comment on phase space representation of quantum state vectors a simple approach to phase space representation of quantum state vectors using the displacement operator formalism is presented although the resulting expressions for the fundamental operators position and momentum are equivalent to those obtained by other methods this approach provides both alternative mathematical foundation as well as physical interpretation of phase space representation of quantum state vectors

orthogonal rational functions for system identification numerical aspects recently there has been a growing interest in the use of orthogonal rational functions orfs in system identification there are many advantages over more classical techniques probably due to a known explicit expression for the basis functions when the orthogonality weight is uniformly equal to 1 the so called malmquist basis the attention has been on the development of methods using this basis however for some discrete identification problems this choice of the orthogonality weight may still lead to serious numerical problems due to the ill conditioning of the linear system of equations to be solved in this note we give an algorithm based on a more general system of orf to overcome the numerical problem and which allows for a fast order update of the estimate

approximation of the hilbert transform on the real line using hermite zeros the authors study the hilbert transform on the real line they introduce some polynomial approximations and some algorithms for its numerical evaluation error estimates in uniform norm are given

rank one perturbations and the zeros of paraorthogonal polynomials on the unit circle we prove several results about zeros of paraorthogonal polynomials using the theory of rank one perturbations of unitary operators in particular we obtain new details on the interlacing of zeros for successive popuc

rational interpolation i least square convergence abstract given a positive bounded borel measure on the interval 1 1 we provide convergence results in l 2 norm to a function f of its sequence of interpolating rational functions at the nodes of rational gauss type quadrature formulas associated with the measure for this we use the connection between rational gauss type quadrature formulas on 1 1 and rational szeg quadrature formulas associated with a positive symmetric borel measure on the complex unit circle

quadrature formulas on the unit circle based on rational functions abstract quadrature formulas on the unit circle were introduced by jones in 1989 on the other hand bultheel also considered such quadratures by giving results concerning error and convergence in other recent papers a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well known szego polynomials in this paper these quadratures are again analyzed and results about convergence given furthermore an application to the poisson integral is also made

convergence and computation of simultaneous rational quadrature formulas we discuss the convergence and numerical evaluation of simultaneous quadrature formulas which are exact for rational functions the problem consists in integrating a single function with respect to different measures using a common set of quadrature nodes given a multi index n the nodes of the integration rule are the zeros of the multi orthogonal hermite pade polynomial with respect to s n where s is a collection of measures and is a polynomial which modifies the measures in s the theory is based on the connection between gauss type simultaneous quadrature formulas of rational type and multipoint hermite pade approximation the numerical treatment relies on the technique of modifying the integrand by means of a change of variable when it has real poles close to the integration interval the output of some tests show the power of this approach in comparison with other ones in use

matrices moments and rational quadrature abstract many problems in science and engineering require the evaluation of functionals of the form f u a u t f a u where a is a large symmetric matrix u a vector and f a nonlinear function a popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the lanczos procedure applied to a with initial vector u and then evaluating pairs of gauss and gauss radau quadrature rules associated with the tridiagonal matrix determined by the lanczos procedure the present paper extends this approach to allow the use of rational gauss quadrature rules

connective steiner 3 eccentricity index and network similarity measure abstract for a set s v g in a network g the steiner distance dg s of s is the minimum size among all connected subnetworks whose vertex sets contain s the steiner k eccentricity k v of a vertex v of g is the maximum steiner distance among all k vertex set s which contains the vertex v i e e k v max d s s v g s k v s based on steiner k eccentricity the connective steiner k eccentricity index is introduced as a newly structural invariant some properties of the connective steiner 3 eccentricity index are investigated firstly we present an o n2 polynomial time algorithm to calculate the connective steiner 3 eccentricity index of trees secondly some optimal problems among some network classes are discussed as its application finally we consider the network similarity measure based on the connective steiner 3 eccentricity index by two different methods we study its advantages numerical results show that the measure based on the connective steiner 3 eccentricity index has more advantages than the ones based on other topological indices graph energy randic index the largest adjacent eigenvalue the largest laplacian eigenvalue

orthogonal rational functions on the real half line with poles in 0 the main objective is to generalize previous results obtained for orthogonal laurent polynomials and their application in the context of stieltjes moment problems to the multipoint case the measure of orthogonality is supposed to have support on 0 while the orthogonal rational functions will have poles that are assumed to be in the neighborhood of 0 and in this way orthogonal laurent polynomials will be a special case obtained when all the poles are at 0 and we shall introduce the restrictions on the measure and the locations of the poles gradually and derive recurrence relations christoffel darboux relations and the solution of the rational stieltjes moment problem under appropriate conditions

some extremal results on the connective eccentricity index of graphs abstract the connective eccentricity index cei of a graph g is defined as ce g v i v g d v i e v i where e v i and d v i are the eccentricity and the degree of vertex v i respectively in g in this paper we obtain some lower and upper bounds on the connective eccentricity index for all trees of order n and with matching number and characterize the corresponding extremal trees and the maximal graphs of order n and with matching number and n edges have been determined which maximize the connective eccentricity index also the extremal graphs with maximal connective eccentricity index are completely characterized among all connected graphs of order n and with matching number moreover we establish some relations between connective eccentricity index and eccentric connectivity index as another eccentricity based invariant of graphs

positive interpolatory quadrature formulas and para orthogonal polynomials we establish a relation between quadrature formulas on the interval 1 1 that approximate integrals of the form j f 11 f x x dx and szego quadrature formulas on the unit circle that approximate integrals of the form i f f ei d the functions x and are assumed to be weight functions on 1 1 and respectively and are related by cos sin it is well known that the nodes of szego formulas are the zeros of the so called para orthogonal polynomials bn z n z n z 1 n z and n z being the orthogonal and reciprocal polynomials respectively with respect to the weight function furthermore for 1 we have recently obtained gauss type quadrature formulas on 1 1 see bultheel et al j comput appl math 132 1 2000 1 in this paper making use of the para orthogonal polynomials with 1 a one parameter family of interpolatory quadrature formulas with positive coefficients for j f is obtained along with error expressions for analytic integrands finally some illustrative numerical examples are also included

application of graph theory relationship of eccentric connectivity index and wiener s index with anti inflammatory activity abstract graph theory was successfully applied in developing a relationship between chemical structure and biological activity the relationship of two graph invariants the eccentric connectivity index and the wiener s index was investigated with regard to anti inflammatory activity for a data set consisting of 76 pyrazole carboxylic acid hydrazide analogues the values of the eccentric connectivity index and the wiener s index of each analogue in the data set were computed and active ranges were identified subsequently each analogue was assigned a biological activity that was compared with the anti inflammatory activity reported as percent reduction in paw swelling prediction with an accuracy of 90 was obtained using the eccentric connectivity index as compared to 84 in the case of wiener s index

on the convergence of certain gauss type quadrature formulas for unbounded intervals we consider the convergence of gauss type quadrature formulas for the integral f 0 f x x dx where w is a weight function on the half line 0 the n point gauss type quadrature formulas are constructed such that they are exact in the set of laurent polynomials p q 1 k p q 1 p a k x k where p p n is a sequence of integers satisfying 0 p n 2n and q q n 2n p n it is proved that under certain carleman type conditions for the weight and when p n or q n goes to oo then convergence holds for all functions f for which fw is integrable on 0 some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical gauss quadrature formulas for the half line

on the convergence of quadrature formulas connected with multipoint pade type approximation abstract let i f 1 1 f x x dx where is a complex valued integrable function we consider quadrature formulas for i f which are exact with respect to rational functions with prescribed poles contained in formula their rate of convergence is studied

computing near best fixed pole rational interpolants we study rational interpolation formulas on the interval 1 1 for a given set of real or complex conjugate poles outside this interval interpolation points which are near best in a chebyshev sense were derived in earlier work the present paper discusses several computation aspects of the interpolation points and the corresponding interpolants we also study a related set of points that includes the end points which is more suitable for applications in rational spectral methods some examples are given at the end of this paper

product integration over infinite intervals i rules based on the zeros of hermite polynomials the paper discusses both theoretical properties and practical implementation of product integration rules of the form n j k x f x dx wnlj xnj o i l where f is continuous k is absolutely integrable the nodes xn are roots of the hermite polynomials hn x and the weights wn are chosen so that the rule is exact if f is any polynomial of degree n convergence of the rule to the exact integral as n oo is proved for a wide class of functions f and k including singular or oscillatory functions k and rates of convergence are estimated the rules are shown to have the property of asymptotic positivity and as a consequence exhibit good numerical stability numerical calculations for some practical cases are presented which show the method to be computationally effective for integrands including highly oscillatory ones that decay suitably at infinity applications of the method to integration over 0 oo are also discussed

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

calculation of gauss kronrod quadrature rules the jacobi matrix of the 2n 1 point gauss kronrod quadrature rule for a given measure is calculated efficiently by a five term recurrence relation the algorithm uses only rational operations and is therefore also useful for obtaining the jacobi kronrod matrix analytically the nodes and weights can then be computed directly by standard software for gaussian quadrature formulas

rational szego quadratures associated with chebyshev weight functions in this paper we characterize rational szeg o quadrature formulas associated with chebyshev weight functions by giving explicit expressions for the corresponding para orthogonal rational functions and weights in the quadratures as an application we give characterizations for szeg o quadrature formulas associated with rational modifications of chebyshev weight functions some numerical experiments are finally presented

quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity in this paper we investigate the szego radau and szego lobatto quadrature formulas on the unit circle these are n m point formulas for which m nodes are fixed in advance with m 1 and m 2 respectively and which have a maximal domain of validity in the space of laurent polynomials this means that the free parameters free nodes and positive weights are chosen such that the quadrature formula is exact for all powers z j email protected email protected p with p p n m as large as possible

interpolatory quadrature formulas on the unit circle for chebyshev weight functions in this paper interpolatory quadrature formulas based upon the roots of unity are studied for certain weight functions positivity of the coefficients in these formulas is deduced along with computable error estimations for analytic integrands a comparison is made with szego quadrature formulas finally an application to the interval 1 1 is also carried out

on the numerical resolution of the generalized airfoil equation with possio kernel in this paper we consider the so called generalized airfoil singular integral equation with a smooth input function and present ad hoc quadrature rules to compute efficiently the elements of collocation and galerkin matrices

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

on computing rational gauss chebyshev quadrature formulas we provide an algorithm to compute the nodes and weights for gauss chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary real poles outside 1 1 contrary to existing rational quadrature formulas the computational effort is very low even for extremely high degrees and under certain conditions on the poles it can be shown that the complexity is of order o n this method is based on the derivation of explicit expressions for chebyshev orthogonal rational functions which are thus far the only examples of explicitly known orthogonal rational functions on 1 1 with arbitrary real poles outside this interval

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

quadrature rules for rational functions summary it is shown how recent ideas on rational gauss type quadrature rules can be extended to gauss kronrod gauss turan and cauchy principal value quadrature rules numerical examples illustrate the advantages in accuracy thus achievable

on the maximal connective eccentricity index of bipartite graphs with some given parameters abstract the connective eccentricity index is a novel graph invariant with vast potential in structure activity property relationships this graph invariant displays high discriminating power with respect to both biological activity and physical properties given a simple connected graph g the connective eccentricity index cei of g is defined as e e g u v e g 1 e g u 1 e g v where e g denotes the eccentricity of the corresponding vertex in this paper we first determine the sharp upper bound on the cei of graphs in the class of all n vertex connected bipartite graphs with matching number q the maximum cei is realized only by the graph k q n q second we characterize the graph with the maximum cei in the class of all the n vertex connected bipartite graphs of given diameter finally all the extremal graphs having the maximum cei in the class of all the connected n vertex bipartite graphs with a given connectivity s are identified as well

on the eccentric distance sum of graphs the eccentric distance sum is a novel topological index that offers a vast potential for structure activity property relationships for a graph g it is defined as d g v ve v d v where e v is the eccentricity of the vertex v and d v u v g d u v is the sum of all distances from the vertex v motivated by g yu l feng a ilic on the eccentric distance sum of trees and unicyclic graphs j math anal appl 375 2011 934 944 in this paper we characterize the extremal trees and graphs with maximal eccentric distance sum various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the wiener index the degree distance eccentric connectivity index independence number connectivity matching number chromatic number and clique number are established in addition we present explicit formulae for the values of eccentric distance sum for the cartesian product applied to some graphs of chemical interest like nanotubes and nanotori

rational quadrature formulae on the unit circle with arbitrary poles interpolatory quadrature rules exactly integrating rational functions on the unit circle are considered the poles are prescribed under the only restriction of not lying on the unit circle a computable upper bound of the error is obtained which is valid for any choice of poles arbitrary weight functions and any degree of exactness provided that the integrand is analytic on a neighborhood of the unit circle a number of numerical examples are given which show the advantages of using such rules as well as the sharpness of the error bound also a comparison is made with other error bounds appearing in the literature

eccentric distance sum a novel graph invariant for predicting biological and physical properties abstract eccentric distance sum a novel graph invariant with vast potential in structure activity property relationships has been conceptualized in the present study this graph invariant displayed high discriminating power with respect to both biological activity and physical properties the structure activity relationship of eccentric distance sum was investigated with regard to anti hiv activity of dihydroseselins the values of eccentric distance sum of each analogue in the data set were computed and active range identified subsequently biological activity was assigned to each analogue in the data set which was then compared with the reported anti hiv activity of dihydroseselin analogues surprisingly the accuracy of prediction was found to be more than 88 with regard to anti hiv activity on the other hand investigations pertaining to quantitative structure property relationship of the novel graph invariant with regard to various physical properties of diverse nature for data sets consisting of primary amines secondary amines and alcohols revealed correlation percentages ranging from 93 to 99 the over all results with regard to structure activity and quantitative structure property studies using eccentric distance sum were better than the corresponding values obtained using wiener s index

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

szegő type quadrature formulas abstract we consider the approximation of integrals with respect to measures supported on the unit circle by means of positive quadrature formulas with maximal domain of exactness and up to three preassigned nodes the quadrature formulas are exact in a new nested sequence of subspaces of laurent polynomials that allows us to extend analogous properties to the usual quadrature formulas on the real line estimations for the error of these quadrature formulas depending on the regularity of the integrand are obtained

on the connective eccentricity index of trees and unicyclic graphs with given diameter abstract the connective eccentricity index cei of a graph g is defined as ce g v v g d v e v where e v d v denote the eccentricity and the degree of v respectively in this paper we investigate the connective eccentricity index of trees and unicyclic graphs on n vertices with diameter d we also characterize the extremal graphs

phase space representation of quantum state vectors the relative state approach and the displacement operator approach phase space representation of quantum state vectors has been recently formulated by means of the relative state method developed by the present author j math phys 39 1744 1998 it is however pointed out by mo ller that the displacement operator method provides another basis of phase space representation of quantum state vectors j math phys to appear hence the relation between the relative state approach and the displacement operator approach is discussed both of which yield equivalent phase space representations

quadrature and orthogonal rational functions abstract classical interpolatory or gaussian quadrature formulas are exact on sets of polynomials the szeg quadrature formulas are the analogs for quadrature on the complex unit circle here the formulas are exact on sets of laurent polynomials in this paper we consider generalizations of these ideas where the laurent polynomials are replaced by rational functions that have prescribed poles these quadrature formulas are closely related to certain multipoint rational approximants of cauchy or riesz herglotz transforms of a positive or general complex measure we consider the construction and properties of these approximants and the corresponding quadrature formulas as well as the convergence and rate of convergence

rational approximants associated with measures supported on the unit circle and the real line abstract as a continuation of the well known connection between the theory of orthogonal polynomials on the unit circle and the interval 1 1 in this paper properties concerning error and convergence of certain rational approximants associated with the measures d t and d d cos supported on 1 1 and the unit circle respectively are deduced numerical illustrations are also given

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

product rules on the unit circle with uniformly distributed nodes error bounds for analytic functions this paper deals with the numerical calculation of integrals over the unit circle in the complex plane by means of quadrature formulas with uniformly distributed nodes on such region several properties of the coefficients as well as error bounds for analytic functions are established

on the eccentric distance sum of trees and unicyclic graphs abstract let g be a simple connected graph with the vertex set v g the eccentric distance sum of g is defined as d g v v g e v d g v where e v is the eccentricity of the vertex v and d g v u v g d u v is the sum of all distances from the vertex v in this paper we characterize the extremal unicyclic graphs among n vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum in addition we characterize the extremal trees with given diameter and minimal eccentric distance sum

characteristics polynomial of normalized laplacian for trees here we find the characteristics polynomial of normalized laplacian of a tree the coefficients of this polynomial are expressed by the higher order general randic indices for matching whose values depend on the structure of the tree we also find the expression of these indices for starlike tree and a double starlike tree hm p q moreover we show that two cospectral hm p q of the same diameter are isomorphic

on the convergence of quadrature formulas for complex weight functions abstract we consider quadrature formulas for complex weights and prove their convergence for a large class of functions

optimized point shifts and poles in the linear rational pseudospectral method for boundary value problems due to their rapid often exponential convergence as the number n of interpolation collocation points is increased polynomial pseudospectral methods are very efficient in solving smooth boundary value problems however when the solution displays boundary layers and or interior fronts this fast convergence will merely occur with very large n to address this difficulty we present a method which replaces the polynomial ansatz with a rational function r and considers the physical domain as the conformal map g of a computational domain g shifts the interpolation points from their classical position in the computational domain to a problem dependent position in the physical domain starting from a map by bayliss and turkel we have constructed a shift that can in principle accomodate an arbitrary number of fronts its parameters as well as the poles of r are optimized numerical results demonstrate how g best accomodates interior fronts while the poles also handle boundary layers

bounds for the sum balaban index and revised szeged index of regular graphs mathematical properties of many topological indices are investigated knor et al gave an upper bound for the balaban index of r regular graphs on n vertices and a better upper bound for fullerene graphs they also suggested exploring similar bounds for other topological indices in this paper we consider the sum balaban index and the revised szeged index and give upper and lower bounds for these three indices of r regular graphs and also the cubic graphs and fullerene graphs respectively

sequences of orthogonal laurent polynomials bi orthogonality and quadrature formulas on the unit circle in this paper the construction of orthogonal bases in the space of laurent polynomials on the unit circle is considered as an application a connection with the so called bi orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on laurent polynomials are studied

a numerical solution of the constrained weighted energy problem a numerical algorithm is presented to solve the constrained weighted energy problem from potential theory as one of the possible applications of this algorithm we study the convergence properties of the rational lanczos iteration method for the symmetric eigenvalue problem the constrained weighted energy problem characterizes the region containing those eigenvalues that are well approximated by the ritz values the region depends on the distribution of the eigenvalues on the distribution of the poles and on the ratio between the size of the matrix and the number of iterations our algorithm gives the possibility of finding the boundary of this region in an effective way we give numerical examples for different distributions of poles and eigenvalues and compare the results of our algorithm with the convergence behavior of the explicitly performed rational lanczos algorithm

maximum of the modulus of kernels of gaussian quadrature formulae for one class of bernstein szegő weight functions abstract we continue with the study of the kernels k n z in the remainder terms r n f of the gaussian quadrature formulae for analytic functions f inside elliptical contours with foci at 1 and a sum of semi axes 1 the weight function w of bernstein szeg type here is w t w 1 2 t 1 1 t 2 1 4 1 2 t 2 1 t 1 1 1 0 sufficient conditions are found ensuring that the kernel attains its maximum absolute value at the intersection point of the contour with either the real or the imaginary axis this leads to effective error bounds of the corresponding gauss quadratures the quality of the derived bounds is demonstrated by a comparison with other error bounds intended for the same class of integrands

error bounds for gauss kronrod quadrature formulae of analytic functions we consider the gauss kronrod quadrature formula for the legendre weight function on certain spaces of analytic functions its error term is a continuous linear functional we derive easy to compute estimates for the norm of the error functional which lead to bounds for the error functional itself the efficiency of these bounds is illustrated with some numerical examples

product integration based on hermite feje r interpolation abstract the use of generalized hermite fejer interpolation polynomials for product integration is studied algorithms for implementing this method are developed and convergence results are proved comparisons are made between this new approach and the classical approach based on lagrange interpolation polynomials it turns out that integration rules based on hermite fejer interpolation converge in more cases but in general more slowly than those based on lagrange interpolation when the latter converge

quadrature formulas for functions with poles near the interval of integration on examine la construction des regles de quadrature de type interpolatoire utilisant uniquement des valeurs de fonctions reelles pour des fonctions ayant des paires complesex conjuguees de poles pres de l intervalle d integration

a connection between quadrature formulas on the unit circle and the interval 1 1 abstract we establish a relation between gauss quadrature formulas on the interval 1 1 that approximate integrals of the form i f 1 1 f x x d x and szeg quadrature formulas on the unit circle of the complex plane that approximate integrals of the form i f f e i d the weight x is positive on 1 1 while the weight is positive on it is shown that if cos sin then there is an intimate relation between the gauss and szeg quadrature formulas moreover as a side result we also obtain an easy derivation for relations between orthogonal polynomials with respect to x and orthogonal szeg polynomials with respect to inclusion of gauss lobatto and gauss radau formulas is natural

szego quadrature formulas for certain jacobi type weight functions in this paper we are concerned with the estimation of integrals on the unit circle of the form 02 f ei d by means of the so called szego quadrature formulas i e formulas of the type j 1n jf xj with distinct nodes on the unit circle exactly integrating laurent polynomials in subspaces of dimension as high as possible when considering certain weight functions related to the jacobi functions for the interval 1 1 nodes xj j 1n and weights j j 1n in szego quadrature formulas are explicitly deduced illustrative numerical examples are also given

further results regarding the sum of domination number and average eccentricity the average eccentricity of a graph g denoted by ecc g is the mean value of eccentricities of all vertices of g let dn i be the n vertex tree obtained from a path p n 1 v 1 v 2 v n 1 by attaching a pendent vertex to vi in 13 it was shown that the maximum value for the sum of domination number and average eccentricity among n vertex connected graphs is attained by dn 3 when n 0 mod 3 and attained by the path pn when n 0 mod 3 in this paper we will further determine the second maximum value for the sum of domination number and average eccentricity among n vertex connected graphs it is interesting that the graphs attaining that second maximum value have three cases which is dn 6 when n 0 mod 3 dn 3 when n 1 mod 3 and tn when n 2 mod 3 where tn is the n vertex tree obtained from a path p n 2 v 1 v 2 v n 2 by attaching a pendent vertex to v3 and a pendent vertex to v n 4

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

a rational spectral collocation method with adaptively transformed chebyshev grid points a spectral collocation method based on rational interpolants and adaptive grid points is presented the rational interpolants approximate analytic functions with exponential accuracy by using prescribed barycentric weights and transformed chebyshev points the locations of the grid points are adapted to singularities of the underlying solution and the locations of these singularities are approximated by the locations of poles of chebyshev pade approximants numerical experiments on two time dependent problems one with finite time blow up and one with a moving front indicate that the method far outperforms the standard chebyshev spectral collocation method for problems whose solutions have singularities in the complex plane close to 1 1

orthogonality interpolation and quadratures on the unit circle and the interval 1 1 given a weight function s x on 1 1 or more generally a positive borel measure the erdos turan theorem assures the convergence in l 2 s norm to a function f of its sequence of interpolating polynomials at the zeros of the orthogonal polynomials or equivalently at the nodes of the gauss christoffel quadrature formulas associated with s in this paper we will extend this result to the nodes of the gauss radau and gauss lobatto quadrature formulas by passing to the unit circle and taking advantage of the results on interpolation by means of laurent polynomials at the zeros of certain para orthogonal polynomials with respect to the weight function w q s cos q sin q on p p as a consequence an application to the construction of certain product integration rules on finite intervals of the real line will be given several numerical experiments are finally carried out

chebyshev series method for computing weighted quadrature formulas abstract in this paper we study convergence and computation of interpolatory quadrature formulas with respect to a wide variety of weight functions the main goal is to evaluate accurately a definite integral whose mass is highly concentrated near some points the numerical implementation of this approach is based on the calculation of chebyshev series and some integration formulas which are exact for polynomials in terms of accuracy the proposed method can be compared with rational gauss quadrature formula

computing orthogonal rational functions with poles near the boundary computing orthogonal rational functions is a far from trivial problem especially for poles close to the boundary of the support of the orthogonality measure in this paper we analyze some of the difficulties involved and present two different approaches for solving this problem

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

a novel method for analyzing inverse problem of topological indices of graphs using competitive agglomeration a novel method for analyzing the inverse problem of topological indices of graph is proposed in this paper which applies the data mining technique to classification of graphs according to topological indices differing from the way of using topological indices to categorize graphs into different isomorphic classes this method aims at dividing molecular graphs with n vertices into several compact classes according to their topological indices then the chemical and physical features of each kind of graph can be researched and new compounds can be found which has great significance on bio medical filed in the experiment three classes of simple connected graphs with 5 6 and 7 vertices respectively are investigated and analyzed using the proposed method the experimental results show the validity of it however there are still some problems needed to be researched further such as selecting the vector of topological indices for clustering selecting the distance between the vectors of topological indices choosing the aspects to analyze the properties of each kind of graph after clustering etc

interpolary product integration in the presence of singularities l 2 theory we study interpolatory product integration rules based on gauss radau and lobatto points with respect to a generalized jacobi weight function as applied to the integration of functions with singularities we give sufficient conditions for convergence of such rules for both endpoint and interior singularities

an interpolation algorithm for orthogonal rational functions let a 1 2 be a sequence of numbers on the extended real line r r and a positive bounded borel measure with support in a subset of r we introduce rational functions n with poles 1 n that are orthogonal with respect to if all poles are at infinity we recover the polynomial situation it is well known that under certain conditions on the location of the poles the system n is regular such that the orthogonal functions satisfy a three term recurrence relation similar to the one for orthogonal polynomials to compute the recurrence coefficients one can use explicit formulas involving inner products we present a theoretical alternative to these explicit formulas that uses certain interpolation properties of the riesz herglotz nevanlinna transform of the measure error bounds are derived and some examples serve as illustration

phase space representation of quantum state vectors phase space representation of quantum state vectors is obtained within the framework of the relative state formulation for this purpose the hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system relative position state and relative momentum state are defined in the extended hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators acting on these states are obtained phase space functions which represent a state vector of the relevant quantum system are obtained in terms of the relative position states and the relative momentum states the absolute square of the phase space function represents the probability distribution of the phase space variables time evolution of a quantum system is investigated in terms of the phase space functions the relations to the phase space representations formulated by the other methods are obtained

quadrature formulas associated with rogers szegő polynomials this paper is concerned with the estimation of integrals of email protected functions with respect to the hermite weight function by passing to the unit circle of the complex plane and considering szego and interpolatory type quadrature formulas with respect now to the rogers szego weight function several numerical experiments are given along with estimations of the error

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

an extended relation between orthogonal rational functions on the unit circle and the interval 1 1 abstract the relation between orthogonal polynomials on the unit circle and the interval has been given by szeg in this paper we generalize this relation to orthogonal rational functions with prescribed complex poles as an application we obtain explicit expressions for the orthogonal rational functions with respect to the weight functions 1 cos on the unit circle respectively 1 x 1 x 1 2 on the interval and use one of them to illustrate the relation

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

rational interpolation ii quadrature and convergence abstract consider an n th rational interpolatory quadrature rule j n f j 1 n j f x j to approximate integrals of the form j f 1 1 f x d x where is a possibly complex bounded measure with infinite support on the interval 1 1 first we discuss the connection of j n f with certain rational interpolatory quadratures on the complex unit circle to approximate integrals of the form f e i d c where c is a possibly complex bounded measure with infinite support on next we provide conditions to ensure the convergence of j n f to j f for n tending to infinity finally an upper bound for the error on the n th approximation and an estimate for the rate of convergence is provided

radau and lobatto type quadratures associated with strong stieltjes distributions in this paper n point quadrature rules associated with a strong distribution on a b 0 a b and exactly integrating certain subspaces of laurent polynomials with the highest possible dimension are first analyzed here the basic ingredients are the theory of orthogonal laurent polynomials for the distribution and an arbitrary ordering accordingly established when considering the case of quadrature rules with one node radau or two nodes lobatto previously fixed in a b the properties of the so called pseudo and quasi orthogonal laurent polynomials will play a fundamental role

computation of gauss kronrod of quadrature rules recently laurie presented a new algorithm for the computation of 2n 1 point gauss kronrod quadrature rules with real nodes and positive weights this algorithm first determines a symmetric tridiagonal matrix of order 2n 1 from certain mixed moments and then computes a partial spectral factorization we describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined and thereby avoids computations that can be sensitive to perturbations our algorithm uses the consolidation phase of a divide and conquer algorithm for the symmetric tridiagonal eigenproblem we also discuss how the algorithm can be applied to compute kronrod extensions of gauss radau and gauss lobatto quadrature rules throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible numerical examples illustrate the performance of the algorithm

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

numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures numerical methods for strongly oscillatory and singular functions are given in this paper beside a summary of standard methods and product integration rules we consider a class of complex integration methods several applications of quadrature processes in problems in telecom munications and physics are also presented 1998 elsevier science ltd all rights reserved integration oscillatory kernel singular kernel orthogonal polynomials product rules gaussian quaclratures error function bessel functions legendre functions

a two parameter family of orthogonal polynomials with respect to a jacobi type weight on the unit circle abstract a two parameter family of polynomials is introduced by a recursion formula the polynomials are orthogonal on the unit circle with respect to the weight 1 z 1 z 2 1 2 z ei explicit representation norm estimates shift identities and explicit connection to jacobi polynomials on the real interval 1 1 is presented

error bounds for interpolatory quadrature rules on the unit circle for the construction of an interpolatory integration rule on the unit circle t with n nodes by means of the laurent polynomials as basis functions for the approximation we have at our disposal two nonnegative integers p n and q n p n q n n 1 which determine the subspace of basis functions the quadrature rule will integrate correctly any function from this subspace in this paper upper bounds for the remainder term of interpolatory integration rules on t are obtained these bounds apply to analytic functions up to a finite number of isolated poles outside t in addition if the integrand function has no poles in the closed unit disc or is a rational function with poles outside t we propose a simple rule to determine the value of p n and hence q n in order to minimize the quadrature error term several numerical examples are given to illustrate the theoretical results

roots of polynomials expressed in terms of orthogonal polynomials a technique is presented for determining the roots of a polynomial p x that is expressed in terms of an expansion in orthogonal polynomials the roots are expressed as the eigenvalues of a nonstandard companion matrix b n whose coefficients depend on the recurrence formula for the orthogonal polynomials and on the coefficients of the orthogonal expansion some questions on the numerical stability of the eigenvalue problem to which they give rise are discussed the problem of finding the roots of a transcendental function f x can be reduced to the problem considered by approximating f x by a chebyshev polynomial we illustrate the effectiveness of this convert to chebyshev strategy by solving several transcendental equations using this plus our new algorithm we analyze the numerical stability through both linear algebra theory and numerical experiments and find that this method is very well conditioned

on the extremal total reciprocal edge eccentricity of trees abstract the total reciprocal edge eccentricity is a novel graph invariant with vast potential in structure activity property relationships this graph invariant displays high discriminating power with respect to both biological activity and physical properties if g v g e g is a simple connected graph then the total reciprocal edge eccentricity ree of g is defined as e e g u v e g 1 e g u 1 e g v where e g v is the eccentricity of the vertex v in this paper we first introduced four edge grafting transformations to study the mathematical properties of the reciprocal edge eccentricity of g using these elegant mathematical properties we characterize the extremal graphs among n vertex trees with given graphic parameters such as pendants matching number domination number diameter vertex bipartition et al some sharp bounds on the reciprocal edge eccentricity of trees are determined

error bounds for rational quadrature formulae of analytic functions we study the error of rational quadrature rules when functions which are analytic on a neighborhood of the set of integration are considered a computable upper bound of the error is presented which is valid for a broad range of rational quadrature formulae and a comparison is made with the exact error for a number of numerical examples

note on two generalizations of the randic index for a given graph g the well known randic index of g introduced by milan randic in 1975 is defined as r g u v e g d u d v 1 2 where the sum is taken over all edges uv and du denotes the degrees of u bollobas and erdos generalized this index by replacing 1 2 with any real number which is called the general randic index dvo ak et al introduced a modified version of randic index r g u v e g max d u d v 1 based on this recently knor et al introduced two generalizations r g u v e g min d u d v and r g u v e g max d u d v for any real number clearly the former is a lower bound for the general randic index and the latter is its upper bound knor et al studied extremal values of r g and r g and concluded some open problems in this paper we consider the open problems and give some comments and results some results for chemical trees are obtained