a central limit theorem for temporally non homogenous markov chains with applications to dynamic programming we prove a central limit theorem for a class of additive processes that arise naturally in the theory of nite horizon markov decision problems the main theorem generalizes a classic result of dobrushin 1956 for tem porally non homogeneous markov chains and the principal innovation is that here the summands are permitted to depend on both the current state and a bounded number of future states of the chain we show through several examples that this added exibility gives one a direct path to asymptotic nor mality of the optimal total reward of nite horizon markov decision problems the same examples also explain why such results are not easily obtained by alternative markovian techniques such as enlargement of the state space

nonconventional limit theorems the polynomial ergodic theorem pet which appeared in bergelson ergod th dynam sys 7 337 349 1987 and attracted substantial attention in ergodic theory studies the limits of expressions having the form 1 n sum n 1 nt q 1 n f 1 cdots t q ell n f ell where t is a weakly mixing measure preserving transformation f i s are bounded measurable functions and q i s are polynomials taking on integer values on the integers motivated partially by this result we obtain a central limit theorem for even more general expressions of the form 1 sqrt n sum n 1 n f x 0 n x 1 q 1 n x 2 q 2 n ldots x ell q ell n overline f where x i s are exponentially fast mixing bounded processes with some stationarity properties f is a lipschitz continuous function overline f int fd mu 0 times mu 1 times cdots times mu ell j is the distribution of x j 0 and q i s are positive functions taking on integer values on integers with some growth conditions which are satisfied for instance when q i s are polynomials of growing degrees when f x 0 x 1 x l x 0 x 1 x 2 x l exponentially fast mixing already suffices this result can be applied in the case when x i n t n f i where t is a mixing subshift of finite type a hyperbolic diffeomorphism or an expanding transformation taken with a gibbs invariant measure as well as in the case when x i n f i n where n is a markov chain satisfying the doeblin condition considered as a stationary process with respect to its invariant measure

linear and sub linear growth and the clt for hitting times of a random walk in random environment on a strip the main goal of this work is to study the asymptotic behaviour of hitting times of a random walk rw in a quenched random environment re on a strip we introduce enlarged random environments in which the traditional hitting time can be presented as a sum of independent random variables whose distribution functions form a stationary random sequence this allows us to obtain conditions stated in terms of properties of random environments for a linear growth of hitting times of relevant random walks in some important cases e g independent random environments these conditions are also necessary for this type of behaviour we also prove the quenched central limit theorem clt for hitting times in the general ergodic setting a particular feature of these ballistic laws in random environment is that whenever they hold under standard normalization the convergence is a convergence with a speed the latter is due to certain properties of moments of hitting times which are also studied in this paper the asymptotic properties of the position of the walk are stated but are not proved in this work since this has been done in goldhseid probab theory relat fields 139 1 41 64 2007

on the quenched central limit theorem for random dynamical systems we provide a necessary and sufficient condition under which the quenched central limit theorem without random centering holds for one dimensional random systems that are uniformly expanding this condition holds in particular when all the maps preserve a common measure we also give a counter example which shows that this condition is not necessarily satisfied when the maps do not preserve a common measure

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

poincare husimi representation of eigenstates in quantum billiards for the representation of eigenstates on a poincare section at the boundary of a billiard different variants have been proposed we compare these poincare husimi functions discuss their properties and based on this select one particularly suited definition for the mean behavior of these poincare husimi functions an asymptotic expression is derived including a uniform approximation we establish the relation between the poincare husimi functions and the husimi function in phase space from which a direct physical interpretation follows using this a quantum ergodicity theorem for the poincare husimi functions in the case of ergodic systems is shown

on level spacing distributions for 2d non normal gaussian random matrices the level spacing distribution of general non normal gaussian random 2d matrices is derived in particular tridiagonal matrices have no level repulsion and show a halfsided gaussian distribution general non normal matrices show strong level repulsion the repulsion exponent is 2 0log

stickiness in generic low dimensional hamiltonian systems a recurrence time statistics approach we analyze the structure and stickiness in the chaotic components of generic hamiltonian systems with divided phase space following the method proposed recently in lozej and robnik phys rev e 98 022220 2018 2470 004510 1103 physreve 98 022220 the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid we perform extensive numerical studies of three example systems the chirikov standard map the family of robnik billiards and the family of lemon billiards the filling of the cells is compared to the random model of chaotic diffusion introduced in robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 for the description of transport in the phase spaces of ergodic systems the model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential in generic systems the stickiness induces correlations in the cell visits the distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping we model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times we introduce the variable s which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness we use s to globally assess the distributions of recurrence times we find that in the bulk of the chaotic sea s 1 while s 1 in areas of stickiness we present the results in the form of animated grayscale plots of the variable s in the largest chaotic component for the three example systems included as supplemental material to this paper

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

quenched decay of correlations for slowly mixing systems we study random towers that are suitable to analyse the statistics of slowly mixing random systems we obtain upper bounds on the rate of quenched correlation decay in a general setting we apply our results to the random family of liveranisaussol vaienti maps with parameters in 0 1 0 1 chosen independently with respect to a distribution on 0 1 and show that the quenched decay of correlation is governed by the fastest mixing map in the family in particular we prove that for every 0 for almost every 0 1 z the upper bound n 1 1 0 holds on the rate of decay of correlation for holder observables on the fibre over for three different distributions on 0 1 discrete uniform quadratic we also derive sharp asymptotics on the measure of return time intervals for the quenched dynamics ranging from n 1 0 to log n 1 0 n 1 0 to log n 2 0 n 1 0 respectively

fuzzy clustering algorithms and their cluster validity abstract a non metric clustering algorithm based on a fuzzy objective function which reflects proximity based on some global dissimilarity measure is proposed the global optimal solution is shown to be difficult to obtain and an alternative iterative procedure is presented this procedure is easily implemented and converges to a local optimum some validity functionals which measure the effectiveness with which cluster structure has been identified are compared in relation with the iterative procedures described in the paper

distribution and fluctuation properties of transition probabilities in a system between integrability and chaos the authors study the statistical properties of transition probabilities or generalized intensities equal to the squared matrix elements of operators having a classical limit they first generalize the shnirelman theorem to include semiclassically integrable states and use this to generalize the feingold peres formula for the average value of generalized intensities they perform an unfolding procedure to separate the smooth mean part of the intensities as a function of frequency from its fluctuating part by applying the generalized feingold peres formula this formula relates the mean value of squared matrix elements to the power spectrum of the given observable over classical trajectories their approach is illustrated numerically by analysing the dipole transition probabilities in a family of billiards between integrability and chaos the average values of the intensities as a function of frequency are excellently described by the generalized feingold peres formula

symbolic dynamics and periodic orbits for the cardioid billiard the periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics the corresponding partition is mapped to a topologically well ordered symbol plane in the symbol plane the pruning front is obtained from orbits running either into or through the cusp we show that all periodic orbits correspond to maxima of the lagrangian and give a complete list up to code length 15 the symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines we show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned all these orbits are related to finite orbits starting and ending in the cusp we obtain an analytical estimate of the kolmogorov sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits furthermore the statistical properties of periodic orbits are investigated

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

clustering data by melting we derive a new clustering algorithm based on information theory and statistical mechanics which is the only algorithm that incorporates scale it also introduces a new concept into clustering cluster independence the cluster centers correspond to the local minima of a thermodynamic free energy which are identified as the fixed points of a one parameter nonlinear map the algorithm works by melting the system to produce a tree of clusters in the scale space melting is also insensitive to variability in cluster densities cluster sizes and ellipsoidal shapes and orientations we tested the algorithm successfully on both simulated data and a synthetic aperture radar image of an agricultural site with 12 attributes for crop identification

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

consolidating boundary methods for finding the eigenstates of billiards the plane wave decomposition method a widely used means of numerically finding eigenstates of the helmholtz equation in billiard systems is described as a variant of the mathematically well established boundary integral method bim a new unified framework encompassing the two methods is discussed furthermore a third numerical method which we call the gauge freedom method is derived from the bim equations this opens the way to further improvements in eigenstate search techniques

implicit restart lanczos as an eigensolver this paper investigates the efficiency of the implicit restart lanczos and simple without reorthogonalization lanczos algorithms as eigensolvers for large scale computations in molecular and chemical physics using the cardioid billiard and the hydrogen cyanide hydrogen isocyanide hcn hnc molecule as model systems we demonstrate superior efficiency of implicit restart lanczos compared to the simple lanczos algorithm a modified implementation of implicit restart lanczos is also presented which works with a smaller krylov space with associated savings in memory and can handle larger basis sets than the usual implicit restart lanczos it also enables getting all eigenpairs of a matrix or all eigenvalues below a threshold where the number of such is not known before hand which is more difficult with the usual implicit restart algorithm

countable alphabet random subhifts of finite type with weakly positive transfer operator we deal with countable alphabet locally compact random subshifts of finite type the latter merely meaning that the symbol space is generated by an incidence matrix under the absence of big images property and under the absence of uniform positivity of the transfer operator we first establish the existence of random conformal measures along with good bounds for the iterates of the perron frobenius operator then using the technique of positive cones and proving a version of bowen s type contraction see equilibrium states and the ergodic theory of anosov diffeomorphisms 1975 we also establish a fairly complete thermodynamical formalism this means that we prove the existence and uniqueness of fiberwise invariant measures giving rise to a global invariant measure equivalent to the fiberwise conformal measures furthermore we establish the existence of a spectral gap for the transfer operators which in the random context precisely means the exponential rate of convergence of the normalized iterated transfer operator this latter property in a relatively straightforward way entails the exponential decay of correlations and the central limit theorem

a note on the level spacings distribution of the hamiltonians in the transition region between integrability and chaos it is well known that there is no strict universality of the spectral fluctuations of quantum hamiltonians whose classical counterparts undergo the transition from integrability to complete chaos the author discusses the level spacings distribution p s and explains why the semiclassical formulae of berry and robnik cannot be correct for small s there is no global universality of p s for nearly integrable systems but the approach to the integrability as the perturbation parameter in goes to zero can be universal this is reflected in the fact that the slope of dp ds at s 0 for small in is universally inversely proportional to in the author gives two models in terms of two dimensional random matrices one of them being based on maximum entropy considerations the author also points out the connection to the statistics of zeros of random functions and discusses the numerical evidence

annealed and quenched limit theorems for random expanding dynamical systems in this paper we investigate annealed and quenched limit theorems for random expanding dynamical systems making use of functional analytic techniques and more probabilistic arguments with martingales we prove annealed versions of a central limit theorem a large deviation principle a local limit theorem and an almost sure invariance principle we also discuss the quenched central limit theorem dynamical borel cantelli lemmas erdos renyi laws and concentration inequalities

vector quantization by deterministic annealing a deterministic annealing approach is suggested to search for the optimal vector quantizer given a set of training data the problem is reformulated within a probabilistic framework no prior knowledge is assumed on the source density and the principle of maximum entropy is used to obtain the association probabilities at a given average distortion the corresponding lagrange multiplier is inversely related to the temperature and is used to control the annealing process in this process as the temperature is lowered the system undergoes a sequence of phase transitions when existing clusters split naturally without use of heuristics the resulting codebook is independent of the codebook used to initialize the iterations

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

maximum norms of chaotic quantum eigenstates and random waves abstract the growth of the maximum norms of quantum eigenstates of classically chaotic systems with increasing energy is investigated the maximum norms provide a measure for localization effects in eigenfunctions an upper bound for the maxima of random superpositions of random functions is derived for the random wave model this gives the bound c ln e in the semiclassical limit e the growth of the maximum norms of random waves is compared with the growth of the maximum norms of the eigenstates of six quantum billiards which are classically chaotic the maximum norms of these systems are numerically shown to be conform with the random wave model furthermore the distribution of the locations of the maximum norms is discussed

failure of semiclassical methods to predict individual energy levels the authors argue that semiclassical methods quite generally cannot predict the individual energy levels not even in the semiclassical limit of small but finite and when the number of energy levels goes to infinity by this they mean that the average relative error of the semiclassical eigenvalues in units of the mean level spacing typically increases indefinitely as the energy goes to infinity or is at least bounded from below this they show for the case of the integrable circular billiard and the one dimensional potential uo cos2 alpha x by comparing the torus quantized semiclassical eigenenergies with the exact results since all the various semiclassical methods such as gutzwiller s and bogomolny s are reduced to the torus quantization in integrable cases they believe that their conclusion is generally valid they have theoretical arguments and strong numerical evidence for the case of the circular billiard that nevertheless the statistical properties of the exact energy spectra are correctly reproduced by the semiclassical approximations it is numerically found that the energy level spacing distribution and the spectral rigidity for the exact spectrum and for the semiclassical spectrum are in excellent agreement even for finite spectra where they both deviate from the limiting poissonian behaviour so they suggest that the nonuniversal approach to the limiting energy level statistics is also correctly described by the semiclassical theory they discuss the validity of semiclassical methods in the light of their negative and positive findings in addition they find the surprising result for the previously mentioned special cases that the error distribution of the semiclassical approximation is stationary i e it is independent of the energy

amplitude distribution of eigenfunctions in mixed systems we study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space for an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and a good agreement with numerical computations for the family of limacon billiards is found the natural extension of our result to more general systems e g with a potential is also discussed

a spectral approach for quenched limit theorems for random hyperbolic dynamical systems we extend the recent spectral approach for quenched limit theorems developed for piecewise expanding dynamics under general random driving to quenched random piecewise hyperbolic dynamics for general ergodic sequences of maps in a neighborhood of a hyperbolic map we prove a quenched large deviations principle ldp central limit theorem clt and local central limit theorem lclt

markov chains in random environments and random iterated function systems we consider random iterated function systems giving rise to markov chains in random stationary environments conditions ensuring unique ergodicity and a pure type characterization of the limiting randomly invariant probability measure are provided we also give a dimension formula and an algorithm for simulating exact samples from the limiting probability measure

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

a spectral approach for quenched limit theorems for random expanding dynamical systems we prove quenched versions of i a large deviations principle ldp ii a central limit theorem clt and iii a local central limit theorem lclt for non autonomous dynamical systems a key advance is the extension of the spectral method commonly used in limit laws for deterministic maps to the general random setting we achieve this via multiplicative ergodic theory and the development of a general framework to control the regularity of lyapunov exponents of twisted transfer operator cocycles with respect to a twist parameter while some versions of the ldp and clt have previously been proved with other techniques the local central limit theorem is to our knowledge a completely new result and one that demonstrates the strength of our method applications include non autonomous piecewise expanding maps defined by random compositions of the form t n 1 t t an important aspect of our results is that we only assume ergodicity and invertibility of the random driving in particular no expansivity or mixing properties are required

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

on the asymptotic moments and edgeworth expansions for some processes in random dynamical environment we prove that certain asymptotic moments exist for some random distance expanding dynamical systems and markov chains in random dynamical environment and compute them in terms of the derivatives at 0 of an appropriate pressure function it will follow that these moments satisfy the relations that the asymptotic moments gamma k lim n rightarrow infty n frac k 2 mathbb e sum i 1 n x i k of sums of independent and identically distributed centered random variables satisfy under certain mixing conditions we will also estimate the convergence rate towards these limits the arguments in the proof of these results yield that the partial sums generated by the random ruelle perron frobenius triplets and all of their parametric derivatives considered as functions on the base corresponding to appropriate random transfer or markov operators satisfy several probabilistic limit theorems such as the central limit theorem we will also obtain certain edgeworth asymptotic expansions related to the central limit theorem for such processes our proofs rely on a parametric random complex ruelle perron frobenius theorem which replaces some of the spectral techniques used in literature in order to obtain limit theorems for deterministic dynamical systems and markov chains

dynamical localization in chaotic systems spectral statistics and localization measure in the kicked rotator as a paradigm for time dependent and time independent systems we study the kicked rotator in the classically fully chaotic regime using izrailev s n dimensional model for various n 4000 which in the limit n tends to the quantized kicked rotator we do treat not only the case k 5 as studied previously but also many different values of the classical kick parameter 5 k 35 and many different values of the quantum parameter ke 5 60 we describe the features of dynamical localization of chaotic eigenstates as a paradigm for other both time periodic and time independent autonomous fully chaotic or and mixed type hamilton systems we generalize the scaling variable l n to the case of anomalous diffusion in the classical phase space by deriving the localization length l for the case of generalized classical diffusion we greatly improve the accuracy and statistical significance of the numerical calculations giving rise to the following conclusions 1 the level spacing distribution of the eigenphases or quasienergies is very well described by the brody distribution systematically better than by other proposed models for various brody exponents br 2 we study the eigenfunctions of the floquet operator and characterize their localization properties using the information entropy measure which after normalization is given by loc in the interval 0 1 the level repulsion parameters br and loc are almost linearly related close to the identity line 3 we show the existence of a scaling law between loc and the relative localization length now including the regimes of anomalous diffusion the above findings are important also for chaotic eigenstates in time independent systems batistic and robnik j phys a math gen 43 215101 2010 arxiv 1302 7174 2013 where the brody distribution is confirmed to a very high degree of precision for dynamically localized chaotic eigenstates even in the mixed type systems after separation of regular and chaotic eigenstates

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards in searching for the manifestations of the sensitivity of the eigenfunctions in quantum billiards with dirichlet boundary conditions with respect to the boundary data the normal derivative we have performed instead various numerical tests for the robnik billiard a quadratic conformal map of the unit disk for 600 shape parameter values where we have looked at the sensitivity of the energy levels with respect to the shape parameter we show the energy level flow diagrams for three stretches of fifty consecutive odd eigenstates each with index 1000 to 2000 in particular we have calculated the unfolded and normalized level curvature distribution and found that it continuously changes from a delta distribution for the integrable case circle to a broad distribution in the classically ergodic regime for some shape parameters the agreement with the goe von oppen formula is very good whereas we have also cases where the deviation from goe is significant and of physical origin in the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the berry robnik 1984 surmise here the agreement with theory is not good partially due to the localization phenomena which are expected to disappear in the semiclassical limit we stress that even for classically ergodic systems there is no global universality for the curvature distribution not even in the semiclassical limit

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

exponential decay of correlations for random lasota yorke maps we consider random piecewise smooth piecewise invertible maps mainly on the interval but also in higher dimensions we assume that on the average and possibly without any stochastic uniformity i the maps expand distances ii do not have too many pieces iii do not have too large a distortion and iv are strongly mixing we assume it no markov property we prove that as in the classical case of the iteration of a fixed piecewise expanding map of the interval we have it exponential decay of random correlations our proof builds on the one given by c liverani for deterministic mixing and piecewise expanding interval maps

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

spectral zeta functions for aharonov bohm quantum billiards this is a study of the sum zeta s alpha identical to ej 1infinity ej alpha s over the eigenvalues ej alpha of schrodinger s equation in a billiard domain d with reflecting walls threaded by a single line of magnetic flux alpha for integer s zeta s alpha is calculated by generalising a green function technique of itzykson et al ibid vol 19 l111 5 1986 based on a conformal transformation between d and the unit disc when the transformation is generated by a polynomial of finite degree an explicit formula enables zeta 2 alpha to be easily computed with high accuracy in conjunction with a semiclassical approximation the exact values of zeta 2 alpha can be used to calculate the ground state e1 alpha for a non integrable billiard with an error of about one per cent

statistical measures for eigenfunctions of nonseparable quantum billiard systems abstract we study two statistical measures for eigenfunctions of several classically pseudointegrable and chaotic billiard systems the directional energy distribution and the nodal line curvature distribution a sharp distinction between the classically pseudointegrable and the chaotic systems is possible with both of these measures they can also be used to distinguish between regular and irregular eigenfunctions of the billiard systems using the same data set additionally the amplitude distribution is calculated for single regular and irregular states and accumulated over several hundred states the well known gaussian prediction holds well for irregular wave functions of the pseudointegrable system too

inside singularity sets of random gibbs measures we evaluate the scale at which the multifractal structure of some random gibbs measures becomes discernible the value of this scale is obtained through what we call the growth speed in holder singularity sets of a borel measure this growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self similar gibbs measures our results are useful to understand the multifractal nature of various heterogeneous jump processes

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

behaviour of boundary functions for quantum billiards we study the behaviour of the normal derivative of eigenfunctions of the helmholtz equation inside billiards with dirichlet boundary condition these boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest therefore they form a reduced representation of the quantum system analogous to the poincare section of the classical system for the normal derivatives we introduce an equivalent to the standard green function and derive an integral equation on the boundary based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies the first term is universal and independent of the shape of the billiard the second one is proportional to the curvature of the boundary the asymptotic behaviour is compared with numerical results for the stadium billiard different limacon billiards and the circle billiard and good agreement is found furthermore we derive an asymptotic completeness relation for the boundary functions

quenched normal approximation for random sequences of transformations we study random compositions of transformations having certain uniform fiberwise properties and prove bounds which in combination with other results yield a quenched central limit theorem equipped with a convergence rate also in the multivariate case assuming fiberwise centering for the most part we work with non stationary randomness and non invariant non product measures independently we believe our work sheds light on the mechanisms that make quenched central limit theorems work by dissecting the problem into three separate parts

semiempirical theory of level spacing distribution beyond the berry robnik regime modeling the localization and the tunneling effects in this work we study the level spacing distribution in the classically mixed type quantum systems which are generic exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions in the asymptotic regime of the sufficiently deep semiclassical limit sufficiently small effective planck constant the berry and robnik 1984 j phys a math gen 17 2413 picture applies which is very well established we present a new quasi universal semiempirical theory of the level spacing distribution in a regime away from the berry robnik regime the near semiclassical limit by describing both the dynamical localization effects of chaotic eigenstates and the tunneling effects which couple regular and chaotic eigenstates the theory works extremely well in the 2d mixed type billiard system introduced by robnik 1983 j phys a math gen 16 3971 and is also tested in other systems mushroom billiard and prosen billiard

dynamical localization of chaotic eigenstates in the mixed type systems spectral statistics in a billiard system after separation of regular and chaotic eigenstates we study the quantum mechanics of a billiard robnik 1983 j phys a math gen 16 3971 in the regime of mixed type classical phase space the shape parameter 0 15 at very high lying eigenstates starting at about 1 000 000th eigenstate and including the consecutive 587654 eigenstates by calculating the normalized poincare husimi functions of the eigenstates and comparing them with the classical phase space structure we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates and the corresponding energies the chaotic eigenstates appear all to be dynamically localized meaning that they do not uniformly occupy the entire available chaotic classical phase space component but are localized on a proper subset we find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the poisson statistics and the chaotic ones obey the brody statistics as anticipated in a recent paper by batistic and robnik 2010 j phys a math theor 43 215101 where the entire spectrum was found to obey the berry robnik brody statistics there are no effects of dynamical tunneling in this regime due to the high energies as they decay exponentially with the inverse effective planck constant which is proportional to the square root of the energy

improved energy level statistics for a family of billiards with analytic boundaries the energy level statistics of a family of classically chaotic billiards with analytic boundaries is studied numerically with a high chi 2 confidence level for the final results which go substantially beyond the quality of existing similar results the hypothesis of bohigas et al 1984 is strongly supported by the results showing that the level statistics of classically ergodic systems in the semiclassical limit can indeed be described by the gaussian orthogonal ensemble of the random matrix theories contrary to some doubts which have been recently raised e g by wilkinson et al 1991 nevertheless some small deviation of the numerical results compared with the gaussian orthogonal ensemble is observed and shown but it is believed to be still a deficiency of the statistical significance rather than of the model

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

a model for the computation of quantum billiards with arbitrary shapes an expansion method for the stationary schrodinger equation of a three dimensional quantum billiard system whose boundary is defined by an arbitrary analytic function is introduced the method is based on a coordinate transformation and an expansion in spherical harmonics the effectiveness is verified and confirmed by a numerical example which is a billiard system depending on a parameter

the algebraic quantisation of the birkhoff gustavson normal form the author develops an algebraic quantisation method for the birkhoff gustavson normal form for this purpose the weyl quantisation rule is used the method developed here for multidimensional systems allows one to calculate the energy levels and the transition probabilities the author gives a brief review of the normal form derives some of its general properties and finds a general analytic solution for the fourth degree normal form for hamiltonians of two degrees of freedom in particular this includes the henon heiles system the author compares the results of specific examples with other works the question of canonically invariant quantisation the relation to the quantum mechanical perturbation theory and the question of chaotic behaviour and quantum stochasticity are discussed the author shows that the operators corresponding to the formal integrals of the motion are also quantum mechanical integrals if the normal form accidentally terminates so that the classical system is integrable then this implies quantum integrability of the normal form hamiltonian

modified fuzzy c means algorithm for cellular manufacturing several studies have used the fuzzy c means fcm algorithm for part machine grouping in cellular manufacturing however the application of the standard fcm algorithm to this problem has a number of drawbacks this paper proposes a modified fcm mfcm algorithm that groups components and machines in parallel and through an annealing process with the weighting exponent arrives at a crisp solution and an objective function value which can be interpreted in terms of the number of voids and intercellular movements of the part machine grouping obtained computational experiences show that although mfcm may sometimes require slightly more computing time than other methods not only is it able to find better solutions but also it has higher discriminating power for determining the number of cells

statistical properties of high lying chaotic eigenstates we study the statistical properties of the high lying chaotic eigenstates 200000 and above which are deep within the semiclassical regime the system we are analysing is the billiard system inside the region defined by the quadratic complex conformal map of the unit disk as introduced by robnik 1983 we are using heller s method of plane wave decomposition of the numerical eigenfunctions and perform extensive statistical analysis with the following conclusions i the local average probability density is in excellent agreement with the microcanonical assumption and all statistical properties are also in excellent agreement with the gaussian random model ii the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with berry s 1977 prediction iii although the scars of unstable classical periodic orbits in such an ergodic regime are expected to exist so far we have not found any around the 200000th state other than a scar like feature resembling the whispering gallery modes

quantum localization of chaotic eigenstates and the level spacing distribution the phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos or wave chaos and thus plays an important role in general quantum mechanics or even in general wave mechanics in this work we propose two different localization measures characterizing the degree of quantum localization and study their relation to another fundamental aspect of quantum chaos namely the energy spectral statistics our approach and method is quite general and we apply it to billiard systems one of the signatures of the localization of chaotic eigenstates is a fractional power law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance s goes like proportionality s for small s where 0 1 and 1 corresponds to completely extended states we show that there is a clear functional relation between the exponent and the two different localization measures one is based on the information entropy and the other one on the correlation properties of the husimi functions we show that the two definitions are surprisingly linearly equivalent the approach is applied in the case of a mixed type billiard system m robnik j phys a math gen 16 3971 1983 in which the separation of regular and chaotic eigenstates is performed

beyond the berry robnik regime a random matrix study of tunneling effects we consider the energy spectra of quantum hamilton systems whose classical dynamics is of mixed type i e regular for some initial conditions in the classical phase space and chaotic for the complementary initial conditions in the strict semiclassical limit when the effective planck constant eff is sufficiently small the berry robnik br statistics applies while at larger values of eff or smaller energies one sees deviations from br due to localization and tunneling effects we derive a two level random matrix model which describes these effects and can be treated analytically in a closed form the coupling between the regular and chaotic levels due to tunneling is assumed to be gaussian distributed this two level model describes most of the features of matrices of large dimensions here n 1000 which we treat numerically and is predicted to apply in mixed type systems at low energies the proposed analytical level spacing distribution function has two parameters the br parameter characterizing the classical phase space and the coupling antenna distortion or tunneling parameter between states localization effects so far are not included in our analysis except in the special case where can be replaced by some effective eff

intermediate statistics in the regime of mixed classical dynamics we explain the arguments in support of the berry robnik 1984 17 2413 picture of the energy level statistics in the asymptotic strict far semiclassical limit of sufficiently small where the entire energy spectrum can be represented as a statistically independent superposition of regular and irregular level sequences the regular ones obeying the poissonian statistics and the irregular ones the rmt statistics goe or gue we generalize the results to describe not only the level spacing distribution the number variance and the delta statistics but also arbitrary statistics the probability to find k levels inside an interval of length l after unfolding very useful and effective approximations for are described we demonstrate very clearly that this regime is excellently described by this picture for k as high as the outer energy scale even when taking into account the regular component and only one the dominant chaotic component this we show numerically for the compactified standard map and for the quartic 2d billiard

comment on energy level statistics in the mixed regime we comment on the recent paper by abul magd 1996 29 1 concerning the energy level statistics in the mixed regime i e such having the mixed classical dynamics where regular and chaotic regions coexist in the phase space we point out that his basic assumption on the additive property of the level repulsion function r s conditional probability density in the sense of dividing it linearly into the regular and chaotic part in proportion to the classical fractional phase space volumes and is not justified since among other things it relies on the type of berry s ergodic assumption which however is only correct in a homogeneous ensemble of ergodic systems not in the neighbourhood of an integrable system thus his resulting distribution cannot be regarded as a theoretically well founded object we point out that the semiclassical limiting energy level spacing distribution must be of berry robnik 1984 type and explain what transitional behaviour of the brody type with fractional power law energy level repulsion we observe in the near semiclassical regime where effective is not yet small enough thus we refer to the derivation arguments and conclusions in our previous paper prosen and robnik 1994 26 8059 and explain again the behaviour in this double transition region