kolmogorov sinai entropy of many body hamiltonian systems the kolmogorov sinai ks entropy is a central measure of complexity and chaos its calculation for many body systems is an interesting and important challenge in this paper the evaluation is formulated by considering n dimensional symplectic maps and deriving a transfer matrix formalism for the stability problem this approach makes explicit a duality relation that is exactly analogous to one found in a generalized anderson tight binding model and leads to a formally exact expression for the finite time ks entropy within this formalism there is a hierarchy of approximations the final one being a diagonal approximation that only makes use of instantaneous hessians of the potential to find the ks entropy by way of a nontrivial illustration the ks entropy of n identically coupled kicked rotors standard maps is investigated the validity of the various approximations with kicking strength particle number and time are elucidated an analytic formula for the ks entropy within the diagonal approximation is derived and its range of validity is also explored

riemannian theory of hamiltonian chaos and lyapunov exponents this paper deals with the problem of analytically computing the largest lyapunov exponent for many degrees of freedom hamiltonian systems this aim is succesfully reached within a theoretical framework that makes use of a geometrization of newtonian dynamics in the language of riemannian geometry a new point of view about the origin of chaos in these systems is obtained independently of homoclinic intersections chaos is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of jacobi equation for geodesic spread under general conditions ane effective stability equation is derived an analytic formula for the growth rate of its solutions is worked out and applied to the fermi pasta ulam beta model and to a chain of coupled rotators an excellent agreement is found the theoretical prediction and the values of the lyapunov exponent obtained by numerical simulations for both models

hydrodynamic lyapunov modes and strong stochasticity threshold in fermi pasta ulam models the existence of a strong stochasticity threshold sst has been detected in many hamiltonian lattice systems including the fermi pasta ulam fpu model which is characterized by a crossover of the system dynamics from weak to strong chaos with increasing energy density correspondingly the relaxation time to energy equipartition and the largest lyapunov exponent exhibit different scaling behavior in the regimes below and beyond the threshold value in this paper we attempt to go one step further in this direction to explore further changes in the energy density dependence of other lyapunov exponents and of hydrodynamic lyapunov modes hlms in particular we find that for the fpu ensuremath beta and fpu ensuremath alpha ensuremath beta models the scalings of the energy density dependence of all lyapunov exponents experience a similar change at the sst as that of the largest lyapunov exponent in addition the threshold values of the crossover of all lyapunov exponents are nearly identical these facts lend support to the point of view that the crossover in the system dynamics at the sst manifests a global change in the geometric structure of phase space they also partially answer the question of why the simple assumption that the ambient manifold representing the system dynamics is quasi isotropic works quite well in the analytical calculation of the largest lyapunov exponent furthermore the fpu ensuremath beta model is used as an example to show that hlms exist in hamiltonian lattice models with continuous symmetries some measures are defined to indicate the significance of hlms numerical simulations demonstrate that there is a smooth transition in the energy density dependence of these variables corresponding to the crossover in lyapunov exponents at the sst in particular our numerical results indicate that strong chaos is essential for the appearance of hlms and those modes become more significant with increasing degree of chaoticity

exit problems associated with affine reflection groups we obtain a formula for the distribution of the first exit time of brownian motion from the alcove of an affine weyl group in most cases the formula is expressed compactly in terms of pfaffians expected exit times are derived in the type widetilde a case the results extend to other markov processes we also give formulas for the real eigenfunctions of the dirichlet and neumann laplacians on alcoves observing that the hot spots conjecture of j rauch is true for alcoves

invariant region preserving dg methods for multi dimensional hyperbolic conservation law systems with an application to compressible euler equations abstract an invariant region preserving irp limiter for multi dimensional hyperbolic conservation law systems is introduced as long as the system admits a global invariant region which is a convex set in the phase space it is shown that the order of approximation accuracy is not destroyed by the irp limiter provided the cell average is away from the boundary of the convex set moreover this limiter is explicit and easy for computer implementation a generic algorithm incorporating the irp limiter is presented for high order finite volume type schemes for arbitrarily high order discontinuous galerkin dg schemes to hyperbolic conservation law systems sufficient conditions are obtained for cell averages to remain in the invariant region provided the projected one dimensional system shares the same invariant region as the full multi dimensional hyperbolic system does the general results are then applied to both one and two dimensional compressible euler equations so to obtain high order irp dg schemes numerical experiments are provided to validate the proven properties of the irp limiter and the performance of irp dg schemes for compressible euler equations

riemannian geometry of hamiltonian chaos hints for a general theory we aim at assessing the validity limits of some simplifying hypotheses that within a riemmannian geometric framework have provided an explanation of the origin of hamiltonian chaos and have made it possible to develop a method of analytically computing the largest lyapunov exponent of hamiltonian systems with many degrees of freedom therefore a numerical hypotheses testing has been performed for the fermi pasta ulam beta model and for a chain of coupled rotators these models for which analytic computations of the largest lyapunov exponents have been carried out in the mentioned riemannian geometric framework appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi isotropy of the mechanical manifolds sometimes breaks down the breakdown is expected whenever the topology of the mechanical manifolds is nontrivial this is an important step forward in view of developing a geometric theory of hamiltonian chaos of general validity

a solution to an open problem for wave equations with generalized wentzell boundary conditions in this paper we solve an open problem put forward by a favini g r goldstein j a goldstein and s romanelli concerning the mixed problem for wave equations with generalized wentzell boundary conditions as a consequence we also develop the previous wellposedness result regarding the mixed problem for heat equations with generalized wentzell boundary conditions

minimal critical exponent of quasiperiodic words abstract a quasiperiodic word is a finite or infinite word that can be obtained by concatenations or overlaps of a given finite word we proved that the least critical exponent of a quasiperiodic binary word is 7 3 for larger alphabets given any 0 we provide a construction of a ternary quasiperiodic word with critical exponent 2

second order boltzmann schemes for compressible euler equations in one and two space dimensions a class of second order numerical schemes for the compressible euler equations is described and their l 1 stability i e rho geqq 0 t geqq 0 is proved following van leer s approach the solution rho u sqrt t here is represented as piecewise linear functions the necessity of a slope limitation appears naturally in the derivation of the schemes but it can be less strict than the slope reconstructions usually used these schemes are written in terms of explicit flux splitting formula and are naturally multidimensional in space the upwinding is obtained through a very generalized notion of characteristics the kinetic one

hydrodynamic lyapunov modes and strong stochasticity threshold in the dynamic xy model an alternative scenario crossover from weak to strong chaos in high dimensional hamiltonian systems at the strong stochasticity threshold sst was anticipated to indicate a global transition in the geometric structure of phase space our recent study of fermi pasta ulam models showed that corresponding to this transition the energy density dependence of all lyapunov exponents is identical apart from a scaling factor the current investigation of the dynamic xy model discovers an alternative scenario for the energy dependence of the system dynamics at ssts though similar in tendency the lyapunov exponents now show individually different energy dependencies except in the near harmonic regime such a finding restricts the use of indices such as the largest lyapunov exponent and the ricci curvatures to characterize the global transition in the dynamics of high dimensional hamiltonian systems these observations are consistent with our conjecture that the quasi isotropy assumption works well only when parametric resonances are the dominant sources of dynamical instabilities moreover numerical simulations demonstrate the existence of hydrodynamical lyapunov modes hlms in the dynamic xy model and show that corresponding to the crossover in the lyapunov exponents there is also a smooth transition in the energy density dependence of significance measures of hlms in particular our numerical results confirm that strong chaos is essential for the appearance of hlms

geometric approach to chaos in the classical dynamics of abelian lattice gauge theory a riemannian geometrization of dynamics is used to study chaoticity in the classical hamiltonian dynamics of a u 1 lattice gauge theory this approach allows one to obtain analytical estimates of the largest lyapunov exponent in terms of time averages of geometric quantities these estimates are compared with the results of numerical simulations and turn out to be very close to the values extrapolated for very large lattice sizes even when the geometric quantities are computed using small lattices the scaling of the lyapunov exponent with the energy density is found to be well described by the law

quantitative analysis of financial system fragility based on manifold curvature abstract manifold curvature is an intrinsic attribute of a dynamical system by analyzing curvature characteristics of embedding attractor manifold resistance and resilience of financial dynamic system are investigated in the face of disturbances thus system fragility is discussed quantitatively from a systemic perspective first financial time series data are reconstructed into a high dimensional phase space topologically equivalent to its original dynamical system then manifold learning is employed to extract attractor manifold embedding in financial dynamic system last by numerical calculation the manifold curvature is obtained and further visualized with spatial mapping in empirical study shanghai composite index and dow jones industrial index are adopted as experimental data and then a more comprehensive understanding about financial systems fragility is obtained by exploring the dynamical properties related to manifold curvature

classes of representable disjoint np pairs for a propositional proof system p we introduce the complexity class dnpp p of all disjoint np pairs for which the disjointness of the pair is efficiently provable in the proof system p we exhibit structural properties of proof systems which make canonical np pairs associated with these proof systems hard or complete for dnpp p moreover we demonstrate that non equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for dnpp p and the reductions between the canonical pairs exist

on dejean s conjecture over large alphabets the maximal exponent of a non empty finite word is the ratio of its length to its period dejean 1972 conjectured that for any n 5 there exists an infinite word over n letters with no factor of its exponent larger than n n 1 we prove that this conjecture is true for n 33

a nonterminating 8 φ 7 summation for the root system c r using multiple q integrals and a determinant evaluation we establish a nonterminating 8 7 summation for the root system cr we also give some important specializations explicitly

infinite horizon backward stochastic differential equations and elliptic equations in hilbert spaces solutions of semilinear elliptic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations the backward equation is considered on an infinite time horizon and a suitable growth condition replaces the final condition elliptic equations are intended in a mild sense suitable also for applications to optimal control we finally notice that due to the lack of smoothing properties the elliptic partial differential equation considered here could not be treated by analytic methods

semiclassical chaos in atomic clusters an exploration with ring polymer molecular dynamics the ring polymer molecular dynamics rpmd method was recently shown to provide a powerful framework to address the time evolution of many particle semiclassical systems its underlying hamiltonian formulation is exploited here to define and compute some measures of deterministic chaos namely the lyapunov characteristic exponent applications are presented for weakly bound neon clusters with the aim of assessing the influence of nuclear delocalization on the nonlinear dynamics it is found that the balance between the additional degrees of freedom in the rpmd description and the harmonic nature of the additional interactions leads to more regular dynamics at high temperature but more chaotic dynamics at low temperature intrinsic features of quantum effects on the isomerization dynamics also have a signature on the lyapunov exponent

classical nonlinear response of a chaotic system i collective resonances we develop a general semiquantitative picture of nonlinear classical response in strongly chaotic systems in contrast to behavior in integrable or almost integrable systems the nonlinear classical response in chaotic systems vanishes at long times the exponential decay of the response functions in the case of strong chaos is attributed to both exponentially decaying and growing elements in the stability matrices we calculate the linear and second order response in one of the simplest chaotic systems free classical motion on a compact surface of constant negative curvature the response reveals certain features of collective resonances which do not correspond to any periodic classical trajectories we demonstrate the relevance of the model for the interpretation of spectroscopic experiments

double centralizing theorems for the alternating groups let v n be the n fold tensor product of a vector space v following i schur we consider the action of the symmetric group sn on v n by permuting coordinates in the super z2 graded case v v0 v1 a sign is added these actions give rise to the corresponding schur algebras s sn v here s sn v is compared with s an v the schur algebra corresponding to the alternating subgroup an sn while in the classical signless case these two schur algebras are the same for n large enough it is proved that in the super case where dim v0 dim v1 s an v is isomorphic to the cross product algebra s an v s sn v z2

a proof of dejean s conjecture we prove dejean s conjecture specifically we show that dejean s conjecture holds for the last remaining open values of n namely 15 n 26

repetiveness of languages generated by morphisms abstract we study the repetition of subwords in languages generated by morphisms fundamental to our approach is the notion of quasi repetitive elements using these elements we present a new characterization for repetitive morphisms from which we derive a simple proof for the fact that a d0l language is repetitive if and only if it is strongly repetitive ehrenfeucht and rozenberg inform and control 59 1983 13 35 from this proof we obtain a structurally simple polynomial time algorithm for deciding whether such a language is repetitive from further results on quasi repetitive elements we obtain as a consequence a complete characterization for all those morphisms on a two letter alphabet that are repetitive a result which is closely related to a result of seebold bull eatcs 36 1988 137 151 on the d0l periodicity problem finally we characterize those morphisms f on a two letter alphabet for which the language l f generated by f or the language sl f of subwords of l f are context free or even regular

chaos suppression in the large size limit for long range systems we consider the class of long range hamiltonian systems first introduced by anteneodo and tsallis and called the xy model this involves n classical rotators on a d dimensional periodic lattice interacting all to all with an attractive coupling whose strength decays as r r being the distance between sites using a recent geometrical approach we estimate for any d dimensional lattice the scaling of the largest lyapunov exponent lle with n as a function of in the large energy regime where rotators behave almost freely we find that the lle vanishes as n with 1 3 for 0 d 1 2 and 2 3 1 d for 1 2 d 1 these analytical results present a nice agreement with numerical results obtained by campa et al including deviations at small n

can chaotic quantum energy levels statistics be characterized using information geometry and inference methods abstract in this paper we review our novel information geometrodynamical approach to chaos igac on curved statistical manifolds and we emphasize the usefulness of our information geometrodynamical entropy ige as an indicator of chaoticity in a simple application furthermore knowing that integrable and chaotic quantum antiferromagnetic ising chains are characterized by asymptotic logarithmic and linear growths of their operator space entanglement entropies respectively we apply our igac to present an alternative characterization of such systems remarkably we show that in the former case the ige exhibits asymptotic logarithmic growth while in the latter case the ige exhibits asymptotic linear growth at this stage of its development igac remains an ambitious unifying information geometric theoretical construct for the study of chaotic dynamics with several unsolved problems however based on our recent findings we believe that it could provide an interesting innovative and potentially powerful way to study and understand the very important and challenging problems of classical and quantum chaos

bounds for the generalized repetition threshold the notion of the repetition threshold which is the object of dejean s conjecture 1972 was generalized by ilie et al 2005 8 to include the lengths of the avoided words we give a lower and an upper bound on this generalized repetition threshold

modular invariant of quantum tori the quantum modular invariant j qt of 2r is defined as a discontin uous pgl2 z invariant multi valued map using the distance to the nearest integer function k k for 2 q it is shown that j qt 1 and for quadratic irrationalities pari gp experiments suggest that j qt is a finite set in the case of the golden mean we produce explicit formulas involving weighted versions of the rogers ramanujan functions for the experimental supremum and infimum we then define a universal modular invariant j a mod c as a continuous and single valued map of ultrasolenoids such that 1 the classical modular invariant is a quotient of the re striction of j to a subsolenoid mod cl a mod fibering over the classical moduli space of elliptic curves and 2 the quantum modular invariant is a quotient of the restric tion of j to a subsolenoid mod qt a mod fibering over the moduli space of elliptic curves equipped with a kronecker foliation

jacobi fields on statistical manifolds of negative curvature abstract two entropic dynamical models are considered the geometric structure of the statistical manifolds underlying these models is studied it is found that in both cases the resulting metric manifolds are negatively curved moreover the geodesics on each manifold are described by hyperbolic trajectories a detailed analysis based on the jacobi equation for geodesic spread is used to show that the hyperbolicity of the manifolds leads to chaotic exponential instability a comparison between the two models leads to a relation among statistical curvature stability of geodesics and relative entropy like quantities finally the jacobi vector field intensity and the entropy like quantity are suggested as possible indicators of chaoticity in the ed models due to their similarity to the conventional chaos indicators based on the riemannian geometric approach and the zurek paz criterion of linear entropy growth respectively

invariant measures and regularity properties of perturbed ornstein uhlenbeck semigroups abstract this paper deals with perturbations of the ornstein uhlenbeck operator on l 2 spaces with respect to a gaussian measure we perturb the generator of the ornstein uhlenbeck semigroup by a certain unbounded non linear drift and show various properties of the perturbed semigroup such as compactness and positivity strong feller property existence and uniqueness of an invariant measure are discussed as well

entropy viscosity method for the single material euler equations in lagrangian frame a new finite element method for solving the euler equations in lagrangian coordinates is proposed the method is stabilized by adding artificial diffusion terms compatible with positivity of mass and internal energy a minimum principle on the specific entropy and all generalized entropy inequalities two options of first order artificial diffusion are considered one is in the spirit of the eulerian based method guermond et al 2011 23 22 zingan et al 2013 and the other is similar to existing viscosity stabilizations in lagrangian frame e g dobrev et al 2012 the method is verified to be high order for smooth solutions even with active viscosity terms this is achieved by using high order finite element spaces and an entropy based viscosity stabilization that degenerates the first order viscous terms this stabilization automatically distinguishes smooth and singular regions the formal accuracy and convergence properties of the proposed methods are tested on a series of benchmark problems this is the first result extending the entropy viscosity methodology to the lagrangian hydrodynamics

entropy flux splittings for hyperbolic conservation laws part i general framework a general framework is proposed for the derivation and analysis of flux splittings and the corresponding flux splitting schemes for systems of conservation laws endowed with a strictly convex entropy the approach leads to several new properties of the existing flux splittings and to a method for the construction of entropy flux splittings for general situations a large family of genuine entropy flux splittings is derived for several significant examples the scalar conservation laws the p system and the euler system of isentropic gas dynamics in particular for the isentropic euler system we obtain a family of splittings that satisfy the entropy inequality associated with the mechanical energy for this system it is proved that there exists a unique genuine entropy flux splitting that satisfies all of the entropy inequalities which is also the unique diagonalizable splitting this splitting can be also derived by the so called kinetic formulation simple and useful difference schemes are derived from the flux splittings for hyperbolic systems such entropy flux splitting schemes are shown to satisfy a discrete cell entropy inequality for the diagonalizable splitting schemes an a priori l estimate is provided by applying the principle of bounded invariant regions the convergence of entropy flux splitting schemes is proved for the 2 2 systems of conservation laws and the isentropic euler system 1995 john wiley sons inc

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

information geometric characterization of the complexity of fractional brownian motions the complexity of the fractional brownian motions is investigated from the viewpoint of information geometry by introducing a riemannian metric on the space of their power spectral densities the geometric structure is achieved based on the general construction for an example whose power spectral density is obtained by use of the normalized mexican hat wavelet we show its information geometric structures e g the dual connections the curvatures and the geodesics furthermore the instability of the geodesic spreads on this manifold is analyzed via the behaviors of the length between two neighboring geodesics the average volume element as well as the divergence or instability of the jacobi vector field finally the lyapunov exponent is obtained

controlling effect of geometrically defined local structural changes on chaotic hamiltonian systems an effective characterization of chaotic conservative hamiltonian systems in terms of the curvature associated with a riemannian metric tensor derived from the structure of the hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric the geodesic equations reproduce the hamilton equations of the original potential model through an inverse map in the tangent space the second covariant derivative of the geodesic deviation in this space generates a dynamical curvature resulting in energy dependent criteria for unstable behavior different from the usual lyapunov criteria we show here that this criterion can be constructively used to modify locally the potential of a chaotic hamiltonian model in such a way that stable motion is achieved since our criterion for instability is local in coordinate space these results provide a minimal method for achieving control of a chaotic system

the effect of microscopic correlations on the information geometric complexity of gaussian statistical models abstract we present an analytical computation of the asymptotic temporal behavior of the information geometric complexity igc of finite dimensional gaussian statistical manifolds in the presence of microcorrelations correlations between microvariables we observe a power law decay of the igc at a rate determined by the correlation coefficient it is found that microcorrelations lead to the emergence of an asymptotic information geometric compression of the statistical macrostates explored by the system at a faster rate than that observed in the absence of microcorrelations this finding uncovers an important connection between micro correlations and macro complexity in gaussian statistical dynamical systems

dejean s conjecture holds for n 30 abstract we extend carpi s results by showing that dejean s conjecture holds for n 30

avoidance bases for formulas with reversal abstract in the interest of studying formulas with reversal of high avoidability index we find n avoidance bases for formulas with reversal for n 1 2 3 we demonstrate that there is a unique formula with reversal in each of these three bases of highest avoidability index n 2 these formulas are xx x y x y r and x y z x y r z r which belong to an infinite family of formulas with reversal that has been the subject of recent study by the authors

repetition avoidance in products of factors we consider a variation on a classical avoidance problem from combinatorics on words that has been introduced by mousavi and shallit at dlt 2013 let texttt pexp i w be the supremum of the exponent over the products of i factors of the word w the repetition threshold texttt rt i k is then the infimum of texttt pexp i w over all words w in sigma omega k moussavi and shallit obtained that texttt rt i 2 2i and texttt rt 2 3 tfrac 13 4 we show that texttt rt i 3 tfrac 3i 2 tfrac14 if i is even and texttt rt i 3 tfrac 3i 2 tfrac16 if i is odd and i ge3

largest lyapunov exponent of long range xy systems abstract we calculate analytically the largest lyapunov exponent of the so called xy hamiltonian in the high energy regime this system consists of a d dimensional lattice of classical spins with interactions that decay with distance following a power law the range being adjustable in disordered regimes the lyapunov exponent can be easily estimated by means of the stochastic approach a theoretical scheme based on van kampen s cumulant expansion the stochastic approach expresses the lyapunov exponent as a function of a few statistical properties of the hessian matrix of the interaction that can be calculated as suitable microcanonical averages we have verified that there is a very good agreement between theory and numerical simulations

correction to maps of convex sets and invariant regions for finite difference systems of conservation laws properties p1 and p2 are trivially satisfied if o is compact we also observe that if the pair g o has either of the properties p1 and p2 then the pair ge o also has the same property for all e 0 where ge g ei the statement of the second part of theorem 4 in 1 should be changed to the following further if f u satisfies 9 and 10 with s u u then f is injective over f is convex and the inequality 11 holds for all u and in the outer normal cone of at provided that in addition one of the following is satisfied i u r and i p u m i for some m 0 ii u r and the pair fe r has the property p1 iii the pair fe has the properties p1 and p2 for all e 0 iv the pair fe has the properties p1 and p3 for all e 0 sufficiently small v the pair f has the properties p1 and p3

generation of localization in a discrete chain with periodic boundary conditions numerical and analytical results we study the generation of localization in a discrete chain composed of n subsystems and periodic boundary conditions strongly localized motions are studied that is time periodic motions where nearly all of the energy is spatially confined to a single subsystem for varying n numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in phase spatially extended solution however in the limit as n rightarrow infty the bifurcation point tends to infinity and a smooth transition from localization to nonlocalization occurs we then present an analytic technique to complement the numerical results it is based on the matching local asymptotic expansions of a solution branch using pade approximants this leads to global analytic representations of the considered solutions valid over the entire range of the control parameter

detecting order and chaos in three dimensional hamiltonian systems by geometrical methods we use a geometrical method to distinguish between ordered and chaotic motion in three dimensional hamiltonian systems we show that this method gives results in agreement with the computation of lyapunov characteristic exponents we discuss some examples of unstable hamiltonian systems in three dimensions giving as a particular illustration detailed results for a potential obtained from a hamiltonian obtained from a yang mills system

reexamination of an information geometric construction of entropic indicators of complexity abstract information geometry and inductive inference methods can be used to model dynamical systems in terms of their probabilistic description on curved statistical manifolds in this article we present a formal conceptual reexamination of the information geometric construction of entropic indicators of complexity for statistical models specifically we present conceptual advances in the interpretation of the information geometric entropy ige a statistical indicator of temporal complexity chaoticity defined on curved statistical manifolds underlying the probabilistic dynamics of physical systems

the fermi pasta ulam problem and its underlying integrable dynamics an approach through lyapunov exponents fpu models in dimension one are perturbations either of the linear model or of the toda model perturbations of the linear model include the usual beta model perturbations of toda include the usual alpha beta model in this paper we explore and compare two families or hierarchies of fpu models closer and closer to either the linear or the toda model by computing numerically for each model the maximal lyapunov exponent chi more precisely we consider statistically typical trajectories and study the asymptotics of chi for large n the number of particles and small varepsilon the specific energy e n and find for all models asymptotic power laws chi simeq c varepsilon a c and a depending on the model the asymptotics turns out to be in general rather slow and producing accurate results requires a great computational effort we also revisit and extend the analytic computation of chi introduced by casetti livi and pettini originally formulated for the beta model with great evidence the theory extends successfully to all models of the linear hierarchy but not to models close to toda

maps of convex sets and invariant regions for finite difference systems of conservation laws general results about maps of convex sets in r n are proved we outline their extensions to an infinite dimensional context such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps here we explore applications only in the finite dimensional context more specifically we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite difference schemes applied to systems of conservation laws in several space variables in particular we establish a final characterization of the invariant regions under the lax friedrichs scheme and also give sufficient conditions for the local invariance further we give sufficient conditions for the global and local invariance of regions under flux splitting finite difference schemes an example of the multi dimensional euler equations for non isentropic gas dynamics is discussed

fast estimation from above of the maximum wave speed in the riemann problem for the euler equations this paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the riemann solution for the euler system of gas dynamics with the co volume equation of state the novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy the convergence rate of the algorithm is cubic and the bound is guaranteed for gasses with the co volume equation of state and the heat capacity ratio in the range 1 5 3

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

asymptotic stability of riemann solutions in bgk approximations to certain multidimensional systems of conservation laws abstract we prove the asymptotic stability of nonplanar two states riemann solutions in bgk approximations of a class of multidimensional systems of conservation laws the latter consists of systems whose flux functions in different directions share a common complete system of riemann invariants the level surfaces of which are hyperplanes the asymptotic stability to which the main result refers is in the sense of the convergence as t in l loc 1 of the space of directions x t that is the solution z t x of the perturbed cauchy problem for the corresponding bgk system satisfies x z t t d r as t in l loc 1 r n where r is the self similar entropy solution of the two states nonplanar riemann problem for the system of conservation laws

new combinatorial interpretations of ramanujan s partition congruences mod 5 7 and 11 soit p n le nombre de partitions non restreintes de n on donne de nouvelles interpretations combinatoires des congruences p 5n 4 p 7n 5 et p 11n 6 0 mod 5 7 et 11 respectivement

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

proof of dejean s conjecture for alphabets with 5 6 7 8 9 10 and 11 letters abstract axel thue proved that overlapping factors could be avoided in arbitrarily long words on a two letter alphabet while on the same alphabet square factors always occur in words longer than 3 francoise dejean stated an analogous result for three letter alphabets every long enough word has a factor which is a fractional power with an exponent at least 7 4 and there exist arbitrary long words in which no factor is a fractional power with an exponent strictly greater than 7 4 the number 7 4 is called the repetition threshold of the three letter alphabets thereafter she proposed the following conjecture the repetition threshold of the k letter alphabets is equal to k k 1 except in the particular cases k 3 where this threshold is 7 4 and k 4 where it is 7 5 for k 4 this conjecture was proved by j j pansiot 1984 in this paper we give a computer aided proof of dejean s conjecture for several other values 5 6 7 8 9 10 and 11

some inequalities for the q beta and the q gamma functions via some q integral inequalities some new inequalities for the q gamma the q beta and the q analogue of the psi functions are established via some q integral inequalities

lyapunov exponent of many particle systems testing the stochastic approach the stochastic approach to the determination of the largest lyapunov exponent of a many particle system is tested in the so called mean field mathrm xy hamiltonians in weakly chaotic regimes the stochastic approach relates the lyapunov exponent to a few statistical properties of the hessian matrix of the interaction which can be calculated as suitable thermal averages we have verified that there is a satisfactory quantitative agreement between theory and simulations in the disordered phases of the mathrm xy models either with attractive or repulsive interactions part of the success of the theory is due to the possibility of predicting the shape of the required correlation functions because this permits the calculation of correlation times as thermal averages

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

quantifying the complexity of geodesic paths on curved statistical manifolds through information geometric entropies and jacobi fields abstract we characterize the complexity of geodesic paths on a curved statistical manifold m s through the asymptotic computation of the information geometric complexity v m s and the jacobi vector field intensity j m s the manifold m s is a 2 l dimensional gaussian model reproduced by an appropriate embedding in a larger 4 l dimensional gaussian manifold and endowed with a fisher rao information metric g with non trivial off diagonal terms these terms emerge due to the presence of a correlational structure embedding constraints among the statistical variables on the larger manifold and are characterized by macroscopic correlational coefficients r k first we observe a power law decay of the information geometric complexity at a rate determined by the coefficients r k and conclude that the non trivial off diagonal terms lead to the emergence of an asymptotic information geometric compression of the explored macrostates on m s finally we observe that the presence of such embedding constraints leads to an attenuation of the asymptotic exponential divergence of the jacobi vector field intensity

the x ray transform and its inversion for the series expansion basis functions in three dimensional tomography a formula for the three dimensional x ray transform and an expression for its inversion have been derived the spherical harmonics and the rotation matrices in terms of euler angles are used in the generalized fourier series expansion some properties of the transformed fourier components are discussed

a numerical method for systems of conservation laws of mixed type admitting hyperbolic flux splitting the present treatment of elliptic regions via hyperbolic flux splitting and high order methods proposes a flux splitting in which the corresponding jacobians have real and positive negative eigenvalues while resembling the flux splitting used in hyperbolic systems the present generalization of such splitting to elliptic regions allows the handling of mixed type systems in a unified and heuristically stable fashion the van der waals fluid dynamics equation is used convergence with good resolution to weak solutions for various riemann problems are observed

generalized lyapunov exponents of the random harmonic oscillator cumulant expansion approach the cumulant expansion is used to estimate generalized lyapunov exponents of the random frequency harmonic oscillator three stochastic processes are considered gaussian white noise ornstein uhlenbeck and poisson shot noise in some cases nontrivial numerical difficulties arise these are mostly solved by implementing an appropriate importance sampling monte carlo scheme we analyze the relation between random frequency oscillators and many particle systems with pairwise interactions like the lennard jones gas

on the number of dejean words over alphabets of 5 6 7 8 9 and 10 letters we give lower bounds on the growth rate of dejean words i e minimally repetitive words over a k letter alphabet for 5 k 10 put together with the known upper bounds we estimate these growth rates with the precision of 0 005 as a consequence we establish the exponential growth of the number of dejean words over a k letter alphabet for 5 k 10

last cases of dejean s conjecture dejean conjectured that the repetition threshold for a k letter alphabet is kk 1 when k 5 dejean s conjecture has already been proved for k 14 and for k 27 we present here a proof for 8 k 38 the same technique is also applied to prove ochem s stronger version of the conjecture for 9 k 38

an improved sectional anosov closing lemma we prove that all nonwandering points of a sectional anosov flow on a compact 3 manifold can be approximated by periodic points or by points for which the omega limit set is a singularity this improves the closing lemma in morales mich math j 56 1 29 53 2008 we also describe a sectional anosov flow for which the recurrent points are not dense in the nonwandering set

n point pade approximants to real valued stieltjes series with nonzero radii of convergence abstract by employing special continued fractions to two stieltjes series with nonzero radii of convergence we extend the inequalities for one point pade approximants reported by baker 1975 corollory 17 1 to the case of two point pade approximants we prove that some convergents of the continued fractions form a monotone sequences of upper and lower bounds converging uniformly to stieltjes function x1 x on compact subsets of r where r is a radius of convergence of an expansion of x f1 x at x 0 for an illustration of theoretical results we provide nontrivial numerical examples as an application to real physical problems second order pade approximants bounds on the effective conductivity of a square array of cylinders are evaluated

letter frequency in infinite repetition free words we estimate the extremal letter frequency in infinite words over a finite alphabet avoiding some repetitions for ternary square free words we improve the bounds of tarannikov on the minimal letter frequency and prove that the maximal letter frequency is 255653 kolpakov et al have studied the function r such that r x is the minimal letter frequency in an infinite binary x free word in particular they have shown that r is discontinuous at 73 and at every integer at least 3 we answer one of their questions by providing some other points of discontinuity for r finally we propose stronger versions of dejean s conjecture on repetition threshold in which unequal letter frequencies are required

applications of geometrical criteria for transition to hamiltonian chaos using a recently developed geometrical method we study the transition from order to chaos in an important class of hamiltonian systems we show agreement between this geometrical method and the surface of section technique applied to detect chaotic behavior we give as a particular illustration detailed results for an important class of potentials obtained from the perturbation of an oscillator hamiltonian by means of higher order polynomials

boltzmann type schemes for gas dynamics and the entropy property let h be a convex function a boltzmann equation is built following the b g k model for which h is an entropy in the fluid limit the compressible euler equations are obtained with a convex entropy naturally associated to h the motivation is to define numerical boltzmann schemes with finite speed of propagation corresponding to an equilibrium density function with bounded support the classical scheme obtained with h f f log f is based on the equilibrium density function exp v 2 2 which has infinite support

second order invariant domain preserving ale approximation of hyperbolic systems abstract in this paper we introduce an invariant domain preserving arbitrary lagrangian eulerian method for solving hyperbolic systems the time stepping is explicit and the approximation in space is done with continuous finite elements the method is second order in space and made invariant domain preserving by using a newly introduced convex limiting technique

on modular homology in the boolean algebra iii abstract let be a set ra ring of characteristicp 0 and denote bymkther module withk element subsets of as basis theset inclusion map mk mk 1is the homomorphism which associates to ak element subset the sum 1 2 kof all its k 1 element subsets i in this paper we study the chain formula arising from we introduce the notion ofp exactness for a sequence if is infinite we show that isp exact for all prime characteristicsp 0 this result can be extended to various submodules and quotient modules and we give general constructions arising from permutation groups with a finitary section two particular applications are the following the orbit module sequence of such a permutation group on isp exact for every primep and we give a formula for thep rank of the orbit inclusion matrix if the group has finitely many orbits onk element subsets

invariant regions for systems of conservation laws on decrit des conditions necessaires et suffisantes pour qu une region dans r n soit invariante pour des solutions du systeme de n lois de conservations u t f u x o