the quasiclassical limit of the modified kp hierarchy in the quasiclassical limit the mkp hierarchy and its lax representation turn into a modified benney hierarchy and its poisson representation a miura map is constructed and shown to be canonical from the modified benney hierarchy into the unmodified one the modified hierarchy is given both hydrodynamical and kinetic representations and the miura map is given a kinetic form explicit combinatorial formulae are proved for the infinite number of conserved densities of the modified benney hierarchy

solution of the dispersionless hirota equations the dispersionless differential pay identity is shown to be equivalent to a kernel expansion providing a universal algebraic characterization and solution of the dispersionless hirota equations some calculations based on d bar data of the action are also indicated

h control of markov jump systems with time varying delay and incomplete transition probabilities this paper addresses the h control of continuous markov jump systems with interval time varying delay and incomplete transition probabilities a linearization method is used to handle unknown transition probabilities meanwhile the wirtinger based integral inequality and the reciprocally convex technique are adopted to deal with the time varying delay additionally a separating technique is employed to tackle the coupling among lyapunov variable system matrix and controller parameter based on these strategies new sufficient conditions for the closed loop system to be stochastically stable are formulated in the framework of linear matrix inequalities finally numerical examples are provided to demonstrate the effectiveness of the proposed method

large deviations upper bounds for the laws of matrix valued processes and non communicative entropies using ito s calculus we study the large deviations properties of the law of the spectral measure of the hermitian brownian motion we extend this strategy to the symmetric unitary and wishart processes this dynamical approach is generalized to the study of the large deviations of the non commutative laws of several independent hermitian brownian motions as a consequence we can bound from above entropies defined in the spirit of the microstates entropy introduced by voiculescu

classical r matrix theory of dispersionless systems ii 2 1 dimension theory a systematic way of constructing 2 1 dimensional dispersionless integrable hamiltonian systems is presented the method is based on the so called central extension procedure and classical r matrix applied to the poisson algebras of formal laurent series results are illustrated with the known and new 2 1 dimensional dispersionless systems

solving the dispersionless dym equation by hodograph transformations the method of hodograph transformation is used to find a class of exact solutions of the dispersionless dym ddym equation and its reductions in particular we discuss hodograph solutions of one variable and two variable reductions of the ddym system and investigate their associated lax as well as hamiltonian formalism

classical r matrices and compatible poisson structures for lax equations on poisson algebras given a classical r matrix on a poisson algebra we show how to construct a natural family of compatible poisson structures for the hamiltonian formulation of lax equations examples for which our formalism applies include the benny hierarchy the dispersionless toda lattice hierarchy the dispersionless kp and modified kp hierarchies the dispersionless dym hierarchy etc

a constructive method to static output stabilisation of markov jump systems this paper is concerned with the static output feedback stabilisation of markov jump systems transition probabilities are assumed to be incomplete namely they may be known uncertain with known lower and upper bounds and unknown to linearise the nonlinearities induced by unknown transition probabilities a linearisation is developed to solve the static output feedback problem by means of linear matrix inequalities a constructive method is proposed to separate the controller gain matrices from lyapunov variables based on the linearisation and the separation sufficient conditions are established to guarantee that the closed loop system is stochastically stable by the designed static output feedback controller the obtained results are further extended to deal with norm bounded or polytopic uncertainties on system matrices numerical examples are given to demonstrate the effectiveness of the proposed method

on dispersionless hirota equations of the dispersionless dym hierarchy we show that under twistor construction the only finite dimensional reduction of the dispersionless dym hierarchy which can be characterized by a free energy is the two reduction model with lax operator of the form up u0 p 1 the primary variables u and u0 can be expressed in terms of second derivatives of the associated free energy and the hierarchy flows can be written as a set of dispersionless hirota equations

robust stability and stabilization of uncertain discrete time markovian jump linear systems this note deals with robust stability and control of uncertain discrete time linear systems with markovian jumping parameters systems with polytopic type parameter uncertainty in either the state space model matrices or in the transition probability matrix of the markov process are considered this note develops methods of robust stability analysis and robust stabilization in the mean square sense which are dependent on the system uncertainty the design of both mode dependent and mode independent control laws is addressed the proposed methods are given in terms of linear matrix inequalities numerical examples are provided to demonstrate the effectiveness of the derived results

robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties this paper studies the static output feedback sof control problem of continuous time linear systems with polytopic uncertainties novel lmi conditions with a line search over a scalar variable for designing robust sof controllers are proposed where the uncertain output matrix of the considered system is allowed to be not of full row rank in particular it is shown that the new method can give less or at least the same conservative results than those methods by inserting a matrix equality constraint between system output matrix and lyapunov matrix furthermore the result is extended to the case of h control numerical examples are given to illustrate the effectiveness of the proposed method

stochastic calculus with respect to free brownian motion and analysis on wigner space we define stochastic integrals with respect to free brownian motion and show that they satisfy burkholder gundy type inequalities in operator norm we prove also a version of ito s predictable representation theorem as well as product form and functional form of ito s formula finally we develop stochastic analysis on the free fock space in analogy with stochastic analysis on the wiener space

s functions reductions and hodograph solutions of the rth dispersionless modified kp and dym hierarchies we introduce an s function formulation for the recently found rth dispersionless modified kp and rth dispersionless dym hierarchies giving also a connection of these s functions with the orlov functions of the hierarchies then we discuss a reduction scheme for the hierarchies that together with the s function formulation leads to hodograph systems for the associated solutions we consider also the connection of these reductions with those of the dispersionless kp hierarchy and with hydrodynamic type systems in particular for the one component and two component reduction we derive for both hierarchies ample sets of examples of explicit solutions

topological landau ginzburg theory with a rational potential and the dispersionless kp hierarchy based on the dispersionless kp dkp theory we study a topological landau ginzburg lg theory characterized by a rational potential writing the dkp hierarchy in a general form treating all the primaries in an equal basis we find that the hierarchy naturally includes the dispersionless continuous limit of toda hierarchy and its generalizations having a finite number of primaries several flat solutions of the topological lg theory are obtained in this formulation and are identified with those discussed by dubrovin we explicitly construct gravitational descendants for all the primary fields giving a residue formulae for the 3 point functions of the fields we show that these 3 point functions satisfy the topological recursion relation the string equation is obtained as the generalized hodograph solutions of the dkp hierarchy which show that all the gravitational effects to the constitutive equations 2 point functions can be renormalized into the coupling constants in the small phase space

static output feedback control synthesis for linear systems with time invariant parametric uncertainties this technical note is concerned with the problem of designing robust static output feedback controllers for linear discrete and continuous time systems with time invariant polytopic uncertainties sufficient conditions for static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities furthermore the results are also extended to h2 static output feedback controller designs numerical examples are given to illustrate the effectiveness of the proposed design methods

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

brief paper stabilization for state input delay systems via static and integral output feedback this paper addresses the static and integral output feedback stabilization problems of continuous time linear systems with an unknown state input delay by combining an augmentation approach and the delay partitioning technique criteria for static and integral output feedback stabilizability are proposed in terms of nonlinear matrix inequalities with a free parameter matrix introduced these new characterizations possess a special structure which leads to linearized iterative computation the effectiveness and merits of the proposed approach are shown through numerical examples

control of markovian jump discrete time systems with norm bounded uncertainty and unknown delay this paper studies the problem of control for discrete time delay linear systems with markovian jump parameters the system under consideration is subjected to both time varying norm bounded parameter uncertainty and unknown time delay in the state and markovian jump parameters in all system matrices we address the problem of robust state feedback control in which both robust stochastic stability and a prescribed h sub spl infin performance are required to be achieved irrespective of the uncertainty and time delay it is shown that the above problem can be solved if a set of coupled linear matrix inequalities has a solution

additional symmetries and solutions of the dispersionless kp hierarchy the dispersionless kp hierarchy is considered from the point of view of the twistor formalism a set of explicit additional symmetries is characterized and its action on the solutions of the twistor equations is studied a method for dealing with the twistor equations by taking advantage of hodograph type equations is proposed this method is applied for determining the orbits of solutions satisfying reduction constraints of gelfand dikii type under the action of additional symmetries

stochastic stability properties of jump linear systems jump linear systems are defined as a family of linear systems with randomly jumping parameters usually governed by a markov jump process and are used to model systems subject to failures or changes in structure the authors study stochastic stability properties in jump linear systems and the relationship among various moment and sample path stability properties it is shown that all second moment stability properties are equivalent and are sufficient for almost sure sample path stability and a testable necessary and sufficient condition for second moment stability is derived the lyapunov exponent method for the study of almost sure sample stability is discussed and a theorem which characterizes the lyapunov exponents of jump linear systems is presented finally for one dimensional jump linear system it is proved that the region for delta moment stability is monotonically converging to the region for almost sure stability at delta down arrow 0 sup

a new integrable equation in the 2 1 dimension dispersionless limit of the bkp equation a new integrable equation is obtained as the dispersionless limit of the bkp equation it is then studied in the framework of kodama s approach 1988 the new equation also turns out to be completely integrable with an infinite number of conservation laws the hodograph transformation method is applied to obtain exact solutions in various cases

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

robust static output feedback control for linear discrete time systems with time varying uncertainties this paper is concerned with the problem of designing robust static output feedback controllers for linear discrete time systems with timevarying polytopic uncertainties sufficient conditions for robust static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities and the results are extended to h2 and h static output feedback controller designs numerical examples are given to illustrate the effectiveness of the proposed design methods 2007 elsevier b v all rights reserved

new results on output feedback h_ infty control for linear discrete time systems this note investigates the problem of output feedback h control for linear discrete time systems three types of h controllers are considered namely static output feedback controllers dynamic output feedback controllers and observer based output feedback controllers new design conditions for the three type of output feedback controllers are introduced in terms of unified linear matrix inequality lmi representations which guarantee the prescribed h performances of the closed loop systems in contrast to the existing lmi conditions for designing the output feedback h controllers the improvement of the proposed results over the existing ones is shown by theoretical proof and numerical example

robust h sub spl infin control of discrete time markovian jump linear systems with mode dependent time delays considers the class of discrete time markovian jump linear system with norm bounded uncertainties and time delay which is dependent on the system mode linear matrix inequality lmi based sufficient conditions for the stability stabilization and h sub spl infin control are developed a numerical example is worked out to show the usefulness of the theoretical results

technical communique static output feedback stabilization an ilmi approach in this note the static output feedback stabilization problem is addressed using the linear matrix inequality technique a necessary and sufficient condition for static output feedback stabilizability for linear time invariant systems is derived in the form of a matrix inequality the extension of the result to h control is studied an iterative lmi ilmi algorithm is proposed to compute the feedback gain numerical examples are employed to demonstrate the effectiveness and the convergence of the algorithm

aging of spherical spin glasses sompolinski and zippelius 1981 propose the study of dynamical systems whose invariant measures are the gibbs measures for hard to analyze statistical physics models of interest in the course of doing so physicists often report of an aging phenomenon for example aging is expected to happen for the sherrington kirkpatrick model a disordered mean field model with a very complex phase transition in equilibrium at low temperature we shall study the langevin dynamics for a simplified spherical version of this model the induced rotational symmetry of the spherical model reduces the dynamics in question to an n dimensional coupled system of ornstein uhlenbeck processes whose random drift parameters are the eigenvalues of certain random matrices we obtain the limiting dynamics for n approaching infinity and by analyzing its long time behavior explain what is aging mathematically speaking what causes this phenomenon and what is its relationship with the phase transition of the corresponding equilibrium invariant measures

technical communique robust h2 control of continuous time markov jump linear systems this paper is concerned with the problem of designing robust h 2 state feedback controllers for continuous time markov jump linear systems subject to polytopic type parameter uncertainty based on the parameter dependent lyapunov function approach a new method for designing robust h 2 controllers is presented in terms of solutions to a set of linear matrix inequalities a numerical example is given to illustrate the effectiveness of the proposed design method

controllability stabilizability and continuous time markovian jump linear quadratic control consideration is given to the control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes the relationship between appropriately defined controllability stabilizability properties and the solution of the infinite time jump linear quadratic jlq optimal control problems is also examined although the solution of the continuous time markov jlq problem with finite or infinite time horizons is known only sufficient conditions for the existence of finite cost constant stabilizing controls for the infinite time problem appear in the literature in this paper necessary and sufficient conditions are established these conditions are based on new definitions of controllability observability stabilizability and detectability that are appropriate for continuous time markovian jump linear systems these definitions play the same role for the jlq problem as the deterministic properties do for the linear quadratic regulator lqr problem

large deviation bounds for matrix brownian motion we prove large deviation bounds for the convergence of hermitian matrix valued brownian motion towards free brownian motion as a consequence we obtain upper and lower bounds on the microstates entropy introduced by voiculescu 24

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

brief paper static output feedback stabilization of discrete time markovian jump linear systems a system augmentation approach this paper studies the static output feedback sof stabilization problem for discrete time markovian jump systems from a novel perspective the closed loop system is represented in a system augmentation form in which input and gain output matrices are separated by virtue of the system augmentation a novel necessary and sufficient condition for the existence of desired controllers is established in terms of a set of nonlinear matrix inequalities which possess a monotonic structure for a linearized computation and a convergent iteration algorithm is given to solve such inequalities in addition a special property of the feasible solutions enables one to further improve the solvability via a simple d k type optimization on the initial values an extension to mode independent sof stabilization is provided as well compared with some existing approaches to sof synthesis the proposed one has several advantages that make it specific for markovian jump systems the effectiveness and merit of the theoretical results are shown through some numerical examples

discussions around voiculescu s free entropies abstract in this article we set two analogous definitions of the free entropies and introduced by voiculescu invent math 118 1994 411 132 1998 189 we discuss their relations improving the preceding results obtained in cabanal duvillard and guionnet ann probab 2001 to appear where a bound on the microstates entropy was established

robust mode independent filtering for discrete time markov jump linear systems with multiplicative noises this paper deals with the robust mode independent linear filtering problem for discrete time markov jump linear systems with multiplicative noises it is assumed that the values of the markov chain are not available and that the parameters of the systems are subject to convex polytopic uncertainties the goal is to design a mode independent that is one that doesn t depend on the markov jump parameter dynamic linear filter such that the closed loop system is mean square stable and minimises an upper bound for the stationary expected value of the square error a linear matrix inequalities lmi formulation based on a parameter dependent lyapunov procedure is proposed to solve the problem the paper concludes with an illustrative example

bihamiltonian hierarchies in 2d topological field theory at one loop approximation we compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2d topological field theory the bihamiltonian structure of the hierarchy is given by a classical w algebra we compute the central charge of this algebra we also express the generating function of elliptic gromov witten invariants via tau function of the isomonodromy deformation problem arising in the theory of wdvv equations of associativity

topological generation and matrix models for quantum reflection groups abstract we prove that the von neumann algebras of quantum permutation groups and quantum reflection groups have the connes embedding property we do this by establishing several new topological generation results for the quantum permutation groups s n and the quantum reflection groups h n s we use these results to show that these quantum groups admit sufficiently many matrix models in particular all of these quantum groups have residually finite discrete duals and are in particular hyperlinear and certain flat matrix models for s n are inner faithful

a separation principle for the continuous time lq problem with markovian jump parameters in this paper we devise a separation principle for the finite horizon quadratic optimal control problem of continuous time markovian jump linear systems driven by a wiener process and with partial observations we assume that the output variable and the jump parameters are available to the controller it is desired to design a dynamic markovian jump controller such that the closed loop system minimizes the quadratic functional cost of the system over a finite horizon period of time as in the case with no jumps we show that an optimal controller can be obtained from two coupled riccati differential equations one associated to the optimal control problem when the state variable is available and the other one associated to the optimal filtering problem this is a separation principle for the finite horizon quadratic optimal control problem for continuous time markovian jump linear systems for the case in which the matrices are all time invariant we analyze the asymptotic behavior of the solution of the derived interconnected riccati differential equations to the solution of the associated set of coupled algebraic riccati equations as well as the mean square stabilizing property of this limiting solution when there is only one mode of operation our results coincide with the traditional ones for the lqg control of continuous time linear systems

wigner chaos and the fourth moment we prove that a normalized sequence of multiple wigner integrals in a fixed order of free wigner chaos converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2 the fourth moment of the semicircular law this extends to the free probabilistic setting some recent results by nualart and peccati on characterizations of central limit theorems in a fixed order of gaussian wiener chaos our proof is combinatorial analyzing the relevant noncrossing partitions that control the moments of the integrals we can also use these techniques to distinguish the first order of chaos from all others in terms of distributions we then use tools from the free malliavin calculus to give quantitative bounds on a distance between different orders of chaos when applied to highly symmetric kernels our results yield a new transfer principle connecting central limit theorems in free wigner chaos to those in gaussian wiener chaos we use this to prove a new free version of an important classical theorem the breuer major theorem

brief continuous time state feedback h2 control of markovian jump linear systems via convex analysis continuous time h 2 control problem for the class of linear systems with markovian jumps mjls using convex analysis is considered in this paper the definition of the h 2 norm for continuous time mjls is presented and related to the appropriate observability and controllability gramians a convex programming formulation for the h 2 control problem of mjls is developed that enables us to tackle the optimization problem of mjls under the assumption that the transition rate matrix p p i j for the markov chain may not be exactly known but belongs to an appropriate convex set an equivalence between the convex formulation when p is exactly known and the usual dynamic programming approach of quadratic optimal control of mjls is established it is shown that there exists a solution for the convex programming problem if and only if there exists the mean square stabilizing solution for a set of coupled algebraic riccati equations these results are compared with other related works in the current literature

on the miura map between the dispersionless kp and dispersionless modified kp hierarchies the miura map between the dispersionless kp and dispersionless modified kp hierarchies is investigated it is shown that the miura map is canonical with respect to their bi hamiltonian structures moreover inspired by the works of takasaki and takebe the twistor construction of solution structure for the dispersionless modified kp hierarchy is given

on the rth dispersionless toda hierarchy factorization problem additional symmetries and some solutions for a family of poisson algebras parametrized by and an associated lie algebraic splitting we consider the factorization of given canonical transformations in this context we rederive the recently found rth dispersionless modified kp hierarchies and rth dispersionless dym hierarchies giving a new miura map among them we also found a new integrable hierarchy which we call the rth dispersionless toda hierarchy moreover additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the rth dispersionless kp rth dispersionless dym and rth dispersionless toda equations some solutions are derived by examining the imposition of a time invariance to the potential rth dispersionless dym equation for which a complete integral is presented and therefore an appropriate envelope leads to a general solution symmetries and miura maps are applied to get new solutions and solutions of the rth dispersionless modified kp equation