generalization of the group decision making using fuzzy sets theory for evaluating the rate of aggregative risk in software development abstract this study is to propose an algorithm of the group decision makers with crisp or fuzzy weights to tackle the rate of aggregative risk in software development in fuzzy circumstances by fuzzy sets theory during any phase of the life cycle the proposed algorithm is more flexible and useful than the ones we have presented before h m lee fuzzy sets and systems 79 3 1996 323 336 80 3 1996 261 271 since the weights against decision makers are considered

the exterior algebra λkrn and area minimization abstract the structure of the exterior algebra k r n is studied in low dimensions and consequences are drawn for k dimensional area minimizing surfaces in r n for a general form o 2 r 4 section 3 gives explicit formulas for the comass o and the face of the grassmannian exposed by o section 4 classifies the faces of the grassmannian of the 3 planes in r 6 and hence the associated geometries of area minimizing surfaces there are four types section 5 establishes an equality involving the comass norm in low dimension and draws implications on when the cartesian product of area minimizing surfaces is area minimizing new examples of area minimizing integral currents with singularities follow

a fusion approach for managing multi granularity linguistic term sets in decision making the aim of this paper is to present a fusion approach of multi granularity linguistic information for managing information assessed in different linguistic term sets multi granularity linguistic term sets together with its application in a decision making problem with multiple information sources assuming that the linguistic performance values given to the alternatives by the different sources are represented in linguistic term sets with different granularity and or semantic in this context a decision process based on two steps is proposed with a view to obtaining the solution set of alternatives first the fusion of the multi granularity linguistic performance values is carried out in order to obtain collective performance evaluations in this step on the one hand the multi granularity linguistic information is made uniform using a linguistic term set as the uniform representation base the basic linguistic term set on the other hand the collective performance evaluations of the alternatives are obtained by means of an aggregation operator being fuzzy sets on the basic linguistic term set second the choice of the best alternative s from the collective performance evaluations is performed to do that a fuzzy preference relation is computed from the collective performance evaluations using a ranking method of pairs of fuzzy sets in the setting of possibility theory applied to fuzzy sets on the basic linguistic term set then a choice degree may be applied on the preference relation in order to rank the alternatives

fuzzy group decision making for evaluating the rate if aggregative risk in software development in this paper we present a new algorithm to evaluate the rate of aggregative risk in software development using fuzzy set theory under the fuzzy group decision making environment the proposed algorithm has the following advantages 1 it does not need to form the fuzzy assessment matrices for attributes 2 it does not need to perform the complicated defuzzification operations of fuzzy numbers using the centroid method the proposed algorithm is more efficient and faster in evaluating the rate of aggregative risk in software development under the fuzzy group decision making environment

error estimates in the numerical evaluation of some bem singular integrals in some applications of galerkin boundary element methods one has to compute integrals which after proper normalization are of the form b 2 x y dxdy a 1 x y y where a b 1 1 or a b a 1 or a b 1 b and f x y is a smooth function in this paper we derive error estimates for a numerical approach recently proposed to evaluate the above integral when a p or h p formulation of a galerkin method is used this approach suggests approximating the inner integral by a quadrature formula of interpolatory type that exactly integrates the cauchy kernel and the outer integral by a rule which takes into account the log endpoint singularities of its integrand some numerical examples are also given

optimal systems of nodes for lagrange interpolation on bounded intervals a survey in this brief survey special attention is paid to some recent procedures for constructing optimal interpolation processes i e with lebesgue constant having logarithmic behaviour a new result on lagrange interpolation based on the zeros of the associated jacobi polynomials and on suitable additional nodes is given

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

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

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

convergence of product integration rules over 0 for functions with weak singularities at the origin in this paper we consider integrals of the form 0 e x k x y f x dx with f c p 0 c q 0 q p 0 and x i f p 1 x c 0 i 1 q p when q p they appear for instance in certain wiener hopf integral equations and are of interest if one wants to solve these by a nystrom method to discretize the integral above we propose to use a product rule of interpolatory type based on the zeros of laguerre polynomials for this rule we derive weighted uniform convergence estimates and present some numerical examples

on mean convergence of hermite fejer and hermite interpolation for erdos weights abstract we investigate convergence of hermite fejer and hermite interpolation polynomials in l p 0 for erdos weights

minimal volume projections of cubes and totally unimodular matrices abstract among all linear projections onto a given linear subspace l in r n we select those that minimize the volume of the image of the cube x xi 1 the paper is devoted to a description of the shape of such images of the cube the shape is characterized in terms of zonotopes spanned by scalar multiples of rows of totally unimodular matrices

stieltjes polynomials and lagrange interpolation bounds are proved for the stieltjes polynomial e n 1 and lower bounds are proved for the distances of consecutive zeros of the stieltjes polynomials and the legendre polynomials p n this sharpens a known interlacing result of szego as a byproduct bounds are obtained for the geronimus polynomials g n applying these results convergence theorems are proved for the lagrange interpolation process with respect to the zeros of e n 1 and for the extended lagrange interpolation process with respect to the zeros of p n e n 1 in the uniform and weighted l p norms the corresponding lebesgue constants are of optimal order

high order methods for weakly singular integral equations with nonsmooth input functions to solve one dimensional linear weakly singular integral equations on bounded intervals with input functions which may be smooth or not we propose to introduce first a simple smoothing change of variable and then to apply classical numerical methods such as product integration and collocation based on global polynomial approximations the advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well conditioned

convergence analysis of the jacobi collocation method for nonlinear weakly singular volterra integral equations in this work we present an efficient spectral collocation method for numerical solution of a class of nonlinear weakly singular volterra integral equations this type of equations typically has a singular behavior at the left endpoint of the interval of integration for overcoming this non smooth behavior we apply the jacobi collocation method the convergence analysis of the proposed method is investigated in the l and the weighted l2 norms and the results of several numerical experiments are presented which support the theoretical results the computed results are compared wherever possible with those already available in the literature

product integration for volterra integral equations of the second kind with weakly singular kernels we introduce a new numerical approach for solving volterra integral equations of the second kind when the kernel contains a mild singularity we give a convergence result we also present numerical examples which show the performance and efficiency of our method

group decision making using fuzzy sets theory for evaluating the rate of aggregative risk in software development abstract the purpose of this study is not only to build a group decision making structure model of risk in software development but also to propose two algorithms to tackle the rate of aggregative risk in a fuzzy environment by fuzzy sets theory during any phase of the life cycle while evaluating the rate of aggregative risk one may adjust or improve the weights or grades of the factors until she he can accept it moreover our result will be more objective and unbiased since it is generated by a group of evaluators

projection methods and condition numbers in uniform norm for fredholm and cauchy singular integral equations in this paper the authors propose a numerical method for the approximate solution of some classes of fredholm and cauchy integral equations including the discrete collocation and collocation methods

noninterpolatory integration rules for cauchy principal value integrals etude de la convergence des regles d integration interpolat pour l approximation des integrales de cauchy

some weighted polynomial inequalities abstract the author considers a weight function of sub exponential type with zeros and singularities in 1 1 some inequalities of remez bernstein markov and nikolsky type are proved the results hold for polynomials and generalized polynomials moreover an upper and a lower bound for the christoffel functions are derived

random matrices nonbacktracking walks and orthogonal polynomials several well known results from the random matrix theory such as wigner s law and the marchenko pastur law can be interpreted and proved in terms of nonbacktracking walks on a certain graph orthogonal polynomials with respect to the limiting spectral measure play a role in this approach

numerical methods for fredholm integral equations on the square abstract in this paper we shall investigate the numerical solution of two dimensional fredholm integral equations by nystrom and collocation methods based on the zeros of jacobi orthogonal polynomials the convergence stability and well conditioning of the method are proved in suitable weighted spaces of functions some numerical examples illustrate the efficiency of the methods

spectral collocation method for the time fractional diffusion wave equation and convergence analysis in this paper we consider the numerical solution of the time fractional diffusion wave equation essentially the time fractional diffusion wave equation differs from the standard diffusion wave equation in the time derivative term we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

algorithms for solving hermite interpolation problems using the fast fourier transform we present a method for computing the hermite interpolation polynomial based on equally spaced nodes on the unit circle with an arbitrary number of derivatives in the case of algebraic and laurent polynomials it is an adaptation of the method of the fast fourier transform fft for this type of problems with the following characteristics easy computation small number of operations and easy implementation in the second part of the paper we adapt the algorithm for computing the hermite interpolation polynomial based on the nodes of the tchebycheff polynomials and we also study hermite trigonometric interpolation problems

hermite and hermite fejer interpolation for stieltjes polynomials let x 1 x 2 1 2 and p n be the ultraspherical polynomials with respect to x then we denote by e n 1 the stieltjes polynomials with respect to x satisfying in this paper we show uniform convergence of the hermite fejer interpolation polynomials h n 1 and h 2n 1 based on the zeros of the stieltjes polynomials e n 1 and the product e n 1 p n for 0 1 and 0 a 1 2 respectively to prove these results we prove that the lebesgue constants of hermite fejer interpolation operators for the stieltjes polynomials e n 1 and the product e n 1 p n are optimal that is the lebesgue constants h n 1 0 a 1 and h 2n 1 0 a 1 2 have optimal order o 1 in the case of the hermite fejer interpolation polynomials h 2n 1 for 1 2 1 we prove weighted uniform convergence moreover we give some convergence theorems of hermite fejer and hermite interpolation polynomials for 0 1 in weighted lp norms

uniform convergence of derivatives of lagrange interpolation abstract uniform convergence of derivatives of lagrange interpolation at the union of zeros of jacobi polynomials and some additional points is investigated

mean convergence of hermite fejer interpolation abstract weighted l p convergence of hermite fejer interpolation based on the zeros of generalized jacobi polynomials is investigated the main result of the paper gives necessary and sufficient conditions for such convergence for all continuous functions

a method for group decision making with multi granularity linguistic assessment information this paper proposes a method to solve the group decision making gdm problems with multi granularity linguistic assessment information in the method the multi granularity linguistic information provided by experts is firstly expressed in the form of fuzzy numbers in order to make the collective opinion close to each expert s opinion a linear goal programming model is constructed to integrate the fuzzy assessment information and to directly compute the collective ranking values of alternatives without the need of information transformation then a fuzzy preference relation on the pairwise comparisons of the collective ranking values of alternatives is constructed using the dominance possibility degree of the comparison between the fuzzy numbers by applying a non dominance choice degree to this fuzzy preference relation the ranking of alternatives is determined and the most desirable alternative s is selected an example is used to illustrate the applicability of the proposed method and its advantages

fuzzy group decision making for facility location selection the selection of a facility location among alternative locations is a multicriteria decision making problem including both quantitative and qualitative criteria the conventional approaches to facility location problem tend to be less effective in dealing with the imprecise or vagueness nature of the linguistic assessment under many situations the values of the qualitative criteria are often imprecisely defined for the decision makers the aim of the paper is to solve facility location problems using different solution approaches of fuzzy multi attribute group decision making the paper includes four different fuzzy multi attribute group decision making approaches the first one is a fuzzy model of group decision proposed by blin the second is the fuzzy synthetic evaluation the third is yager s weighted goals method and the last one is fuzzy analytic hierarchy process although four approaches have the same objective of selecting the best facility location alternative they come from different theoretic backgrounds and relate differently to the discipline of multi attribute group decision making these approaches are extended to select the best facility location alternative by taking into account quantitative and qualitative criteria a short comparative analysis among the approaches and a numeric example to each approach are given

mean convergence of derivatives of lagrange interpolation abstract weighted lp convergence of derivatives of lagrange interpolation at the union of zeros of generalized jacobi polynomials and some additional points is investigated

uniform convergence of derivatives of extended lagrange interpolation the authors construct some extended interpolation formulae to approximate the derivatives of a function in uniform norm they prove theorems on uniform convergence and give estimates of pointwise type and of simultaneous approximation

convergence of product integration rules for functions with interior and end point singularities over bounded and unbounded intervals the convergence of product integration rules based on gaussian quadrature points is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals the investigation is based on a new convergence result for lagrangian interpolation and gaussian quadrature of singular integrands

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 survey on mean convergence of interpolatory processes abstract a detailed account of what happened in the theory of mean convergence of lagrange and hermite fejer type interpolation in the last fifty five years is given starting from the classical result of erd s and turan hopefully all important developments are considered in some cases with an indication of the method of proof even some yet unpublished results are included a list of references helps to orientate those interested in the details

rates of convergence of gauss lobattom and radau integration rules for singular integrands rates of convergence or divergence are obtained in the application of gauss lobatto and radau integration rules to functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a generalized jacobi weight function on 1 1 the error in applying an n point rule to f x ix vi 6 is o n 2 28 if y 1 and o n 1 6 if y e 1 1 providedwe avoid the singularity if we ignore the singularity y the error is o n 1 28 log n 6 log log n 1 e for almost all choices of y these assertions are sharp with respect to order 1 introduction this paper is a sequel to that of lubinsky and myself 2 on the rates of convergence of gauss integration rules for singular integrands in this paper we extend the results in 2 in several directions on the one hand we extend the results for gauss integration rules to results for lobatto and radau rules this extension follows easily from the generalized markov stieltjes inequality for lobatto and radau rules given in lemma 3 2 and from a representation of the coefficients or weights or cotes numbers in the lobatto and radau rules in terms of the coefficients in the gauss rule with respect to a related weight function a second extension is to the special case of gauss jacobi integration rules as well as lobatto jacobi and radau jacobi rules for certain values of the parameters defining the jacobi weight function

nystrom method for cauchy singular integral equations with negative index in this paper the authors propose a nystrom method to approximate the solutions of cauchy singular integral equations with constant coefficients having a negative index they consider the equations in spaces of continuous functions with weighted uniform norm they prove the stability and the convergence of the method and show some numerical tests that confirm the error estimates

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

applying fuzzy set theory to evaluate the rate of aggregative risk in software development abstract the purpose of this study is not only to build a structure model of risk in software development but also evaluate the rate of aggregative risk by fuzzy set theory while evaluating the rate of aggregative risk we may adjust the weights or grades of the factors until we can accept it we also show that the rate of aggregative risk is reasonable

numerical solutions for solving time fractional fokker planck equations based on spectral collocation methods abstract in this paper we consider the numerical solution of the time fractional fokker planck equation we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

on lagrange interpolation at disturbed roots of unity let z nk e itnk 0 t n0 0 such that l n f f p 0 f 1 n whenever n p n 1 it must be emphasized that p necessarily depends on p in the sense that there exists a family z nk k 0 n with n 2 n 1 and such that l n f f 2 o f 1 n for all f a but sup l n f p f a f 1 diverges for sufficiently large values of p