the ngp stability of runge kutta methods for systems of neutral delay differential equations this paper deals with the stability analysis of implicit runge kutta methods for the numerical solutions of the systems of neutral delay differential equations we focus on the behavior of such methods with respect to the linear test equations where tau 0 l m and n are d times d complex matrices we show that an implicit runge kutta method is ngp stable if and only if it is a stable

class groups of number fields numerical heuristics extending previous work of h w lenstra jr and the first author we give quantitative conjectures for the statistical behavior of class groups and class numbers for every type of field of degree less than or equal to four given the signature and the galois group of the galois closure the theoretical justifications for these conjectures will appear elsewhere but the agreement with the existing tables is quite good 1 introduction and notations in 3 h w lenstra jr and the first author developed a method for conjecturing quantitative results on class groups of quadratic fields and cyclic extensions of prime degree in a forthcoming paper 4 we shall show that this technique can be extended to a much wider class of number fields and also to relative extensions the aim of the present paper is to rapidly make available the numerical conjec tures obtained for people not really interested in our heuristic reasoning or not wanting to wait for 4 to appear hence apart from a total lack of justifications for the conjectures that we present this paper is essentially self contained the plan is as follows in the rest of this section we present the notations used in the sequel some of them being nonstandard and in general differing from the notations of 3 we urge the reader to read the notations carefully before applying the conjectures in the next section we present templates for the subsequent conjectures and then the conjectures themselves illustrated by numerical examples first for their own sake and second as a double check for the reader to understand the templates these conjectures are given for all types of fields of degree less than or equal to four in the final section we comment on the consistency of the conjectures with existing tables which is quite good combinatorial notations if x is a set xi denotes its cardinality

a primal partitioning approach for single and non simultaneous multicommodity flow problems abstract in the first part of this paper we study the minimum linear cost multicommodity flow problem where the given traffic demand is satisfied through routes having less than a given maximum number of edges we propose a simplex based solution method which uses a refined primal partitioning of the basis matrix our program has been tested on some real world instances given by the national french telecommunication operator france telecom and also on randomly generated multicommodity flow problems the second part of our paper generalizes the techniques described previously to handle additional survivability constraints major telecommunication operators want to design multicommodity survivable networks i e which are able to route all traffic demands even if any node or edge is damaged we present a simple model of this problem based on non simultaneous multicommodity flows

existence and approximate lp and continuous solutions of nonlinear integral equations of the hammerstein and volterra types in this paper we consider a compact interval a b a positive real number p 1 and give different existence results for l p a b and c a b solutions of some nonlinear integral equations of the hammerstein and volterra types the main ingredients of our existence results are the shaefer s and schauder s fixed point theorems combined with a general version of gronwall s inequality moreover we give a numerical method for the approximation of the solutions of the volterra hammerstein integral equations finally to illustrate the results of this work we provide the reader with some numerical examples

a multistep legendre gauss spectral collocation method for nonlinear volterra integral equations we introduce a multistep legendre gauss spectral collocation method for the nonlinear volterra integral equations of the second kind this method is easy to implement and possesses high order accuracy in addition it is very suitable for long time calculations we also derive the optimal convergence of the hp version of the multistep collocation method under the l 2 norm numerical experiments confirm the theoretical expectations

an extension of numerical stability criteria for linear neutral multidelay integro differential equations in this paper the underlying general linear methods glms are adapted to linear neutral multidelay integro differential equations nmides in order to obtain stability criteria of the extended glms the corresponding results in paper of zhang and vandewalle 2008 are generalized based on the concepts of a stability and a stability of the underlying glms a serial of asymptotic stability criteria of the extended glms are obtained

formulating a mixed integer programming problem to improve solvability a standard formulation of a real world distribution problem could not be solved even for a good solution by a commercial mixed integer programming code however after reformulating it by reducing the number of 0 1 variables and tightening the linear programming relaxation an optimal solution could be found efficiently the purpose of this paper is to demonstrate with a real application the practical importance of the need for good formulations in solving mixed integer programming problems

stability analysis of one step methods for neutral delay differential equations in this paper stability properties of one step methods for neutral functional differential equations are investigate stability regions are characterized for runge kutta methods with respect to the linear test equation begin gathered y left t right ay left t right by left t tau right cy left t tau right t geqq 0 hfill y left t right g left t right tau leqq t leqq 0 hfill end gathered 0 where a b andc are complex parameters in particular it is shown that everya stable collocation method for ordinary differential equations can be extended to a method for neutrals delay differential equations with analogous stability properties the so called np stable method we also investigate how the approximation to the derivative of the solution affects stability properties of numerical methods for neutral equations

improving discrete model representations via symmetry considerations in this paper we focus on a useful modeling concept that is frequently ignored while formu lating discrete optimization problems very often there exists a natural symmetry inherent in the problem itself that if propagated to the model can hopelessly mire a branch and bound solver by burdening it to explore and eliminate such alternative symmetric solutions we discuss three applications where such a symmetry arises a telecommunications network design problem a noise pollution problem and a machine procurement and operation problem for each case we identify the indistinguishable objects in the model that create the problem symmetry and show how imposing certain decision hierarchies within the model significantly enhances its solvability while using a popular modern day commercial branch and cut software cplex 6 5

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

on spectral methods for volterra type integro differential equations this paper considers the spectral methods for a volterra type integro differential equation firstly the volterra type integro differential equation is equivalently restated as two integral equations of the second kind secondly a legendre collocation method is used to solve them then the error analysis is conducted based on the l norm in addition numerical results are presented to confirm our analysis

chebyshev spectral solution of nonlinear volterra hammerstein integral equations abstract in this paper the chebyshev spectral cs method for the approximate solution of nonlinear volterra hammerstein integral equations f o k r g r y r d r 0 t is investigated the method is applied to approximate the solution not to the equation in its original form but rather to an equivalent equation z t g t y t t 1 1 the function z is approximated by the nth degree interpolating polynomial zn with coefficients determined by discretizing g t y t at the chebyshev gauss labatto nodes we then define the approximation to y to be of the form y n t f t 1 1 k t s z n s d s 1 1 and establish that under suitable conditions limn yn t y t uniformly in t finally a numerical experiment for a nonlinear volterra hammerstein integral equation is presented which confirms the convergence demonstrates the applicability and the accuracy of the chebyshev spectral cs method

ngp α stability of general linear methods for nddes abstract this paper deals with the asymptotic behavior of general linear methods for a class of neutral delay differential equations nddes new equivalent stability relations between the ode methods and the induced ndde methods are obtained moreover the obtained results are also extended to the case of nddes with distributed delays

a new collocation type method for hammerstein integral equations on presente une methode du type de collocation pour la resolution numerique des equations integrales d hammerstein on etablit la convergence de l approximation desiree vers la solution exacte

spectral and pseudo spectral approximations of the navier stokes equations galerkin and collocation approximations by trigonometric polynomials to the stationary navier stokes equations with periodic boundary conditions are analyzed stability results and optimal rates of convergence for both velocity and pressure in sobolev spaces are proven essential tools to carry out this analysis are some abstract results concerning general polynomial approximations to nonsingular solutions of nonlinear problems brezzi rappag raviart numer math 36 1980 pp 1 25 maday quarteroni rairo numer anal 1980

adams methods for neutral functional differential equations in this paper adams type methods for the special case of neutral functional differential equations are examined it is shown thatk step methods maintain orderk 1 for sufficiently small step size in a sufficiently smooth situation however when these methods are applied to an equation with a non smooth solution the order of convergence is only one some computational considerations are given and numerical experiments are presented

stability analysis of lmms for systems of neutral multidelay differential equations this paper deals with the asymptotic stability of theoretical and numerical solutions for systems of neutral multidelay differential equations nmdes in particular it is shown that a stability of the linear multistep methods lmms for odes is equivalent to ngpk stability of the induced methods for nmdes under the suitable conditions

p stability properties of runge kutta methods for delay differential equations recently the author and others have studied a class of runge kutta methods for ddes these methods make use of certain constrained meshes which allow to get optimal order results in this paper we study their asymptotic stability properties using the test equation begin gathered y t ay t by t tau t 0 hfill y t g t for tau leqq t leqq 0 hfill end gathered wherea b 0 andg t is continuous and complex valued it is known thaty t 0 ast if b re a in particular we show that any astable one step collocation method for odes inherits the same property when it is applied to ddes with a constrained mesh i e it is p stable

nonlinear stability of general linear methods for neutral delay differential equations this paper is concerned with the numerical solution of neutral delay differential equations nddes we focus on the stability of general linear methods with piecewise linear interpolation the new concepts of gs p stability gas p stability and weak gas p stability are introduced these stability properties for k p 0 algebraically stable general linear methods glms are further investigated some extant results are unified

the spectral methods for parabolic volterra integro differential equations in this paper we study the numerical solutions to parabolic volterra integro differential equations in one dimensional bounded and unbounded spatial domains in a bounded domain the given parabolic volterra integro differential equation is converted to two equivalent equations then a legendre collocation method is used to solve them and finally a linear algebraic system is obtained for an unbounded case we use the algebraic mapping to transfer the problem on a bounded domain and then apply the same presented approach for the bounded domain in both cases some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method

evolution and state of the art in integer programming under a unifying theme of exploiting both algebraic and polyhedral special structures present in integer linear programming problems we discuss the evolution of both technique and philosophy leading to the current state of the art for modeling and solving this challenging class of problems integrated throughout the discussion are insights into the rationale and motivation that have contributed in a large part to the past and present direction of research in this fascinating field

legendre and chebyshev spectral approximations of burgers equation we give error estimates for both legendre and chebyshev spectral approximations of the steady state burgers problem u xx lambda uu x f 0 in i 1 1 u 1 u 1 0 to do that we prove some abstract approximation properties of orthogonal projection operators in some weighted sobolev spacesh s i for both legendre and chebyshev weights