non connected toric hilbert schemes we construct small 50 and 26 points respectively point sets in dimension 5 whose graphs of triangulations are not connected these examples improve our construction in j amer math soc 13 3 2000 611 637 not only in size but also in that the associated toric hilbert schemes are not connected either a question left open in that article additionally the point sets can easily be put into convex position providing examples of 5 dimensional polytopes with non connected graph of triangulations

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

a new efficient method for cases of the singular integral equation of the first kind various cases of cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties these equations are usually difficult to solve analytically and it is required to obtain approximate solutions this paper investigates the numerical solution of various cases of cauchy type singular integral equations using reproducing kernel hilbert space rkhs method the solution u x is represented in the form of a series in the reproducing kernel space afterwards the n term approximate solution u n x is obtained and it is proved to converge to the exact solution u x the major advantage of the method is that it can produce good globally smooth approximate solutions moreover in this paper an efficient error estimation of the rkhs method is introduced finally numerical experiments show that our reproducing kernel method is efficient

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

numerical solution of systems of cauchy singular integral equations with constant coefficients abstract this paper deals with the numerical solution of a class of systems of cauchy singular integral equations with constant coefficients the proposed procedure consists of two basic steps the first one is to consider a modified problem equivalent to the original one under suitable conditions the second one is to approximate its solution by means of a vector of polynomial functions such array is constructed by applying a quadrature type method based on gaussian rules that leads to solve a determined and well conditioned linear system the convergence and stability of the method are proved in weighted l 2 spaces some numerical tests are also shown

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

hilbert scheme strata defined by bounding cohomology abstract let hilb p be the hilbert scheme parametrizing the closed subschemes of p k n with hilbert polynomial p q t over a field k of characteristic zero by bounding below the cohomological hilbert functions of the points of hilb p we define locally closed subspaces of the hilbert scheme the aim of this paper is to show that some of these subspaces are connected for this we exploit the edge ideals constructed by d mall in d mall connectedness of hilbert function strata and other connectedness results j pure appl algebra 150 2000 175 205 it turns out that these ideals are sequentially cohen macaulay and that their initial ideals with respect to the reverse lexicographic term order are generic initial ideals

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 the consistency of numerical solutions of singular integral equations of the second kind with negative index using jacobi polynomials abstract singular integral equations of second kind with negative index possess bounded solutions whenever the forcing function satisfies the orthogonality conditions numerically the discretization of the integrals via gauss jacobi quadrature does not ensure the preservation of a consistent system it is shown that for gauss jacobi quadrature the overdetermined system of equations is consistent if and only if the discretized compatibility condition is satisfied the solutions of inconsistent systems via generalized inverses are considered error estimates are established which are shown to be independent of which equation is excluded in the solution of the overdetermined system

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

an approximate numerical method for solving cauchy singular integral equations composed of multiple implicit parameter functions with unknown integral limits in contact mechanics abstract cauchy singular integral equations are widely used in physics and mathematics especially in solid contact mechanics the solution of cauchy singular integral equations composed of explicit functions has already been achieved in existing research however when dealing with the contact problem between two solid bodies with irregular surfaces described by implicit parametric functions difficulties arise when trying to solve the cauchy singular integral because it is composed of multiple implicit parameter functions moreover the integral limits are unknown and constrained by physical characteristics to solve this kind of problem an approximate calculation method with high accuracy will be provided in this paper specifically based on the quadrature method and taking the constraint function of the boundary as the convergence criterion both the integral limits satisfying the physical characteristics condition and the solution of the cauchy singular integral equations composed of multiple parameter functions can be derived by an iterative method finally five different examples are calculated using the new method and the absolute errors between the approximate values provided by the new method and the true values are analysed

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

cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution cauchy singular integral equations on an interval are studied in weighted spaces of continuous functions convergence rates for a general method for their numerical solution are proved in weighted uniform norms applications of these results to the collocation method and the quadrature method are given

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

a comparative analysis of painleve lax pair and similarity transformation methods in obtaining the integrability conditions of nonlinear schrodinger equations we derive the integrability conditions of nonautonomous nonlinear schrodinger equations using the lax pair and similarity transformation methods we present a comparative analysis of these integrability conditions with those of the painleve method we show that while the painleve integrability conditions restrict the dispersion nonlinearity and dissipation gain coefficients to be space independent and the external potential to be only a quadratic function of position the lax pair and the similarity transformation methods allow for space dependent coefficients and an external potential that is not restricted to the quadratic form the integrability conditions of the painleve method are retrieved as a special case of our general integrability conditions we also derive the integrability conditions of nonautonomous nonlinear schrodinger equations for two and three spacial dimensions

murphy s law in algebraic geometry badly behaved deformation spaces we consider the question how bad can the deformation space of an object be the answer seems to be unless there is some a priori reason otherwise the deformation space may be as bad as possible we show this for a number of important moduli spaces more precisely every singularity of finite type over up to smooth parameters appears on the hilbert scheme of curves in projective space and the moduli spaces of smooth projective general type surfaces or higher dimensional varieties plane curves with nodes and cusps stable sheaves isolated threefold singularities and more the objects themselves are not pathological and are in fact as nice as can be the curves are smooth the surfaces are automorphism free and have very ample canonical bundle the stable sheaves are torsion free of rank 1 the singularities are normal and cohen macaulay etc this justifies mumford s philosophy that even moduli spaces of well behaved objects should be arbitrarily bad unless there is an a priori reason otherwise thus one can construct a smooth curve in projective space whose deformation space has any given number of components each with any given singularity type with any given non reduced behavior similarly one can give a surface over mathbb f p that lifts to p 7 but not p 8 of course the results hold in the holomorphic category as well it is usually difficult to compute deformation spaces directly from obstruction theories we circumvent this by relating them to more tractable deformation spaces via smooth morphisms the essential starting point is mnev s universality theorem

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

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

controlling temporal solitary waves in the generalized inhomogeneous coupled nonlinear schrodinger equations with varying source terms in this paper we study the families of solitary wave solutions to the inhomogeneous coupled nonlinear schrodinger equations with space and time modulated coefficients and source terms by means of the similarity reduction method and mobius transformations many types of novel temporal solitary wave solutions of this nonlinear dynamical system are analytically found under some constraint conditions such as the bright bright bright dark dark dark periodic periodic w shaped and rational wave solutions in particular we find that the localized rational type solutions can exhibit both bright bright and bright dark wave profiles by choosing different families of free parameters moreover we analyze the relationships among the group velocity dispersion profiles gain or loss distributions external potentials and inhomogeneous source profiles which provide the necessary constraint conditions to control the emerging wave dynamics finally a series of numerical simulations are performed to show the ro

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

nonautonomous mixed mkdv sinh gordon hierarchy the construction of a nonautonomous mixed mkdv sine gordon model is proposed by employing an infinite dimensional affine lie algebraic structure within the zero curvature representation a systematic construction of soliton solutions is provided by an adaptation of the dressing method which takes into account arbitrary time dependent functions a particular choice of those arbitrary functions provides an interesting solution describing the transition of a pure mkdv system into a pure sine gordon soliton

analytical solitary wave solutions of the generalized nonautonomous cubic quintic nonlinear schrodinger equation with different external potentials a large family of analytical solitary wave solutions to the generalized nonautonomous cubic quintic nonlinear schr odinger equation with time and space dependent distributed coefficients and external potentials are obtained by using a similarity transformation technique we use the cubic nonlinearity as an independent parameter function where a simple procedure is established to obtain different classes of potentials and solutions the solutions exist under certain conditions and impose constraints on the coefficients depicting dispersion cubic and quintic nonlinearities and gain or loss we investigate the space quadratic potential optical lattice potential flying bird potential and potential barrier well some interesting periodic solitary wave solutions corresponding to these potentials are then studied also properties of a few solutions and physical applications of interest to the field are discussed finally the stability of the solitary wave solutions under slight disturbance of the constraint conditions and initial perturbation of white noise is discussed numerically the results reveal that the solitary waves can propagate in a stable way under slight disturbance of the constraint conditions and the initial perturbation of white noise

equivalence groupoid of a class of variable coefficient korteweg de vries equations we classify the admissible transformations in a class of variable coefficient korteweg de vries equations as a result a full description of the structure of the equivalence groupoid of the class is given the class under study is partitioned into six disjoint normalized subclasses the widest possible equivalence group for each subclass is found which appears to be generalized extended in five cases ways for improvement of transformational properties of the subclasses are proposed using gaugings of arbitrary elements and mapping between classes the group classification of one of the subclasses is carried out as an illustrative example

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

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

integrable nonautonomous nonlinear schrodinger equations are equivalent to the standard autonomous equation a class of nonautonomous nonlinear schrodinger equations claiming to be novel integrable systems with rich properties continues to appear in the literature all such equations are shown to be not new but equivalent to the standard autonomous equation which trivially explains their integrability features

uniform convergence of a fast algorithm for cauchy singular integral equations abstract error estimates for a fast algorithm applied to a cauchy singular integral equation are proved in weighted uniform norms this algorithm is essentially based on the application of discrete sine transformations numerical results are presented

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

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

numerical solution of various cases of cauchy type singular integral equation abstract in this paper a robust but very simple numerical method is developed to solve various cases of cauchy type singular integral equation for this first bernstein polynomials are defined which are used for approximation of solution of the given singular integral equation then numerical method is introduced by using bernstein polynomials this ultimately leads to solution of a system of linear algebraic equations examples are illustrated to demonstrate simplicity of proposed method results are also compared with those present in literature to claim better efficiency of the method introduced

irreducible components of the equivariant punctual hilbert schemes abstract let h ab be the equivariant hilbert scheme parameterizing the zero dimensional subschemes of the affine plane invariant under the natural action of the one dimensional torus t ab t b t a t k we compute the irreducible components of h ab they are in one to one correspondence with the set of possible hilbert functions as a by product of the proof we give new proofs of results by ellingsrud and stromme namely the main lemma of the computation of the betti numbers of the hilbert scheme h l parametrizing the zero dimensional subschemes of the affine plane of length l invent math 87 1988 343 and a description of bialynicki birula cells on h l by means of explicit flat families invent math 91 1988 365 in particular we give precise conditions when this last description applies

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

smooth and irreducible multigraded hilbert schemes abstract the multigraded hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed hilbert function we prove that any multigraded hilbert scheme is smooth and irreducible when the polynomial ring is z x y which establishes a conjecture of haiman and sturmfels