how to generate unknown orthogonal polynomials out of known orthogonal polynomials abstract we consider the problem of generating the three term recursion coefficients of orthogonal polynomials for a weight function v t r t w t obtained by modifying a given weight function w by a rational function algorithms for the construction of the orthogonal polynomials for the new weight v in terms of those for the old weight w are presented all the methods are based on modified moments as applications we present gaussian quadrature rules for integrals in which the integrant has singularities close to the interval of integration and the generation of orthogonal polynomials for the finite hermite weight e t 2 supported on a finite interval b b

neumann boundary value problems for a time fractional diffusion wave equation in a half plane the time fractional diffusion wave equation with the caputo derivative of the order 0 a 2 is considered in a half plane two types of neumann boundary condition are examined the mathematical condition with the prescribed boundary value of the normal derivative and the physical one with the prescribed boundary value of the matter flux

summations on trigonometric functions a brief maple program of one sentence is designed to compute some striking summations of trigonometric functions then our human imagination figures out the generating functions by means of formal power series and binomial convolution which are confirmed through partial fraction expansions on trigonometric polynomials some related trigonometric sums are treated as consequences

brief paper robustness of exponential stability of a class of stochastic functional differential equations with infinite delay we regard the stochastic functional differential equation with infinite delay dx t f x t dt g x t dw t as the result of the effects of stochastic perturbation to the deterministic functional differential equation email protected t f x t where x t x t q c 0 r n is defined by x t q x email protected email protected 0 we assume that the deterministic system with infinite delay is exponentially stable in this paper we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable

computations with half range chebyshev polynomials an efficient construction of two non classical families of orthogonal polynomials is presented in the paper the so called half range chebyshev polynomials of the first and second kinds were first introduced by huybrechs in huybrechs 2010 5 some properties of these polynomials are also shown every integrable function can be represented as an infinite series of sines and cosines of these polynomials the so called half range chebyshev fourier hcf series the second part of the paper is devoted to the efficient computation of derivatives and multiplication of the truncated hcf series where two matrices are constructed for this purpose the differentiation and the multiplication matrix

the closure of the range of an elementary operator abstract let b h denote the algebra of operators on a hilbert space h and let b b h be the elementary operator defined by x axb cxd a necessary condition for 1 0 b h b h is that 0 is an isolated point of the spectrum of we prove a sufficient condition for 1 0 b h b h applied to the case in which the hyponormal a b and normal c d satisfy certain conditions it is seen that the condition 0 is isolated in the set s a b c and d is sufficient for 1 0 b h b h

some classes of infinite series associated with the riemann zeta and polygamma functions and generalized harmonic numbers the authors aim at presenting a systematic investigation of several families of infinite series which are associated with the riemann zeta function the digamma and polygamma functions the harmonic and generalized harmonic numbers and the stirling numbers of the first kind relevant connections of the results derived here with those considered in many earlier works on this subject are also indicated

remarks on the existence and uniqueness of the solutions to stochastic functional differential equations with infinite delay in this paper we obtain some results on the existence and uniqueness of solutions to stochastic functional differential equations with infinite delay at phase space bc 0 r d which denotes the family of bounded continuous r d value functions f defined on 0 with norm f sup q 0 f q under non lipschitz condition with lipschitz condition being considered as a special case and a weakened linear growth condition the solution is constructed by the successive approximation

existence and uniqueness theorems for periodic markov process and applications to stochastic functional differential equations abstract this work is devoted to an m valued markov process with a periodic transition probability function where m is a polish space sufficient conditions for the existence and uniqueness of the periodic markov process are given these existence theorems are rather general and the corresponding results in 1 11 15 23 are improved and generalized the uniqueness theorems for the periodic markov process are original as applications we study the existence uniqueness and global attractivity of periodic solution for the periodic stochastic n species lotka voltrra competitive model with bounded delays and the periodic stochastic neural networks with infinite delay respectively numerical simulations are presented to illustrate the main results

the improved lasalle type theorems for stochastic functional differential equations abstract the main aim of this paper is to improve some results obtained by mao x mao the lasalle type theorems for stochastic functional differential equations nonlinear stud 7 2000 307 328 our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above for example we need only the local lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions to guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation sfde under the weaker conditions imposed in this paper we establish a generalised existence and uniqueness theorem which covers a wider class of nonlinear sfdes as demonstrated by the examples discussed in this paper moreover from our improved results follow some new criteria on the stochastic asymptotic stability for sfdes

ergodicity for functional stochastic differential equations and applications abstract in this paper using the remote start or dissipative method we investigate ergodicity for several kinds of functional stochastic equations including functional stochastic differential equations sdes with variable delays neutral functional sdes functional sdes driven by jump processes and semi linear functional stochastic partial differential equations spdes using the ergodicity derived we then treat a couple of applications in stochastic approximation and optimization problems

minimal solutions of three term recurrence relations and orthogonal polynomials we observe that the well known recurrence relation pn x z z a pn z bnp l z for orthogonal polynomials admits a minimal solution if z is outside the spectrum of the mass distribution ds t with respect to which the polynomials are orthogonal and if the moment problem for this distribution is determined the minimal solution indeed is f z m ds t z t and can be computed accurately by means of the author s continued fraction algorithm an application is made to special gauss type quadrature formulas 1 introduction minimal solutions of three term recurrence relations and their computational implications are discussed systematically in 7 effective algorithms for computing minimal solutions have been developed in 7 16 17 and continue to be the subject of further study see e g 14 26 5 2 24 25 3 with these powerful algorithms at hand it seems desirable to delineate large classes of recurrence relations for which the presence of minimal solutions can be ascertained and the minimal solution itself identified even more desirable is an understanding of the deeper reasons for the occurrence of minimal solutions in terms of intrinsic features of the subject area in which they arise few attempts have been made along these lines the work of thacher 22 23 on power series solution of linear differential equations with polynomial coeffi cients may be considered a beginning inasmuch as the presence of minimal solutions of the recurrence relation satisfied by the expansion coefficients is conjectured to be related in a specified way to the type of singularities in the differential equation here we wish to consider the recurrence relation

stochastic lotka volterra population dynamics with infinite delay we discover in this paper that when environmental noise is strongly dependent on the population size this noise may suppress the population explosion in a finite time and guarantee the global positive solution when the environmental noise is weakly dependent on the population size the conditions that guarantee the global positive solution are independent of this environmental noise we also discuss the pth moment boundedness stochastic ultimate boundedness and moment average boundedness in time under two classes of conditions these properties are natural requirements from the biological point of view

on the calculation of jacobi matrices abstract given a jacobi matrix the problem in question is to find the jacobi matrix corresponding to the weight function modified by a polynomial r galant and gautschi derived algorithms based on the generalized christoffel theorem of uvarov applicable when the roots of r are known in this paper we present two methods not requiring the explicit knowledge of the roots of r we also obtain various properties of the similarity transformations between jacobi matrices which we prove by simple matrix calculus without using the generalized christoffel theorem

totally hereditarily normaloid operators and weyl s theorem for an elementary operator abstract a hilbert space operator t b h is hereditarily normaloid notation t hn if every part of t is normaloid an operator t hn is totally hereditarily normaloid notation t thn if every invertible part of t is normaloid we prove that thn operators with bishop s property also thn contractions with a compact defect operator such that t 1 0 t 1 0 and non zero isolated eigenvalues of t are normal are not supercyclic take a and b in thn and let d a b denote either of the elementary operators in b b h a b and a b where a b x a x b x and a b x a x x b we prove that if non zero isolated eigenvalues of a and b are normal and b 1 0 b 1 0 then d a b is an isoloid operator such that the quasi nilpotent part h 0 d a b of d a b equals d a b 1 0 for every complex number which is isolated in d a b if additionally d a b has the single valued extension property at all points not in the weyl spectrum of d a b then d a b and the conjugate operator d a b satisfy weyl s theorem

robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay this paper establishes the existence and uniqueness theorem of neutral stochastic functional differential equations with infinite delay and examines the almost sure stability of this solution with general decay rate this result may be used to examine almost sure robust stability to illustrate our idea more carefully we carefully discuss a scalar stochastic integro differential equation with neutral type and its asymptotic stability including the exponential stability and the polynomial stability

a certain generalized pochhammer symbol and its applications to hypergeometric functions in this article we first introduce an interesting new generalization of the familiar pochhammer symbol by means of a certain one parameter family of generalized gamma functions with the help of this new generalized pochhammer symbol we then introduce an extension of the generalized hypergeometric function rf s with r numerator and s denominator parameters finally we present a systematic study of the various fundamental properties of the class of the generalized hypergeometric functions introduced here

global stability for infinite delay lotka volterra type systems abstract in this paper sufficient conditions are established for the global stability of the saturated equilibrium of an infinite delay nonautonomous lotka volterra type system the present work is distinguished from previous work principally be allowing the system to be nonautonomous and relaxing the traditional requirement that the undelayed intraspecific competition dominates both the delayed intraspecific competition as well as the interspecific interactions we require the undelayed intraspecific competition to dominate the latter but not the former

inequalities for the median of the gamma distribution we provide several new inequalities involving l n the median of the gamma distribution of order n 1 with parameter 1 among others we present sharp upper and lower bounds for the arithmetic mean of l 1 l 2 l n for all integers n 1 we have an 76 n2 8405h nn email protected email protected k bn 76 n2 8405h nn with the best possible constants email protected k 1 l k k 23 8405k email protected email protected 1 683405 0 0080 here h n denotes the nth harmonic number

orthogonal rational functions and quadrature on an interval rational functions with real poles and poles in the complex lower half plane orthogonal on the real line are well known quadrature formulas similar to the gauss formulas for orthogonal polynomials have been studied we generalize to the case of arbitrary complex poles and study orthogonality on a finite interval the zeros of the orthogonal rational functions are shown to satisfy a quadratic eigenvalue problem in the case of real poles these zeros are used as nodes in the quadrature formulas

a domain decomposition method for generating orthogonal polynomials for a gaussian weight on a finite interval abstract a domain decomposition method has been developed for generating orthogonal polynomials for a gaussian weight on 1 1 the method takes advantage of the underlying asymptotic structure of the orthogonal polynomials and hence it is effective in the sense that it makes maximal use of the analytic properties of the solution to increase accuracy and efficiency these polynomials are necessary for constructing gaussian quadrature formulas that are encountered in large quantum chemistry computational packages and in calculating the compton scattering kernel and its associated angular moments

computation of gauss type quadrature formulas gaussian formulas for a linear functional l such as a weighted integral are best computed from the recursion coefficients relating the monic polynomials orthogonal with respect to l in gauss type formulas one or more extraneous conditions such as pre assigning certain nodes replace some of the equations expressing exactness when applied to high order polynomials these extraneous conditions may be applied by modifying the same number of recursion coefficients we survey the methods of computing formulas from recursion coefficients methods of obtaining recursion coefficients and modifying them for gauss type formulas and questions of existence and numerical accuracy associated with those computations

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

on generating polynomials which are orthogonal over several intervals we consider the problem of generating the recursion coefficients of orthogonal polynomials for a given weight function the weight function is assumed to be the weighted sum of weight functions each supported on its own interval some of these intervals may coincide overlap or are contiguous we discuss three algorithms two of them are based on modified moments whereas the other is based on an explicit expression for the desired coefficients several examples illustrating the numerical performance of the various methods are presented

equations with infinite delay blending the abstract and the concrete using perturbation theory for adjoint semigroups a modification of sun star calculus we prove in the case of infinite delay the principle of linearized stability for nonlinear renewal equations delay differential equations and coupled systems of these two types of equations our results extend those of diekmann et al 1995 13 and diekmann et al 2007 14 to the case of infinite delay

attractors for differential equations with unbounded delays we prove the existence of attractors for some types of dierential problems contain ing in nite delays applications and examples are provided to illustrate the theory which is valid for both cases with and without explicit dependence of time and with or without uniqueness of solutions as well

the existence and construction of rational gauss type quadrature rules abstract consider a hermitian positive definite linear functional f and assume we have m distinct nodes fixed in advance anywhere on the real line in this paper we then study the existence and construction of n th rational gauss radau m 1 and gauss lobatto m 2 quadrature formulas that approximate f f these are quadrature formulas with n positive weights and with the n m remaining nodes real and distinct so that the quadrature is exact in a 2 n m dimensional space of rational functions further we also consider the case in which the functional is defined by a positive bounded borel measure on an interval for which it is required in addition that the nodes are all in the support of the measure

orthogonal polynomials in matlab a suite of matlab programs has been developed as part of the book orthogonal polynomials computation and approximation oxford university press oxford 2004 by gautschi the package contains routines for generating orthogonal polynomials as well as routines dealing with applications in this paper a brief review of the first part of the package is given dealing with procedures for generating the three term recurrence relation for orthogonal polynomials and more general recurrence relations for sobolev orthogonal polynomials moment based methods and discretization methods and their implementation in matlab are among the principal topics discussed

integral inequalities of systems and the estimate for solutions of certain nonlinear two dimensional fractional differential systems this paper generalizes the results for the constructions of explicit bounds and the qualitative properties for the solutions of certain two dimensional fractional differential systems established in a recent paper of the authors the main generalizations come from an elementary inequality and by means of the modification of medved s de singular approach

stability in distribution of stochastic functional differential equations abstract in this paper we investigate the stability in distribution for a class of stochastic functional differential equations sfdes which include stochastic differential delay equations sddes although stability in distribution has been studied by several authors recently there is so far no stability in distribution criterion on sfdes where the terms involved the delay components are highly nonlinear not bounded by linear functions in this paper we will establish the sufficient criteria on the stability in distribution for a class of highly nonlinear sfdes two examples will be given to illustrate our new results we also explain the reason why the existing stability in distribution criteria are not applicable by these two examples

quadrature rules for weakly singular strongly singular and hypersingular integrals in boundary integral equation methods we present n point gauss gegenbauer quadrature rules for weakly singular strongly singular and hypersingular integrals that arise in integral equation formulations of potential problems in domains with edges and corners the rules are tailored to weight functions with algebraic endpoint singularities related to the geometrical singularities of the domain each rule has two different expressions involving legendre functions and hypergeometric functions respectively numerical examples amply demonstrate the accuracy and stability of the proposed algorithms application to the solution of a singular integral equation is exemplified

fractional integration of the h function of several variables abstract the main object of the present paper is to derive a number of key formulas for the fractional integration of the multivariable h function which is defined by a multiple contour integral of mellin barnes type each of the general eulerian integral formulas obtained in this paper are shown to yield interesting new results for various families of generalized hypergeometric functions of several variables some of these applications of the key formulas would provide potentially useful generalizations of known results in the theory of fractional calculus

the existence and uniqueness of the solution for stochastic functional differential equations with infinite delay abstract this paper is devoted to build the existence and uniqueness theorem of solutions to stochastic functional differential equations with infinite delay short for isfdes at phase space bc 0 r d under the uniform lipschitz condition the linear growth condition is weaked to obtain the moment estimate of the solution for isfdes furthermore the existence and uniqueness theorem of the solution for isfdes is derived and the estimate for the error between approximate solution and accurate solution is given on the other hand under the linear growth condition the uniform lipschitz condition is replaced by the local lipschitz condition the existence and uniqueness theorem is also valid for isfdes on t 0 t moreover the existence and uniqueness theorem still holds on interval t 0 where t 0 r is an arbitrary real number

rational gauss chebyshev quadrature formulas for complex poles outside 1 1 in this paper we provide an extension of the chebyshev orthogonal rational functions with arbitrary real poles outside 1 1 to arbitrary complex poles outside 1 1 the zeros of these orthogonal rational functions are not necessarily real anymore by using the related para orthogonal functions however we obtain an expression for the nodes and weights for rational gauss chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside 1 1

weighted integration of periodic functions on the real line integration of periodic functions on the real line with an even rational weight function is considered a transformation method of such integrals to the integrals on 1 1 with respect to the email protected bernstein weights and a construction of the corresponding gaussian quadrature formulas are given the recursion coefficients in the three term recurrence relation for the corresponding orthogonal polynomials were obtained in an analytic form numerical examples are also included

a generalized eigenvalue problem for quasi orthogonal rational functions in general the zeros of an orthogonal rational function orf on a subset of the real line with poles among alpha 1 ldots alpha n subset mathbb c 0 cup infty are not all real unless alpha n is real and hence they are not suitable to construct a rational gaussian quadrature rule rgq for this reason the zeros of a so called quasi orf or a so called para orf are used instead these zeros depend on one single parameter tau in mathbb c cup infty which can always be chosen in such a way that the zeros are all real and simple in this paper we provide a generalized eigenvalue problem to compute the zeros of a quasi orf and the corresponding weights in the rgq first we study the connection between quasi orfs para orfs and orfs next a condition is given for the parameter so that the zeros are all real and simple finally some illustrative and numerical examples are given

an averaging principle for two time scale stochastic functional differential equations abstract delays are ubiquitous pervasive and entrenched in everyday life thus taking it into consideration is necessary dupire recently developed a functional ito formula which has changed the landscape of the study of stochastic functional differential equations and encouraged a reconsideration of many problems and applications based on the new development this work examines functional diffusions with two time scales in which the slow varying process includes path dependent functionals and the fast varying process is a rapidly changing diffusion the gene expression of biochemical reactions occurring in living cells in the introduction of this paper is such a motivating example this paper establishes mixed functional ito formulas and the corresponding martingale representation then it develops an averaging principle using weak convergence methods by treating the fast varying process as a random noise under appropriate conditions it is shown that the slow varying process converges weakly to a stochastic functional differential equation whose coefficients are averages of that of the original slow varying process with respect to the invariant measure of the fast varying process

quadrature discretization method in tethered satellite control a pseudospectral method for solving the tethered satellite retrieval problem based on nonclassical orthogonal and weighted interpolating polynomials is presented traditional pseudospectral methods expand the state and control trajectories using global lagrange interpolating polynomials based on a specific class of orthogonal polynomials from the jacobi family such as legendre or chebyshev polynomials which are orthogonal with respect to a specific weight function over a fixed interval although these methods have many advantages the location of the collocation points are more or less fixed the method presented in this paper generalizes the existing methods and allows a much more flexible selection of grid points by the arbitrary selection of the orthogonal weight function and interval the trajectory optimization problem is converted to set of algebraic equations by discretization of the necessary conditions using a nonclassical pseudospectral method

stochastic functional differential equations with infinite delay existence and uniqueness of solutions solution maps markov properties and ergodicity abstract this work is devoted to stochastic functional differential equations sfdes with infinite delay first existence and uniqueness of the solutions of such equations are examined because the solutions of the delay equations are not markov a viable alternative for studying further asymptotic properties is to use solution maps or segment processes by examining solution maps this work investigates the markov properties as well as the strong markov properties also obtained are adaptivity and continuity mean square boundedness and convergence of solution maps from different initial data this paper then examines the ergodicity of underlying processes and establishes existence of the invariant measure for sfdes with infinite delay under suitable conditions

gaussian rational quadrature formulas for ill scaled integrands a flexible treatment of gaussian quadrature formulas based on rational functions is given to evaluate the integral if x w x dx when f is meromorphic in a neighborhood v of the interval i and w x is an ill scaled weight function some numerical tests illustrate the power of this approach in comparison with gautschi s method

orthogonal rational functions with complex poles the favard theorem abstract let n be a sequence of rational functions with arbitrary complex poles generated by a certain three term recurrence relation in this paper we show that under some mild conditions the rational functions n form an orthonormal system with respect to a hermitian positive definite inner product

multiple gamma and related functions the authors give several new and potentially useful relationships between the multiple gamma functions and other mathematical functions and constants as by products of some of these relationships a classical definite integral due to euler and other definite integrals are also considered together with closed form evaluations of some series involving the riemann and hurwitz zeta functions

an elementary operator with log hyponormal p hyponormal entries a hilbert space operator a b h is p hyponormal a p h if a 2p a 2p an invertible operator a b h is log hyponormal a l h if log tt log t t let dab ab or ab where ab b b h is the generalised derivation ab x ax xb and ab b b h is the elementary operator ab x axb x it is proved that if a b l h p h then for all complex dab 1 0 da b 1 0 the ascent of dab 1 and dab satisfies the range kernel orthogonality inequality x x dab y for all x dab 1 0 and y b h furthermore isolated points of dab are simple poles of the resolvent of dab a version of the elementary operator e x a1xa2 b1xb2 and perturbations of dab by quasi nilpotent operators are considered and weyl s theorem is proved for dab

comparison of newton cotes and gaussian methods of quadrature we compare newton cotes and gauss methods of various orders we give two matlab programs that evaluates integrals numerically for given order with given number of points we make extensive tests with various functions and intervals using same number of points for each method and compare errors