intrinsic noise in game dynamical learning demographic noise has profound effects on evolutionary and population dynamics as well as on chemical reaction systems and models of epidemiology such noise is intrinsic and due to the discreteness of the dynamics in finite populations we here show that similar noise sustained trajectories arise in game dynamical learning where the stochasticity has a different origin agents sample a finite number of moves of their opponents in between adaptation events the limit of infinite batches results in deterministic modified replicator equations whereas finite sampling leads to a stochastic dynamics the characteristics of these fluctuations can be computed analytically using methods from statistical physics and such noise can affect the attractors significantly leading to noise sustained cycling or removing periodic orbits of the standard replicator dynamics

deterministic evolutionary game dynamics in finite populations evolutionary game dynamics describes the spreading of successful strategies in a population of reproducing individuals typically the microscopic definition of strategy spreading is stochastic such that the dynamics becomes deterministic only in infinitely large populations here we present a microscopic birth death process that has a fully deterministic strong selection limit in well mixed populations of any size additionally under weak selection from this process the frequency dependent moran process is recovered this makes it a natural extension of the usual evolutionary dynamics under weak selection we find simple expressions for the fixation probabilities and average fixation times of the process in evolutionary games with two players and two strategies for cyclic games with two players and three strategies we show that the resulting deterministic dynamics crucially depends on the initial condition in a nontrivial way

fuzzy proximity ordered spaces in 1976 singal and lal extended the study of nearness between sets to the nachbin s 1965 topological and order structures also in 1978 dimitrijevic developed such an idea in the same manner with more study of its properties the aim of this paper is to introduce the idea of order in a fuzzy proximity space katsaras 1979 1980 1980 and construct new spaces in the area of fuzzy topological and order structures the so called fuzzy proximity ordered spaces also we study some properties of these spaces

stochastic turing patterns in the brusselator model a stochastic version of the brusselator model is proposed and studied via the system size expansion the mean field equations are derived and shown to yield to organized turing patterns within a specific parameters region when determining the turing condition for instability we pay particular attention to the role of cross diffusive terms often neglected in the heuristic derivation of reaction diffusion schemes stochastic fluctuations are shown to give rise to spatially ordered solutions sharing the same quantitative characteristic of the mean field based turing scenario in term of excited wavelengths interestingly the region of parameter yielding to the stochastic self organization is wider than that determined via the conventional turing approach suggesting that the condition for spatial order to appear can be less stringent than customarily believed

traveling wave solutions and the coupled korteweg de vries equation some coupled nonlinear equations are considered for studying traveling wave solutions by introducing a stream function it is shown that if one of the solutions is of the form v v x ct the other also must be of the form u u x ct in addition the possibility of including cubic nonlinear terms has been considered and such a system assuming that the solutions are of the traveling wave type has been solved

evolutionary dynamics intrinsic noise and cycles of cooperation we use analytical techniques based on an expansion in the inverse system size to study the stochastic evolutionary dynamics of finite populations of players interacting in a repeated prisoner s dilemma game we show that a mechanism of amplification of demographic noise can give rise to coherent oscillations in parameter regimes where deterministic descriptions converge to fixed points with complex eigenvalues these quasicycles between cooperation and defection have previously been observed in computer simulations here we provide a systematic and comprehensive analytical characterization of their properties we are able to predict their power spectra as a function of the mutation rate and other model parameters and to compare the relative magnitude of the cycles induced by different types of underlying microscopic dynamics we also extend our analysis to the iterated prisoner s dilemma game with a win stay lose shift strategy appropriate in situations where players are subject to errors of the trembling hand type

fuzzy proximities and totally bounded fuzzy uniformities after the classical papers of zadeh 181 and chang 3 many concepts of general topology have been extended to fuzzy set theory in this paper we are interested to develop the study of fuzzy uniformities fuzzy proximities and of the connexions between such structures fuzzy uniformities have been introduced by lowen in 9 and with slight modifications in lo and by hutton in 5 the two approaches are quite different however the one proposed by hutton seems to suit much better to fuzzy set theory also because it allows to generalize a lot of results in the most proper way therefore in this paper we shall deal with hutton fuzzy uniformities the concept of fuzzy proximity used up to now see e g 7 8 15 1 is quite unsatisfactory indeed its fuzzyness is rather poor since these fuzzy proximities are in a canonical l 1 correspondence with the usual proximities see remark 2 6 moreover the open sets of the induced topologies are crisp and though every lowen fuzzy uniformity induces a fuzzy proximity this correspondence cannot work well since the two structures do not give the same fuzzy topology for these reasons we propose a new definition of fuzzy proximity it differs from the old one only in axiom p5 in short we could say that we privilege complementation with respect to intersection an agreement between the two operations being problematic in fuzzy set theory we point out that as a consequence a fuzzy set may be far from itself however this fact ought not to amaze anyone since it may happen just for pale sets which in a certain sense resemble the empty set nevertheless this modification enables us to associate a topology in a completely different way moreover every fuzzy uniformity induces a fuzzy proximity and vice versa we succeed to construct a fuzzy uniformity starting with a given fuzzy proximity and the induced topologies do not change at any step this implies that the topologies which admit a fuzzy proximity are exactly the completely regular ones moreover we prove that there exists a l l correspondence between fuzzy proximity spaces and a

gradations of uniformity and gradations of proximity abstract in 1992 a new definition of a fuzzy topological space was given by introducing the concept of a gradation of openness in this paper a new definition of a fuzzy uniform resp proximity space is given by introducing the concept of a gradation of uniformity resp proximity on a non empty set x how to construct a gradation of openness induced by a gradation of uniformity resp proximity is explained the connexions between gradations of proximity and gradations of uniformity are investigated

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

pointwise uniformities in fuzzy set theory a theory of pointwise quasi uniformities is based on fuzzy sets many theorems corresponding to the usual theorems in general topology are generalized on fuzzy sets in particular each fuzzy topological space is pointwise quasi uniformizable a pointwise uniformity is characterized in terms of a kind of complete regularity a uniformity on the fuzzy unit interval is constructed

the category of pointwise s proximity spaces first it is proved that the category of pointwise quasi uniform spaces and pointwise quasi uniformly continuous morphisms is topological then the concept of pointwise s quasi proximity on lattice valued fuzzy set theory is introduced in such a way as to be compatible with pointwise quasi uniformity and pointwise p q metric each co topology can be induced by a pointwise s quasi proximity when a valued lattice is a completely distributive lattices equipped with an order reversing involution the concept of pointwise s proximity can be defined by means of pointwise s quasi proximity the relation between pointwise s quasi proximities and pointwise quasi uniformities is discussed it is proved that the category of pointwise s quasi proximity spaces is isomorphic to the category of totally bounded pointwise quasi uniform spaces the category of pointwise s proximity spaces is isomorphic to a subcategory of totally bounded pointwise uniform spaces hence they are topological

fluctuations and oscillations in a simple epidemic model we show that the simplest stochastic epidemiological models with spatial correlations exhibit two types of oscillatory behavior in the endemic phase in a large parameter range the oscillations are due to resonant amplification of stochastic fluctuations a general mechanism first reported for predator prey dynamics in a narrow range of parameters that includes many infectious diseases which confer long lasting immunity the oscillations persist for infinite populations this effect is apparent in simulations of the stochastic process in systems of variable size and can be understood from the phase diagram of the deterministic pair approximation equations the two mechanisms combined play a central role in explaining the ubiquity of oscillatory behavior in real data and in simulation results of epidemic and other related models

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

coevolutionary dynamics from finite to infinite populations traditionally frequency dependent evolutionary dynamics is described by deterministic replicator dynamics assuming implicitly infinite population sizes only recently have stochastic processes been introduced to study evolutionary dynamics in finite populations however the relationship between deterministic and stochastic approaches remained unclear here we solve this problem by explicitly considering large populations in particular we identify different microscopic stochastic processes that lead to the standard or the adjusted replicator dynamics moreover differences on the individual level can lead to qualitatively different dynamics in asymmetric conflicts and depending on the population size can even invert the direction of the evolutionary process

solutions of an equation that lead to solutions of a modified emden equation and a coupled korteweg devries equation in a recent paper moreira res rep if ufrj 83 25 universidade federal do rio de janeiro inst de fisica cidade univ ilha do fundao rio de janeiro brazil obtained a nonlinear second order differential equation that leads to the first integral of a modified emden equation he also obtained two particular solutions of his equation this paper completely integrates moreira s equation and uses it to get a class of solutions of a coupled korteweg devries kdv equation recently studied by guha ray bagchi and sinha j math phys 27 2558 1986

robust ecological pattern formation induced by demographic noise we demonstrate that demographic noise can induce persistent spatial pattern formation and temporal oscillations in the levin segel predator prey model for plankton herbivore population dynamics although the model exhibits a turing instability in mean field theory demographic noise greatly enlarges the region of parameter space where pattern formation occurs to distinguish between patterns generated by fluctuations and those present at the mean field level in real ecosystems we calculate the power spectrum in the noise driven case and predict the presence of fat tails not present in the mean field case these results may account for the prevalence of large scale ecological patterns beyond that expected from traditional nonstochastic approaches

fuzzy uniformities induced by fuzzy proximities abstract given a fuzzy proximity we construct a fuzzy uniformity in the sense of lowen which turns out to be the coarse stone among those which induce we show that some basic theorems of general topology about precompact uniformities have a nice extension to fuzzy set topology

l proximities and totally bounded pointwise l uniformities in this paper we shall develop the study of connections between l quasi proximities in the sense of katsaras liu artico moresco and pointwise l quasi uniformities in the sense of shi we shall prove that there exists a one to one correspondence between l quasi proximity spaces and totally bounded pointwise l quasi uniform spaces

non gaussian fluctuations arising from finite populations exact results for the evolutionary moran process the appropriate description of fluctuations within the framework of evolutionary game theory is a fundamental unsolved problem in the case of finite populations the moran process recently introduced into this context in nowak et al nature london 428 646 2004 defines a promising standard model of evolutionary game theory in finite populations for which analytical results are accessible in this paper we derive the stationary distribution of the moran process population dynamics for arbitrary 2 ifmmode times else texttimes fi 2 games for the finite size case we show that a nonvanishing background fitness can be transformed to the vanishing case by rescaling the payoff matrix in contrast to the common approach to mimic finite size fluctuations by gaussian distributed noise the finite size fluctuations can deviate significantly from a gaussian distribution

pointwise pseudo metrics in l fuzzy set theory a theory of pointwise pseudo metrics is based on l fuzzy sets a pointwise pseudo metric is characterized by its remote neighborhood maps many theorems in general topology are generalized on l fuzzy sets the l fuzzy real line and the l fuzzy unit interval are pointwise pseudo metrizable a pseudo metrization theorem of pointwise uniformity is obtained relations between erceg s p metric pseudometric and pointwise p metric are discussed

fuzzy quasi proximities and fuzzy quasi uniformities abstract some of the properties of the fuzzy quasi proximities and the fuzzy quasi uniformities are investigated it is shown that there exists a one to one correspondence between the family of all fuzzy quasi proximities on set x and the family of all totally bounded fuzzy quasi uniformities on x some of the properties of this correspondence are studied

extension of shi s quasi uniformities in a kubiak sostak sense abstract the aim of this paper is to study the extension of shi fu gui s quasi uniformities in a kubiak sostak sense the relationship between this extension of shi fu gui s quasi uniform molecular lattices shi s quml and the corresponding kubiak sostak extension of wang guo jun s topological molecular lattices wang s tml is discussed quml and tml denote the categories of shi s quml and wang s tml respectively cd is the category of completely distributive lattices with complete lattice morphisms as morphisms we prove that fquml the category of fuzzy quasi uniform molecular lattices the extension of shi s quml is a topological category over cd op and ftml the category of fuzzy topological molecular lattices the extension of wang s tml can be embedded in fquml we also prove that fquml is isomorphic to quml c m when m is a completely distributive lattice with multiplicative property where quml c m is the co tower extension of quml finally we study the kubiak sostak extension of hutton s quasi uniformities

on s quasi fuzzy proximity spaces in 1992 a new definition for a fuzzy topological space was given by introducing the concept of a gradation of openness chattopadhyay et al 1992 in 1996 we gave a new definition for a fuzzy proximity space by introducing the concept of a gradation of proximity ghanim et al 1996 in this paper we introduce the concept of a gradation of s quasi proximity we show that each gradation of s quasi proximity induces two gradations of openness some relations between a gradation of s quasi proximity and its induced gradations of openness are also investigated

fuzzy proximity spaces redefined abstract we use the concept l quasi coincident in fuzzy set theory to redefine l fuzzy proximity space and we study the following problems proximity of l fuzzy normal spaces l fuzzy filters proximity neighborhoods and the l fuzzy topology induced by the l fuzzy proximity

generalized quasi proximities this paper studies several kinds of fuzzy quasi proximities firstly we introduce generalized pointwise s quasi proximities and prove that s fpqps the category of generalized pointwise s quasi proximity spaces is isomorphic to tfrns the category of topological fuzzy remote neighborhood spaces secondly we introduce generalized s quasi proximities and prove that s fqps the category of generalized s quasi proximity spaces is isomorphic to s fpqps finally we study generalized quasi proximities and discuss the relationships between this kind of quasi proximities and generalized pointwise quasi uniformities

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve

uniformity and proximity of l fuzzy topological spaces abstract in 3 4 a k katsaras defined fuzzy proximity spaces and studied some of their properties in 5 we used the concept l quasi coincident in fuzzy set theory and fuzzy topology to redefine l fuzzy proximity space in such a way that it takes fuzzy proximity space as a special case also some pathologic properties were avoided for example proposition 2 1 in 4 in 5 we proved also some results analogous to those that hold for ordinary proximity spaces in this paper we continue the study of l fuzzy proximity spaces we shall prove that the partially ordered set x of all l fuzzy proximities on x is a complete lattice theorem 2 5 and we shall discuss the relationship between l fuzzy uniform spaces and l fuzzy proximity spaces theorem 3 4 4 1