quasi polynomials linear diophantine equations and semi linear sets we investigate the family of semi linear sets of n t and z t we study the growth function of semi linear sets and we prove that such a function is a piecewise quasi polynomial on a polyhedral partition of n t moreover we give a new proof of combinatorial character of a famous theorem by dahmen and micchelli on the partition function of a system of diophantine linear equations

adjoint based optimal control using meshfree discretizations the paper at hand presents a combination of optimal control approaches for pdes with meshless discretizations applying a classical lagrangian type particle method to optimization problems with hyperbolic constraints several adjoint based strategies differing in the sequential order of optimization and discretization of the lagrangian or eulerian problem formulation are proposed and compared the numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior moreover they exemplify the superiority of meshless methods over the conventional mesh based approaches for the numerical handling and optimization of problems with time dependent geometries and freely moving boundaries

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

modified moving least squares with polynomial bases for scattered data approximation one common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes to tackle such problems we present a modified robust moving least squares mls method for scattered data smoothing and approximation the error functional used in the derivation of the classical mls approximation is augmented with additional terms based on the coefficients of the polynomial base functions this allows quadratic base functions to be used with the same size of the support domain as linear base functions resulting in better approximation capability the increased robustness of the modified mls method to irregular nodal distributions makes it suitable for use across many fields the analysis is supported by several univariate and bivariate examples

an adaptive procedure based on background cells for meshless methods in practical implementations meshless methods using moving least squares mls approximation and global galerkin formulation require a background mesh for domain integration an adaptive procedure based on background mesh is developed for meshless methods using mls it comprises a cell energy error estimate and a local domain refinement technique the error estimate differs from conventional pointwise approaches in that it evaluates error based on individual cells instead of points for each cell a computed cell energy and a reference cell energy are generated based on a same stress field by using two different integration schemes the difference between the two energy values is used as the basic measure of error estimation the advantage of this strategy is that it requires only one stress field and no reference field needs to be furnished this has significance as the stress field generated by meshless methods is already very smooth and traditional error estimates based on stress smoothing techniques for finite element methods are not applicable to achieve high efficiency in domain refinement a local technique based on the delaunay algorithm is implemented in this technique each node is assigned a scaling factor to control local nodal density refinement of the neighborhood of a node is accomplished simply by adjusting its scaling factor the numerical experiments in this paper show that the proposed adaptive procedure is simple effective and efficient the limitations of the approach are also discussed

new boundary condition treatments in meshfree computation of contact problems due to the loss of kronecker delta properties in the meshfree shape functions the imposition of essential boundary conditions consumes significant cpu time in meshfree computation in this work two boundary condition treatments are proposed to enhance the computational efficiency of meshfree methods for contact problems the mixed transformation method is modified from the previous full transformation method by introducing a node partitioning and a mixed coordinate so that the matrix inversion and multiplication of coordinate transformation involves operations only on the sub degrees of freedom associated with the boundary group the boundary singular kernel method introduces singularities to the kernel functions of the essential and contact boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values and consequently kinematic constraints can be imposed directly the effectiveness of the proposed methods is demonstrated in several numerical examples

angle theorems for the lagrangian mean curvature flow we prove that symplectic maps between riemann surfaces l m of constant nonpositive and equal curvature converge to minimal symplectic maps if the lagrangian angle alpha for the corresponding lagrangian submanifold in the cross product space l times m satisfies text osc alpha le pi if one considers a 4 dimensional kahler einstein manifold overline m of nonpositive scalar curvature that admits two complex structures j k which commute and assumes that l subset overline m is a compact oriented lagrangian submanifold w r t j such that the kahler form overline kappa w r t k restricted to l is positive and text osc alpha le pi then l converges under the mean curvature flow to a minimal lagrangian submanifold which is calibrated w r t overline kappa

a coupled element free galerkin boundary element method for stress analysis of two dimensional solids element free galerkin efg method is a newly developed meshless method for solving partial differential equations using moving least squares interpolants it is however computationally expensive for many problems a coupled efg boundary element be method is proposed in this paper to improve the solution efficiency a procedure is developed for the coupled efg be method so that the continuity and compatibility are preserved on the interface of the two domains where the efg and be methods are applied the present coupled efg be method has been coded in fortran the validity and efficiency of the efg be method are demonstrated through a number of examples it is found that the present method can take the full advantages of both efg and be methods it is very easy to implement and very flexible for computing displacements and stresses of desired accuracy in solids with or without infinite domains

composite mesh concept based fem error estimation and solution improvement abstract the use of composite finite element models where meshes with different granularities are allowed to share the same problem domain is introduced the idea of this method was taken from the mixture theory of multiphase materials where the properties of each component enter into the formulation affected by a participation factor the different behaviors of each component mesh due to differences in the model accuracy rather than to the material properties we perform estimation of local and global errors for some problems in planar domains that are drawn from engineering problems structural elastic deformations or heat and mass transfer in catalytic chemical reactors the main examples in this work are elliptic problems with boundary conditions of dirichlet neumann and mixed types the method is not restricted to 2d being also applicable to 3d problems in this paper we also develop the use of composite models with the target of improving the numerical solution of finite element models by using extrapolation like concepts improved solutions may be obtained without increasing the computational cost of the problem or requiring postprocessing of the results some examples are shown for which exact solution is available in order to validate the method

enhancing the approximation order of local shepard operators by hermite polynomials we show how to combine local shepard operators with hermite polynomials on the simplex c k chui m j lai multivariate vertex splines and finite elements j approx theory 60 1990 245 343 so as to raise the algebraic precision of the shepard taylor operators r farwig rate of convergence of shepard s global interpolation formula math comp 46 1986 577 590 that use the same data and contemporaneously maintain the interpolation properties at each sample point derivative data included and a good accuracy of approximation numerical results are provided

on donaldson s flow of surfaces in a hyperkahler four manifold we prove some basic properties of donaldson s flow of surfaces in a hyperkahler 4 manifold when the initial submanifold is symplectic with respect to one kahler form and lagrangian with respect to another we show that certain kinds of singularities cannot form and we prove a convergence result under a condition related to one considered by m t wang for the mean curvature flow

an interval for the shape parameter in radial basis function approximation the radial basis functions rbfs depend on an auxiliary parameter called the shape parameter great theoretical and numerical efforts have been made to find the relationship between the accuracy of the rbf approximations and the value of the shape parameter in many cases the numerical approaches are based on minimization of an estimation of the error function such as rippa s approach and its modifications these approaches determine a value for the shape parameter in this paper we propose a practical approach to determine an interval instead of a value without any minimization and estimation of error function the idea is based on adding a loop on the shape parameter but not to minimize an error norm suitable values of the shape parameter are determined by take into account the practical convergence behavior of the problem the proposed method is applied to some illustrative examples including one and two dimensional interpolation and partial differential equations the results of the method are compared with those of some other approaches to confirm the reliability of the method furthermore numerical stability of the method of line respect to the shape parameter is considered for time dependent pdes the results show that the proposed method can be used as an independent method to find a shape parameter and also as an alternative for investigating the validity of the values of the other methods

an adaptive finite element meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems in this paper an automatic adaptive coupling procedure is proposed for the finite element method fem and the element free galerkin method efgm for linear elasticity and for problems with both material and geometrical nonlinearities in this new procedure initially the whole of the problem domain is modelled using the fem during an analysis those finite elements which violate a predefined error measure are automatically converted to an efg zone this efg zone can be further refined by adding nodes thus avoiding computationally expensive fe remeshing local maximum entropy shape functions are used in the efg zone of the problem domain for two reasons their weak kronecker delta property at the boundaries allows straightforward imposition of essential boundary conditions and also provides a natural way to couple the efg and fe regions compared to the use of moving least squares basis functions the zienkiewicz and zhu error estimation procedure with the superconvergent patch method for strains and stresses recovery is used in the fe region of the problem domain while the chung and belytschko error estimation procedure is used in the efg region

hermite type moving least squares approximations the moving least squares approach first presented by mclain 1 is a method for approximating multivariate functions using scattered data information the method is using local polynomial approximations incorporating weight functions of different types some weights with certain singularities induce c interpolation approximation in n in this work we present a way of generalizing the method to enable hermite type interpolation namely interpolation to derivatives data as well the essence of the method is the use of an appropriate metric in the construction of the local polynomial approximations

analytic models and ambiguity of context free languages we establish that several classical context free languages are inherently ambiguous by proving that their counting generating functions when considered as analytic functions exhibit some characteristic form of transcendental behaviour to that purpose we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages in particular the paper contains a general density theorem for unambiguous context free languages

a meshless interpolation algorithm using a cell based searching procedure in this paper we propose a fast algorithm for bivariate interpolation of large scattered data sets it is based on the partition of unity method for constructing a global interpolant by blending radial basis functions as local approximants and using locally supported weight functions the partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cell based searching procedure more precisely we construct a cell structure which partitions the domain and strictly depends on the dimension of the subdomains thus providing a meaningful improvement in the searching process compared to the nearest neighbour searching techniques presented in allasia et al 2011 and cavoretto and de rossi 2010 2012 in fact this efficient algorithm and in particular the new searching procedure enable us a fast computation also in several applications where the amount of data to be interpolated is often very large up to many thousands or even millions of points analysis of computational complexity shows the high efficiency of the proposed interpolation algorithm this is also supported by numerical experiments

a perturbation method for stochastic meshless analysis in elastostatics a stochastic meshless method is presented for solving boundary value problems in linear elasticity that involves random material properties the material property was modelled as a homogeneous random field a meshless formulation was developed to predict stochastic structural response unlike the finite element method the meshless method requires no structured mesh since only a scattered set of nodal points is required in the domain of interest there is no need for fixed connectivities between nodes in conjunction with the meshless equations classical perturbation expansions were derived to predict second moment characteristics of response numerical examples based on one and two dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method a good agreement is obtained between the results of the proposed method and monte carlo simulation since mesh generation of complex structures can be a far more time consuming and costly effort than the solution of a discrete set of equations the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems copyright 2001 john wiley sons ltd

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

a meshfree enriched finite element method for compressible and near incompressible elasticity summary in this paper a two dimensional displacement based meshfree enriched fem me fem is presented for the linear analysis of compressible and near incompressible planar elasticity the me fem element is established by injecting a first order convex meshfree approximation into a low order finite element with an additional node the convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the kronecker delta property on the element boundaries the gradient matrix of me fem element satisfies the integration constraint for nodal integration and the resultant me fem formulation is shown to pass the constant stress test for the compressible media the me fem interpolation is an element wise meshfree interpolation and is proven to be discrete divergence free in the incompressible limit to prevent possible pressure oscillation in the near incompressible problems an area weighted strain smoothing scheme incorporated with the divergence free me fem interpolation is introduced to provide the smoothing on strains and pressure with this smoothed strain field the discrete equations are derived based on a modified hu washizu variational principle several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near incompressible problems copyright 2012 john wiley sons ltd

a trivariate interpolation algorithm using a cube partition searching procedure in this paper we propose a fast algorithm for trivariate interpolation which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions the partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube partition searching procedure more precisely we construct a cube structure which partitions the domain and strictly depends on the size of its subdomains so that the new searching procedure and accordingly the resulting algorithm enable us to efficiently deal with a large number of nodes complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm

frictional contact analysis of functionally graded materials with lagrange finite block method summary based on the one dimensional differential matrix derived from the lagrange series expansion the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials in this paper the formulation of the lagrange finite block method with boundary type in the strong form is presented and applied to non conforming contact problems for the functionally graded materials subjected to either static or dynamic loads the first order partial differential matrices are only needed both in the governing equations and in the neumann boundary condition by introducing the mapping technique a block of quadratic type is transformed from the cartesian coordinate of global system to the normalized coordinate with eight seeds time dependent partial differential equations are analyzed in the laplace transformed domain and the durbin s inversion method is applied to determine all the physical values in the time domain conforming and non conforming contacts are investigated by using the iterative algorithm with full load technique illustrative numerical examples are given and comparisons have been made with analytical solutions copyright 2015 john wiley sons ltd

mean curvature flows and isotopy of maps between spheres let f be a smooth map between unit spheres of possibly different dimensions we prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions a corollary is that any area decreasing map between unit spheres of possibly different dimensions is isotopic to a constant map c 2004 wiley periodicals inc

on the commutative equivalence of semi linear sets of n k given two subsets s 1 s 2 of n k we say that s 1 is commutatively equivalent to s 2 if there exists a bijection f s 1 s s 2 from s 1 onto s 2 such that for every v s 1 v f v where v denotes the sum of the components of v we prove that every semi linear set of n k is commutatively equivalent to a recognizable subset of n k

efficient computation of partition of unity interpolants through a block based searching technique in this paper we propose a new efficient interpolation tool extremely suitable for large scattered data sets the partition of unity method is used and performed by blending radial basis functions rbfs as local approximants and using locally supported weight functions in particular we present a new space partitioning data structure based on a partition of the underlying generic domain in blocks this approach allows us to examine only a reduced number of blocks in the search process of the nearest neighbour points leading to an optimized searching routine complexity analysis and numerical experiments in two and three dimensional interpolation support our findings some applications to geometric modelling are also considered moreover the associated software package written in matlab is here discussed and made available to the scientific community

enrichment and coupling of the finite element and meshless methods a mixed hierarchical approximation based on finite elements and meshless methods is presented two cases are considered the first one couples regions where finite elements or meshless methods are used to interpolate continuity and consistency is preserved the second one enriches a finite element mesh with particles thus there is no need to remesh in adaptive refinement processes in both cases the same formulation is used convergence is studied and examples are shown copyright 2000 john wiley sons ltd

the number of solutions to linear diophantine equations and multivariate splines etude de la variation du nombre des solutions entieres non negatives de s equations diophantiennes lineaires en fonction des termes non homogenes

coupling finite element method with meshless finite difference method in thermomechanical problems this paper focuses on coupling two different computational approaches namely finite element method fem and meshless finite difference method mfdm in one domain the coupled approach is applied in solving thermomechanical initialboundary value problem where the heat transport in the domain is non stationary in this method the domain is divided into two subdomains for fem and mfdm respectively contrary to other coupling techniques the approach presented in this paper is defined in terms of mathematical problem formulation rather than at the approximation level in the weak form of thermomechanical initialboundary value problem variational principle the appropriate additional coupling integrals are defined a priori subsequently the fem and the mfdm approximations which may differ from each other are provided to the formulation it is assumed that there exists a very thin layer of material between the subdomains which is not spatially discretized the width of this layer may be considered the coupling parameter and it is the same for both thermal and mechanical parts similar approach is applied to essential boundary conditions e g prescribed temperature and displacements consequently the consistent formulation of the mixed problem for the coupled femmfdm method is derived the analysis is illustrated with two and three dimensional examples of mechanical and thermomechanical problems

a novel element differential method for solid mechanical problems using isoparametric triangular and tetrahedral elements abstract a novel strong form numerical method element differential method edm is developed to solve geometrically complex mechanics problems based on triangular or tetrahedral meshes the discretization of the structure under investigation has been based on lagrange isoparametric quadrilateral or hexahedral elements while applying edm in this paper a new family of isoparametric triangular and tetrahedral elements with a central node is proposed for edm a set of shape functions with analytical expressions for their first and second order partial derivatives is constructed for these triangular and tetrahedral elements respectively moreover a new element collocation scheme is proposed to establish a system of equations directly from the governing differential equations for internal nodes and traction equilibrium equations for nodes on edges of an element in this collocation scheme no variational principles or virtual energy principles are required to set up the solution scheme while no integration is needed when forming the coefficients of the system of equations numerical examples including standard patch tests and more practical problems are given to demonstrate the correctness of the constructed elements and the efficiency of the proposed element collocation method

on the commutative equivalence of bounded context free and regular languages the semi linear case this is the third paper of a group of three where we prove the following result let a be an alphabet of t letters and let a nt be the corresponding parikh morphism given two languages l1 l2 a we say that l1 is commutatively equivalent to l2 if there exists a bijection f l1 l2 from l1 onto l2 such that for every u l1 u f u then every bounded context free language is commutatively equivalent to a regular language

on the structure of the counting function of sparse context free languages we give an exact description of the counting function of a sparse context free language let l be a sparse context free language and let fl be its counting function then there exist polynomials p0 p1 pk 1 with rational coefficients and an integer constant k0 such that for any n k0 one has fl n pj n where j is such that j n mod k as a consequence one can easily show the decidability of some questions concerning sparse context free languages finally we show that for any sparse context free language l there exists a regular language l such that for any n 0 one has fl n fl n and therefore fl is rational

the effective interface approach for coupling of the fe and meshless fd methods and applying essential boundary conditions the paper presents a new effective technique for coupling two computational methods with different types of discretization and approximation it is based on a concept of two adjacent subdomains which are connected with each other by means of a thin layer of material each of the subdomain may have a different discretization structure and approximation base the standard finite element method fem as well as the meshless finite difference method mfdm are applied here to be coupled however the coupling can be used for any other methods the same concept is used for applying the essential boundary conditions for both of the methods the width of the interface layer depending on discretization density is evaluated by means of several heuristic assumptions the paper is illustrated with selected two and three dimensional examples

the parikh counting functions of sparse context free languages are quasi polynomials let l be a sparse context free language over an alphabet of t letters and let f l n t n be its parikh counting function we prove the following two results 1 there exists a partition of n t into a finite family of polyhedra such that the function f l is a quasi polynomial on each polyhedron of the partition 2 there exists a partition of n t into a finite family of rational subsets such that the function f l is a polynomial on each set of the partition

radial basis functions for the multivariate interpolation of large scattered data sets an efficient method for the multivariate interpolation of very large scattered data sets is presented it is based on the local use of radial basis functions and represents a further improvement of the well known shepard s method although the latter is simple and well suited for multivariate interpolation it does not share the good reproduction quality of other methods widely used for bivariate interpolation on the other band radial basis functions which have proven to be highly useful for multivariate scattered data interpolation have a severe drawback they are unable to interpolate large sets in an efficient and numerically stable way and maintain a good level of reproduction quality at the same time both problems have been circumvented using radial basis functions to evaluate the nodal function of the modified shepard s method this approach exploits the flexibility of the method and improves its reproduction quality the proposed algorithm has been implemented and numerical results confirm its efficiency

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

efg approximation with discontinuous derivatives a technique for incorporating discontinuities in derivatives into meshless methods is presented the technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives the special shape functions have compact support which results in banded matrix equations the technique is described in element free galerkin context but is easily applicable to other meshless methods and projections numerical results for problems in one and two dimensions are reported 1998 john wiley sons ltd

stable computations with gaussian radial basis functions radial basis function rbf approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh free and can be spectrally accurate a perceived practical obstacle is that the interpolation matrix becomes increasingly ill conditioned as the rbf shape parameter becomes small corresponding to flat rbfs two stable approaches that overcome this problem exist the contour pade method and the rbf qr method however the former is limited to small node sets and the latter has until now been formulated only for the surface of the sphere this paper focuses on an rbf qr formulation for node sets in one two and three dimensions the algorithm is stable for arbitrarily small shape parameters it can be used for thousands of node points in two dimensions and still more in three dimensions a sample matlab code for the two dimensional case is provided

a least squares radial basis function partition of unity method for solving pdes recently collocation based radial basis function rbf partition of unity methods pums for solving partial differential equations have been formulated and investigated numerically and theoretically when combined with stable evaluation methods such as the rbf qr method high order convergence rates can be achieved and sustained under refinement however some numerical issues remain the method is sensitive to the node layout and condition numbers increase with the refinement level here we propose a modified formulation based on least squares approximation we show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling we derive theoretical error estimates both for the collocation and least squares rbf pums numerical experiments are performed for the poisson equation in two and three space dimensions for regular and irregular geometries the convergence experiments confirm the theoretical estimates and the least squares formulation is shown to

mean curvature flow of surfaces in einstein four manifolds let be a compact oriented surface immersed in a four dimensional kahler einstein manifold m w we consider the evolution of in the direction of its mean curvature vector it is proved that being symplectic is preserved along the flow and the flow does not develop type i singularity when m has two parallel kahler forms w and w that determine different orientations and is symplectic with respect to both w and w we prove the mean curvature flow of exists smoothly for all time in the positive curvature case the flow indeed converges at infinity

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

on the commutative equivalence of bounded context free and regular languages this is the first paper of a group of three where we prove the following result let a be an alphabet of t letters and let a s n t be the corresponding parikh morphism given two languages l 1 l 2 a we say that l 1 is commutatively equivalent to l 2 if there exists a bijection f l 1 s l 2 from l 1 onto l 2 such that for every u l 1 u f u then every bounded context free language is commutatively equivalent to a regular language

singularity of mean curvature flow of lagrangian submanifolds in this article we study the tangent cones at first time singularity of a lagrangian mean curvature flow if the initial compact submanifold 0 is lagrangian and almost calibrated by re in a calabi yau n fold m and t 0 is the first blow up time of the mean curvature flow then the tangent cone of the mean curvature flow at a singular point x 0 t is a stationary lagrangian integer multiplicity current in r 2n with volume density greater than one at x 0 when n 2 the tangent cone is a finite union of at least two 2 planes in r 4 which are complex in a complex structure on r 4

rbf pu method for pricing options under the jump diffusion model with local volatility abstract meshfree methods based on radial basis functions rbfs are of general interest for solving partial differential equations pdes because they can provide high order or spectral convergence for smooth solutions in complex geometries for global rbf methods one of the major disadvantages is the computational cost associated with the dense linear systems that arise therefore this paper is currently directed toward localized rbf approximations known as the rbf partition of unity rbf pu method for partial integro differential equation pide arisen in option pricing problems in jump diffusion model rbf pu method produces algebraic systems with sparse matrices which have small condition number also for comparison some stable time discretization schemes are combined with the operator splitting method to get a fully discrete problem numerical examples are presented to illustrate the convergence and stability of the proposed algorithms for pricing european and american options with merton and kou models

a meshless singular boundary method for transient heat conduction problems in layered materials abstract the transient heat conduction problems in layered materials are common encountered in practical application but bring great challenge to most numerical algorithms in this study the singular boundary method sbm a meshless boundary collocation technique is first employed to numerically simulate such problems based on the time dependent fundamental solution of diffusion equation taking the boundary conditions and interface conditions into account the computing system of the sbm for layered materials is established and then is solved the proposed method fully inherits the merits of conventional boundary type methods while possessing its distinctive advantages furthermore it is simple straightforward computationally efficient and stable since it does not need to discretize temporal derivative term three numerical experiments are performed to verify the efficiency and accuracy of the proposed scheme numerical results clearly indicate the efficiency accuracy and stability of the presented sbm for solving transient heat conduction problems in layered materials

fast and accurate interpolation of large scattered data sets on the sphere in this paper a new efficient algorithm for spherical interpolation of large scattered data sets is presented the solution method is local and involves a modified spherical shepard s interpolant which uses zonal basis functions as local approximants the associated algorithm is implemented and optimized by applying a nearest neighbour searching procedure on the sphere specifically this technique is mainly based on the partition of the sphere in a suitable number of spherical zones the construction of spherical caps as local neighbourhoods for each node and finally the employment of a spherical zone searching procedure computational cost and storage requirements of the spherical algorithm are analyzed moreover several numerical results show the good accuracy of the method and the high efficiency of the proposed algorithm

on the asymptotic solutions of a differential equation with multiple singular points abstract a singularly perturbed model problem with multiple distinct regular singular points is studied the uniform approximation of wazwaz and hanson 1986 is effectively extended to many singular points thus establishing a generalized version of that theorem where the classic inner and outer expansions are not employed a leading order general asymptotic solution is correctly represented by a set of matched exponential asymptotic expansions where each approximation contains dominant and recessive terms the resonance criteria due to the influence of multiple singular points are discussed for an even number of singular points the eigenvalues were found to be positive with a minimal eigenvalue however negative eigenvalues with a maximal eigenvalue arise for an odd number of singular points the maximal and the minimal eigenvalue may not be unique and each is 0 1

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

scattered and track data interpolation using an efficient strip searching procedure abstract a new local algorithm for bivariate interpolation of large sets of scattered and track data is presented the method which changes partially depending on the kind of data is based on the partition of the interpolation domain in a suitable number of parallel strips and starting from these on the construction for any data point of a square neighbourhood containing a convenient number of data points then the well known modified shepard s formula for surface interpolation is applied with some effective improvements the proposed algorithm is very fast owing to the optimal nearest neighbour searching and achieves good accuracy computational cost and storage requirements are analyzed moreover the efficiency and reliability of the algorithm are shown by several numerical tests also performed by renka s algorithm for a comparison

voronoi cell finite difference method for the diffusion operator on arbitrary unstructured grids voronoi cells and the notion of natural neighbours are used to develop a finite difference method for the diffusion operator on arbitrary unstructured grids natural neighbours are based on the voronoi diagram which partitions space into closest point regions the sibson and the laplace non sibsonian interpolants which are based on natural neighbours have shown promise within a galerkin framework for the solution of partial differential equations in this paper we focus on the laplace interpolant with a two fold objective first to unify the previous developments related to the laplace interpolant and to indicate its ties to some well known numerical methods and secondly to propose a voronoi cell finite difference scheme for the diffusion operator on arbitrary unstructured grids a conservation law in integral form is discretized on voronoi cells to derive a finite difference scheme for the diffusion operator on irregular grids the proposed scheme can also be viewed as a point collocation technique a detailed study on consistency is conducted and the satisfaction of the discrete maximum principle stability is established owing to symmetry of the laplace weight a symmetric positive definite stiffness matrix is realized which permits the use of efficient linear solvers on a regular rectangular or hexagonal grid the difference scheme reduces to the classical finite difference method numerical examples for the poisson equation with dirichlet boundary conditions are presented to demonstrate the accuracy and convergence of the finite difference scheme copyright 2003 john wiley sons ltd

mls moving least square based finite elements for three dimensional nonmatching meshes and adaptive mesh refinement abstract with the aid of moving least square mls approximation a new class of three dimensional finite elements are proposed for treating nonmatching meshes and adaptive mesh refinement for which the existing finite elements are hardly efficient with a special choice of the weight function supports and the base functions the method results in useful elements with the polynomial shape function for which the c 1 continuity breaks down on the boundaries between the neighboring subdomains comprising one element the effectiveness of the new elements in handling the discontinuities due to nonmatching interfaces and automatic mesh refinement is demonstrated via three dimensional examples

element free galerkin methods in combination with finite element approaches element free galerkin efg methods are methods for solving partial differential equations with the help of shape functions coming from moving least squares approximation the efg method is more flexible than the finite element fe method since it requires only nodal data and no element connectivity is needed because the efg method is computationally expensive combinations of the efg method and the fe method are considered implementations based on several weak forms of the problem description are studied numerical examples in elastostatics and fracture mechanics show that the implementations are able to compute both displacement and stress fields accurately

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

mean curvature flows of lagrangian submanifolds with convex potentials this article studies the mean curvature flow of lagrangian submanifolds in particular we prove the following global existence and convergence theorem if the potential function of a lagrangian graph in t 2n is convex then the flow exists for all time and converges smoothly to a flat lagrangian submanifold we also discuss various conditions on the potential function that guarantee global existence and convergence

a mortar finite element method using dual spaces for the lagrange multiplier the mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries in the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of lagrange multipliers in this paper this lagrange multiplier space is replaced by a dual space without losing the optimality of the method the advantage of this new approach is that the matching condition is much easier to realize in particular all the basis functions of the new method are supported in a few elements the mortar map can be represented by a diagonal matrix in the standard mortar method a linear system of equations must be solved the problem is considered in a positive definite nonconforming variational as well as an equivalent saddle point formulation

contracting convex hypersurfaces in riemannian manifolds by their mean curvature which shrink towards the center of the initial sphere in finite time it was shown in 3 that this behaviour is very typical if the initial hypersurface m o o r 1 is uniformly convex then the surfaces m t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction if the ambient space n is a general riemannian manifold the curvature of n will interfere with the mot ion of the surfaces m r we want to show here that the contract ion first to a small sphere and then to a single point is still

length considerations in context free languages abstract in this paper we investigate languages containing at most a bounded number of words of each length we first show that the context free languages for which the number of words of every length is bounded by a fixed polynomial are exactly the bounded context free languages in the sense of ginsburg 1966 thus we present a length characterization for bounded context free languages we then study slender context free languages i e those containing at most a constant number of words of each length recently ilie proved that every such language can be described by a finite union of terms of the form uv i wx i y ilie 1994 we provide a completely different proof of this using constructive methods this enables us to prove that thinness and slenderness are decidable our proofs are based upon a novel characterization of languages in terms of the structure of the infinite paths in their prefix closure this characterization seems to be interesting in itself and can be expanded to more general families of languages

higher order meshless schemes applied to the finite element method in elliptic problems abstract this paper presents selected approximation techniques typical for the meshless finite difference method mfdm although applied to the finite element method fem finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes based upon the correction terms of a simple difference operator those terms consist of higher order derivatives which are evaluated by means of the appropriate formulas composition as well as a numerical solution which corresponds to the primary interpolation order assigned to element shape functions correction terms modify the right hand sides of algebraic fe equations only yielding an iterative procedure therefore neither re generation of the stiffness matrix nor introduction of any additional nodes and or degrees of freedom is required such improved fe mfd solution approach allows for the optimal application of advantages of both methods for instance a high accuracy of the nodal fe solution and a derivatives super convergence phenomenon at arbitrary domain points typical for the meshless fdm existing and proposed higher order techniques applied in the fem are compared with each other in terms of the solution accuracy algorithm efficiency and computational complexity in order to examine the considered algorithms numerical results of several two dimensional benchmark elliptic problems are presented both the accuracy of a solution and the solution s derivatives as well as their convergence rates evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes are taken into account

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

combination of the meshless finite difference approach with the monte carlo random walk technique for solution of elliptic problems abstract this paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations it is based upon the well known monte carlo mc method with a random walk rw technique carried out on the grid of points this method allows for accurate estimation of the solution of the differential equation at selected point s of the domain and or its boundary it extends the standard formulation of the monte carlo random walk mc rw approach by means of its appropriate combination with the meshless version of the finite difference method in this manner the proposed approach may deal with elliptic equations in more general non homogeneous form as well as boundary conditions of both essential and natural types moreover arbitrarily irregular clouds of nodes may be used with no a priori imposed nodes structure therefore the meshless mc rw approach may be applied to the significantly wider class of problems with more complex geometry this concept was examined on variety of 2d boundary value problems selected numerical results are presented and discussed a simple matlab code is included as well

a coupled interpolating meshfree method for computing sound radiation in infinite domain summary in this paper the coupling of the improved interpolating element free galerkin iiefg method and the variable order infinite acoustic wave envelope element wee method is studied a coupled iiefg wee method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages the coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the iiefg and wee domains in the iiefg domain the improved interpolating moving least squares iimls method is employed to form the shape functions satisfying the kronecker delta condition while nonsingular weight functions can be used the impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated the numerical results show that the coupled iiefg wee method can take full advantage of both the iiefg and wee methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the wee the experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain this article is protected by copyright all rights reserved

a novel meshless scheme for solving surface integral equations with flat integral domains numerical solutions for electromagnetic em integral equations rely on the discretization of integral domains and the use of meshes for geometric description meshing geometries is very tedious especially for complicated structures with many details tiny parts and geometric discontinuities corners or edges and remeshing could be required in many scenarios to reduce the costs of generating quality meshes meshless or mesh free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in em community the meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally in this work we first address the traditional meshless scheme for solving em integral equations based on the moving least square mls approximation for unknown currents and the use of background meshes for integrations and then develop a novel meshless scheme by applying the green s lemma to the em surface integral equations with flat domains the novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain since only the domain boundaries are discretized and no background meshes are needed the scheme is truly meshless numerical examples for em scattering by flat surface objects are presented to demonstrate the effectiveness of the novel scheme

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper