the qr decomposition and the singular value decomposition in the symmetrized max plus algebra revisited this paper is an updated and extended version of the paper the qr decomposition and the singular value decomposition in the symmetrized max plus algebra b de schutter and b de moor siam j matrix anal appl 19 1998 pp 378 406 the max plus algebra which has maximization and addition as its basic operations can be used to describe and analyze certain classes of discrete event systems such as flexible manufacturing systems railway networks and parallel processor systems in contrast to conventional algebra and conventional linear system theory the max plus algebra and the max plus algebraic system theory for discrete event systems are at present far from fully developed and many fundamental problems still have to be solved currently much research is going on to deal with these problems and to further extend the max plus algebra and to develop a complete max plus algebraic system theory for discrete event systems in this paper we address one of the remaining gaps in the max plus algebra by considering matrix decompositions in the symmetrized max plus algebra the symmetrized max plus algebra is an extension of the max plus algebra obtained by introducing a max plus algebraic analogue of the operator we show that we can use well known linear algebra algorithms to prove the existence of max plus algebraic analogues of basic matrix decompositions from linear algebra such as the qr decomposition the singular value decomposition the hessenberg decomposition the lu decomposition and so on these max plus algebraic matrix decompositions could play an important role in the max plus algebraic system theory for discrete event systems

cyclical interactions with alliance specific heterogeneous invasion rates we study a six species lotka volterra type system on different two dimensional lattices when each species has two superior and two inferior partners the invasion rates from predator sites to a randomly chosen neighboring prey site depend on the predator prey pair whereby cyclic symmetries within the two three species defensive alliances are conserved monte carlo simulations reveal an unexpected nonmonotonous dependence of alliance survival on the difference of alliance specific invasion rates this behavior is qualitatively reproduced by a four point mean field approximation the study addresses fundamental problems of stability for the competition of two defensive alliances and thus has important implications in natural and social sciences

a maximum principle for evolution hamilton jacobi equations on riemannian manifolds abstract we establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form u t f t d x u 0 u 0 x u 0 x where u 0 m r is a bounded uniformly continuous function m is a riemannian manifold and f 0 t m r this yields uniqueness of the viscosity solutions of such hamilton jacobi equations

von neummann s and related scaling laws in rock paper scissors type games we introduce a family of rock paper scissors type models with z n symmetry n is the number of species and we show that it has a very rich structure with many completely different phases we study realizations that lead to the formation of domains where individuals of one or more species coexist separated by interfaces whose average dynamics is curvature driven this type of behavior which might be relevant for the development of biological complexity leads to an interface network evolution and pattern formation similar to the ones of several other nonlinear systems in condensed matter and cosmology

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

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

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

evolutionary prisoner s dilemma game on newman watts networks maintenance of cooperation was studied for a two strategy evolutionary prisoner s dilemma game where the players are located on a one dimensional chain and their payoff comes from games with the nearest and next nearest neighbor interactions the applied host geometry makes it possible to study the impacts of two conflicting topological features the evolutionary rule involves some noise affecting the strategy adoptions between the interacting players using monte carlo simulations and the extended versions of dynamical mean field theory we determined the phase diagram as a function of noise level and a payoff parameter the peculiar feature of the diagram is changed significantly when the connectivity structure is extended by extra links as suggested by newman and watts

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

hamilton jacobi equations in infinite dimensions i uniqueness of viscosity solutions soient un espace de banach v son dual v et des solutions d une equation h x u du 0 dans etablies dans un sous ensemble de v h v r v r est continue on demontre que des solutions classiques sont determinees de facon unique par leurs valeurs limites on montre que l extension de la notion de solutions viscosite aux espaces de banach reussit dans des espaces munis de la propriete de radon nikodym

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

polynomial convolutions in max plus algebra abstract recently in a work that grew out of their exploration of interlacing polynomials marcus spielman and srivastava 21 and marcus 20 studied certain combinatorial polynomial convolutions these convolutions preserve real rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices thus providing a link to free probability we explore analogues of these types of convolutions in the setting of max plus algebra in this setting the max permanent replaces the determinant the maximum is the analogue of the expected value and real rootedness is replaced by full canonical form our results resemble those of marcus et al however in contrast to the classical setting we obtain an exact and simple description of all roots of the convolution of p x and q x in terms of the roots of p x and q x

coexistence in a predator prey system we propose a lattice model of two populations predators and prey the model is solved via monte carlo simulations each species moves randomly on the lattice and can live only a certain time without eating the lattice cells are either grass eaten by prey or tree giving cover for prey each animal has a reserve of food that is increased by eating prey or grass and decreased after each monte carlo step to breed a pair of animals must be adjacent and have a certain minimum of food supply the number of offspring produced depends on the number of available empty sites we show that such a predator prey system may finally reach one of the following three steady states coexisting with predators and prey pure prey or an empty one in which both populations become extinct we demonstrate that the probability of arriving at one of the above states depends on the initial densities of the prey and predator populations the amount of cover and the way it is spatially distributed

spirals and coarsening patterns in the competition of many species a complex ginzburg landau approach in order to model real ecological systems one has to consider many species that interact in complex ways however most of the recent theoretical studies have been restricted to few species systems with rather trivial interactions the few studies dealing with larger number of species and or more complex interaction schemes are mostly restricted to numerical explorations in this paper we determine starting from the deterministic mean field rate equations for large classes of systems the space of coexistence fixed points at which biodiversity is maximal for systems with a single coexistence fixed point we derive complex ginzburg landau equations that allow to describe space time pattern realized in two space dimensions for selected cases we compare the theoretical predictions with the pattern observed in numerical simulations

hamilton jacobi equations in infinite dimensions iii on etudie des equations de la forme f x u du 0 dans ou est un sous ensemble ouvert d un espace de banach v la fonction inconnue u r est continue du x est la derivee de frechet de u en x et f est une fonction non lineaire qui est une application continue de r v dans r on introduit la notion de solution viscosite stricte afin d eliminer l hypothese que v a la propriete de radon nikodym

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

three and four state rock paper scissors games with diffusion cyclic dominance of three species is a commonly occurring interaction dynamics often denoted the rock paper scissors rps game such a type of interactions is known to promote species coexistence here we generalize recent results of reichenbach et al nature london 448 1046 2007 of a four state variant of the rps game we show that spiral formation takes place only without a conservation law for the total density nevertheless in general fast diffusion can destroy species coexistence we also generalize the four state model to slightly varying reaction rates this is shown both analytically and numerically not to change pattern formation or the effective wavelength of the spirals and therefore not to alter the qualitative properties of the crossover to extinction

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

junctions and spiral patterns in generalized rock paper scissors models we investigate the population dynamics in generalized rock paper scissors models with an arbitrary number of species n we show that spiral patterns with n arms may develop both for odd and even n in particular in models where a bidirectional predation interaction of equal strength between all species is modified to include one n cyclic predator prey rule while the former case gives rise to an interface network with y type junctions obeying the scaling law l ensuremath propto t 1 2 where l is the characteristic length of the network and t is the time the latter can lead to a population network with n armed spiral patterns having a roughly constant characteristic length scale we explicitly demonstrate the connection between interface junctions and spiral patterns in these models and compute the corresponding scaling laws this work significantly extends the results of previous studies of population dynamics and could have profound implications for the understanding of biological complexity in systems with a large number of species

a walk on max plus algebra abstract max plus algebra is a kind of idempotent semiring over r max r with two operations max and in this paper we introduce a new model of a walk on one dimensional lattice on z as an analogue of the quantum walk over the max plus algebra and we call it max plus walk in the conventional quantum walk the summation of the l 2 norm of the states over all the positions is a conserved quantity in contrast the summation of eigenvalues of state decision matrices is a conserved quantity in the max plus walk moreover spectral analysis on the total time evolution operator is also given

fractal properties of the lattice lotka volterra model the lattice lotka volterra llv model is studied using mean field analysis and monte carlo simulations while the mean field phase portrait consists of a center surrounded by an infinity of closed trajectories when the process is restricted to a two dimensional 2d square lattice local inhomogeneities fluctuations appear spontaneous local clustering is observed on lattice and homogeneous initial distributions turn into clustered structures reactions take place only at the interfaces between different species and the borders adopt locally fractal structure intercluster surface reactions are responsible for the formation of local fluctuations of the species concentrations the box counting fractal dimension of the llv dynamics on a 2d support is found to depend on the reaction constants while the upper bound of fractality determines the size of the local oscillators lacunarity analysis is used to determine the degree of clustering of homologous species besides the spontaneous clustering that takes place on a regular 2d lattice the effects of fractal supports on the dynamics of the llv are studied for supports of dimensionality d s 2 the lattice can for certain domains of the reaction constants adopt a poisoned state where only one of the species survives by appropriately selecting the fractal dimension of the substrate it is possible to direct the system into a poisoned or oscillatory steady state at will

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

max plus singular values in this paper we prove a new characterization of the max plus singular values of a max plus matrix as the max plus eigenvalues of an associated max plus matrix pencil this new characterization allows us to compute max plus singu lar values quickly and accurately as well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially pa rameterized we show experimentally that max plus singular values give order of magnitude approximations to the classical singular values of parameter inde pendent classical matrices we also discuss hungarian scaling which is a diagonal scaling strategy for preprocessing classical linear systems we show that hungarian scaling can dramatically reduce the 2 norm condition number and that this action can be explained using our new theory for max plus singular values keywords tropical algebra max plus algebra singular values diagonal scaling condition number optimal assignment problem we also discuss hungarian scaling which is a diagonal scaling strategy for preprocessing classical linear systems we show that hungarian scaling can dramatically reduce the d norm condition number and that this action can be explained using our new theory for max plus singular values