stone coalgebras we argue that the category of stone spaces forms an interesting base category for coalgebras in particular if one considers the vietoris functor as an analogue to the power set functor on the category of sets we prove that the so called descriptive general frames which play a fundamental role in the semantics of modal logics can be seen as stone coalgebras in a natural way this yields a duality between modal algebras and coalgebras for the vietoris functor building on this idea we introduce the notion of a vietoris polynomial functor over the category of stone spaces for each such functor t we provide an adjunction between t sorted boolean algebras with operators and the stone coalgebras for t we also identify the subcategory of algebras on which the adjunction restricts to an equivalence and show that the final t coalgebra is the dual of the initial t bao

global continuation of nonlinear waves in a ring of neurons ms received 14 may 2003 accepted 23 march 2005 in this paper we consider a ring of neurons with self feedback and delays as a resultof our approach based on global bifurcation theorems of delay dierential equationscoupled with representation theory of lie groups the coexistence of its asynchronousperiodic solutions i e mirror re ecting waves standing waves and discrete waves bifurcated simultaneously from the trivial solution at some critical values of thedelay will be established for delay not only near to but also far away from the criticalvalues therefore we can obtain wave solutions of large amplitudes in addition weconsider the coincidence of these periodic solutions

families of curve congruences on lorentzian surfaces and pencils of quadratic forms we dene and study in this paper families of conjugate and reected curve congruences associated to a self adjoint operator a on a smooth and oriented surface m endowed with a lorentzian metric g these families trace parts of the pencil joining the equations of the a asymptotic and the a principal curves and the pencil joining the a characteristic and the a principal curves respectively the binary dierential equations bdes of these curves can be viewed as points in the projective plane using the polar lines of various bdes with respect to the conic of degenerate quadratic forms we obtain geometric results on the above pencils and their relation with the metric g on the type of solutions of a given bde and of its a conjugate equation and on bdes with orthogonal roots

approximating markov processes through filtration in this paper we define a probabilistic version of filtration and use it to provide a finite approximation of markov processes in order to measure the approximation we employ probability logic to construct the final markov process and define a metric on the set of markov processes through this logic moreover we show that the set endowed with this metric is a polish space finally we point to some questions connecting approximation to uniformity and approximate bisimilarity as topics for future research

specifying coalgebras with modal logic abstract we propose to use modal logic as a logic for coalgebras and discuss it in view of the work done on coalgebras as a semantics of object oriented programming two approaches are taken first standard concepts of modal logic are applied to coalgebras for a certain kind of functor it is shown that the logic exactly captures the notion of bisimulation and a complete calculus is given examples of verifications of object properties are given second we discuss the relationship of this approach with the coalgebraic logic of moss coalgebraic logic ann pure appl logic 96 1999 277 317

delay induced periodicity in a neural netlet of excitation and inhibition abstract the dynamical behaviour of a two neuron netlet of excitation and inhibition with a transmission delay is investigated it is shown that in the absence of delay the netlet relaxes to the trivial resting state if the delay is of sufficient magnitude the network is excited to a temporally periodic cyclic behaviour the analytical mechanism for the onset of cyclic behaviour is through a hopf type bifurcation approximate solusions to the periodic output of the netlet is calculated stability of the temporally periodic cycle is investigated it is shown that the bifurcation is supercritical a related discrete version of the continuous time system is formulated it is found that the discrete system also displays a cyclic behaviour results of a number of computer simulations are displayed graphically the article concludes with a brief neurobiological discussion

bifurcations of implicit differential equations in this paper we give a local classi cation of the integral curves of implicit di erential equations f x y p 0 where f is a smooth function and p dy dx at points where fp 0 fpp 6 0 and where the discriminant f x y f fp 0g has a morse singularity we also produce models for generic bifurcations of such equations and apply the results to the di erential geometry of smooth surfaces this completes the local classi cation of generic one parameter families of binary di erential equations bdes

dupin indicatrices and families of curve congruences we study a number of natural families of binary differential equations bde s on a smooth surface m in r 3 one introduced by g j fletcher in 1996 interpolates between the asymptotic and principal bde s another between the characteristic and principal bde s the locus of singular points of the members of these families determine curves on the surface in these two cases they are the tangency points of the discriminant sets given by a fixed ratio of principle curvatures with the characteristic resp asymptotic bde more generally we consider a natural class of bde s on such a surface m and show how the pencil of bde s joining certain pairs are related to a third bde of the given class the so called polar bde this explains in particular why the principal asymptotic and characteristic bde s are intimately related

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

positive quadratic differential forms and foliations with singularities on surfaces to every positive cr quadratic differential form defined on an oriented two manifold is associated a pair of transversal one dimensional crfoliations with common singularities an open set of positive cr quadratic differential forms with structural stable associated foliations is characterized and it is proved that this set is dense in the space of positive c quadratic differential forms with c2 topology also a realization theorem is established

rank 1 codimension one singularities of positive quadratic differential forms abstract we complete the study of first order structural stability at singular points of positive quadratic differencial forms on two manifolds for this we consider the generic 1 parameter bifurcation of a d 23 singular point this situation consists in having before the bifurcation two locally stable singular points one of type d 2 and the other of type d 3 which collapse at the d 23 singular point when the bifurcation parameter is reached and afterwards disappear in local x y coordinates such a point appears at the origin of a planar differential equation of the form a x y dy 2 2 b x y dx dy c x y dx 2 with b 2 ac x y 0 such that 1 the first jet of the map a b c at the origin is t 1 a b c 0 0 y 0 y and 2 2 b x 2 0

a hierarchy of probabilistic system types we arrange various classes of probabilistic systems studied in the literature in an expressiveness hierarchy our expressiveness criterion is the existence of a system translation from the less expressive type into the more expressive type that preserves and reflects probabilistic bisimilarity we model the different system types as coalgebras of suitable behaviour functors and argue that the corresponding coalgebraic bisimilarity coincides with probabilistic bisimilarity for the classes for which the latter notion has been proposed in the literature the theory of coalgebras provides a unified framework for the presentation of the different classes and the system translations we needed to establish the hierarchy all these translations arise in a standard way from natural transformations between the two behaviour functors involved such a translation generally preserves coalgebraic bisimilarity we exploit a new result that under mild assumptions on the behaviour functors a system translation induced by a natural transformation with injective components also reflects bisimilarity

games with incomplete information played by bayesian players i iii this article originally appeared in management science november 1967 volume 14 number 3 pp 159 182 published by the institute of management sciences the paper develops a new theory for the analysis of games with incomplete information where the players are uncertain about some important parameters of the game situation such as the payoff functions the strategies available to various players the information other players have about the game etc however each player has a subjective probability distribution over the alternative possibilities in most of the paper it is assumed that these probability distributions entertained by the different players are mutually consistent in the sense that they can be regarded as conditional probability distributions derived from a certain basic probability distribution over the parameters unknown to the various players but later the theory is extended also to cases where the different players subjective probability distributions fail to satisfy this consistency assumption in cases where the consistency assumption holds the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution and the outcome of this lottery will decide which particular subgame will be played i e what the actual values of the relevant parameters will be in the game yet each player will receive only partial information about the outcome of the lottery and about the values of these parameters however every player will know the basic probability distribution governing the lottery thus technically the resulting game will be a game with complete information it is called the bayes equivalent of the original game part i of the paper describes the basic model and discusses various intuitive interpretations for the latter part ii shows that the nash equilibrium points of the bayes equivalent game yield bayesian equilibrium points for the original game finally part iii considers the main properties of the basic probability distribution

coalgebraic semantics of modal logics coalgebras can be seen as a natural abstraction of kripke frames in the same sense coalgebraic logics are generalised modal logics in this paper we give an overview of the basic tools techniques and results that connect coalgebras and modal logic we argue that coalgebras unify the semantics of a large range of different modal logics such as probabilistic graded relational conditional and discuss unifying approaches to reasoning at this level of generality we review languages defined in terms of the so called cover modality languages induced by predicate liftings as well as their common categorical abstraction and present abstract results on completeness expressiveness and complexity in these settings both for basic languages as well as a number of extensions such as hybrid languages and fixpoints

frustration stability and delay induced oscillations in a neural network model the effect of time delays on the linear stability of equilibria in an artificial neural network of hopfield type is analyzed the possibility of delay induced oscillations occurring is characterized in terms of properties of the not necessarily symmetric connection matrix of the network such oscillations are possible exactly when the network is frustrated equivalently when the signed digraph of the matrix does not require the perron property nonlinear analysis centre manifold computation of a three unit frustrated network is presented giving the nature of the bifurcations taking place a supercritical hopf bifurcation is shown to occur and a codimension two bifurcation is unfolded

final coalgebras for functors on measurable spaces we prove that every functor on the category meas of measurable spaces built from the identity and constant functors using products coproducts and the probability measure functor d has a final coalgebra our work builds on the construction of the universal harsanyi type spaces by heifetz and samet and papers by rosziger and jacobs on coalgebraic modal logic we construct logical languages probabilistic logics of transition systems and interpret them on coalgebras the final coalgebra is carried by the set of descriptions of all points in all coalgebras for the category set we work with the functor d of discrete probability measures we prove that every functor on set built from d and the expected functors has a final coalgebra the work for set differs from the work for meas negation in needed for final coalgebras on set but not for meas

hopf bifurcating periodic orbits in a ring of neurons with delays in this paper we consider a ring of neurons with self feedback and delays the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation based on the normal form approach and the center manifold theory we derive the formula for determining the properties of hopf bifurcating slowly oscillating periodic orbits for a ring of neurons with delays including the direction of hopf bifurcation stability of the hopf bifurcating slowly oscillating periodic orbits and so on moreover by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups we not only investigate the effect of synaptic delay of signal transmission on the pattern formation but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio temporal patterns

from modal logic to terminal coalgebras we define a modal logic whose models are coalgebras of a polynomial functor bisimilarity turns out to be the same as logical equivalence ideas and concepts of modal logic are directly applied to the theory of coalgebras we give an axiomatization and define canonical coalgebras that leads to a completeness result each canonical coalgebra proves to be terminal in a certain class of coalgebras the approach also yields a functional characterization of the terminal coalgebra of all coalgebras with respect to a given polynomial functor

stability analysis of a delayed hopfield neural network in this paper we study a class of neural networks which includes bidirectional associative memory networks and cellular neural networks as its special cases by brouwer s fixed point theorem a continuation theorem based on gains and mawhin s coincidence degree matrix theory and inequality analysis we not only obtain some different sufficient conditions ensuring the existence uniqueness and global exponential stability of the equilibrium but also estimate the exponentially convergent rate our results are less restrictive than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity