l 1 stability estimates for n x n conservation laws let u t f u x 0 be a strictly hyperbolic n times n system of conservation laws each characteristic field being linearly degenerate or genuinely nonlinear in this paper we explicitly define a functional phi phi u v equivalent to the l 1 distance which is almost decreasing i e phi big u t v t big phi big u s v s big leq o ve cdot t s quad hbox for all t s geq 0 for every pair of e approximate solutions u v with small total variation generated by a wave front tracking algorithm the small parameter e here controls the errors in the wave speeds the maximum size of rarefaction fronts and the total strength of all non physical waves in u and in v from the above estimate it follows that front tracking approximations converge to a unique limit solution depending lipschitz continuously on the initial data in the vec l 1 norm this provides a new proof of the existence of the standard riemann semigroup generated by a n n system of conservation laws

vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

weak solutions of general systems of hyperbolic conservation laws in this paper we establish the existence theory for general system of hyperbolic conservation laws and obtain the uniform l1 boundness for the solutions the existence theory generalizes the classical glimm theory for systems for which each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of lax we construct the solutions by the glimm scheme through the wave tracing method one of the key elements is a new way of measuring the potential interaction of the waves of the same characteristic family involving the angle between waves a new analysis is introduced to verify the consistency of the wave tracing procedure the entropy functional is used to study the l1 boundedness

the unique limit of the glimm scheme we introduce the definitions of a standard riemann semigroup and of a viscosity solution for a nonlinear hyperbolic system of conservation laws for a class including general 2 2 systems it is proved that the solutions obtained by a wavefront tracking algorithm or by the glimm scheme are precisely the semigroup trajectories in particular these solutions are unique and depend lipschitz continuously on the initial data in the l1 norm

existence theory for hyperbolic systems of conservation laws with general flux functions to begin with we assume that the flux function f u is piecewise genuinely nonlinear in the sense that it exhibits finitely many at most p say points of lack of genuine nonlinearity along each wave curve importantly our analysis applies to arbitrary large p in the sense that the constant c restricting the total variation is independent of p second by an approximation argument we prove that the existence theory above extends to general flux functionsf u that can be approached by a sequence of piecewise genuinely nonlinear flux functions fe u

sharp convergence rate of the glimm scheme for general nonlinear hyperbolic systems consider a general strictly hyperbolic quasilinear system in one space dimesion u t a u u x 0 quad quad quad quad quad quad 1 where u mapsto a u u in omega subset mathbb r n is a smooth matrix valued map given an initial datum u 0 with small total variation let u t be the corresponding unique vanishing viscosity solution of 1 obtained as a limit of solutions to the viscous parabolic approximation u t a u u x u xx as 0 for every t 0 we prove the a priori bound left u varepsilon t cdot u t cdot right mathbb l 1 o 1 cdot sqrt varepsilon log varepsilon quad quad quad quad quad quad 1 for an approximate solution u varepsilon of 1 constructed by the glimm scheme with mesh size delta x delta t varepsilon and with a suitable choice of the sampling sequence this result provides for general hyperbolic systems the same type of error estimates valid for glimm approximate solutions of hyperbolic systems of conservation laws u t f u x 0 satisfying the classical lax or liu assumptions on the eigenvalues k u and on the eigenvectors r k u of the jacobian matrix a u df u

a note on asymptotic evaluation of some hankel transforms asymptotic behavior of the integral if w ejo wx f x2 xdx is investigated where jo x is the bessel function of the first kind and w is a large positive parameter it is shown that 1f w decays exponentially like ewy2 y 0 when f z is an entire function subject to a suitable growth condition a complete asymptotic expansion is obtained when f z is a meromorphic function satisfying the same growth condition similar results are given when f z has some specific branch point singularities

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

well posedness theory for hyperbolic conservation laws the paper presents a well posedness theory for the initial value problem for a general system of hyperbolic conservation laws we will start with the refinement of glimm s existence theory and discuss the principle of nonlinear through wave tracing our main goal is to introduce a nonlinear functional for two solutions with the property that it is equivalent to the l1 x distance between the two solutions and is time decreasing moreover the functional is constructed explicitly in terms of the wave patterns of the solutions through the nonlinear superposition it consists of a linear term measuring the l1 x distance a quadratic term measuring the coupling of waves and distance and a generalized entropy functional 1999 john wiley sons inc

error bounds for a deterministic version of the glimm scheme consider the hyperbolic system of conservation laws u t f u x 0 let u be the unique viscosity solution with initial condition u 0 x bar u x and let ue be an approximate solution constructed by the glimm scheme corresponding to the mesh sizes delta x delta t o delta x with a suitable choice of the sampling sequence we prove the estimate big u ve t cdot u t cdot big bf l 1 strut l 1 o 1 cdot sqrt delta x big ln delta x big

on the riemann problem for non conservative hyperbolic systems we consider the construction and the properties of the riemann solver for the hyperbolic system

bayesian models for multiple local sequence alignment and gibbs sampling strategies abstract a wealth of data concerning life s basic molecules proteins and nucleic acids has emerged from the biotechnology revolution the human genome project has accelerated the growth of these data multiple observations of homologous protein or nucleic acid sequences from different organisms are often available but because mutations and sequence errors misalign these data multiple sequence alignment has become an essential and valuable tool for understanding structures and functions of these molecules a recently developed gibbs sampling algorithm has been applied with substantial advantage in this setting in this article we develop a full bayesian foundation for this algorithm and present extensions that permit relaxation of two important restrictions we also present a rank test for the assessment of the significance of multiple sequence alignment as an example we study the set of dinucleotide binding proteins and predict binding segments for dozens of its members

weighted finite population sampling to maximize entropy summary attention is drawn to a method of sampling a finite population of n units with unequal probabilities and without replacement the method was originally proposed by stern cover 1989 as a model for lotteries the method can be characterized as maximizing entropy given coverage probabilities 7ci or equivalently as having the probability of a selected sample proportional to the product of a set of weights wi we show the essential uniqueness of the wi given the 7i and describe practical geometrically convergent algorithms for computing the wi from the 7i we present two methods for stepwise selection of sampling units and corresponding schemes for removal of units that can be used in connection with sample rotation inclusion probabilities of any order can be written explicitly in closed form second order inclusion probabilities 7rij satisfy the condition 0 ij 7ic 7j which guarantees yates grundy s variance estimator to be unbiased definable for all samples and always nonnegative for any sample size

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

a new glimm functional and convergence rate of glimm scheme for general systems of hyperbolic conservation laws in this paper we introduce a new glimm functional for general systems of hyperbolic conservation laws this new functional is consistent with the classical glimm functional for the case when each characteristic field is either genuinely nonlinear or linearly degenerate so that it can be viewed as optimal in some sense with this new functional the consistency of the glimm scheme is proved clearly for general systems moreover the convergence rate of the glimm scheme is shown to be the same as the one obtained in bressan marson arch ration mech anal 142 2 155 176 1998 for systems with each characteristic field being genuinely nonlinear or linearly degenerate

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

nonparametric bayes modeling of multivariate categorical data modeling of multivariate unordered categorical nominal data is a challenging problem particularly in high dimensions and cases in which one wishes to avoid strong assumptions about the dependence structure commonly used approaches rely on the incorporation of latent gaussian random variables or parametric latent class models the goal of this article is to develop a nonparametric bayes approach which defines a prior with full support on the space of distributions for multiple unordered categorical variables this support condition ensures that we are not restricting the dependence structure a priori we show this can be accomplished through a dirichlet process mixture of product multinomial distributions which is also a convenient form for posterior computation methods for nonparametric testing of violations of independence are proposed and the methods are applied to model positional dependence within transcription factor binding motifs

markovian structures in biological sequence alignments abstract the alignment of multiple homologous biopolymer sequences is crucial in research on protein modeling and engineering molecular evolution and prediction in terms of both gene function and gene product structure in this article we provide a coherent view of the two recent models used for multiple sequence alignment the hidden markov model hmm and the block based motif model to develop a set of new algorithms that have both the sensitivity of the block based model and the flexibility of the hmm in particular we decompose the standard hmm into two components the insertion component which is captured by the so called propagation model and the deletion component which is described by a deletion vector such a decomposition serves as a basis for rational compromise between biological specificity and model flexibility furthermore we introduce a bayesian model selection criterion that in combination with the propagation model genetic algorithm and other computational aspects forms the cor

exactification of the poincare asymptotic expansion of the hankel integral spectacularly accurate asymptotic expansions and non asymptotic scales we obtain an exactification of the poincare asymptotic expansion pae of the hankel integral as using the distributional approach of mcclure wong we find that for half integer orders of the bessel function the exactified asymptotic series terminates so that it gives an exact finite sum representation of the hankel integral for other orders the asymptotic series does not terminate and is generally divergent but is amenable to superasymptotic summation i e by optimal truncation for specific examples we compare the accuracy of the optimally truncated asymptotic series owing to the mcclure wong distributional method with owing to the mellin barnes integral method we find that the former is spectacularly more accurate than the latter by in some cases more than 70 orders of magnitude for the same moderate value of b moreover the exactification can lead to a resummation of the pae when it is exact with the resummed poincare series exhibiting again the same spectacular accuracy more importantly the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences

convergence rate of markov chain methods for genomic motif discovery we analyze the convergence rate of a popular gibbs sampling method used for statistical discovery of gene regulatory binding motifs in dna sequences this sampler satisfies a very strong form of ergodicity uniform however we show that due to multimodality of the posterior distribution the rate of convergence often decreases exponentially as a function of the length of the dna sequence specifically we show that this occurs whenever there is more than one true repeating pattern in the data in practice there are typically multiple even numerous such patterns in biological data the goal being to detect the most well conserved and frequently occurring of these our findings match empirical results in which the motif discovery gibbs sampler has exhibited such poor convergence that it is used only for finding modes of the posterior distribution candidate motifs rather than for obtaining samples from that distribution ours appear to be the first meaningful bounds on the convergence rate of a markov chain method for sampling from a multimodal posterior distribution as a function of statistical quantities like the number of observations

the collapsed gibbs sampler in bayesian computations with applications to a gene regulation problem abstract this article describes a method of grouping and collapsing in using the gibbs sampler and proves from an operator theory viewpoint that the method is in general beneficial the norms of the forward operators associated with the corresponding nonreversible markov chains are used to discriminate among different simulation schemes when applied to bayesian missing data problems the idea of collapsing suggests skipping the steps of sampling parameter s values in standard data augmentation by doing this we obtain a predictive update version of the gibbs sampler a procedure of calculating the posterior odds ratio via the collapsed gibbs sampler when incomplete observations are involved is presented as an illustration of possible applications three examples along with a bayesian treatment for identifying common protein binding sites in unaligned dna sequences are provided

a locally quadratic glimm functional and sharp convergence rate of the glimm scheme for nonlinear hyperbolic systems consider the cauchy problem for a strictly hyperbolic n n quasilinear system in one space dimension u t a u u x 0 qquad u 0 x bar u x quad quad quad quad 1 where u mapsto a u is a smooth matrix valued map and the initial data overline u is assumed to have small total variation we investigate the rate of convergence of approximate solutions of 1 constructed by the glimm scheme under the assumption that letting k u rk u denote the k th eigenvalue and a corresponding eigenvector of a u respectively for each k th characteristic family the linearly degenerate manifold mathcal m k doteq left u in omega nabla lambda k u cdot r k u 0 right is either the whole space or it is empty or it consists of a finite number of smooth n 1 dimensional connected manifolds that are transversal to the characteristic vector field rk we introduce a glimm type functional which is the sum of the cubic interaction potential defined in bianchini discrete contin dyn syst 9 133 166 2003 and of a quadratic term that takes into account interactions of waves of the same family with strength smaller than some fixed threshold parameter relying on an adapted wave tracing method and on the decrease amount of such a functional we obtain the same type of error estimates valid for glimm approximate solutions of hyperbolic systems satisfying the classical lax assumptions of genuine nonlinearity or linear degeneracy of the characteristic families

on integral transforms whose kernels are solutions of singular sturm liouville problems on etudie des transformees integrales du type f x 0 f s x s d ou x s est la solution du probleme de sturm liouville y s 2 q x y 0 0 x avec y 0 cos y 0 sin 0 y x est bornee a l infini et d est la mesure spectrale

a bayesian insertion deletion algorithm for distant protein motif searching via entropy filtering bayesian models have been developed that find ungapped motifs in multiple protein sequences in this article we extend the model to allow for deletions and insertions in motifs direct generalization of the ungapped algorithm based on gibbs sampling proved unsuccessful because the configuration space became much larger to alleviate the convergence difficulty a two stage procedure is introduced at the first stage we develop a method called entropy filtering which quickly searchs good starting points for the alignment approach without the concern of deletion insertion patterns at the second stage we switch to an algorithm that generates both a random vector that represents insertion deletion patterns and a random variable of motif locations after the two steps gapped motif alignments are obtained for multiple sequences when applied to datasets that consist of helix loop helix proteins and high mobility group proteins respectively our methods show great improvements over those that produce u

an improved convergence rate of glimm scheme for general systems of hyperbolic conservation laws we study the convergence rate of glimm scheme for general systems of hyperbolic conservation laws without the assumption that each characteristic field is either genuinely nonlinear or linearly degenerate we first give a sharper estimate of the error arising from the wave tracing argument by a careful analysis of the interaction between small waves with this key estimate the convergence rate is shown to be o 1 s13 lns 1 which is sharper compared to o 1 s14 lns given in t yang convergence rate of glimm scheme for general systems of hyperbolic conservation laws taiwanese j math 7 2 2003 195 205 however it is still slower than o 1 s12 lns given in a bressan a marson error bounds for a deterministic version of the glimm scheme arch ration mech anal 142 2 1998 155 176 for systems with each characteristic field being genuinely nonlinear or linearly degenerate here s is the mesh size and is any positive constant

asymptotic expansions of some integral transforms by using generalized functions the technique devised by wong to derive the asymptotic expansions of multiple fourier transforms by using the theory of schwartz distributions is extended to a large class of integral transforms the extension requires establishing a general procedure to extend these integral transforms to generalized functions wong s technique is then applied to some of these integral transforms to obtain their asymptotic expansions this class of integral transforms encompasses among others the laplace the airy the k and the hankel transforms

covariance structure and convergence rate of the gibbs sampler with various scans this paper presents results on covariance structure and convergence for the gibbs sampler with both systematic and random scans it is shown that under conditions that guarantee the compactness of the markov forward operator and irreducibility of the corresponding chain the gibbs sampling scheme converges geometrically in terms of pearson 2 distance in particular for the random scan the autocovariance can be expressed as variances of iterative conditional expectations as a consequence the autocorrelations are all positive and decrease monotonically

variable selection in regression mixture modeling for the discovery of gene regulatory networks the profusion of genomic data through genome sequencing and gene expression microarray technology has facilitated statistical research in determining gene interactions regulating a biological process current methods generally consist of a two stage procedure clustering gene expression measurements and searching for regulatory switches typically short conserved sequence patterns motifs in the dna sequence adjacent to the genes this process often leads to misleading conclusions as incorrect cluster selection may lead to missing important regulatory motifs or making many false discoveries treating cluster memberships as known rather than estimated introduces bias into analyses preventing uncertainty about cluster parameters further there is underutilization of the available data as the sequence information is ignored for purposes of expression clustering and vice versa we propose a way to address these issues by combining gene clustering and motif discovery in a unified framework a mixture

covariance structure of the gibbs sampler with applications to the comparisons of estimators and augmentation schemes summary we study the covariance structure of a markov chain generated by the gibbs sampler with emphasis on data augmentation when applied to a bayesian missing data problem the gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector the empirical distribution based on the sampled values of the parameter vector and a mixture of complete data posteriors we prove that rao blackwellization causes a one lag delay for the autocovariances among dependent samples obtained from data augmentation and consequently the mixture approximation produces estimates with smaller variances than the empirical approximation the covariance structure results are used to compare different augmentation schemes it is shown that collapsing and grouping random components in a gibbs sampler with two or three components usually result in more efficient sampling schemes

bv solutions of the semidiscrete upwind scheme consider the strictly hyperbolic system of conservation laws u t f u x 0 1 where u n f n n smooth the system is said to be strictly hyperbolic provided that the jacobian matrix a u df u has n distinct eigenvalues 1 u n u

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