detecting the presence of mixing with multiscale maximum likelihood a test of homogeneity tries to decide whether observations come from a single distribution or from a mixture of several distributions a powerful theory has been developed for the case where the component distributions are members of an exponential family when no parametric assumptions are appropriate the standard approach is to test for bimodality which is known not to be very sensitive for detecting heterogeneity to develop a more sensitive procedure this article builds on an approach employed in sampling literature and models the component distributions as logarithmically concave densities it is shown how this leads to a special semiparametric model in which the homogeneity problem is equivalent to testing whether a parameter c equals zero versus the alternative that c 0 this setup leads naturally to a novel multiscale maximum likelihood procedure where the multiscale character reflects the desirable property of adaptivity to the unknown value of the parameter c under the alternative to en

some consequences of a symmetry in strong distributions some polynomials and interpolatory quadrature rules associated with strong stieltjes distributions are considered especially when the distributions satisfy a certain symmetric property

conditions on unimodality and logconcavity for densities of coherent systems with an application to bernstein operators abstract in this note the distribution of the lifetime of a coherent system with independent and identically distributed component lifetimes is considered conditions yielding unimodality or logconcavity of the density function of the system lifetime are obtained in the conditions only assumptions on the density function of the components and on the signature of the system are imposed the results are illustrated with several examples additionally a problem on preservation of logconcavity under the classical bernstein operator is solved

inequalities for the tails of some elementary series we prove that if dn k x n k 2 n k 2 in k x in k x n 2 in x 2 with formula then we have for all x 0 dn k x k 0 and dn 1 k x 2 dn k x dn 2 k x n k 1 0

asymptotic formulas for generalized szasz mirakyan operators in the present paper we consider the general szasz mirakyan operators and investigate their asymptotic behaviours we obtain quantitative voronovskaya and quantitative gruss type voronovskaya theorems using the weighted modulus of continuity the particular cases are presented for classical szasz mirakyan operators

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

on the preservation of log convexity and log concavity under some classical bernstein type operators this paper analyzes the preservation of both the log convexity and the log concavity under certain bernstein type operators some results are provided for the bernstein szasz baskakov the gamma type and the weierstrass operators probabilistic methods support the proofs of these results

orthogonal polynomials and some q beta integrals of ramanujan abstract two integrals of ramanujan are used to define a q analogue of the euler beta integral on the real line and of the cauchy beta integral on the complex unit circle such integrals are connected to orthogonal biorthogonal and laurent polynomials explicit examples of laurent orthogonal polynomials are given on the real line and on the circle

log concavity preserving property of bernstein operators and bernstein semigroup abstract we prove that bernstein operators preserve log concavity of functions we give a proof which bases solely on elementary properties of the binomial coefficients we prove that the bernstein semigroup t t t 0 also has the log concavity preserving property we also solve two problems presented recently by i rasa

maximum likelihood estimation of a multi dimensional log concave density fn is shown to have smaller mean integrated squared error compared with kernel based methods even when we allow the use of a theoret ical optimal fixed bandwidth for the kernel estimator that would not be available in practice we also present a real data clustering example which shows that our methodology can be used in conjunction with the expectation maximization algorithm to fit finite mixtures of log concave densities

on uniform approximation by some classical bernstein type operators abstract we investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval the discussed examples include the szasz operators the szasz durrmeyer operators the gamma operators the baskakov operators and the meyer konig and zeller operators we show that some results of totik remain valid for unbounded functions at the same time that we give simple rates of convergence in terms of the usual modulus of continuity we also show by a counterexample that the result for meyer konig and zeller operators does not extend to cheney and sharma operators

another quadrature rule of highest algebraic degree of precision we consider certain quadrature rules of highest algebraic degree of precision that involve strong stieltjes distributions i e strong distributions on the positive real axis the behavior of the parameters of these quadrature rules when the distributions are strong c inversive stieltjes distributions is given a quadrature rule whose parameters have explicit expressions for their determination is presented an application of this quadrature rule for the evaluation of a certain type of integrals is also given

multiscale maximum likelihood analysis of a semiparametric model with applications a special semiparametric model for a univariate density is introduced that allows analyzing a number of problems via appropriate transformations two problems treated in some detail are testing for the presence of a mixture and detecting a wear out trend in a failure rate the analysis of the semiparametric model leads to an approach that advances the maximum likelihood theory of the grenander estimator to a multiscale analysis the construction of the corresponding test statistic rests on an extension of a result on a two sided brownian motion with quadratic drift to the simultaneous control of excursions under parabolas at various scales of a brownian bridge the resulting test is shown to be asymptotically optimal in the minimax sense regarding both rate and constant and adaptive with respect to the unknown parameter in the semiparametric model the performance of the method is illustrated with a simulation study for the failure rate problem and with data from a flow cytometry experiment for the mixture analysis

the weak solution of a functional differential equation in a general banach space abstract the existence of a unique weak lipschitz continuous solution of the abstract functional differential equation u t a t u t g t u t t 0 t fde u t t t r 0 is obtained by a fixed point argument here x is a general banach space a t d a t x x is m accretive t 0 t and g is lipschitz continuous in both variables stability properties of this weak solution are given and a first order hyperbolic pde to which the theory can be applied is considered

convergence of two point pade approximants to series of stieltjes abstract in this paper we study the convergence of two point pade approximants to stieltjes series a carleman type condition given by gragg 1980 in the context of convergence theory for continued fractions is here established in the framework of orthogonal laurent polynomials jones njastad and thron 1984 jones and thron 1981 njastad and thron 1983 1986 a bound for the error is also given

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

companion orthogonal polynomials we give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions email protected 1 x and d 2 x such that email protected 2 x 1 kx 2 d 1 x as applications of properties recurrence relations for many interesting systems of orthogonal polynomials are obtained

inequalities for residuals of power expansions for the exponential function and completely monotone functions letin x be the residual after thenth term of power series for a functionf in the casef x ex x 0 inequalities forinhave been studied by many authors but there are almost no results forx 0 in this paper we present some bounds forinwithf x e xand we give some results for completely monotone functions

markov chain monte carlo convergence diagnostics a comparative review abstract a critical issue for users of markov chain monte carlo mcmc methods in applications is how to determine when it is safe to stop sampling and use the samples to estimate characteristics of the distribution of interest research into methods of computing theoretical convergence bounds holds promise for the future but to date has yielded relatively little of practical use in applied work consequently most mcmc users address the convergence problem by applying diagnostic tools to the output produced by running their samplers after giving a brief overview of the area we provide an expository review of 13 convergence diagnostics describing the theoretical basis and practical implementation of each we then compare their performance in two simple models and conclude that all of the methods can fail to detect the sorts of convergence failure that they were designed to identify we thus recommend a combination of strategies aimed at evaluating and accelerating mcmc sampler convergence including ap

unimodality convexity and applications properties of univariate unimodal distributions concepts of multivariate unimodality some more notions of unimodality unimodality for discrete distributions unimodality of infinitely divisible distributions unimodality and notions of dependence ordering of distributions by peakedness applications of unimodality in statistical inference convexity in reliability theory appendix

on certain applications of two point pade type approximants with a single pole in this paper several examples belonging to different topics in numerical analysis are considered making use of certain two point pade type approximants with a single pole results about convergence and error estimations are given

on an approximation process of integral type abstract in this paper we study a class of integral type positive linear operators depending on a parameter 0 1 approximation properties of this class are explored the rate of convergence in terms of the usual moduli of smoothness is given the uniform approximation over unbounded intervals is established the convergence in certain weighted spaces is investigated for particular case 0 some previous results are recaptured

functional inequalities for the incomplete gamma function abstract we present several inequalities for f a x a x a 0 a 0 x 0 where a x is the incomplete gamma function one of our theorems states that the inequalities f a s p x 1 x n f a x 1 f a x n f a s q x 1 x n p q 0 hold for all nonnegative real numbers x 1 x n n 2 if and only if p m i n a 1 and q m a x a 1 here s t x 1 x n denotes the power sum of order t this extends and complements a result published by ismail and laforgia in 2006

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

convergence of difference approximations of quasilinear evolution equations abstract the very successful theory of quasilinear evolution equations which applies to many problems of mathematical physics has been developed by t kato the theory obtains solutions of quasilinear problems via contraction mappings which are defined by means of a theory of linear evolution equations also developed by kato in the current work we show how the existence and continuous dependence theorems obtained by kato can be proved by discretization in time as opposed to earlier work in this direction the current results are much sharper concerning the continuity properties of the solutions of the discretized problem and the strength of the norms in which they converge

ordered olps and cf nth numerators abstract definitions theorems and examples are established for a general model of laurent polynomial spaces and ordered orthogonal laurent polynomial sequences ordered with respect to ordered bases and orthogonal with respect to inner products l decomposed into transition functional and strong moment functional or more generally sample functional l couplings under this formulation that is shown to subsume those in the existing literature new fundamental results are produced including necessary and sufficient conditions for ordered olps to be sequences of nth numerators of continued fractions in contrast to the classical result concerning nth denominators which is shown to hold only in special cases

foreword to the proceedings of the orthoquad 2014 conference orthoquad 2014 was an international symposium on orthogonality quadrature and related topics held on january 20 24 2014 in puerto de la cruz tenerife spain it was held in memory of professor pablo gonzalez vera 1955 2012

approximating szasz and gamma operators by baskakov operators abstract it is known that both szasz and gamma operators are limits in an appropriate sense of baskakov operators in this paper we obtain rates of convergence for these approximations we also show that convergences are monotonic if the involved function is convex

two point pade expansions for a family of analytic functions abstract each member g z of a family of analytic functions defined by stieltjes transforms is shown to be represented by a positive t fraction the approximants of which form the main diagonal in the two point pade table of g z the positive t fraction is shown to converge to g z throughout a domain d a b z z b a uniformly on compact subsets in addition truncation error bounds are given for the approximants of the continued function these bounds supplement previously known bounds and apply in part of the domain of g z not covered by other bounds the proofs of our results employ properties of orthogonal l polynomials laurent polynomials and l gaussian quadrature which are of some interest in themselves a number of examples are considered

log concavity for bernstein type operators using stochastic orders abstract this paper aims to study the preservation of log concavity for bernstein type operators in particular attention is focused on positive linear operators defined on the positive semi axis admitting a probabilistic representation in terms of a process with independent increments this class includes the classical gamma szasz and szasz durrmeyer operators with respect to the first and second operators the results of this paper correct two erroneous counterexamples in 10 as a main tool in our results we use stochastic order techniques our results include as a particular case the log concavity of certain functions related to the incomplete gamma function

inequalities for the chi square distribution function abstract letpa x 1 x a0 e ttx 1 dtbe the chi square distribution function and letmt u v be the weighted power mean of ordert we prove the following ifx y 0 x y and 0 1 are fixed real numbers then the inequalities formula are valid for all real numbersa 0 if and only ifr 0 ands in particular we obtain that for alla 0 the functionx pa x is log concave on 0 this proves a conjecture of m merkle

quadrature on the half line and two point pade approximants to stieltjes functions ii convergence abstract let be a distribution function on a b 0 a c k b a d x exist for all the integers k the main course of the paper is to give convergence results both for sequences of two point pade approximants to the cauchy transform of the distribution stieltjes function and for quadrature formulas based on exactly integrating laurent polynomials the case when d x k x dx k x being a possibly complex function is also considered and estimates of the rate of convergence are given

shape preserving approximations by polynomials and splines a review of shape preserving approximation methods and algorithms for approxi mating univariate functions or discrete data is given the notion of shape refers to the geometrical behavior of a function s or approximant s graph and usually includes positivity monotonicity and or convexity but in the recent literature the broader concept of shape also includes symmetry gen eralized convexity unimodality lipschitz property possessing peaks or discontinuities etc special stress is put on shape preserving interpolation methods by polynomials and splines of course this text has no pretensions to be complete geywords shape preserving approximation approximation of univariate functions approxima tion of discrete data shape preserving interpolation polynomial spline positivity monotonicity convexity generalized convexity unimodality lipschitz property

bernstein durrmeyer operators abstract in this paper we consider the durrmeyer type modifications of the classical bernstein szasz and baskakov operators and deal with two different kind of problems firstly we obtain new results concerning preservation of shape properties lipschitz constants and global smoothness as well as monotonic convergence under convexity to do this we use a probabilistic approach based on representations of these operators in terms of stochastic processes having a s nondecreasing paths and satisfying a suitable martingale type condition secondly we show that the szasz durrmeyer operator is the limit in an appropriate sense of both the bernstein durrmeyer and the baskakov durrmeyer operators we provide rates of convergence which are derived from the bounds for the total variation distance between the probability measures involved

on a symmetry in strong distributions a strong stieltjes distribution d t is called symmetric if it satises the property t d 2 t 2 t d t for t 2 a b 0 1 2 2 z and 0 in this article some consequences of symmetry on the moments the orthogonal l polynomials and the quadrature formulae associated with the distribution are given c 1999 elsevier science b v all rights reserved msc 30e05 30b70 41a55

quadrature formulas based on rational interpolation we consider quadrature formulas based on interpolation using the ba sis functions 1 1 tkx k 1 2 3 on 1 1 where tk are parameters on the interval 1 1 we investigate two types of quadratures quadrature formulas of maximum accuracy which correctly integrate as many basis functions as possible gaussian quadrature and quadrature formulas whose nodes are the zeros of the orthogonal functions obtained by orthogonalizing the system of basis functions or thogonal quadrature we show that both approaches involve orthogonal polynomials with modified or varying weights which depend on the number of quadrature nodes the asymptotic distribution of the nodes is obtained as well as various interlacing properties and monotonicity results for the nodes

theoretical guarantees for approximate sampling from smooth and log concave densities sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators test procedures or confidence intervals in many situations the exact sampling from a given distribution is impossible or computationally expensive and therefore one needs to resort to approximate sampling strategies however there is no well developed theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures especially in the high dimensional problems this paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log concave density defined on rp for some integer p 0 we establish nonasymptotic bounds for the error of approximating the true distribution by the one obtained by the langevin monte carlo method and its variants we illustrate the effectiveness of the established guarantees with various experiments underlying our analysis are insights from the theory of continuous time diffusion processes which may be of interest beyond the framework of distributions with log concave densities considered in the present work

the quotient difference algorithm and the pade table an alternative form and a general continued fraction the quotient difference algorithm is applied to a given power series in a modified way and various continued fractions provided by the algorithm are described in terms of their relationships with the pade table for the power series in particular a general continued fraction whose convergents form any chosen combination of horizontal or vertical connected sequences of pade approximants is introduced 1 introduction in 4 gragg gives a substantial review of some of the properties of the pade table for a given power series ek o k and the relations between pade approximants and other areas of numerical analysis the earliest and as gragg suggests perhaps the most significant of these related areas is the theory of continued fractions in particular it is the so called corresponding fractions for the power series that then provide the link between the pade table and the quotient dif ference algorithm of rutishauser one of the many applications of the quotient difference algorithm is to obtain the coefficients of the corresponding fractions from those of the power series and the convergents of these continued fractions form staircases in the pade table for the series in this paper the quotient difference algorithm is taken in a form that is slightly different from the usual and is then used to provide very simply the coefficients of six types of continued fractions whose convergents form ordered sequences of pade approximants including those mentioned above it is then shown that the table of coefficients generated by the revised quotient difference algorithm also provide the partial numerators and denominators of a general continued fraction whose convergents are a sequence of pade approximants that is made up of any chosen horizontal or vertical connected subsequences this freedom in choosing the path in the pade table means that some of the existing algorithms based on particular paths can be regarded as special cases of the revised quotient difference algorithm two such instances are described the revised quotient difference algorithm used in this paper is essentially the algorithm that has already been used to construct the continued fractions whose convergents form the so called two point pade table for two given power series the only difference being the necessity for an additional rule for defining certain coefficients in the absence of a second series however by looking at the case where

the method of lines and the approximation of zeros of m accretive operators in general banach spaces abstract it is shown that a zero of an m accretive operator t d t x 2 x in a general banach space x can be approximated via methods of lines for associated evolution equations results of browder for single valued locally defined continuous accretive operators t in spaces x with uniformly convex duals or uniformly continuous accretive operators t in general banach spaces x are extended to the present case results of reich for m accretive operators in reflexive banach spaces are also extended to the present setting unlike browder s results our estimates do not use the modulus of continuity of the operators t or the duality mapping of x kobayashi type estimates are used for the bounds involving the normed differences between the endpoints of the methods of lines and a zero of t

the method of lines applied to nonlinear nonlocal functional differential equations this paper discusses nonlinear functional differential equation in a real reflexive banach space with nonlocal history condition by using the method of lines the existence and uniqueness of a strong solution are established finally some applications of the abstract results are presented

on the connection between the existence of zeros and the asymptotic behavior of resolvents of maximal monotone operators in reflexive banach spaces a more systematic approach is introduced in the theory of zeros of maximal monotone operators t x d d t 2x where x is a real banach space a basic pair of necessary and sufficient boundary conditions is given for the existence of a zero of such an operator t these conditions are then shown to be equivalent to a certain asymptotic behavior of the resolvents or the yosida resolvents of t furthermore several interesting corollaries are given a nd the extendability of the necessary and sufficient conditions to the existence of zeros of locally defined demicontinuous monotone mappings is demonstrated a result of guan about a pathwise connected set lying in the range of a morlotone operator is improved by including non convex domains a partial answer to nirenberg s problem is also given namely it is shown that a continzuous expansive mapping t on a real hilbert space 1t is surjective if there exists a constant ae g 0 1 such that ktx ty x y cax y 112 x y e h the methods for these results do not involve explicit use of any degree theory

method of lyapunov functionals construction in stability of delay evolution equations the investigation of stability for hereditary systems is often related to the construction of lyapunov functionals the general method of lyapunov functionals construction which was proposed by v kolmanovskii and l shaikhet and successfully used already for functional differential equations for difference equations with discrete time for difference equations with continuous time is used here to investigate the stability of delay evolution equations in particular partial differential equations

rates of convergence of the hastings and metropolis algorithms we apply recent results in markov chain theory to hastings and metropolis algorithms with either independent or symmetric candidate distributions and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi in the independence case in mathbb r k these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi in the symmetric case in mathbb r only we show geometric convergence essentially occurs if and only if pi has geometric tails we also evaluate recently developed computable bounds on the rates of convergence in this context examples show that these theoretical bounds can be inherently extremely conservative although when the chain is stochastically monotone the bounds may well be effective

best constants for tensor products of bernstein type operators abstract for the tensor product of k copies of the same one dimensional bernstein type operator l we consider the problem of finding the best constant in preservation of the usual modulus of continuity for the l p norm on r k two main results are obtained the first one gives both necessary and sufficient conditions in order that 1 k 1 1 p is the best uniform constant for a single operator the second one gives sufficient conditions in order that 1 k 1 1 p is the best uniform constant for a family of operators the general results are applied to several classical families of operators usually considered in approximation theory throughout the paper probabilistic concepts and methods play an important role

optimization of two point pade type approximants abstract in this paper we first give characterization theorems for the best two point pade type approximants 2ptas in the uniform norm secondly we consider sequences of 2ptas in a domain of the complex plane from the viewpoint of the asymptotic degree of convergence and we also give conditions for geometric convergence

new results in the perturbation theory of maximal monotone and accretive operators in banach spaces let x be a real banach space and g a bounded open and convex subset of x the solvability of the fixed point problem tx cx 3 x in d t n g is considered where t x d d t 2x is a possibly discontinuous m dissipative operator and c g x is completely continuous it is assumed that x is uniformly convex d t n g 0 and t c d t n og c g a result of browder concerning single valued operators t that are either uniformly continuous or continuous with x uniformly convex is extended to the present case browder s method cannot be applied in this setting even in the single valued case because there is no class of permissible homeomorphisms let if 7z r 3 r o 0 as r oo the effect of a weak boundary condition of the type u cx x 3 11xfl flxl12 on the range of operators t c is studied for m accretive and maximal monotone operators t here b e f x c d t with sufficiently large norm and u c tx various new eigenvalue results are given involving the solvability of tx acx 3 0 with respect to a x e 0 oo x d t several results do not require the continuity of the operator c four open problems are also given the solution of which would improve upon certain results of the paper

properties of the probability density function of the non central chi squared distribution abstract in this paper we consider the probability density function pdf of a non central 2 distribution with arbitrary number of degrees of freedom for this function we prove that can be represented as a finite sum and we deduce a partial derivative formula moreover we show that the pdf is log concave when the degrees of freedom is greater or equal than 2 at the end of this paper we present some turan type inequalities for this function and an elegant application of the monotone form of l hospital s rule in probability theory is given

on the approximation of convex functions by bernstein type operators abstract as a consequence of jensen s inequality centered operators of probabilistic type also called bernstein type operators approximate convex functions from above starting from this fact we consider several pairs of classical operators and determine in each case which one is better to approximate convex functions in almost all the discussed examples the conclusion follows from a simple argument concerning composition of operators however when comparing szasz mirakyan operators with bernstein operators over the positive semi axis the result is derived from the convex ordering of the involved probability distributions analogous results for non centered operators are also considered

mcmc convergence diagnosis via multivariate bounds on log concave densities we begin by showing how piecewise linear bounds may be devised which bound both above and below any concave log density in general dimensions we then show how these bounds may be used to gain an upper bound to the volume in the tails outside the convex hull of the sample path in order to assess how well the sampler has explored the target distribution this method can be used as a stand alone diagnostic to determine when the sampler output provides a reliable basis for inference on the stationary density or in conjunction with existing convergence diagnostics to ensure that they are based upon good sampler output we provide an example and briefly discuss possible extensions to the method and alternative applications of the bounds 1 introduction the use of markov chain monte carlo mcmc methods has increased dramatically within the statistical community since their

a study of delayed cooperation diffusion system with dirichlet boundary conditions abstract in this paper we study the existence and uniqueness of strong solution of a delayed cooperation diffusion system with dirichlet boundary conditions using the method of semidiscretization in time

nonlinear partial functional differential equations existence and stability existence and uniqueness of solutions for a class of nonlinear functional differential equations in hilbert spaces are established sufficient conditions which guarantee the transference of exponential stability from partial differential equations to partial functional differential equations are studied the stability results derived are applied also to ordinary differential equations with hereditary characteristics

quadrature on the half line and two point pade approximants to stieltjes functions part i algebraic aspects abstract let f z be a stieltjes function with asymptotic expansions l0 and l at z 0 and z respectively let k n denote the two point pade approximant of type m n which matches k of the coefficients of the series l0 and which takes its remaining interpolation conditions from l we discuss in this paper the algebraic aspects of this problem we shall emphasize the relation between quadrature formulas and two point pade approximants and derive expressions for the error in a subsequent paper we shall consider the convergence aspects of these approximants for example the positivity of the error which is obtained here will result in the monotonic convergence of certain sequences of two point pade approximants a property which is well known in the case of classical pade approximants

on the existence of solutions for functional evolution equations soit x un espace de banach de norme soit pc l espace des fonctions continues par morceaux r 0 x pour un r fixe 0 pour pc on pose pc sup r 0 on considere l equation d evolution fonctionnelle non lineaire abstraite g t u t u t a t u t u 0 t 0 t

on linear combinations of l orthogonal polynomials associated with distributions belonging to symmetric classes this paper deals with the classes s 3 w b b of strong distribution functions defined on the interval b 2 b b 0 b

preservation properties by bernstein type operators a probabilistic approach abstract in this paper we are concerned with preservation properties of first and second order by bernstein type operators which preserve monotone functions we obtain characterizations of the preservation of nondecreasing right continuous functions first and second order modulus of smoothness lipschitz classes of first and second order uniform and absolute continuity and convexity these kinds of problems lead us to consider the notions of dual and derived operators we give a simple unified approach based on stochastic orders and probabilistic coupling techniques in the sense that we represent the operators under consideration in terms of stochastic processes the preceding results are illustrated by considering well known bernstein type operators such as generalized bernstein kantorovich generalized szasz kantorovich gamma and beta operators

a method of lines for a nonlinear abstract functional evolution equation let x be a real banach space with x uniformly convex a method of lines is introduced and developed for the abstract functional problem lipschitzian like function in its two variables further conditions are given for the convergence of the method to a strong solution of e recent results for perturbed abstract ordinary equations are substantially improved the method applies also to large classes of functional parabolic problems as well as problems of integral perturbations the method is straightforward because it avoids the introduction of the operators a t and the corresponding use of nonlinear evolution operator theory 1 introduction preliminaries let x be a real banach space with norm 11 let