holomorphic quantization on the torus and finite quantum mechanics we explicitly construct the quantization of classical linear maps of on toroidal phase space of arbitrary modulus using the holomorphic chiral version of the metaplectic representation we show that finite quantum mechanics fqm on tori of arbitrary integer discretization is a consistent restriction of the holomorphic quantization of to the subgroup being the principal congruent subgroup on a finite dimensional hilbert space the generators of the rotation group for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates

a likelihood ratio test for mtp_2 within binary variables multivariate totally positive mtp 2 binary distributions have been studied in many fields such as statistical mechanics computer storage and latent variable models we show that mtp 2 is equivalent to the requirement that the parameters of a saturated log linear model belong to a convex cone and we provide a fisher scoring algorithm for maximum likelihood estimation we also show that the asymptotic distribution of the log likelihood ratio is a mixture of chi squares a distribution known as chi bar squared in the literature on order restricted inference for this we derive tight bounds which turn out to have very simple forms a potential application of this method is for item response theory irt models which are used in educational assessment to analyse the responses of a group of subjects to a collection of questions items an important issue within irt is whether the joint distribution of the manifest variables is compatible with a single latent variable representation satisfying local independence and monotonicity which in turn imply that the joint distribution of item responses is mtp 2

periodic orbit maslov indices for systems with time reversal symmetry we show that in time reversal invariant systems a pair of periodic orbits related by time reversal symmetry have the same maslov index previously this result had been implicitly assumed in the semiclassical derivation of the gaussian orthogonal ensemble spectral form factor

association arrays in assessing forms of dependencies between bivariate random variables abstract the bivariate distribution of pairs of random variables x y is said to be associated with respect to the classes of functions unk and unk if the product moment correlation r x y 0 for all unk and unk in the case in which both unk unk unk consist of all increasing functions then the bivariate distribution of x y is said to be positive quadrant dependent to apply the concept to data i examine the correlations for classes of extremal functions that span by positive combinations the totality of functions unk and unk to investigate whether the pair of random variables x y are associated with respect to unk and unk and to assess the relative degree or strength of association when comparing two sets of random variables x y and z w

information theoretic characterization of quantum chaos hypersensitivity to perturbation is a criterion for chaos based on the question of how much information about a perturbing environment is needed to keep the entropy of a hamiltonian system from increasing we demonstrate numerically that hypersensitivity to perturbation is present in the following quantum maps the quantum kicked top the quantum baker s map the quantum lazy baker s map and the quantum sawtooth and cat maps

commensurability effects on the spectra of integrable systems the eigenvalue spectra of two simple integrable systems a three dimensional anisotropic harmonic oscillator aho and a particle in a cuboid are studied as the aspect ratios are changed from rational to irrational values for rational aspect ratios commensurable case there are large number theoretic degeneracies which grow with energy these degeneracies disappear when the aspect ratios are irrational incommensurable case however for a given set of rational aspect ratios commensurable behavior ensues only at large energies the number of distinct eigenvalues per unit energy interval which is indicative of the degree of degeneracies is seen to approach a constant asymptotically for rational aspect ratios the asymptotic constant is determined for all sets of aspect ratios for the aho and is estimated semi analytically for several sets of aspect ratios for the cuboid the crossover from incommensurable to commensurable behaviour is studied by following a sequence of rational aspect ratios which approaches an irrational limit a scaling form the crossover is suggested and explored numerically a clear indication of scaling is seen for the aho while the evidence is suggestive but not definitive for the cuboid the distribution of energy level spacings in the commensurable regime is determined for the aho and a sequence of cuboids

mobius mellin and mathematical physics abstract we examine some results and techniques of analytic number theory which have application or potential application in mathematical physics we consider inversion formulae for lattice sums various transformations of infinite series and products functional equations and scaling relations with selected applications in electrostatics and statistical mechanics in the analysis the mellin transform and the riemann zeta function play a key role

on the quantization of the three particle toda lattice the authors compare the einstein brillouin keller quantization procedure and the canonical quantization of a three particle toda chain with periodic boundary conditions in particular the transition from very low energies at which the system may be approximated by harmonic oscillators to intermediate energies is investigated this is the regime of a general integrable nonlinear system for which they find a poissonian statistics for the energy levels in the limit of very high energies they exploit the fact that the system may be described essentially by a triangular billiard and thus can derive some exact results

beyond the heisenberg time semiclassical treatment of spectral correlations in chaotic systems with spin 1 2 the two point correlation function of chaotic systems with spin 1 2 is evaluated using periodic orbits the spectral form factor for all times thus becomes accessible equivalence with the predictions of random matrix theory for the gaussian symplectic ensemble is demonstrated a duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established

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

flux quantization and quantum mechanics on riemann surfaces in an external magnetic field the authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non compact two dimensional manifold of constant negative gaussian curvature and of finite volume for the motion on compact riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized as it is proportional to the integrated first chern class of a certain complex line bundle over the manifold in the case of non compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non compact riemann surface is holomorphically trivial

experimental versus numerical eigenvalues of a bunimovich stadium billiard a comparison we compare the statistical properties of eigenvalue sequences for a gamma 1 bunimovich stadium billiard the eigenvalues have been obtained in two ways one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator real system and the other set is calculated numerically ideal system we show influence of mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system we also discuss the influence of a family of marginally stable orbits the bouncing ball orbits in two microwave stadium billiards with different geometrical dimensions

random matrices generalized zeta functions and self similarity of zero distributions there is growing evidence for a connection between random matrix theories used in physics and the theory of the riemann zeta function and l functions the theory underlying the location of the zeros of these generalized zeta functions is one of the key unsolved problems physicists are interested because of the hilbert polya conjecture that the non trivial zeros of the zeta function correspond to the eigenvalues of some positive operator to complement the continuing theoretical work it would be useful to study empirically the locations of the zeros by different methods in this paper we use the rescaled range analysis to study the spacings between successive zeros of these functions over large ranges of the zeros the spacings have a hurst exponent of about 0 095 using sample sizes of 10 000 zeros this implies that the distribution has a high fractal dimension 1 9 and shows a lot of detailed structure the distribution is of the anti persistent fractional brownian motion type with a significant degree of anti persistence thus the high order zeros of these functions show a remarkable self similarity in their distribution over fifteen orders of magnitude for the riemann zeta function we find that the hurst exponents for the random matrix theories show a different behaviour a heuristic study of the effect of low order primes seems to show that this effect is a promising candidate to explain the results that we observe in this study we study the distribution of zeros for l functions of conductors 3 and 4 and find that the distribution is similar to that of the riemann zeta functions

properties of time delay and s matrix for chaotic scattering on a leaky surface of constant negative curvature properties of a model quantum scattering system displaying hard classical chaos elastic scattering on a leaky surface of constant negative curvature are analysed theoretically and serve to interpret previously obtained numerical results the low energy scattering behaviour is shown to be influenced in the usual fashion by a bound state just below the scattering continuum threshold a connection between the widths of the infinite number of simple poles of the s matrix and the lyapunov exponent for classical trajectories is analysed at high energies the scattering is characterized by fluctuations in the s matrix via its phase and the time delay analytic expressions for the autocorrelation function of the s matrix via its phase and the time delay analytic expressions for the autocorrelation function of the s matrix and of the time delay are obtained using montgomery s conjecture for the pair correlation function of the celebrated riemann zeros whose values correspond to the positions of the s matrix poles in momentum space the autocorrelation function for the s matrix is found to be lorentzian asymptotically at large energy differences delta e that is to decrease as delta e 2 but that for the time delay is not the distribution of fluctuations of s matrix phases is likely to be gaussian

likelihood inference on the underlying structure of irt models the assumptions underlying item response theory irt models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model it is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which however are necessary but not sufficient in this paper we describe how the theory for likelihood based inference under equality and inequality constraints may be used to test the underlying assumptions of irt models it turns out that the analysis based directly on the latent structure is simpler and more flexible in particular we indicate how several interesting extensions of the rasch model may be obtained by partial relaxation of the basic constraints an application to a data set provided by educational testing service is used to illustrate the approach

periodic orbits and semiclassical quantization of dispersing billiards periodic orbits in a dispersing billiard system consisting of three circular arcs are studied numerically by using a partial coding rule together with an efficient method for enumerating periodic orbits on the real billiard plane by examining several statistical measures it is shown that the length spectrum and the stability exponents are highly uncorrelated the validity of the semiclassical trace formula is also tested and a remarkable agreement of the semiclassical and quantum density of states is obtained at least for about the lower 15 levels

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

resummation and the semiclassical theory of spectral statistics we address the question as to why in the semiclassical limit classically chaotic systems generically exhibit universal quantum spectral statistics coincident with those of random matrix theory to do so we use a semiclassical resummation formalism that explicitly preserves the unitarity of the quantum time evolution by incorporating duality relations between short and long classical orbits this allows us to obtain both the non oscillatory and the oscillatory contributions to spectral correlation functions within a unified framework thus overcoming a significant problem in previous approaches in addition our results extend beyond the universal regime to describe the system specific approach to the semiclassical limit

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

orbit bifurcations and spectral statistics systems whose phase space is mixed have been conjectured to exhibit quantum spectral correlations that are in the semiclassical limit a combination of poisson and random matrix with relative weightings determined by the corresponding measures of regular and chaotic orbits we here identify an additional component in long range spectral statistics associated with periodic orbit bifurcations which can be semiclassically large this is illustrated for a family of perturbed cat maps it has been conjectured that in the semiclassical limit the quantum spectral statistics of classically integrable systems are generically poissonian and that those of classically chaotic systems are generically given by the average over an appropriate random matrix ensemble 1 4 in between these two extremes lie systems whose phase space is mixed that is in which regular and irregular motion coexist such systems are said to exhibit soft chaos 5 for these it has been suggested that the quantum spectral statistics are a combination of poisson and random matrix with relative weightings determined by the corresponding measures of the regular and chaotic orbits 6 our purpose here is to identify in this case an additional component in the long range statistics that is associated with periodic orbit bifurcations and which can be semiclassically large a semiclassical theory for long range spectral statistics has been developed 1 7 9 based on gutzwiller s trace formula 10 which relates the quantum density of states d e d p n e en to classical periodic orbits

energy level statistics of an integrable billiard system in a rectangle in the hyperbolic plane a separable billiard system in the hyperbolic strip a particular realization of the hyperbolic plane is investigated the author considers a rectangle bounded by straight lines in the hyperbolic strip as an approximation of either a regular octagon with area a 4 pi or of a hyperbolic rectangle bounded by geodesics respectively due to the simple geometry of the approximation the schrodinger equation is separable and leads to simple transcendental equations for the eigenvalues in terms of legendre functions the statistical properties of the eigenvalue spectrum are investigated and checked with respect to weyl s 1912 law level spacing number variance and spectral rigidity respectively in particular the numerical results for the spectral rigidity are in close agreement with the semi classical theory of berry 1985

multivariate logistic models summary when data composed of several categorical responses together with categorical or continuous predictors are observed the multivariate logistic transform introduced by mccullagh and nelder can be used to define a class of regression models that is in many applications particularly suitable for relating the joint distribution of the responses to predictors in this paper we give a general definition of this class of models and study their properties a computational scheme for performing maximum likelihood estimation for data sets of moderate size is described and a system of model formulae that succinctly define particular models is introduced applications of these models to longitudinal problems are illustrated by numerical examples

neutrino billiards time reversal symmetry breaking without magnetic fields a dirac hamiltonian describing massless spin half particles neutrinos moving in the plane r x y under the action of a 4 scalar not electric potential v r is in position representation h i h c v r z where x y and z are the pauli matrices acts on two component column spinor wavefunctions r 1 2 and has eigen values ck n does not possess time reversal symmetry t if v r describes a hard wall bounding a finite domain d billiards this is equivalent to a novel boundary condition for 2 1 t breaking is interpreted semiclassically as a difference of between the phases accumulated by waves travelling in opposite senses round closed geo desics in d with odd numbers of reflections the semiclassical large k asymptotics of the eigenvalue counting function spectral staircase n k are shown to have the weyl leading term ak 2 4 where a is the area of d but zero perimeter correction the dirac equation is transformed to an integral equation round the boundary of d and forms the basis of a numerical method for computing the k n when d is the unit disc geodesics are integrable and the eigenvalues which satisfy j l k n j l 1 k n are locally poisson distributed when d is an africa shape cubic conformal map of the unit disc the eigenvalues are locally distributed according to the statistics of the gaussian unitary ensemble of random matrix theory as predicted on the basis of t breaking and lack of geometric symmetry

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

parabolic equations for curves on surfaces 2 intersections blow up and generalized solutions we describe a theory for parabolic equations for immersed curves on surfaces which generalizes the curve shortening or flow by mean curvature problem as well as several models in the theory of phase transitions in two dimensions a class of equations is described for which the initial value problem is well posed for rough initial data for which one can give a description of the way a smooth solution becomes singular and for which one can define generalized solutions i e solutions which are smooth except at a discrete set of times the methods which are used in this paper are more geometrical than those of part i by comparing arbitrary solutions with certain special solutions and by considering the way they intersect we derive estimates for the curvature and the tangent which allow one to study the initial value problem and the way solutions become singular

quantum mechanics as an approximation to classical mechanics in hilbert space classical mechanics is formulated in complex hilbert space with the introduction of a commutative product of operators an antisymmetric bracket and a quasidensity operator that is not positive definite these are analogues of the star product the moyal bracket and the wigner function in the phase space formulation of quantum mechanics quantum mechanics is then viewed as a limiting form of classical mechanics as planck s constant approaches zero rather than the other way around the forms of semiquantum approximations to classical mechanics analogous to semiclassical approximations to quantum mechanics are indicated

a class of parameter dependent commuting matrices we present a novel class of real symmetric matrices in arbitrary dimension d linearly dependent on a parameter x the matrix elements satisfy a set of nontrivial constraints that arise from asking for commutation of pairs of such matrices for all x and an intuitive sufficiency condition for the solvability of certain linear equations that arise therefrom this class of matrices generically violates the wigner von neumann non crossing rule and is argued to be intimately connected with finite dimensional hamiltonians of quantum integrable systems

statistics of resonances for a class of billiards on the poincare half plane the lower boundary of artin s billiard on the poincar half plane is continuously deformed to generate a class of billiards with classical dynamics varying from fully integrable to completely chaotic the quantum scattering problem in these open billiards is described and the statistics of both real and imaginary parts of the resonant momenta are investigated the evolution of the resonance positions is followed as the boundary is varied which leads to large changes in their distribution the transition to arithmetic chaos in artin s billiard which is responsible for the poissonian level spacing statistics of the bound states in the continuum cusp forms at the same time as the formation of a set of resonances all with width and real parts determined by the zeros of riemann s zeta function is closely examined regimes are found which obey the universal predictions of random matrix theory rmt as well as exhibiting non universal long range correlations the brody parameter is used to describe the transitions between different regimes

spectral statistics on zoll surfaces the subject of this article is a family of partly conjectural spectral invariants of a laplace operator with a discrete spectrum j known as its k level correlation functions k and level spacings distribution the 2 level or pair correlation function 2 for instance measures the asymptotic number of spacings i j between normalized eigenvalues i as the eigenvalues tend to infinity the normalization is so that the mean level spacing i 1 i is one these invariants have been studied by many physicists including wigner dyson and berry and are conjectured to have certain universal forms depending on the qualitative behaviour of the corresponding classical geodesic flow

global cross ratio models for bivariate discrete ordered responses summary a family of statistical models is presented for bivariate discrete response to a regressor when both components of the response have ordered categories association between components is expressed in terms of global cross ratios cross product ratios of quadrant probabilities for each double dichotomy of the response table of probabilities into quadrants pearson and heron 1913 biometrika 9 159 315 these models are extensions to the work of plackett 1965 journal of the american statistical association 60 516 522 and mantel and brown 1973 biometrics 29 649 665 the marginal cumulative probabilities may satisfy linear logistic or other generalized linear models mccullagh 1980 journal of the royal statistical society series b 42 109 142 an analysis of patients postoperative pain level and medication frequency illustrates these methods

the n level spectral correlations for chaotic systems we study the n level spectral correlation functions of classically chaotic quantum systems without time reversal symmetry according to bohigas giannoni and schmit s universality conjecture it is expected that the correlation functions are in agreement with the prediction of the circular unitary ensemble cue of random matrices a semiclassical resummation formalism allows us to express the correlation functions as sums over pseudo orbits using an extended version of the diagonal approximation on the pseudo orbit sums we derive the n level correlation functions identical to the n n determinantal correlation functions of the cue

on selberg s trace formula chaos resonances and time delays the quantization of the chaotic geodesic motion on riemann surfaces g of constant negative curvature with genus g and a finite number of points infinitely far away cusps describing scattering channels is investigated it is shown that terms in selberg s trace formula describing scattering states can be expressed in terms of a renormalized time delay this quantity is the time delay associated with the surface in question minus the time delay corresponding to the scattering problem on the poincare upper half plane uniformizing our surface poles in these quantities give rise to resonances reflecting the chaos of the underlying classical dynamics our results are illustrated for the surfaces 1 1 gutzwiller s leaky torus 0 3 pants and a class of g 2 surfaces the generalization covering the inclusion of an integer b 2 magnetic field is also presented it is shown that the renormalized time delay is not dependent on the magnetic field this shows that the semiclassical dynamics with an integer magnetic field is the same as the free dynamics

nearest neighbour spacing distributions of the β hermite ensemble of random matrices the evolution with of the distributions of the spacing s between nearest neighbour levels of unfolded spectra of random matrices from the hermite ensemble he is investigated by monte carlo simulations the random matrices from the he are real symmetric and tridiagonal where which can take any positive value is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues the distribution of eigenvalues coincide with those of the three classical gaussian ensembles for 1 2 4 the use of the he ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices generalized gamma distributions are shown to be excellent approximations of the nearest neighbor spacing nns distributions for any while being still simple they account both for the level repulsion in s when s 0 and for the whole shape of the nns distributions in the range of s which is accessible to experiment or to most numerical simulations the exact nns distribution of the goe 1 is in particular significantly better described by a generalized gamma distribution than it is by the wigner surmise while the best generalized gamma approximation coincides essentially with the wigner surmise for 2 they describe too the evolution of the level repulsion between that of a poisson distribution and that of a goe distribution when increases from 0 to 1 the distribution of ln s related to the electrostatic interaction energy between neighbouring charges is accordingly well approximated by a generalized gumbel distribution for any 0 the distributions of the minimum nn spacing between eigenvalues of matrices from the he obtained both from as calculated eigenvalues and from unfolded eigenvalues are brody distributions which are classically used to characterize the spectral fluctuations of various physical systems

computational strategies for the riemann zeta function we provide a compendium of evaluation methods for the riemann zeta function presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new we concentrate primarily on practical computational issues such issues depending on the domain of the argument the desired speed of computation and the incidence of what we call value recycling c 2000 elsevier science b v all rights reserved

positive quadrant dependence and marginal modeling in two way tables with ordered margins for a collection of two way tables where subjects are cross classified according to the same pair of ordinal categorical variables conditionally on the value of one or more discrete explanatory variables we propose a general approach to likelihood inference that combines marginal modeling with fitting and testing of inequality constraints such as those implied by the assumption that one marginal distribution is stochastically larger than the other positive dependence and stronger positive dependence the approach is based on parameterizing bivariate conditional distributions with global logits and global log odds ratios and we provide a general framework for handling models defined by equality and inequality constraints on these parameters in this way such models as marginal homogeneity proportional odds among row or columns margins and plackett distribution may be treated together with various models defined by inequality constraints on the same parameters such as for instance those implied by

proposed solution of the quantum measurement problem a discussion of quantum measurement theory is presented it is argued that the quantum measurement problem consists in the failure of any formalism based on unitary evolution to describe fully the observed behaviour in measurement like processes it is shown that the problem is essentially concerned with the phase of the coherence between orthogonal eigenstates of an object measuring device system it is then proposed that the observed behaviour is fully described if the value of this phase is formally undecidable

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

universal twinkling exponents for spectral fluctuations associated with mixed chaology for systems that are neither fully integrable nor fully chaotic bifurcations of periodic orbits give rise to semiclassically emergent singularities in the fluctuating part n fl of the energy level counting function the bifurcations dominate the spectral moments m m n fl 2 m in the limit 0 we argue that m m const vm and calculate the twinkling exponents v m as the result of a competition between bifurcations with different codimensions and repetition numbers

maximum likelihood methods for a generalized class of log linear models we discuss maximum likelihood methods for fitting a broad class of multivariate categorical response data models in particular we derive the large sample distributions for maximum likelihood estimators of parameters of product multinomial generalized log linear models the large sample behavior of other relevant likelihood based statistics such as goodness of fit statistics and adjusted residuals is also described the asymptotic results are derived within the framework of the constraint specification rather than the more common freedom specification of the model we also outline an improved fitting algorithm for computing parameter maximum likelihood estimates and other relevant statistics the broad class of multivariate categorical response data models which are referred to as generalized log linear models can imply structure on several response configuration distributions e g joint and marginal distributions these models which include as special cases log linear logit and cumulative logit models enjoy a wide breadth of application including longitudinal rater agreement and crossover data analyses

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

the local nature of hypothesis tests involving inequality constraints in nonlinear models this paper derives several properties unique to nonlinear model hypothesis testing problems involving linear or nonlinear inequality constraints in the null or alternative hypothesis the paper is organized around a lemma which characterizes the set containing the least favorable parameter value for a nonlinear model inequality constraints hypothesis test we then present two examples which illustrate several implications of this lemma we also discuss the impact of these properties on the empirical implementation and interpretation of these test procedures

multifractal eigenstates of quantum chaos and the thue morse sequence we analyze certain eigenstates of the quantum baker s map and demonstrate using the walsh hadamard transform the emergence of the ubiquitous thue morse sequence a simple sequence that is at the border between quasiperiodicity and chaos and hence is a good paradigm for quantum chaotic states we show a family of states that are also simply related to the thue morse sequence and are strongly scarred by short periodic orbits and their homoclinic excursions we give approximate expressions for these states and provide evidence that these and other generic states are multifractal

the behaviour of resonances in hecke triangular billiards under deformation the right hand boundary of artin s billiard on the poincare half plane is continuously deformed to generate a class of chaotic billiards which includes fundamental domains of the hecke groups 2 n at certain values of the deformation parameter the quantum scattering problem in these open chaotic billiards is described and the distributions of both real and imaginary parts of the resonant eigenvalues are investigated the transitions to arithmetic chaos in the cases n 4 6 are closely examined and the explicit analytic form for the scattering matrix is given together with the fourier coefficients for the scattered wavefunction the n 4 and 6 cases have an additional set of regular equally spaced resonances compared to artin s billiard n 3 for a general deformation a numerical procedure is presented which generates the resonance eigenvalues and the evolution of the eigenvalues is followed as the boundary is varied continuously which leads to dramatic changes in their distribution for deformations away from the non generic arithmetic cases including that of the tiling hecke triangular billiard n 5 the distributions of the positions and widths of the resonances are consistent with the predictions of a random matrix theory

analytic representation of finite quantum systems a transform between functions in r and functions in zd is used to define the analogue of number and coherent states in the context of finite d dimensional quantum systems the coherent states are used to define an analytic representation in terms of theta functions all states are represented by entire functions with growth of order 2 which have exactly d zeros in each cell the analytic function of a state is constructed from its zeros results about the completeness of finite sets of coherent states within a cell are derived

two point spectral correlations for the square billiard we investigate the two point correlations in the quantum spectrum of the square billiard this system is unusual in that the degeneracy of the energy levels increases in the semiclassical limit in such a way that the average level separation is not given by the inverse of the mean density of states hence for example the standard level spacings distribution does not tend to the poissonian limit expected for integrable systems in this paper we calculate the leading order asymptotic form of a degeneracy weighted two point correlation function using a combination of probabilistic techniques and classical number theory the result exhibits number theoretical fluctuations about a mean which is a sum of two terms one having the usual constant poissonian form and the second representing a small correction which decays as the inverse of the correlation distance this is confirmed by numerical computations

a unified approach to likelihood inference on stochastic orderings in a nonparametric context abstract for data in a two way contingency table with ordered margins we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities this leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log likelihood ratio tests the usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases in particular for each of these three orderings we determine the transformation required to apply the estimation algorithm we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered as well as testing each stochastic order against an u

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

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

a rule for quantizing chaos the authors find a real function delta e whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories delta e is a finite sum over combinations of classical periodic orbits it is obtained from gutzwiller s infinite and divergent sum 1982 representing the spectral density in terms of periodic orbits by means of a resummation conjectured by analogy with a derivation of the riemann siegel formula for the riemann zeros they assess the practicality of the quantization condition

convergence and the length spectrum abstract the author defines and analyzes the 1 k length spectra l 1 k m whose union over all k n is the classical length spectrum these new length spectra are shown to converge in the sense that lim k k 1 k m i l 1 k m 0 as m i m in the gromov hausdorff sense energy methods are introduced to estimate the shortest element of l 1 k as well as a concept called the minimizing index which may be used to estimate the length of the shortest closed geodesic of a simply connected manifold in any dimension a number of gap theorems are proven including one for manifolds m n with ricci n 1 and volume close to vol s n many results in this paper hold on compact length spaces in addition to riemannian manifolds

periodic orbit action correlations in the baker map periodic orbit action correlations are studied for the piecewise linear area preserving baker map semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum baker map suggest the existence of universal periodic orbit correlations the calculation of periodic orbit sums for the baker map can be performed with the help of a perron frobenius type operator this makes it possible to study periodic orbit correlations for orbits with period up to 500 iterations of the map periodic orbit correlations are found to agree quantitatively with the predictions from random matrix theory up to a critical length determined by the semiclassical error exponentially increasing terms dominate the correlations for longer orbits which are due to the violation of unitarity in the semiclassical approximation

properties of ideal point classification models for bivariate binary data the ideal point classification ipc model was originally proposed for analysing multinomial data in the presence of predictors in this paper we studied properties of the ipc model for analysing bivariate binary data with a specific focus on three quantities 1 the marginal probabilities 2 the association structure between the two binary responses and 3 the joint probabilities we found that the ipc model with a specific class point configuration represents either the marginal probabilities or the association structure however the ipc model is not able to represent both quantities at the same time we then derived a new parametrization of the model the bivariate ipc bipc model which is able to represent both the marginal probabilities and the association structure like the standard ipc model the results of the bipc model can be displayed in a biplot from which the effects of predictors on the binary responses and on their association can be read we will illustrate our findings with a psychological example relating personality traits to depression and anxiety disorders

asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints summary the analysis of moment structural models has become an important tool of investigation in behavioural educational and economic studies the chi squared largesample test is routinely employed to assess the goodness of fit of the model considered however in order to invoke the standard asymptotic distribution theory certain regularity conditions have to be met here we consider the case where the population value of the parameter vector is a boundary point of the feasible region we show that in this case the asymptotic distribution of test statistic is a mixture of chi squared distributions the problem of finding the corresponding weights is discussed

disheveled arnold s cat and the problem of quantum classic correspondence abstract quantum arnold s cat map is studied for a case of perfect square inverse planck s constant n m 2 the classic limit is analyzed on a subset of numbers n increasing as 4 k the quantum problem in this case allows exact reduction to the classic cat map defined on a discrete lattice of size m m and supplemented by evolution of a phase variable a link between the classic periodic orbits and spectrum of eigenvalues of the quantum evolution operator is outlined for m growing as 2 k genetic analysis is developed for periodic orbits and they are classified by means of a tree like graph a phase shift accumulated over a period of the orbits evolves from level to level of the graph according to a certain rule governed by non periodic binary code representation of a localized gaussian wave packet in a basis of eigenvectors of the evolution operator gives rise to a probability measure distributed on a unit circle where the eigenvalues are located this measure looks like spectrum of a finite time sample of a stationary random process periodogram 1 majority of the eigenstates have intensities of comparable order of magnitude 2 the spectral distribution is of locally random like nature i e statistical variance of the amplitudes has the same order as the amplitudes themselves this combination of properties in very straightforward manner follows from chaotic nature of the classic map and is conjectured to be the most fundamental attribute of quantum chaos

extended rc association models allowing for order restrictions and marginal modeling in the context of two way contingency tables the rc model may be seen as a way of modeling the set of log odds ratios for adjacent 2 2 subtables through row and column scores this article presents an extended class of rc models that allows simultaneous modeling of the marginal distributions and the association between two categorical variables these components are parameterized with logits and log odds ratios that may be of different types e g global or continuation to suit the nature of the data and the type of dependence of interest an algorithm for maximum likelihood estimation under equality and inequality constraints is presented the asymptotic distribution of the likelihood ratio statistic to test some interesting hypotheses is derived and extensions to multiway tables are outlined three examples are presented to highlight the features of the new approach

marginal modeling of correlated ordinal data using a multivariate plackett distribution abstract an extension of the bivariate model suggested by dale is proposed for the analysis of dependent ordinal categorical data the so called multivariate dale model is constructed by first generalizing the bivariate plackett distribution to any dimensions because the approach is likelihood based it satisfies properties that are not fulfilled by other popular methods such as the generalized estimating equations approach the proposed method models both the marginal and the association structure in a flexible way the attractiveness of the multivariate dale model is illustrated in three key examples covering areas such as crossover trials longitudinal studies with patients dropping out from the study and discriminant analysis applications the differences and similarities with the generalized estimating approach are highlighted

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

statistical inference for uniform stochastic ordering in several populations tion for k population problems under uniform stochastic ordering restrictions we derive closed form estimates even with right censored data by a reparameterization which reduces the problem to a well known isotonic regression problem we also derive the asymptotic distribution of the likelihood ratio statistic for testing equality of the k populations against the uniform stochastic ordering restriction this asymptotic distribution is of the chi bar square type as discussed by robertson wright and dykstra these distributional results are obtained by appealing to elegant results from empirical process theory and showing that the proposed test is asymptotically distribution free recurrence formulas are derived for the weights of the chi bar square distribution for particular cases the theory developed in this paper is illustrated by an example involving data for survival times for carcinoma of the oropharynx

discrete symmetries and spectral statistics we calculate the 2 point spectral statistics associated with a given irreducible representation i e symmetry class for time reversal invariant systems possessing discrete symmetries using semiclassical periodic orbit theory when the representation in question is real or pseudo real our results conform to those of the gaussian orthogonal ensemble goe of random matrices when it is complex we find instead gaussian unitary ensemble gue behaviour this provides a direct semiclassical explanation for the recent observation by leyvraz et al 1996 of gue correlations in the desymmetrized spectra of certain symmetric billiards in the absence of any time reversal invariance breaking e g magnetic fields

time delay resonances riemann zeros and chaos in a model quantum scattering system the quantum treatment of an intrinsically chaotic model scattering system originally studied by gutzwiller 1983 is extended to include the time delay and to make explicit the zeros of the riemann zeta function in the mathematical expressions for the scattering matrix s and the time delay the system consists of a particle moving on a two dimensional surface of constant negative curvature the scattering in this unusual system is dominated by resonances associated with poles of s in the complex momentum plane the real parts of these poles are one half of the imaginary parts of the riemann zeros the resonances have a constant width but their average spacing varies with the momentum the focal point is the manifestation of chaotic behaviour in the time delay features considered in this regard include the momentum dependence of the time delay in regions of overlapping and isolated resonances the decomposition of the time delay into an average dynamical component and a fluctuating chaotic component and characterisation of the fluctuating component by its autocorrelation function

some geometrical models of chaotic dynamics the free motion of a particle on a surface of constant negative curvature a pseudosphere was one of the first models of chaotic motion it became the prototype for the theory of hyperbolic systems developed by bowen and sinai in these models geometry suggests a symbolic coding which already exhibits fully chaotic behaviour one can return to these models to seek possible manifestations of quantum chaos here the mathematical technique is harmonic analysis on hyperbolic space chaotic behaviour seems to appear both in the behaviour of individual eigenfunctions and in the sequence of spectral values

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

on the quantum reconstruction of the riemann zeros we discuss a possible spectral realization of the riemann zeros based on the hamiltonian h xp perturbed by a term that depends on two potentials which are related to the berry keating semiclassical constraints we find perturbatively the potentials whose jost function is given by the zeta function it for 1 for 1 2 we find the potentials that yield the smooth approximation to the zeros we show that the existence of potentials realizing the zeta function at 1 2 as a jost function would imply that the riemann zeros are point like spectra embedded in the continuum resolving in that way the emission spectral interpretation

inference for likelihood ratio ordering in the two sample problem abstract we obtain the maximum likelihood estimators of two multinomial probability vectors under the constraint that they are likelihood ratio ordered we extend this estimation approach to the case of two univariate distributions and show strong consistency of the estimators we also derive and study the asymptotic distribution of the likelihood ratio statistic for testing the equality of two discrete probability distributions against the alternative that one distribution is greater than the other in the likelihood ratio ordering sense finally we examine a data set pertaining to average daily insulin dose from the boston collaborative drug surveillance program and compare our testing procedure to testing procedures for other stochastic orderings