on the generalized bi skew symmetric solutions of a linear matrix equation and its procrust problems in this paper the solvability conditions and the explicit expressions of the generalized bisymmetric and bi skew symmetric solutions of the matrix equation ax b are respectively established by applying two methods then the maximal and minimal ranks of the solutions are derived if the solvability conditions are not satisfied the generalized bisymmetric and bi skew symmetric least squares solutions of the matrix equation are considered and the generalized bisymmetric and bi skew symmetric least squares solutions with the minimum norm are also obtained in addition two algorithms are provided to compute the generalized bi skew symmetric least squares solution and some examples are given to illustrate that the algorithms are feasible

non commutative calculus optimal transport and functional inequalities in dissipative quantum systems we study dynamical optimal transport metrics between density matrices associated to symmetric dirichlet forms on finite dimensional c algebras our setting covers arbitrary skew derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for markov chains lindblad equations and the fermi ornstein uhlenbeck semigroup we develop a non nommutative differential calculus that allows us to obtain non commutative ricci curvature bounds logarithmic sobolev inequalities transport entropy inequalities and spectral gap estimates

how should one define a weak crystal we compare two proposals for the study of positional long range order one in terms of the spectrum of the translation operator the other in terms of the fourier spectrum we point out that only the first one allows for the consideration of molecular as opposed to atomic weakly periodic structures we illustrate this point on the thue morse system

diffractive point sets with entropy after a brief historical survey this paper introduces the notion of entropic model sets cut and project sets and more generally the notion of diffractive point sets with entropy such sets may be thought of as generalizations of lattice gases we show that taking the site occupation of a model set stochastically results with probabilistic certainty in well defined diffractive properties augmented by a constant diffuse background we discuss both the case of independent but identically distributed i i d random variables and that of independent but different i e site dependent random variables several examples are shown

brief paper the open loop linear quadratic differential game for index one descriptor systems in this paper we consider the linear quadratic differential game for descriptor systems that have index one we derive both necessary and sufficient conditions for existence of an open loop nash equilibrium

quantum logarithmic sobolev inequalities and rapid mixing a family of logarithmic sobolev inequalities on finite dimensional quantum state spaces is introduced the framework of non commutative lp spaces is reviewed and the relationship between quantum logarithmic sobolev inequalities and the hypercontractivity of quantum semigroups is discussed this relationship is central for the derivation of lower bounds for the logarithmic sobolev ls constants essential results for the family of inequalities are proved and we show an upper bound to the generalized ls constant in terms of the spectral gap of the generator of the semigroup these inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups when the ls constant and the spectral gap are of the same order convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory we provide a number of examples where improved bounds on the mixing time of several semigroups are obtained incl

impulse controllability and observability of rectangular descriptor systems the presence of impulsive responses in descriptor systems and how it relates to impulse controllability and impulse observability is considered it is shown that the equivalence between impulse controllability observability and the existence of an impulse eliminating semistate feedback output injection gain although true for square descriptor systems does not hold for rectangular descriptor systems in general hence necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback output injection gain are presented

causal observability of descriptor systems observability analysis for descriptor systems by using a matrix pencil approach is presented systems under consideration are the most general class of linear continuous descriptor systems a few practically relevant notions of observability particularly the causal observability which corresponds to the existence of a luenberger observer and the corresponding necessary and sufficient test conditions are provided the interpretation of various observabilities in terms of matrix pencils offers insights into the physical meaning of these concepts

hypercontractivity for semigroups of unital qubit channels hypercontractivity is proved for products of qubit channels that belong to self adjoint semigroups the hypercontractive bound gives necessary and sufficient conditions for a product of the form e t 1 h 1 otimes cdots otimes e t n h n to be a contraction from l p to l q where l p is the algebra of 2 n dimensional matrices equipped with the normalized schatten norm and each generator h j is a self adjoint positive semidefinite operator on the algebra of 2 dimensional matrices as a particular case the result establishes the hypercontractive bound for a product of qubit depolarizing channels

noncommutative maximal ergodic theorems this paper is devoted to the study of various maximal ergodic theorems in noncommutative spaces in particular we prove the noncommutative analogue of the classical dunford schwartz maximal ergodic inequality for positive contractions on and the analogue of stein s maximal inequality for symmetric positive contractions we also obtain the corresponding individual ergodic theorems we apply these results to a family of natural examples which frequently appear in von neumann algebra theory and in quantum probability

quantum reverse hypercontractivity we develop reverse versions of hypercontractive inequalities for quantum channels by generalizing classical techniques we prove a reverse hypercontractive inequality for tensor products of qubit depolarizing channels we apply this to obtain a rapid mixing result for depolarizing noise applied to large subspaces and to prove bounds on a quantum generalization of non interactive correlation distillation

extremals of functionals with competing symmetries abstract we present a new method of producing optimizing sequences for highly symmetric functionals the sequences have good convergence properties built in we apply the method in different settings to give elementary proofs of some classical inequalities such as the hardy littlewood sobolev and the logarithmic sobolev inequality in their sharp form

on the computation of the fundamental subspaces for descriptor systems abstractin this paper we investigate several theoretical and computational aspects of fundamental subspaces for linear time invariant descriptor systems which appear in the solution of many control and estimation problems different types of reachability and controllability for descriptor systems are described and discussed the rosenbrock system matrix pencil is employed for the computation of supremal output nulling subspaces and supremal output nulling reachability subspaces for descriptor systems

stability analysis of discrete singular fuzzy systems abstract this work mainly develops a discrete singular fuzzy model and discusses its stability the standard discrete t s model is generalized into the so called discrete singular takagi sugeno dst s model which can represent a larger class of nonlinear systems and the stability criteria of this system are then investigated these criteria are described by non strict linear matrix inequalities and we thus involve the projection method for ensuring stability based on some derived projection operators a projection algorithm is proposed for solving a feasible solution an illustrating example demonstrates the validity and effectiveness of the proposed method

an analog of the 2 wasserstein metric in non commutative probability under which the fermionic fokker planck equation is gradient flow for the entropy let mathfrak c denote the clifford algebra over mathbb r n which is the von neumann algebra generated by n self adjoint operators qj j 1 n satisfying the canonical anticommutation relations qiqj qjqi 2 iji and let denote the normalized trace on mathfrak c this algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom let mathfrak p denote the set of all positive operators rho in mathfrak c such that 1 these are the non commutative analogs of probability densities in the non commutative probability space mathfrak c tau the fermionic fokker planck equation is a quantum mechanical analog of the classical fokker planck equation with which it has much in common such as the same optimal hypercontractivity properties in this paper we construct a riemannian metric on mathfrak p that we show to be a natural analog of the classical 2 wasserstein metric and we show that in analogy with the classical case the fermionic fokker planck equation is gradient flow in this metric for the relative entropy with respect to the ground state we derive a number of consequences of this such as a sharp talagrand inequality for this metric and we prove a number of results pertaining to this metric several open problems are raised

riesz transforms associated with the number operator on the walsh system and the fermions abstract we prove that for 2 p f l p 1 1 n 1 2 f p is equivalent to n j 1 f f e j 2 1 2 p where is the discrete laplacian and the constants do not depend on f nor n such an equivalence is false for 1 p 1 2 f p similar results are obtained for fermions t viewed as some 2 n 2 n matrices equipped with their schatten norm c p comparing n 1 2 t p with n j 1 d j t 2 s 1 2 p where the d j s are the annihi lation operators on the fermions and n n j 1 d j d j is the number operator

direct transcription solution of high index optimal control problems and regular euler lagrange equations direct transcription methods are used to solve optimal control problems in many industrial settings models for physical systems often take the form of differential algebraic equations daes the index of the dae traditionally is viewed as an important factor in deciding whether a particular numerical approach should be used recently it has been observed that what the user thinks is the index of the dae may not be the same as the index available to the optimization software an investigation of this fact is underway in order to develop guidelines to assist users of various numerical optimal control packages this paper develops some theoretical results that will be needed for this development

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

impulsive mode elimination for descriptor systems by a structured p d feedback the problem of impulsive mode elimination for descriptor systems via a structured output proportional and derivative p d feedback is investigated in this technical note the motivation of solving this problem arises from an observation that the impulse behavior for some descriptor systems can not be eliminated completely by a parameterized output p d feedback but it can be removed by a structured p d feedback elegantly designed in order to explain this phenomenon explicitly a concept of the structured p d feedback is first introduced then an explicit necessary and sufficient condition is constructively derived for the closed loop systems to be regular and impulse free by the structured p d feedback the main result is represented in terms of the original system parameters instead of decomposed subsystem parameters finally an example is provided to illustrate the effectiveness of the proposed result

random fields on model sets with localized dependency and their diffraction for a random field on a general discrete set we introduce a condition that the range of the correlation from each site is within a predefined compact set d for such a random field defined on the model set that satisfies a natural geometric condition we develop a method to calculate the diffraction measure of the random field the method partitions the random field into a finite number of random fields each being independent and admitting the law of large numbers the diffraction measure of consists almost surely of a pure point component and an absolutely continuous component the former is the diffraction measure of the expectation e while the inverse fourier transform of the absolutely continuous component of turns out to be a weighted dirac comb which satisfies a simple formula moreover the pure point component will be understood quantitatively in a simple exact formula if the weights are continuous over the internal space of then we provide a sufficient condition that the diffraction measure of a random field on a model set is still pure point

necessary and sufficient conditions for the output feedback regularization of descriptor systems the problem of output feedback regularization of descriptor systems is studied necessary and sufficient conditions are given to regulate the descriptor systems by derivative and proportional output feedback into a closed loop regular system which is of index at most one

a system of matrix equations with five variables in this paper we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations a 1 x c 1 y b 1 d 1 a 2 w c 2 z b 2 d 2 a 3 v c 3 v b 3 c 4 a 4 v b 4 c 5 a 5 x y b 5 c 6 w z d 6 e 6 v f 6 g 6 and constitute an expression of the general solution to the system when it is solvable the outcomes of this paper encompass some recognized results in the collected works in addition we establish an algorithm and a numerical example to illustrate the theory constructed in the paper

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

on martingale inequalities in non commutative stochastic analysis we develop a non commutativelpstochastic calculus for the clifford stochastic integral anl2theory of which has been developed by barnett streater and wilde the main results are certain non commutativelpinequalities relating clifford integrals and their integrands these results are applied to extend the domain of the clifford integral froml2tol1integrands and we give applications to optional stopping of clifford martingales proving an analog of a theorem of burkholder the stopped clifford processfthas zero expectation providedet in proving these results we establish a number of results relating the clifford integral to the differential calculus in the clifford algebra in particular we show that the clifford integral is given by the divergence operator and we prove an explicit martingale representation theorem both of these results correspond closely to basic results for stochastic analysis on wiener space thus furthering the analogy between the clifford process and brownian motion

impulsive modes and causality in singular systems the concepts of impulsive modes and causality in singular systems are considered their properties relations and applications are studied matrix rank criteria are presented thus all controllabilities and observabilities and related concepts have matrix rank criteria based on the original system coefficient matrices

hypercontractivity in finite dimensional matrix algebras we obtain hypercontractivity estimates for a large class of semigroups defined on finite dimensional matrix algebras mn these semigroups arise from poisson like length functions on zn zn and provide new hypercontractive families of quantum channels when is conditionally negative we also study the optimality of our estimates

hypercontractivity for a quantum ornstein uhlenbeck semigroup we prove hypercontractivity for a quantum ornstein uhlenbeck semigroup on the entire algebra mathcal b h of bounded operators on a separable hilbert space h we exploit the particular structure of the spectrum together with hypercontractivity of the corresponding birth and death process and a proper decomposition of the domain then we deduce a logarithmic sobolev inequality for the semigroup and gain an elementary estimate of the best constant

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

inclusion principle for descriptor systems the purpose of this paper is to propose an expansion contraction framework for linear constant descriptor systems within the inclusion principle for dynamic systems our primary objective is to provide an explicit characterization of the expansion process whereby a given descriptor system is expanded into the larger space where all its solutions are reproducible by the expanded descriptor system if appropriate initial conditions are selected when a control law is formulated in the expanded space the proposed characterizations provide contractibility conditions for implementation of the control law in the original system a full freedom is provided for selecting appropriate matrices in the proposed expansion contraction control scheme in particular the derived theoretical framework serves as a flexible environment for expansion contraction control design of descriptor systems under overlapping information structure constraints

hypercontractivity in noncommutative lpspaces we introduce and study a setup for hypercontractivity in the interpolating family of noncommutative lpspaces associated to a locally normal state on an inductive limitc algebra

an inequality for hilbert schmidt norm for the absolute value c c c 1 2 and the hilbert schmidt norm c hs trc c 1 2 of an operatorc the following inequality is proved for any bounded linear operatorsa andb on a hilbert space a b hs leqq 2 1 2 a b hs the corresponding inequality for two normal state and of a von neumann algebram is also proved in the following form d varphi psi leqq xi varphi xi psi leqq 2 1 2 d varphi psi here denotes the unique vector representative of a state in a natural positive conep form andd denotes the bures distance defined as the infimum which is also the minimum of the distance of vector representatives of and in particular xi varphi 1 xi varphi 2 leqq 2 1 2 xi 1 xi 2 for any vector representatives j of j j 1 2

sharp uniform convexity and smoothness inequalities for trace norms we prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the schatten trace ideals c p which are the analogs of the lebesgue spaces l p in non commutative integration the inequalities are all precise analogs of results which had been known in l p but were only known in c p for special values of p in the course of our treatment of uniform convexity and smoothness inequalities for c p we obtain new and simple proofs of the known inequalities for l p

hypercontractivity of quasi free quantum semigroups hypercontractivity of a quantum dynamical semigroup has strong implications for its convergence behavior and entropy decay rate a logarithmic sobolev inequality and the corresponding logarithmic sobolev constant can be inferred from the semigroup s hypercontractive norm bound we consider completely positive quantum mechanical semigroups described by a lindblad master equation to prove the norm bound we follow an approach which has its roots in the study of classical rate equations we use interpolation theorems for non commutative spaces to obtain a general hypercontractive inequality from a particular norm bound then we derive a bound on the norm from an analysis of the block diagonal structure of the semigroup s spectrum we show that the dynamics of an n qubit graph state hamiltonian weakly coupled to a thermal environment is hypercontractive as a consequence this allows for the efficient preparation of graph states in time by coupling at sufficiently low temperature furthermore we extend our results to gapped liouvillians arising from a weak linear coupling of a free fermion systems

controllability and elimination of impulsive modes in descriptor systems as a refined notion impulsive mode controllability is suggested for the general class of linear continuous descriptor systems this notion reveals rich properties of the general descriptor systems in terms of admissibility of initial conditions admissibility of feedback control and impulsive mode elimination the condition for impulsive mode controllability is derived and proved to be necessary and sufficient for the existence of admissible feedback controls eliminating impulsive modes under impulsive mode controllability admissible feedback controls are constructed for eliminating impulsive modes on the basis of numerically reliable decompositions of the system s coefficient matrices

gaussian hypercontractivity revisited abstract recent work on convolution type products on fock space allow an elementary treatment of hypercontractivity for second quantised contractions in the fermi case new results are obtained lending further support to a conjecture of l gross

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

derivations as square roots of dirichlet forms abstract the aim of this work is to analyze the structure of a tracially symmetric dirichlet form on a c algebra in terms of a killing weight and a closable derivation taking values in a hilbert space with a bimodule structure it is shown that the generator of the associate markovian semigroup always appears in a natural way as the divergence of a closable derivation applications are shown to the decomposition of dirichlet forms and to the construction of differential calculus on metric spaces

the full information and state feedback h2 optimal controllers for descriptor systems in this paper we consider the full information and the state feedback h 2 control problems for continuous time descriptor systems an explicit characterization of all optimal controllers is given for these cases as a consequence we are able to present a parameterization of all optimal static and dynamic controllers for the lqr problem via state feedback we also show that for the general state feedback h 2 optimization the usual gain matrix defined as a linear function of a solution of a generalized riccati equation can be non optimal

optimal hypercontractivity for fermi fields and related non commutative integration inequalities optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained these are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by nelson in the process several results of independent interest in the theory of non commutative integration are established

matrix poincare φ sobolev inequalities and quantum ensembles sobolev type inequalities have been extensively studied in the frameworks of real valued functions and non commutative lp spaces and have proven useful in bounding the time evolution of classical quantum markov processes among many other applications in this paper we consider yet another fundamental setting matrix valued functions and prove new sobolev type inequalities for them our technical contributions are two fold i we establish a series of matrix poincare inequalities for separably convex functions and general functions with gaussian unitary ensembles inputs and ii we derive sobolev inequalities for matrix valued functions defined on boolean hypercubes and for those with gaussian distributions our results recover the corresponding classical inequalities i e real valued functions when the matrix has one dimension finally as an application of our technical outcomes we derive the upper bounds for a fundamental entropic quantity the holevo quantity in quantum information science since classical quantum channels are a special instance of matrix valued functions this is obtained through the equivalence between the constants in the strong data processing inequality and the sobolev inequality

non fragile pd state h control for a class of uncertain descriptor systems abstract this paper is concerned with the problem of non fragile proportional plus derivative pd state h control for a class of uncertain descriptor systems the parameter uncertainties are assumed to be time varying norm bounded appearing not only in the state matrix but also in the derivative matrix a new version of bounded real lemma for descriptor systems is proposed which guarantees a descriptor system to be normal and stable ns with disturbance rejection based on this we address the problem of designing non fragile pd state h controllers with additive and multiplicative controller uncertainties respectively sufficient conditions are presented to synthesize two classes of such non fragile controllers in terms of lmis finally a numerical example is provided to illustrate the effectiveness of the proposed design method

diffraction of stochastic point sets explicitly computable examples stochastic point processes relevant to the theory of long range aperiodic order are considered that display diffraction spectra of mixed type with special emphasis on explicitly computable cases together with a unified approach of reasonable generality the latter is based on the classical theory of point processes and the palm distribution several pairs of autocorrelation and diffraction measures are discussed which show a duality structure analogous to that of the poisson summation formula for lattice dirac combs

on the lyapunov theorem for singular systems in this paper we revisit the lyapunov theory for singular systems there are basically two well known generalized lyapunov equations used to characterize stability for singular systems we start with the lyapunov theorem of the work by lewis we show that the lyapunov equation of that theorem can lead to incorrect conclusion about stability some cases where that equation can be used are clarified we also show that an attempt to correct that theorem with a generalized lyapunov equation similar to the original one leads naturally to the generalized equation of takaba et al

technical communique regularizability of linear time invariant descriptor systems under decentralized control regularizability conditions for linear time invariant descriptor systems under decentralized output feedback are presented in this paper the regularizability is characterized by rank conditions that involve the original system matrices only new rank conditions for regularity of the system are also derived

remarks on the non commutative khintchine inequalities for 0 p 2 abstract we show that the validity of the non commutative khintchine inequality for some q with 1 q 2 implies its validity with another constant for all 1 p q we prove this for the inequality involving the rademacher functions but also for more general lacunary sequences or even non commutative analogues of the rademacher functions for instance we may apply it to the z 2 sequences previously considered by harcharras the result appears to be new in that case it implies that the space l 1 n contains as an operator space a large subspace uniformly isomorphic as an operator space to r k c k with k n 1 2 this naturally raises several interesting questions concerning the best possible such k unfortunately we cannot settle the validity of the non commutative khintchine inequality for 0 p 1 but we can prove several would be corollaries for instance given an infinite scalar matrix x i j we give a necessary and sufficient condition for x i j to be in the schatten class s p for almost all independent choices of signs 1 we also characterize the bounded schur multipliers from s 2 to s p the latter two characterizations extend to 0 p 1 results already known for 1 p 2 in addition we observe that the hypercontractive inequalities proved by carlen and lieb for the fermionic case remain valid for operator space valued functions and hence the kahane inequalities are valid in this setting

hypercontractivity on the q araki woods algebras extending a work of carlen and lieb biane has obtained the optimal hypercontractivity of the q ornstein uhlenbeck semigroup on the q deformation of the free group algebra in this note we look for an extension of this result to the type iii situation that is for the q araki woods algebras we show that hypercontractivity from l p to l 2 can occur if and only if the generator of the deformation is bounded

hypercontractivity in non commutative holomorphic spaces we prove an analog of janson s strong hypercontractivity inequality in a class of non commutative holomorphic algebras our setting is the q gaussian algebras q associated to the q fock spaces of bozejko kummerer and speicher for q 1 1 we construct subalgebras open image in new window a q segal bargmann transform and prove janson s strong hypercontractivity open image in new window for r an even integer

the p q skew symmetric extremal rank solutions to a system of quaternion matrix equations abstract let h m n denote the set of all m n matrices over the quaternion algebra h and p h m m q h n n be involutions we say that a h m n is p q symmetric or p q skewsymmetric if a paq or a paq we in this paper mainly investigate the p q skew symmetric maximal and minimal rank solutions to the system of quaternion matrix equations ax b xc d we present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with p q symmetry and p q skewsymmetry to the system the expressions of such solutions to this system are also given when the solvability conditions are satisfied a numerical example is presented to illustrate our results the findings of this paper extend some known results in this literature

on diffraction by aperiodic structures this paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals it analyses diffraction in the limit of the infinite system through an appropriately defined autocorrelation the main results are a justification of the standard way of calculating the diffraction spectrum of tilings obtained by the projection method and a proof of a variation on a conjecture by bombieri and taylor

contractivity properties of ornstein uhlenbeck semigroup for general commutation relations we study a certain class of von neumann algebras generated by selfadjoint elements i a i a i where a i a i satisfy the general commutation relations

variation of the parameters formula and the problem of bibo for singular measure differential systems with impulse effect recently we first presented and studied singular measure differential systems with impulsive solutions some fundamental theory was established in the present paper with the help of the drazin inverse the variation of the parameters formula of singular measure differential systems is obtained as an application the problem of boundary input and boundary output bibo is investigated

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

disturbance impulse controllability in descriptor system the descriptor system ex ax gu hv with external disturbance v is considered to characterize the possibility to cancel the impulses produced by external disturbance v through using control u the concept of disturbance impulse controllability is introduced

non commutative martingale inequalities we prove the analogue of the classical burkholder gundy inequalites for non commutative martingales as applications we give a characterization for an ito clifford integral to be an lp martingale via its integrand and then extend the ito clifford integral theory in l2 developed by barnett streater and wilde to lp for all 1 p we include an appendix on the non commutative analogue of the classical fefferman duality between h1 and bmo

absence of singular continuous diffraction for discrete multi component particle models particle models with finitely many types of particles are considered both on d and on discrete point sets of finite local complexity such sets include many standard examples of aperiodic order such as model sets or certain substitution systems the particle gas is defined by an interaction potential and a corresponding gibbs measure under some reasonable conditions on the underlying point set and the potential we show that the corresponding diffraction measure almost surely exists and consists of a pure point part and an absolutely continuous part with continuous density in particular no singular continuous part is present

concentration inequalities for functions of gibbs fields with application to diffraction and random gibbs measures we derive useful general concentration inequalities for functions of gibbs fields in the uniqueness regime we also consider expectations of random gibbs measures that depend on an additional disorder field and prove concentration w r t the disorder field both fields are assumed to be in the uniqueness regime allowing in particular for non independent disorder fields the modification of the bounds compared to the case of an independent field can be expressed in terms of constants that resemble the dobrushin contraction coefficient and are explicitly computable on the basis of these inequalities we obtain bounds on the deviation of a diffraction pattern created by random scatterers located on a general discrete point set in euclidean space restricted to a finite volume here we also allow for thermal dislocations of the scatterers around their equilibrium positions extending recent results for independent scatterers we give a universal upper bound on the probability of a deviation of the random scattering measures applied to an observable from its mean the bound is exponential in the number of scatterers with a rate that involves only the minimal distance between points in the point set

regularization of descriptor systems by output feedback conditions are given under which a descriptor or generalized state space system can be regularized by output feedback it is shown that under these conditions proportional and derivative output feedback controls can be constructed such that the closed loop system is regular and has index at most one this property ensures the solvability of the resulting system of dynamic algebraic equations a reduced form is given that allows the system properties as well as the feedback to be determined the construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way

new insight into reachable set estimation for uncertain singular time delay systems abstract this paper investigates the problem of reachable set estimation for a class of uncertain singular systems with time varying delays from a new point of view our consideration is centered on the design of a proportional derivative state feedback controller pdsfc such that the considered singular system is robustly normalizable and all the states of the closed loop system can be contained by a bounded set under zero initial conditions first a nominal singular time delay system is considered and sufficient conditions are obtained in terms of matrix inequalities for the existence of a pdsfc and an ellipsoid in this case the considered system is guaranteed to be normalizable and the reachable set of the closed loop systems is contained by the ellipsoid then the result is extended to the case of singular time delay systems with polytopic uncertainties and relaxed conditions are derived by introducing some weighting matrix variables furthermore based on the obtained results the reachable set of the considered closed loop singular system can be contained in a prescribed ellipsoid finally the effectiveness of our results are demonstrated by two numerical examples

observer design for descriptor systems two detectability notions are introduced for the most general class of linear descriptor systems necessary and sufficient conditions are derived for the detectabilities and for the existence of observers depending on existence conditions satisfied the observers can be either in the familiar luenberger form or in a generalized form i e in descriptor form as the original system itself is a unified procedure is provided for the design of observers in both forms

technical communique on impulsive algebraic multiplicities of linear time invariant singular systems under feedback impulsive algebraic multiplicities of linear time invariant singular systems under output feedback or decentralized output feedback are characterized in terms of system matrices directly the multiplicities describe the number of impulsive modes that can be eliminated and indicate the number of finite poles that can be assigned by the respective feedback these characterizations also lead to new regularizability conditions for singular systems

positive definite solutions of the nonlinear matrix equation x a h x 1 a i this paper is concerned with the positive definite solutions to the nonlinear matrix equation x a email protected 1a i where x is the unknown and a is a given complex matrix by introducing and studying a matrix operator on complex matrices it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation w b tw 1b i which has been extensively studied in the literature where b is a real matrix and is uniquely determined by a it is also shown that if the considered nonlinear matrix equation has a positive definite solution then it has the maximal and minimal solutions bounds of the positive definite solutions are also established in terms of matrix a finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

decentralized stabilization of singular and time delay large scale control systems with impulsive solutions decentralized stabilization of singular and time delay large scale control systems with impulsive solutions which has not been studied up to now is introduced and studied the control problem considered is the construction of a decentralized controller which ensures that a certain singular impulsive and time delay closed loop large scale system is exponentially stable by means of lumped picard and gauss seidel iteration methods which enable the authors to avoid the difficulties of constructing lyapunov functions explicit algebraic criteria of stabilization for such systems are established an example is given to illustrate the effectiveness of results obtained

feedback design for regularizing descriptor systems this paper surveys numerical techniques for the regularization of descriptor generalized state space systems by proportional and derivative feedback we review generalizations of controllability and observability to descriptor systems along with definitions of regularity and index in terms of the weierstras canonical form three condensed forms display the controllability and observability properties of a descriptor system the condensed forms are obtained through orthogonal equivalence transformations and rank decisions so they may be computed by numerically stable algorithms in addition the condensed forms display whether a descriptor system is regularizable i e when the system pencil can be made to be regular by derivative and or proportional output feedback and if so what index can be achieved also included is a a new characterization of descriptor systems that can be made to be regular with index 1 by proportional and derivative output feedback

on regularizing singular systems by decentralized output feedback this paper considers linear time invariant decentralized singular systems which are either nonregular or if they are regular they have impulsive modes it derives algebraic necessary and sufficient conditions for making a singular system both regular and impulse free by decentralized output feedback control laws and decentralized proportional plus derivative output feedback control laws

a simultaneous decomposition for seven matrices with applications abstract let h m n be the set of all m n matrices over the quaternion algebra h in this paper we construct a simultaneous decomposition for seven matrices with compatible sizes a h m n b h m p 1 c h m p 2 d h m p 3 e h q 1 n f h q 2 n and g h q 3 n as applications of the simultaneous matrix decomposition we give some solvability conditions general solutions as well as the range of ranks of the general solutions to the following two generalized sylvester matrix equations b x e c y f d z g a and b x w e c y f d z g a where a b c d e f and g are given quaternion matrices moreover we provide some numerical examples to illustrate the results of this paper

feedback controller norm optimization for linear time invariant descriptor systems with pole region constraint an algorithm is proposed to compute a state feedback gain matrix for a linear time invariant regular descriptor system which ensures that i the closed loop system is impulse free and ii all the finite poles are placed within a pre defined stability region in the complex plane the associated design flexibilities are exploited in minimizing upper bounds of the frobenius norm of respective gain matrices a class of linear matrix inequality lmi regions in the complex plane are chosen as constraints for the closed loop finite poles by representing a subset of the nonsingular matrices through solution of an lmi and linearizing the set of matrix inequalities arise in the regional pole finite poles assignment the associated optimizations are formulated as semidefinite programs the effectiveness of the developed algorithm is demonstrated through numerical examples significant reduction in the norm of feedback gain matrix is achieved in a two generator six bus power system

hypercontractivity and the logarithmic sobolev inequality for the completely bounded norm we develop the notions of hypercontractivity hc and the log sobolev ls inequality for completely bounded norms of one parameter semigroups of super operators acting on matrix algebras we prove the equivalence of the completely bounded versions of hc and ls under suitable hypotheses we also prove a version of the gross lemma which allows ls at general q to be deduced from ls at q 2

minimum norm regularization of descriptor systems by mixed output feedback we study the regularization problem for linear constant coefficient descriptor systems ex ax bu y1 cx y2 x by proportional and derivative mixed output feedback necessary and sufficient conditions are given which guarantee that there exist output feedbacks such that the closed loop system is regular has index at most one and e bg has a desired rank i e there is a desired number of differential and algebraic equations to resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of e is decreased numerical procedures are derived to construct the desired feedback gains these numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way

unknown input proportional multiple integral observer design for linear descriptor systems application to state and fault estimation in this note the problem of observer design for linear descriptor systems with faults and unknown inputs is considered first it is considered that the fault vector function f is s times piecewise continuously differentiable if the s th time derivative of f is null then s integral actions are included into a luenberger observer which is designed such that it estimates simultaneously the state the fault and its finite derivatives face to unknown inputs second when the fault is not time piecewise continuously differentiable but bounded like actuator noise or s th time derivative of fault is not null but bounded too a high gain observer is derived to attenuate the fault impact in estimation errors the considered faults may be unbounded may not be determinist and faults and unknown inputs may affect the state dynamic and plant outputs sufficient conditions for the existence of such observer are given results are illustrated with a differential algebraic power system

information filtering and array algorithms for descriptor systems subject to parameter uncertainties this paper presents information filters in riccati recursions and in array algorithms for descriptor systems subject to parameters uncertainties the filters are developed in filtered and predicted forms the inversion of the state matrix is avoided in the new information recursive formulas therefore it turns out clear that the invertibility of the state matrix usually considered in the state space information recursions is not necessary a numerical example is provided to illustrate the performance of the proposed robust array algorithms

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

poincare type inequalities on the discrete cube and in the car algebra we prove l p poincare inequalities with suitable dimension free constants for functions on the discrete cube 1 1 n as well known such inequalities for p an even integer allow to recover an exponential inequality hence the concentration phenomenon first obtained by bobkov and gotze we also get inequalities between the l p norms of left vert nabla f right vert and delta alpha f alpha 0 moreover l p spaces may be replaced by more general ones similar results hold true replacing functions on the cube by matrices in the algebra spanned by n fermions and the l p norm by the schatten norm c p

universal bounds on the selfaveraging of random diffraction measures we consider diffraction at random point scatterers on general discrete point sets in restricted to a finite volume we allow for random amplitudes and random dislocations of the scatterers we investigate the speed of convergence of the random scattering measures applied to an observable towards its mean when the finite volume tends to infinity we give an explicit universal large deviation upper bound that is exponential in the number of scatterers the rate is given in terms of a universal function that depends on the point set only through the minimal distance between points and on the observable only through a suitable sobolev norm our proof uses a cluster expansion and also provides a central limit theorem

solutions to optimization problems on ranks and inertias of a matrix function with applications abstract consider the hermitian matrix function f x a 3 b 3 x b 3 x subject to a consistent system of matrix equations 0 1 a 1 x c 1 a 2 xb 2 c 2 where means conjugate transpose in this paper we first establish explicit expansion formulas to calculate the global maximal and minimal ranks and inertias of the hermitian matrix function f x then we use the derived formulas to give necessary and sufficient conditions for system 0 1 to have re nonnegative definite re nonpositive definite re positive definite and re negative definite solutions moreover as another application of the derived formulas we establish necessary and sufficient conditions for the solvability to the system of matrix equations 0 2 a 1 x c 1 a 2 xb 2 c 2 b 3 x b 3 x a 3 and provide an expression of the general solution to 0 2 when it is solvable the findings of this paper widely extend the known results in the literature

some applications of hypercontractive inequalities in quantum information theory hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation in this note we discuss how hypercontractive inequalities in various settings can be used to obtain fairly concise proofs of several results in quantum information theory a recent lower bound of lancien and winter on the bias achievable by local measurements which are 4 designs spectral concentration bounds for k local hamiltonians and a recent result of pellegrino and seoane sepulveda giving general lower bounds on the classical bias obtainable in multiplayer xor games

brief paper recursive linear estimation for general discrete time descriptor systems this paper considers the optimal linear estimates recursion problem for discrete time linear systems in its more general formulation the system is allowed to be in descriptor form rectangular time variant and with the dynamical and measurement noises correlated we propose a new expression for the filter recursive equations which presents an interesting simple and symmetric structure convergence of the associated riccati recursion and stability properties of the steady state filter are provided

type b gaussian statistics as noncommutative central limits abstract we show that the noncommutative central limit theorem of speicher can be adapted to produce the gaussian statistics associated to coxeter groups of type b in the sense of bozejko ejsmont and hasebe viewed through the lens of central limits the passage from q gaussian statistics associated with symmetric groups to the gaussian statistics associated with coxeter groups of type b is precisely the passage from a sequence of independent elements that pairwise commute or anticommute to a coupled pair of such sequences the results pave the way for the transfer of known results from the bosonic fermionic settings to such broader contexts

observers for discrete singular systems the author considers the design of observers for the discrete singular system ex k 1 ax k bu k y k cx k placing special emphasis on the problems of state reconstruction and minimal time state reconstruction it is shown that for a singular system finite poles can be moved to infinity by state feedback and the state can be reconstructed by causal observers