an algebraic proof of rabin s tree theorem we continue the development of a theory of recognisability for infinite trees by introducing the equivalent of a wilke algebra as an application we give a new proof of the decidability of the monadic second order theory of the infinite binary tree a proof that does not use automata or games

on labeled birooted tree languages with an aim to developing expressive language theoretical tools applicable to inverse semigroup languages that is subsets of inverse semigroups this paper explores the language theory of finite labeled birooted trees munn s birooted trees extended with vertex labeling to this purpose we define a notion of finite state birooted tree automata that simply extends finite state word automata semantics this notion is shown to capture the class of languages that are definable in monadic second order logic and upward closed with respect to the natural order then we derive from these automata the notion of quasi recognizable languages that is languages recognizable by means of adequate premorphisms into finite adequately ordered monoids this notion is shown to capture finite boolean combinations of languages as above a class that contains classical regular languages of finite mono rooted trees this contrasts with the known collapse of classical algebraic tools when applied to inverse semigroups

positive varieties of tree languages pin s variety theorem for positive varieties of string languages and varieties of finite ordered semigroups is proved for trees i e a bijective correspondence between positive varieties of tree languages and varieties of finite ordered algebras is established this in turn is extended to generalized varieties of finite ordered algebras which corresponds to steinby s generalized variety theorem also families of tree languages and classes of ordered algebras that are definable by ordered syntactic or translation monoids are characterized

crossed modules for lie rinehart algebras abstract we introduce the notion of crossed module for lie rinehart algebras and prove that they are classified by the third cohomology of lie rinehart algebras developed in j huebschmann j reine angew math 408 1990 57 113 g s rinehart trans amer math soc 108 1963 195 222

lichnerowicz jacobi cohomology in this paper we extend the notion of lichnerowicz poisson cohomology to jacobi manifolds we study the relation of the so called lichnerowicz jacobi cohomology with the basic de rham cohomology and the cohomology of the lie algebra of functions relative to the representation defined by the hamiltonian vector fields a natural pairing with the canonical homology is constructed the relation between the lichnerowicz poisson cohomology of a quantizable poisson manifold and the lichnerowicz jacobi cohomology of the total space of a prequantization bundle is obtained particular cases of cosymplectic contact and locally conformal symplectic manifolds are discussed finally the lichnerowicz jacobi cohomology of a non transitive example is studied

phase transitions in coupled map lattices we clarify some aspects related to critical phenomena in 1d coupled map lattices in particular we explain why a model studied by chate and manneville did not seem to be in the same universality class as directed percolation contrary to first expectations we claim that this model is actually in this universality class but that scaling is extremely late we present a slightly modified model which seems to be in this universality class with respect to some properties but not all in contrast to directed percolation and to the model studied by chate and manneville our model shows a very complex sequence of singularities vaguely reminiscent of the richness of bifurcations in 1d maps

phase transitions in 2d linearly stable coupled map lattices abstract interface dynamics separating homogeneous phases is shown to be the main mechanism underlying irregular evolution in 2d linearly stable coupled map lattices in a fully deterministic model belonging to this class we find evidence of at least two different regimes that we call weak and strong turbulence the transition between the two regimes is carefully investigated revealing a direct connection with the destabilization of the interfaces separating homogeneous phases the critical behaviour is analysed and compared with that of stochastic models like directed percolation

cohomology of restricted lie rinehart algebras and the brauer group abstract we give an interpretation of the brauer group of a purely inseparable extension of exponent 1 in terms of restricted lie rinehart cohomology in particular we define and study the category p lr a of restricted lie rinehart algebras over a commutative algebra a we define cotriple cohomology groups h p l r l m for l p lr a and m a beck l module we classify restricted lie rinehart extensions thus we obtain a classification theorem for regular extensions considered by hochschild

spatio temporal intermittency in coupled map lattices abstract the transition to turbulence in a one dimensional array of maps coupled by diffusion is shown to display critical properties resembling those of directed percolation the analogy is supported by the reconstruction of a probabilistic cellular automation with closely similar statistical properties numerical results suggest however that spatio temporal intermittency does not belong to the same universality class as directed percolation

characterizing ctl like logics on finite trees we associate a modal operator with each language belonging to a given class of regular tree languages and use the cascade product of tree automata to give an algebraic characterization of the expressive power of the resulting logic

spatiotemporal chaos in a 2d coupled map lattice with pinning like force abstract spatiotemporal chaos stc has been observed in a coupled map lattice cml which describes magnetic vortex pinning in superconductors for the single site map a transition to chaos is realized by period doubling with feigenbaum scaling in the developed chaos region intermittent laminar and turbulent behaviors of the map take place in this region the natural invariant density and lyapunov exponent are deduced in analytical form analytical expressions of a correlation function and a distribution function of laminar phase durations are represented as well for the cml a state density and correlation functions characterizing stc are calculated

lie rinehart cohomology and integrable connections on modules of rank one abstract let k be an algebraically closed field of characteristic 0 let r be a commutative k algebra and let m be a torsion free r module of rank one with a connection we consider the lie rinehart cohomology with values in end r m with its induced connection and give an interpretation of this cohomology in terms of the integrable connections on m when r is an isolated singularity of dimension d 2 we relate the lie rinehart cohomology to the topological cohomology of the link of the singularity and when r is a quasi homogenous hypersurface of dimension two we give a complete computation of the cohomology

conversion of local transient chaos into global laminar states in coupled map lattices with long range interactions abstract spatially extended dynamical systems may exhibit intermittent behavior in both spatial and temporal scales characterized by repeated conversions from spatially localized transient chaos into global laminar patterns a simple model yet retaining some features of more complex systems consists of a lattice of a class of tent maps with an escaping region the coupling prescription we adopt in this work considers the interaction of a site with all its neighbors the corresponding strength decaying with the lattice distance as a power law this makes possible to pass continuously from a local nearest neighbor to a global kind of coupling we investigate statistical properties of both the chaotic transient bursts and the periodic laminar states with respect to the coupling parameters

ising type and other transitions in one dimensional coupled map lattices with sign symmetry we consider a one dimensional lattice of expanding antisymmetric maps 1 1 1 1 with nearest neighbor diffusive coupling for such systems it is known that if the coupling parameter e is small there is unique stationary in time state which is chaotic in space time a disputed question is whether such systems can exhibit ising type phase transitions as e grows beyond some critical value ec we present results from computer experiments which give definite indication that such a transition takes place the mean square magnetization appears to diverge as e approaches some critical value with a critical exponent around 0 9 we also study other properties of the coupled map system

universality classes of spatiotemporal intermittency abstract we discuss the spatiotemporal intermittency sti seen in the coupled sine circle map lattice the phase diagram of this system when updated with random initial conditions shows very rich behaviour including synchronised solutions and sti of various kinds these behaviours are organised around the bifurcation boundary of the synchronised solutions as well as an infection line which separates the lower part of the phase diagram into a spreading and a non spreading regime the sti seen at the bifurcation boundary in the spreading regime belongs convincingly to the directed percolation dp universality class in the non spreading regime spatial intermittency si with temporally regular bursts is seen at the bifurcation boundary the laminar length distribution scales as a power law with an exponent which is quite distinct from dp behaviour therefore both dp and non dp universality classes are seen in this system when the coupled map lattice is mapped to a cellular automaton via coarse graining a transition from a probabilistic cellular automaton to a deterministic cellular automaton at the infection line signals the transition from spreading to non spreading behaviour

lie algebroid cohomology and lie algebroid extensions abstract we consider the extension problem for lie algebroids over schemes over a field given a locally free lie algebroid q over a scheme x and a coherent sheaf of lie o x algebras l we determine the obstruction to the existence of extensions 0 l e q 0 and classify the extensions in terms of a suitable lie algebroid hypercohomology group in the preliminary sections we study free lie algebroids and recall some basic facts about lie algebroid hypercohomology

dynamic characterizers of spatiotemporal intermittency we study spatiotemporal intermittency sti in a system of coupled sine circle maps the phase diagram of the system shows parameter regimes where the sti lies in the directed percolation dp class as well as regimes which show pure spatial intermittency where the temporal behavior is regular which do not belong to the dp class thus both dp and non dp behavior can be seen in the same system the signature of dp and non dp behavior can be seen in the dynamic characterizers viz the spectrum of eigenvalues of the linear stability matrix of the evolution equation as well as in the multifractal spectrum of the eigenvalue distribution the eigenvalue spectrum of the system in the dp regimes is continuous whereas it shows evidence of level repulsion in the form of gaps in the spectrum in the non dp regime the multifractal spectrum of the eigenvalue distribution also shows the signature of dp and non dp behavior these results have implications for the manner in which correlations build up in extended systems

the fixed scale transformation approach to spatiotemporal intermittency in coupled map lattices abstract a one dimensional array of nonlinear maps with laplacian couplings is studied at the phase transition between the asymptotically turbulent and laminar phases along the critical boundary where spatiotemporal intermittency is observed turbulent sites form a fractal set the fixed scale transformation approach allows us to simultaneously determine the fractal dimension of the turbulent sites and the invariant asymptotic measure defining scale invariant conditional probabilities and taking into account the nearest neighbour correlations exactly we determine an asymptotic distribution function for the variables pertaining to sites on or neighbouring the turbulent clusters at any given level of coarse graining and obtain the fractal dimension analytically

can the ising critical behaviour survive in non equilibrium synchronous cellular automata abstract universality classes of ising like phase transitions are investigated in a series of two dimensional synchronously updated probabilistic cellular automata pcas whose time evolution rules are either of glauber type or of majority vote type and degrees of anisotropy are varied although early works showed that coupled map lattices and pcas with synchronously updating rules belong to a universality class distinct from the ising class careful calculations reveal that synchronous glauber pcas should be categorized into the ising class regardless of the degree of anisotropy majority vote pcas for the system size considered yield exponents which are between those of the two classes closer to the ising value with slight dependence on the anisotropy the results indicate that ising critical behaviour is robust with respect to anisotropy and synchronism for those types of non equilibrium pcas and conclusions made in earlier studies on them should be reconsidered

robust space time intermittency and 1 f noise abstract we investigate a type of intermittency that occurs in space as well as time studying a one dimensional lattice of coupled quadratic maps this system naturally forms spatial domains motion of the domain walls causes spatially localized changes from chaotic to almost periodic behavior the almost periodic phases have eigenvalues quite close to one resulting in long lived laminar bursts with a 1 f low frequency spectrum this behavior has aspects of both crutchfield and pomeau manneville intermittency unlike pomeau manneville however the behavior that we observe here is quite robust under changes of parameters

ising type transitions in coupled map lattices we study by means of computer simulations some models of coupled map lattices cml with symmetry subject to diffusive nearest neighbor coupling with the purpose of providing a better understanding of the occurrence of isingtype transitions of the type found by miller and huse we argue on the basis of numerical evidence that such transitions are connected to the appearance of a minimum in the lyapunov dimension of the system as a function of the coupling parameter two dimensional cmls similar to the one in miller and huse but with no minimum in the lyapunov dimension plot have no ising transition the condition seems to be necessary though by no means sufficient we also argue relying on the analysis of bunimovich and sinai that coupled map lattices should behave differently with respect to dimension than ising models

triple cohomology of lie rinehart algebras and the canonical class of associative algebras abstract we introduce a bicomplex which computes the triple cohomology of lie rinehart algebras we prove that the triple cohomology is isomorphic to the rinehart cohomology provided the lie rinehart algebra is projective over the corresponding commutative algebra as an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra and extend sridharan s result on almost commutative algebras

on two dimensional directed percolation extended series expansions for the mean size and the first and second moments of the pair connectedness for both bond and site percolation on the directed square and triangular lattices have been obtained analysis based on differential approximants allows the critical percolation probabilities and exponents to be estimated and as a result the critical exponents are conjectured to be gamma 41 18 nu perpendicular to 79 72 and nu sub 26 15 scaling then gives beta 199 720 alpha 299 360 and delta 1839 199

multiscaling of the graph length in coupled map lattices we consider generalized scaling exponents for the graph of a scalar function to allow for multiscaling of the graph length these generalized exponents are related to the moments of the distribution of roughness over the support of the graph we report numerical computations of these exponents for the graphs of coupled map lattices exhibiting phase transitions between laminar and turbulent behaviour we observe multiscaling of the graph in the transition region furthermore the generalized dimensions d q are computed for a conserved measure of graph roughness

chaotic traveling waves in a coupled map lattice traveling waves triggered by a phase slip are studied in a coupled map lattice a local phase slip affects globally the system which is in strong contrast with kink propagation attractors with different velocities coexist and form quantized bands determined by the number of phase slips if the system size is not far from an integer multiple of the selected wavelength attractors are tori otherwise weak chaos induces modulation of waves or a chaotic itinerancy of traveling states with a long ranged temporal correlation supertransients before the formation of traveling waves are noted in the high nonlinearity regime in the weaker nonlinearity there are fluctuations of domain sizes and brownian like motion of domains propagation of chaotic domains by phase slips is also found

fundamental study tree algebras and varieties of tree languages we consider several aspects of wilke s t wilke an algebraic characterization of frontier testable tree languages theoret comput sci 154 1996 85 106 tree algebra formalism for representing binary labelled trees and compare it with approaches that represent trees as terms in the traditional way a convergent term rewriting system yields normal form representations of binary trees and contexts as well as a new completeness proof and a computational decision method for the axiomatization of tree algebras varieties of binary tree languages are compared with varieties of tree languages studied earlier in the literature we also prove a variety theorem thus solving a problem noted by several authors syntactic tree algebras are studied and compared with ordinary syntactic algebras the expressive power of the language of tree algebras is demonstrated by giving equational definitions for some well known varieties of binary tree languages

recognisability for algebras of infinite trees we develop an algebraic language theory for languages of infinite trees we define a class of algebras called w hyperclones and we show that a language of infinite trees is regular if and only if it is recognised by a finitary path continuous w hyperclone

on directed interacting animals and directed percolation we study the phase diagram of fully directed lattice animals with nearest neighbour interactions on the square lattice this model comprises several interesting ensembles directed site and bond trees bond animals strongly embeddable animals as special cases and its collapse transition is equivalent to a directed bond percolation threshold precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n 17 the crossover region in the vicinity of the collapse transition is analysed in detail and the crossover exponent is determined directly from the singular part of the free energy we show using scaling arguments and an exact relation due to dhar that is equal to the fisher exponent governing the size distribution of large directed percolation clusters

power law persistence characterizes traveling waves in coupled circle maps with repulsive coupling we study persistence in coupled circle maps with repulsive inhibitory coupling and find that it offers an effective way to characterize the synchronous traveling wave and spatiotemporally chaotic states of the system in the traveling wave state persistence decays as a power law and in contrast to earlier observations in dynamical systems this power law scaling does not occur at the transition point alone but over the entire dynamical phase with the same exponent we give a cellular automata model displaying the qualitative features of the traveling wave regime and provide an argument based on the theory of motzkin numbers in combinatorics to explain the observed scaling

general varieties of tree languages we consider varieties of tree languages which are not restricted to a fixed ranked alphabet varieties of finite algebras that contain algebras of all finite types and a matching notion of varieties of congruences a variety theorem that yields isomorphisms between the lattices formed by these three types of varieties is proved to achieve this some basic universal algebra is suitably generalized and we define syntactic algebras so that two tree languages over any alphabets belong to the same varieties exactly in case their syntactic algebras are isomorphic many families of regular tree languages are shown to be varieties in our sense in particular we prove that every family of regular tree languages which can be characterized by syntactic monoids is such a variety but that the converse does not hold

are damage spreading transitions generically in the universality class of directed percolation we present numerical evidence for the fact that the damage spreading transition in the domany kinzel automaton found by martinset al is in the same universality class as directed percolation we conjecture that also other damage spreading transitions should be in this universality class unless they coincide with other transitions as in the ising model with glauber dynamics and provided the probability for a locally damaged state to become healed is not zero

algebraic recognizability of regular tree languages we propose a new algebraic framework to discuss and classify recognizable tree languages and to characterize interesting classes of such languages our algebraic tool called preclones encompasses the classical notion of syntactic algebra or minimal tree automaton but adds new expressivity to it the main result in this paper is a variety theorem a la eilenberg but we also discuss important examples of logically defined classes of recognizable tree languages whose characterization and decidability was established in recent papers by benedikt and segoufin and by bojanczyk and walukiewicz and can be naturally formulated in terms of pseudovarieties of preclones finally this paper constitutes the foundation for another paper by the same authors where first order definable tree languages receive an algebraic characterization