eigenvalue asymptotics of layered media and their applications to the inverse problem we compute the asymptotics of the eigenvalues of the classical sturm liouville problem with a piecewise smooth coefficient q this means that q and or its derivatives can have jump discontinuities the boundary conditions are arbitrary our results extend the classical work of h hochstadt see comm pure appl math 14 1961 pp 749 764 and some related formulas discovered by g borg see acta math 78 1946 pp 1 96 then we apply our results to the inverse problem of determining the interfaces in a layered medium from acoustic data since the index of refraction of such a medium can be considered piecewise smooth

discontinuous inverse eigenvalue problems on considere des problemes de sturm liouville inverses dans lesquels les fonctions propres ont une discontinuite dans un point interieur on montre que si le potentiel est connu sur la moitie de l intervalle et si une condition limite est donnee alors le potentiel et l autre condition limite sont determines de facon unique par les valeurs propres

an algorithm for discontinuous inverse sturm liouville problems with symmetric potentials abstract in this paper we consider two sturm liouville problems with symmetric potentials and symmetric discontinuities satisfying symmetric boundary and jump conditions in the first section we derive a simple expression for the difference of potentials when only a finite number of eigenvalues differ in the second section this result is used to construct an algorithm for solving the discontinuous inverse sturm liouville problem numerically

reconstruction techniques for classical inverse sturm liouville problems this paper gives constructive algorithms for the classical inverse sturm liouville problem it is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator two methods of solving this latter problem are then provided and numerical examples are presented

titchmarsh s asymptotic formula for periodic eigenvalues and an extension to the p laplacian an asymptotic formula for periodic eigenvalues due to titchmarsh is taken to a higher degree of accuracy using a prufer transformation method not depending on floquet theory with the idea of the rotation number this method is then extended to the p laplacian

efficient construction of reversible jump markov chain monte carlo proposal distributions the major implementational problem for reversible jump markov chain monte carlo methods is that there is commonly no natural way to choose jump proposals since there is no euclidean structure in the parameter space to guide our choice we consider mechanisms for guiding the choice of proposal the first group of methods is based on an analysis of acceptance probabilities for jumps essentially these methods involve a taylor series expansion of the acceptance probability around certain canonical jumps and turn out to have close connections to langevin algorithms the second group of methods generalizes the reversible jump algorithm by using the so called saturated space approach these allow the chain to retain some degree of memory so that when proposing to move from a smaller to a larger model information is borrowed from the last time that the reverse move was performed the main motivation for this paper is that in complex problems the probability that the markov chain moves between such spaces may be prohibitively small as the probability mass can be very thinly spread across the space therefore finding reasonable jump proposals becomes extremely important we illustrate the procedure by using several examples of reversible jump markov chain monte carlo applications including the analysis of autoregressive time series graphical gaussian modelling and mixture modelling

sturm liouville eigenvalue problems with finitely many singularities with boundary conditions ay 0 q by9 0 s 0 and cy p q dy9 p s 0 one generally needs the eigenvalue sequences for two sets of boundary conditions or one spectrum and a sequence of norming constants to w x uniquely determine p thanks to 15 16 there is even an elegant geometric description of the isospectral sets when only one set of boundary conditions is given if p is an even function or satisfies some analogous constraint then for some sets of boundary conditions p may be recovered w x from the eigenvalues of a single operator 15 p 55 14 p 55 there is additional work considering the effects of regularity assump w x tions on p see 2 6 7 8 12 for instance if p g c the eigenvalues

a beta mixture model for assessing genetic population structure an important fraction of recently generated molecular data is dominant markers they contain substantial information about genetic variation but dominance makes it impossible to apply standard techniques to calculate measures of genetic differentiation such as f statistics in this article we propose a new bayesian beta mixture model that more accurately describes the genetic structure from dominant markers and estimates multiple f st s from the sample the model also has important application for codominant markers and single nucleotide polymorphism snp data the number of f st is assumed unknown beforehand and follows a random distribution the reversible jump algorithm is used to estimate the unknown number of multiple f st s we evaluate the performance of three split proposals and the overall performance of the proposed model based on simulated dominant marker data the model could reliably identify and estimate a spectrum of degrees of genetic differentiation present in multiple loci the estimates of f st s also incorporate uncertainty about the magnitude of within population inbreeding coefficient we illustrate the method with two examples one using dominant marker data from a rare orchid and the other using codominant marker data from human populations

optimization of congestion pricing road network with variable demands this paper addresses a new optimization scheme designed for a congestion pricing road network with variable demands cpram a congestion pricing road network with variable demands can be formulated as a mathematical program with equilibrium constraints mpec where the user equilibrium with variable demands is expressed as a variational inequality problem due to the non differentiability of the perturbed solutions in equilibrium constraints a non smooth optimization model is established a generalized bundle subgradient projection method gsp is presented to solve the cpram with global convergence numerical calculations are conducted on a small scale road network and encouraging results are reported

solution of the inverse spectral problem for an impedance with integrable derivative part ii the inverse eigenvalue problem for a sturm liouville equation in impedance form with dirichlet boundary conditions on a unit interval is solved the solution of this nonlinear problem requires the investigation of a combined map from the logarithmic derivative of the impedance assumed to be square integrable to two sequences of spectral data the first is the sequence of dirichlet eigenvalues shown to be locally bounded with square roots differing from the sequence of integral multiples of by a square summable sequence the second sequence has as first term the mean of the logarithmic derivative of the impedance each term in the remainder of the sequence is the logarithm of the product of the impedance and the derivative of an eigenfunction evaluated at an endpoint it is shown that this second sequence is a locally bounded square summable sequence the combined map is a real analytic isomorphism a uniqueness result follows from the asymptotics and analyticity shown in the first paper and properties of riesz bases the existence is shown by giving a closed form of the solution it is also shown that the same map solves the inverse sturm liouville problem in impedance form with neumann boundary conditions 1993 john wiley sons inc

a model selection approach for the identification of quantitative trait loci in experimental crosses summary we consider the problem of identifying the genetic loci called quantitative trait loci qtls contributing to variation in a quantitative trait with data on an experimental cross a large number of different statistical approaches to this problem have been described most make use of multiple tests of hypotheses and many consider models allowing only a single qtl we feel that the problem is best viewed as one of model selection we discuss the use of model selection ideas to identify qtls in experimental crosses we focus on a back cross experiment with strictly additive qtls and concentrate on identifying qtls considering the estimation of their effects and precise locations of secondary importance we present the results of a simulation study to compare the performances of the more prominent methods

eigenfunction expansions a discontinuous version in this paper it is shown that any absolutely continuous function can be expanded in terms of the eigenfunctions of a sturrn liouville problem where the eigenfunctions have two symmetrically located discontinuities satisfying symmetric jump conditions the result extends titchmarsch s classical work on eigenfunctions expansions associated with second order differential equations next the orthogonality property of the eigenfunctions is shown and a relationship between the derivative of the wronskian and the l 2 norm of the eigenfunctions is established in the first of two corollaries a variation of the eigenfunction expansion is presented this result has been used to construct an algorithm for the discontinuous inverse sturrn liouville problem with symmetric potentials and symmetric discontinuities in the second corollary the sturrn liouville expansion is given in a form that may be of more use to the general reader this expression is analogous to the standard fourier series expansion finally a

assessing population differentiation and isolation from single nucleotide polymorphism data summary we introduce a new hierarchical model for single nucleotide polymorphism allele frequencies in a structured population which is naturally fitted via markov chain monte carlo methods there is one parameter for each population closely analogous to a population specific version of wright s fst which can be interpreted as measuring how isolated the relevant population has been our model includes the effects of single nucleotide polymorphism ascertainment and is motivated by population genetics considerations explicitly in the transient setting after divergence of populations rather than as the equilibrium of a stochastic model as is traditionally the case for the sizes of data set that we consider the method provides good parameter estimates and considerably outperforms estimation methods analogous to those currently used in practice we apply the method to one new and one existing human data set each with rather different characteristics the first consisting of three rather close european populations the second of four populations taken from across the globe a novelty of our framework is that the fit of the underlying model can be assessed easily and these results are encouraging for both data sets analysed our analysis suggests that iceland is more differentiated than the other two european populations france and utah a finding which is consistent with the historical record but not obvious from comparisons of simple summary statistics

forensic identification with imperfect evidence mortera uniroma3 it summary we study the problem of forensic identification when the trace evidence from the scene of the crime is imperfect for example it might be measured with error or be partially missing a general framework for imperfect data is developed and applied in particular to the following cases singly and in combination measurement error in the recorded information at the scene of the crime binning i e discretisation of an originally continuous crime measurement paternity testing in which the dna profiles of the child and of the mother provide partial information on the true father s dna two types of laboratory error one in which the error is equally distributed among all possible results and a second in which there is a bias in the error mechanism producing a false match and partial data as when there is information on the offender s dna profile but not on his racial group

bounds and higher order estimates for sturm liouville eigenvalues in 2 an algorithm is described for computing upper and lower bounds for the eigenvalues of the regular sturm liouville problem on a finite interval a b the method which deals easily with coefficients having jump discontinuities replaces the coefftcients by constants on each subinterval of a partition rc a x0 x x b see i for a survey of the related literature the bounds converge to the true eigenvalues as h maxi hi 0 where hi xi xi1 the rate of convergence being of the first order in h 1 21 however it was noted in 2 that numerical results suggested that the arithmetic mean of the upper and lower bounds exhibits second order convergence in section 3 we give a proof of this second order convergence and also explain a remarkable numerical result noted in 2 although the main purpose of the algorithm of 2 is to give bounds the fact that it simultaneously gives higher order estimates is an important bonus

eigenvalues of discontinuous sturm liouville problems with symmetric potentials abstract in this paper we consider three examples of discontinuous sturm liouville problems with symmetric potentials the eigenvalues of the systems were determined using the classical fourth order runge kutta method these eigenvalues are used to reconstruct the potential function using an algorithm presented in kobayashi 1 2 the results of our numerical experiments are discussed

on bayesian analysis of mixtures with an unknown number of components discussion author s reply new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

discussion on the meeting on statistical modelling and analysis of genetic data david j balding imperial college school of medicine london i extendmy apologies to the authors that an unavoidable commitment arising unexpectedly in the 18 hours before the meeting robbed me of my final preparation so that my comments at the meeting were poorly presented i shall try to do a better job in this written version and to leave enough space i shall omit my introductory comments about the role of statisticians in bioinformatics

detection of multilayered media in the acoustic waveguide this work is concerned with the inverse problem for ocean acoustics modeled by a multilayered waveguide with a nite depth we provide explicit formulae to locate the layers including the seabed and reconstruct the speed of sound and the densities in each layer from measurements collected on the surface of the waveguide we proceed in two steps first we use gaussian type excitations on the upper surface of the waveguide and then from the corresponding scattered elds collected on the same surface we recover the boundary spectral data of the related 1d spectral problem second from these spectral data we reconstruct the values of the normal derivatives of the singular solutions of the original waveguide problem on that upper surface finally we derive formulae to reconstruct the layers from these values based on the asymptotic expansion of these singular solutions in terms of the source points

an mpec approach to second best toll pricing this paper addresses two second best toll pricing problems one with fixed and the other with elastic travel demands as mathematical programs with equilibrium constraints several equivalent nonlinear programming formulations for the two problems are discussed one formulation leads to properties that are of interest to transportation economists another produces an algorithm that is capable of solving large problems and easy to implement with existing software for linear and nonlinear programming problems numerical results using transportation networks from the literature are also presented

high order embedded runge kutta formulae abstract the criteria to be satisfied by embedded runge kutta pairs of formulae are reviewed two new formulae of orders 6 and 8 are presented together with tests on their efficiency relative to other high order formulae in current use

measuring heterogeneity in forensic databases using hierarchical bayes models summary as currently defined dna fingerprint profiles do not uniquely identify individuals for criminal cases involving dna evidence forensic scientists evaluate the conditional probability that an unknown but distinct individual matches the crime sample given that the defendant matches estimates of the conditional probability of observing matching profiles are based on reference populations maintained by forensic testing laboratories each of these databases is heterogeneous being composed of subpopulations of different heritages this heterogeneity has an impact on the weight of the evidence a hierarchical bayes model is formulated that incorporates the key physical characteristics inherent in these data with the help of markov chain monte carlo sampling levels of heterogeneity are estimated for three major ethnic groups in the database of lifecodes corporation

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

coherent analysis of forensic identification evidence a murder has been committed and there is a known population of possible suspects the identification evidence available based on information at the scene of the crime is that the criminal may have a certain characteristic information may also be available on a set of suspects about which of them have the characteristic we discuss the problem of assessing who is the guilty party taking particular account of the following complicating features the evidence at the scene of the crime may be unreliable the suspects examined may have been chosen by means of a search process which might itself be informative we also examine the effect of the assumed population dependence structure for the relevant identification characteristics

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12