on institutions for modular coalgebraic specifications abstract we present an algebraic extension of standard coalgebraic specification techniques for state based systems which allows us to integrate constants and n ary operations in a smooth way and which leads to institutions enabling the use of modular specification techniques a sound and complete proof system for first order observational properties of modular specifications is given the framework of structures that we present can be considered as the result of a transformation of concepts of observational logic as in hennicker and bidoit in a haeberer ed algebraic methodology and software technology amast 98 lecture notes in computer science vol 1548 springer berlin 1999 into the coalgebraic world moreover it is shown that the features of structures that make them suitable models for an observational approach to specifications can be categorically expressed by the fact that the operation mapping an structure to its behaviour is a fibred idempotent monad

closure properties for the class of behavioral models hidden k logics can be considered as the underlying logics of program specification they constitute natural generalizations of k deductive systems and encompass deductive systems as well as hidden equational logics and inequational logics in our abstract algebraic approach the data structures are sorted algebras endowed with a designated subset of their visible parts called filter which represents a set of truth values we present a hierarchy of classes of hidden k logics the hidden k logics in each class are characterized by three different kinds of conditions namely properties of their leibniz operators closure properties of the class of their behavioral models and properties of their equivalence systems using equivalence systems we obtain a new and more complete analysis of the axiomatization of the behavioral models this is achieved by means of the leibniz operator and its combinatorial properties

resonant suppression of turing patterns by periodic illumination we study the resonant behavior of turing pattern suppression in a model of the chlorine dioxide iodine malonic acid reaction with periodic illumination the results of simulations based on integration of partial differential equations display resonance at the frequency of autonomous oscillations in the corresponding well stirred system the resonance in turing pattern suppression is sharper at lower complexing agent concentration and is affected by the waveform of the periodic driving force square wave on off periodic forcing is more effective in suppressing turing patterns than sinusoidal forcing we compare the dynamics of periodically forced turing patterns with the dynamics of periodically forced nonhomogeneous states in a system of two identical coupled cells bifurcation analysis based on numerical continuation of the latter system gives good predictions for the boundaries of the major resonance regions of the periodically forced patterns

behavioural theories and the proof of behavioural properties abstract behavioural theories are a generalization of first order theories where the equality predicate symbol is interpreted by a behavioural equality of objects and not by their identity in this paper we first consider arbitrary behavioural equalities determined by partial congruence relations and we show how to reduce the behavioural theory of any class of algebras to a subset of the standard theory of some corresponding class of algebras this reduction is the basis of a method for proving behavioural theorems whenever an axiomatization of the behavioural equality is provided then we focus on the important special case of partial observational equalities where two elements are observationally equal if they cannot be distinguished by observable computations over some set of input values we provide general conditions under which an obvious infinite axiomatization of the observational equality can be replaced by a finitary one and we provide methodological guidelines for finding such finitary axiomatizations as a consequence any proof system for first order logic can be used to prove the behavioural validity of first order formulas with respect to a given partial observational equality

structured algebraic specifications a kernel language abstract a language called asl for describing structured algebraic specifications is presented asl is a declarative higher order language it contains constructs for building possibly infinite signatures sets of terms and sets of formulas as well as constructs embodying primitive operations on algebraic specifications in particular asl includes a very general observability operation which can be used to behaviourally abstract from a specification the expressive power of these operations allows the choice of a simple notion of implementation which is transitive and monotonic syntax and two different denotational semantics a presentation semantics and a model class semantics are given the presentation semantics is used for showing the existence of a complete semiformal proof system for specifications whereas the model class semantics is fully abstract with respect to specification expressions both semantics are related by a homomorphism moreover computability questions are studied it is shown that any recursively enumerable signature and any class of algebras which is pseudo axiomatizable in a recursively enumerable way are definable in asl also every computable transformation of specifications can be expressed in asl

spatial synchronization of regular optical patterns we investigate an extended nonlinear optical experiment exhibiting the spontaneous formation of hexagonal patterns out of a stationary bifurcation the system is exposed to a two dimensional spatially periodic forcing namely static hexagonal patterns under variation of their spatial periodicity parameters are the strength of the forcing and the distance to pattern forming threshold the system response is quantitatively characterized with different methods we observe several locking regimes where the system is entrained by the forcing most of the locking regimes can be related to resonances between the different critical wave numbers and the forcing wave number or its spatial harmonics one particular locking appears to result from two of these simple resonances in a kind of generalized order m n synchronization the width of the locking regimes increases with forcing strength apparently representing a spatial analog of arnold tongues

bifurcation in a spherical reaction diffusion system with imposed gradient abstract bifurcation of steady states is studied for the system t d d h f g where d d are p x p matrices h is a self adjoint operator on a hilbert space h of functions on a d dimensional domain is a given function h p and are real parameters considering as a perturbation parameter we develop a general method for successive construction of primary secondary and further bifurcation points and branches motivated by a previously studied biological problem connected to cytokinesis cell division in early blastulas we examine the branching of solutions from a homogeneous state in the three sphere with h the laplacian operator and a simple gradient from one pole of the sphere to the other f and g are nonlinear functions describing autocatalytic biochemical reaction kinetics and in such systems spontaneous pattern formation may occur turing system of the second kind we examine the splitting of bifurcation points for parameters which yields j n nl y nm with n 2 in the unperturbed 0 problem previous studie s have shown a bipolar solution as the only one in this case i e m 0 but with imposed gradient we show here that although such a branch still emerges new branches arise with m 0 confirming previous numerical studies

observational logic constructor based logic and their duality observability and reachability are important concepts for formal software development while observability concepts are used to specify the required observable behavior of a program or system reachability concepts are used to describe the underlying data in terms of datatype constructors in this paper we first reconsider the observational logic institution which provides a logical framework for dealing with observability then we develop in a completely analogous way the constructor based logic institution which formalizes a novel treatment of reachability both institutions are tailored to capture the semantically correct realizations of a specification from either the observational or the reachability point of view we show that there is a methodological and even formal duality between both frameworks in particular we establish a correspondence between observer operations and datatype constructors observational and constructor based algebras fully abstract and reachable algebras and observational and inductive consequences of specifications the formal duality between the observability and reachability concepts is established in a category theoretic setting

universal coalgebra a theory of systems in the semantics of programming finite data types such as finite lists have traditionally been modelled by initial algebras later final coalgebras were used in order to deal with infinite data types coalgebras which are the dual of algebras turned out to be suited moreover as models for certain types of automata and more generally for transition and dynamical systems an important property of initial algebras is that they satisfy the familiar principle of induction such a principle was missing for coalgebras until the work of aczel 1988 on a theory of non wellfounded sets in which he introduced a proof principle nowadays called coinduction it was formulated in terms of bisimulation a notion originally stemming from the world of concurrent programming languages milner 1980 park 1981 using the notion of coalgebra homomorphism the definition of bisimulation on coalgebras can be shown to be formally dual to that of congruence on algebras aczel and mendler 1989 thus the three basic notions of universal algebra algebra homomorphism of algebras and congruence turn out to correspond to coalgebra homomorphism of coalgebras and bisimulation respectively in this paper the latter are taken as the basic ingredients of a theory called universal coalgebra some standard results from universal algebra are reformulated using the afore mentioned correspondence and proved for a large class of coalgebras leading to a series of results on e g the lattices of subcoalgebras and bisimulations simple coalgebras and coinduction and a covariety theorem for coalgebras similar to birkhoff s variety theorem

observational proofs by rewriting observability concepts contribute to a better understanding of software correctness in order to prove observational properties the concept of context induction has been developed by hennicker hennicker formal aspects of computing 3 4 1991 326 345 we propose in this paper to embed context induction in the implicit induction framework of bouhoula and rusinowitch journal of automated reasoning 14 2 1995 189 235 the proof system we obtain applies to conditional specifications it allows for many rewriting techniques and for the refutation of false observational conjectures under reasonable assumptions our method is refutationally complete i e it can refute any conjecture which is not observationally valid moreover this proof system is operational it has been implemented within the spike prover and interesting computer experiments are reported

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

three dimensional dissipative structures in reaction diffusion systems abstract we study analytically and numerically the tridimensional pattern selection problem for reaction diffusion systems qualitative agreement is found with the recent experimental results

symmetric asymmetric and antiphase turing patterns in a model system with two identical coupled layers we study turing pattern formation in a model reaction diffusion system with two coupled identical layers the coupling creates a pitchfork bifurcation which unfolds the symmetric steady state via primary turing instability into a pair of distinct unstable asymmetric steady states a ss the a ss gain stability at a reverse turing bifurcation the multiple stabilities created by the coupling generate a corresponding multiplicity of structures including symmetric asymmetric antiphase and localized turing patterns coexistence and competition of the different types of turing patterns are studied a one dimensional localized structure exhibits striking curvature effects

competition in ramped turing structures abstract stationary pattern selection and competition in the uniform brusselator in two 2d and three 3d dimensions are reviewed including reentrant hexagonal and striped zig zag phases influences of linear or chain like profiles of the pool chemicals on this selection are presented in the form of numerical experiments the relation with the recent experimental patterns obtained with the cima reaction is discussed

the random version of the kirzbraun valentine extension theorem abstract we obtain a measurable kirzbraun valentine extension theorem as applications we prove a random approximation theorem and a random fixed point theorem our results extend several earlier ones existing in the literature

remarks on the extension of random contractions for x1 x2 d where d is a nonempty subset of x the problem of the existence of an extension of any contraction mapping is an interesting one this problem was studied for euclidean spaces by kirzbraun 7 and extended to hilbert spaces by valentine 15 16 recently myjak and zygadlewicz 11 beg and shahzad 3 and shahzad 14 gave stochastic versions of the kirzbraun valentine extension theorem let be a measurable space with a algebra of subsets of denote by i and f the sets of in nite and nite sequences of positive integers respectively let g be a family of sets and k f g be a map for each r ri i 1 i and n n we shall denote r n by r1 rn then r i n 1 k r n is said to be obtained from g by the souslin family if every set obtained from g in this way is also in g then g is called a souslin family if is an outer measure on the is a souslin family in particular if is a complete measurable space then is a souslin family see 17 let 2x be the family of all subsets of x and c x the family of all nonempty closed subsets of x a multifunction f 2x is said to be measurable resp weakly measurable if for every closed resp open subset a of x the set f 1 a f a 6 note that if x is a metric space measurability implies weak

numerical studies of turing patterns selection in a two dimensional system abstract the turing bifurcations of the schnackenberg model are studied by numerical simulations the relative stability of the striped and hexagonal patterns are determined for a large range of control parameters we show that the stability of the triangular and honeycomb forms of hexagonal patterns are exchanged at a critical point striped patterns are favoured in the vicinity of this point we also show that one of the hexagonal phase is reentrant far from the bifurcation point finally we show that striped patterns develop preferentially in directions orthogonal to the walls when no flux boundary conditions are applied

from quasi 2d to 3d turing patterns in ramped systems abstract we elaborate on the transition from quasi two dimensional to three dimensional turing patterns in a chemical reaction diffusion system confined in gradients of chemicals between two feed boundaries this transition is observed in open spatial reactors specially designed to make possible the unfolding of a pattern sequence in one direction of the plane of observation in this direction the confinement of the structure is progressively relaxed complementary observations from two reactor geometries allow the dimensionality of the structure to be elucidated quasi two dimensional and three dimensional patterns respectively correspond to patterns developing in monolayers and in bilayers beyond the now classical hexagonal and stripe patterns various new stable planforms are shown to result from the coupling of these two classical pattern modes which develop in two adjacent layers with well defined phase relations between the two pattern modes

turing patterns in three dimensions we investigate three dimensional turing patterns in two component reaction diffusion systems the fitzhugh nagumo equation the brusselator and the gray scott model are solved numerically in three dimensions several interconnected structures of domains as well as lamellar hexagonal and spherical domains are obtained as stable motionless equilibrium patterns the relative stability of these structures is studied analytically based on the reduction approximation the relation with the microphase separated structures in block copolymers is also discussed

reaction diffusion patterns in confined chemical systems we review on the effects of the feed mode on pattern selection observed in chemical systems operated in open spatial reactors in two side fed reactors strong parameter ramps naturally confine patterns in a stratum the reactor thickness acts both as a genuine bifurcation parameter and on the pattern dimensionality depending on that thickness standard 2d hexagon and stripe turing patterns or more complex 3d planforms are observed in thin one side fed reactors patterning process must escape the imposed fixed boundary conditions either by devices introducing mixed boundary conditions or by an intrinsic phenomenon dubbed spatial bistability we show that in most cases for a comprehensive understanding of experimental observations the full 3d aspects have to be taken into account

pattern selection and localized structures in reaction diffusion systems abstract we present some theoretical concepts that have been used in the study of chemical disspative structures together with a brief description of recent experimental work on turing patterns and their interaction with travelling waves

forced patterns near a turing hopf bifurcation we study time periodic forcing of spatially extended patterns near a turing hopf bifurcation point a symmetry based normal form analysis yields several predictions including that i weak forcing near the intrinsic hopf frequency enhances or suppresses the turing amplitude by an amount that scales quadratically with the forcing strength and ii the strongest effect is seen for forcing that is detuned from the hopf frequency to apply our results to specific models we perform a perturbation analysis on general two component reaction diffusion systems which reveals whether the forcing suppresses or enhances the spatial pattern for the suppressing case our results are consistent with features of previous experiments on the chlorine dioxide iodine malonic acid chemical reaction however we also find examples of the enhancing case which has not yet been observed in experiment numerical simulations verify the predicted dependence on the forcing parameters

proof systems for structured specifications with observability operators abstract an asl like kernel language for structured specifications is presented which contains observability operators for determining the observable behaviour the observational quotient and as a derived operator the observational abstraction of a specification sound and complete infinitary proof systems for first order properties of specifications and for proving implementation relations between specifications are provided the relationships to approaches based on an observational satisfaction relation are discussed

coupling leads to chaos abstract the main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two brusselators linearly coupled by diffusion we will show that such a family contains a generic unfolding of a 4 dimensional nilpotent singularity of codimension 4 on the other hand we will prove that in any generic unfolding x of an n dimensional nilpotent singularity of codimension n there are bifurcation curves of n 1 dimensional nilpotent singularities of codimension n 1 which are in turn generically unfolded by x arguments conclude recalling that any generic unfolding of the 3 dimensional nilpotent singularity of codimension 3 exhibits strange attractors

on behavioural abstraction and behavioural satisfaction in higher order logic abstract the behavioural semantics of specifications with higher order logical formulae as axioms is analyzed a characterization of behavioural abstraction via behavioural satisfaction of formulae in which the equality symbol is interpreted as indistinguishability which is due to reichel and was recently generalized to the case of first order logic by bidoit et al is further generalized to this case the fact that higher order logic is powerful enough to express the indistinguishability relation is used to characterize behavioural satisfaction in terms of ordinary satisfaction and to develop new methods for reasoning about specifications under behavioural semantics

applications of the proximity map to random fixed point theorems in hilbert spaces a random approximation theorem for 1 set contractive random operators is proved as applications some random fixed point theorems are derived

influence of boundary conditions on the dynamics of oscillatory media abstract we study the pattern dynamics in a reaction diffusion model of the activator inhibitor type in the oscillatory regime we consider finite systems with partially absorptive boundary conditions analysing examples in different geometries in one and two dimensions we observe that the boundary conditions have important effects in the pattern forming properties of the systems in all the studied cases the arising complex behaviour is found to be dependent on the absorption parameter by changing this parameter we can control the asymptotic behaviour to be stationary periodic quasiperiodic or chaotic

morphological transitions and bistability in turing systems it is well known that in two dimensions turing systems produce spots stripes and labyrinthine patterns and in three dimensions lamellar and spherical structures or their combinations are observed in this paper we study transitions between these states in both two and three dimensions first we derive the regions of stability for different patterns using nonlinear bifurcation analysis then we apply large scale computer simulations to analyze the pattern selection in a bistable system by studying the effect of parameter selection on morphological clustering and the appearance of topological defects the method elaborated in this paper presents a probabilistic approach for studying pattern selection in a bistable reaction diffusion system

universal algebra in higher types abstract we develop the elementary theory of higher order universal algebra using the nonstandard approach to finite type theory introduced by henkin basic results include existence theorems for free and initial higher type algebras a complete higher type equational calculus and characterisation theorems for higher type equational and horn classes

on the random dugundji extension theorem soit 7b b un espace mesurable x un espace metrique a un sous ensemble clos non vide de x et y un espace norme separable soit f a y de caratheodory alors il existe une fonction de caratheodory f x y telle que 1 f restreinte a a est egale a f et 2 f x cof a pour chaque

travelling stripe forcing of turing patterns we report on the effect of a spatio temporal forcing applied on turing stripe patterns under spatial resonance experiments conducted with the photosensitive cdima reaction unveil a large variety of dynamical responses of striped patterns from entrained to waving when changing the forcing velocity from small to large values in between and quite unexpectedly we also observe a symmetry breaking phenomenon leading to hexagonal lattices either entrained with the forcing or oscillating we propose a new set of coupled amplitude equations 1 which provides a theoretical framework where the experimental phenomena can be interpreted

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