design of digital pi dynamic controllers using projective controls extension of the projective controls approach to the design of discrete multivariable pi dynamic controllers for systems with external disturbances and controls feeding directly to the outputs is presented the procedure is applied to a ship boiler control problem to provide comparisons with the usual observer based approach

set valued functions in frechet spaces continuity hukuhara differentiability and applications to set differential equations abstract the set k c f of compact convex subsets of a frechet space f is studied in detail and is realized as a projective limit of metric spaces the notion of hausdorff metric on it is replaced by a family of corresponding semi metrics which provide the necessary background for the support of continuity and lipschitz continuity finally the notion of hukuhara derivative suited to our study is developed the proposed approach forms the appropriate environment within which the study of set differential equations for frechet spaces can be developed a first example on r is included

interconnection between set and fuzzy differential equations abstract the original formulation of fuzzy differential equations suffers from the disadvantage since the solutions increases as time increases in this paper employing the results of set differential equations obtained in v laksmikantham s leela a s vatsala setvalued hybrid differential equations and stability in terms of two measures j hybrid systems 2 2 2002 169 188 a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness

brief transfer equivalence and realization of nonlinear higher order input output difference equations two fundamental modelling problems in nonlinear discrete time control systems are studied using the language of differential forms the discrete time nonlinear single input single output systems to be studied are described by input output i o difference equations i e a high order difference equation relating the input the output and a finite number of their time shifts a new definition of equivalence is introduced which generalizes the notion of transfer equivalence well known for the linear case our definition is based upon the notion of an irreducible differential form of the system and was inspired by the analogous definition for continuous time systems the second problem to be addressed is the realization problem the i o difference equation is assumed to be in the irreducible form so that one can obtain an accessible and observable realization necessary and sufficient conditions are given for the existence of a local state space realization of the irreducible i o difference equation these conditions are formulated in terms of the integrability of certain subspaces of one forms classified according to their relative degree the sufficiency part of the proof gives a constructive procedure up to finding the integrating factors and integration of the set of one forms for obtaining a locally observable and accessible state space system if the system is not in the irreducible form one has first to apply the reduction procedure to transform the system into the irreducible form

zero one survival behavior of cyclically competing species the coexistence of competing species is due to unavoidable fluctuations always transient in this letter we investigate the ultimate survival probabilities characterizing different species in cyclic competition we show that they often obey a surprisingly simple though nontrivial behavior within a model where coexistence is neutrally stable we demonstrate a robust zero one law when the interactions between the three species are generically asymmetric the weakest species survives at a probability that tends to one for large population sizes while the other two are guaranteed to extinction we rationalize our findings from stochastic simulations by an analytic approach

survival behavior in the cyclic lotka volterra model with a randomly switching reaction rate we study the influence of a randomly switching reproduction predation rate on the survival behavior of the nonspatial cyclic lotka volterra model also known as the zero sum rock paper scissors game used to metaphorically describe the cyclic competition between three species in large and finite populations demographic fluctuations internal noise drive two species to extinction in a finite time while the species with the smallest reproduction predation rate is the most likely to be the surviving one law of the weakest here we model environmental external noise by assuming that the reproduction predation rate of the strongest species the fastest to reproduce and predate in a given static environment randomly switches between two values corresponding to more and less favorable external conditions we study the joint effect of environmental and demographic noise on the species survival probabilities and on the mean extinction time in particular we investigate whether the survival probabilities follow the law of the weakest and analyze their dependence on the external noise intensity and switching rate remarkably when on average there is a finite number of switches prior to extinction the survival probability of the predator of the species whose reaction rate switches typically varies nonmonotonically with the external noise intensity with optimal survival about a critical noise strength we also outline the relationship with the case where all reaction rates switch on markedly different time scales

saddles arrows and spirals deterministic trajectories in cyclic competition of four species population dynamics in systems composed of cyclically competing species has been of increasing interest recently here we investigate a system with four or more species using mean field theory we study in detail the trajectories in configuration space of the population fractions we discover a variety of orbits shaped like saddles spirals and straight lines many of their properties are found explicitly most remarkably we identify a collective variable that evolves simply as an exponential q e t where is a function of the reaction rates it provides information on the state of the system for late times as well as for t we discuss implications of these results for the evolution of a finite stochastic system a generalization to an arbitrary number of cyclically competing species yields valuable insights into universal properties of such systems

output feedback pole placement in the design of suboptimal linear quadratic regulators a design oriented methodology is presented for the construction of low order suboptimal output feedback compensators the design is suboptimal in the sense that it retains an l dimensional eigenspace from a reference optimal state feedback linear quadratic regulator problem if r is the number of outputs then for fixed l r it is known that the design requires a dynamic compensator of dimension l r the parameters of the compensator are determined by the solution of an associated output feedback pole placement problem using iterative dyadic pole placement a procedure is given which determines the solution of this problem without a priori assumptions on the dimension of the compensator the design is also extended to the class of stabilizable systems and the resulting compensator is shown to possess a separation property

coexistence versus extinction in the stochastic cyclic lotka volterra model cyclic dominance of species has been identified as a potential mechanism to maintain biodiversity see e g b kerr m a riley m w feldman and b j m bohannan nature 418 171 2002 and b kirkup and m a riley nature 428 412 2004 through analytical methods supported by numerical simulations we address this issue by studying the properties of a paradigmatic non spatial three species stochastic system namely the rock paper scissors or cyclic lotka volterra model while the deterministic approach rate equations predicts the coexistence of the species resulting in regular yet neutrally stable oscillations of the population densities we demonstrate that fluctuations arising in the system with a finite number of agents drastically alter this picture and are responsible for extinction after long enough time two of the three species die out as main findings we provide analytic estimates and numerical computation of the extinction probability at a given time we also discuss the implications of our results for a broad class of competing population systems

second type hukuhara differentiable solutions to the delay set valued differential equations abstract in this paper we consider the delay set valued differential equations with a recently introduced notion of a second type hukuhara derivative under condition that the right hand side of the equation is lipschitzian in the functional variable we obtain the existence and uniqueness of the solution to such the equations some properties of the solutions are established existence of at least one solution is also proved application of the second type hukuhara differentiability concept to the problems of set valued differential equations implies that the diameter of the solution values is a function nonincreasing in time

some results on sheaf solutions of sheaf set control problems abstract in this paper we study the existence and some properties of solutions of the so called set control differential equations scde and sheaf solutions of sheaf set control problems

theory of invasion extinction dynamics in minimal food webs when food webs are exposed to species invasion secondary extinction cascades may be set off although much work has gone into characterizing the structure of food webs systematic predictions on their evolutionary dynamics are still scarce here we present a theoretical framework that predicts extinctions in terms of an alternating sequence of two basic processes resource depletion by or competitive exclusion between consumers we first propose a conceptual invasion extinction model iem involving random fitness coefficients we bolster this iem by an analytical recursive procedure for calculating idealized extinction cascades after any species addition and simulate the long time evolution our procedure describes minimal food webs where each species interacts with only a single resource through the generalized lotka volterra equations for such food webs ex tinction cascades are determined uniquely and the system always relaxes to a stable steady state the dynamics and scale invariant species life time resemble the behavior of the iem and correctly predict an upper limit for trophic levels as observed in the field

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

pfaffian orientations and perfect matchings of scale free networks counting perfect matchings is of great interest in theoretical computer science mathematics among other disciplines however explicitly determining the number of perfect matchings of general graphs is a challenge in this paper we present a first analytical study of the enumeration problem of perfect matchings on scale free networks which are ubiquitous in real life systems we focus on two iteratively growing scale free networks one is fractal while the other is non fractal for each network we construct an orientation and prove that it is pfaffian then based on the connection between the number of perfect matchings and determinant of the skew adjacency matrix corresponding to a pfaffian orientation we derive the recursive expressions for the number of perfect matchings of the networks at two successive iterations furthermore using the recursive relations we show that for both scale free networks their entropy is zero which is in sharp contrast to the previously obtained results for other networks without scale free behavior but having the same average degree as the studied networks such as square lattice and honeycomb lattice whose entropies are larger than zero our results indicate from another angle that scale free networks have a distinct architecture in comparison with those lattices lacking the power law property

fuzzy differential equations without fuzzy convexity classical fuzzy differential equations defined in terms of the hukuhara derivative depend critically on the convexity of the level sets and result in expanding level sets here hullermeier s suggestion of defining fuzzy differential equations at each level set via differential inclusions is combined with ideas of aubin on morphological equations which allow nonlocal set evolution to remove the assumption of fuzzy convexity and thus to allow fuzzy differential equations to be defined for non convex level sets this approach uses reachable sets as a more general form of set integration and in contrast to the aumann set integral does not necessarily give rise to convex sets the results presented in this paper are even more general since they concern fuzzy sets that need to be only closed without additional assumptions of convexity compactness or even normality in particular an existence and uniqueness theorem is established under the assumption that the right hand sides satisfy a one sided lipschitz condition rather than a much stronger lipschitz condition fuzzy delay differential equations are also considered from this new perspective

coexistence and survival in conservative lotka volterra networks analyzing coexistence and survival scenarios of lotka volterra lv networks in which the total biomass is conserved is of vital importance for the characterization of long term dynamics of ecological communities here we introduce a classification scheme for coexistence scenarios in these conservative lv models and quantify the extinction process by employing the pfaffian of the network s interaction matrix we illustrate our findings on global stability properties for general systems of four and five species and find a generalized scaling law for the extinction time

random matrices and condensation into multiple states in the present work we employ methods from statistical mechanics of disordered systems to investigate static properties of condensation into multiple states in a general framework we aim at showing how typical properties of random interaction matrices play a vital role in manifesting the statistics of condensate states in particular an analytical expression for the fraction of condensate states in the thermodynamic limit is provided that confirms the result of the mean number of coexisting species in a random tournament game we also study the interplay between the condensation problem and zero sum games with correlated random payoff matrices

optimal strategies for node selection games on oriented graphs skew matrices and symmetric games abstract in the node selection game d each of the two players simultaneously selects a node from the oriented graph d if there is an arc between the selected nodes then there is a payoff from the dominated player to the dominating player we investigate the set of optimal strategies for the players in the node selection game d we point out that a classical theorem from game theory relates the dimension of the polytope of optimal strategies for d to the nullity of certain skew submatrix of the payoff matrix for d we show that if d is bipartite with at least two nodes in each partite set then an optimal strategy for the node selection game d is never unique our work also implies that if d is a tournament then there is a unique optimal strategy for each player a result obtained by fisher and ryan optimal strategies for a generalized scissors paper and stone game amer math monthly 99 1992 935 942 and independently by laffond laslier and le breton the bipartisan set of a tournament game games econom behav 5 1993 182 201

a golden point rule in rock paper scissors lizard spock game abstract we study a novel five species system on two dimensional lattices when each species have two superior and two inferior partners here we simplify the huge parameter space of predation probability to only two parameters both of monte carlo simulation and mean field theory reveal that two of strategies may die out when the ratio of the two parameters is close to the golden point 0 618 and the remaining three strategies are provided a cyclic dominance system

dimer statistics on the klein bottle abstract the problem of enumerating close packed dimers or perfect matchings on a quadratic lattice embedded on the klein bottle is considered thomassen c thomassen tilings of the torus and the klein bottle and vertex transitive graphs on a fixed surface trans amer math soc 323 2 1991 605 635 characterized that there are six quadrilateral lattices embedded on the klein bottle lu and wu w t lu and f y wu close packed dimers on nonorientable surfaces phys lett a 293 2002 235 246 had obtained a expression for the number of close packed dimers on one of them in this paper we investigate four other embeddings and obtain explicit expressions of the numbers of close packed dimers and free energy per dimer by enumerating pfaffians

evolutionary game theory theoretical concepts and applications to microbial communities ecological systems are complex assemblies of large numbers of individuals interacting competitively under multifaceted environmental conditions recent studies using microbial laboratory communities have revealed some of the self organization principles underneath the complexity of these systems a major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established it ensures the viability of microbial colonies by allowing for species diversity cooperative behavior and other kinds of social behavior a synthesis of evolutionary game theory nonlinear dynamics and the theory of stochastic processes provides the mathematical tools and a conceptual framework for a deeper understanding of these ecological systems we give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial systems intrinsic fluctuations stemming from the discreteness of individuals are ubiquitous and can have an important impact on the stability of ecosystems in the absence of speciation extinction of species is unavoidable it may however take very long times we provide a general concept for defining survival and extinction on ecological time scales spatial degrees of freedom come with a certain mobility of individuals when the latter is sufficiently high bacterial community structures can be understood through mapping individual based models in a continuum approach onto stochastic partial differential equations these allow progress using methods of nonlinear dynamics such as bifurcation analysis and invariant manifolds we conclude with a perspective on the current challenges in quantifying bacterial pattern formation and how this might have an impact on fundamental research in non equilibrium physics

nonlinear variation of parameters formula for set differential equations in a metric space nonlinear variation of parameters formula for set differential equations in a metric space is developed investigating necessary tools such as differentiability of solutions relative to initial values other needed results are also proven

dimer problem on the cylinder and torus institute of mathematics academia sinica nsfc 10771086 10671162 fujian province university nsc nsc96 2115 m 001 005

dimers on the 33 42 lattice in this work we obtain explicit expression of the number of close packed dimers perfect matchings of the 33 42 lattice with cylindrical boundary condition particularly we show that the entropy of 33 42 lattice is the same for cylindrical and toroidal boundary conditions

dimers on the kagome lattice i finite lattices abstract we report exact results on the enumeration of close packed dimers on a finite kagome lattice with general asymmetric dimer weights under periodic and cylindrical boundary conditions for symmetric dimer weights the resulting dimer generating functions reduce to very simple expressions and we show how the simple expressions can be obtained from the consideration of a spin variable mapping

topologically robust zero sum games and pfaffian orientation how network topology determines the long time dynamics of the antisymmetric lotka volterra equation to explore how the topology of interaction networks determines the robustness of dynamical systems we study the antisymmetric lotka volterra equation alve the alve is the replicator equation of zero sum games in evolutionary game theory in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time here we show that there also exist topologically robust zero sum games such as the rock paper scissors game for which all strategies coexist for all choices of interaction strengths we refer to such zero sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies by mapping the long time dynamics of the alve to the algebra of antisymmetric matrices we identify simple graph theoretical rules by which coexistence networks are constructed examples are triangulations of cycles characterized by the golden ratio phi 1 6180 cycles with complete subnetworks and non hamiltonian networks in graph theoretical terms we extend the concept of a pfaffian orientation from even sized to odd sized networks our results show that the topology of interaction networks alone can determine the long time behavior of nonlinear dynamical systems and may help to identify robust network motifs arising for example in ecology

stability criteria for set control differential equations abstract the existence and comparison results on solutions of set control differential equation were studied in n d phu t t tung some results on sheaf solutions of sheaf set control problems nonlinear analysis 67 2007 1309 1315 in this paper we present the stability criteria for solutions of set control differential equation

maximum matchings in scale free networks with identical degree distribution the size and number of maximum matchings in a network have found a large variety of applications in many fields as a ubiquitous property of diverse real systems power law degree distribution was shown to have a profound influence on size of maximum matchings in scale free networks where the size of maximum matchings is small and a perfect matching often does not exist in this paper we study analytically the maximum matchings in two scale free networks with identical degree sequence and show that the first network has no perfect matchings while the second one has many for the first network we determine explicitly the size of maximum matchings and provide an exact recursive solution for the number of maximum matchings for the second one we design an orientation and prove that it is pfaffian based on which we derive a closed form expression for the number of perfect matchings moreover we demonstrate that the entropy for perfect matchings is equal to that corresponding to the extended sierpiski graph with the same average degree as both studied scale free networks our results indicate that power law degree distribution alone is not sufficient to characterize the size and number of maximum matchings in scale free networks

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

interval cauchy problem with a second type hukuhara derivative in this paper we consider interval differential equations such the equations can be appropriate in modeling of dynamical systems under presence of uncertainty of parameters we study an interval initial value problem with a second type hukuhara derivative by an example of real world application we indicate the advantages of the usage of such a kind of interval valued derivative a continuous dependence of the solution on initial value and right hand side of the equation is shown the existence of approximate local solutions is proven and then it is used in the derivation of existence of at least one local solution to interval cauchy problem with second type hukuhara derivative the compactness of solutions set is also stated finally the explicit formulae for local solutions to linear interval differential equations are provided

synchronization and extinction in cyclic games with mixed strategies we consider cyclic lotka volterra models with three and four strategies where at every interaction agents play a strategy using a time dependent probability distribution agents learn from a loss by reducing the probability to play a losing strategy at the next interaction for that an agent is described as an urn containing balls of three and four types respectively where after a loss one of the balls corresponding to the losing strategy is replaced by a ball representing the winning strategy using both mean field rate equations and numerical simulations we investigate a range of quantities that allows us to characterize the properties of these cyclic models with time dependent probability distributions for the three strategy case in a spatial setting we observe a transition from neutrally stable to stable when changing the level of discretization of the probability distribution for large values of yielding a good approximation to a continuous distribution spatially synchronized temporal oscillations dominate the system for the four strategy game the system is always neutrally stable but different regimes emerge depending on the size of the system and the level of discretization

uncertain systems and fuzzy differential equations the study of fuzzy differential equations is initiated and some typical results are given in order to understand the intricacies involved in incorporating fuzziness in the terms of differential equations

design of coupled decentralized output regulators decentralized static output regulators are designed via projective controls in cases where the controllers are directly coupled i e when the controls of other decision makers appear in the local outputs for two decision makers the solution is reduced to the problem of solving a general non square algebraic riccati equation the existence and uniqueness of the solution is proved and then determined analytically these results are then generalized to an arbitrary number of decision makers finally the possibilities of the sequential design of static and dynamic regulators for this type of decentralized information structure are investigated a numerical example illustrates the design procedure

the design of optimal output regulators for linear multivariable systems with constant disturbances a method recently developed for solving the output regulator problem is extended to linear systems with constant disturbances the design of static and low order dynamic regulators for linear systems based on the use of projective controls is reviewed first the proposed method determines through a unified procedure the order and the parameters of the compensator together with the feedback parameters this methodology is then applied to systems with external disturbances and various control structures incorporating integral action are investigated in the light of the proposed methodology

extinction in neutrally stable stochastic lotka volterra models populations of competing biological species exhibit a fascinating interplay between the nonlinear dynamics of evolutionary selection forces and random fluctuations arising from the stochastic nature of the interactions the processes leading to extinction of species whose understanding is a key component in the study of evolution and biodiversity are influenced by both of these factors here we investigate a class of stochastic population dynamics models based on generalized lotka volterra systems in the case of neutral stability of the underlying deterministic model the impact of intrinsic noise on the survival of species is dramatic it destroys coexistence of interacting species on a time scale proportional to the population size we introduce a new method based on stochastic averaging which allows one to understand this extinction process quantitatively by reduction to a lower dimensional effective dynamics this is performed analytically for two highly symmetrical models and can be generalized numerically to more complex situations the extinction probability distributions and other quantities of interest we obtain show excellent agreement with simulations

stability of set differential equations and applications abstract the paper introduces some results on stability of sheaf solutions exponential stability of solutions and sheaf solutions of set differential equations and their applications to set control differential equations

existence and interrelation between set and fuzzy differential equations abstract an existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general then the connection between the solutions of fuzzy differential equation and the set differential equation that is generated from it is studied

a contribution to von neumann s theory of games ii abstract let a be the skew symmetric matrix attached to a symmetric two person zero sum game g then g is completely mixed i e both players must make some use of every strategy if and only if the principal pfaffians of a alternate in sign

exact solution of close packed dimers on the kagome lattice it is well known that exact enumerations of close packed dimers can be carried out for two dimensional lattices while details of results are now known for most lattices due to the unique nature of the lattice structure there has been no complete analysis for the kagom e lattice here we derive the close form expression 1 3 mathrm ln 4xyz for the free energy of close packed dimers on the kagom e lattice where x y and z are dimer weights we use two different approaches the kasteleyn method of evaluating a pfaffian and an alternative vertex model formulation both methods lead to the same final expression the correlation function between two dimers at a distance equal or greater than two lattice spacings is found to vanish identically

set differential equations versus fuzzy differential equations the study of fuzzy differential equations fdes forms a suitable setting for a mathematical modeling of real world problems in which uncertainties or vagueness pervades in recent years the theory of fdes has been investigated extensively in the original formulation as well as in an alternative framework which leads to ordinary multivalued differential inclusions it has recently been realized that initiating the study of set differential equations in a metric space has several advantages in addition to providing a natural setting for considering fdes in this paper we present some interesting results in this direction with the necessary background material