an automatic quadrature for cauchy principal value integrals an automatic quadrature is presented for computing cauchy principal value integrals q f c faf t t c dt a c b for smooth functions f t after subtracting out the singularity we approximate the function f t by a sum of chebyshev polynomials whose coefficients are computed using the ftt the evaluations of q f c for a set of values of c in a b are efficiently accomplished with the same number of function evaluations numerical examples are also given

some results on generalized residual entropy shannon s entropy plays an important role in the context of information theory since this entropy is not applicable to a system which has survived for some units of time the concept of residual entropy has been developed in the literature here we generalize the residual entropy by choosing a convex function f with f 1 0 in this paper some orderings and aging properties have been defined in terms of the generalized residual entropy function and their properties have been studied quite a few results available in the literature have been generalized and some distributions viz uniform exponential pareto power series finite range have been characterized through the generalized residual entropy

information properties of order statistics and spacings we explore properties of the entropy kullback leibler information and mutual information for order statistics the probability integral transformation plays a pivotal role in developing our results we provide bounds for the entropy of order statistics and some results that relate entropy ordering of order statistics to other well known orderings of random variables we show that the discrimination information between order statistics and data distribution the discrimination information among the order statistics and the mutual information between order statistics are all distribution free and are computable using the distributions of the order statistics of the samples from the uniform distribution we also discuss information properties of spacings for uniform and exponential samples and provide a large sample distribution free result on the entropy of spacings the results show interesting symmetries of information orderings among order statistics

is gauss quadrature optimal for analytic functions we consider the problem of optimal quadratures for integrandsf 1 1 which have an analytic extension bar f to an open diskd r of radiusr about the origin such that left bar f right 1 on bar d r ifr 1 we show that the penalty for sampling the integrand at zeros of the legendre polynomial of degreen rather than at optimal points tends to infinity withn in particular there is an infinite penalty for using gauss quadrature on the other hand ifr 1 gauss quadrature is almost optimal these results hold for both the worst case and asymptotic settings

orthogonal polynomials and gaussian quadrature rules related to oscillatory weight functions in this paper we consider polynomials orthogonal with respect to an oscillatory weight function w x xeim x on 1 1 where m is an integer the existence of such polynomials as well as several of their properties three term recurrence relation differential equation etc are proved we also consider related quadrature rules and give applications of such quadrature rules to some classes of integrals involving highly oscillatory integrands

uniform approximations to finite hilbert transform and its derivative interpolatory integration rules of numerical stability are presented for approximating cauchy principal value p v integrals 11 f t t c dt and hadamard finite part f p integrals 11f t t c 2 dt 1 c 1 respectively for a given smooth function f t present quadrature rules consist of interpolating f t at abscissae in the interval of integration 1 1 except for the pole c where neither the function value f c nor its derivative f c is required followed by subtracting out the singularities we demonstrate that the use of both endpoints 1 as abscissae in interpolating f t is essential for uniformly approximating the integrals namely for bounding the approximation errors independently of the values of c in fact for the f p integrals the use of double abscissae at both endpoints 1 as well as simple abscissae in 1 1 enables the uniform approximations while the use of simple abscissae at both endpoints 1 and those in 1 1 is sufficient for the p v integrals these facts suggest that finite hilbert transforms p v integrals and their derivatives f p integrals with varied values of c could be approximated efficiently with the same number of abscissae respectively some numerical examples are given

the entropy of ordered sequences and order statistics the entropy of a sequence of random variables under order restrictions is examined a theorem that shows the amount of entropy reduction when the sequence is ordered is presented upper and lower bounds to the entropy reduction and conditions under which they are achieved are derived some interesting properties of the entropy of the individual order statistics are also presented it is shown that the difference between the average entropy of the individual order statistics and the entropy of a member of the original independent identically distributed iid population is a constant regardless of the original distribution finally the entropies of the individual order statistics are found to be symmetric about the median when the probability density function pdf of the original iid sequence is symmetric about its mean

results on residual renyi entropy of order statistics and record values this paper explores properties of the residual renyi entropy of some ordered random variables the residual renyi entropy of the kth order statistic from a continuous distribution function is represented in terms of the residual renyi entropy of the kth order statistic from uniform distribution the monotone behavior of the residual renyi entropy of order statistic under various conditions is discussed analogues results for the residual renyi entropy of record values are also given

cumulative tsallis entropy for maximum ranked set sampling with unequal samples abstract in this paper we consider the information content of maximum ranked set sampling procedure with unequal samples mrssu in terms of tsallis entropy which is a non additive generalization of shannon entropy we obtain several results of tsallis entropy including bounds monotonic properties stochastic orders and sharp bounds under some assumptions we also compare the uncertainty and information content of mrssu with its counterpart in the simple random sampling srs data finally we develop some characterization results in terms of cumulative tsallis entropy and residual tsallis entropy of mrssu and srs data

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

on the remainder term of gauss radau quadratures for analytic functions for analytic functions the remainder term of gauss radau quadrature formulae can be represented as a contour integral with a complex kernel we study the kernel on elliptic contours with foci at the points 1 and a sum of semi axes r 1 for the chebyshev weight function of the second kind starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined the corresponding gautschi s conjecture from on the remainder term for analytic functions of gauss lobatto and gauss radau quadratures rocky mountain j math 21 1991 209 226 is proved

the effect of imperfect judgment rankings on properties of procedures based on the ranked set samples analog of the mann whitney wilcoxon statistic abstract in this article we address the issue of imperfect judgment rankings in ranked set sampling and in particular their effect on the properties of test procedures based on the ranked set samples analog of the mann whitney wilcoxon statistic u rss we consider the impact of these imperfect rankings on the null distribution of the statistic and the resulting effect on the nominal level of associated hypothesis tests we propose a model for the probabilities of imperfect judgment rankings based on the concept of expected spacings and use this model to study the properties of tests based on u rss this investigation includes both small sample monte carlo power simulations and a detailed analysis of the asymptotic relative efficiency properties of the u rss procedure we also examine as an indication of the merits of using ranked set sampling the relative cost of measuring the value of a sample item as compared to obtaining a judgment ordering of a set of sample items

stieltjes polynomials and related quadrature rules in this paper we present a survey of theoretical results some of them new and numerical evidence of others concerning stieltjes polynomials kronrod schemes and their generalizations applied to integrals with a classical nonnegative weight function the use of these schemes in automatic integration routines is briefly outlined

on polynomials orthogonal on the semicircle and applications abstract polynomials n orthogonal on the semicircle z c z e i 0 with respect to the inner product f g f z g z w z iz 1 dz where z w z is a complex weight function have been introduced in 1986 1987 by gautschi landau and the author in this paper we introduce the functions of the second kind as well as the corresponding associated polynomials and prove some recurrence relations for gauss gegenbauer quadrature formulae over the semicircle applied to analytic functions we develop error bounds from contour integral representations of the remainder term and give some numerical results

on polynomials orthogonal on a circular arc abstract polynomials rk orthogonal on a circular arc with respect to the complex inner product f g f1 g1 w1 d where 0 1 2 and for f z the function f1 is defined by f1 f ir ei r2 1 1 2 r tan have been introduced by de bruin 1990 in this paper the functions of the second kind as well as the corresponding associated polynomials are introduced some recurrence relations and identities of christoffel darboux type are proved also the corresponding stieltjes polynomials which are orthogonal to all lower degree polynomials with respect to a complex measure on r z c z ir ei r2 1 1 2 tan r are investigated a class of polynomials orthogonal on a symmetrical circular arc in the down half plane is also introduced finally in the jacobi case w z 1 z 1 z 1 a linear second order differential equation for rn z is obtained

the remainder term for analytic functions of gauss radau and gauss lobatto quadrature with multiple end points a study is undertaken of the kernels in the contour integral representations of the remainder terms for gauss radau and gauss lobatto quadrature rules over the interval 1 1 it is assumed that the respective end points in these rules have multiplicity two and that integration is with respect to one of the four chebyshev weight functions of particular interest is the location on the contour where the modulus of the kernel attains its maximum value only elliptic contours are considered having foci at the points 1

error bounds and estimates for gauss turan quadrature formulae of analytic functions we study the error of gauss turan quadrature formulae 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 arbitrary weight functions a comparison is made with the exact error and a number of numerical examples for arbitrary weight functions 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 the other effective error bounds for some special weight functions appearing in the literature asymptotic error estimates when the number of nodes in the quadrature increases are presented a couple of numerical examples which show their efficiency are included

fehlerabschatzungen fur nichtsymmetrische gauβ quadraturformeln we consider gaussian quadrature formulaeq n n approximating the integral i f int limits 1 1 w x f x dx wherew is a weight function in certain spaces of analytic functions the error functionalr n i q n is continuous previously one of the authors deduced estimates for r n for symmetric gaussian quadrature formulae in this paper we extend these results to nonsymmetric gaussian formulae using a recent result of gautschi concerning the sign ofr n q k q k x x k for a wide class of weight functions including the jacobi weights

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

formulas for renyi information and related measures for univariate distributions in a recent paper song j stat plan infer 93 2001 51 considered renyi information of order and established its connection to the loglikelihood from this relation an intrinsic distribution measure was proposed and analytic expressions of this measure and renyi information were derived for some standard continuous distributions in this paper we derive analytical formulas for renyi and shannon entropies as well as for song s measure for 26 flexible families of univariate continuous distributions we believe that the results presented here will serve as an important reference for scientists and engineers in many areas

liouville field theory ist and poisson bracket structure the general solution for the liouville equation with any given number n of singularities is considered with the help of the inverse scattering transform method ist the set of regular continuous and discrete canonical variables is derived the dynamical generators of poincare and dilatation groups and n soliton solutions are constructed in terms of these variables

on some entropy functionals derived from renyi information divergence we consider the maximum entropy problems associated with renyi q entropy subject to two kinds of constraints on expected values the constraints considered are a constraint on the standard expectation and a constraint on the generalized expectation as encountered in nonextensive statistics the optimum maximum entropy probability distributions which can exhibit a power law behaviour are derived and characterized the renyi entropy of the optimum distributions can be viewed as a function of the constraint this defines two families of entropy functionals in the space of possible expected values general properties of these functionals including nonnegativity minimum convexity are documented their relationships as well as numerical aspects are also discussed finally we work out some specific cases for the reference measure q x and recover in a limit case some well known entropies

maximum of the modulus of kernels in gauss turan quadratures we study the kernels k n s z in the remainder terms r n s f of the gauss turan quadrature formulae tor analytic functions on elliptical contours with foci at 1 when the weight is a generalized chebyshev weight function for the generalized chebyshev weight of the first third kind it is shown that the modulus of the kernel k n s z attains its maximum on the real axis positive real semi axis for each n no no n 0 s it was stated as a conjecture in math comp 72 2003 1855 1872 for the generalized chebyshev weight of the second kind in the case when the number of the nodes n in the corresponding gauss turan quadrature formula is even it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each n no no n 0 p s numerical examples are included

error bounds of certain gaussian quadrature formulae we study the kernel of the remainder term of gauss quadrature rules for analytic functions with respect to one class of bernstein szego weight functions the location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated this leads to effective error bounds of the corresponding gauss quadratures

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

on the remainder term of gauss radau quadrature with chebyshev weight of the third kind for analytic functions abstract for analytic functions the remainder term of quadrature formulae can be represented as a contour integral with a complex kernel we study the kernel on elliptic contours with foci at the points 1 and a sum of semi axes 1 for gauss radau quadrature formula with chebyshev weight function of the third kind starting from the explicit expression of the corresponding kernel derived by gautschi we determine the locations on the ellipses where maximum modulus of the kernel is attained the obtained values confirm the corresponding conjectured values given by gautschi in his paper w gautschi on the remainder term for analytic functions of gauss lobatto and gauss radau quadratures rocky mounatin j math 21 1991 209 206 in this way the last unproved conjecture from the mentioned paper is now verified

ranked set sampling for replicated sampling designs summary in practical ecological sampling studies a certain design such as plot sampling or line intercept sampling is usually replicated more than once for each replication the horvitz thompson estimation of the objective parameter is considered finally an overall estimator is achieved by averaging the single horvitz thompson estimators because the design replications are drawn independently and under the same conditions the overall estimator is simply the sample mean of the horvitz thompson estimators under simple random sampling this procedure may be wisely improved by using ranked set sampling hence we propose the replicated protocol under ranked set sampling which gives rise to a more accurate estimation than the replicated protocol under simple random sampling

the remainder term for analytic functions of symmetric gaussian quadratures for analytic functions the remainder term of gaussian quadrature rules can be expressed as a contour integral with kernel k n in this paper the kernel is studied on elliptic contours for a great variety of symmetric weight functions including especially gegenbauer weight functions first a new series representation of the kernel is developed and analyzed then the location of the maximum modulus of the kernel on suitable ellipses is determined depending on the weight function the maximum modulus is attained at the intersection point of the ellipse with either the real or imaginary axis finally a detailed discussion for some special weight functions is given

error bounds of the micchelli sharma quadrature formula for analytic functions micchelli and sharma constructed in their paper on a problem of turan multiple node gaussian quadrature rend mat 3 1983 529 552 a quadrature formula for the fourier chebyshev coefficients which has the highest possible precision for analytic functions the remainder term of this quadrature formula can be represented as a contour integral with a complex kernel we study the kernel on elliptic contours with foci at the points 1 and a sum of semi axes r 1 for the quoted quadrature formula starting from the explicit expression of the kernel we determine the location on the ellipses where the maximum modulus of the kernel is attained and derive effective error bounds for this quadrature formula numerical examples are included

quadrature rule for indefinite integral of algebraic logarithmic singular integrands an automatic quadrature method is presented for approximating the indefinite integral of functions having algebraic logarithmic singularities q x y c f email protected x yf t t c alog t c dt 1 1 within a finite range 1 1 for some smooth function f t that is approximated by a finite sum of chebyshev polynomials we expand the given indefinite integral in terms of chebyshev polynomials by using auxiliary algebraic logarithmic functions present scheme approximates the indefinite integral q x y c f uniformly namely bounds the approximation error independently of the value c as well x and y this fact enables us to evaluate the integral transform q x y c f with varied values of x y and c efficiently some numerical examples illustrate the performance of the present quadrature scheme

the remainder term for analytic functions of gauss radau and gauss lobatto quadrature rules with multiple end points a study is undertaken of the kernels in the contour integral representations of the remainder terms for gauss radau and gauss lobatto quadrature rules over the interval 1 1 it is assumed that

polynomials orthogonal on a circular arc in this paper complex polynomials p k p k z z k orthogonal with respect to the complex inner product f g email protected p f ff 1 q g 1 qw 1 q email protected will be studied here email protected 0 frcase 1 2 p and for f z the function f 1 q is defined by f 1 q f ir e i qr 2 1 r tan q under suitable integrability conditions on w and assuming the existence of an analytic continuation to the region m email protected c z ir 0 these orthogonal polynomials always exist in contrast to the case f 0 the semicircle treated by gautschi 1989 gautschi et al 1987 and gautschi and milovanovic 1986 where an extra condition on re p 0w 1 q email protected is needed and they will be expressed in terms of the real polynomials orthogonal on 1 1 with respect tow x this shows the connection between orthogonality on an arc and orthogonality on the spanning chord on the arc there is no conjugate taken for the second factor actor

error bounds for gauss turan quadrature formulae of analytic functions we study the kernels of the remainder term rn s f of gauss thran quadrature formulas 11f t w t dt ai vf i v rn s f n n s n0 for classes of analytic functions on elliptical contours with foci at 1 when the weight w is one of the special jacobi weights w t 1 t 1 t 1 2 1 2 s 1 2 1 2 s 1 2 s 1 2 we investigate the location on the contour where the modulus of the kernel attains its maximum value some numerical examples are included

classical scattering in liouville field theory a detailed description of classical scattering in liouville field theory lft is presented contrary to widespread belief lft scattering is shown to be non trivial although it is finite dimensional in some sense in particular for certain phase spaces the lft s matrix is represented as a transformation of the poisson group sl 2 r the completeness and conformal invariance of the scattering are also indicated singular fields are treated on an equal footing with regular ones except that only the latter are given a consistent hamiltonian interpretation a number of unexpected peculiarities of lft scattering are revealed first for some exceptional field configurations the asymptotic fields are not solutions of d alembert s equation rather they are a sum of the d alembert and liouville components second the scattering occurs in two or three spaces and last depending on the choice of the algebra of observables the conventional splitting of the d alembert field into leftward and rightward components is either in general impossible or essentially non unique

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

exact quantum s matrix in the liouville field theory classical scattering in the liouville field theory lft is essentially finite dimensional and in some cases the classical s matrix can be represented as a transformation of the poisson group sl 2 r motivated by this and using conjectural quantum analogues of some ingredients of the classical model we find an exact quantum s matrix without constructing the quantum lft in full the quantum s matrix is explicitly represented as a transformation of the quantum group slq 2 r and for a particular implementation the s matrix is shown to be unitary unitarily generated

asymptotic error estimates for gaussian quadrature formulas this paper gives derivative free asymptotic error estimates for the gaussian quadrature formula with the nonnegative weight function w x belonging to a certain class numerical examples are presented

hilbert and hadamard transforms by generalized chebyshev expansion abstract an automatic quadrature is presented for approximating hadamard finite part fp integrals of a smooth function with a double pole singularity within the range of integration the quadrature rule is derived from the differentiation of an approximation to a cauchy principal value integral or the hilbert transform the approximation to the fp integral is represented as a function of the value of pole by using chebyshev polynomials of the second kind since the error can be estimated independently of the value of pole a set of integrals for a set of values of pole can be efficiently approximated to a required tolerance with the same number of function evaluations numerical examples are also included to illustrate the performance of the methods

the error norm of certain gaussian quadrature formulae we consider gauss quadrature formulae q n e n approximating the integral j f f1l w x f x dx w w pi i 1 2 with w x 1x a 1 x a i 1 2 and pl x 1 a2 2ax p2 x 2b 1 x2 b2 b 0 in certain spaces of analytic functions the error functional r i q is continuous in 1 and 2 estimates for jir j are given for a wide class of weight functions here for a restricted class of weight functions we calculate the norm of r explicitly

application of a modified fft to product type integration abstract an automatic integration scheme is proposed for evaluating the so called product type indefinite integral q k y x k t t dt 1 x y where t is assumed to be a smooth function and k t are some singular or badly behaved functions typical examples for k t are 1n t c t c 1 cauchy principal value 1 t c and ei t 1 the function t is approximated by a truncated chebyshev series pn t of degree n whose coefficients are efficiently computed using the fft the approximation qn k to the integral q k is given by q k pn the sequence pn t is recursively generated until the required tolerance for the integral is satisfied to enhance the efficiency of the automatic quadrature the degree n is increased more slowly than doubling which is usually the case the evaluations of qn k q k pn for a set of x y c can be efficiently made by using recurrence relations for the singular kernels k t above numerical examples for the algebraic singular kernel k t t c 1 are included

renyi information measure for a used item recently in the literature a measure of tail heaviness has been proposed based on renyi entropy this measure is very useful in the sense that it can be used to measure tail heaviness even for the distributions for which the kurtosis measure b 2 m 4 m 2 2 does not exist nadarajah and zografos nadarajah zografos information sciences 153 2003 119 138 have derived the expression for this measure for different univariate continuous distributions but this measure can only be used for the lifetime of a new item in case of used item this measure needs some modification in this paper we have modified the measure accordingly so that it can be used in the case of used item and also for the new item we have also derived expressions for this measure for sixteen different univariate distributions and ten other standard distributions derived from the general distributions used in reliability and survival analysis the most of the results obtained in the literature in this direction can be obtained as particular cases of our general results

ordering univariate distributions by entropy and variance this paper examines the role of variance and entropy in ordering distributions and random prospects there is no universal relation between entropy and variance orderings of distributions but we place their relationship in the context of a stronger ordering relation known as dispersion ordering further some conditions are identified under which variance and entropy order similarly when continuous variables are transformed we also analyze parametric changes which do not disturb the agreement between these rankings the results are conveniently tabulated in terms of distribution parameters

on the error term of symmetric gauss lobatto quadrature formulae for analytic functions gauss lobatto quadrature formulae associated with symmetric weight functions are considered the kernel of the remainder term for classes of analytic functions is investigated on elliptical contours sufficient conditions are found ensuring that the kernel attains its maximal absolute value at the intersection point of the contour with either the real or the imaginary axis the results obtained here are an analogue of some recent results of t schira concerning gaussian quadratures

asymptotics on laguerre or hermite polynomial expansions and their applications in gauss quadrature abstract in this paper we present asymptotic analysis on the coefficients of functions expanded in forms of laguerre or hermite polynomial series which shows the decay of the coefficients and derives new error bounds on the truncated series moreover by applying the asymptotics new estimates on the errors for gauss laguerre radau laguerre and gauss hermite quadrature are deduced these results show that gauss laguerre type and gauss hermite type quadratures are nearly of same convergence rates

bounds of the error of gauss turan type quadratures we consider the remainder term of the gauss turan quadrature formulae rn s f 11w t f t dt v 1n i 02sai vf i v for analytic functions in some region of the complex plane containing the interval 1 1 in its interior the remainder term is presented in the form of a contour integral over confocal ellipses or circles a strong error analysis is given for the case with a generalized class of weight functions introduced recently by gori and micchelli also we discuss a general case with an even weight function defined on 1 1 numerical results are included

the entropy of consecutive order statistics calculation of the entropy of a set of consecutive order statistics is relatively more complicated than that of the entropy of the individual order statistic which has been studied by wong and chan 1990 we provide some fundamental relations occurring in the entropy of consecutive order statistics which are very useful for computations we first consider the decomposition of the entropy of order statistics and derive some recurrence relations in the first r order statistics we also establish a dual principle for the entropy of order statistics which yields a dual relation from a given relation in the entropy of order statistics

error estimates of anti gaussian quadrature formulae anti gauss quadrature formulae associated with four classical chebyshev weight functions are considered complex variable methods are used to obtain expansions of the error in anti gaussian quadrature formulae over the interval 1 1 the kernel of the remainder term in anti gaussian quadrature formulae is analyzed the location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated this leads to effective l error bounds of anti gauss quadratures moreover the effective l 1 error estimates are also derived the results obtained here are an analogue of some results of gautschi and varga 1983 11 gautschi et al 1990 9 and hunter 1995 10 concerning gaussian quadratures

the error bounds of gauss radau quadrature formulae with bernstein szegő weight functions we consider the gauss radau quadrature formulae begin aligned int 1 1f t w t dt sum nu 1 n lambda nu f tau nu lambda n 1 f c r n f end aligned 11f t w t dt 1n f n 1f c rn f with c 1 c 1 or c 1 c 1 for the bernstein szeg weight functions consisting of anyone of the four chebyshev weights divided by the polynomial rho t 1 frac 4 gamma 1 gamma 2 t 2 t in 1 1 1 1 1 for the given quadrature formula starting from the explicit expression of the kernel we determine the locations on the ellipses where maximum modulus of the kernel is attained so we derive effective error bounds for this quadrature formula an alternative approach which has initiated this research has been proposed recently by notaris math comp 10 1090 mcom 2944 2015

estimating quadrature errors for analytic functions using kernel representations and biorthogonal systems summary for analytic functions the remainder term of quadrature rules can be represented as a contour integral with a complex kernel function from this representation different remainder term estimates involving the kernel are obtained it is studied in detail how polynomial biorthogonal systems can be applied to derive sharp bounds for the kernel function it is shown that these bounds are practical to use and can easily be computed finally various numerical examples are presented

poisson structure of the liouville field theory a new analysis of the poisson structure of the liouville field theory lft in an infinite volume is presented the second poisson structure of the korteweg de vries equation is thoroughly investigated as an essential part of the approach and a variety of correct poisson brackets is found this along with other causes leads to a variety of correct lft poisson structures special attention is paid to the most important lft property of conformal invariance in particular a maximal conformal group suitable for the adopted lft phase spaces is found and various hamiltonian representations of the conformal algebra are described the properties of local commutativity and canonicity are proved

the error bounds of gauss quadrature formulae for the modified weight functions of chebyshev type abstract in this paper we consider the gauss quadrature formulae corresponding to some modifications of each of the four chebyshev weights considered by gautschi and li in 4 as it is well known in the case of analytic integrands the error of these quadrature formulas can be represented as a contour integral with a complex kernel we study the kernel of the mentioned quadrature formulas on suitable elliptic contours in such a way that the behavior of its modulus is analyzed in a rather simple manner allowing us to derive some effective error bounds in addition some numerical examples checking the accuracy of such error bounds are included

numerical differentiation of analytic functions using quadratures on the semicircle abstract using quadrature formulas on the semicircle of gauss christoffel type and an integral representation of derivatives we consider differentiation formulas for higher derivatives of an analytic function an error analysis and some numerical experiments are included

orthogonal polynomials constructive theory and applications we consider polynomials orthogonal with respect to some measure on the real line a basic problem in the constructive theory of such polynomials is the determination of their three term recurrence relation given the measure in question depending on what is known about this measure there are different ways to proceed if as is typical in applications in the physical sciences one knows the measure only through moment information the appropriate procedure is an algorithm that goes back to chebyshev the algorithm in effect implements the nonlinear map from the given moments or modified moments to the desired recursion coefficients the numerical effectiveness of this procedure is determined in an essential way by the numerical condition of this map it is known that the map is ill conditioned in the case of ordinary moments for modified moments it may or may not be well conditioned the matter depending on certain properties of the given measure and the additional measure defining modified moments a theorem to this effect will be given and illustrated on some typical examples if more is known about the given measure for example if it is absolutely continuous and can be evaluated pointwise then a procedure can be employed which is based on an observation of stieltjes stieltjes remarked that the desired recursion coefficients can be successively built up by alternating between known inner product formulae for these coefficients and the initial sections of the recurrence relation already obtained an effective implementation of this idea requires a suitable discretization of the inner product this requirement while possibly a weakness of the method also accounts for its beauty since it leaves room for imagination and ingenuity used skillfully the discretized stieltjes procedure is among the most widely applicable and effective methods for generating orthogonal polynomials some examples will be given to illustrate its use finally we show how our newly acquired capability of generating nonstandard orthogonal polynomials can be used to solve some special problems in approximation theory and in the summation of slowly convergent series a novel set of polynomials orthogonal on the semicircle will also be mentioned briefly in connection with cauchy principal value integrals

on uncertainty and information properties of ranked set samples ranked set sampling is a sampling design which has a wide range of applications in industrial statistics economics and environmental and ecological studies etc it is well known that ranked set samples provide more fisher information than simple random samples of the same size about the unknown parameters of the underlying distribution in parametric inferences in this paper we consider the uncertainty and information content of ranked set samples in both perfect and imperfect ranking scenarios in terms of shannon entropy renyi and kullback leibler kl information measures it is proved that under these information measures ranked set sampling design performs better than its simple random sampling counterpart of the same size the information content is also a monotone function of the set size in ranked set sampling moreover the effect of ranking error on the information content of the data is investigated

sharp error bounds for jacobi expansions and gegenbauer gauss quadrature of analytic functions this paper provides a rigorous and delicate analysis for exponential decay of jacobi polynomial expansions of analytic functions associated with the bernstein ellipse using an argument that can recover the best estimate for the chebyshev expansion we derive various new and sharp bounds of the expansion coefficients which are featured with explicit dependence of all related parameters and valid for degree n ge 1 we demonstrate the sharpness of the estimates by comparing with existing ones in particular the very recent results in siam j numer anal 50 2012 pp 1240 1263 we also extend this argument to estimate the gegenbauer gauss quadrature remainder of analytic functions which leads to some new tight bounds for quadrature errors

some observations on gauss legendre quadrature error estimates for analytic functions abstract this paper is concerned with estimates for the error when a gauss legendre quadrature rule is used to numerically integrate an analytic function the error in a standard approximation to the kernel function which appears in a contour integral representation for the quadrature error is investigated for this purpose applications to meromorphic functions with simple poles are discussed

numerical integration of functions with poles near the interval of integration abstract an automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole or poles of order 2 i e a double pole on the real axis or a complex conjugate pair of double poles near the interval of integration the present scheme is based on product integration rules of interpolatory type using function values of the abscissae only in the interval of integration the integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the chebyshev polynomials the fast fourier transform fft technique is used to generate efficiently the sequence of the finite chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error numerical examples are included to illustrate the performance of the method