a stability result for linear parameter varying systems abstract a negative real part eigenvalue assumption on a linear parameter varying system implies exponential stability under a condition that includes either slowly varying parameters or slowly accelerating parameters

extension of noncommutative soliton hierarchies a linear system which generates a moyal deformed two dimensional soliton equation as an integrability condition can be extended to a three dimensional linear system treating the deformation parameter as an additional coordinate the supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter the flow of which commutes with the flow of the deformed soliton equation in this way a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations we prove the extended hierarchy properties for the deformed akns hierarchy and specialize to the cases of deformed nls kdv and mkdv hierarchies corresponding results are also obtained for the deformed kp hierarchy a deformation equation determines a kind of seiberg witten map from classical solutions to solutions of the respective noncommutative deformed equation

brief paper a numerical sufficiency test for the asymptotic stability of linear time varying systems a numerical sufficiency test for the asymptotic stability of linear time varying hurwitz systems is proposed the algorithmic procedure constructs a bounding tube in which the state is guaranteed to stay the continuous time system is evaluated at discrete time instants for which successive quadratic lyapunov functions are generated the tube is constructed based on i a conservative estimate of the state evolution from a discrete time instant to the next obtained from the corresponding lyapunov function and ii re evaluation of the tube diameter at each discrete time instant to account for variations in the plant matrix the numerical test is illustrated in simulation via both a stable and an unstable system

new stability conditions for a class of linear time varying systems this paper considers the problem of asymptotic stability for linear time varying systems of the form x t a t x t some new stability conditions are proposed first two stability conditions for nonlinear time varying systems are given by using non monotonic lyapunov functions then the results obtained are extended to the linear case two stability conditions with infinite integral are derived furthermore by using the top floor function a linear matrix inequalities lmi condition and an eigenvalue criterion for asymptotic stability of systems are presented comparing with the existing results the conditions obtained allow both a t and a t are unbounded at t some numerical examples are provided to show the effectiveness of the theoretical results

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

non local continuity equations and binary darboux transformation of noncommutative anti self dual yang mills equations we present an infinite number of non local continuity equations of noncommutative anti self dual yang mills nc a sdym equations using the induction method of brezin et al 1979 phys lett b 82 442 and relate it to the lax pair and the parametric backlund transformation of the system from the lax pair we derive a binary darboux transformation to generate solutions of the nc a sdym equations

subspace identification of bilinear and lpv systems for open and closed loop data in this paper we present a novel algorithm to identify lpv systems with affine parameter dependence operating under open and closed loop conditions a factorization is introduced which makes it possible to form a predictor that predicts the output which is based on past inputs outputs and scheduling data the predictor contains the lpv equivalent of the markov parameters using this predictor ideas from closed loop lti identification are developed to estimate the state sequence from which the lpv system matrices can be constructed a numerically efficient implementation is presented using the kernel method it turns out that if structure is present in the scheduling sequence the computational complexity reduces even more

walks on groups counting reducible matrices polynomials and surface and free group automorphisms we prove sharp limit theorems on random walks on graphs with values in finite groups we then apply these re sults together with some elementary algebraic geometry number theory and representation theory to finite quotients of lattices in semisimple lie groups specifically sl n z and sp 2n z to show that a random element in one of these lattices has irre ducible characteristic polynomials overz the term random can be defined in at least two ways in terms of height and also in terms of word length in terms of a generating set we show the result using both definitions we use the above results to show that a random in terms of word length element of the mapping class group of a surface is pseudo anosov and that a a random free group automorphism is irreducible with irreducible powers or strongly irreducible 1

a direct characterization of l sub 2 gain controllers for lpv systems in this paper a class of linear parameter dependent output feedback controllers that satisfy quadratic stability and an induced l sub 2 norm bound for a given linear parameter varying lpv plant is considered by using a parameter independent common lyapunov function the solvability conditions are expressed in terms of finite dimensional linear matrix inequalities lmi evaluated at the extreme points of the admissible parameter set conditions under which strictly proper controllers can be used are obtained by restricting some of the controller matrices to be constant the input and output matrices can be parameter varying without destroying the problem s convexity cases where the controller matrices can be obtained without interpolation are also discussed thereby simplifying the implementation of the controller a numerical example is included which demonstrates the application of the results

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

assessing asymptotic stability of linear continuous time varying systems by computing the envelope of all trajectories in this note necessary and sufficient numerical conditions for asymptotic stability and for uniform asymptotic stability of linear continuous time varying systems are derived for a given set of initial conditions a tube containing all the trajectories of the system is constructed in the state space at each instant of time there exists an initial condition inside the set such that the resulting trajectory attains the border of the tube based on the above formulation necessary and sufficient conditions for asymptotic stability and for uniform asymptotic stability are expressed through the solution of a linear differential lyapunov equation the conditions can deal with the stability of periodic systems as well one of the main characteristics of the proposed necessary and sufficient conditions is that the only assumption on the dynamical matrix of the time varying system is continuity examples from the literature illustrate the superiority of the proposed conditions when compared to other methods

counting problems in apollonian packings an apollonian circle packing is a classical construction which is made by repeatedly inscribing circles into the triangular interstices in a descartes configuration of four mutually tangent circles remarkably if the original four circles have integer curvature all of the circles in the packing will have integer curvature making the packings of interest from a number theoretic point of view many of the natural arithmetic problems have required new and sophisticated tools to solve them the reason for this difficulty is that the study of apollonian packings reduces to the study of a subgroup of gl4 z which is thin in a sense that we describe in this article and arithmetic problems involving thin groups have only recently become approachable in broad generality in this article we report on what is currently known about apollonian packings in which all circles have integer curvature and how these results are obtained this survey is also meant to illustrate how to treat arithmetic problems related to other thin groups

random walks kleinian groups and bifurcation currents let be a holomorphic family of representations of a finitely generated group g into psl 2 parameterized by a complex manifold we define a notion of bifurcation current in this context that is a positive closed current on describing the bifurcations of this family of representations in a quantitative sense it is the analogue of the bifurcation current introduced by demarco for holomorphic families of rational mappings on 1 our definition relies on the theory of random products of matrices so it depends on the choice of a probability measure on g

a general fuzzy scheme for nonlinear processes with stability analysis abstract in this paper we develop a general fuzzy control scheme for nonlinear processes assuming little knowledge about the dynamics of the controlled process the proposed scheme starts by probing the process at different points in its operating region to generate a fuzzy quantisation a simple local controller is then designed at each fuzzy locality a fuzzy inference mechanism then links up tje local controllers to form a global controller which can be further refined by the learning algorithm by employing a newly developed structure adaptive fuzzy modelling scheme the appropriate fuzzy rule base for the inference mechanism can be extracted stably and efficiently the conditions for the stability of the global controller are rigourously established simulation results are presented to illustrate the effectiveness of the scheme

noncommutative burgers equation we present a noncommutative version of the burgers equation which possesses the lax representation and discuss the integrability in detail we find a noncommutative version of the cole hopf transformation and succeed in the linearization of it the linearized equation is the noncommutative diffusion equation and exactly solved we also discuss the properties of some exact solutions the result shows that the noncommutative burgers equation is completely integrable even though it contains infinite number of time derivatives furthermore we derive the noncommutative burgers equation from the noncommutative anti self dual yang mills equation by reduction which is an evidence for the noncommutative ward conjecture finally we present a noncommutative version of the burgers hierarchy by both the lax pair generating technique and sato s approach

bounded amplitude performance of switched lpv systems with applications to hybrid systems this paper discusses recent results on multiple linear agent control for systems satisfying a bounded amplitude performance constraint the plant is assumed to be a linear parameter varying lpv system scheduled along a nominal parameter trajectory in this respect the control problem represents a plant operating between a prespecified set of operating conditions linear controllers are designed at setpoints along this scheduling trajectory to satisfy bounded amplitude performance constraints this paper discusses an approach to analyze the switched system behavior under practical assumptions on the structure of the switching rule the approach combines the scheduling parameter with lpv system properties to derive bounds on the switching behavior of the system these estimates are then used to construct a logical model of the switched system behavior in the form of a timed automaton in this respect this paper presents a way of extracting logical models of continuous time system behavior

about the self dual chern simons system and toda field theories on the noncommutative plane the relation of the noncommutative self dual chern simons ncsdcs system to the noncommutative generalizations of toda and of affine toda field theories is investigated more deeply this paper continues the programme initiated in cabrera carnero i 2005 j high energy phys jhep10 2005 071 where it was presented how it is possible to define toda field theories through second order differential equation systems starting from the ncsdcs system here we show that using the connection of the ncsdcs to the noncommutative chiral model exact solutions of the toda field theories can also be constructed by means of the noncommutative extension of the uniton method proposed in lee k m 2004 j high energy phys jhep08 2004 054 by lee particularly some specific solutions of the nc liouville model are explicitly constructed

design of controllers for linear parameter varying systems by the gain scheduling technique abstract design of controllers for linear multi input multi output systems whose dynamics depend on a time varying parameter is studied design algorithms for the state feedback control law based on the gain scheduling technique are proposed state feedback control laws are designed so that one of the following goals is achieved i the poles of the closed loop system for all fixed parameter values are placed in a small neighborhood of a set of desired closed loop poles ii the stability of the closed loop system for all fixed parameter values guarantees that the of the system with the time varying parameter independent of the rate of change of the parameter rigorously justified and computationally appealing design algorithms are developed

an approach for computing the exact stability domain for a class of lti parameter dependent systems in this paper the complete stability domain for both single and multi parameter dependent lti systems is synthesized by extending existing results in the literature this domain is calculated through a guardian map which involves the determinant of the kronecker sum of a matrix with itself the single parameter case is easily computable whereas the multi parameter case is more involved the determinant of the bialternate sum of a matrix with itself is also exploited to reduce the computational complexity of the results presented in this paper

noncommutative instantons revisited we find a new gauge in which u 1 noncommutative instantons are explicitly non singular on noncommutative r 4 we also present a pedagogical introduction into noncommutative gauge theories

noncommutative integrable coupling system of tc equation hierarchy a noncommutative version of the tc soliton equation hierarchy is presented which possesses the zero curvature representation then we show that noncommutative nc tc equation can be derived from the noncommutative anti self dual yang mills equation by reduction finally an integrable coupling system of the nc tc equation hierarchy is constructed by using of the enlarged lax pairs

on noncommutative merons and instantons the yang mills ym and self dual yang mills sdym equations on the noncommutative euclidean four dimensional space are considered we introduce an ansatz for a gauge potential reducing the noncommutative sdym equations to a difference form of the nahm equations by constructing solutions to the difference nahm equations we obtain solutions of the noncommutative sdym equations they are noncommutative generalizations of the known solutions to the sdym equations such as the minkowski solution the one instanton solution and others using the noncommutative deformation of the corrigan fairlie t hooft wilzek ansatz we reduce the noncommutative ym equations to equations on a scalar field which have meron solutions in the commutative limit and show that they have no such solutions in the noncommutative case to overcome this difficulty another ansatz reducing the noncommutative ym equations to a system of difference equations on matrix valued functions is used for self dual configurations this system is reduced to the difference nahm equations

a topological tits alternative let k be a local eld and gln k a linear group over k we prove that either contains a relatively open solvable subgroup or it contains a relatively dense free subgroup this result has applications in dynamics riemannian foliations and pronite groups

integrable evolution equations on associative algebras this paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra which includes matrix clifford and group algebra valued systems a variety of new examples of integrable systems possessing higher order symmetries are presented symmetry reductions lead to an associative algebra valued version of the painleve transcendent equations the basic theory of hamiltonian structures for associative algebra valued systems is developed and the bihamiltonian structures for several examples are found

existence of oscillatory solutions along the path of longitudinal flight equilibriums of an unmanned aircraft when the automatic flight control system fails abstract the motion around the center of mass of a rigid unmanned aircraft whose flight control system fails in an aero data model in a research environment is described by a set of nine nonlinear ordinary differential equations the longitudinal flight with constant forward velocity is described by a subset of three nonlinear differential equations obtained from the general system in this paper the existence of oscillatory solutions of this system of three differential equations is proved by means of coincidence degree theory and mawhin s continuation theorem

conserved quantities in the noncommutative principal chiral model with wess zumino term we construct a noncommutative extension of the u n principal chiral model with wess zumino term and obtain an infinite set of local and non local conserved quantities for the model using the iterative procedure of brezin et al 1979 phys lett b 82 442 we also present the equivalent description as a lax formalism of the model we expand the fields perturbatively and derive zeroth and first order equations of motion zero curvature condition iteration method lax formalism local and non local conserved quantities

2 generation of simple lie algebras and free dense subgroups of algebraic groups abstract we construct generating pairs of simple lie algebras in characteristic zero we apply this construction to exhibit infinite series of 2 generator zariski dense subgroups that are free of rank 2 of the simple algebraic groups sl n c sp n c g 2 c

bi differential calculi and integrable models the existence of an infinite set of conserved currents in completely integrable classical models including chiral and toda models as well as the kp and self dual yang mills equations is traced back to a simple construction of an infinite chain of closed respectively covariantly constant 1 forms in a gauged bi differential calculus the latter consists of a differential algebra on which two differential maps act in a gauged bi differential calculus these maps are extended to flat covariant derivatives

bicomplexes and backlund transformations a bicomplex is a simple mathematical structure in particular associated with completely integrable models the conditions defining a bicomplex are a special form of a parameter dependent zero curvature condition we generalize the concept of a darboux matrix to bicomplexes and use it to derive backlund transformations for several models the method also works for moyal deformed equations with a corresponding deformed bicomplex

commuting flows and conservation laws for noncommutative lax hierarchies we discuss commuting flows and conservation laws for lax hierarchies on noncommutative spaces in the framework of the sato theory on commutative spaces the sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the lax hierarchies in terms of pseudodifferential operators noncommutative extension of the sato theory has been already studied by the author and toda and the existence of various noncommutative lax hierarchies are guaranteed in this paper we present conservation laws for the noncommutative lax hierarchies with both space space and space time noncommutativities and prove the existence of infinite number of conserved densities we also give the explicit representations of them in terms of lax operators our results include noncommutative versions of kp kdv boussinesq coupled kdv sawada kotera modified kdv equation and so on

the gain scheduling and the robust state feedback stabilization problems we consider dynamical uncertain systems of the form x spl dot a x w b x w u where w t spl isin w is an unknown but bounded time varying parameter we consider the robust state feedback stabilization problem in which we consider a control of the form u spl phi x and the gain scheduling stabilization problem in which a control of the form u spl phi x w full information control is admitted we show that for convex processes those systems in which for fixed x the set of all a x w b x w is convex the two problems are equivalent i e if there exists a locally lipschitz gain scheduling stabilizing control then there exists a robustly stabilizing control continuous everywhere except possibly at the origin thus for convex process stabilization knowledge of w t is not an advantage for the compensator then we consider the special class of polytopic lpv systems and show that there is no loss of regularity as in the general case if we pass from a gain scheduling controller to a state feedback controller in particular no discontinuity at the origin may occur we show that the existence of a dynamic controller always implies the existence of a static one finally we show that we can always find a linear gain scheduling controller for a stabilizable system this means that a possible advantage of the online measurement of the parameter w t is that this always allows for linear compensators whose implementation can be easier than that of nonlinear ones

reduced systems of 2 2 pseudo euclidean noncommutative self dual yang mills theories self dual yang mills equations on noncommutative spaces associated with pseudo euclidean space of signature 2 2 are shown to be related via dimensional reductions to noncommutative formulations of toda equations of generalized nonlinear schrodinger ns equations of the super korteweg de vries super kdv as well as of the matrix kdv equations the noncommutative extensions of their linear systems and bicomplexes associated with conserved quantities are discussed as well a q plane version of the kdv equation with linear system is also shown

random equations in nilpotent groups abstract in this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group g let sat g k be the set of all satisfiable equations over g in k variables for a free abelian group a m of rank m we show that the asymptotic density sat a m k of the set sat a m k in the whole space of all equations in k variables over g is equal to 0 for k 1 and is equal to k m k for k 2 more generally if g is a finitely generated nilpotent infinite group then again the asymptotic density of the set sat g 1 is 0 for k 2 we give robust estimates for the upper and lower asymptotic densities of the set sat g k namely we prove that these densities lie in the interval from 1 t g k h g k to k m k where h g is the hirsch length of g t g is the order of lower central torsion of g and m is the torsion free rank hirsch length of the abelianization of g this allows one to describe the asymptotic behavior of the set sat g k for different parameters k m h g t g

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

the investigation into the schwarz korteweg de vries equation and the schwarz derivative in 2 1 dimensions in this note we shall introduce a new integrable equation and the schwarz derivative in 2 1 dimensions first we show the existence of the lax pair for an equation which has the relation to the schwarz korteweg de vries skdv equation next we derive a new equation in 2 1 dimensions by using a well known higher dimensional manner to the lax pair for the skdv equation the 2 1 dimensional schwarz derivative is defined here finally we briefly discuss various results which we have obtained about the new equation

bicomplex formulation and moyal deformation of 2 1 dimensional fordy kulish systems using bicomplex formalism we construct generalizations of fordy kulish systems of matrix nonlinear schrodinger equations on two dimensional space time in two respects firstly we obtain corresponding equations in three space time dimensions secondly a moyal deformation is applied to the space time coordinates and the ordinary product of functions replaced by the moyal product in a suitable way both generalizations preserve the existence of an infinite set of conservation laws

on dense free subgroups of lie groups abstract we give a method for constructing dense and free subgroups in real lie groups in particular we show that any dense subgroup of a connected semisimple real lie group g contains a free group on two generators which is still dense in g and that any finitely generated dense subgroup in a connected non solvable lie group h contains a dense free subgroup of rank 2 dim h this answers a question of carriere and ghys and it gives an elementary proof to a conjecture of connes and sullivan on amenable actions which was first proved by zimmer

factorization methods for noncommutative kp and toda hierarchy we show that the solution space of the noncommutative kp hierarchy is the same as that of the commutative kp hierarchy owing to the birkhoff decomposition of groups over the noncommutative algebra the noncommutative toda hierarchy is introduced we derive the bilinear identities for the baker akhiezer functions and calculate the n soliton solutions of the noncommutative toda hierarchy

a necessary condition for quantitative exponential stability of linear state space systems for an arbitrary n nconstant matrix a the two following facts are well known 1 n re tracea maxj 1 nre j a 0 if u is a unitary matrix one can always find a skew hermitian matrix a so that u ea in this note we present the extension of these two facts to the context of linear time varying dynamical systems x a t x t 0 as a by product this result suggests that the notion of slowly varying state space systems commonly used in literature is mathematically not natural to the problem of exponential stability

lyapunov exponents for surface group representations let rho lambda lambda in lambda be a holomorphic family of representations of a surface group pi 1 s into mathrm psl 2 mathbb c where s is a topological possibly punctured surface with negative euler characteristic given a structure of riemann surface of finite type on s we construct a bifurcation current on the parameter space that is a 1 1 positive closed current attached to the bifurcations of the family it is defined as the dd c of the lyapunov exponent of the representation with respect to the brownian motion on the riemann surface s endowed with its poincare metric we show that this bifurcation current describes the asymptotic distribution of various codimension 1 phenomena in for instance the random hypersurfaces of defined by the condition that a random closed geodesic on s is mapped under to a parabolic element or the identity are asymptotically equidistributed with respect to the bifurcation current the proofs are based on our previous work deroin and dujardin invent math 190 57 118 2012 and on a careful control of a discretization procedure of the brownian motion

guaranteed properties of gain scheduled control for linear parameter varying plants abstract gain scheduling has proven to be a successful design methodology in many engineering applications however in the absence of a sound theoretical analysis these designs come with no guarantees on the robustness performance or even nominal stability of the overall gain scheduled design this paper presents such an analysis for one type of gain scheduled system namely a linear parameter varying plant scheduling on its exogenous parameters conditions are given which guarantee that the stability robustness and performance properties of the fixed operating point designs carry over to the global gain scheduled design these conditions confirm and formalize popular notions regarding gain scheduled design such as the scheduling variable should vary slowly

brief stability and l2 gain properties of lpv systems stability and l 2 gain properties of linear parameter varying systems are obtained under assumed bounds on either the maximum or average value of the parameter rate