regular dynamics in a delayed network of two neurons with all or none activation functions we consider a delayed network of two neurons with both self feedback and interaction described by an all or none threshold function the model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity we show that the dynamics of the network can be understood in terms of the iteration of a one dimensional map and we obtain simple criteria for the convergence of solutions the existence multiplicity and attractivity of periodic solutions

near equality near convexity sums of maximally monotone operators and averages of firmly nonexpansive mappings we study nearly equal and nearly convex sets ranges of maximally monotone operators and ranges and fixed points of convex combinations of firmly nonexpansive mappings the main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings a striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular throughout examples are provided to illustrate the theory we also obtain detailed information on the domain and range of the resolvent average

existence and stability of periodic solutions for parabolic systems with time delays this paper is concerned with the existence and stability time periodic solutions for a class of coupled parabolic equations with time delay and time delays may appear in the nonlinear reaction functions the existence of time periodic solutions is for a class of locally lipschitz continuous reaction functions without any quasimonotone requirement our approach to the problem is by the method of upper and lower solution and using schauder fixed point theorem some methods for proving the stability of the periodic solution are also given the results for the general system can be applied to the standard parabolic equations without time delay and corresponding ordinary differential system finally a model arising from chemistry is used to illustrate the obtained results

compositions and averages of two resolvents relative geometry of fixed points sets and a partial answer to a question by c byrne abstract we show that the set of fixed points of the average of two resolvents can be found from the set of fixed points for compositions of two resolvents associated with scaled monotone operators recently the proximal average has attracted considerable attention in convex analysis our results imply that the minimizers of proximal average functions can be found from the set of fixed points for compositions of two proximal mappings associated with scaled convex functions when both convex functions in the proximal average are indicator functions of convex sets least squares solutions can be completely recovered from the limiting cycles given by compositions of two projection mappings this provides a partial answer to a question posed by c byrne a novelty of our approach is to use the notion of resolvent average and proximal average

convergence to equilibria in scalar nonquasimonotone functional differential equations abstract we consider a class of scalar functional differential equations generating a strongly order preserving semiflow with respect to the exponential ordering introduced by smith and thieme it is shown that the boundedness of all solutions and the stability properties of an equilibrium are exactly the same as for the ordinary differential equation which is obtained by ignoring the delays the result on the boundedness of the solutions combined with a convergence theorem due to smith and thieme leads to explicit necessary and sufficient conditions for the convergence of all solutions starting from a dense subset of initial data under stronger conditions guaranteeing that the functional differential equation is asymptotically equivalent to a scalar ordinary differential equation a similar result is proved for the convergence of all solutions

exponential stability of general tracking algorithms tracking and adaptation algorithms are from a formal point of view nonlinear systems which depend on stochastic variables in a fairly complicated way the analysis of such algorithms is thus quite complicated a first step is to establish the exponential stability of these systems this is of interest in its own right and a prerequisite for the practical use of the algorithm it is also a necessary starting point to analyze the performance in terms of tracking and adaptation because that is how close the estimated parameters are to the time varying true ones in this paper we establish some general conditions for the exponential stability of a wide and common class of tracking algorithms this includes least mean squares recursive least squares and kalman filter based adaptation algorithms we show how stability of an averaged linear and deterministic equation and stability of the actual algorithm are linked to each other under weak conditions on the involved stochastic processes we also give explicit conditions for exponential stability of the most common algorithms the tracking performance of the algorithms is studied in a companion paper

the resolvent average on symmetric cones of jb algebras abstract in this paper we introduce the resolvent averages on symmetric cones of jb algebra and derive important properties such as self duality monotonicity and subhomogeneity this leads to the conclusion that each resolvent average contracts the thompson metric

traveling wave solutions in delayed reaction diffusion systems with mixed monotonicity this paper deals with the existence of traveling wave solutions in delayed reaction diffusion systems with mixed monotonicity based on two different mixed quasi monotonicity reaction terms we propose new conditions on the reaction terms and new definitions of upper and lower solutions by using schauder s fixed point theorem and a new cross iteration scheme we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions the general results obtained have been applied to type k monotone and type k competitive diffusive lotka volterra systems

asymptotic constancy for pseudo monotone dynamical systems on function spaces abstract a pseudo monotone dynamical system is a dynamical system which preserves the order relation between initial points and equilibrium points the purpose of this paper is to present some convergence oscillation and order stability criteria for pseudo monotone dynamical systems on function spaces for which each constant function is an equilibrium point some applications to neutral functional differential equations and semilinear parabolic partial differential equations with neumann boundary condition are given

self dual regularization of monotone operators via the resolvent average we give two self dual regularizations of maximal monotone operators on hilbert spaces these regularizations and their set valued inverses are strongly monotone single valued and lipschitz with full domain moreover these regularizations graphically converge to the original monotone operator if a maximal monotone operator has nonempty zeros these self dual regularizations can be used to find its least norm solution when the maximal monotone operator is the subdifferential of a proper lower semicontinuous convex function with nonempty minimizers this translates to finding the least norm minimizer

discrete order preserving semigroups and stability for periodic parabolic differential equations on etudie la stabilite globale des solutions des equations paraboliques dans une forme non auto adjointe avec des coefficients periodiques dans le temps la structure mathematique est celle des semigroupes preservant l ordre a temps discret s n n e n sur un reseau de banach

complicated dynamics of scalar reaction diffusion equations with a nonlocal term we consider the dynamics of scalar equations ut uxx f x u c x u 0 x l where denotes some weighted spatial average and dinchlet boundary conditions are assumed prescribing f c appropriately it is shown that complicated dynamics can occur specifically linearisations at equilibria can have any number of purely imaginary eigenvalues moreover the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily up to any finite order these results are in marked contrast with the case 0 where bounded solutions are known to converge to equilibrium

stability convergence and performance of an adaptive control algorithm applied to a randomly varying system the stability and performance of a stochastic adaptive control algorithm applied to a randomly varying linear system are investigated the authors demonstrate that loss functions on the input output process converge to their expectation with respect to an invariant probability at a geometric rate and hence a form of stochastic exponential asymptotic stability is established and when the parameter variation and measurement noise are small it is shown that the performance is nearly optimal and if an excitation signal is added in the control law near consistency of the parameter estimates is obtained further results include central limit theorems and the law of large numbers of the input output and parameter processes

existence stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations abstract in this paper we study the existence stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise we first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions we then examine the existence and stability of random complete quasi solutions and establish the relations of these solutions and the structures of tempered attractors when the stochastic equations are incorporated with periodic forcing we obtain the existence and stability of random periodic solutions for the stochastic chafee infante equation we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions

periodic solutions in an inhibitory two neuron network in this paper we consider a delayed network of two neurons with self feedback and interaction described by an all or none threshold function the discontinuity of signal function makes it difficult to apply directly dynamical system we show that the dynamics of the network can be understood in terms of the iterations of a one dimensional map and we obtain the existence and attractivity of periodic solutions moreover because the network is a limiting case of the corresponding smooth system as the parameter tends to infinity the above results can act as the guide to the rich dynamics of the smooth system therefore our results have important significance in both theory and applications

effects of a degeneracy in the competition model part i classical and generalized steady state solutions abstract we study the competition model where the coefficient functions are strictly positive over the underlying spatial region except b x which vanishes in a nontrivial subdomain of and is positive in the rest of we show that there exists a critical number such that if then the model behaves similarly to the well studied classical competition model where all the coefficient functions are positive constants but when new phenomena occur our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative but can be qualitative as well in part i here we mainly study two kinds of steady state solutions which determine the dynamics of the model one consists of finite functions while the other consists of generalized functions which satisfy u v 0 on the part of the domain that b x vanishes but are positive and finite on the rest of the domain and are determined by certain boundary blow up systems the research is continued in part ii where these two kinds of steady state solutions will be used to determine the dynamics of the model

the resolvent average on symmetric cones abstract recently bauschke et al introduced a new notion of proximal average in the context of convex analysis and studied this subject systemically in bauschke et al 2004 2007 2008 2009 3 7 from various viewpoints in addition this new concept was applied to positive semidefinite matrices under the name of resolvent average and basic properties of the resolvent average are successfully established by themselves from a totally different view and techniques of convex analysis rather than the classical matrix analysis 8 bauschke et al 2010 inspired by their works and the well known fact that the convex cone of positive definite matrices is a typical example of a symmetric cone self dual homogeneous cone we study the resolvent average on symmetric cones and derive corresponding results in a different manner based on a purely jordan algebraic technique

monotone dynamical systems with dense periodic points abstract in this paper we prove a recent conjecture by m hirsch which says that if f is a discrete time monotone dynamical system with f a homeomorphism on an open connected subset of a finite dimensional vector space and the periodic points of f are dense in then f is periodic

existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays the existence of periodic quasisolutions and the dynamics for a couple system of semilinear parabolic equations with discrete time delays and periodic coefficients are investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the periodic boundary value system has a pair of coupled t upper and lower solutions then the periodic boundary value problem has a pair of periodic quasisolutions and the sector between the quasisolutions is an attractor of the delayed periodic parabolic system under some additional conditions the periodic quasisolutions are exactly true periodic solutions of the periodic boundary value system finally three models arising from ecology are given to illustrate the obtained results

asymptotic periodicity of a food limited diffusive population model with time delay abstract in this paper a general reaction diffusion food limited population model with time delay is proposed accordingly the existence and uniqueness of the periodic solutions for the boundary value problem and the asymptotic periodicity of the initial boundary value problem are considered finally the effect of the time delay on the asymptotic behavior of the solutions is given

the resolvent average of monotone operators dominant and recessive properties within convex analysis a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago when one considers the subdifferential of the proximal average a natural averaging operation of the subdifferentials of the averaged functions emerges in the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in hilbert spaces transforming it into the resolvent average the theory of resolvent averages contains many desirable properties in particular we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average as a consequence we not only recover a significant part of the theory of proximal averages but also obtain results and properties which have not been previously available

information divergence and risk for binary experiments we unify f divergences bregman divergences surrogate regret bounds proper scoring rules cost curves roc curves and statistical information we do this by systematically studying integral and variational representations of these objects and in so doing identify their representation primitives which all are related to cost sensitive binary classification as well as developing relationships between generative and discriminative views of learning the new machinery leads to tight and more general surrogate regret bounds and generalised pinsker inequalities relating f divergences to variational divergence the new viewpoint also illuminates existing algorithms it provides a new derivation of support vector machines in terms of divergences and relates maximum mean discrepancy to fisher linear discriminants

the optimal value and optimal solutions of the proximal average of convex functions abstract the proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection this paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter concavity of the optimal value is established and implies further regularity properties of the optimal value boundedness outer semicontinuity single valuedness continuity and lipschitz continuity of the minimizer mapping are concluded under various assumptions sharp minimizers are given further attention several examples are given to illustrate our results

stability and attractivity of periodic solutions of parabolic systems with time delays abstract this paper is concerned with the existence stability and global attractivity of time periodic solutions for a class of coupled parabolic equations in a bounded domain the problem under consideration includes coupled system of parabolic and ordinary differential equations and time delays may appear in the nonlinear reaction functions our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations the existence of time periodic solutions is for a class of locally lipschitz continuous reaction functions without any quasimonotone requirement using schauder fixed point theorem while the stability and attractivity analysis is for quasimonotone nondecreasing and mixed quasimonotone reaction functions using the monotone iterative scheme the results for the general system are applied to the standard parabolic equations without time delay and to the corresponding ordinary differential system applications are also given to three lotka volterra reaction diffusion model problems and in each problem a sufficient condition on the reaction rates is obtained to ensure the stability and global attractivity of positive periodic solutions

periodic solutions of parabolic systems with time delays existence of maximal and minimal periodic solutions of a coupled system of parabolic equations with time delays and with nonlinear boundary conditions is discussed the proof of the existence theorem is based on the method of upper and lower solutions and its associated monotone iterations this method is constructive and can be used to develop a computational algorithm for numerical solutions of the periodic parabolic system an application is given to a competitor competitor mutualist model which consists of a coupled system of three reaction diffusion equations with time delays

a proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in the presence of computational errors abstract in a hilbert space we study the convergence of a proximal point method to a common zero of a finite family of maximal monotone operators under the presence of computational errors most results known in the literature establish the convergence of proximal point methods when computational errors are summable in the present paper the convergence of the method is established for nonsummable computational errors we show that the proximal point method generates a good approximate solution if the sequence of computational errors is bounded from above by a constant

fitzpatrick functions cyclic monotonicity and rockafellar s antiderivative several deeper results on maximal monotone operators have recently found simpler proofs using fitzpatrick functions in this paper we study a sequence of fitzpatrick functions associated with a monotone operator the first term of this sequence coincides with the original fitzpatrick function and the other terms turn out to be useful for the identification and characterization of cyclic monotonicity properties it is shown that for any maximal cyclically monotone operator the pointwise supremum of the sequence of fitzpatrick functions is closely related to rockafellar s antiderivative several examples are explicitly computed for the purpose of illustration in contrast to rockafellar s result a maximal 3 cyclically monotone operator need not be maximal monotone a simplified proof of asplund s observation that the rotation in the euclidean plane by n is n cyclically monotone but not n 1 cyclically monotone is provided the fitzpatrick family of the subdifferential operator of a sublinear and of an indicator function is studied in detail we conclude with a new proof of moreau s result concerning the convexity of the set of proximal mappings

the aastrom wittenmark self tuning regulator revisited and els based adaptive trackers although considerable progress has been made in stochastic adaptive control the problem concerning the convergence of the original self tuning regulator proposed by k j aastrom and b wittenmark 1973 is still open since it is attractive in theory and important in applications this problem is examined and a rigorous proof of its stability and optimality is given related problems such as convergence of the extended least squares els based adaptive tracker are also considered

discrete infinite dimensional type k monotone dynamical systems and time periodic reaction diffusion systems abstract the asymptotic behavior of discrete type k monotone dynamical systems and reaction diffusion equations is investigated the studying content includes the index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is locally asymptotically stable with respect to the face it belongs to and at this point the principal eigenvalue of the diagonal partial derivative about any component not belonging to the face is not one a nice result presented is the sufficient and necessary conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergent result for all orbits applications are made to time periodic lotka volterra systems with diffusion and sufficient conditions for such systems to have a unique positive periodic solution attracting all positive initial value functions are given for more general time periodic type k monotone reaction diffusion systems with spatial homogeneity a simple condition is given to guarantee the convergence of all positive solutions

autoconjugate representers for linear monotone operators monotone operators are of central importance in modern optimization and nonlinear analysis their study has been revolutionized lately due to the systematic use of the fitzpatrick function pioneered by penot and svaiter a topic of recent interest has been the representation of maximal monotone operators by so called autoconjugate functions two explicit constructions were proposed the first by penot and z linescu in 2005 and another by bauschke and wang in 2007 the former requires a mild constraint qualification while the latter is based on the proximal average we show that these two autoconjugate representers must coincide for continuous linear monotone operators on reflexive spaces the continuity and the linearity assumption are both essential as examples of discontinuous linear operators and of subdifferential operators illustrate furthermore we also construct an infinite family of autoconjugate representers for the identity operator on the real line

the resolvent average for positive semidefinite matrices we define a new average termed the resolvent average for positive semidefinite matrices for positive definite matrices the resolvent average enjoys self duality and it interpolates between the harmonic and the arithmetic averages which it approaches when taking appropriate limits we compare the resolvent average to the geometric mean some applications to matrix functions are also given

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

phase translation group actions on strongly monotone skew product semiflows we establish a convergence property for pseudo bounded forward orbits of strongly monotone skew product semiflows with invariant phasetranslation group actions the results are then applied to obtain global convergence of certain chemical reaction networks whose associated systems in reaction coordinates are monotone as well as the dynamics of certain reactiondiffusion systems in time recurrent structure including periodicity almost periodicity and almost automorphy

a reaction diffusion system of a competitor competitor mutualist model we investigate the homogeneous dirichlet problem and neumann problem to a reactiondiffusion system of a competitor competitor mutualist model the existence uniqueness and boundedness of the solutions are established by means of the comparison principle and the monotonicity method for the dirichlet problem we study the existence of trivial and nontrivial nonnegative equilibrium solutions and their stabilities for the neumann problem we analyze the contant equilibrium solutions and their stabilities the main method used in studying of the stabilities is the spectral analysis to the linearized operators the o d e problem to the same model was proposed and studied by b rai h i freedman and j f addicott

contractive barycentric maps and l1 ergodic theorems on the cone of positive definite matrices abstract we are concerned with contractive with respect to the wasserstein metric barycenters of probability measures with bounded support on the convex cone of positive definite matrices equipped with the thompson metric based on the important construction schemes of multivariate matrix means namely the proximal average and the cartan mean the least squares average for the cartan hadamard metric we construct a one parameter family of contractive barycentric maps interpolating continuously and monotonically the harmonic arithmetic and cartan barycenters we show that each contractive barycentric map is monotonic for the stochastic order induced by the cone and establish stochastic approximations and l 1 ergodic theorems for the parameterized contractive barycenters

convergence and logarithm laws of self tuning regulators abstract this paper starts with a survey of some recent results on least squares ls and ls based self tuning regulators str several long standing problems concerning the basic properties of ls based str such as stability optimality consistency and the best convergence rate are solved within a unified framework some previously related results are also subsumed and extended various new techniques for analysing ls based adaptive tracking systems are presented which may also be useful for analysing other adaptive control problems

dynamics of the periodic type k competitive kolmogorov systems abstract for time periodic dissipative and irreducible type k competitive kolmogorov systems it is proved that there is a canonically defined countable family f of unordered disjoint invariant sets with the property that for every persistent trajectory whose limit set is not a cycle there exists a unique trajectory in some element of f such that these two trajectories are asymptotic and the corresponding points in these two trajectories are k related

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

asymptotic behaviour of comparable skew product semiflows with applications the almost 1 cover lifting property of omega limit sets is established for nonmonotone skew product semiflows which are comparable to uniformly stable eventually strongly monotone skew product semiflows these results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of odes reaction diffusion systems differential systems with time delays and semilinear parabolic equations

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems ii finite dimensional systems abstract this is the second part in a series of papers concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part focuses on applications of general theory developed in the authors paper j mierczy ski w shen principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory trans amer math soc in press http dx doi org 10 1090 s0002 9947 2013 05814 x preprint available at http arxiv org abs 1209 3475 to positive random dynamical systems on finite dimensional ordered banach spaces it is shown under some quite general assumptions that measurable linear skew product semidynamical systems generated by measurable families of positive matrices and by strongly cooperative or type k strongly monotone systems of linear ordinary differential equations admit measurable families of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations

monotone semiflows in scalar non quasi monotone functional differential equations abstract by introducing a stronger than pointwise ordering conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi monotone typically the maximum delay must be the smaller the more quasi monotonicity is violated the theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations

how to transform one convex function continuously into another the proximal average operator provides a parametric family of convex functions that continuously transform one convex function into another even when the domains of the two functions do not intersect we prove that the proximal average operator is a homotopy with respect to the epi topology study its properties and present several explicit formulas for specific classes of functions the parametric family inherits desirable properties such as differentiability and strict convexity from the given functions the results illustrate the powerful tools available in convex and variational analysis from both a theoretical and a computational point of view

dynamics integrability and topology for some classes of kolmogorov hamiltonian systems in r 4 abstract in this paper we first give the sufficient and necessary conditions in order that two classes of polynomial kolmogorov systems in r 4 are hamiltonian systems then we study the integrability of these hamiltonian systems in the liouville sense finally we investigate the global dynamics of the completely integrable lotka volterra hamiltonian systems in r 4 as an application of the invariant subsets of these systems we obtain topological classifications of the 3 submanifolds in r 4 defined by the hypersurfaces a x y b z w c x 2 y d x y 2 e z 2 w f z w 2 h where a b c d e f w and h are real constants

structure preserving matrix means on the marcus minc stochastic matrices abstract in this paper we show that the set ps m of all m m positive definite stochastic matrices with diagonal entries bounded above by 1 m 1 is stable under the weighted geometric mean operation it is further shown that ps m is stable for some well known multivariable matrix means the least squares means for the riemannian trace metric and the kullback leibler divergence on the convex cone of m m positive definite matrices and alm ando li mathias and bmp bini meini poloni geometric means

nonexpansiveness of the resolvent average abstract we show that the resolvent average for positive definite matrices which interpolates between the harmonic and the arithmetic means contracts the thompson metric on the convex cone of positive definite matrices classical results depending on the nonexpansive property of the arithmetic average are considered in the context of the resolvent average in particular resolvent power mean approximation to the karcher mean and ferrante and levy like matrix equations are investigated

periodic solutions of parabolic systems with nonlinear boundary conditions this paper is concerned with the existence and stability of periodic solutions for a coupled system of nonlinear parabolic equations under nonlinear boundary conditions the approach to the problem is by the method of upper and lower solutions and its associated monotone iterations this method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for parabolic initial boundary value problems a sufficient condition for the stability of a periodic solution is also given these results are applied to three model problems arising from chemical kinetics ecology and population biology

the matrix geometric mean of parameterized weighted arithmetic and harmonic means we define a new family of matrix means l a r where and a vary over all positive probability vectors in rm and m tuples of positive definite matrices resp each of these means interpolates between the weighted harmonic mean and the arithmetic mean of the same weight with l l for each has a variational characterization as the unique minimizer of the weighted sum for the symmetrized parameterized kullback leibler divergence furthermore each can be realized as the common limit of the mean iteration by arithmetic and harmonic means in the unparameterized case or more generally the arithmetic and resolvent means other basic typical properties for a multivariable mean are derived

new generic quasi convergence principles with applications in this paper essentially strongly order preserving and conditionally set condensing semiflows are considered obtained is a new type of generic quasi convergence principles implying the existence of an open and dense set of stable quasi convergent points when the state space is order bounded the generic quasi convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations giving some results of theoretical and practical significance

computing the conjugate of convex piecewise linear quadratic bivariate functions we present a new algorithm to compute the legendre fenchel conjugate of convex piecewise linear quadratic plq bivariate functions the algorithm stores a function using a primal planar arrangement it then computes the dual arrangement associated with the conjugate by looping through vertices edges and faces in the primal arrangement and building associated dual vertices edges and faces using optimal computational geometry data structures the algorithm has a linear time worst case complexity we present the algorithm and illustrate it with numerical examples we proceed to build a toolbox for convex bivariate plq functions by implementing the addition and scalar multiplication operations finally we compose these operators to compute classical convex analysis operators such as the moreau envelope and the proximal average

periodic solutions of a class of degenerate parabolic system with delays this paper is concerned with a class of periodic degenerate parabolic system with time delays in a bounded domain under mixed boundary condition under locally lipschitz condition on reaction functions we apply schauder fixed point theorem to obtain the existence of periodic solutions of the periodic problem with quasi monotonicity in addition we also show that the periodic problem has a maximal and a minimal periodic solutions applications of the obtained results are also given to some nonlinear diffusion models arising from ecology

a system of coupled oscillators can have arbitrary prescribed attractors a system of coupled nonlinear oscillators is considered if the size n of this oscillator network is large enough then the model can have arbitrary prescribed attractors which can be defined by some finite dimensional flows one can effectively find a simple interaction between the oscillators which gives these prescribed attractors

multi valued characteristics and morse decompositions abstract a rapid growth of molecular and systems biology in recent years challenges mathematicians to develop robust modeling and analytical tools for this area we combine a theory of monotone input output systems with a classical theory of morse decompositions in the context of ordinary differential equations models of biochemical reactions we show that a multi valued input output characteristic can be used to define non trivial morse decompositions which provide information about a global structure of the attractor the previous work on input output characteristics is shown to apply locally to individual morse sets and is seamlessly incorporated into our global theory we apply our tools to a model of cell cycle maintenance we show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions

complicated dynamics in scalar semilinear parabolic equations in higher space dimension abstract we study the dynamics of the boundary value problem ut lu g x u u x 1 u 0 2 where l is a second order uniformly elliptic operator and r n is diffeomorphic to the ball in r n n 2 the main result asserts that given any ck vector field v on r n 1 with v 0 0 one can adjust coefficients of l and the function g such that the corresponding problem 1 2 has an n 1 dimensional invariant manifold through the equilibrium u 0 and the taylor expansion at u 0 of the vector field representing the flow on this manifold coincides in appropriate coordinates with the taylor expansion of v up to k th order terms this result implies that a hyperbolic invariant n torus can be found in 1 2 if l and g are appropriately chosen this result also indicates that chaotic dynamics is likely to occur for some choices of l and g

on the finite dimensional dynamical systems with limited competition the asymptotic behavior of dynamical systems with limited competition is investigated we study index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to a nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits applications are made to time periodic ordinary differential equations and reaction diffusion equations

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

on an abstract competition model and applications is a map such that the coordinates of f x x2 represent the population or population densities which have evolved from the state xi x2 at a fixed time later setting yi y2 f x x2 it is natural to suppose that an increase in x coupled with a decrease in x2 results in an increase in y and a decrease in y2 and symmetrically a decrease in x1 coupled with an increase in x2 results in a decrease in yr and an increase in yz we define an order in e x e2 as follows if x xi e e x2 xi e e x1 i xi and xi i x2 then we write x1 7 1 5 4 9 x

the global attractor of a competitor competitor mutualist reaction diffusion system with time delays abstract the aim of this paper is to investigate the asymptotic behavior of time dependent solutions of a three species reaction diffusion system in a bounded domain under a neumann boundary condition the system governs the population densities of a competitor a competitor mutualist and a mutualist and time delays may appear in the reaction mechanism it is shown under a very simple condition on the reaction rates that the reaction diffusion system has a unique constant positive steady state solution and for any nontrivial nonnegative initial function the corresponding time dependent solution converges to the positive steady state solution an immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable moreover the state state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion all the conclusions for the time delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays

attractors in strongly monotone flows where f is a c vector field in r and all the off diagonal terms of its jacobian matrix are nonnegative the well known kamke theorem implies that the flow 4 i ir0 generated by 1 has the property it preserves the partial ordering 0 another system which has this property is the semilinear parabolic equation on a compact n dimensional submanifold solutions of its dirichlet or neumann problem determine a monotone flow in appropriate function space see 11 or 51 in general a flow which preserves a partial ordering on the state space is called monotone m w hirsch has studied this flow in a series of papers 14 io he 2 31 has studied the limiting behavior and convergence of solutions of system 1 he 1 4 lo has investigated this flow in the considerably abstract state space in chapter iii of cl he has used two sections to discuss the properties of attractors for monotone flows and obtained some very important results he has shown that a monotone flow cannot have an attracting cycle see l corollary 2 41 and that every attractor must contain an equilibrium see cl theorem 3 11 in particular if the flow is strongly monotone he has shown that every attractor k contains an order stable equilibrium see 1 theorem 4 11 meanwhile if we suppose z is attracted to k but is not quasiconvergent then k contains two order stable equilibria p q such that p o z 4 q see cl theorem 4 31 the purpose of this paper is further to discuss the properties of attractors for strongly monotone flows we shall study when an o limit set can be an 210 0022 247x 91 3 00

towards understanding the capability of adaptation for time varying systems the information uncertainty coupled with the structural complexity is more pertinent than the rate of parameter changes in characterizing the capability of adaptation for time varying systems

dynamics of lotka volterra competition reaction diffusion systems with degenerate diffusion abstract this paper is dealt with a class of lotka volterra competition reaction diffusion system and a competitor competitor mutualist system with density dependent diffusion in a bounded domain under the three basic types of dirichlet neumann and robin boundary conditions the governing equations for the competition system consist of an arbitrary number of degenerate quasilinear parabolic equations while the competitor competitor mutualist system involves three degenerate equations the goal of the paper is to show the existence of positive steady state solutions or quasi solutions the existence and uniqueness of a classical global time dependent solution and the asymptotic behavior of the time dependent solution in relation to the steady state solutions or quasi solutions the above goals are achieved under a very simple condition on the reaction rates of the reaction functions and these results yield a global attractor and the coexistence of the competing species in the case of neumann boundary condition the system has a unique constant positive steady state solution which is a global attractor of all the competing species the above conclusions lead to some interesting distinct dynamic behavior between degenerate quasilinear reaction diffusion systems and the corresponding semilinear reaction diffusion systems in which some or all of the competing species may be in extinction

the kernel average for two convex functions and its application to the extension and representation of monotone operators we provide and analyze an average for two convex functions based on a kernel function it covers several known averages such as the arithmetic average epigraphical average and the proximal average when applied to the fitzpatrick function and the conjugate of fitzpatrick function associated with a monotone operator our average produces an autoconjugate also known as selfdual lagrangian which can be used for finding an explicit maximal monotone extension of the given monotone operator this completely settles one of the open problems posed by fitzpatrick in the setting of reflexive banach spaces

performance analysis of general tracking algorithms a general family of tracking algorithms for linear regression models is studied it includes the familiar least mean square gradient approach recursive least squares and kalman filter based estimators the exact expressions for the quality of the obtained estimates are complicated approximate and easy to use expressions for the covariance matrix of the parameter tracking error are developed these are applicable over the whole time interval including the transient and the approximation error can be explicitly calculated

projection and proximal point methods convergence results and counterexamples recently hundal has constructed a hyperplane h a cone k and a starting point y0 in 2 such that the sequence of alternating projections pk ph n y0 n n converges weakly to some point in h k but not in norm we show how this construction results in a counterexample to norm convergence for iterates of averaged projections hence we give an a rmative answer to a question raised by reich two decades ago furthermore new counterexamples to norm convergence for iterates of rmly nonexpansive maps a la genel and lindenstrauss and for the proximal point algorithm al a g uler are provided we also present a counterexample along with some weak and norm convergence results for the new framework of string averaging projection methods introduced by censor et al extensions to banach spaces and the situation for the hilbert ball are discussed as well 2003 elsevier ltd all rights reserved msc 47h09 47j25 90c25

the proximal average basic theory the recently introduced proximal average of two convex functions is a convex function with many useful properties in this paper we introduce and systematically study the proximal average for finitely many convex functions the basic properties of the proximal average with respect to the standard convex analytical notions domain fenchel conjugate subdifferential proximal mapping epi continuity and others are provided and illustrated by several examples

on adaptive stabilization of time varying stochastic systems the basic stability issue of time varying stochastic systems under adaptive control is studied a difficulty arising from treating the stochastic case as compared to the deterministic case is the lack of an a priori upper bound on the sample paths of the random noise sequence a projected gradient algorithm with small stepsize is used avoiding possible large deviations of the estimates it is shown that if the unknown parameters vary slowly in some sense then an adaptive control law can be designed so that the closed loop system is stable issues of performance and robustness are also discussed

rapid fluctuations of chaotic maps on rn the iterates f n of a chaotic map f display heightened oscillations or fluctuations as n i ff is a chaotic interval map in one dimension then it is now known that the total variation of f n on that interval grows exponentially with respect to n g chen t huang y huang chaotic behavior of interval maps and total variations of iterates internat j bifur chaos 14 2004 2161 2186 however the characterization of chaotic behavior of maps in multi dimensional spaces is generally much more challenging here we generalize the definition of bounded variations for vector valued maps in terms of the hausdorff measure and then use it to study what we call rapid fluctuations on fractal sets in multi dimensional chaotic discrete dynamical systems the relations among rapid fluctuations strict turbulence and positive entropy are established for lipschitz continuous systems on general n dimensional euclidean spaces applications to planar monotone or competitive systems and triangular systems on the square are also given

adaptive tracking games for coupled stochastic linear multi agent systems stability optimality and robustness distributed adaptive tracking type games are investigated for a class of coupled stochastic linear multi agent systems with uncertainties of unknown structure parameters external stochastic disturbances unmodeled dynamics and unknown agents interactions the control goal is to make the states of all the agents converge to a desired function of the population state average psa due to the fact that only local information is available for each agent the control is distributed for the time invariant parameter case the extended least squares algorithm nash certainty equivalence nce principle and certainty equivalence ce principle are used to estimate the unknown parameters and the psa term and to design adaptive control respectively under some mild conditions it is shown that the closed loop system is almost surely uniformly stable with respect to the population number n the estimate for the psa term is strongly consistent the adaptive control is almost surely an asymptotic nash equilibrium when the dynamics of each agent contains time varying parameters and unmodeled dynamics the projected least mean square lms algorithm nce principle and ce principle are adopted to estimate the unknown time varying parameters and the unknown psa term and to design robust adaptive control respectively in addition to stability of the closed loop system and consistency of the psa estimate the control law is shown to be robust nash equilibrium with respect to the unmodeled dynamics the variation of the unknown parameters and the external disturbances two numerical examples are given to illustrate the methods and results of this paper

primal dual symmetric intrinsic methods for finding antiderivatives of cyclically monotone operators a fundamental result due to rockafellar states that every cyclically monotone operator a admits an antiderivative f in the sense that the graph of a is contained in the graph of the subdifferential operator partial f given a method mathfrak m that assigns every finite cyclically monotone operator a some antiderivative mathfrak m a we say that the method is primal dual symmetric if mathfrak m applied to the inverse of a produces the fenchel conjugate of mathfrak m a rockafellar s antiderivatives do not possess this property utilizing fitzpatrick functions and the proximal average we present novel primal dual symmetric intrinsic methods the antiderivatives produced by these methods provide a solution to a problem posed by rockafellar in 2005 the results leading to this solution are illustrated by various examples

asymptotic tracking for constrained monotone systems this note addresses tracking control of nonlinear discrete time monotone systems subject to input and state constraints forcing saturation on a previously designed controller may in general lead to destabilization or at least result in constraint violation and performance losses hereby it is shown that for a certain class of nonlinear monotone systems it is possible to design a static nonlinear output feedback which saturated among suitable state dependent bounds is able to guarantee constraint satisfaction and asymptotic tracking of piecewise constant references with a moderate online computational burden simulation experiments concerning the synthesis of a protein demonstrate the effectiveness of the proposed control strategy