level dynamics in pseudointegrable billiards an experimental study abstract the level dynamics of pseudointegrable systems with different genus numbers g is studied experimentally using microwave cavities for higher energies the distribution of the eigenvalue velocities is gaussian as it is expected for chaotic systems with time reversal symmetry and shows no dependence on g also the curvature distribution p k for large k is decaying as it is expected for chaotic systems i e p k k 3 for small k an intermediate behavior is found where p k changes from integrable towards chaotic behavior with growing g

universal equivalence of partially commutative metabelian lie algebras abstract in this paper we find a criterion for universal equivalence of partially commutative lie algebras whose defining graphs are trees besides we obtain bases for partially commutative metabelian lie algebras

quantum algorithmic integrability the metaphor of classical polygonal billiards we study the algorithmic complexity of motions in classical polygonal billiards which as the number of sides increases tend to curved billiards both regular and chaotic this study unveils the equivalence of this problem to the procedure of quantization the average complexity of symbolic trajectories in polygonal billiards features the same scaling relations with respect to the number of sides that govern quantum systems when a semiclassical parameter is varied two cases the polygonal approximations of the circle and of the stadium are examined in detail and are presented as paradigms of quantization of integrable and chaotic systems

algebraic invariants for right angled artin groups a finite simplicial graph determines a right angled artin group g with generators corresponding to the vertices of and with a relation w w for each pair of adjacent vertices we compute the lower central series quotients the chen quotients and the first resonance variety of g directly from the graph

asymptotic analysis of a random walk with a history dependent step length we study an unbiased discrete time random walk on the nonnegative integers with the origin absorbing and a history dependent step length letting y denote the maximum distance the walker has ever been from the origin steps that do not change y have length v while those that increase y taking the walker to a site that has never been visited have length n the process serves as a simplified model of spreading in systems with an infinite number of absorbing configurations asymptotic analysis of the probability generating function shows that for large t the survival probability decays as s t t with v 2n our expression for the decay exponent is in agreement with results obtained via numerical iteration of the transition matrix

toric complexes and artin kernels abstract a simplicial complex l on n vertices determines a subcomplex t l of the n torus with fundamental group the right angled artin group g l given an epimorphism g l z let t l be the corresponding cover with fundamental group the artin kernel n we compute the cohomology jumping loci of the toric complex t l as well as the homology groups of t l with coefficients in a field k viewed as modules over the group algebra k z we give combinatorial conditions for h r t l k to have trivial z action allowing us to compute the truncated cohomology ring h r t l k we also determine several lie algebras associated to artin kernels under certain triviality assumptions on the monodromy z action and establish the 1 formality of these not necessarily finitely presentable groups

continuously variable survival exponent for random walks with movable partial reflectors we study a one dimensional lattice random walk with an absorbing boundary at the origin and a movable partial reflector on encountering the reflector at site x the walker is reflected with probability r to x 1 and the reflector is simultaneously pushed to x 1 iteration of the transition matrix and asymptotic analysis of the probability generating function show that the critical exponent delta governing the survival probability varies continuously between 1 2 and 1 as r varies between 0 and 1 our study suggests a mechanism for nonuniversal kinetic critical behavior observed in models with an infinite number of absorbing configurations

root multiplicities for borcherds algebras and graph coloring abstract we establish a connection between root multiplicities for borcherds kac moody algebras and graph coloring we show that the generalized chromatic polynomial of the graph associated to a given borcherds algebra can be used to give a closed formula for certain root multiplicities using this connection we give a second interpretation namely that the root multiplicity of a given root coincides with the number of acyclic orientations with a unique sink of a certain graph depending on the root finally using the combinatorics of lyndon words we construct a basis for the root spaces corresponding to these roots and determine the hilbert series in the case when all simple roots are imaginary as an application we give a lie theoretic proof of stanley s reciprocity theorem of chromatic polynomials

the autocorrelation function of a pseudointegrable system abstract the autocorrelation function of a pseudointegrable system is considered the system consists of billiards on plane surface formed out of three squares arranged in an l shape this system has the important property of being constructed from copies of an integrable subsystem the single square the motion can be decomposed into a continuous and a discrete part the unpredictability in the system being associated with the latter a discrete autocorrelation function is calculated and its decay properties investigated structure found in this autocorrelation function is associated with the continued fraction expansion of the ratio of velocity components for repeating continued fractions such as the golden mean the autocorrelation function exhibits a selfsimilar structure for the general case of a randomly chosen velocity ratio we derive the time dependence of the number of occurences of large autocorrelation values which differs from the behavior in integrable and chaotic systems

level statistics and eigenfunctions of square torus billiards manifesting the transition from regular to chaotic behaviors we thoroughly analyze the level statistics and eigenfunctions in concentric as well as nonconcentric square torus billiards we confirm the characteristics of quantum and classical correspondence and the existence of scarred and superscarred modes in concentric square torus billiards furthermore we not only verify that the transition from regular to chaotic behaviors can be manifested in nonconcentric square torus billiards but also develop an analytical distribution to excellently fit the numerical level statistics finally we intriguingly observe that numerous eigenstates commonly exhibit the wave patterns to be an ensemble of classical diamond trajectories as the effective wavelengths are considerably shorter than the size of internal hole

isoperimetric bounds for higher eigenvalue ratios for the n dimensional fixed membrane problem we give several results which extend our recent proof of the payne polya weinberger conjecture to ratios of higher eigenvalues in particular we show that for a bounded domain n the eigenvalues of its dirichlet laplacian obey where m denotes the m th eigenvalue and j p k denotes the k th positive zero of the bessel function j p x certain extensions of this result are given the most general being the bound where k 2 and l m denotes the number of nodal domains of an m th eigenfunction our results imply certain further conjectures of payne polya and weinberger concerning 3 2 and 4 3 in addition we find a resonably good bound on 4 1 we also briefly discuss extensions to schrodinger operators and other elliptic eigenvalue problems

free partially commutative structures abstract in this paper we study algebraic structures defined by a presentation of the form a ab ba for a b i we show that these relators are the only one that can be interpreted as both monoid and lie relators we also study the free partially commutative monoids lie algebras and groups in all of these cases we show how to obtain decomposition results for these structures into the corresponding free structures

analytically solvable dynamical systems which are not integrable abstract the central two fold aim of this paper is to expose a class of nonintegrable systems whose solutions can be analytically expressed in terms of computationally tractable algorithms and to explicitly construct a demonstrably meaningful general solution for one member of this class toward this end information theoretic results are reviewed which show that dynamical systems of null metric entropy have general solutions of null algorithmic complexity hence all systems exhibiting null metric entropy including ergodic only and mixing only systems have orbits which can be specified by algorithmically meaningful analytic expressions whose information content is very much less than that of the orbits they describe in consequence we may quite legitimately define the class of systems having null metric entropy ot be algorithmically integrable called a integrable herein after establishing the existence of a integrability this paper turns to the problem of exposing a specific nonintegrable but nonetheless a integrable system whose algorithmically meaningful solution can be explicitly derived the billiard moving within a plane polygon having the shape of a 60 120 rhombus is shown to be a simple example indeed the whole class of billiards moving in polygons whose angles are all rational multiples of have null metric entropy and are therefore a integrable systems moreover all rational billiards appear to have explicitly derivable solutions as is discussed herein and in a recent preprint by bjorn birnir because rational billiards are dense in the set of all plane billiards including chaotic ones the present paper suggests that a integrable billiards may provide a long sought analytic route to chaos finally it is pointed out that null entropied a integrable systems are perhaps the broadest category of systems which can reasonably be termed integrable for systems having positive metric entropy are truly nonintegrable due to the presence of orbits which are their own simplest description and their own fastest computer

spacing distributions for rhombus billiards we show that the spacing distributions of rational rhombus billiards fall in a family of universality classes distinctly different from the wigner dyson family of random matrix theory and the poisson distribution some of the distributions find explanation in a recent work of bogomolny gerland and schmit for the irrational billiards despite ergodicity we get the same distribution for the examples considered once again distinct from the wigner dyson distributions all the results are obtained numerically by a method that allows us to reach very high energies

on the support of graph lie algebras we deal with the support of graph lie algebras and we characterize the set of independence alphabets for which the noncommutative result can naturally be extended we also provide an algorithm which decides if an independence alphabet fulfills the preceding condition

stochastic one dimensional lorentz gas on a lattice we stud ya one dimensional stochastic lorentz gas wher ea light particle moves i na fixed array of nonidentical random scatterers arranged i na lattice each scatterer is characterized b ya random transmissionreflection coefficient we consider the case when the transmission coefficients of the scatterers are independent identically distributed random variables a symbolic program is presented which generates the exact velocity autocorrelation functio n vacf in terms of the moments of the transmission coefficients the vacf is found for different types of disorder for times up to 20 collision times we then conside ra specific type of disorder a two state lorentz gas in which two types of scatterers are arranged randomly i na lattice the na lattice point is occupied b ya scatterer whose transmission coefficient is with probability p or with probability 1 p a perturbation expansion with respect to is derived the 2 term in this expansion shows that the vacf oscillates with time the period of oscillation being twice the time of flight from one scatterer to its nearest neighbor the coarse grained vacf decays for long times like t 32 which is similar to the decay of the vacf of the random lorentz gas wit ha single type of scatterer the perturbation results and the exact one s found up to 20 collision times show good agreement

lyapunov exponents and anomalous diffusion of a lorentz gas with infinite horizon using approximate zeta functions we compute the lyapunov exponent the generalized lyapunov exponents and the diffusion constant for a lorentz gas on a square lattice thus having infinite horizon approximate zeta functions written in terms of probabilities rather than periodic orbits are used in order to avoid the convergence problems of cycle expansions the emphasis is on the relation between the analytic structure of the zeta function where a branch cut plays an important role and the asymptotic dynamics of the system the lyapunov exponent for the corresponding map agrees with the conjectured limit map 2 log r c o r and we derive an approximate value for the constantc in good agreement with numerical simulations we also find a diverging diffusion constantd t logt and a phase transition for the generalized lyapunov exponents

generalized semigroup semirings which are zero divisor free or multiplicatively left cancellative abstract let x o be a set pc x one of the various free partially commutative monoids on x including the free monoid x and the free commutative monoid c x as the extreme cases cf cartier and foata 1969 and s pc x the polynomial semiring in the corresponding partially commuting indeterminates x i x over a suitable semiring s recently it was shown duchamp and thibon 1988 that each of these semirings s pc x is zero divisor free iff s is so and is additively cancellative and multiplicatively left cancellative iff s has the same properties in this paper we refine these results and generalize them to the semirings s pc x of power series in those indeterminates this is done in the context of more general investigations concerning the above properties for semigroup semirings s u and generalized semigroup semirings a s u for arbitrary semigroups u

on the time evolution of macroscopic systems autodiffusion a basse densite et theoreme de lanford correlations dans l etat initial auto diffusion a densite finie le gaz de lorentz et le theoreme de bunimovich et sinai

asymptotic distribution of global errors in the numerical computations of dynamical systems abstract we propose an analysis of the effects introduced by finite accuracy and round off arithmetic on numerical computations of discrete dynamical systems our method which uses the statistical tool of the decay of fidelity computes the error by directly comparing the numerical orbit with the exact one or more precisely with another numerical orbit computed with a much higher accuracy furthermore as a model of the effects of round off arithmetic on the map we also consider a random perturbation of the exact orbit with an additive noise for which exact results can be obtained for some prototype maps we investigate the decay laws of fidelity and their relationship with the error probability distribution for regular and chaotic maps for both additive and numerical noise in particular for regular maps we find an exponential decay for additive noise and a power law decay for numerical noise for chaotic maps numerical noise is equivalent to additive noise and our method is suitable for identifying a threshold for the reliability of numerical results i e the number of iterations below which global errors can be ignored this threshold grows linearly with the number of bits used to represent real numbers

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

billiards in polygons survey of recent results in the ten or so years since the publication of ref 19 polygonal billiards have remained an active subject of research in the mathematics and physics literature as a result our understanding of the subject although still far from being complete is much better than it was ten years ago this survey attempts to give a broad overview of the dynamics of billiards in polygons with an emphasis on the material that was not in the literature in 1985 this is not a survey of the publications on polygonal billiards and we apologize to authors whose work has not been included the selection of topics has been strongly influenced by the personal taste of the author and by space limitations thus we do not discuss quantum polygonal billiards in particular quantum chaos see e g refs 41 44 and 45 we hope that somebody will write a survey on this important subject in the near future a fascinating aspect of the subject is the interplay between the geometric shape of the billiard table i e a planar curve and the qualitative features of the billiard dynamics hence it is instructive to compare polygonal billiard tables with other classes of billiard tables in particular the smooth strictly convex tables and the dispersing or sinai tables produce strikingly different types of billiard dynamics we recommend refs 46 and 15 for a general overview of the subject and a comparative study of billiard dynamics for various types of billiard tables

evanescent wave approach to diffractive phenomena in convex billiards with corners what we are going to call in this paper diffractive phenomena in billiards is far from being deeply understood these are sorts of singularities that for example some kind of corners introduce in the energy eigenfunctions in this paper we use the well known scaling quantization procedure to study them we show how the scaling method can be applied to convex billiards with corners taking into account the strong diffraction at them and the techniques needed to solve their helmholtz equation as an example we study a classically pseudointegrable billiard the truncated triangle then we focus our attention on the spectral behavior a numerical study of the statistical properties of high lying energy levels is carried out it is found that all computed statistical quantities are roughly described by the so called semi poisson statistics but it is not clear whether the semi poisson statistics is the correct one in the semiclassical limit

finite size lyapunov exponent review on applications in dynamical systems the growth of infinitesimal perturbations is well characterized by the lyapunov exponents in many situations of interest however important phenomena involve finite amplitude perturbations which are ruled by nonlinear dynamics out of tangent space and thus cannot be captured by the standard lyapunov exponents we review the application of the finite size lyapunov exponent fsle for the characterization of non infinitesimal perturbations in a variety of systems in particular we illustrate their usage in the context of predictability of systems with multiple spatio temporal scales of geophysical relevance in the characterization of nonlinear instabilities and in some aspects of transport in fluid flows we also discuss the application of the fsle to more general aspects such as chaos noise detection and coarse grained descriptions of signals this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

integrity of formal power series on a partially commutative alphabet resume nous demontrons que l anneau de series formelles sur un alphabet partiellement commutatif est integre

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

escape exponent for transient chaos and chaotic scattering in non hyperbolic hamiltonian systems it is shown that in non hyperbolic hamiltonian systems where correlations decay as a power law transient chaos and chaotic scattering demonstrate different power laws numerical evidence of this effect for the stadium billiard is presented

bounds for ratios of the membrane eigenvalues abstract for a bounded planar region in r 2 we obtain the ratios of lower order eigenvalues of laplace operator combining our results with the recursive formula in cheng and yang 2007 11 we can obtain better upper bound of the k 1 th k 3 membrane eigenvalues

on infinite volume mixing in the context of the long standing issue of mixing in infinite ergodic theory we introduce the idea of mixing for observables possessing an infinite volume average the idea is borrowed from statistical mechanics and appears to be relevant at least for extended systems with a direct physical interpretation we discuss the pros and cons of a few mathematical definitions that can be devised testing them on a prototypical class of infinite measure preserving dynamical systems namely the random walks

on survival dynamics of classical systems non chaotic open billiards abstract we report on decay problem of classical systems mesoscopic level consideration is given on the basis of transient dynamics of non interacting classical particles bounded in billiards three distinct decay channels are distinguished through the long tailed memory effects revealed by temporal behavior of survival probability t i the universal independent of geometry initial conditions and space dimension channel with 1 of brownian relaxation of non trapped regular parabolic trajectories and ii the non brownian channel 1 due to highly chaotic bouncing ball trajectories we develop a statistical approach to the problem earlier proposed for chaotic classical systems physica a 275 2000 70 a systematic coarse graining procedure is introduced for non chaotic systems exemplified by circle and square geometry which are characterized by a certain finite characteristic collision time we demonstrate how the transient dynamics is related to the intrinsic dynamics driven by the preserved liouville measure the detailed behavior of the late time survival probability including a role of the initial conditions and a system geometry is studied in detail both theoretically and numerically

billiards correlation functions we discuss various experiments on the time decay of velocity autocorrelation functions in billiards we perform new experiments and find results which are compatible with an exponential mixing hypothesis first put forward by friedman and martin fm they do not seem compatible with the stretched exponentials believed in spite of fm and more recently of chernov to describe the mixing the analysis leads to several byproducts we obtain information about the normal diffusive nature of the motion and we consider the probability distribution of the number of collisions in timetm astm finding a strong dependence on some geometric characteristics of the locus of the billiard obstacles

periodic orbit theory and spectral rigidity in pseudointegrable systems we calculate numerically the periodic orbits of pseudointegrable systems of low genus numbers g that arise from rectangular systems with one or two salient corners from the periodic orbits we calculate the spectral rigidity delta3 l using semiclassical quantum mechanics with l reaching up to quite large values we find that the diagonal approximation is applicable when averaging over a suitable energy interval comparing systems of various shapes we find that our results agree well with delta3 calculated directly from the eigenvalues by spectral statistics therefore additional terms such as e g diffraction terms seem to be small in the case of the systems investigated in this work by reducing the size of the corners the spectral statistics of our pseudointegrable systems approaches that of an integrable system whereas very large differences between integrable and pseudointegrable systems occur when the salient corners are large both types of behavior can be well understood by the properties of the periodic orbits in the system

deformations of the tracy widom distribution in random matrix theory the tracy widom tw distribution describes the behavior of the largest eigenvalue we consider here two models in which tw undergoes transformations in the first one disorder is introduced in the gaussian ensembles by superimposing an external source of randomness a competition between tw and a normal gaussian distribution results depending on the spreading of the disorder the second model consists of removing at random a fraction of correlated eigenvalues of a random matrix the usual formalism of fredholm determinants extends naturally a continuous transition from tw to the weilbull distribution characteristic of extreme values of an uncorrelated sequence is obtained

barrier billiards a simple pseudo integrable system a hard line segment barrier lies in a periodic cell in the plane parallel to one edge and half its length this constitutes perhaps the simplest pseudo integrable system and the classical motion of a particle within it admits concise analysis the hitting and missing of the barrier defines a quasi periodic doubly periodic sequence with the period ratio depending on the invariant gradient of the trajectory the consequent motion is expressed in terms of its continued fraction expansion rational gradients correspond to periodic orbits which can be classified in full quadratic irrational gradients lead to trajectories of calculable fractal dimension or power law correlation decay

theoremes de transfert pour les polynomes partiellement commutatifs abstract we prove that the properties of a semiring k to be without zero divisors and to be regular and simplifiable transfer to the k algebras of the free partially commutative monoids we also give an algorithm which tests the divisibility of partially commutative polynomials and computes the quotient in the case of a positive answer

the shuffle algebra and its derivations abstract the shuffle algebra on a partially commutative alphabet and its derivations are studied the derivation of the shuffle algebra is characterized in terms of admissible mappings the module of all derivations of the shuffle algebra is proved to be isomorphic to the free partially commutative power series lie algebra with the coefficient ring extended the orthogonality between the lie elements and the proper shuffles is established

eigenvalue dependence on small pseudointegrable perturbation of a two dimensional box the eigenvalues of a hard walled parallelogram shaped enclosure were investigated using quantum perturbation theory the perturbation being a small angle distortion of a pair of parallel sides of the rectangle a comparison of perturbation theory results and exact computations for low lying states showed good agreement the distribution of eigenvalues grossly disturbed from their value in the nearby regular system was studied and found to be systematic

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

on the boundary element method for billiards with corners the boundary element method is one of the reliable numerical schemes to solve the eigenvalue problem of the helmholtz equation which is justified by the fredholm theory for domains with a smooth boundary when a domain has corners however the corresponding integral equation is singular so that the boundary element method lacks its well established base employing a cutoff technique we here formulate a well grounded version of the boundary element method and also give a certain justification to the standard boundary element method even for domains with corners

physical insight into superdiffusive dynamics of sinai billiard through collision statistics abstract we report on distinct steady motion dynamic regimes in chaotic sinai billiard sb a numerical study on elastic reflections from the sb boundary square wall of length l and circle obstacle of radius r is carried out for different r l the research is based on the exploration of the generalized diffusion equation and on the analysis of wall collision and the circle collision distributions observed at late times the asymptotes for the diffusion coefficient d r and the corresponding diffusion exponent z r are established for all geometries the universal r independent diffusion with d 1 t 1 3 and z 1 1 5 replaces the ballistic motion regime z 0 1 attributed to square billiard r 0 geometrically this superdiffusive regime is bounded by small radii 0 r r 1 r 1 l 2 4 when both diagonal and non diagonal bleher s corridors are open in the correspondent square lattice of lorentz gas lg model the relaxation dynamics observed is ensured by the universal diffusive propagation of the regular type and the bouncing ball type orbits within the random walk scheme this superdiffusive regime is due to the levy flights between the long distant scatterers with the increase in circle radius r 1 r r 2 r 2 l 2 the diagonal corridor closes but the arc touching effects continue to bring the long living bouncing ball orbits back to the non diagonal infinite corridors this transient non universal dynamics with 1 5 z r 2 also associated with the trapping of regular orbits seems to be characteristic of non fully hyperbolic billiard systems in sb with finite horizon r r 2 all the principal corridors are closed and the interplay between square and circle boundaries generates the known chaotic dynamics attributed to the fully hyperbolic systems this is also observed through the normal brownian diffusion z 2 2 and the gaussian statistics proved for both kinds of collisions

level statistics and eigenfunctions of pseudointegrable systems dependence on energy and genus number we study the level statistics second half moment i 0 and rigidity ensuremath delta 3 and the eigenfunctions of pseudointegrable systems with rough boundaries of different genus numbers g we find that the levels form energy intervals with a characteristic behavior of the level statistics and the eigenfunctions in each interval at low enough energies the boundary roughness is not resolved and accordingly the eigenfunctions are quite regular functions and the level statistics shows poisson like behavior at higher energies the level statistics of most systems moves from poisson like toward wigner like behavior with increasing g on investigating the wave functions we find many chaotic functions that can be described as a random superposition of regular wave functions the amplitude distribution p ensuremath psi of these chaotic functions was found to be gaussian with the typical value of the localization volume v mathrm loc ensuremath approx 0 33 for systems with periodic boundaries we find several additional energy regimes where i 0 is relatively close to the poisson limit in these regimes the eigenfunctions are either regular or localized functions where p ensuremath psi is close to the distribution of a sine or cosine function in the first case and strongly peaked in the second case an interesting intermediate case between chaotic and localized eigenfunctions also appears

slow relaxation in weakly open rational polygons the interplay between the regular piecewise linear and irregular vertex angle boundary effects in nonintegrable rational polygonal billiards of m equal sides is discussed decay dynamics in polygons of perimeter p m and small opening delta is analyzed through the late time survival probability s m approximately equal t delta two distinct slow relaxation channels are established the primary universal channel exhibits relaxation of regular sliding orbits with delta 1 the secondary channel is given by delta 1 and becomes open when m p m delta it originates from vertex order disorder dual effects and is due to relaxation of chaoticlike excitations

generic spectral properties of right triangle billiards this article presents a new method to calculate eigenvalues of right triangle billiards its efficiency is comparable to the boundary integral method and more recently developed variants its simplicity and explicitness however allow new insight into the statistical properties of the spectra we analyse numerically the correlations in level sequences at high level numbers 10 5 for several examples of right triangle billiards we find that the strength of the correlations is closely related to the genus of the invariant surface of the classical billiard flow surprisingly the genus plays an important role at the quantum level also based on this observation a mechanism is discussed which may explain the particular quantum classical correspondence in right triangle billiards though this class of systems is rather small it contains examples for integrable pseudo integrable and non integrable ergodic weakly mixing dynamics so that the results might be relevant in a more general context

lyndon bases of partially commutative free lie algebras abstract les empilements de lyndon forment un analogue partiellement commutatif des mots de lyndon on construit une base de l algebre de lie libre partiellement commutative correspondante en utilisant les proprietes combinatoires des empilements de lyndon

optimal bounds for ratios of eigenvalues of one dimensional schrodinger operators with dirichlet boundary conditions and positive potentials consider the schrodinger equation u v x u u on the intervali wherev x 0 forx i and where dirichlet boundary conditions are imposed at the endpoints ofi we prove the optimal bound frac lambda n lambda 1 leqq n 2 for n 2 3 4 on the ratio of thenth eigenvalue to the first eigenvalue for this problem this leads to a complete treatment of bounds on ratios of eigenvalues for such problems extensions of these results to singular problems are also presented a modified prufer transformation and comparison techniques are the key elements of the proof

fluctuation of the entropy production for the lorentz gas under small external forces in this paper we study the physical and statistical properties of the periodic lorentz gas with finite horizon driven to a non equilibrium steady state by the combination of non conservative external forces and deterministic thermostats a version of this model was introduced by chernov eyink lebowitz and sinai and subsequently generalized by chernov and the third author non equilibrium steady states for these models are srb measures and they are characterized by the positivity of the steady state entropy production rate our main result is to establish that the entropy production in this context equal to the phase space contraction satisfies the gallavotti cohen fluctuation relation the main tool needed in the proof is the family of anisotropic banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques

escape probability for classically chaotic systems abstract a statistical approach to study the escape of an ensemble of particles from classical chaotic systems is proposed the universal kinetic decay laws exponential and algebraic are found through the velocity angle distribution and shown to be specific of distinct motion regimes in the billiards with a small opening on the basis of the particular case of the dispersing sinai billiard the escape probability is given in explicit form and the temporal and geometrical conditions are enunciated to make controlled observations of decay dynamics in numerical experiments

quantum aspects of triangular billiards abstract the energy spectrum of a particle enclosed in a two dimensional triangular box can in general not be calculated analytically therefore in order to establish the degree of intergrability using some well known criteria this spectrum must be calculated numerically for more than 800 triangles these calculations were performed and then used to check a number of conjectures found in the literature for almost all of them we find counter examples our own suggestion is that in general acute triangles are more integrable than obtuse ones however also this rule has its exceptions which cannot be explained away by some triangles being more rational than others

properties making a chaotic system a good pseudo random number generator we discuss the properties making a deterministic algorithm suitable to generate a pseudo random sequence of numbers high value of kolmogorov sinai entropy high dimensionality of the parent dynamical system and very large period of the generated sequence we propose the multidimensional anosov symplectic cat map as a pseudo random number generator we show what chaotic features of this map are useful for generating pseudo random numbers and investigate numerically which of them survive in the discrete state version of the map testing and comparisons with other generators are performed

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

kolmogorov sinai entropy lyapunov exponents and mean free time in billiard systems we perform new experiments on the kolmogorov sinai entropy lyapunov exponents and the mean free time in billiards we study their dependence on the geometry of the scatterers made up of two interpenetrating square lattices each one with circular scatterers with different radius we find in particular that the above quantities are continuous functions of the ratio of the scatterer radius however it seems that their derivative is discontinuous around the radius ratio which separates the diffusive and nondiffusive types of geometries

nonunitary transformations and hydrodynamic limit in the one dimensional hard point gas abstract the ergodic theoretical formalism of nonunitary transformations is compared with the derivation of kinetic equations by the hydrodynamic scaling limit the one dimensional hard point system is used as an illustration both methods rest on the k mixing property of the physical systems considered we show that the specific form of the operator is important only on the microscopic time scale on the macroscopic scale it acts like a projector with broken time reversal symmetry we also discuss the connection between the kolmogorov sinai k s entropy of the model dynamical system the relative entropy production rate in the nonunitary formalism and the macroscopic entropy production rate

the statistics of the trajectory of a certain billiard in a flat two torus we consider a billiard in the punctured torus obtained by removing a small disk of radius 0 from the flat torus 2 with trajectory starting from the center of the puncture in this case the phase space is given by the range of the velocity only let and respectively r denote the first exit time length of the trajectory and respectively the number of collisions with the side cushions when 2 is being identified with 0 1 2 we prove that the probability measures on 0 associated with the random variables and r are weakly convergent as varepsilon rightarrow 0 and explicitly compute the densities of the limits

periodic orbits contribution to the 2 point correlation form factor for pseudo integrable systems using heuristic arguments based on the trace formulas we relate the 2 point correlation form factor k2 at small values of with sums over classical periodic orbits for typical examples of pseudo integrable systems the later sums have been explicitly calculated for the following models i plane billiards in the form of right triangles with one angle n and ii rectangular billiards with the aharonov bohm flux line in the first model using the properties of the veech structure it is shown that k2 0 n e n 3 n 2 where e n 0 for odd n e n 2 for even n not divisible by 3 and e n 6 for even n divisible by 3 for completeness we also recall informally the main features of the veech construction in the second model the answer depends on arithmetical properties of ratios of flux line coordinates to the corresponding sides of the rectangle when these ratios are non commensurable irrational numbers k2 0 1 3 where is the fractional part of the flux through the rectangle when and it is symmetric with respect to the line when the comparison of these results with numerical calculations of the form factor is discussed in detail the above values of k2 0 differ from all known examples of spectral statistics thus confirming analytically the peculiarities of statistical properties of the energy levels in pseudo integrable systems

the free partially commutative lie algebra bases and ranks abstract in this paper we study the free partially commutative lie k algebra l a defined by a commutation relation on an alphabet a its behavior is very similar to that of the free lie algebra indeed we obtain in particular a partially commutative version of lazard s elimination process which allows us to prove that the k module l a is free and to construct explicitly k bases for it we show also how the classical witt s calculus can be extended to l a

on some equations in free partially commutative monoids abstract we characterize the solutions of all the equations in two unknowns in the free partially commutative monoids and we show that the connected solutions are cyclic next we define the transposition relation m and n are transposed iff m xy and n yx and the conjugacy relation m and n are conjugate iff m n we show that the conjugacy is the transitive closure of the transposition and that the set of conjugacy factors is a recognizable subset of the free partially commutative monoid

recurrence properties of a special type of heavy tailed random walk in the proof of the invariance principle for locally perturbed periodic lorentz process with finite horizon a lot of delicate results were needed concerning the recurrence properties of its unperturbed version these were analogous to the similar properties of simple symmetric random walk however in the case of lorentz process with infinite horizon the analogous results for the corresponding random walk are not known either in this paper these properties are ascertained for the appropriate random walk this happens to be in the non normal domain of attraction of the normal law as a tool an estimation of the remainder term in the local limit theorem for the corresponding random walk is computed

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces

hopf algebras and identities in free partially commutative monoids abstract we define a hopf algebra structure on the algebra of polynomials over a partially commutative alphabet by considering formulas for the antipodes of this hopf algebra and its dual we obtain several new identities on partially commutative words

two new weyl type bounds for the dirichlet laplacian in this paper we prove two new weyl type upper estimates for the eigenvalues of the dirichlet laplacian as a consequence we obtain the following lower bounds for its counting function for 1 one has n 2 n 2 1 h n 1 n 2 1 n 2 and n n 2 n 4 n 2 1 h n 1 4 n 1 n 2 1 n 2 where h n 2n j 2 n 2 1 1j 2 n 2 jn 2 1 1 is a constant which depends on n the dimension of the underlying space and bessel functions and their zeros