linearly stable orbits in 3 dimensional billiards we construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart it shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing lyapunov exponents cannot be easily extended to 3 dimensions

a system of one dimensional balls with gravity we introduce a hamiltonian system with many degrees of freedom for which the nonvanishing of some lyapunov exponents almost everywhere can be established analytically

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

caustics for inner and outer billiards with a plane closed convex curve t we associate two area preserving twist maps the classical inner billiard int and the outer billiard in the exterior oft the invariant circles of these twist maps correspond to certain plane curves the inner and the outer caustics oft we investigate how the shape oft determines the possible location of caustics establish the existence of open regions which are free of caustics and estimate fro below the size of these regions in terms of the geometry oft

survival probability for the stadium billiard abstract we consider the open stadium billiard consisting of two semicircles joined by parallel straight sides with one hole situated somewhere on one of the sides due to the hyperbolic nature of the stadium billiard the initial decay of trajectories due to loss through the hole appears exponential however some trajectories bouncing ball orbits persist and survive for long times and therefore form the main contribution to the survival probability function at long times using both numerical and analytical methods we concur with previous studies that the long time survival probability for a reasonably small hole drops like c o n s t a n t t i m e 1 here we obtain an explicit expression for the constant

hyperbolic billiards on surfaces of constant curvature we establish sufficient conditions for the hyperbolicity of the billiard dynamics on surfaces of constant curvature this extends known results for planar billiards using these conditions we construct large classes of billiard tables with positive lyapunov exponents on the sphere and on the hyperbolic plane

diagnostics for a cumulative multinomial generalized linear model with applications to grouped toxicological mortality data abstract toxicologists frequently conduct toxicity experiments in which different treatment conditions are applied to groups of animals and the resulting mortality in each group is measured at a number of discrete time points over the course of the experiment in this article we develop and extend a number of diagnostic tools for the detection of mean misspecification or systematic departures of the mean link specification in cumulative multinomial generalized linear models fit to such data several real data sets are used to illustrate these diagnostics these tools help the analyst to differentiate between two sources of lack of fit in such models mean misspecification and extra multinomial variation or overdispersion

on power and sample size calculations for likelihood ratio tests in generalized linear models summary a direct extension of the approach described in self mauritsen and ohara 1992 biometrics 48 31 39 for power and sample size calculations in generalized linear models is presented the major feature of the proposed approach is that the modification accommodates both a finite and an infinite number of covariate configurations furthermore for the approximation of the noncentrality of the noncentral chi square distribution for the likelihood ratio statistic a simplification is provided that not only reduces substantial computation but also maintains the accuracy simulation studies are conducted to assess the accuracy for various model configurations and covariate distributions

hard chaos in magnetic billiards on the hyperbolic plane in this paper some results on the local and global stability analysis of magnetic billiard systems established on two dimensional riemannian manifolds of constant curvature are presented with particular emphasis on the hyperbolic plane for special billiards possessing a discrete group of rotational or translational symmetry a geometrical theorem illustrated by numerical simulations is given on the stability of trajectories with the same symmetry we also present sufficient criteria for the global hyperbolicity of the dynamics hard chaos and give lower estimations for the lyapunov exponent in terms of the shape of the billiard

bifurcations and complete chaos for the diamagnetic kepler problem we describe the structure of bifurcations in the unbounded classical diamagnetic kepler problem we conjecture that this system does not have any stable orbits and that the non wandering set is described by a complete trinary symbolic dynamics for scaled energies larger then epsilon c 0 328782 ldots

optimal designs for binary data abstract this article is concerned with the problem of selecting the values of the explanatory variable in logistic regression to obtain likelihood based confidence regions of minimum area one way of finding analytic solutions is by replacing the log likelihood by a quadratic surface around the maximum approximating in this way the likelihood regions by ellipses the optimal allocation for this local approximation is derived without unnecessary restrictions since the implementation of the optimal allocation requires accurate initial estimates a two stage procedure that uses the second stage to complement the first is recommended an alternative approach is to deal directly with the likelihood regions in this case the identification of the globally optimal allocation calls for numerical integration and optimization the results presented here attempt to strengthen those introduced by abdelbasit and plackett 1983 who derived optimal allocation for the local criterion under the restriction of symmet

a practical asymptotical optimal sor method abstract this paper presents a practical asymptotical optimal successive over relaxation sor method for solving the large sparse linear system based on two optimization models asymptotically optimal relaxation factors are given which are computed by solving the low order polynomial equations in each iteration the coefficients of the two polynomials are determined by the residual vector and the coefficient matrix a of the real linear system the numerical examples show that the new methods are more feasible and effective than the classical sor method

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

the ergodicity of a hamiltonian system of two particles in an external field abstract for a hamiltonian system of two one dimensional balls in an external field we prove the ergodicity if the total energy of the system is fixed our proof is an extension of that applied earlier to dispersing billiard systems

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

nonnull asymptotic distributions of three classic criteria in generalised linear models summary this paper develops simple formulae for the asymptotic expansions up to order n of the distributions of the likelihood ratio wald and score statistics in generalised linear models under a sequence of pitman alternatives the powers of all three criteria which are equivalent to first order are compared under specific conditions based on second order approximations the asymptotic distributions of all three statistics are obtained for testing a subset of linear parameters when the dispersion parameter is known for testing the dispersion parameter and for testing the whole set of linear parameters assuming that the dispersion parameter is unknown the formulae derived are simple enough to be used analytically to obtain closed form expressions for these expansions in special models with a fisher information matrix in closed form

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

the optimal ω is not best for the sor iteration method abstract the sor iteration method is popular for solving many of the large sparse systems of linear algebraic equations which are used to approximate many of the partial differential equations which arise in engineering we consider the matrix equation au w when david young s sor theory is applicable for example when the matrix a is a symmetric block tridiagonal matrix it is well known that using the optimal relaxation factor b produces the smallest possible spectral radius of the associated sor iteration matrix l it is also known that no polynomial acceleration can reduce the size of the spectral radius when the relaxation factor b is used therefore it has been assumed that it is best to use the combination of the unaccelerated sor iteration together with the optimal relaxation factor b the spectral radius is b 1 but we show that a smaller average spectral radius can be achieved by using a polynomial acceleration together with a suboptimal relaxation factor b

ergodic properties of plane billiards with symmetric potentials we study chaotic behaviour of the motion of a particle moving like in a billiard table outside some disks where a symmetric potential acts quadratic forms introduced in markarian 1988 to study non vanishing lyapunov exponents are used

big islands in dispersing billiard like potentials abstract we derive a rigorous estimate of the size of islands in both phase space and parameter space appearing in smooth hamiltonian approximations of scattering billiards the derivation includes the construction of a local return map near singular periodic orbits for an arbitrary scattering billiard and for the general smooth billiard potentials thus universality classes for the local behavior are found moreover for all scattering geometries and for many types of natural potentials which limit to the billiard flow as a parameter 0 islands of polynomial size in appear this suggests that the loss of ergodicity via the introduction of the physically relevant effect of smoothening of the potential in modeling for example scattering molecules may be of physically noticeable effect

optimal semi iterative methods for complex sor with results from potential theory we consider the application of semi iterative methods sim to the standard sor method with complex relaxation parameter under the following two assumptions 1 the associated jacobi matrix j is consistently ordered and weakly cyclic of index 2 and 2 the spectrum j of j belongs to a compact subset of the complex plane mathbb c which is symmetric with respect to the origin by using results from potential theory we determine the region of optimal choice of omega in mathbb c for the combination sim sor and settle for a large class of compact sets the classical problem of characterising completely all the cases for which the use of the sim sor is advantageous over the sole use of sor under the hypothesis that sigma j subset sigma in particular our results show that unless the outer boundary of is an ellipse sim sor is always better and furthermore one of the best possible choices is an asymptotically optimal sim applied to the gauss seidel method in addition we derive the optimal complex sor parameters for all ellipses which are symmetric with respect to the origin our work was motivated by recent results of m eiermann and r s varga

is the optimal ω best for the sor iteration method abstract the successive overrelaxation sor iterative method for linear systems is well understood if the associated jacobi matrix b is consistently ordered and weakly cyclic of index 2 if in addition b 2 has only nonnegative eigenvalues and if b the spectral radius of b is strictly less than unity then by d m young s classical theorem the optimal relaxation parameter for the sor method is given by b 2 1 1 2 b young derived this result assuming that b 2 0 2 with b is the only information available about the spectrum b 2 of b 2 it is also well known that no polynomial acceleration can improve the asymptotic rate of convergence of the sor scheme if the optimal relaxation parameter has been selected the recent claim by j dancis that a smaller average spectral radius can be achieved by using a polynomial acceleration together with a suboptimal relaxation factor b therefore comes as a surprise a closer look however reveals that this improvement can only be achieved if more profound information on b 2 of the form b 2 0 2 2 with is at hand we show that no polynomial acceleration of the sor method for any real is asymptotically faster than the sor scheme with b under the assumption 08 thereby answering the question in the title of this paper in the affirmative as well as solving an old related conjecture of d m young we also carefully investigate the question of what can be gained from the additional information

billiards with pesin region of measure one we give a description of a large class of plane billiards with pesin region of measure one open conditions including properly those founded by wojtkowski w1 forc4 focusing boundaries are obtained lyapunov s forms introduced by lewowicz are used

improved likelihood ratio statistics for exponential family nonlinear models summary this paper gives a general formula for the expected likelihood ratio statistic for exponential family nonlinear models corrected up to order n 1 where n is the sample size this result generalizes previous work by cordeiro 1983 various general bartlett adjustment formulae are given to some special models and improved likelihood ratio tests are discussed

semi focusing billiards hyperbolicity in this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of 3 we also prove that a related class of semi focusing billiards has mixed dynamics i e their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other these billiards are the first rigorous examples of billiards in domains of 3 with divided phase space

stability in high dimensional steep repelling potentials the appearance of elliptic periodic orbits in families of n dimensional smooth repelling billiard like potentials that are arbitrarily steep and limit to sinai billiards is established for any finite n for typical potentials the stability regions in the parameter space scale as a power law in 1 n and in the steepness parameter thus it is shown that even though these systems have a uniformly hyperbolic albeit singular limit the ergodicity of this limit system is destroyed in the more realistic smooth setting the considered example is highly symmetric and is not directly linked to the smooth many particle problem nonetheless the possibility of explicitly constructing stable motion in smooth n degrees of freedom systems limiting to strictly dispersing billiards is now established

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

on the non integrability of some generalized toda lattices abstract we consider a one parameter family of two degrees of freedom generalized toda hamiltonians using ziglin s theorem with some extensions we prove the non existence of an additional single valued analytic integral except for special values of the parameter

design of hyperbolic billiards we formulate a general framework for the construction of hyperbolic billiards spherical symmetry is exploited for a simple treatment of billiards with spherical caps and soft billiards in higher dimensions other examples include the papenbrock stadium

improved likelihood ratio statistics for generalized linear models summary this paper gives a general formula for the expected value of the likelihood ratio statistic for generalized linear models corrected up to terms of order n 1 where n is the size of the sample the application of this formula to some models is illustrated

a rapidly convergent series for computing and its derivatives we derive a series expansion for ap z in which the terms of the expansion are simple rational functions of z from a computational viewpoint the new series is of interest in that it converges for all z not necessarily real valued and is particularly rapid for values of z near the origin from a mathematical viewpoint the series is of interest in that although 4i z has poles at the negative integers and zero the series is uniformly convergent in any finite interval a re z b a series expansion for a z the derivative of the log gamma function usually denoted by a z d log r z dz is a regular function with simple poles at the negative integers and zero no power series can therefore have infinite radius of convergence and this constraint limits the speed at which the coefficients in such a power series decrease in this paper we give a series expansion for ab z that has infinite radius of convergence despite the poles the expansion is not a power series but converges uniformly to a z in any finite interval convergence is particularly rapid for small z say jzj 1 0 and in this range ten terms of the expansion give accuracy to at least ten decimal places for computational purposes the recurrence relation 4 z 1 a z 1 z can be used for real z to obtain a value of z near the origin extremely large values of z are best dealt with by asymptotic expansions the taylor series expansion for 4p 1 z about z 0 abramowitz and stegun 1 eq 6 3 14 is 1 4p 1 z y 2 1 r id r 1 zr zi 1 r 1 where y is euler s constant and t r 1 ew ik ri is riemann s zeta function the coefficients of the power series 1 do not tend to zero so that the speed of convergence depends entirely on the magnitude of izl to obtain a more rapidly convergent series we rewrite 1 as a double series and regroup to obtain 2 ip 1 z y i 1 r zr cr ar z r jzj o r 1 where ar r r and cr k i r ik r l it is readily shown that lim rr r e 1 1 0 582 so that both ar and cr diminish rapidly to zero the series 2 therefore converges pointwise for all finite z and it is therefore the analytic continuation of 1 to establish uniform convergence in any finite interval a re z b let gj z be the sum of the first n 1 terms in 2 counting the constant received march 3 1980 1980 mathematics subject classification primary 33a15 secondary 65d20 i 1981 american mathematical society 0025 571 8 81 oooq 0020 01 50 247 this content downloaded from 157 55 39 185 on fri 16 sep 2016 05 35 55 utc all use subject to http about jstor org terms

is the improved likelihood ratio statistic really improved in the discrete case summary on the basis of some numerical studies we argue that bartlett adjustment of the likelihood ratio statistic for discrete data does not in general yield an improvement in the asymptotic error rate of the chi squared approximation also we argue that the small sample effect of the bartlett adjustment is unclear and that it is in no way comparable with the improvement obtained for continuous data

a note on likelihood ratio tests for models with latent variables the likelihood ratio test lrt is widely used for comparing the relative fit of nested latent variable models following wilks theorem the lrt is conducted by comparing the lrt statistic with its asymptotic distribution under the restricted model a formula see text distribution with degrees of freedom equal to the difference in the number of free parameters between the two nested models under comparison for models with latent variables such as factor analysis structural equation models and random effects models however it is often found that the formula see text approximation does not hold in this note we show how the regularity conditions of wilks theorem may be violated using three examples of models with latent variables in addition a more general theory for lrt is given that provides the correct asymptotic theory for these lrts this general theory was first established in chernoff j r stat soc ser b methodol 45 404 413 1954 and discussed in both van der vaart asymptotic statistics cambridge cambridge university press 2000 and drton ann stat 37 979 1012 2009 but it does not seem to have received enough attention we illustrate this general theory with the three examples

chaotic scattering by steep repelling potentials consider a classical two dimensional scattering problem a ray is scattered by a potential composed of several tall repelling steep mountains of arbitrary shape we study when the traditional approximation of this nonlinear far from integrable problem by the corresponding simpler billiard problem of scattering by hard wall obstacles of similar shape is justified for one class of chaotic scatterers named here regular sinai scatterers the scattering properties of the smooth system indeed limit to those of the billiards for another class the singular sinai scatterers these two scattering problems have essential differences though the invariant set of such singular scatterers is hyperbolic possibly with singularities that of the smooth flow may have stable periodic orbits even when the potential is arbitrarily steep it follows that the fractal dimension of the scattering function of the smooth flow may be significantly altered by changing the ratio between the steepness parameter and a parameter which measures the billiards deviation from a singular scatterer thus even in this singular case the billiard scattering problem is utilized as a skeleton for studying the properties of the smooth flow finally we see that corners have nontrivial and significant impact on the scattering functions

laplace approximation of high dimensional integrals summary it is shown that the usual laplace approximation is not a valid asymptotic approximation when the dimension of the integral is comparable with the limiting parameter n the formal laplace expansion for multidimensional integrals is given and used to construct asymptotic approximations for high dimensional integrals one example is considered in which the dimension of the integral is o n 1 2 and the relative error of the unmodified laplace approximation is 0 1 nevertheless it is possible to construct a valid asymptotic expansion by regrouping terms in the formal expansion according to asymptotic order in n

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

intrinsic stickiness and chaos in open integrable billiards tiny border effects rounding border effects at the escape point of open integrable billiards are analyzed via the escape time statistics and emission angles the model is the rectangular billiard and the shape of the escape point is assumed to have a semicircular form stickiness chaos and self similar structures for the escape times and emission angles are generated inside backgammon like stripes of initial conditions these stripes are born at the boundary between two different emission angles but with the same escape times and when rounding effects increase they start to overlap generating a very rich dynamics tiny rounded borders around 0 1 from the whole billiard size are shown to be sufficient to generate the sticky motion with power law decay esc 1 27 while borders larger than 10 are enough to produce escape times related to the chaotic motion escape exponents in the interval 1 esc 2 are generated due to marginal unstable periodic orbits trapping alternately in time regular and chaotic trajectories

using integrability to produce chaos billiards with positive entropy a new open class of convex 2 dimensional planar billiards with positive lyapunov exponent almost everywhere is constructed we introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive lyapunov exponents we prove that under smallc6 perturbations focusing arcs remain focusing and thereby show that perturbations of the bunimovich stadium billiard have positive lyapunov exponents

integrable hamiltonians with exponential potential abstract we construct a large class of integrable hamiltonian systems with n degrees of freedom this class naturally extends the nonperiodic hamiltonians of toda lattice type

a theorem on ergodicity of two dimensional hyperbolic billiards ergodicity of two dimensional billiards which satisfy some general conditions is proved this theorem is applied to one concrete class of billiards that contains in particular billiards in the stadium

a modern maximum likelihood theory for high dimensional logistic regression students in statistics or data science usually learn early on that when the sample size n is large relative to the number of variables p fitting a logistic model by the method of maximum likelihood produces estimates that are consistent and that there are well known formulas that quantify the variability of these estimates which are used for the purpose of statistical inference we are often told that these calculations are approximately valid if we have 5 to 10 observations per unknown parameter this paper shows that this is far from the case and consequently inferences produced by common software packages are often unreliable consider a logistic model with independent features in which n and p become increasingly large in a fixed ratio we prove that i the maximum likelihood estimate mle is biased ii the variability of the mle is far greater than classically estimated and iii the likelihood ratio test lrt is not distributed as a 2 the bias of the mle yields wrong predictions for the probability of a case based on observed values of the covariates we present a theory which provides explicit expressions for the asymptotic bias and variance of the mle and the asymptotic distribution of the lrt we empirically demonstrate that these results are accurate in finite samples our results depend only on a single measure of signal strength which leads to concrete proposals for obtaining accurate inference in finite samples through the estimate of this measure

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

power law behavior of lyapunov exponents in some conservative dynamical systems abstract three billiards whose border depends on a parameter e are considered for e 0 they are integrable while for e 0 they are known to be k systems the maximal lyapunov exponent 1 is numerically computed and a power law behavior 1 e is found with 1 2 for all billiards a related abstract problem is then considered precisely the case of an infinite product of conservative 2 2 random matrices which are perturbations of commuting parabolic ones analogous computations give here two power laws with 1 2 or 2 3 depending on the probability law used to construct random matrices all these phenomena seem to have an universal character a heuristic mean field theory is also proposed based on the formal interpretation of the onset of stochasticity in our dynamical systems as a phase transition

block cyclic sor for markov chains with p cyclic infinitesimal generator we introduce a new application of p cyclic iterations for arbitrary p 2 the block sor method for the computation of the steady state distribution of finite markov chains that possess p cyclic infinitesimal generators is considered it is shown that convergence in a sense more general than the usual may be obtained even if the sor iteration violates the usual conditions for semiconvergence necessary and sufficient conditions for convergence in this extended sense are derived they are then applied in the case where the pth power of the associated jacobi matrix of the system to be solved possesses only nonnegative eigenvalues exact convergence intervals and the optimal values are derived for this case in addition to the usual optimal in the interval 1 p p 1 other values that yield convergence in the extended sense are found to achieve the same optimal convergence rate numerical tests indicate that small perturbations of around the optimal value affect the convergence factor much less if these newly introduced optimal values are used

on the bernoulli property of planar hyperbolic billiards we consider billiards in non polygonal domains of the plane with boundary consisting of curves of three different types straight segments strictly convex inward curves and strictly convex outward curves of a special kind the billiard map for these domains is known to have non vanishing lyapunov exponents a e provided that the distance between the curved components of the boundary is sufficiently large and the set of orbits having collisions only with the flat part of the boundary has zero measure under a few additional conditions we prove that there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained up to a zero measure set in one bernoulli component of the billiard map using this result we show that there exists a large class of planar hyperbolic billiards that have the bernoulli property this class includes the billiards in convex domains bounded by straight segments and strictly convex inward arcs constructed by donnay

a note on the calculation of bartlett adjustments summary we describe for the numerical calculation of bartlett adjustments a method which may be of use when the cumulants of the log likelihood derivatives are easy to determine in one parametrization while the hypotheses to be tested are all linear in some other parametrization the method relies on the cartesian tensorial nature of those cumulants and should be particularly convenient in connection with statistical packages with structure similar to glim

focusing components in typical chaotic billiards should be absolutely focusing we demonstrate that the defocusing mechanism fails to work if not all focusing components of the boundary are absolutely focusing more precisely we construct billiard tables with arbitrary long free path away from a non absolutely focusing component such that a nonlinearly stable periodic orbit exists therefore the only known standard procedure of constructing chaotic ergodic billiards works in general only if all focusing boundary components are absolutely focusing

hyperbolic billiards with nearly flat focusing boundaries i abstract the standard wojtkowski markarian donnay bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary this is due to the physical principle that is used in the proofs the so called defocusing mechanism of geometrical optics in this paper we construct examples of hyperbolic billiards with a focusing boundary component of arbitrarily small curvature whose diameter is bounded by a constant independent of that curvature our proof employs a nonstandard cone bundle that does not solely use the familiar dispersing and defocusing mechanisms

second order asymptotics for score tests in generalised linear models summary this paper develops finite sample corrections for score tests in generalised linear models when the dispersion parameter is unknown thus generalising the results of cordeiro ferrari paula 1993 we show that the coefficients which define the edgeworth expansion for the null distribution of the score statistic when the dispersion parameter is unknown are the coefficients obtained by these authors plus some extra terms we also give closed form expressions for such coefficients an important special case of our results is the normal linear regression model simulation results for this model show that the corrections can be very effective reducing the probability of conflict with other asymptotically equivalent testing criteria

many dimensional nowhere dispersing billiards with chaotic behavior abstract we consider some examples of domains in the d dimensional euclidean space d 3 that generate billiards with chaotic behavior time correlations in these billiards tend to zero in the limit t the rate of decay of the correlations is a power law for billiards in discrete time and could be a power law or quasiexponential for billiards with continuous time

soft billiards with corners we develop a framework for dealing with smooth approximations to billiards with corners in the two dimensional setting let a polygonal trajectory in a billiard start and end up at the same billiard s corner point we prove that smooth hamiltonian flows which limit to this billiard have a nearby periodic orbit if and only if the polygon angles at the corner are acceptable the criterion for a corner polygon to be acceptable depends on the smooth potential behavior at the corners which is expressed in terms of a scattering function we define such an asymptotic scattering function and prove the existence of it explain how can it be calculated and predict some of its properties in particular we show that it is non monotone for some potentials in some phase space regions we prove that when the smooth system has a limiting periodic orbit it is hyperbolic provided the scattering function is not extremal there we then prove that if the scattering function is extremal the smooth system has elliptic periodic orbits limiting to the corner polygon and furthermore that the return map near these periodic orbits is conjugate to a small perturbation of the henon map and therefore has elliptic islands we find from the scaling that the island size is typically algebraic in the smoothing parameter and exponentially small in the number of reflections of the polygon orbit

hard chaos in magnetic billiards on the euclidean plane this work presents local and global results on the stability of the dynamics of classical magnetic billiard systems with homogeneous magnetic field established on the euclidean plane in the first part of the paper our previous results concerning the properties of the stability matrices on curved riemannian manifolds are rederived in a simpler elementary way in the euclidean case as applications the stability regions for special symmetric orbits are determined analytically and numerically using a new technique necessary conditions for hard chaos and lower estimations for the lyapunov exponent are given for planar magnetic billiards with dispersing and focusing boundary segments too it is also shown that in the investigated billiard types hard chaos is structurally stable below a certain threshold magnetic field

power calculations for likelihood ratio tests in generalized linear models summary recently self and mauritsen 1988 biometrics 44 79 86 described an approach for sample size power calculations within the framework of generalized linear models that is based on a noncentral chi square approximation to the distribution of score statistics in this work we present another approach to this same problem that is based on a noncentral chi square approximation to the distribution of the likelihood ratio statistic this approach is easier to implement than the previous one and simulation studies indicate that it is accurate over a much wider range of parameter values and data configurations than the earlier method the new method is also applied to the problem of power calculations for 1 k matched case control studies which is related to but formally outside of the generalized linear model framework

improved likelihood ratio statistics for covariance selection models summary the expected value of the likelihood ratio statistic for covariance selection models when this statistic has a closed form is derived correct to terms of order n 2 n being the sample size then a bartlett type correction factor is available for such statistics by simulation the practical use of such a correction factor is illustrated and is compared with an alternative correction factor in a special case the null density of the corrected statistic is also considered

static and time dependent perturbations of the classical elliptical billiard the elliptical billiard problem defines a two dimensional integrable discrete dynamical system integrability not being a robust property we study some static and time dependent perturbations of this problem for the static case we observe the transition from integrability to chaos on some perturbations of the ellipse then we study time dependent perturbations supposing that the boundary deforms periodically with the time remaining always an ellipse we investigate numerically the now four dimensional phase space looking mainly at the question of whether or not the velocity of a given trajectory may increase indefinitely

a modified score test statistic having chi squared distribution to order n 1 summary we give a general modified score test statistic whose nuil distribution is chi squared to order n 1 where n is the sample size the modified statistic depends on the joint cumulants of log likelihood derivatives for the full data some applications are discussed following cox reid 1987a we also derive a general formula for bartlett type corrections to improve test statistics whose null asymptotic distributions are chi squared

wavefunctions expectation values and scars on poincare sections a scattering approach using the scattering approach to quantization we define an appropriate representation of the billiard wavefunction on the scattering poincar section the expectation values of smooth operators in terms of these poincar section wavefunctions are expressed as sums over periodic orbits a special operator is used to define scars on the section and the relation to scars in configuration space is discussed and demonstrated numerically

bias correction in generalized linear models summary in this paper we derive general formulae for first order biases of maximum likelihood estimates of the linear parameters linear predictors the dispersion parameter and fitted values in generalized linear models these formulae may be implemented in the glim program to compute bias corrected maximum likelihood estimates to order n 1 where n is the sample size with minimal effort by means of a supplementary weighted regression for linear logistic models it is shown that the asymptotic bias vector of j8 is almost collinear with f3 the approximate formula flp m for the bias of s in logistic models where p dim and m e mi is the sum of the binomial indices is derived and checked numerically

improved score tests for generalized linear models summary we present in matrix notation an asymptotic expansion for the null distribution of the score statistic in generalized linear models corrected up to order n 1 where n is the sample size the expansion has advantages for numerical purposes because it requires only simple operations on matrices it is also sufficiently simple to be used analytically to obtain several closed form expressions analogous to bartlett corrections in a variety of important score tests improved score tests for these models are discussed and various applications to some special tests are given we show by simulation that two asymptotically equivalent adjusted score tests seem to improve at least for continuous data on the usual score test