the em algorithm an old folk song sung to a fast new tune celebrating the 20th anniversary of the presentation of the paper by dempster laird and rubin which popularized the em algorithm we investigate after a brief historical account strategies that aim to make the em algorithm converge faster while maintaining its simplicity and stability e g automatic monotone convergence in likelihood first we introduce the idea of a working parameter to facilitate the search for efficient data augmentation schemes and thus fast em implementations second summarizing various recent extensions of the em algorithm we formulate a general alternating expectation conditional maximization algorithm aecm that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations we illustrate these methods using multivariate t models with known or unknown degrees of freedom and poisson models for image reconstruction we show through both empirical and theoretical evidence the potential for a dramatic reduction in computational time with little increase in human effort we also discuss the intrinsic connection between em type algorithms and the gibbs sampler and the possibility of using the techniques presented here to speed up the latter the main conclusion of the paper is that with the help of statistical considerations it is possible to construct algorithms that are simple stable and fast

application of supersymmetric wkb method to cyclic shape invariant potentials we examine the accuracy and the validity of the lowest order supersymmetric wkb swkb formula for cyclic shape invariant potentials csips for period 2 csips we show analytically that the swkb formula can yield exact eigenenergies for either all the even states or all the odd states such alternate exactness of the swkb formula is due to the fact that period 2 csips can also be considered as a kind of translational shape invariant potential for csips with periods greater than 2 we note that the accuracy of the swkb formula also demonstrates similar alternating patterns however as a consequence of the rapid oscillations in the potential at large distances the swkb quantization formula fails to produce highly accurate results even in the high energy limit

parameter expansion to accelerate em the px em algorithm summary the em algorithm and its extensions are popular tools for modal estimation but ar e often criticised for their slow convergence we propose a new method that can often make em much faster the intuitive idea is to use a covariance adjustment to correct the analysis of the m step capitalising on extra information captured in the imputed complete data the way we accomplish this is by parameter expansion we expand the complete data model while preserving the observed data model and use the expanded complete data model to generate em this parameter expanded ei m px em algorithm shares the simplicity and stability of ordinary em but has a faster rate of convergence since its m step performs a more efficient analysis the px em algorithm is illustrated for the multivariate t distribution a random effects model factor analysis probit regression and a poisson imaging model

a multivariate random effects model with restricted parameters application to assessing radiation therapy for brain tumours in clinical studies multiple endpoints are often measured for each patient longitudinally the multivariate random effects or random coefficient model has been a useful method for analysis however medical research problems may impose restrictions on the model parameters of interests for example in a paediatric brain tumour study on radiation therapy there is a natural ordering in the white matter relaxation time of brain tissues among different regions surrounding the primary tumour i e the closer a specific region of brain tissues is to the centre of primary tumour the shorter is the relaxation time such parameter constraints should be accounted for in the analysis this article proposes a class of multivariate random coefficient models with restricted parameters and derives its maximum likelihood estimates mle we propose a modified em algorithm for the quadratic optimalization with linear inequality constraints necessary in deriving the mle the method is applied to analysing the paediatric brain tumour study

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

iterative image reconstruction algorithms based on cross entropy minimization the related problems of minimizing the functionals f x alpha kl y px 1 alpha kl p x and g x alpha kl px y 1 alpha kl x p respectively over the set of vectors x or 0 are considered kl a b is the cross entropy or kullback leibler distance between two nonnegative vectors a and b iterative algorithms for minimizing both functionals using the method of alternating projections are derived a simultaneous version of the multiplicative algebraic reconstruction technique mart algorithm called smart is introduced and its convergence is proved

on the oscillation of some linear difference equations with periodic coefficients abstract some linear difference equations with periodic coefficients not necessarily nonnegative are considered necessary conditions and sufficient conditions for the oscillation of the solutions are established conditions under which all nonoscillatory solutions tend to zero at are also presented the results obtained are the discrete analogues of the oscillation results for some linear delay differential equations with periodic coefficients which were given earlier by the second author oscillations of some delay differential equations with periodic coefficients j math anal appl 162 1991 452 475

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

on the bias in estimating genetic length and other quantities in simplex constrained models the genetic distance between two loci on a chromosome is defined as the mean number of crossovers between the loci the parameters of the crossover distribution are constrained by the parameters of the distribution of chiasmata ott 1996 derived the maximum likelihood estimator mle of the parameters of the crossover distribution and the mle of the mean we demonstrate that the mle of the mean is pointwise less than or equal to the empirical mean number of crossovers it follows that the mle is negatively biased for small sample sizes the bias can be nonnegligible we recommend reduced bias estimators generalizations to many other problems involving linear constraints on parameters are made included in the generalizations are a variety of problems involving simplex constraints as studied recently by liu 2000

the global holder continuous solution of isentropic gas dynamics this paper considers the cauchy problem for the isentropic equations of gas dynamics in euler coordinates t u x 0 u t u 2 p x 0 and gives the holder continuous solution by applying the method of vanishing viscosity

low energy levels and matrix elements in the large n small h cross asymptotic region the asymptotic form of the froman and froman solutions 1977 to the schrodinger equation has been used to calculate the low energy spectra and corresponding matrix elements for many well potentials it has been proved that the bohr sommerfeld quantisation condition is also valid in the low energy region independently for each well it has also been shown that the corresponding matrix elements are dominated by the energy levels belonging to the same well the general results have then been used to show that in n dimensions with spherically symmetric hamiltonians quantum systems approach their classical configurations with constraints when n to infinity

symmetries and global solvability of the isothermal gas dynamics equations we study the cauchy problem associated with the system of two conservation laws arising in isothermal gas dynamics in which the pressure and the density are related by the law equation p with 1 our results complete those obtained earlier for 1 we prove the global existence and compactness of entropy solutions generated by the vanishing viscosity method the proof relies on compensated compactness arguments and symmetry group analysis interestingly we make use here of the fact that the isothermal gas dynamics system is invariant modulo a linear scaling of the density this property enables us to reduce our problem to that with a small initial density

existence and asymptotic behavior of solutions to inhomogeneous equations of elasticity with little viscosity we study the existence and asymptotic behavior of the solutions to inhomogeneous equations of elasticity with little viscosity u t x v f u v eu xx v t u x g u v ev xx with initial value u v x 0 u 0 x v 0 x here v is a given smooth strongly monotone function on r v a 2 v a v 0 ver and f u v and g u v are given smooth functions on r 2 and e is a positive constant

quasiconvexity and relaxation of nonconvex problems in the calculus of variations keywords quasiconvexity relaxation reference caa article 1982 002doi 10 1016 0022 1236 82 90046 5view record in web of science record created on 2008 11 25 modified on 2017 05 12

singular limits of stiff relaxation and dominant diffusion for nonlinear systems abstract we are concerned with singular limits of stiff relaxation and dominant diffusion for general 2 2 nonlinear systems of conservation laws that is the relaxation time tends to zero faster than the diffusion parameter e o e e 0 we establish the following general framework if there exists an a priori l bound that is uniformly with respect to e for the solutions of a system then the solution sequence converges to the corresponding equilibrium solution of this system our results indicate that the convergent behavior of such a limit is independent of either the stability criterion or the hyperbolicity of the corresponding inviscid quasilinear systems which is not the case for other type of limits this framework applies to some important nonlinear systems with relaxation terms such as the system of elasticity the system of isentropic fluid dynamics in eulerian coordinates and the extended models of traffic flows the singular limits are also considered for some physical models without l bounded estimates including the system of isentropic fluid dynamics in lagrangian coordinates and the models of traffic flows with stiff relaxation terms the convergence of solutions in l p to the equilibrium solutions of these systems is established provided that the relaxation time tends to zero faster than e

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

exactness of supersymmetric wkb method for translational shape invariant potentials by examining the generic form of the superpotential of translational shape invariant potentials tsips we explicitly show the exactness of the lowest order supersymmetric wkb swkb formula for tsips remarkably our method applies to both unbroken and broken supersymmetric systems we also demonstrate the equivalence of one parameter and multi parameter tsips thus establishing the exactness of the swkb formula for all tsips

quantum anharmonic oscillators a new approach the determination of the eigenenergies of a quantum anharmonic oscillator consists merely in finding the zeros of a function of the energy namely the wronskian of two solutions of the schr dinger equation which are regular respectively at the origin and at infinity we show in this paper how to evaluate that wronskian exactly except for numerical rounding errors the procedure is illustrated by application to the gx2 x2n n a positive integer oscillator

approximate solutions of functional equations with small delay using the wkb method a class of functional equations containing delay terms in which the delay is small compared with an overall scale or difference equations with slowly varying coefficients is analysed using the wkb method a particular example in which a periodic solution is required is analysed in detail results analogous to quantization conditions are discussed as are turning points the approximate solutions obtained by the wkb method are compared with exact solutions where available in simple cases and with numerical integration and excellent agreement is obtained

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

convergence of the viscosity solutions for the system of nonlinear elasticity abstract in this paper some special entropy pairs of lax type are constructed for the system of nonlinear elasticity in which the progressive terms are functions of a single variable the necessity estimates for the major terms are obtained by using the theory of singular perturbation of the ordinary differential equations the special entropy pairs are used to prove that the family of young measures uniquely determined by the viscosity solution sequence is a family of dirac measures hence a convergence theorem on the viscosity solutions is established by applying the method of compensated compactness

stark wannier ladders we study the schrodinger equation for an electron in a one dimensional crystal submitted to a constant electric field we prove the existence of ladders of resonances the imaginary part of which is exponentially small with the field

a relaxation theorem and its application to the equilibrium of gases keywords relaxed problem lower convex envelope gas equilibrium van der waal s equation reference caa article 1981 001doi 10 1007 bf00280643view record in web of science record created on 2008 11 25 modified on 2017 05 12

wkb approach to zero distribution of solutions of linear second order differential equations given a second order linear differential equation y z s z y z 0 the distribution of zeros of its solutions is defined by v y z 0 z where z stands for the dirac delta at the point z some techniques of approximation of the restriction of v to r directly from s z are considered in particular for the wkb method error bounds are provided and some related results established in the second part formulas for the appropriate scaling in the holonomic case are given as an illustration we obtain the asymptotic distribution of the real zeros of some families of polynomials

the comparison of the riemann solutions in gas dynamics in this paper we compare two riemann solution constructions of isentropic and nonisentropic gas dynamic equations with the same initial values we obtain that when rarefaction waves occur in the non isentropic solution the same family rarefaction waves occur in the isentropic solution however when advancing or reflected shock waves occur in the isentropic case the same kind shock waves occur in the nonisentropic case moreover by numerical calculation it can be found that the wave intensities of the two solutions are quite close our conclusion reflects that almost non isentropic gas dynamic process can be simulated by isentropic gas dynamic process in practice c 2015 elsevier inc all rights reserved

statistical mechanics of the periodic benjamin ono equation the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations the periodic benjamin ono equation is an autonomous hamiltonian system with a gibbs measure on l2 t this paper shows that the gibbs measures on bounded balls of l2 satisfy some logarithmic sobolev inequalities the space of n soliton solutions of the periodic benjamin ono equation as discovered by case is a hamiltonian system with an invariant gibbs measure as n these gibbs measures exhibit a concentration of measure phenomenon case introduced soliton solutions that are parameterized by atomic measures in the complex plane the limiting distributions of these measures give the density of a compressible gas that satisfies the isentropic euler equations

exactness of conventional and supersymmetric jwkb formulae and their relation to the shape invariance property of potentials a mechanism for the jwkb formulae in one dimensional quantum mechanics to quantize energy levels exactly is discussed this mechanism is most easily applied when considered potentials are represented by a suitable change of variable as periodic ones to be successful the mechanism demands potentials with no more than one second order pole and no more than two occasionally four turning points in the basic period strips it then selects 11 potentials which have been known in the literature for a long time it is also shown that the exactness of the supersymmetric susy jwkb formulae for these 11 potentials follows directly from the corresponding exactness of the conventional ones being a consequence of the singularity structures of the potentials a relation of these singularity structures to the shape invariance symmetry of the quantized potentials is shown to guarantee the conventional and susy jwkb formulae to be exact simultaneously the relevant two non perturbative theorems describing these facts are formulated and proved

hyperasymptotics for integrals with saddles integrals involving exp kf z where k is a large parameter and the contour passes through a saddle of f z are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles through which the contour does not pass these contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent the refinement is achieved by means of an exact resurgence relation expressing the original integral as its truncated saddle point asymptotic expansion plus a remainder involving the integrals through certain adjacent saddles determined by a topological rule iteration of the resurgence relation and choice of truncation near the least term of the original series leads to a representation of the integral as a sum of contributions associated with multiple scattering paths among the saddles no resummation of divergent series is involved each path gives a hyperseries depending on the terms in the asymptotic expansions for each saddle these depend on the particular integral being studied and so are non universal and certain hyperterminant functions defined by integrals these are always the same and hence universal successive hyperseries get shorter so the scheme naturally halts for two saddles the ultimate error is approximately varepsilon 2 386 where varepsilon proportional to exp a k where a is a positive constant is the error in optimal truncation of the original series as a numerical example an integral with three saddles is computed hyperasymptotically

a theoretical study of some maximum likelihood algorithms for emission and transmission tomography this paper has the dual purpose of introducing some new algorithms for emission and transmission tomography and proving mathematically that these algorithms and related antecedent algorithms converge like the em algorithms for positron single photon and transmission tomography the algorithms provide maximum likelihood estimates of pixel concentration or linear attenuation parameters one particular innovation we discuss is a computationally practical scheme for modifying the em algorithms to include a bayesian prior the bayesian versions of the em algorithms are shown to have superior convergence properties in a vicinity of the maximum we anticipate that some of the other algorithms will also converge faster than the em algorithms

estimation of discrete distributions with a class of simplex constraints abstract simplex constraints such as monotonicity and convexity or concavity on the probabilities of a set of discrete distributions are useful for modeling and analyzing discrete data this article considers both maximum likelihood estimation and bayesian estimation of discrete distribution with a class of simplex constraints using the expectation maximization em algorithm and the data augmentation da algorithm the formulation and implementation of em and da for binomial poisson hierarchical poisson binomial multinomial and hierarchical multinomial distributions are considered in detail and illustrated with examples

conjecture on the interlacing of zeros in complex sturm liouville problems the zeros of the eigenfunctions of self adjoint sturm liouville eigenvalue problems interlace for these problems interlacing is crucial for completeness for the complex sturm liouville problem associated with the schrodinger equation for a non hermitian pt symmetric hamiltonian completeness and interlacing of zeros have never been examined this paper reports a numerical study of the sturm liouville problems for three complex potentials the large n limit of a ix n potential a quasiexactly solvable x4 potential and an ix3 potential in all cases the complex zeros of the eigenfunctions exhibit a similar pattern of interlacing and it is conjectured that this pattern is universal understanding this pattern could provide insight into whether the eigenfunctions of complex sturm liouville problems form a complete set

superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum during the last years several theoretical and experimental works have been devoted to different aspects of the adiabatic regime of quantum dynamics we consider here two aspects of this physically important regime the construction of approximate evolutions of the schriidinger equation called superadiabatic evolutions and the explicit computation of the probability of a transition between two nondegenerate energy levels isolated in the spectrum of the hamiltonian from the mathematical point of view the adiabatic limit for time dependent systems is of the same kind as the semiclassical limit for stationary systems it is a singular limit which cannot be tackled by usual perturbation methods recently a mathematical theory of the two considered aspects has been developed when the time dependent hamiltonian is analytic in time the transition probability for two level systems has been studied in refs l 3 and was shown to be exponentially small in the adiabaticity parameter for general systems in ref 4 on the other hand the existence of superadiabatic evolutions has been shown in ref 5 however no general formula for the transition probability between isolated levels in the adiabatic limit can be found in the literature it is the goal of this paper to establish such a formula for general unbounded hamiltonians the paper is divided into two parts we first construct a superadiabatic evolution which approximates the true evolution of the schrijdinger equation up to exponentially small correction terms with respect to the parameter controlling the adiabatic regime once the superadiabatic evolution is defined we show that the computation of the transition probability of the initial problem reduces to an effective two dimensional problem which we analyze as in ref 1 the concept of superadiabatic evolution can be traced back up to a paper of garrido 6 who considered evolutions different from the usual adiabatic evolution which he used to obtain higher order corrections to the adiabatic theorem of quantum mechanics but it is berry7v8 who emphasized the importance of such evolutions in the present problem and introduced the

borel summable solutions to one dimensional schrodinger equation it is shown that so called fundamental solutions the semiclassical expansions of which have been established earlier to be borel summable to the solutions themselves appear also to be the unique solutions to the one dimensional 1d schrodinger equation having this property namely it is shown in this paper that for the polynomial potentials the borel function defined by the fundamental solutions can be considered as the canonical one the latter means that any borel summable solution can be obtained by the borel transformation of this unique canonical borel function multiplied by some dependent and borel summable constant this justifies the exceptional role the fundamental solutions play in 1d quantum mechanics and completes the relevant semiclassical theory relied on the borel resummation technique and developed in our other papers

eulerian droplet model delta shock waves and solution of the riemann problem abstract we study an eulerian droplet model which can be seen as the pressureless gas system with a source term a subsystem of this model and the inviscid burgers equation with source term the condition for loss of regularity of a solution to burgers equation with source term is established the same condition applies to the eulerian droplet model and its subsystem the riemann problem for the eulerian droplet model is constructively solved by going through the solution of the riemann problems for the inviscid burgers equation with a source term and the subsystem respectively under suitable generalized rankine hugoniot relations and entropy condition the existence of delta shock solution is established the existence of a solution to the generalized rankine hugoniot conditions is proven some numerical illustrations are presented

existence of global weak solutions to a special system of euler equation with a source ii general case abstract in this paper we apply the maximum principle and the theory of compensated compactness to establish an existence theorem for global weak solutions to the cauchy problem of the non strictly hyperbolic system a special system of euler equation with a general source term

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

the semiclassical small ℏ limit of loci of roots of subdominant solutions for polynomial potentials in this paper a description of the small limit of loci of zeros of subdominant solutions for polynomial potentials is given called as fundamental solutions in our earlier papers the considered potentials are those which provide us with the simple turning points only three types of stokes graphs sg s associated with the potentials are investigated the general non critical ones the general critical ones but with only single internal stokes line sl and the stokes graphs corresponding to arbitrary multiple well real even degree polynomial potentials with internal stokes lines distributed on the real axis only all these cases are considered in their both versions of the quantized and not quantized in particular due to the fact that the small limit is semiclassical it is shown that loci of roots of subdominant solutions in the cases considered are collected along stokes lines there are infinitely many roots of subdominant solutions on such lines escaping to infinity and a finite number of th

cauchy problem for hyperbolic conservation laws with a relaxation term this paper considers the cauchy problem for hyperbolic conservation laws arising in chromatography

topological expansion and exponential asymptotics in 1d quantum mechanics borel summable semiclassical expansions in 1d quantum mechanics are considered these are the borel summable expansions of fundamental solutions and of quantities constructed with their help an expansion called topological is constructed for the corresponding borel functions this allows us to study the borel plane singularity structure in a systematic way examples of such structures are considered for linear harmonic and anharmonic potentials together with the best approximation provided by the semiclassical series the exponentially small contributions completing the approximation are considered a natural method of constructing such exponential asymptotics based on the borel plane singularity structures provided by the topological expansion is developed the method is used to form the semiclassical series including exponential contributions for the energy levels of the anharmonic oscillator

viscous limits for piecewise smooth solutions of the p system abstract we study the zero dissipation problem for a one dimensional model system for the isentropic flow of a compressible viscous gas the so called p system with viscosity when the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof is given by a matched asymptotic analysis and an elementary energy method and we do not need the smallness condition on the shock strength

from image deblurring to optimal investments maximum likelihood solutions for positive linear inverse problems the problem of recovering an input signal from a blurred output in an input output system with linear distortion is ubiquitous in science and technology when the blurred output is not degraded by statistical noise the problem is entirely deterministic and amounts to a mathematical inversion of a linear system with positive parameters subject to positivity constraints on the solution we show that all such linear inverse problems with positivity restrictions lininpos problems for short can be interpreted as statistical estimation problems from incomplete data based on infinitely large samples and that maximum likelihood ml estimation and the em algorithm provide a straightforward method of solution for such problems

on the coulomb green function the connection between the green function for the isotropic harmonic oscillator in two dimensional complex space and the coulomb green function is established using the method of path integrals the representations for the coulomb green function are proposed to be useful in concrete applications

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

waves near stokes lines the large k asymptotics of d 2 u z d z 2 k 2 r 2 z u z are studied near a stokes line z z 0 r d z real where z 0 is a zero of r 2 z of any order on which there is greatest disparity between the dominant and subdominant exponential waves in the phase integral wkb approximations the aim is to establish precisely how the multiplier b of the subdominant wave varies across the stokes line although b always has a total change proportional to i times the multiplier of the dominant wave the stokes phenomenon the form of the change depends on the convention used to define the two waves the optimal convention for which the variation is maximally compact and smooth is to define them by the phase integral approximation truncated at its least term whose order is proportional to k and therefore large asymptotics of asymptotics then the variation of b is proportional to the error function of the natural stokes crossing variable im k re this result is obtained without resumming divergent series thereby avoiding asymptotics of asymptotics of asymptotics an application is given to the birth of exponentially weak reflected waves in media with smoothly varying refractive index

existence of global entropy solutions of a nonstrictly hyperbolic system in this paper we use the theory of compensated compactness coupled with some basic ideas of the kinetic formulation by lions perthame souganidis tadmor lps lpt to establish an existence theorem for global entropy solutions of the nonstrictly hyperbolic system 1

em algorithms for ml factor analysis the details of em algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models the algorithm is essentially the same for both cases and involves only simple least squares regression operations the largest matrix inversion required is for aq q symmetric matrix whereq is the matrix of factors the example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation

asymptotic stability of rarefaction waves for 2 2 viscous hyperbolic conservation laws this paper concerns the asymptotic behavior toward rarefaction waves of the solution of a genera 2 x 2 hyperbolic conservation laws with positive viscosity we prove that if the initial data is close to a constant state and its values at a3 lie on the kth rarefaction curve for the corresponding hyperbolic conservation laws then the solution tends as i co to the rarefaction wave determined by these states

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

maximum likelihood estimation for mixed continuous and categorical data with missing values summary maximum likelihood procedures for analysing mixed continuous and categorical data with missing values are presented the general location model of olkin tate 1961 and extensions introduced by krzanowski 1980 1982 form the basis for our methods maximum likelihood estimation with incomplete data is achieved by an application of the em algorithm dempster laird rubin 1977 special cases of the algorithm include orchard woodbury s 1972 algorithm for incomplete normal samples fuchs s 1982 algorithms for log linear modelling of partially classified contingency tables and day s 1969 algorithm for multivariate normal mixtures applications include a imputation of missing values b logistic regression and discriminant analysis with missing predictors and unclassified observations c linear regression with missing continuous and categorical predictors and d parametric cluster analysis with incomplete data methods are illustrated using data from the st louis risk research project some key word8 cluster analysis discriminant analysis em algorithm incomplete data linear regression logistic regression log linear model mixture model

analytic calculation of energies and wave functions of the quartic and pure quartic oscillators ground state energies and wave functions of quartic and pure quartic oscillators are calculated by first casting the schrodinger equation into a nonlinear riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method qlm in the qlm the nonlinear differential equation is solved by approximating the nonlinear terms by a sequence of linear expressions the qlm is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter our explicit analytic results are then compared with exact numerical and also with wkb solutions and it is found that our ground state wave functions using a range of small to large coupling constants yield a precision of between 0 1 and 1 percent and are more accurate than wkb solutions by two to three orders of magnitude in addition our qlm wave functions are devoid of unphysical turning point singularities and thus allow one to make analytic

kinetic formulation of the isentropic gas dynamics and p systems we consider the 2 x 2 hyperbolic system of isentropic gas dynamics in both eulerian or lagrangian variables also called the p system we show that they can be reformulated as a kinetic equation using an additional kinetic variable such a formulation was first obtained by the authors in the case of multidimensio nal scalar conservation laws a new phenomenon occurs here namely that the advection velocity is now a combination of the macroscopic and kinetic velocities various applications are given we recover the invariant regions deduce new l estimates using moments lemma and prove l w stability for 7 3

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

convergence of the viscosity method for a nonstrictly hyperbolic conservation law a convergence theorem for the method of artificial viscosity applied to the nonstrictly hyperbolic systemvt vu x 0 u t left frac 1 2 u 2 mathop smallint limits v s s delta r 3 ds right x 0 0 r 3 is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

global solutions to the compressible euler equations with geometrical structure we prove the existence of global solutions to the euler equations of compressible isentropic gas dynamics with geometrical structure including transonic nozzle flow and spherically symmetric flow due to the presence of the geometrical source terms the existence results themselves are new especially as they pertain to radial flow in an unbounded region vec x geqq 1 and to transonic nozzle flow arbitrary data withl bounds are allowed in these results a shock capturing numerical scheme is introduced to compute such flows and to construct approximate solutions the convergence and consistency of the approximate solutions generated from this scheme to the global solutions are proved with the aid of a compensated compactness framework

nonstrictly hyperbolic systems with stiff relaxation terms abstract in this paper a zero factor idea is introduced to extend the convergence framework in g q chen c d levermore t p liu hyperbolic conservation laws with stiff relaxation terms and entropy comm pure appl math 47 1994 787 830 for the singular limits of stiff relaxation from general 2 2 hyperbolic conservation systems to nonstrictly hyperbolic systems and an application of this framework on the so called system of extended traffic flow is obtained

kinetic formulation for systems of two conservation laws and elastodynamics for scalar conservation laws the kinetic formulation makes it possible to generate all the entropies from a simple kernel we show how this concept replaces and simplifies greatly the concept of young measures avoiding the difficulties encountered when working in lp the general construction of the two kinetic functions that generate the entropies of 2 2 strictly hyperbolic systems is also developed here we show that it amounts to building a universal entropy i e one that can be truncated by a kinetic value along riemann invariants for elastodynamics this construction can be completed and specialized using the additional galilean invariance this allows a full characterization of convex entropies it yields a kinetic formulation consisting of two semi kinetic equations which as usual are equivalent to the infinite family of all the entropy inequalities

complex zeros of real ergodic eigenfunctions we determine the limit distribution as of complex zeros for holomorphic continuations to grauert tubes of real eigenfunctions of the laplacian on a real analytic compact riemannian manifold m g with ergodic geodesic flow if phi j k is an ergodic sequence of eigenfunctions we prove the weak limit formula frac 1 lambda j z phi j k mathbb c to frac i pi partial bar partial xi g where z phi j k mathbb c is the current of integration over the complex zeros and where overline partial is with respect to the adapted complex structure of lempert szoke and guillemin stenzel

generalised wkbj formulae an infinite sequence of new representations having froman and froman s form 1974 has been found for the solutions to the one dimensional schrodinger equation se this enables the author to construct in compact wkbj like form approximate solutions to se the new approximate solutions have the property that their asymptotic expansions in h cross coincide with the corresponding asymptotic expansions of the exact solutions to any a priori given order in h cross they generalise the well known wkbj formula being in this scheme the zeroth order approximation the generalised wkbj formulae are then tested by the quantisation of the simplest quantum mechanical systems of the harmonic and anharmonic quartic oscillators as well as a more complicated one of a double well potential the approximate quantisation rules a generalisation of the well known bohr sommerfeld quantisation conditions for the general polynomial potential is finally obtained and the formulae for calculating the necessary corrections are derived the limiting formal solutions to se having the wkbj form are also presented their asymptotic expansions in h cross reproduce those found by froman 1966

viscous solutions of quadratic conservation laws with umbilic points viscous solutions of quadratic conservation laws with umbilic points ltj yunguang zhu changjiang and zhao huijiang wuhan institute of mathematical sciences academia sinica wuhan 430071 people s republic of china received 10 january 1991 received in revised form 4 august 1992 received for publication 1 march 1993

existence of solutions to hyperbolic conservation laws with a source existence of solutions to three different systems of eqs 1 1 1 2 and 1 3 coming from physically relevant models is shown each needing a different proof which are given in sects 2 3 and 4 the unifying theme is the presence of source terms and the general method of proof is vanishing viscosity together with compensated compactness for system 1 2 entropy entropy flux pairs of lax type are constructed and estimates from singular perturbation theory of odes are used for 1 1 and 1 3 weak entropy entropy flux pairs are constructed following the compensated compactness framework set up by diperna 4

exact semiclassical expansions for one dimensional quantum oscillators a set of rules is given for dealing with wkb expansions in the one dimensional analytic case whereby such expansions are not considered as approximations but as exact encodings of wave functions thus allowing for analytic continuation with respect to whichever parameters the potential function depends on with an exact control of small exponential effects these rules which include also the case when there are double turning points are illustrated on various examples and applied to the study of bound state or resonance spectra in the case of simple oscillators it is thus shown that the rayleigh schrodinger series is borel resummable yielding the exact energy levels in the case of the symmetrical anharmonic oscillator one gets a simple and rigorous justification of the zinn justin quantization condition and of its solution in terms of multi instanton expansions

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

stark ladder resonances for small electric fields we prove the existence of resonances in the semi classical regime of smallh for stark ladder hamiltonians h h f equiv h 2 frac d 2 dx 2 v fx in one dimension the potentialv is a real periodic function with period which is the restriction to of a function analytic in a strip about the electric field strengthf satisfies the bounds v f 0 in general the imaginary part of the resonances are bounded above by open image in new window for some 0 0 independent ofh andf forh sufficiently small this is an extension of the oppenheimer formula for the stark effect to the case of periodic potentials

jwkb connection formula problem revisited again via borel summation silverstone phys rev lett 55 2523 2526 1985 has expressed the opinion that the traditional version of the jwkb connection formulas associated with a classical turning point is incorrect and

isometric immersions of surfaces with two classes of metrics and negative gauss curvature the isometric immersion of two dimensional riemannian manifolds or surfaces with negative gauss curvature into the three dimensional euclidean space is studied in this paper the global weak solutions to the gauss codazzi equations with large data in l infty are obtained through the vanishing viscosity method and the compensated compactness framework the l infty uniform estimate and h 1 compactness are established through a transformation of state variables and construction of proper invariant regions for two types of given metrics including the catenoid type and the helicoid type the global weak solutions in l infty to the gauss codazzi equations yield the c1 1 isometric immersions of surfaces with the given metrics

change of variables as borel resummation of semiclassical series the 1d schr odinger equation with meromorphic potentials are considered it is shown that any fundamental solution constructed for such potentials has a property to depend in some particular way on an arbitrary meromorphic function f x a family of fundamental solutions created in this way can be reproduced by multiplying some fixed member of the family by appropriate dependent constants this freedom allows for a proper construction of fundamental solutions at simple and double poles of considered potentials it expresses the fact that any change of variablex ydy x keeping the form of 1d schr odinger equation keeps invariant the form of the fundamental solutions as well in fact there is one to one correspondence between the functions f x and the schwarzian ofy x since any fundamental solution is borel summable it means also that the effect of the change of variable in fundamental solutions can be recoverd by multiplying the initial fundamental solutions by suitably chosenh dependent constants and borel resumming the corresponding semiclassical expansions this explains why a change of variable can improve jwkb formulae however it is shown also that a change of variable itself cannot provide us with the exact jwkb formulae

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea