error analysis for moving least squares approximation in 2d space abstract in this paper we provide a theoretical analysis of the moving least squares mls approximation which belongs to the family of meshless methods first the non matrix form of the mls shape function in two dimensional space is obtained by using consistency conditions the error estimates for mls approximation in sobolev space are presented when u x y c m 1 and u x y w m 1 q respectively we establish the error estimates for interpolating element free galerkin iefg method when it is used for solving poisson s equation the error bound is related to the radii of the weight functions and the bound of the norm of derivatives of shape functions three numerical examples are selected to confirm our analysis

the homomorphism problem for trace monoids it is proved to be decidable for any given finite subset f of x dependency alphabet y and mapping f y whether or not can be extended to a monoid homomorphism f m y this contrasts with the undecidability of the symmetric problem when we consider the dependency alphabet x instead some other particular cases of homomorphism and isomorphism problems among trace monoids are also discussed

error analysis in sobolev spaces for the improved moving least square approximation and the improved element free galerkin method the improved moving least square imls approximation is a method to form shape functions in meshless methods for the application of imls based meshless methods to the numerical solution of boundary value problems it is fundamental to analyze error of the imls approximation in sobolev spaces this paper begins by discussing properties of the imls shape function under appropriate assumption on weight functions error estimates for the imls approximation are then established in sobolev spaces in multiple dimensions the improved element free galerkin iefg method is a typical meshless galerkin method based on coupling the imls approximation and galerkin weak form error analysis of the iefg method is also provided numerical examples are finally presented to prove the theoretical error results

asymptotic behaviour of a class of stochastic approximation procedures letb be a real banach space considering stochastic approximation procedures of robbins monro and kiefer wolfowitz type one is led to recursion formulas of the form u k 1 u k k 1 gamma k u k k rho v k k eta t k and positive real numbers uk vk tkb valued random variables k random linear operatorsb b where in addition to some further assumptions k almost surely andtk t say almost surely letting y k n k 1 n n n be an array ofb valued random variables such that the partial sums sum limits k 1 m v k and sum limits k 1 m y k n do not differ too much uniformly inm 1 n we shall investigate the distance of u m m 1 n and u n m m 1 n where u k 1 left n right u k left n right k 1 gamma u k left n right k rho y k n k eta t fork 1 n 1 the proofs are based on a deterministic argument that enables us to present a unified approach for proving central limit theorems weak and almost sure invariance principles and variants of the bounded and the functional law of the iterated logarithm for the sequence un

linear numeration systems of order two abstract a numeration system is a sequence of integers such that any integer can be represented by means of the sequence using integers of bounded size we study numeration systems defined by linear recurrences of order two we give a necessary and sufficient condition on the system such that every integer has a canonical representation we show that this canonical representation can be computed from any representation by a rational function this rational function is the composition of two subsequential functions that are simply obtained from the system the addition of two integers represented in the system can be performed by a subsequential machine

modeling of biological population problems using the element free kp ritz method a degenerate parabolic equation arising in the spatial diffusion of biological population is analyzed using the mesh free kp ritz method the mesh free kernel particle estimate is employed to approximate the 2d displacement field a system of discrete equations is obtained through application of the ritz minimization procedure to the energy expressions to validate the accuracy of the results and stability of the present method convergence studies were carried out based on influences of support size and number of nodes effectiveness of the mesh free kp ritz method for biological population model is investigated by numerical examples in this paper the present results were compared with results reported in extant literature and were found to be in good agreement with the literature

error estimates for the interpolating moving least squares method in this paper the interpolating moving least squares imls method presented by lancaster and salkauskas is discussed in details the advantage of the imls method is that the meshless method which is constructed based on the imls method can apply the essential boundary conditions directly and easily a simpler expression of the approximation function of the imls method is obtained then the error estimate of the approximation function and its first and second order derivatives of the imls method are presented in one dimensional case in this paper the theoretical results show that if the order of the polynomial basis functions is big enough and the original function is sufficiently smooth then the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes for the purpose of demonstration some selected numerical examples are given to prove the theories in this paper

a consistently coupled isogeometric meshfree method abstract a consistently coupled isogeometric meshfree method is presented this method takes advantage of the geometry exactness of isogeometric analysis and the refinement flexibility of meshfree method the coupling of isogeometric approximation and meshfree approximation is based upon the reproducing or consistency conditions which are crucial for the coupled method to achieve the expected optimal convergence rates it is shown that unlike the reproducing kernel meshfree shape functions which satisfy the reproducing conditions with the nodal points as the reproducing locations the monomial reproducing points for different orders of b spline basis functions in isogeometric analysis are different and consequently a rational method is proposed to compute these reproducing points both theoretical proof and computational justification of the reproducing conditions for b spline basis functions are given subsequently within the framework of reproducing conditions a mixed reproducing point vector is proposed to ensure arbitrary order monomial reproducibility for both b spline basis functions and reproducing kernel meshfree shape functions which leads to a consistently coupled approximation with smoothing transition between b spline basis functions and reproducing kernel meshfree shape functions consequently a coupled isogeometric meshfree method is established with the galerkin formulation the effectiveness of the proposed coupled isogeometric meshfree method is demonstrated through a series of benchmark numerical examples

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

semigroup automata with rational initial and terminal sets we consider a natural extension of the usual definition of m automata also known as extended automata or valence automata which permits the automaton to utilise more of the structure of each monoid and additionally allows us to define s automata for s an arbitrary semigroup in the monoid case the resulting automata are equivalent to the valence automata with rational target sets which arise in the theory of regulated rewriting systems we study these automata in the case where the register semigroup is completely simple or completely 0 simple obtaining a complete characterisation of the classes of languages corresponding to such semigroups in terms of their maximal subgroups in the process we obtain a number of results about rational subsets of rees matrix semigroups which are likely to be of independent interest

the approximation power of moving least squares a general method for near best approximations to functionals on r d using scattered data information is discussed the method is actually the moving least squares method presented by the backus gilbert approach it is shown that the method works very well for interpolation smoothing and derivatives approximations for the interpolation problem this approach gives mclain s method the method is near best in the sense that the local error is bounded in terms of the error of a local best polynomial approximation the interpolation approximation in r d is shown to be a c function and an approximation order result is proven for quasi uniform sets of data points

jackson type inequalities for h p clouds and error estimates abstract the interest in meshfree methods for solving boundary value problems has grown rapidly in recent years a meshless method that has attracted much interest in the community of computational mechanics is the h p clouds method for this kind of applications it is fundamental to analyze the orders of approximation in this paper we prove jackson type inequalities for h p cloud functions these inequalities set up a general framework for the theoretical analysis of high order error estimates of the h p clouds method with the same remarkable features of finite element theory

a hybrid boundary node method a new variational formulation for boundary node method bnm using a hybrid displacement functional is presented here the formulation is expressed in terms of domain and boundary variables and the domain variables are interpolated by classical fundamental solution while the boundary variables are interpolated by moving least squares mls the main idea is to retain the dimensionality advantages of the bnm and get a truly meshless method which does not require a boundary element mesh either for the purpose of interpolation of the solution variables or for the integration of the energy all integrals can be easily evaluated over regular shaped domains in general semi sphere in the 3 d problem and their boundaries numerical examples presented in this paper for the solution of laplace s equation in 2 d show that high rates of convergence with mesh refinement are achievable and the computational results for unknown variables are most accurate no further integrations are required to compute the unknown variables inside the domain as in the conventional bem and bnm copyright 2001 john wiley sons ltd

basic notions of universal algebra for language theory and graph grammars abstract this paper reviews the basic properties of the equational and recognizable subsets of general algebras these sets can be seen as generalizations of the context free and regular languages respectively this approach based on universal algebra facilitates the development of the theory of formal languages so as to include the description of sets of finite trees finite graphs finite hypergraphs tuples of words partially commutative words also called traces and other similar finite objects

a rbf partition of unity collocation method based on finite difference for initial boundary value problems abstract meshfree radial basis function rbf methods are popular tools used to numerically solve partial differential equations pdes they take advantage of being flexible with respect to geometry easy to implement in higher dimensions and can also provide high order convergence since one of the main disadvantages of global rbf based methods is generally the computational cost associated with the solution of large linear systems in this paper we focus on a localizing rbf partition of unity method rbf pum based on a finite difference fd scheme specifically we propose a new rbf pum fd collocation method which can successfully be applied to solve time dependent pdes this approach allows to significantly decrease ill conditioning of traditional rbf based methods moreover the rbf pum fd scheme results in a sparse matrix system reducing the computational effort but maintaining at the same time a high level of accuracy numerical experiments show performances of our collocation scheme on two benchmark problems involving unsteady convection diffusion and pseudo parabolic equations

on u statistics and v mise statistics for weakly dependent processes some probabilistic limit theorems for hoeffding s u statistic 13 and v mises functional are established when the underlying processes are not necessarily independent we consider absolutely regular processes 24 and processes x n n 1 which are uniformly mixing 14 as well as their time reversal x n n 1 called uniformly mixing in both directions of time many authors have weakened the hypothesis of independence in statistical limit theorems and considered m dependent markov or weakly dependent processes in particular for u statistics under weak dependence sen 22 has considered mixing processes and derived a central limit theorem and a law of the iterated logarithm while yoshihara 26 proved central limit theorems and a s results in the absolutely regular and uniformly mixing case here we extend these results considerably and prove central limit theorems and their rate of convergence in the prohorov metric and a berry esseen type theorem functional central limit theorems and a s approximation by a brownian motion extensions to multisample versions and other extensions are briefly discussed

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

interpolating stabilized moving least squares mls approximation for 2d elliptic interface problems abstract the main aim of the current paper is to propose a new truly meshless numerical technique to solve the one and two dimensional elliptic interface problems the employed meshless approach is based on a new class of mls approximation the numerical procedure is based on the interpolating stabilized mls ismls approximation the new shape functions that have been made by the ismls technique have the kronecker property thus the dirichlet boundary conditions can be applied directly in the current investigation we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ismls technique several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure moreover test problems show the acceptable accuracy and efficiency of the proposed scheme

analysis of moving least squares approximation revisited in this article the error estimation of the moving least squares approximation is provided for functions in fractional order sobolev spaces the analysis presented in this paper extends the previous estimations and explains some unnoticed mathematical details an application to galerkin method for partial differential equations is also supplied

principes d invariance faible pour la mesure empirique d une suite de variables aleatoires melangeante we give weak invariance principles for the empirical measure of a stationary strongly mixing sequence xi k k geqq 0 x n f frac 1 sqrt n sum limits k 1 n f xi k ef xi k for the case where f b s the unit ball of the sobolev space h s x of a riemannian compact manifold and f is a lip function frac 1 2 0 such that mathbb p text mathop text sup limits text x in k text nh n d text text 1 2 hat g n x g x y n x geqq cn a leqq cn a if h n n b finally we give estimates of the kind e mathop sup limits x in k text hat g n x g x 2 leqq cn a text if h n n b here h n is the window of the kernel estimate hat g n

limit theorems for sums of weakly dependent banach space valued random variables we prove an estimate for the prohorov distance in the central limit theorem for strong mixing banach space valued random variables using a recent variant of an approximation theorem of berkes and philipp 1979 we obtain as a corollary a strong invariance principle for absolutely regular sequences with error term t tfrac 1 2 gamma for strong mixing sequences we prove a strong invariance principle with error term o t log logt 1 2

gaussian approximation of suprema of empirical processes we develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of gaussian processes without taking the route of approximating empirical processes themselves in the sup norm we prove an abstract approximation theorem that is applicable to a wide variety of problems primarily in statistics especially the bound in the main approximation theorem is non asymptotic and the theorem does not require uniform boundedness of the class of functions the proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors the proof of which depends on an effective use of stein s method for normal approximation and some new empirical processes techniques we study applications of this approximation theorem to local empirical processes and series estimation in nonparametric regression where the classes of functions change with the sample size and are not donsker type importantly our new technique is able to prove the gaussian approximation for the supremum type statistics under considerably weak regularity conditions especially concerning the bandwidth and the number of series functions in those examples

an improved nurbs based isogeometric analysis with enhanced treatment of essential boundary conditions the nurbs based isogeometric analysis offers a seamless integration between the cad and subsequent finite element analysis and has been shown to be very effective for wide classes of problems due to its accurate geometric description however similar to the moving least square or reproducing kernel meshfree shape functions the nurbs basis functions generally are not interpolatory functions in this work it is shown that the direct imposition of the inhomogeneous essential boundary conditions to the nurbs control points may lead to significant errors with deteriorated rates of convergence consequently an improved formulation for nurbs based isogeometric analysis is proposed this is fulfilled by employing a transformation method that relates the control variables to the collocated nodal values at the essential boundary by using open knot vectors the resulting nurbs basis functions associated with the interior control points vanish at the boundary thus unlike the meshfree approximation the nurbs control points can be clearly partitioned into boundary and interior groups therefore the transformation method can be only applied to the boundary control points via invoking collocation of physical values at a set of boundary points the nurbs approximation enhanced with transformed physical boundary variables is kinematically admissible and the essential boundary conditions can be enforced straightforwardly like the finite element method several potential and elasticity problems evince that much higher solution accuracy with optimal convergence rates can be achieved by the present improved formulation compared with the method with direct imposition of essential boundary conditions to control variables

small overlap monoids i the word problem abstract we develop a combinatorial approach to the study of semigroups and monoids with finite presentations satisfying small overlap conditions in contrast to existing geometric methods our approach facilitates a sequential left right analysis of words which lends itself to the development of practical efficient computational algorithms in particular we obtain a highly practical linear time solution to the word problem for monoids and semigroups with finite presentations satisfying the condition c 4 and a polynomial time solution to the uniform word problem for presentations satisfying the same condition

a matrix triangularization algorithm for the polynomial point interpolation method a novel matrix triangularization algorithm mta is proposed to overcome the singularity problem in the point interpolation method pim using the polynomial basis and to ensure stable and reliable construction of pim shape functions the present algorithm is validated using several examples and implemented in the local point interpolation method lpim that is a truly meshfree method based on a local weak form numerical examples demonstrate that lpim using the present mta are very easy to implement and very robust for solving problems of computational mechanics it is shown that pim with the present mta is very effective in constructing shape functions most importantly pim shape functions possess kronecker delta function properties parameters that influence the performance of them are studied in detail the convergence and efficiency of them are thoroughly investigated

which kleene semigroups are finite the set of idempotents in any given d class of a kleene semigroup is rational we use simon s theorem to obtain some finitude results

boundary element free method befm and its application to two dimensional elasticity problems in this study we first discuss the moving least square approximation mls method in some cases the mls may form an ill conditioned system of equations so that the solution cannot be correctly obtained hence in this paper we propose an improved moving least square approximation imls method in the imls method the orthogonal function system with a weight function is used as the basis function the imls has higher computational efficiency and precision than the mls and will not lead to an ill conditioned system of equations combining the boundary integral equation bie method and the imls approximation method a direct meshless bie method the boundary element free method befm for two dimensional elasticity is presented compared to other meshless bie methods befm is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables and the boundary conditions can be applied easily hence it has higher computational precision for demonstration purpose selected numerical examples are given

complex variable moving least squares method a meshless approximation technique based on the moving least squares mls approximation we propose a new approximation method the complex variable moving least squares cvmls approximation with the cvmls approximation the trial function of a two dimensional problem is formed with a one dimensional basis function the number of unknown coefficients in the trial function of the cvmls approximation is less than in the trial function of the mls approximation and we can thus select fewer nodes in the meshless method that is formed from the cvmls approximation than are required in the meshless method of the mls approximation with no loss of precision the meshless method that is derived from the cvmls approximation also has a greater computational efficiency from the cvmls approximation we propose a new meshless method for two dimensional elasticity problems the complex variable meshless method cvmm and the formulae of the cvmm for two dimensional elasticity problems are obtained compared with the conventional meshless method the cvmm has a greater precision and computational efficiency for the purposes of demonstration some selected numerical examples are solved using the cvmm

finite transducers for divisibility monoids divisibility monoids are a natural lattice theoretical generalization of mazurkiewicz trace monoids namely monoids in which the distributivity of the involved divisibility lattices is kept as an hypothesis but the relations between the generators are not supposed to necessarily be commutations here we show that every divisibility monoid admits an explicit finite transducer which allows to compute normal forms in quadratic time in addition we prove that every divisibility monoid is biautomatic

an adaptive orthogonal improved interpolating moving least square method and a new boundary element free method abstract the improved interpolating moving least square iimls method has been widely used in data fitting and meshfree methods and the obtained shape functions have the property of the delta function compared with those obtained by the moving least square mls method however the moment matrix in iimls may be singular or ill conditioned because of the ill quality of the point sets used in this paper the weighted orthogonal basis functions are applied in iimls to obtain a diagonal moment matrix which can overcome the difficulty caused by directly inversing singular or ill conditioned matrices however the weighted orthogonal basis functions cannot change the nature of the singular or ill conditioned moment matrix since the diagonal elements of the new moment matrix may be zero or close to zero thus an adaptive scheme is further employed to resolve this problem by ignoring the contribution from the zero or very small diagonal elements in the diagonal moment matrix combined with shifted and scaled polynomial basis functions a stabilized adaptive orthogonal iimls sao iimls approximation is obtained based on this approximation a new boundary element free method is proposed for solving elasticity problems numerical results for curve fitting surface fitting and the new boundary element free method have shown that the proposed sao iimls approximation is accurate stable and performs well for ill quality point sets

meshless techniques for anisotropic diffusion abstract meshless local petrov galerkin mlpg methods are pure meshless techniques for solving partial differential equations they have been successfully used for solving many real life problems their efficiency is downgraded by the requirement of performing numerical multi dimensional integration of tricky non polynomial factors recently direct mlpg dmlpg methods have been proposed dmlpg techniques require lower computational costs with respect to their mlpg counterparts the dmlpg accuracy has been initially analyzed in few papers but its performance is quite unexplored in this paper we perform numerical comparisons between mlpg and dmlpg accuracy and efficiency in solving anisotropic diffusion problems in particular we set different boundary conditions in order to check if and when mlpg and or dmlpg suffer locking effects

strong approximations of semimartingales by processes with independent increments strong approximation theorems for continuous time semimartingales are obtained by combining some techniques of the general theory of stochastic processes with some of the direct approximation of dependent random variables by independent ones continuous processes with independent increments whose variance functions increase polynomially or exponentially are considered as approximating processes the basic assumptions of the main results only contain rates of convergence for certain probabilities in particular moment assumptions are not required some almost sure invariance principles for partial sum processes with nonlinear growth of variance and for functionals of markov processes are derived by applying the main results

small overlap monoids ii automatic structures and normal forms abstract we show that any finite monoid or semigroup presentation satisfying the small overlap condition c 4 has word problem which is a deterministic rational relation it follows that the set of lexicographically minimal words forms a regular language of normal forms and that these normal forms can be computed in linear time we also deduce that c 4 monoids and semigroups are rational in the sense of sakarovitch asynchronous automatic and word hyperbolic in the sense of duncan and gilman from this it follows that c 4 monoids satisfy analogues of kleene s theorem and admit decision algorithms for the rational subset and finitely generated submonoid membership problems we also prove some automata theoretic results which may be of independent interest

renewal theory properties of ion channels this paper uses a type of data to attack the problem of deciding whether or not the channels in a membrane are operating independently and if so estimating the distribution of the sojourn periods in emitting and nonomitting states

on the stability of the moving least squares approximation and the element free galerkin method in this paper the stability of the moving least squares mls approximation and a stabilized mls approximation is analyzed theoretically and verified numerically it is shown that the stability of the mls approximation deteriorates severely as the nodal spacing decreases while the stability of the stabilized mls approximation is not affected by the nodal spacing the stabilized mls approximation is introduced into the element free galerkin efg method to produce a stabilized efg method theoretical error analysis of the stabilized efg method is provided for boundary value problems with mixed boundary conditions of dirichlet and robin type numerical examples confirm the theoretical results and show that the stabilized efg method has higher computational precision and better stability than the original efg method

easy multiplications i the realm of kleene s theorem abstract we study here the properties of a family of monoids which we call the rational monoids and which are monoids with a multiplication of low complexity a monoid is rational if its multiplication may be described by a rational function from a free monoid into itself the main results are that rational monoids like finite ones have the properties that green s relations d and f are equal and that kleene s theorem holds in rational monoids as in free ones every monoid described so far in which kleene s theorem holds is a rational monoid the closure of the family of rational monoids under rees quotient direct product and free product is then studied extensions of rational monoids will be considered in a forthcoming paper

a two dimensional isogeometric boundary element method for elastostatic analysis the concept of isogeometric analysis where functions that are used to describe geometry in cad software are used to approximate the unknown fields in numerical simulations has received great attention in recent years the method has the potential to have profound impact on engineering design since the task of meshing which in some cases can add significant overhead has been circumvented much of the research effort has been focused on finite element implementations of the isogeometric concept but at present little has been seen on the application to the boundary element method the current paper proposes an isogeometric boundary element method bem which we term igabem applied to two dimensional elastostatic problems using non uniform rational b splines nurbs we find it is a natural fit with the isogeometric concept since both the nurbs approximation and bem deal with quantities entirely on the boundary the method is verified against analytical solutions where it is seen that superior accuracies are achieved over a conventional quadratic isoparametric bem implementation

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

error analysis of the reproducing kernel particle method interest in meshfree or meshless methods has grown rapidly in recent years in solving boundary value problems bvps arising in mechanics especially in dealing with difficult problems involving large deformation moving discontinuities etc in this paper we provide a theoretical analysis of the reproducing kernel particle method rkpm which belongs to the family of meshfree methods one goal of the paper is to set up a framework for error estimates of rkpm we introduce the concept of a regular family of particle distributions and derive optimal order error estimates for rkp interpolants on a regular family of particle distributions the interpolation error estimates can be used to yield error estimates for rkp solutions of bvps

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

a new implementation of the element free galerkin method abstract element free galerkin methods efg are methods for solving partial differential equations with moving least squares interpolants efg methods require only nodal data no element connectivity is needed in a previous implementation of the efg method lagrange multipliers were used to enforce the essential boundary condition however the use of lagrange multipliers increases the cost of solving the linear algebraic equations a new implementation is developed based on a modified variational principle in which the lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded in addition weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes the latter makes it very advantageous for modelling progressive cracking

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

improved rates and asymptotic normality for nonparametric neural network estimators we obtain an improved approximation rate in sobolev norm of r sup 1 2 spl alpha d 1 for a large class of single hidden layer feedforward artificial neural networks ann with r hidden units and possibly nonsigmoid activation functions when the target function satisfies certain smoothness conditions here d is the dimension of the domain of the target function and spl alpha spl isin 0 1 is related to the smoothness of the activation function when applying this class of anns to nonparametrically estimate train a general target function using the method of sieves we obtain new root mean square convergence rates of op n log n sup 1 2 spl alpha d 1 4 1 spl alpha d 1 op n sup 1 4 by letting the number of hidden units spl tau sub n increase appropriately with the sample size number of training examples n these rates are valid for i i d data as well as for uniform mixing and absolutely regular spl beta mixing stationary time series data in addition the rates are fast enough to deliver root n asymptotic normality for plug in estimates of smooth functionals using general ann sieve estimators as interesting applications to nonlinear time series we establish rates for ann sieve estimators of four different multivariate target functions a conditional mean a conditional quantile a joint density and a conditional density we also obtain root n asymptotic normality results for semiparametric model coefficient and average derivative estimators

slightly commutative kleene semigroups abstract commutative kleene semigroups are known to be rational but pelletier constructed a nonrational weakly commutative kleene semigroup we introduce slightly commutative kleene semigroups a class of weakly commutative kleene semigroups and prove that every slightly commutative kleene semigroup is rational

the boundary node method for potential problems the element free galerkin efg method allows one to use a nodal data structure usually with an underlying cell structure within the domain of a body of arbitrary shape the usual efg combines moving least squares mls interpolants with a variational principle weak form and has been used to solve two dimensional 2 d boundary value problems in mechanics such as in potential theory elasticity and fracture this paper proposes a combination of mls interpolants with boundary integral equations bie in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter this new method called the boundary node method bnm only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself an underlying cell structure is again used for numerical integration in principle the bnm for 3 d problems should be extremely powerful since one would only need to put nodes points on the surface of a solid model for an object numerical results are presented in this paper for the solution of laplace s equation in 2 d dirichlet neumann and mixed problems have been solved some on bodies with piecewise straight and others with curved boundaries results from these numerical examples are extremely encouraging 1997 by john wiley sons ltd

the meshless galerkin boundary node method for stokes problems in three dimensions the galerkin boundary node method gbnm is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least squares mls approximations for generating the trial and test functions in this approach boundary conditions can be implemented directly and easily despite of the fact that the mls shape functions lack the delta function property besides the resulting formulation inherits the symmetry and positive definiteness of the variational problems the gbnm is developed in this paper for solving three dimensional stationary incompressible stokes flows in primitive variables the numerical scheme is based on variational formulations for the first kind integral equations which are valid for both interior and exterior problems simultaneously a rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in sobolev spaces the capability of the method is also illustrated and assessed through some selected numerical examples copyright 2011 john wiley sons ltd

regularized moving least square method and regularized improved interpolating moving least square method with nonsingular moment matrices abstract the moving least square mls method has been popular applied in surface construction and meshless methods however the moment matrix in mls method may be singular for ill quality point sets and the computation of the inverse of the singular moment matrix is difficult to overcome this problem a regularized moving least square method with nonsingular moment matrix is proposed the shape functions obtained from the regularized mls method still do not have the delta function property and may result in difficulty for imposing boundary conditions in regularized mls based meshless method to overcome this problem a regularized improved interpolating moving least square iimls method based on the iimls method is also proposed compared with the regularized mls method the regularized iimls not only has nonsingular moment matrices but also obtains shape functions with delta function property shape functions of the proposed methods are compared in 1d and 2d cases and the methods have been applied in curve fitting surface fitting and meshless method in numerical examples

easy multiplications ii extensions of rational semigroups abstract in our first part of this paper we defined and studied the family of rational monoids which are monoids with an easy multiplication by this we mean indeed that the multiplication can be realized by a finite automaton in this second part of our work we consider rational semigroups instead of monoids and investigate the properties of two operations on these semigroups the redei extensions and the ideal extensions we give necessary and sufficient conditions for such extensions of rational semigroups to be rational semigroups the theory of ideal extensions is also used to build a non rational monoid in which kleene s theorem holds which answers a question asked in the first part a last application of our work on redei and ideal extensions is the construction of a family of monoids too simple to be syntactic monoids of context free languages

singular boundary method for solving plane strain elastostatic problems this study documents the first attempt to apply the singular boundary method sbm a novel boundary only collocation method to two dimensional 2d elasticity problems unlike the method of fundamental solutions mfs the source points coincide with the collocation points on the physical boundary by using an inverse interpolation technique to regularize the singularity of the fundamental solution of the equation governing the problems of interest three benchmark elasticity problems are tested to demonstrate the feasibility and accuracy of the proposed method through detailed comparisons with the mfs boundary element method bem and finite element method fem

a galerkin boundary node method and its convergence analysis the boundary node method bnm exploits the dimensionality of the boundary integral equation bie and the meshless attribute of the moving least square mls approximations however since mls shape functions lack the property of a delta function it is difficult to exactly satisfy boundary conditions in bnm besides the system matrices of bnm are non symmetric a galerkin boundary node method gbnm is proposed in this paper for solving boundary value problems in this approach an equivalent variational form of a bie is used for representing the governing equation and the trial and test functions of the variational formulation are generated by the mls approximation as a result boundary conditions can be implemented accurately and the system matrices are symmetric total details of numerical implementation and error analysis are given for a general bie taking the dirichlet problem of laplace equation as an example we set up a framework for error estimates of gbnm some numerical examples are also given to demonstrate the efficacity of the method

a boundary face method for potential problems in three dimensions this work presents a new implementation of the boundary node method bnm for numerical solution of laplace s equation by coupling the boundary integral equations and the moving least squares mls approximation the bnm is a boundary type meshless method however it still uses the standard elements for boundary integration and approximation of the geometry thus loses the advantages of the meshless methods in our implementation here called the boundary face method the boundary integration is performed on boundary faces which are represented in parametric form exactly as the boundary representation data structure in solid modeling the integrand quantities such as the coordinates of gauss integration points jacobian and out normal are calculated directly from the faces rather than from elements in order to deal with thin structures a mixed variable interpolation scheme of 1 d mls and lagrange polynomial for long and narrow faces an adaptive integration scheme for nearly singular integrals has been developed numerical examples show that our implementation can provide much more accurate results than the bnm and keep reasonable accuracy in some extreme cases such as very irregular distribution of nodes and thin shells copyright 2009 john wiley sons ltd

a posteriori error estimates for nonlinear problems l r o t p ω error estimates for finite element discretizations of parabolic equations using the abstract framework of 9 we analyze a residual a posteriori error estimator for space time finite element discretizations of quasilinear parabolic pdes the estimator gives global upper and local lower bounds on the error of the numerical solution the finite element discretizations in particular cover the so called 0 scheme which includes the implicit and explicit euler methods and the crank nicholson scheme

invariance principles for sums of banach space valued random elements and empirical processes almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable banach space it is then shown that empirical processes readily fit into this general framework thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes

law of the iterated logarithm for a constant gain linear stochastic gradient algorithm we study almost sure limiting properties taken as varepsilon searrow 0 of the finite horizon sequence of random estimates theta 0 varepsilon theta 1 varepsilon theta 2 varepsilon ldots theta left lfloor t varepsilon right rfloor varepsilon for the linear stochastic gradient algorithm theta varepsilon n 1 theta varepsilon n varepsilon left a n 1 theta varepsilon n prime x n 1 right x n 1 theta 0 varepsilon mbox stackrel triangle theta mbox nonrandom noindent where t in 0 infty is an arbitrary constant varepsilon in 0 1 is a small adaptation gain and an and xn are data sequences which drive the algorithm these limiting properties are expressed in the form of a functional law of the iterated logarithm