non expandable non overlapping sets of pictures the non overlapping sets of pictures are sets such that no two pictures in the set properly overlap they are the generalization to two dimensions of the cross bifix free sets of strings non overlapping sets of pictures are non expandable when no other picture can be added without violating the property we propose a general construction method for non expandable non overlapping neno sets based on some structural properties of neno sets as an application we show a first example of a family of neno sets

homoclinic orbits on compact hypersufaces in 293 1293 1293 1 of restricted contact type consider a smooth hamiltonian system in 2n dot x jh x the energy surface x h x h 0 being compact and 0 being a hyperbolic equilibrium we assume moreover that 0 is of restricted contact type these conditions are symplectically invariant by a variational method we prove the existence of an orbit homoclinic i e non constant and doubly asymptotic to 0

on nonautonomous dirac equation we construct the fundamental solution of time dependent linear ordinary dirac system in terms of unknown phase functions this construction gives approximate representation of solutions which is useful for the study of asymptotic behavior introducing analog of rayleigh quotient for differential equations we generalize hartman wintner asymptotic integration theorems with the error estimates for applications to the dirac system we also introduce the adiabatic invariants for the dirac system which are similar to the adiabatic invariant of lorentz s pendulum using a small parameter method it is shown that the change in the adiabatic invariants approaches zero with the power speed as a small parameter approaches zero as another application we calculate the transition probabilities for the dirac system we show that for the special choice of electromagnetic field the only transition of an electron to the positron with the opposite spin orientation is possible

nekhoroshev theorem for small amplitude solutions in nonlinear schrodinger equations we prove a nekhoroshev type result 26 27 for the nonlinear schrodinger equation with vanishing or periodic boundary conditions on 0 pi here m in mathbb r is a parameter and varphi mathbb r to mathbb r is a function analytic in a neighborhood of the origin and such that varphi 0 0 varphi 0 not 0 more precisely we consider the cauchy problem for 0 1 with initial data which extend to analytic entire functions of finite order and prove that all the actions of the linearized system are approximate constants of motion up to times growing faster than any negative power of the size of the initial datum the proof is obtained by a method which applies to hamiltonian perturbations of linear pde s with the following properties i the linear dynamics is periodic ii there exists a finite order birkhoff normal form which is integrable and quasi convex as a function of the action variables eq 0 1 satisfies i and ii when restricted to a level surface of left vert u l 2 right vert which is an integral of motion the main technical tool used in the proof is a normal form lemma for systems with symmetry which is also proved here

an averaging theorem for hamiltonian dynamical systems in the thermodynamic limit it is shown how to perform some steps of perturbation theory if one assumes a measure theoretic point of view i e if one renounces to control the evolution of the single trajectories and the attention is restricted to controlling the evolution of the measure of some meaningful subsets of phase space for a system of coupled rotators estimates uniform in n for finite specific energy can be obtained in quite a direct way this is achieved by making reference not to the sup norm but rather following koopman and von neumann to the much weaker l2 norm

some properties of wkb series we investigate some properties of the wkb series for arbitrary analytic potentials and then specifically for potentials xn n even where more explicit formulae for the wkb terms are derived our main new results are i we find the explicit functional form for the general wkb terms k where one has only to solve a general recursion relation for the rational coefficients ii we give a systematic algorithm for an essential simplification of the integrated wkb terms k dx that enter the energy eigenvalue equation and iii we derive almost explicit formulae for the wkb terms for the energy eigenvalues of the homogeneous power law potentials v x xn where n is even

nonconvergence of formal integrals ii improved estimates for the optimal order of truncation we investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 dof hamiltonian systems in particular we study the dependence of the optimal order of truncation nopt on the distance from the elliptic equilibrium by numerical and analytical means the function nopt determines the region of nekhoroshev stability of the orbits and the time of practical stability we find that the function nopt decreases by abrupt steps the decrease is roughly approximated with an average power law nopt o a with a 1 we find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via lie series and in the direct construction of first integrals precisely we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity our results agree with those obtained by servizi g et al 1983 phys lett a 95 11 for a conservative map of the plane moreover our analytical considerations allow us to explain the results of our previous paper contopoulos g et al 2003 j phys a math gen 36 8639 including in particular the different behaviour observed for low order and higher order resonances

the steep nekhoroshev s theorem revising nekhoroshev s geometry of resonances we provide a fully constructive and quantitative proof of nekhoroshev s theorem for steep hamiltonian systems proving in particular that the exponential stability exponent can be taken to be 1 2n alpha 1 cdots alpha n 2 alpha i s being nekhoroshev s steepness indices and n ge 3 the number of degrees of freedom on the base of a heuristic argument we conjecture that the new stability exponent is optimal

stability properties of the riemann ellipsoids the riemann ellipsoids are steady motions of an ideal incompressible self gravitating fluid that retain an ellipsoidal shape the existence properties and stability of these steady motions have been investigated since newton s time most of the stability results are due to riemann who studied lyapunov stability and to chandrasekhar who studied primarily numerically spectral stability thus obtaining lyapunov instability results this article addresses the nekhoroshev stability stability for finite but very long time scales of those riemann ellipsoids that are spectrally stable but of unknown lyapunov stability we base our analysis on a hamiltonian formulation of the problem derived from riemann s original formulation which we interpret here as a formulation on a covering space using recent results from hamiltonian perturbation theory given the complexity of the system we resort to numerical calculations at certain steps of the stability analysis as a prerequisite to our analysis we repeat the lyapunov and spectral stability analyses finding important discrepancies with chandrasekhar s findings we provide numerical evidence that

long time stability estimates for the non periodic littlewood boundedness problem we consider the nonlinear oscillator equation for 3 and non periodic forcing p for any solution x which is unbounded in the phase plane it is shown that for e 0 small enough there is a solution x e which is bounded in the phase plane and such that the actions of x e and x remain close on a time interval of length e 2 further precise information is available on the location of the zeros of x e in addition it is possible to take the bounded solution x e from a fixed countable family of such solutions

cross bifix free sets in two dimensions a bidimensional bifix in short bibifix of a square matrix t is a square submatrix of t which occurs in the top left and bottom right corners of t this allows us to extend the definition of bifix free words and cross bifix free set of words to bidimensional structures in this paper we exhaustively generate all the bibifix free square matrices and we construct a particular non expandable cross bibifix free set of square matrices moreover we provide a gray code for listing this set

wkb approach applied to 1d time dependent nonlinear hamiltonian oscillators we present a simple wkb like approach to obtain approximate analytic solutions to a certain class of time dependent nonlinear 1d hamiltonian oscillators the case of homogeneous power law potentials is solved explicitly in a closed form in the leading order the accuracy of the approximation is surprisingly good and we illustrate it in the case of the quartic oscillator

the hamiltonian p2 m2 1 2 α r near the critical value αc 2 π we elucidate the behavior of the operator p2 m2 1 2 r near the critical value c where it ceases to be bounded below by obtaining a family h z of operators which is self adjoint holomorphic in a domain including all real z c c and such that h c c is just the operator p2 m2 1 2 r or its friedrich extension while h c c c we show that these eigenvalues cannot vanish but that the lowest eigenvalue of h c goes to zero when c the l 0 eigenvalues are analytic in at c

super exponential stability for generic real analytic elliptic equilibrium points we consider the dynamics in a neighborhood of an elliptic equilibrium point with a diophantine frequency of a symplectic real analytic vector field and we prove the following result of effective stability generically both in a topological and measure theoretical sense any solution starting sufficiently close to the equilibrium point remains close to it for an interval of time which is doubly exponentially large with respect to the inverse of the distance to the equilibrium point

estimates on invariant tori near an elliptic equilibrium point of a hamiltonian system abstract we give a precise statement for the kam theorem in a neighbourhood of an elliptic equilibrium point of a hamiltonian system if the frequencies of the elliptic point are nonresonant up to a certain order k 4 and a nondegeneracy condition is fulfilled we get an estimate for the measure of the complement of the kam tori in a neighbourhood of given radius moreover if the frequencies satisfy a diophantine condition with exponent we show that in a neighbourhood of radius r the measure of the complement is exponentially small in 1 r 1 1 we also give a related result for quasi diophantine frequencies which is more useful for practical purposes the results are obtained by putting the system in birkhoff normal form up to an appropiate order and the key point relies on giving accurate bounds for its terms

cross bifix free codes within a constant factor of optimality a cross bifix free code is a set of words in which no prefix of any length of any word is the suffix of any word in the set cross bifix free codes arise in the study of distributed sequences for frame synchronization we provide a new construction of cross bifix free codes which generalizes the construction by bajic to longer code lengths and to any alphabet size the codes are shown to be nearly optimal in size we also establish new results on fibonacci sequences which are used in estimating the size of the cross bifix free codes

variational methods in relativistic quantum mechanics this review is devoted to the study of stationary solutions of lin ear and nonlinear equations from relativistic quantum mechanics involving the dirac operator the solutions are found as critical points of an energy func tional contrary to the laplacian appearing in the equations of nonrelativistic quantum mechanics the dirac operator has a negative continuous spectrum which is not bounded from below this has two main consequences first the energy functional is strongly indefinite second the euler lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap between the negative and positive continuous spectra moreover since we work in the space domain r 3 the palais smale condition is not satisfied for these reasons the problems discussed in this review pose a challenge in the calculus of variations the existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems in the first part we consider the fixed eigenvalue problem for models of a free self interacting relativistic particle they allow to describe the localized state of a spin 1 2 particle a fermion which propagates without changing its shape this includes the soler models and the maxwell dirac or klein gordon dirac equations the second part is devoted to the presentation of min max principles al lowing to characterize and compute the eigenvalues of linear dirac operators with an external potential in the gap of their essential spectrum many con sequences of these min max characterizations are presented among them a new kind of hardy like inequalities and a stable algorithm to compute the eigenvalues in the third part we look for normalized solutions of nonlinear eigenvalue problems the eigenvalues are lagrange multipliers lying in a spectral gap we review the results that have been obtained on the dirac fock model which is a nonlinear theory describing the behavior of n interacting electrons in an external electrostatic field in particular we focus on the problematic definition of the ground state and its nonrelativistic limit in the last part we present a more involved relativistic model from quan tum electrodynamics in which the behavior of the vacuum is taken into ac count it being coupled to the real particles the main interesting feature of this model is that the energy functional is now bounded from below providing us with a good definition of a ground state

the spectrum of relativistic one electron atoms according to bethe and salpeter bethe and salpeter introduced a relativistic equation different from the bethe salpeter equation which describes relativistic multi particle systems here we will begin some basic work concerning its mathematical structure in particular we show self adjointness of the one particle operator which will be a consequence of a sharp sobolev type inequality yielding semi boundedness of the corresponding sesquilinear form moreover we locate the essential spectrum of the operator and show the absence of singular continuous spectrum

nekhoroshev type stability results for hamiltonian systems with an additional transversal component abstract we prove exponential stability theorems of nekhoroshev type for motion in the neighbourhood of an elliptic fixed point in hamiltonian systems having an additional transverse component of arbitrary dimension under certain conditions on this transverse component we consider both the cases i of a strongly constrained motion and ii of a weak perturbation

superadiabatic tracking of quantum evolution the adiabatic quantum evolution of a two state system without energy level crossings is an example of the stokes phenomenon in the latter a small subdominant exponential is an asymptotic expansion appears when a stokes line is crossed truncating the dominant asymptotic series at its least term causes the multiplier of the subdominant term to rise in a smooth compact and universal manner across the stokes line in quantum evolution this corresponds to a smooth transition universal in form between superadiabatic basis states high order wkb approximate solutions of the time dependent schrodinger equation the authors give a numerical demonstration of this previously predicted universality by constructing for two hamiltonians the superadiabatic quantum bases asymptotic to the actual evolving state universality when a stokes line is crossed is seen in the changing probability that the system makes a transition away from the superadiabatic state and occurs at that order of superadiabatic approximation corresponding to truncating the asymptotic series at its least term

stability of some waves in the dissipative boussinesq system abstract in this paper we study analytically a class of waves in the variant of the classical dissipative boussinesq system given by t u x v x x x v x x t u x u v t v x u c x x x u x x t v v x v where c 0 is a small parameter and 0 1 this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every c and almost every 1 it contains solutions that are defined for large values of time and they are very close of order o to a linear torus for long times of order o 1 the proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates

asymptotic analysis of a class of three degree of freedom hamiltonian systems near stable equilibria abstract a conservative near integrable hamiltonian dynamical system is examined which to leading order consists of three uncoupled harmonic oscillators with constant frequencies in the ratio 1 2 for certain rational formally the problem considered can arise by perturbing any three degree of freedom hamiltonian near a stable equilibrium point so that the hamiltonian consists of a power series expansion in a small parameter where successive terms are homogeneous polynomials of increasing degree in the coordinates and the momenta the special case of two exact simultaneous resonances one in the first perturbation term and one in the second is examined and explicit asymptotic solutions are obtained the solution procedure involves reducing the original hamiltonian to two degrees of freedom using one integral of the motion then transforming to standard form to find two additional adiabatic invariants by near identity averaging canonical transformations a specific example is studied numerically to verify the asymptotic validity of the results over long times

on the problem of energy equipartition for large systems of the fermi pasta ulam type analytical and numerical estimates abstract we report on some analytical and numerical results on the exchanges of energy in systems of the fermi pasta ulam type in the light of nekhoroshev s theorem with particular attention to the dependence of the estimates on the number n of degrees of freedom for the ordinary fpu problem we look for a control of the single normal mode energies and we find both the analytical and numerical estimates to agree in predicting that the energy exchanges of the single modes cannot be controlled in the thermodynamic limit we consider then a modified fpu model with alternating light and heavy particles which appears as composed of two subsystems of low acoustic frequency and of high optical frequency respectively we try to control the exchange of the total energy of the high frequency modes up to times increasing exponentially with the frequency in this case the numerical estimates are stronger than the available analytical ones and give indications for nonequipartition with constants essentially independent of the number n of degrees of freedom

an averaging technique for highly oscillatory hamiltonian problems in this paper we are concerned with the numerical solution of highly oscillatory hamiltonian systems with a stiff linear part we construct an averaged system whose solution remains close to the exact one over bounded time intervals possesses the same adiabatic and hamiltonian invariants as the original system and is nonstiff we then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in e cances et al numer math 100 2005 pp 211 232 eventually we demonstrate the efficiency of our approach on two test problems with one or several frequencies

on the eigenvalues of operators with gaps application to dirac operators this paper is devoted to a general min max characterization of the eigenvalues in a gap of the essential spectrum of a self adjoint unbounded operator we prove an abstract theorem then we apply it to the case of dirac operators with a coulomb like potential the result is optimal for the coulomb potential

extensive adiabatic invariants for nonlinear chains we look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit considering the quadratic part of the klein gordon hamiltonian by a linear change of variables we transform it into a sum of two parts in involution at variance with the usual method of introducing normal modes our constructive procedure allows us to exploit the complete resonance while keeping the extensive nature of the system next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear schrodinger model the fluctuations of this quantity are controlled via gibbs measure estimates independent of the system size for a large set of initial data at low specific energy finally by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory with fluctuation much smaller than expected according to standard statistical estimates

classical freezing of plane rotations a proof of the boltzmann jeans conjecture using simple known methods and results of classical perturbation theory especially those due to nekhoroshev and neishtadt we study the energy exchanges between the rotational and the translational degrees of freedom in a particular model representing the planar motion of a rigid body in a bounded analytic potential we prove that if the angular velocity co is initially large then the energy exchanges are small o co 1 for times growing exponentially with co ill exp co we also deduce that in a scattering process from a smooth potential barrier the overall change in the rotational energy of the incoming body is exponentially small in co g exp co the results are interpreted in the light of an old conjecture by boltzmann and jeans on the existence of very large time scales for equilibrium in statistical systems containing high frequency degrees of freedom purely classical freezing of the high frequency degrees of freedom the rotating object is in this interpretation a classical molecule which moves in an external field or collides with the wall of a container two different limits of large co are considered namely the limit of large rotational energy and as is interesting for the molecular interpretation the limit of point mass at finite rotational energy

non overlapping codes we say that a q ary length n code is non overlapping if the set of non trivial prefixes of codewords and the set of non trivial suffices of codewords are disjoint these codes were first studied by levenshtein in 1964 motivated by applications in synchronization more recently these codes were independently invented under the name cross bifix free codes by bajic and stojanovic we provide a simple construction for a class of non overlapping codes which has optimal cardinality whenever n divides q moreover for all parameters n and q we show that a code from this class is close to optimal in the sense that it has cardinality within a constant factor of an upper bound due to levenshtein from 1970 previous constructions have cardinality within a constant factor of the upper bound only when q is fixed chee kiah purkayastha and wang showed that a q ary length n non overlapping code contains at most q n 2n 1 codewords this bound is weaker than the levenshtein bound their proof appealed to the application in synchronization we provide a direct combinatorial argument to establish the bound of chee et al we also consider codes of short length finding the leading term of the maximal cardinality of a non overlapping code when n is fixed and q rightarrow infty the largest cardinality of non overlapping codes of lengths 3 or less is determined exactly

simple dynamical system with discrete bound states we study numerically the dynamical system of a two electron atom with the darwin interaction as a model to investigate scale dependent effects of the relativistic action at a distance electrodynamics this dynamical system consists of a small perturbation of the coulomb dynamics for energies in the atomic range the key properties of the coulomb dynamics are i a peculiar mixed type phase space with sparse families of stable nonionizing orbits and ii scale invariance symmetry with all orbits defined by an arbitrary scale parameter the combination of this peculiar chaotic dynamics i and ii with the scale dependent relativistic corrections darwin interaction generates the phenomenon of scale dependent stability we find numerical evidence that stable nonionizing orbits can exist only for a discrete set of resonant energies the fourier transform of these nonionizing orbits is a set of sharp frequencies the energies and sharp frequencies of the nonionizing orbits we study are in the quantum atomic range

non trivial prefactors in adiabatic transition probabilities induced by high order complex degeneracies the adiabatic limit of the transition probability for two level systems driven by real symmetric time dependent hamiltonians is considered when the hamiltonian depends analytically on time the transition probability is governed in simple cases by a complex eigenvalue crossing point which is generically a square root branching point for the eigenvalues in such a case the transition probability is given by the dykhne formula the recently discovered geometric prefactor being equal to one in this paper we deal with the general situation where the relevant eigenvalue crossing point is a branching point of order n 2 n or 1 for the eigenvalues and a zero of order m or 0 for the hamiltonian itself the analysis shows that the dykhne formula must be completed by a novel prefactor which depends on both n and m in particular this prefactor can take the value zero in contrast to the geometrical prefactor we also consider the case where the transition probability is governed by n complex eigenvalue crossing points of different orders nj and mj j 1 n the end result displays an interference phenomenon between the individual prefactors similar to the case of generic eigenvalue crossing points nj 1 and mj 0 considered earlier

smoothed langevin dynamics of highly oscillatory systems abstract we consider hamiltonian systems containing a strong constraining potential such systems were first investigated by rubin and ungar who derived the corresponding reduced equations of motion for the slow degree of freedom the purpose of this paper is two fold i we transform the highly oscillatory degree of freedom to action angle variables and use recent results on the truncation error of the corresponding normal form transformations to derive an estimate on the lifespan of the underlying adiabatic invariant ii we investigate thermally embedded hamiltonian systems i e langevin dynamics containing a strong constraining potential using again a transformation to action angle variables and applying the principle of averaging we derive reduced equations of motion for the slowly varying degrees of freedom this includes in particular the derivation of a correcting force term that stands for the coupling of the slow and fast degrees of motion in case of strong thermal coupling our reduced equations of motion suggest that the correcting force term can be approximated by the gradient of the fixman potential as used in statistical mechanics we also discuss smoothing in the context of constant temperature molecular dynamics and provide a numerical example

geometry of arnold diffusion the goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of arnold diffusion we choose to speak of a specific model that of geometric rays in a periodic optical medium this model is equivalent to that of a particle in a periodic potential in mathbb r n with energy prescribed and to the geodesic flow in a riemannian metric on mathbb r n

arnold diffusion in a priori chaotic symplectic maps we assume that a symplectic real analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder we prove that generically in the real analytic category the boundaries of the invariant cylinder are connected by trajectories of the map

semiclassically weak reflections above analytic and non analytic potential barriers the coefficient r for reflection above a barrier v x is computed semiclassically i e as h cross to 0 employing an exact multiple reflection series whose mth term is a 2m 1 fold integral if v x is analytic all terms have the same semiclassical order exp h cross 1 the multiple integrals are evaluated exactly and the series summed if v x has a discontinuous nth derivative the term m 1 dominates semiclassically and gives r approximately h cross n if v x has all derivatives continuous but possesses an essential singularity on the real axis the term m 1 again dominates semiclassically and for v approximately exp mod x mod n gives r approximately exp h cross n n 1 with an oscillatory factor corresponding to transmission resonances the formulae are illustrated by computations of mod r mod 2 for four potentials with different continuity properties and show the limiting asymptotics emerging only when the de broglie wavelength is less than 1 of the barrier width and mod r mod 2 approximately 10 1000

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

counting bordered and primitive words with a fixed weight a word w is primitive if it is not a proper power of another word and w is unbordered if it has no prefix that is also a suffix of w we study the number of primitive and unbordered words w with a fixed weight that is words for which the parikh vector of w is a fixed vector moreover we estimate the number of words that have a unique border

a property of exponential stability in nonlinear wave equations near the fundamental linear mode abstract we consider the nonlinear wave equation utt c2uxx u u t 0 u t 0 where is an analytic function satisfying 0 0 and 00 0 for each value of the harmonic energy e i e energy when 0 we consider the closed curve e which is the phase space trajectory of the linear mode with lowest frequency if the perturbation parameter and the harmonic energy e are small enough then near e we construct by means of a suitable simplified equation a closed curve e with the following property provided the distance between the initial datum and such a curve is small enough the corresponding solution remains close to e up to times exponentially long with 1 eve cannot expect the curves e to be trajectories of solutions of such curves depend smoothly on the parameters e and the above result is deduced from a general theorem on generic hamiltonian perturbations of completely resonant linear systems

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

long term stability of proper rotations of the perturbed euler rigid body we study the long term stability of the proper rotations of the perturbed euler rigid body in the framework of nekhoroshev theory for simplicity we treat here in detail only the kinetically symmetric case the potential needs not to be symmetric but we indicate how to extend the results to the triaxial case we show that the proper rotations around the symmetry axis are nekhoroshev stable more precisely if the initial datum is sufficiently close to a proper rotation then for a very long time it remains such and the tip of the unit vector parallel to the angular momentum precesses up to a small noise along the level curves of a regular function on the unit sphere if the proper rotations are resonant chaotic motions with positive lyapunov exponents are possible but chaos unlike the case of ordinary motions that is motions not close to proper rotations is always localized and does not affect in an essential way the motion of the angular momentum in space preliminary numerical results indicate that the theory is in many aspects optimal although in some points it can still be improved

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

energy localization on q tori long term stability and the interpretation of fermi pasta ulam recurrences we focus on two approaches that have been proposed in recent years for the explanation of the so called fermi pasta ulam fpu paradox i e the persistence of energy localization in the low q fourier modes of fermi pasta ulam nonlinear lattices preventing equipartition among all modes at low energies in the first approach a low frequency fraction of the spectrum is initially excited leading to the formation of natural packets exhibiting exponential stability while in the second emphasis is placed on the existence of q breathers i e periodic continuations of the linear modes of the lattice which are exponentially localized in fourier space following ideas of the latter we introduce in this paper the concept of q tori representing exponentially localized solutions on low dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the fpu paradox

exact energy distribution function in a time dependent harmonic oscillator following a recent work by robnik and romanovski 2006 j phys a math gen 39 l35 2006 open syst inf dyn 13 197 222 we derive an explicit formula for the universal distribution function of the final energies in a time dependent 1d harmonic oscillator whose functional form does not depend on the details of the frequency t and is closely related to the conservation of the adiabatic invariant the normalized distribution function is where e1 is the final energy is its average value and 2 is the variance of e1 and 2 can be calculated exactly using the wkb approach to all orders

homogenization of hamiltonian systems with a strong constraining potential abstract the paper studies hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale in particular we are interested in the limit situation where the size of this small time scale tends to zero but the velocity components remain oscillating with an amplitude variation of the order o 1 the process of establishing an effective initial value problem for the limit positions will be called homogenization of the hamiltonian system this problem occurs in mechanics as the problem of realization of holonomic constraints as various singular limits in fluid flow problems in plasma physics as the problem of guiding center motion and in the simulation of biomolecules as the so called smoothing problem we suggest the systematic use of the notion of weak convergence in order to approach this problem this methodology helps to establish unified and short proofs of many known results which throw light on the inherent structure of the problem moreover we give a careful and critical review of the literature

stability of motions near resonances in quasi integrable hamiltonian systems nekhoroshev s theorem on the stability of motions in quasi integrable hamiltonian systems is revisited at variance with the proofs already available in the literature we explicitly consider the case of weakly perturbed harmonic oscillators furthermore we prove the confinement of orbits in resonant regions in the general case of nonisochronous systems by using the elementary idea of energy conservation instead of more complicated mechanisms an application of nekhoroshev s theorem to the study of perturbed motions inside resonances is also provided

effective stability for gevrey and finitely differentiable prevalent hamiltonians for perturbations of integrable hamiltonian systems the nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time provided the integrable part satisfies a steepness condition and the system is analytic this fundamental result has been extended in two distinct directions the first one is due to niederman who showed that under the analyticity assumption the result holds true for a prevalent class of integrable systems which is much wider than the steep systems the second one is due to marco sauzin but it is limited to quasi convex integrable systems for which they showed exponential stability if the system is assumed to be only gevrey regular if the system is finitely differentiable we showed polynomial stability still in the quasi convex case the goal of this work is to generalize all these results in a unified way by proving exponential or polynomial stability for gevrey or finitely differentiable perturbations of prevalent integrable hamiltonian systems

first order elliptic systems and the existence of homoclinic orbits in hamiltonian systems and h r x r 2 r is smooth by a homoclinic or a doubly asymptotic orbit to y where y is a periodic solution of 1 we mean a solution x y of 1 such that i x 0 y t l 0 as i tl oo finding homoclinic orbits in systems like 1 can be quite difficult in the case when y is a hyperbolic orbit this is equivalent to showing that the stable and unstable manifold of y intersect in some situation this can be done by melnikov s method recently some progress has been made by applying variational methods coti zelati and ekeland showed in 3 using dual variational methods that 1 has at least one homoclinic orbit if h satisfies some convexity and growth assumptions in their proof the convexity assumption enters in a crucial way already in the set up of the variational problem variational methods have been used by benci and giannoni 2 who studied homoclinic orbits on compact riemannian manifolds and by p rabinowitz in 9 who proved existence of heteroclinic orbits on the n dimensional torus as well as their multiplicity in 2 and 9 only special hamiltonian of lagrangian type are studied the associated variational problem is then at least semi definite the

on the formulation of new explicit conditions for steepness from a former result of n n nekhoroshev the application of the nekhoroshev theorem to many problems arising in different fields of physics and astronomy depends on a non degeneracy property called steepness that a suitable hamiltonian approximation must satisfy since steepness is implicitly defined we have the problem of recognizing whether a given function is steep or not for this purpose we here consider some sufficient conditions for steepness provided by nekhoroshev in 1979 based on the solvability of a collection of systems depending on the number n of degrees of freedom the derivatives of the function up to a certain order r and some auxiliary parameters these conditions are really explicit only for r 2 corresponding to quasi convexity and for r 3 instead for r 4 the conditions are implicit since they require an elaborate computation of the closure of a certain set in this paper we first revisit nekhoroshev s result and we show that the number of parameters in the collections of systems can be suitably reduced th

realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory part ii the so called problem of the realization of the holonomic constraints of classical mechanics is here revisited in the light of nekhoroshev like classical perturbation theory precisely if constraints are physically represented by very steep potential wells with associated high frequency transversal vibrations then one shows that within suitable assumptions the vibrational energy and the energy associated to the constrained motion are separately almost constant for a very long time scale growing exponentially with the frequency i e with the rigidity of the constraint one aims to realize this result can also be applied to microscopic physics providing a possible entirely classical mechanism for the freezing of the high frequency degrees of freedom in terms of non equilibrium statistical mechanics according to some ideas expressed by boltzmann and jeans at the turn of the century in this part i we introduce the problem and prove a first theorem concerning the realization of a single constraint within a system of any number of degrees of freedom the problem of the realization of many constraints will be considered in a forthcoming part ii

averaging slaving and balance dynamics in a simple atmospheric model we report numerical results from a study of balance dynamics using a simple model of atmospheric motion that is designed to help address the question of why balance dynamics is so stable the non autonomous hamiltonian model has a chaotic slow degree of freedom representing vortical modes coupled to one or two linear fast oscillators representing inertia gravity waves the system is said to be balanced when the fast and slow degrees of freedom are separated we find adiabatic invariants that drift slowly in time this drift is consistent with a random walk behaviour at a speed which qualitatively scales even for modest time scale separations as the upper bound given by neishtadt s and nekhoroshev s theorems moreover a similar type of scaling is observed for solutions obtained using a singular perturbation slaving technique in resonant cases where nekhoroshev s theorem does not apply we present evidence that the smaller lyapunov exponents of the system scale exponentially as well the results suggest that the observed stability of nearly slow motion is a consequence of the approximate adiabatic invariance of the fast motion

on the search for a sequence from a predefined set of sequences in random and framed data streams in this paper we present a statistical analysis of the search for a set of predefined sequences in non equiprobable data streams both random and framed and derive a number of corresponding statistical parameters most importantly we derive the probability of occurrence of some sequence from the set at any given position from which all other statistical quantities of interest can be obtained the presented analysis is based on descriptors called cross bifices which express similarities among sequences in the set based on the obtained results we examine the problem of frame synchronization in synchronous transmission and derive parameters that could be used for the evaluation of the properties of frame synchronization sequences we introduce a survival probability of the search in a frame and a list miss probability for the list synchronizer which give exact bounds on the probability of correct frame synchronization when hard correlation is used as a sequence detection metric

adiabatic invariants of some time dependent oscillators two types of time dependent 1d oscillators are considered parametrically excited described by the mathieu equation and parametrically pumped with a periodic driving force the adopted field method approach is applied and the complete sets of their linear invariants as well as the corresponding quadratic invariants are derived

statistical properties of 1d time dependent hamiltonian systems from the adiabatic limit to the parametrically kicked systems we consider 1d time dependent hamiltonian systems and their statistical properties namely the time evolution of microcanonical distributions whose properties are very closely related to the existence and preservation of the adiabatic invariants we review the elements of the recent developments by robnik and romanovski during 2006 2008 for the entirely general 1d time dependent linear oscillator and try to generalize the results to the 1d nonlinear hamilton oscillators in particular the power law potentials like e g the quartic oscillator furthermore we consider the limit opposite to the adiabatic limit namely parametrically kicked 1d hamiltonian systems even for the linear oscillator interesting properties are revealed an initial kick disperses the microcanonical distribution to a spread one but an anti kick at an appropriate moment of time can annihilate it kicking it back to the microcanonical distribution the case of periodic parametric kicking is also interesting finally we propose that in the parametric kicking of a general 1d hamilton system the average value of the adiabatic invariant always increases which we prove for the power law potentials we find that the approximation of kicking is good for quite long times of the parameter variation up to the order of not much less than one period of the oscillator we also look at the behavior of the quartic oscillator for the case of the kick and anti kick and also the periodic kicking

nekhoroshev stability of elliptic equilibria of hamiltonian systems we prove a conjecture by n n nekhoroshev about the long time stability of elliptic equilibria of hamiltonian systems without any diophantine condition on the frequencies higher order terms of the hamiltonian are used to provide convexity the singularity of the action angle coordinates at the origin is overcome by working in cartesian coordinates

effective stability for a hamiltonian system near an elliptic equilibrium point with an application to the restricted three body problem abstract we consider an n degrees of freedom hamiltonian system near an elliptic equilibrium point the system is put in normal form up to an arbitrary order and with respect to some resonance module and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form a bound to the rate of diffusion is thus found and by optimizing the order of normalization exponential estimates of nekhoroshev s type are obtained this provides explicit estimates for the stability properties of the elliptic point and leads in some cases to effective stability i e stability up to finite but long times an application to the stability of the triangular libration points in the spatial restricted three body is also given

a new approach to cross bifix free sets cross bifix free sets are sets of words such that no prefix of any word is a suffix of any other word in this paper we introduce a general constructive method for the sets of cross bifix free binary words of fixed length it enables us to determine a cross bifix free words subset which has the property to be non expandable

statistical properties of the energy in time dependent homogeneous power law potentials we study classical 1d hamilton systems with a homogeneous power law potential and their statistical behavior assuming a microcanonical distribution of the initial conditions and describing its change under a monotonically increasing time dependent function a t the prefactor of the potential using the nonlinear wentzel kramers brillouin like method of papamikos and robnik 2012 j phys a math theor 44 315102 and following a previous work by papamikos and robnik 2011 j phys a math theor 45 015206 we specifically analyze the mean energy the variance and the adiabatic invariant action of the systems for large time and we show that the mean energy and variance increase as powers of a t while the action oscillates and finally remains constant by means of a number of detailed case studies we show that the theoretical prediction is excellent which demonstrates the usefulness of the method in such applications

slow dynamics via degenerate variational asymptotics we introduce the method of degenerate variational asymptotics for a class of singularly perturbed ordinary differential equations in the limit of strong gyroscopic forces such systems exhibit dyna

asymptotic stability and asymptotic solutions of second order differential equations abstract we improve simplify and extend on quasi linear case some results on asymptotical stability of ordinary second order differential equations with complex valued coefficients obtained in our previous paper g r hovhannisyan asymptotic stability for second order differential equations with complex coefficients electron j differential equations 2004 85 2004 1 20 to prove asymptotic stability of second order differential equations we establish stability estimates using integral representations of solutions via asymptotic solutions and error estimates several examples are discussed

nekhoroshev estimates for finitely differentiable quasi convex hamiltonians abstract a major result about perturbations of integrable hamiltonian systems is the nekhoroshev theorem which gives exponential stability for all solutions provided the system is analytic and the integrable hamiltonian is generic in the particular but important case where the latter is quasi convex these exponential estimates have been generalized by marco and sauzin if the hamiltonian is gevrey regular using a method introduced by lochak in the analytic case in this paper using the same approach we investigate the situation where the hamiltonian is assumed to be only finitely differentiable for which it is known that exponential stability does not hold but nevertheless we prove estimates of polynomial stability

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

non overlapping matrices abstract two matrices are said non overlapping if one of them cannot be put on the other one in a way such that the corresponding entries coincide we provide a set of non overlapping binary matrices and a formula to enumerate it which involves the k generalized fibonacci numbers moreover the generating function for the enumerating sequence is easily seen to be rational

asymptotic stability estimates near an equilibrium point abstract we use the error bounds for adiabatic invariants found in the work of chartier murua and sanz serna 3 to bound the solutions of a hamiltonian system near an equilibrium over exponentially long times our estimates depend only on the linearized system and not on the higher order terms as in kam theory nor do we require any steepness or convexity conditions as in nekhoroshev theory we require that the equilibrium point where our estimate applies satisfy a type of formal stability called lie stability

stability of some solutions for elliptic equations on a cylindrical domain we study analytically a class of solutions for the elliptic equation formula 1 where 0 and e is a small parameter this equation is ill posed and most initial conditions do not lead to solutions nevertheless we show that for almost every 0 it contains solutions that are defined for large values of time and they are very close of order o e to a linear torus for long times of order o e 1 the proof uses the fact that the equation leaves invariant a smooth centre manifold and for the restriction of the system to the centre manifold uses arguments of classical perturbation theory by considering the hamiltonian formulation of the problem the birkhoff normal form and neckhoroshev type estimates 1 embed graphic 1 gif

towards a rigorous treatment of the jeans landau teller method for the energy exchanges of harmonic oscillators for classical hamiltonian systems containing an harmonic oscillator of high frequency one has the problem of controlling the energy exchange between the oscillator and the remaining slow degrees of freedom under very general conditions such an exchange turns out to be exponentially small with the fre quency of the oscillator in the jeans landau teller method one aims to prove the exponential dependence and to estimate the coefficient of the exponential by exploiting the analyticity properties of the solution of the differential equations describing the motion of the system however in practice since the exact solution is not known such properties are inferred from those of an approximate solution with no control of the difference this fact might a priori even invalidate the exponential dependence itself in the present paper a rigorous treatment is given for a particular model of interest in the domain of atomic collisions by keeping control of the difference between the exact and the approximate solution

exact analysis of adiabatic invariants in the time dependent harmonic oscillator the theory of adiabatic invariants has a long history and important applications in physics here we treat exactly the general time dependent 1d harmonic oscillator which cannot be solved in general we follow the time evolution of an initial ensemble of phase points with sharply defined energy e0 and calculate rigorously the distribution of energy e1 after time t and all its moments especially its average value and variance 2 using wkb theory we get the exact result for the leading asymptotic behaviour of 2

existence of localized solutions for a classical nonlinear dirac field we prove the existence of stationary states for nonlinear dirac equations of the form i gamma mu partial mu psi m psi f bar psi psi psi 0 we seek solutions which are separable in spherical coordinates and we use a shooting method for solving the associated problem of ordinary differential equations