towards a quantum analog of weak kam theory we discuss a quantum analogue of mather s minimization principle for lagrangian dynamics and provide some formal calculations suggesting the corresponding euler lagrange equation we then rigorously construct from the dual eigenfunctions of a certain non selfadjoint operator a candidate for a minimizer and recover aspects of weak kam theory in the limit as h 0 regarding our state as a quasimode we furthermore derive some error estimates although it remains an open problem to improve these bounds

boundary nlc graph grammars basic definitions normal forms and complexity node label controlled nlc grammars are graph grammars operating on node labeled undirected graphs which rewrite single nodes only and establish connections between the embedded graph and the neighbors of the rewritten node on the basis of the labels of the involved nodes only they define possibly infinite languages of undirected node labeled graphs or if we just omit the labels languages of unlabeled graphs here we consider a restriction of nlc grammars so called boundary nlc bnlc grammars distinguished by the property that whenever in a graph already generated two nodes may be rewritten then these nodes are not adjacent the graph languages generated by this type of grammars are called bnlc languages although we show that this restriction leads to a smaller class of languages still enough generative power remains to define interesting graph languages for example trees complete bipartite graphs maximal outerplanar graphs k trees graphs of bandwidth k graphs of cyclic bandwidth k graphs of binary tree bandwidth k graphs of cutwidth k always for a fixed positive integer k turn out all to be bnlc languages we prove a number of normal forms for bnlc grammars and then we indicate their usefulness by various applications in particular we show that for connected graphs of bounded degree the membership problem for bnlc languages is solvable in deterministic polynomial time

a local level set method for paraxial geometrical optics we propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two dimensional 2 d acoustic wave equations the geometrical optics term consists of two multivalued functions a travel time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally the multivalued travel times are obtained by solving a level set equation and a travel time equation with a forcing term the multivalued amplitudes are computed by a neweulerian formula based on the gradients of travel times and takeoff angles as a byproduct the method is also able to capture the caustic locations the proposed eulerian method has complexity of o n 2 rm log n rather than o n 4 as typically seen in the lagrangian ray tracing method several examples including the well known marmousi synthetic model illustrate the accuracy and efficiency of the eulerian method

individual based predator prey model for biological coevolution fluctuations stability and community structure we study an individual based predator prey model of biological coevolution using linear stability analysis and large scale kinetic monte carlo simulations the model exhibits approximate 1 f noise in diversity and population size fluctuations and it generates a sequence of quasisteady communities in the form of simple food webs these communities are quite resilient toward the loss of one or a few species which is reflected in different power law exponents for the durations of communities and the lifetimes of species the exponent for the former is near ensuremath 1 while the latter is close to ensuremath 2 statistical characteristics of the evolving communities including degree predator and prey distributions and proportions of basal intermediate and top species compare reasonably with data for real food webs

aubry mather theory and periodic solutions of the forced burgers equation consider a hamiltonian system with hamiltonian of the form h x t p where h is convex in p and periodic in x a ndt and x2 r 1 it is well known that its smooth invariant curves correspond to smoothz 2 periodic solutions of the pde ut h x t u x 0 in this paper we establish a connection between the aubry mather theory of invariant sets of the hamiltonian system andz 2 periodic weak solutions of this pde by realizing the aubry mather sets as closed subsets of the graphs of these weak solutions we show that the complement of the aubry mather set on the graph can be viewed as a subset of the generalized unstable manifold of the aubry mather set defined in 2 24 the graph itself is a backward invariant set of the hamiltonian system the basic idea is to embed the globally minimizing orbits used in the aubry mather theory into the characteristic fields of the above pde this is done by making use of one and two sided minimizers a notion introduced in 12 and inspired by the work of morse on geodesics of type a 26 the asymptotic slope of the minimizers also known as the rotation number is given by the derivative of the homogenized hamiltonian defined in 21 as an application we prove that thez 2 periodic weak solution of the above pde with given irrational asymptotic slope is unique a similar connection also exists in multidimensional problems with the convex hamiltonian except that in higher dimensions two sided minimizers with a specified asymptotic slope may not exist c 1999 john wiley sons inc

periodic homogenisation of hamilton jacobi equations 2 eikonal equations homogenisation of the first order hamilton jacobi equations when h is periodic in the second variable leads to an effective hamiltonian h satisfying u e converges as e 0 to the solution u of u t h du 0 in our first paper we assumed that h is convex and we derived a variational formula giving h in this second paper we consider eikonal equations i e h p x p 2 v x using our variational formula we compute explicitly the effective hamiltonian in several cases and we study precisely the lack of strict convexity for h flat part around the origin

stochastic calculus of variations abstract a theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes generalizations of the euler equation and noether s theorem are obtained and several conservation laws are discussed an application to nelson s probabilistic framework of quantum mechanics is also given

effective hamiltonians and averaging for hamiltonian dynamics i this paper building upon ideas of mather moser fathi e and others applies pde partial differential equation methods to understand the structure of certain hamiltonian flows the main point is that the cell or corrector pde introduced and solved in a weak sense by lions papanicolaou and varadhan in their study of periodic homogenization for hamilton jacobi equations formally induces a canonical change of variables in terms of which the dynamics are trivial we investigate to what extent this observation can be made rigorous in the case that the hamiltonian is strictly convex in the momenta given that the relevant pde does not usually in fact admit a smooth solution

asymmetric dynamics and critical behavior in the bak sneppen model abstract we investigate using mean field theory and simulation the effect of asymmetry on the critical behavior and probability density of bak sneppen models two kinds of anisotropy are investigated i different numbers of sites to the left and right of the central minimum site are updated and ii sites to the left and right of the central site are renewed in different ways of particular interest is the crossover from symmetric to asymmetric scaling for weakly asymmetric dynamics and the collapse of data with different numbers of updated sites but the same degree of asymmetry all non symmetric rules studied fall independent of the degree of asymmetry in the same universality class conversely symmetric variants reproduce the exponents of the original model our results confirm the existence of two symmetry based universality classes for extremal dynamics

schrodinger spectra and the effective hamiltonian of weak kam theory on the flat torus in this paper we investigate the link between the spectrum of some periodic schrodinger type operators and the effective hamiltonian of the weak kam theory we show that the extension of some local quasimodes is linked to the localization of the schrodinger spectrum such a result provides additional information with respect to the well known bohr sommerfeld quantization rules here in a more general setting than the integrable or quasi integrable ones

homogenization of a neutronic critical diffusion problem with drift in this paper we study the homogenization of an eigenvalue problem for a cooperative system of weakly coupled elliptic partial differential equations called the neutronic multigroup diffusion model in a periodic heterogeneous domain such a model is used for studying the criticality of nuclear reactor cores in a recent work in collaboration with gregoire allaire it is proved that under a symmetry assumption the first eigenvector of the multigroup system in the periodicity cell controls the oscillatory behaviour of the solutions whereas the global trend is asymptotically given by a homogenized diffusion eigenvalue problem it is shown here that when this symmetry condition is not fulfilled the asymptotic behaviour of the neutron flux corresponding to the first eigenvector of the multigroup system is dramatically different this result enables to consider new types of geometrical configurations in practical nuclear reactor core computations

bak sneppen model near zero dimension we consider the bak sneppen model near zero dimension where the avalanche exponent tau is close to 1 and the exponents m and sigma are close to 0 we demonstrate that tau 1 m sigma exp m 1 gamma small middle dot small middle dot small middle dot in this limit where gamma is euler s constant the avalanche hierarchy equation is rewritten in a form that makes it possible to find the relation between the critical exponents sigma and m with high accuracy we obtain precise values of the critical exponents for the one and two dimensional bak sneppen model and for the one dimensional anisotropic bak sneppen model

restrictions on nlc graph grammars several models of graph grammars have been studied with the objective of generating graphs from graphs using a finite set of derivation rules in this way possibly infinite sets of graphs called graph languages can be finitely defined one aspect that must be addressed in any such model is the embedding problem that is when a production is applied how does the new subgraph get reconnected to the original graph node label controlled nlc grammars solve this problem in an elegant way that depends only on the labels of nodes in the new and original graphs this paper examines certain restrictions on nlc grammars similar to the chomsky or greibach normal forms for context free string grammars for example one restriction we consider requires each production to produce a terminal labeled node similar to greibach normal form we also consider restrictions on the form which the embedding mechanism can take our result is that each of the restrictions we examine causes a reduction in generating power for the grammars finally we discuss some directions for future research on nlc grammars

on singular probability densities generated by extremal dynamics abstract extremal dynamics is the mechanism that drives the bak sneppen model into a self organized critical state marked by a singular stationary probability density p x with the aim of understanding this phenomenon we study the bs model and several variants via mean field theory and simulation in all cases we find that p x is singular at one or more points as a consequence of extremal dynamics furthermore we show that the extremal barrier x i always belongs to the prohibited interval in which p x 0 our simulations indicate that the bak sneppen universality class is robust with regard to changes in the updating rule we find the same value for the exponent for all variants mean field theory which furnishes an exact description for the model on a complete graph reproduces the character of the probability distribution found in simulations for the modified processes mean field theory takes the form of a functional equation for p x

on the thresholds probability densities and critical exponents of bak sneppen like models abstract we report a simple method to accurately determine the threshold and the exponent of the bak sneppen bs model and also investigate the bs universality class for the random neighbor version of the bs model we find the threshold x 0 33332 3 in agreement with the exact result x 1 3 given by mean field theory for the one dimensional original model we find x 0 6672 2 in good agreement with the results reported in the literature for the anisotropic bs model we obtain x 0 7240 1 we study the finite size effect x l x l l observed in a system with l sites and find 1 00 1 for the random neighbor version 1 40 1 for the original model and 1 58 1 for the anisotropic case finally we discuss the effect of defining the extremal site as the one which minimizes a general function f x instead of simply f x x as in the original updating rule we emphasize that models with extremal dynamics have singular stationary probability distributions p x our simulations indicate the existence of two symmetry based universality classes

the generating power of boundary nlc graph grammars and cycle graphs it is known that a can be generated by a node label controlled nlc graph grammar sa cannot be generated by nlc graph grammars and a cannot be generated by boundary nlc bnlc graph grammars where a is the set of all graphs labeled by the single label a and sa is the set of all cycle graphs labeled by the single label a in this paper we consider the class lc of all graph languages including sa we show that the class lc and the class of bnlc graph languages are mutually disjoint furthermore we provide a new simple graph language that can be generated by an nlc graph grammar but cannot be generated by bnlc graph grammars using a method different from rozenberg and welzl

branching process in a stochastic extremal model we considered a stochastic version of the bak sneppen model sbsm of ecological evolution where the number m of sites mutated in a mutation event is restricted to only two here the mutation zone consists of only one site and this site is randomly selected from the neighboring sites at every mutation event in an annealed fashion the critical behavior of the sbsm is found to be the same as the bs model in dimensions d 1 and 2 however on the scale free graphs the critical fitness value is nonzero even in the thermodynamic limit but the critical behavior is mean field like finally m has been made even smaller than two by probabilistically updating the mutation zone which also shows the original bs model behavior we conjecture that a sbsm on any arbitrary graph with any small branching factor greater than unity will lead to a self organized critical state

a pumping lemma and the structure of derivations in the boundary nlc graph languages abstract a pumping lemma is known in the nlc node label controlled graph languages 4 however the pumping lemma has an unnatural point at the first pumping that is zero time pumping is not allowed in the pumping lemma we consider the boundary nlc bnlc languages 10 then we obtain a pumping lemma without the unnatural point similar to one for the nlc graph languages using the church rosser property of the bnlc graph grammars we also construct the pumped graphs and provide a constructive proof for our pumping lemma as a result we provide the structure of derivations in the bnlc graph grammars and a role of the church rosser property in the nlc graph grammars