spanwise pressure gradient and the associated spanwise mass flux the study of the reduced problem shows that there are two different kinds of solutions spirals and ribbons which are 2 p periodic in the spanwise direction as in the couette taylor problem and both of them bifurcate in the same direction

minimum variance prediction for linear time varying systems in this paper we study the problem of minimum variance prediction for linear time varying systems we consider the standard time varying autoregression moving average arma model and develop a predictor which guarantees minimum variance prediction for a large class of linear time varying systems the predictor is developed based on a pseudocommutation technique for dealing with noncommutativity of linear time varying operators in a transfer operator framework we also show connections between this input output predictor and the kalman predictor via an example

asymptotic compactness and absorbing sets for 2d stochastic navier stokes equations on some unbounded domains we introduce a notion of an asymptotically compact ac random dynamical system rds we prove that for an ac rds the limit set b of any bounded set b is nonempty compact strictly invariant and attracts the set b we establish that the 2d navier stokes equations nses in a domain satisfying the poincare inequality perturbed by an additive irregular noise generate an ac rds in the energy space h as a consequence we deduce existence of an invariant measure for such nses our study generalizes on the one hand the earlier results by flandoli crauel 1994 and schmalfuss 1992 obtained in the case of bounded domains and regular noise and on the other hand the results by rosa 1998 for the deterministic nses

kawahara dynamics in dispersive media abstract in a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled kdv equations here we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes the long wave limit can be rigorously approximated by two decoupled kawahara equations

the poles and zeros of a linear time varying system abstract for linear time varying discrete time and continuous time systems a notion of poles and zeros is developed in terms of factorizations of operator polynomials with time varying coefficients in the discrete time case it is shown that the poles can be computed by solving a nonlinear recursion with time varying coefficients in the continuous time case the poles can be calculated by solving a nonlinear differential equation with time varying coefficients the theory is applied to the study of the zero input response and asymptotic stability it is shown that if a time varying analogue of the vandermonde matrix is invertible the zero input response can be decomposed into a sum of modes associated with the poles stability is then studied in terms of the components of the modal decomposition

time varying polynomial matrix systems abstract differential polynomial matrices with coefficients meromorphic on an open interval i r are studied first the subclass of full matrices p having the properties p is non singular antl every local solution of pf 0 extends analytically to i forms a sublattice with respect to right division of square matrices which is anti isomorphic to the lattice of the corresponding solution spaces this and further properties are then exploited in the study of time varying systems in differential operator representation results on equivalence state space models controllability observability and transfer functions are derived

modulation equations stochastic bifurcation in large domains we consider the stochastic swift hohenberg equation on a large domain near its change of stability we show that under the appropriate scaling its solutions can be approximated by a periodic wave which is modulated by the solutions to a stochastic ginzburg landau equation we then proceed to show that this approximation also extends to the invariant measures of these equations

model reference adaptive and nonadaptive control of linear time varying plants we address the adaptive and nonadaptive model reference control problem for a class of multivariable linear time varying plants namely index invariant ones we show that under appropriate controllability and observability conditions this class of plants admits a fractional description in terms of polynomial differential operators and as such allows for a polynomial equation based controller design we also show that for a model reference control objective the controller can be designed by solving a set of algebraic equations further when the plant parameters are only partially known we employ a gradient based adaptive law with projection and normalization to update the controller parameters and establish the stability and tracking properties of adaptive closed loop plant finally we present a simple example to illustrate the design and realization of both the adaptive and nonadaptive control laws

a behavioral approach to time varying linear systems part 2 descriptor systems in the sequel to a ilchmann and v mehrmann em siam j control optim 44 2005 pp 1725 1747 we discuss a behavioral approach for linear time varying differential algebraic descriptor systems with real analytic coefficients the analysis is almost global in the sense that the analysis is not restricted to an interval i subset r but is allowed for the time axis r setminus t where t is a discrete set of critical points at which the solution may exhibit a finite escape time controllable observable and autonomous behavior for linear time varying descriptor systems is characterized

linearized control systems and applications to smooth stabilization for a control system dot x f x u the author proves that for generic feedback laws u such that f x u x does not vanish the linearized control systems around the trajectories of dot x f x u x have the same strong accessibility algebra as f applications are given to the smooth stabilization problem

diffusive mixing of stable states in the ginzburg landau equation the ginzburg landau equation on the real line has spatially periodic steady states of the form with and for we construct solutions which converge for all t 0 to the limiting pattern as these solutions are stable with respect to sufficiently small perturbations and behave asymptotically in time like where is uniquely determined by the boundary conditions this extends a previous result of brk92 by removing the assumption that should be close to zero the existence of the limiting profile is obtained as an application of the theory of monotone operators and the long time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term

the existence of dendritic fronts in this paper we study a fourth order semilinear parabolic equation on the infinite real line we show that in a certain parameter range this equation has propagating front solutions solutions tending to 0 at and advancing to the right with a speedc which leave behind them aperiodic pattern in the laboratory frame this is thus an example of spontaneous pattern formation

global existence via ginzburg landau formalism and pseudo orbits of ginzburg landau approximations the so called ginzburg landau formalism applies for parabolic systems which are defined on cylindrical domains which are close to the threshold of instability and for which the unstable fourier modes belong to non zero wave numbers this formalism allows to describe an attracting set of solutions by a modulation equation here the ginzburg landau equation if the coefficient in front of the cubic term of the formally derived ginzburg landau equation has negative real part the method allows to show global existence in time in the original system of all solutions belonging to small initial conditions inl another aim of this paper is to construct a pseudo orbit of ginzburg landau approximations which is close to a solution of the original system up tot we consider here as an example the socalled kuramoto shivashinsky equation to explain the methods but it applies also to a wide class of other problems like e g hydrodynamical problems or reaction diffusion equations too

multivariate survival trees a maximum likelihood approach based on frailty models summary a method of constructing trees for correlated failure times is put forward it adopts the backfitting idea of classification and regression trees cart breiman et al 1984 in classification and regression trees the tree method is developed based on the maximized likelihoods associated with the gamma frailty model and standard likelihood related techniques are incorporated the proposed method is assessed through simulations conducted under a variety of model configurations and illustrated using the chronic granulomatous disease cgd study data

modelling marginal hazards in multivariate failure time data we consider the problem of estimating relative risk parameters in the marginal distribution of failure times from a sample in which individuals are grouped into clusters and failure times for individuals within clusters may be dependent we assume that the marginal hazard of failure for individuals within a cluster is specified by a relative risk regression model and the structure of dependence between individuals within a cluster is left unspecified estimating equations for the regression parameters and an estimator for the cumulative base line marginal hazard are described

a behavioral approach to time varying linear systems part 1 general theory we develop a behavioral approach to linear time varying differential algebraic systems the analysis is almost everywhere in the sense that the statements hold on mathbb r setminus mathbb t where mathbb t is a discrete set controllability observability and autonomy are introduced and related to the behavior of the system classical results on the behavior of time invariant systems are studied in the context of time varying systems

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

the validity of phase diffusion equations and of cahn hilliard equations for the modulation of pattern in reaction diffusion systems abstract so called phase diffusion equations and cahn hilliard equations can be derived via multiple scaling analysis in order to describe slow modulations in time and space of stable and slightly unstable spatially periodic pattern it is the purpose of this paper to explain the extent to which these formal approximations are valid by proving estimates between the formal approximations and true solutions of the original system the proof is given for an abstract reaction diffusion system as original system

the time dependent amplitude equation for the swift hohenberg problem precise estimates for the validity of the amplitude approximation for the swift hohenberg equation are given in a fully time dependent framework it is shown that small solutions of ordero which are modulated like stationary solutions have an evolution which is well described in the amplitude approximation for a time of ordero 2 for the proofs we use techniques for nonlinear semigroups and oscillatory integrals

time varying linear systems and invariants of system equivalence in the paper we consider time varying systems with coefficients depending mero morphically on time in differential operator representations these systems are described by matrices over a skew polynomial ring with coefficients in the field of real meromorphic functions different kinds of indices controllability minimal geometric and dynamical are introduced and it is proved that they essentially coincide the input module and the formal transfer matrix are defined and used for an algebraic description of time varying systems a characterization of system equivalence is given in these terms and also a complete list of invariants of similarity for time varying state space systems

attractivity of the ginzburg landau mode distribution for a pattern forming system with marginally stable long modes abstract the mathematical theory of the description of pattern forming systems close to the first instability via the ginzburg landau equation is based on approximation and attractivity results this theory allowed to prove global existence results and upper semicontinuity of attractors for classical hydrodynamical stability problems such as the couette taylor problem recently approximation results for the ginzburg landau approximation for pattern forming systems with marginally stable long modes such as the benard marangoni system have been shown it is the purpose of this paper to prove the second fundamental property namely the attractivity of the ginzburg landau mode distribution for such systems

the validity of modulation equations for extended systems with cubic nonlinearities modulation equations play an essential role in the understanding of complicated systems near the threshold of instability here we show that the modulation equation dominates the dynamics of the full problem locally at least over a long time scale for systems with no quadratic interaction term we develop a method which is much simpler than previous ones it involves a careful bookkeeping of errors and an estimate of gronwall type as an example for the dissipative case we find that the ginzburg landau equation is the modulation equation for the swift hohenberg problem moreover the method also enables us to handle hyperbolic problems the nonlinear schrodinger equation is shown to describe the modulation of wave packets in the sine gordon equation

adjoint and hamiltonian input output differential equations based on developments in the theory of variational and hamiltonian control systems by crouch and van der schaft 1987 this paper answers two questions given an input output differential equation description of a nonlinear system what is the adjoint variational system in input output differential form and what are the conditions for the system to be hamiltonian i e such that the variational and the adjoint variational systems coincide this resulting set of conditions is then used to generalize classical conditions such as the well known helmholtz conditions for the inverse problem in classical mechanics

space time invariant measures entropy and dimension for stochastic ginzburg landau equations we consider a randomly forced ginzburg landau equation on an unbounded domain the forcing is smooth and homogeneous in space and white noise in time we prove existence and smoothness of solutions existence of an invariant measure for the corresponding markov process and we define the spatial densities of topological entropy of measure theoretic entropy and of upper box counting dimension we prove inequalities relating these different quantities the proof of existence of an invariant measure uses the compact embedding of some space of uniformly smooth functions into the space of locally square integrable functions and a priori bounds on the semi flow in these spaces the bounds on the entropy follow from spatially localised estimates on the rate of divergence of nearby orbits and on the smoothing effect of the evolution