the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

hopf algebras having a dense big cell we discuss some axioms that ensure that a hopf algebra has its simple comodules classified using an analogue of the borel weil construction more precisely we show that a hopf algebra having a dense big cell satisfies the above requirement this method has its roots in the work of parshall and wang in the case of q deformed quantum groups gl and sl here we examine the example of universal cosovereign hopf algebras for which the weight group is the free group on two generators

hopf algebras of low dimension the main aim of this paper is to classify all types of hopf algebras of dimension less thn or equal to 11 over an algebraically closed field of characteristic 0 if a is such a hopf algebra that is not semisimple then we shall prove that a or a is pointed this property will result from the fact that under some assumptions any hopf algebra that is generated as an algebra by a four dimensional simple subcoalgebra is a hopf quotient of the coordinate ring of quantum sl 2 k the first result allows us to reduce the classification to the case of pointed hopf algebras of dimension 8 we shall describe their types in the last part of the paper

examples of compact matrix pseudogroups arising from the twisting operation we construct multiparameter compact matrix pseudogroups of all types and show their representation theories are the same as their classical analogs

large deviation probabilities and dominating points for open convex sets nonlogarithmic behavior the existence of a dominating point for an open convex set and a corresponding representation formula for large deviation probabilities are established in the infinite dimensional setting under conditions which are both necessary and sufficient and follow from those used previously in mathbb r d a precise nonlogarithmic estimate of large deviation probabilities applicable to gaussian measures is also included

quantum subgroups of glα β n abstract let c 0 and l n be odd with l 3 we determine all hopf algebra quotients of the quantized coordinate algebra o gl n when 1 is a primitive l th root of unity and satisfy certain mild conditions and we characterize all finite dimensional quotients when 1 is not a root of unity as a byproduct we find a new family of non semisimple and non pointed hopf algebras with non pointed duals which are quotients of o gl n

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

influence of perturbations on chaotic behavior of the parabolic ray system this paper points out that the realistic sound speed is not only a periodic function of range there are also many other factors such as random perturbation modulating the periodic perturbation which had been investigated in previous papers by the calculations of phase portraits power spectra lyapunov exponents etc the influence of random perturbation on the dynamic behavior of the ray system is investigated the study demonstrates that the random perturbation lowers the threshold from quasiperiod to chaos and reinforces the irregularity of the trajectories which were regular originally however for the trajectories which were chaotic originally the random perturbation makes their irregularity decrease slightly finally the probability distribution of the lyapunov exponent is given

some topics on glq n abstract the quantum groups gl q n and sl q n are defined as pairs of hopf algebras and it is shown they are absolutely connected when q is a root of 1 some quantum analogue of the frobenius map and some related finite quantum groups are introduced and their simplicity is studied

periodic solution finder for an impact oscillator with a drift in this paper an efficient semi analytical method is developed to compute periodic solutions for a new model of an impact oscillator with a drift which explains the progression mechanism in vibro impact systems and can be used to optimize their performance the method constructs a periodic response assuming that each period is comprised of a sequence of distinct phases for which analytical solutions are known for example a period may consist of the following sequential phases i contact with progression ii contact without progression iii no contact and iv contact without progression using this information a system of four piecewise linear first order differential equations is transformed to a system of non linear algebraic equations the method allows one to accurately predict a range of control parameters for which the best progression rates are obtained

stochastic response of vibro impact duffing oscillators under external and parametric gaussian white noises abstract this study presents a solution procedure for the stationary probability density function pdf of the response of vibro impact duffing oscillators under external and parametric gaussian white noises first the zhuravlev non smooth coordinate transformation is adopted to convert a vibro impact oscillator into an oscillator without barriers the stationary pdf of the converted oscillator is governed by the fokker planck fp equation the fp equation is solved by the exponential polynomial closure epc method illustrative examples are presented with vibro impact duffing oscillators under external and parametric gaussian white noises to show the effectiveness of the solution procedure the parametric excitation is acting in displacement and the constraint is a unilateral zero offset barrier the restitution coefficient of impacts is taken as 0 90 comparison with the simulated results shows that the proposed solution procedure can provide good approximate pdfs for displacement and velocity although a little difference exists in the tail of these pdfs this difference may be due to the weak approximation on the response of the vibro impact oscillators using a continuous markov process when the restitution coefficient is not very close to unity

some topics on frobenius lusztig kernels i abstract an alternative description of the frobenius lusztig kernels and restricted specializations is presented and used to find their coradical filtrations automorphisms and imbeddings moreover factor hopf algebras of the frobenius lusztig kernels are determined

analysis of a random repeated impact process a simple random repeated impact process similar to the classical random walk process is analyzed the process which consists of a ball bouncing on a randomly vibrating surface is analogous to loss of contact situations which can occur in linkages and vibrating tools it also has relevance to rolling contact where the rolling element may separate from the surface and sustain repeated impacts a coefficient of restitution is used to describe impact and some important assumptions limit the analysis to values of this coefficient greater than 0 8 the markov nature of the process is demonstrated by its one step memory it may be regarded as a discrete markov process imbedded in continuous time a difference equation governing the process is developed and analyzed for the case where the vibrating surface has a gaussian distribution of velocities with the one step transition probability density function a linear weighting function is used to account for the weighting effect of the ball s velocity on the distribution of table velocities at impact the resulting integral equation is solved iteratively to yield the probability density function of ball velocities after impact this information may then be used to predict the magnitudes of the impacts and the time between them

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

bouncing ball problem stability of the periodic modes exploring all its ramifications we give an overview of the simple yet fundamental bouncing ball problem which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity the dynamics is modeled on the basis of a discrete map of difference equations which numerically solved fully reveals a rich variety of nonlinear behaviors encompassing irregular nonperiodic orbits subharmonic and chaotic motions chattering mechanisms and also unbounded nonperiodic orbits for periodic motions the corresponding conditions for stability and bifurcation are determined from analytical considerations of a reduced map through numerical examples it is shown that a slight change in the initial conditions makes the ball motion switch from periodic to chaotic orbits bounded by a velocity strip v gamma 1 epsilon where gamma is the nondimensionalized shaking acceleration and epsilon the coefficient of restitution which quantifies the amount of energy lost in the ball table collision

large deviation lower bounds for arbitrary additive functionals of a markov chain a universal large deviation lower bound is proved for sums of banach space valued functions of an irreducible general state space markov chain there are no restrictions on the functions other than measurability

chaos in a noisy mechanical system with stress relaxation the influence of periodic and random external excitation on the chaotic behaviour of a self excited system is considered the lyapunov exponents of the noisy system have been defined as random variables the properties of their distribution which allow one to quantify chaos have been determined

on nonlinear boundary value problems ghosts parasites and discretizations ghosts are a new class of approximate solutions to nonlinear boundary value problems bvps they are bounded away from true solutions but solve an initial value problem ivp for the differential equations exactly and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the ivp depends sensitively on initial data they correspond to shifted localized solutions and they can be observed in imperfect physical experiments and be approximated by convergent discretization schemes the latter property suggests that the concept of the numerically irrelevant solution introduced for discretized second order bvps should be re examined

large deviation lower bounds for additive functionals of markov processes let x 1 x 2 ldots be a markov process with state space e a polish space let l n omega a n 1 sum n 1 j 0 1 a x j omega denote the normalized occupation time measure if mu is a probability measure on e g is a weak neighborhood of mu and if v subset e then we obtain asymptotic lower bounds for probabilities p x lbrack l n omega cdot in g x j omega in v 0 leq j leq n 1 rbrack in terms of i mu the rate function of donsker and varadhan our assumptions are weaker than those imposed by donsker and varadhan and the proof works without any essential change in the continuous time case as well in fact the same proofs apply to certain bounded additive functionals let r geq 0 and let f omega rightarrow mathbf b be bounded mathscr f 0 r measurable where omega is the sample space with the product topology skorohod topology in the continuous time case and mathbf b is a separable banach space let theta k omega rightarrow omega be the shift operator i e theta k omega j omega k j then we get lower bounds for probabilities involving n 1 f omega f theta 1 omega cdots f theta n 1 omega in place of l n omega cdot in this latter situation the rate function has to be the entropy function h q of donsker and varadhan

large deviations for vector valued functionals of a markov chain lower bounds on obtient des bornes inferieures que l on exprime en termes des parametres de convergence de certains noyaux on donne une application aux mesures empiriques de chaines de markov

consistent multiparameter quantisation of gl n the author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl n and on its dual the space of polynomials in n2 variables the dimension of the manifold is n2 n 2 2

chaotic dynamics of a bouncing ball abstract a detailed study of a mapping on a two dimensional manifold is made the mapping describes a system subject to periodic forcing in particular an imperfectly elastic ball bouncing on a vibrating platform quasiperiodic motion on a one dimensional manifold is proven and observed numerically at low forcing while at higher forcing smale horseshoes are present we examine the evolution of the attracting set with changing parameter spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises a new type of chaos in which a trajectory alternates between two distinct chaotic regions is described and explained in terms of manifold collisions throughout we are concerned to examine the behaviour of lyapunov exponents typical behaviour of lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed at higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given

symmetries of quantum spaces subgroups and quotient spaces of quantum su 2 and so 3 groups we prove that each action of a compact matrix quantum group on a compact quantum space can be decomposed into irreducible representations of the group we give the formula for the corresponding multiplicities in the case of the quotient quantum spaces we describe the subgroups and the quotient spaces of quantumsu 2 andso 3 groups

identifying a large deviation rate function assume a sequence of probabilities pn has a large deviation rate function i it is proved that i takes a form analogous to a convex conjugate if i is also assumed convex then i is a convex conjugate of an explicitly defined function ft the results are applied to the empirical law of a markov chain yielding universal bounds on i examples are given of markov chains in which the empirical law has a large deviation rate strictly between the given bounds

cocycle deformations of coordinate rings of quantum matrices abstract a negative answer is given to the natural question whether o q gl n the coordinate ring of quantum gl n is a cocycle deformation of o gl n it is shown that this occurs only ifq 1 the main results are expressed more generally in terms of two parameter quantizations o m n

some dynamical properties of a classical dissipative bouncing ball model with two nonlinearities abstract some dynamical properties for a bouncing ball model are studied we show that when dissipation is introduced the structure of the phase space is changed and attractors appear increasing the amount of dissipation the edges of the basins of attraction of an attracting fixed point touch the chaotic attractor consequently the chaotic attractor and its basin of attraction are destroyed given place to a transient described by a power law with exponent 2 the parameter space is also studied and we show that it presents a rich structure with infinite self similar structures of shrimp shape

quantum groups and quantum determinants abstract we construct function coordinate algebras of multi parameter quantum deformations of g sl n gl n so n o n and sp n when g is of rank l the obtained function algebra has 1 l l 1 2 continuous parameters and some discrete parameters whose values are roots of unity to do this we introduce a notion of co quasitriangular bialgebras and study their group like elements quantum determinants

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

dynamical analysis of a two parameter family for a vibro impact system in resonance cases abstract dynamics of a two degree of freedom vibro impact system in resonance is considered the dynamical model and poincare maps are established when a pair of complex conjugate eigenvalues of the jacobian matrix cross the unit circle and satisfy the resonant condition 0 4 1 or 0 3 1 the four dimensional map is reduced to a two dimensional one by the center manifold theorem and the reduced map is put into its normal form by the method of normal form the two parameter unfoldings of local dynamical behavior studied in this paper develop the results of one parameter family analysis the hopf and subharmonic bifurcation conditions of period n 1 motion are given the numerical simulation method confirms the theoretical analysis it is shown that there exists an invariant torus via hopf bifurcation and period 4 4 motions via subharmonic bifurcation as two controlling parameters varying near the critical point for the resonance 0 4 1 and that there exists an invariant circle and unstable fixed points of order 3 bifurcating from the fixed point and the system leads eventually to chaos in the resonance 0 3 1

drinfel d doubles and lusztig s symmetries of two parameter quantum groups we flnd the deflning structures of two parameter quantum groups ur s g corre sponding to the orthogonal and the symplectic lie algebras which are realized as drinfel d doubles we further investigate the environment conditions upon which the lusztig s sym metries exist between ur s g h i and its associated object u si1 ri1 g hji

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

repeated impacts on a sinusoidally vibrating table reappraised the traditional simplification for one commonly used example of repeated collisions is shown to be frequently erroneous for a wide range of coefficients of restitution bounded errors arise from an artificial speed constraint whose circumvention is largely but not exclusively unpredictable therefore exact equations whose periodic solutions are detailed should be generally used

the periodicity of chaotic impact oscillators in hausdorff phase spaces it is well known that non periodic behavior is one of the most puzzling characteristics of chaotic oscillators so far chaotic dynamical systems have been investigated in euclidean spaces in this paper the concept of non autonomous dynamical systems and that of hausdorff phase spaces are proposed the behavior of chaotic impact oscillators is investigated in hausdorff phase spaces it is discovered that although the non autonomous dynamical systems described by chaotic impact oscillators are non periodic in euclidean phase spaces they are periodic in hausdorff phase spaces this shows that euclidean spaces in which we stayed for hundreds of years may no longer be suitable for the investigation into chaotic phenomena in addition the periodicity of chaotic dynamical systems in hausdorff metric spaces induces a new class of strange invariant sets in euclidean spaces such strange invariant sets may be an ideal symbol of chaotic dynamical systems