spherical functions on symmetric cones in this note we obtain a recursive formula for the spherical functions associated with the symmetric cone of a formally real jordan algebra we use this formula as an inspiration for a similar recursive formula involving the jack polynomials

centers and limit cycles of polynomial differential systems of degree 4 via averaging theory abstract in this paper we classify the phase portraits in the poincare disc of the centers of the generalized class of kukles systems x y y x a x 3 y b x y 3 symmetric with respect to the y axis and we study using the averaging theory up to sixth order the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential systems of degree 4

invariant functions on lie groups and hamiltonian flows of surface group representations si est le groupe fondamental d une surface orientee fermee s et g est un groupe de lie satisfaisant des conditions tres generales alors l espace hom g g des classes de conjugaison de representation g a une structure symplectique naturelle on etudie la geometrie de cette structure symplectique a l aide d une famille naturelle de fonctions sur hom g g

stability analysis for continuous time positive systems with time varying delays this note addresses the stability problem of continuous time positive systems with time varying delays it is shown that such a system is asymptotically stable for any continuous and bounded delay if and only if the sum of all the system matrices is a hurwitz matrix the result is a time varying version of the widely known asymptotic stability criterion for constant delay positive systems a numerical example illustrates the correctness of our result

on exponential stability of switched homogeneous positive systems of degree one in this paper we investigate the stability problem of switched homogeneous positive systems of degree one we first show that the positiveness of the considered system is guaranteed when each switched subsystem is cooperative then an integral excitation condition is formulated to ensure the exponential stability of the switched homogeneous positive system of degree one in particular the proposed condition allows that some or even all switched subsystems are only lyapunov stable rather than asymptotically stable in the second part of this paper we further consider the cases of sub homogeneous systems time varying delay systems and discrete time systems in the sub homogeneous case we first show that the previous integral excitation condition is also sufficient such that the exponential stability is achieved we also derive an exponential stability result by relaxing the cooperative and integral excitation conditions in the time varying delay case a delay independent exponential stability result is shown under the extended integral excitation condition finally in the discrete time case based on the fact that each switched subsystem is order preserving instead of cooperative the exponential stability of the considered system is shown under a weak excitation condition

the centers and their cyclicity for a class of polynomial differential systems of degree 7 abstract we classify the global phase portraits in the poincare disc of the generalized kukles systems x y y x a x y 6 b x 3 y 4 c x 5 y 2 d x 7 which are symmetric with respect to both axes of coordinates moreover using the averaging theory up to sixth order we study the cyclicity of the center located at the origin of coordinates i e how many limit cycles can bifurcate from the origin of coordinates of the previous differential system when we perturb it inside the class of all polynomial differential systems of degree 7

stability of continuous time positive switched linear systems a weak common copositive lyapunov functions approach in this paper we study the problem of asymptotic stability of continuous time positive switched linear systems under both arbitrary and restricted switchings it is well known that asymptotic stability under arbitrary switching can be implied by several classes of strong common copositive lyapunov functions clfs i e functions whose derivative along the nontrivial system trajectories is negative however asymptotically stable positive switched systems may not admit strong common clfs the main contribution of this paper is to study the stability problem by requiring only weak common clfs firstly necessary and sufficient conditions are established for asymptotic stability under arbitrary switching among them an easily verifiable graphical stability criterion based on the connectivity of the digraphs associated with the subsystem matrices is proposed secondly we further relax the obtained graphical condition to derive a relaxed weak excitation condition for asymptotic stability under dwell time switching finally two examples are provided to illustrate the effectiveness of our theoretical results

on delay independent stability of a class of nonlinear positive time delay systems we present a condition for delay independent stability of a class of nonlinear positive systems this result applies to systems that are not necessarily monotone and extends recent work on cooperative nonlinear systems

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

exponential stability of homogeneous positive systems of degree one with time varying delays while the asymptotic stability of positive linear systems in the presence of bounded time delays has been thoroughly investigated the theory for nonlinear positive systems is considerably less well developed this technical note presents a set of conditions for establishing delay independent stability and bounding the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case specifically when the time delays have a known upper bound we derive necessary and sufficient conditions for exponential stability of a continuous time positive systems whose vector fields are homogeneous and cooperative and b discrete time positive systems whose vector fields are homogeneous and order preserving we then present explicit expressions that allow us to quantify the impact of delays on the decay rate and show that the best decay rate of positive linear systems that our bounds provide can be found via convex optimization finally we extend the results to general linear systems with time varying delays

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

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

stability of positive switched linear systems weak excitation and robustness to time varying delay this article investigates the stability of positive switched linear systems we start from motivating examples and focus on the case when each switched subsystem is marginally stable in the sense that all the eigenvalues of the subsystem matrix are in the closed left half plane with those on the imaginary axis simple instead of asymptotically stable a weak excitation condition is first proposed such that the considered positive switched linear system is exponentially stable an extension to the case without dwell time assumption is also presented then we study the influence of time varying delay on the stability of the considered positive switched linear system we show that the proposed weak excitation condition for the delay free case is also sufficient for the asymptotic stability of the positive switched linear system under unbounded time varying delay in addition it is shown that the convergence rate is exponential if there exists an upper bound for the delay irrespective of the magnitude of this bound the motivating examples are revisited to illustrate the theoretical results

stabilization of discrete time switched positive linear systems via weak switched linear copositive lyapunov function abstract by introducing a class of weak switched linear copositive lyapunov functions this paper designs new time dependent switching signals guaranteeing asymptotic stability of discrete time switched positive linear systems whose subsystems are only marginally stable our results have advantages over some existing ones in the literature moreover the main results have been extended to the general switched linear time varying system with disturbance numerical examples well demonstrate the superiority of the obtained results

observer based discrete time nonnegative edge synchronization of networked systems this paper studies the multi input and multi output discrete time nonnegative edge synchronization of networked systems based on neighbors output information the communication relationship among the edges of networked systems is modeled by well known line graph two observer based edge synchronization algorithms are designed for which some necessary and sufficient synchronization conditions are derived moreover some computable sufficient synchronization conditions are obtained in which the feedback matrix and the observer matrix are computed by solving the linear programming problems we finally design several simulation examples to demonstrate the validity of the given nonnegative edge synchronization algorithms

special functions of matrix argument i algebraic induction zonal polynomials and hypergeometric functions hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis multivariate statistics quantum physics molecular chemistry and number theory this paper presents a general theory of such functions for real division algebras these functions which generalize the classical hypergeometric functions are defined by infinite series on the space s s n f of all n x n hermitian matrices over the division algebra f the theory depends intrinsically upon the representation theory of the general linear group g gl n f of invertible n x n matrices over f and the theme of this work is the full exploitation of the inherent group theory the main technique is the use of the method of algebraic induction to realize explicitly the appropriate representations of g to decompose the space of polynomial functions on s and to describe the zonal polynomials from which the hypergeometric functions are constructed detailed descriptions of the convergence properties of the series expansions are given and integral representations are provided future papers in this series will develop the fine structure of these functions 0 introduction we begin a series of articles in which we develop the fine structure of generalized hypergeometric functions of matrix argument by fine structure we allude to the analogues of such classical results as series expansions integral formulas asymptotics differential equations summation formulas addition theorems composition formulas and recurrence relations this first paper in which we simultaneously treat real complex and quaternionic analysis is the result of our desire to present a complete theory of hypergeometric functions of matrix argument over real division algebras not only as a framework for the body of detailed results to follow in later papers but also to clarify the representationtheoretic foundation for such a theory although these hypergeometric functions are of interest on purely analytic grounds they arise in a wide range of applications indeed various classes of received by the editors may 19 1986 1980 malhema1zics s ject claszficain 1985 ren primary 22e30 22e45 33a75 43a85 43a90 62h10 secondary 20g20 32a07 32m15 44a10 62e15 key w and phrases generalized hypergeometric functions zonal polynomials representation theory algebraic induction multivariate statistics general linear group generalized gamma functions pochhammer symbols laplace transforms maximal compact subgroup invariant polynomials positive cones symmetric spaces schur functions special functions of matrix argument the first author is on leave from the university of wyoming the second author was partially supported by the national science foundation under grant mcs 8403381 and by the research council of the university of north carolina b1987 american mathematical society 0002 9947 87 1 00 25 per page

characterization and stability of autonomous positive descriptor systems this note addresses two problems the characterization of positivity of autonomous descriptor systems and the problem of finding conditions for their stability for these fundamental issues we provide necessary and sufficient conditions complemented with a numerically reliable computational approach the given conditions can be checked by means of linear programming lp also we derive a necessary and sufficient condition for stability that is connected to the underlying eigenstructure of the system

stabilization of switched positive system with impulse and marginally stable subsystems a mode dependent dwell time method abstract in this article the problem of stabilization for the switched positive linear system spls with impulse and marginally stable subsystems is studied in which both the continuous time case and the discrete time case are taken into account by the usage of a piecewise weak linear copositive lyapunov function lclf and the mode dependent dwell time mdt time dependent switching control laws are designed to guarantee the asymptotic stability of the system furthermore based on the comparison principal for positive systems the main results are generalized to the time variant spls two examples are worked out to indicate that our results have advantages over some known results in the literature

convexity of the cost functional in an optimal control problem for a class of positive switched systems abstract this paper deals with the optimal control of a class of positive switched systems the main feature of this class is that switching alters only the diagonal entries of the dynamic matrix the control input is represented by the switching signal itself and the optimal control problem is that of minimizing a positive linear combination of the final state variable first the switched system is embedded in the class of bilinear systems with control variables living in a simplex for each time point the main result is that the cost is convex with respect to the control variables this ensures that any pontryagin solution is optimal algorithms to find the optimal solution are then presented and an example taken from a simplified model for hiv mutation mitigation is discussed

stability and positivity of equilibria for subhomogeneous cooperative systems building on recent work on homogeneous cooperative systems we extend results concerning stability of such systems to subhomogeneous systems we also consider subhomogeneous cooperative systems with constant input and relate the global asymptotic stability of the unforced system to the existence and stability of positive equilibria for the system with input

stability analysis of positive switched linear systems with delays this technical note addresses the stability problem of delayed positive switched linear systems whose subsystems are all positive both discrete time systems and continuous time systems are studied in our analysis the delays in systems can be unbounded under certain conditions several stability results are established by constructing a sequence of functions that are positive monotonically decreasing and convergent to zero as time tends to infinity additionally continuous for continuous time systems it turns out that these functions can serve as an upper bound of the systems trajectories starting from a particular region finally a numerical example is presented to illustrate the obtained results

stability of discrete time positive switched linear systems with stable and marginally stable subsystems abstract in this paper we investigate the stability of discrete time positive switched linear systems the highlight of this work is that each subsystem is only assumed to be stable or marginally stable some sufficient conditions imposed on the subsystem matrices and switching signals are established to guarantee exponential stability of the considered positive switched linear systems an extension to the case of time varying and unbounded delay is also given compared with the continuous time counterparts in the literature our discrete time case is much more complicated involving the transitive connectivity between the subsystems specified by the switching signals

stability analysis of linear delay systems with cone invariance this paper is concerned with the stability and input output gain analysis of linear delay systems with cone invariance based on the partial ordering over a cone the monotonicity of the trajectory of the cone preserving systems with constant delays is first studied then by comparing the trajectory of the constant delay systems and that of time varying delay systems we prove that a cone preserving system with interval time varying delays is asymptotically stable if and only if the corresponding delay free system is asymptotically stable this implies that the stability of a cone preserving system is insensitive to the magnitude of the delays moreover based on the cone induced norms an explicit characterization on the cone induced gain of an input output cone preserving system is given in terms of system matrices finally numerical examples are provided to illustrate the theoretical results

stability theory for nonnegative and compartmental dynamical systems with time delay abstract nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations and involve the exchange of nonnegative quantities between subsystems or compartments these models are widespread in biological and physical sciences and play a key role in understanding these processes a key physical limitation of such systems is that transfers between compartments is not instantaneous and realistic models for capturing the dynamics of such systems should account for material in transit between compartments in this paper we present necessary and sufficient conditions for stability of nonnegative and compartmental dynamical systems with time delay specifically asymptotic stability conditions for linear and nonlinear nonnegative dynamical systems with time delay are established using linear lyapunov krasovskii functionals

asymptotic stability and decay rates of homogeneous positive systems with bounded and unbounded delays there are several results on the stability of nonlinear positive systems in the presence of time delays however most of them assume that the delays are constant this paper considers time varying possibly unbounded delays and establishes asymptotic stability and bounds the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case specifically we present a necessary and sufficient condition for delay independent stability of continuous time positive systems whose vector fields are cooperative and homogeneous we show that global asymptotic stability of such systems is independent of the magnitude and variation of the time delays for various classes of time delays we are able to derive explicit expressions that quantify the decay rates of positive systems we also provide the corresponding counterparts for discrete time positive systems whose vector fields are nondecreasing and homogeneous

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

generalized binomial expansion on lorentz cones we study some properties of generalized binomial coefficients for symmetric cones and we obtain a generalized binomial expansion formula for lorentz cones

stability properties of a class of positive switched systems with rank one difference abstract given a single input continuous time positive system described by a pair a b with a a diagonal matrix we investigate under what conditions there exists a state feedback law u t c x t that makes the resulting controlled system positive and asymptotically stable by this meaning that a b c is metzler and hurwitz in the second part of this note we assume that the state space model switches among different state feedback laws c i i 1 2 p each of them ensuring the positivity and show that the asymptotic stability of this type of switched system is equivalent to the asymptotic stability of all its subsystems while its stabilizability is equivalent to the existence of an asymptotically stable subsystem

conditions for the existence of a center for the kukles homogeneous systems abstract in this work we study necessary and sufficient conditions for the existence of centers in two families of linear centers with homogeneous quartic and quintic nonlinearities systems of this class are called kukles homogeneous systems systems of this type were studied for the first time by kukles who studied a linear center with cubic nonhomogeneous nonlinearities

almost sure estimation and synthesis of ultimate bounds for discrete time markov jump linear systems this paper is concerned with the problems of almost sure ultimate bound estimation and controller design for markov jump linear systems with bounded stochastic disturbances by utilising a lyapunov based scheme proposed in this paper an almost sure estimation of ellipsoidal ultimate bound eub of the system is obtained through tractable matrix inequalities on the basis of the estimation results the problem of designing mode dependent state feedback controllers that make the closed loop system admit a prescribed ellipsoid as an eub is considered the obtained results on estimation and synthesis are then extended to the case of systems with deficient mode information finally a practical example in dc motor devices is presented to demonstrate the applicability of the obtained results

on wishart distribution some extensions this paper proposes a unified approach that enables the wishart distribution to be studied simultaneously in the real complex quaternion and octonion cases under elliptical models in particular the matrix multivariate elliptical distribution the noncentral generalised wishart distribution the joint density of the eigenvalues and the distribution of the maximum eigenvalue are obtained for real normed division algebras

on the stability of positive linear switched systems under arbitrary switching laws we consider n dimensional positive linear switched systems a necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is hurwitz several researchers conjectured that for positive linear switched systems this condition is also sufficient recently gurvits shorten and mason showed that this conjecture is true for the case n 2 but is not true in general their results imply that there exists some minimal integer np such that the conjecture is true for all n np but is not true for n np we show that np 3

positivity of discrete singular systems and their stability in this paper we present an efficient approach to the analysis of discrete positive singular systems one of our main objectives is to investigate the problem of characterizing positivity of such systems previously this issue was not completely addressed we provide easily checkable necessary and sufficient conditions for such problem to be solved on the other hand we study the stability of discrete positive singular systems note that this is not a trivial problem since the set of admissible initial conditions is not the whole space but it is represented by a special cone all the conditions we provide are necessary and sufficient and are based on a reliable computational approach via linear programming

on the stability of switched positive linear systems it was recently conjectured that the hurwitz stability of the convex hull of a set of metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching in this note we show that 1 this conjecture is true for systems constructed from a pair of second order metzler matrices 2 the conjecture is true for systems constructed from an arbitrary finite number of second order metzler matrices and 3 the conjecture is in general false for higher order systems the implications of our results both for the design of switched positive linear systems and for research directions that arise as a result of our work are discussed toward the end of the note