evaluation of the integral terms in reproducing kernel methods reproducing kernel method rkm has its origins in wavelets and it is based on convolution theory being their continuous version rkm is often referred as the general framework for meshless methods in fact since in real computation discretization is inevitable these integrals need to be evaluated numerically leading to the creation of reproducing kernel particle method rkpm and moving least squares mls approximation nevertheless in this paper the integrals in rkm are explicitly evaluated for polynomials basis function and simple geometries in one two and three dimensions even with conforming holes moreover a general formula is provided for complicated shapes also for multiple connected domains this is possible through a boundary formulation where domain integrals involved in rkm are transformed by gauss theorem in circular or flux integrals parallelization is readily enabled since no preliminary arrangements of nodes is needed for the moments matrix furthermore using symbolic inversion computation of shape functions in rkm is considerably speeded up

free vibration analysis of plates using least square based finite difference method in this paper we apply the two dimensional least square based finite difference lsfd method for solving free vibration problems of isotropic thin arbitrarily shaped plates with simply supported and clamped edges using the chain rule we show how the fourth order derivatives of the plate governing equation can be discretized in two or three steps as well as how the boundary conditions can be implemented directly into the governing equation by analyzing vibrating plates of various shapes and comparing the solutions obtained against existing results we clearly demonstrate the effectiveness of lsfd as a mesh free method for computing vibration frequencies of generally shaped plates accurately

fracture mechanics analysis using the wavelet galerkin method and extended finite element method summary this paper presents fracture mechanics analysis using the wavelet galerkin method and extended finite element method the wavelet galerkin method is a new methodology to solve partial differential equations where scaling wavelet functions are used as basis functions in solid structural analyses the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain to improve accuracy wavelet functions are superposed on the scaling functions within a region having a high stress concentration such as near a hole or notch thus the method can be considered a refinement technique in fixed grid approaches however because the basis functions are assumed to be continuous in applications of the wavelet galerkin method there are difficulties in treating displacement discontinuities across the crack surface in the present research we introduce enrichment functions in the wavelet galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method this paper presents the mathematical formulation and numerical implementation of the proposed technique as numerical examples stress intensity factor evaluations and crack propagation analyses for two dimensional cracks are presented copyright 2012 john wiley sons ltd

a unified approach to the mathematical analysis of generalized rkpm gradient rkpm and gmls abstract it is well known that the conventional reproducing kernel particle method rkpm is unfavorable when dealing with the derivative type essential boundary conditions 1 2 3 to remedy this issue a group of meshless methods in which the derivatives of a function can be incorporated in the formulation of the corresponding interpolation operator will be discussed formulation of generalized moving least squares gmls on a domain and gmls on a finite set of points will be presented the generalized rkpm will be introduced as the discretized form of gmls on a domain another method that helps to deal with derivative type essential boundary conditions is the gradient rkpm which incorporates the first gradients of the function in the reproducing equation in present work the formulation of gradient rkpm will be derived in a more general framework some important properties of the shape functions for the group of methods under consideration are discussed moreover error estimates for the corresponding interpolants are derived by generalizing the concept of corrected collocation method it will be seen that in the case of employing each of the proposed methods to a bvp not only the essential boundary conditions involving the function but also the essential boundary conditions which involve the derivatives could be satisfied exactly at particles which are located on the boundary

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

high order central difference scheme for caputo fractional derivative abstract in this paper we propose a class of central difference schemes for resolving the caputo fractional derivative the accuracy may reach any selected integer order more precisely the caputo fractional derivative operator is decomposed into symmetric and antisymmetric components starting from difference schemes of lower order accuracy for each component we enhance the accuracy by a weighted average of shifted differences the weights are calculated by matching the symbols of the scheme and the operators we further illustrate the application of the proposed schemes to a fractional advection diffusion equation together with the crank nicolson algorithm it reaches designed accuracy order and is unconditionally stable numerical tests are presented to demonstrate the nice features

the invariant charges of the nambu goto string and canonical quantization it is shown that the algebra of diffeomorphism invariant charges of the nambu goto string cannot be quantized in the framework of canonical quantization the argument is shown to be independent of the dimension of the underlying minkowski space

extra dof free and linearly independent enrichments in gfem abstract a gfem without extra dof is developed without the extra dof the resulting linear system becomes independent in size of the order of local approximation the resulting pu approximation becomes free of linear dependence and the resulting stiffness matrix becomes good conditioned numerical studies show the new gfem s excellent convergence properties and excellent stability

scattering states of plektons particles with braid group statistics in 2 1 dimensional quantum field theory a haag ruelle scattering theory for particles with braid group statistics is developed and the arising structure of the hilbert space of multiparticle states is analyzed

moving least square reproducing kernel method part ii fourier analysis in part i of this work the moving least square reproducing kernel mlsrk method is formulated and implemented based on its generic construction an m consistency structure is discovered and the convergence theorems are established in this part of the work a systematic fourier analysis is employed to evaluate and further establish the method the preliminary fourier analysis reveals that the mlsrk method is stable for sufficiently dense non degenerated particle distribution in the sense that the kernel function family satisfies the riesz bound one of the novelties of the current approach is to treat the mlsrk method as a variant of the standard finite element method and depart from there to make a connection with the multiresolution approximation in the spirits of multiresolution analysis we propose the following mlsrk transformation fm k hu i 1npu k ik h x xi x w1 the highlight of this paper is to embrace the mlsrk formulation with the notion of the controlled fp approximation based on its characterization the strang fix condition for example a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the mlsrk algorithm the main effort here is to obtain a constant correction function in the interior region of a general domain i e c h 1 this can create a leap in the approximation order of the mlsrk algorithm significantly if a highly smooth window function is embedded within the kernel one consequence of this development is the synchronized convergence phenomenon a unique convergence mechanism for the mlsrk method i e by properly tuning the dilation parameter the convergence rate of higher order error norms will approach the same order convergence rate of the l2 error norm they are synchronized

reproducing kernel element method part iv globally compatible cn n 1 triangular hierarchy in this part of the work a globally compatible c n x triangular element hierarchy is constructed in the framework of reproducing kernel element method rkem for arbitrary two dimensional domains in principle the smoothness of the globally conforming element can be made arbitrarily high np 1 the triangle interpolation field can interpolate the derivatives of an unknown function up to arbitrary mth order i m and it can reproduce complete kth order polynomials with k p m this is the first interpolation hierarchical structure that has ever been constructed with both minimal degrees of freedom and higher order smoothness and continuity over discretizations of a multiple dimensional domain the performance of the newly constructed compatible element is evaluated in solving several kirchhoff plate problems

arbitrarily smooth generalized finite element approximations this paper presents a procedure to build ck k arbitrarily large generalized finite element fe shape functions defined on non structured finite element meshes the functions have the same support as corresponding global fe lagrangian shape functions meshes with both convex and non convex clouds set of elements sharing a vertex node can be used the so called r functions are used to build ck fe based partition of unity functions with non convex support a technique to combine c0 lagrangian fe shape functions with the proposed ck partition of unity is presented the technique allows the use of ck generalized fe shape functions in parts of the computational domain where their high smoothness is required as in the case of problems with distributional boundary conditions and the less computationally demanding c0 generalized fe shape functions elsewhere in the domain a linear elasticity problem with a concentrated moment is solved using the proposed ck generalized fe method higher order distributional boundary conditions can also be handled by the method a detailed convergence analysis is presented for this class of problems as well as for problems in energy space the integrability of the functions using standard gauss legendre rules is also investigated

geometry representation modification and iterative design using rkem we present a new method for representing smooth geometry using the reproducing kernel element method further the formulation provides a simple method to modify geometry that does not require modification of the underlying rkem mesh we develop a method for determining the representation from a discrete point set that is applicable to both traditional engineering domains as well as applications where only discrete point sets are available such as biology and medicine geometry representation and modification are demonstrated using examples of analytic functions and on a digitized shark tooth finally we develop an isogeometric analysis capability for modified and un modified geometry the isogeometric capability on modified geometry may be useful in an iterative design analyze cycle or shape optimization

string localized quantum fields and modular localization we study free covariant quantum bose fields that are associated with irreducible representations of the poincare group and localized in semi infinite strings extending to spacelike infinity among these are fields that generate the irreducible representations of mass zero and infinite spin that are known to be incompatible with point like localized fields for the massive representations and the massless representations of finite helicity all string localized free fields can be written as an integral along the string of point localized tensor or spinor fields as a special case we discuss the string localized vector fields associated with the point like electromagnetic field and their relation to the axial gauge condition in the usual setting

a jost schroer theorem for string fields it is shown that the truncated wightman functions of three or more string localized fields vanish if they are solutions of a klein gordon equation in each variable as an application it is shown that a string field is a free field if its two point functions are those of a free field another application to perturbation theory is pointed out

new conditions for non uniform behaviour of linear cocycles over flows abstract we give a characterization of tempered exponential dichotomies for linear cocycles over flows in terms of the spectral properties of certain linear operators we consider noninvertible linear cocycles acting on infinite dimensional spaces and our approach avoids the use of lyapunov norms finally we apply obtained results to give new conditions for uniform exponential stability of linear cocycles

element free galerkin approach based on the reproducing kernel particle method for solving 2d fractional tricomi type equation with robin boundary condition the traditional element free galerkin efg approach is constructed on variational weak form that the test and trial functions are shape functions of moving least squares mls approximation in the current paper we propose a new version of the efg method based on the shape functions of reproducing kernel particle method rkpm in other words based on the developed approach in han and meng 2001 the fractional tricomi type equation will be solved using the new technique the fractional derivative has been introduced in the caputos sense and is approximated by a finite difference plan of order o 3 1 2 we use the efg rkpm to discrete the spatial direction we illustrate some numerical results on non rectangular domains the unconditional stability and convergence of the new technique have been proved numerical examples display the theoretical results and the efficiency of the proposed approach also the numerical results are compared with the finite element method fem and efg mls procedure

string and brane localized causal fields in a strongly nonlocal model we study a weakly local but nonlocal model in spacetime dimension d 2 and prove that it is maximally nonlocal in a certain specific quantitative sense nevertheless depending on the number of dimensions d it has string localized or brane localized operators which commute at spatial distances in two spacetime dimensions the model even comprises a covariant and local subnet of operators localized in bounded subsets of minkowski space which has a nontrivial scattering matrix the model thus exemplifies the algebraic construction of local operators from algebras associated with nonlocal fields

adaptivity for structured meshfree particle methods in 2d and 3d the implementation of h adaptivity for meshfree particle methods within a structured framework is described in this framework the initial particle arrangement is structured along with a background mesh and outside boundaries and interior interfaces are described by implicit functions the advantage of meshfree approximations in this framework lies in the ease of implementing h adaptivity and the simplicity of the data structures particles can easily be added and removed without complications in the data structure although there are some issues in the quadrature an a posteriori error estimation is used for the adaptive refinement an adaptive refinement strategy is applied to several linear elastic problems with high stress and strain gradients and singularities several non linear examples are also given copyright 2005 john wiley sons ltd

discretization errors associated with reproducing kernel methods one dimensional domains abstract the reproducing kernel particle method rkpm is a discretization technique for partial differential equations that uses the method of weighted residuals classical reproducing kernel theory and modified kernels to produce either mesh free or mesh full methods although rkpm has many appealing attributes the method is new and its numerical performance is just beginning to be quantified in order to address the numerical performance of rkpm von neumann analysis is performed for semi discretizations of three model one dimensional pdes the von neumann analyses results are used to examine the global and asymptotic behavior of the semi discretizations the model pdes considered for this analysis include the parabolic and hyperbolic first and second order wave equations numerical diffusivity for the former and phase speed for the latter are presented over the range of discrete wavenumbers and in an asymptotic sense as the particle spacing tends to zero group speed is also presented for the hyperbolic problems excellent diffusive and dispersive characteristics are observed when a consistent mass matrix formulation is used with the proper choice of refinement parameter in contrast the row sum lumped mass matrix formulation severely degraded performance the asymptotic analysis indicates that very good rates of convergence o x 6 o x 8 are possible when the consistent mass matrix formulation is used with an appropriate choice of refinement parameter and quadrature rule

reproducing kernel element method part ii globally conforming im cn hierarchies in this part of the work a minimal degrees of freedom arbitrary smooth globally compatible i m c n interpolation hierarchy is constructed in the framework of reproducing kernel element method rkem for arbitrary multiple dimensional domains this is the first interpolation hierarchical structure that has been constructed with both minimal degrees of freedom and higher order smoothness or continuity over multi dimensional domain the proposed hierarchical structure possesses the generalized kronecker property i e o a w b i ox a x j dijdab jaj jbj 6 m

asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic navier equation abstract in this work we study reproducing kernel rk collocation method for peridynamic navier equation in the first part we apply a linear rk approximation to both displacement and dilatation and then back substitute dilatation and solve the peridynamic navier equation in a pure displacement form the rk collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish the stability is shown by comparing the collocation scheme with the standard galerkin scheme using fourier analysis in the second part we apply the rk collocation to the quasi discrete peridynamic navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed the analysis is carried out on a special family of rectilinear cartesian grids for the rk collocation method with a designated kernel with finite support we assume the lame parameters satisfy to avoid extra assumptions on the nonlocal kernel finally numerical experiments are conducted to validate the theoretical results

consistent pseudo derivatives in meshless methods a meshless petrov galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of shepard functions a key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo derivatives for petrov galerkin method which pass the patch test numerical results show that the resulting method is substantially more accurate than the galerkin method with shepard approximants and exceeds the rate of convergence of linear finite elements

maximum principle and convergence analysis for the meshfree point collocation method the discrete laplacian operator is considered in the sense of the meshfree point collocation method which will be called the strong meshfree laplacian operator to define the strong meshfree laplacian operator we use the fast version of the generalized moving least square approximation which can calculate the approximated derivatives of shape functions some types of the locally layered node distribution are defined in this paper and two specific domains are constructed onto which we can distribute locally layered nodes at such nodes the discrete maximum principle can be shown to hold through the representation formula for the strong meshfree laplacian operator the discrete maximum principle together with the reproducing property of the meshfree approximations results in an a priori estimate for the strong meshfree laplacian operator in the nodal solution space furthermore the a priori estimate we have obtained guarantees the existence and the uniqueness of the numerical solution and plays a central role in achieving converged results for the poisson problem with dirichlet boundary conditions in the nodal solution space the order of convergence of the nodal solutions can be raised up to o h2 at the proposed type of nodes in specific domains for generally shaped domains immersed in the previously mentioned domains we can obtain the first order convergence result of o h

reproducing kernel triangular b spline based fem for solving pdes abstract we propose a reproducing kernel triangular b spline based finite element method fem as an improvement to the conventional triangular b spline element for solving partial differential equations pdes in the latter unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the b spline the performance therefore becomes unstable and cannot be controlled in a desirable way to address this issue the proposed improvement adopts the reproducing kernel approximation in the calculation of b spline kernel function three types of pde problems are tested to validate the present element and compare against the conventional triangular b spline it has been shown that the improved triangular b spline satisfies the partition of unity condition even for extreme conditions including corners and holes

gradient enhanced damage modeling of cracked bodies by reproducing kernel element method abstract in crack modeling by extended finite element method the enrichment by the asymptotic analytical solution works excellent for linear elastic fracture mechanics however for the more sophisticated condition of a gradient enhanced damage model firstly the analytical solution is not available and secondly higher order continuity of the shape functions is required in this study the extended finite element model of the cracked body is augmented with shape functions of higher order continuity and higher order consistency generated by reproducing kernel element method they are used in a region near the crack tip to have an appropriate interpolation of the damaged zone while for the rest of the solution domain standard shape functions are used to reduce the computational cost no new nodes are needed when defining the higher order shape functions by reproducing kernel element method and thus remeshing is avoided during the crack growth after describing the formulations several numerical examples are implemented to investigate the ability of the proposed method for both the linear elastic fracture mechanics and the gradient enhanced damage models

construction of quantum field theories with factorizing s matrices the subject of this thesis is a novel construction method for interacting relativistic quantum field theories on two dimensional minkowski space employing the algebraic framework of quantum field theory it is shown under which conditions an algebra of observables localized in a wedge shaped region of spacetime can be used to construct model theories a crucial input in this context is the modular nuclearity condition for wedge algebras which implies the existence of local observables as an application of the new method a rigorous construction of a large family of models with factorizing s matrices is obtained in an inverse scattering approach a given factorizing scattering operator is used to define certain semi localized wightman fields associated to it with the help of these fields a wedge algebra can be defined which determines the local observable content of a well defined quantum field theory in this approach the modular nuclearity condition translates to certain analyticity and boundedness conditions on the formfactors of wedge local observables these conditions are shown to hold for a large class of underlying s matrices including the scattering operators of the sinh gordon model and the scaling ising model as special examples the so constructed models are investigated with respect to their scattering properties they are shown to solve the inverse scattering problem for the underlying s matrices and a proof of asymptotic completeness for these models is given

an extension of a theorem of r datko to the case of non uniform exponential stability of linear skew product semiflows abstract in 1970 r datko proved that the trajectories of a c 0 semigroup t t t 0 on a hilbert space x exhibit an exponential decay if and only if they stay in l 2 r x datko s theorem was crucial in extending the lyapunov operator equation to the case of autonomous systems x a x with unbounded a extensions have been done to the case of uniform exponential stability of evolution families and more recently of lsps linear skew product semiflow the aim of the present paper is to extend datko s result to the general case of non uniform exponential stability of the lsps that does not necessarily possess a uniform exponential growth

locality in free string field theory free string field operators are constructed for the open bosonic string in the light cone gauge in any dimension these are naturally localized by the center of mass coordinate relative to this localization they are shown to have a causal commutator provided there are no tachyons for the critical string in d 26 the result still holds if the tachyon is suppressed we also show a causal commutator relative to the string light cone

the generalized product partition of unity for the meshless methods the partition of unity is an essential ingredient for meshless methods named by gfem pufem partition of unity fem xfem extended fem rkpm reproducing kernel particle method rppm reproducing polynomial particle method the method of hp clouds in the literature there are two popular choices for partition of unity a piecewise linear fem mesh and the shepard type partition of unity however the partition of unity pu by a fem mesh leads to the singular or nearly singular matrices and non smooth approximation functions the shepard type partition of unity requires lengthy computing time and its implementation is difficult in order to alleviate these difficulties oh et al introduced the smooth piecewise polynomial pu functions with flat top that lead to small matrix condition numbers and almost everywhere partition of unity that can handle essential boundary conditions nevertheless we could not have the smooth closed form pu functions with flat top for general polygonal patches 2d and general polyhedral patches 3d in this paper we introduce one of the most simple and efficient partition of unity called the generalized product partition of unity the product pu functions constructed by this method are the closed form smooth piecewise polynomials with flat top and could handle background meshes general polygonal patches as well as general polyhedral patches arising in practical applications of meshless methods

reproducing kernel element method part i theoretical formulation in this paper and its sequels we introduce and analyze a new class of methods collectively called the reproducing kernel element method rkem the central idea in the development of the new method is to combine the strengths of both finite element methods fem and meshfree methods two distinguished features of rkem are the arbitrarily high order smoothness and the interpolation property of the shape functions these properties are desirable especially in solving galerkin weak forms of higher order partial differential equations and in treating dirichlet boundary conditions so unlike the fem there is no need for special treatment with the rkem in solving high order equations compared to meshfree methods dirichlet boundary conditions do not present any difficulty in using the rkem a rigorous error analysis and convergence study of the method are presented the performance of the method is illustrated and assessed through some numerical examples

causal independence and the energy level density of states in local quantum field theory within the general framework of local quantum field theory a physically motivated condition on the energy level density of well localized states is proposed and discussed it is shown that any model satisfying this condition obeys a strong form of the principle of causal statistical independence which manifests itself in a specific algebraic structure of the local algebras split property it is also shown that the proposed condition holds in a free field theory

a moving kriging interpolation based meshless method for numerical simulation of kirchhoff plate problems this paper mainly proposes an alternative way for numerical implementation of thin plates bending based on a new improvement of meshless method which is combined between the standard element free galerkin method and one different shape functions building technique the moving kriging mk interpolation is applied instead of the traditional moving least square approximation in order to overcome kronecker s delta property where the standard method does not satisfy obviously the deflection of the thin plates is approximated via the mk interpolation to illustrate this approach numerical analysis is examined in both regular and irregular systems three examples with different geometric shapes of thin plates undergoing a simply supported boundary are performed in addition two important parameters of the present method are also analyzed a good agreement can be found among the proposed analytical and finite element methods copyright 2008 john wiley sons ltd

dichotomies for linear evolutionary equations in banach spaces abstract in this paper we present a characterization for the existence of an exponential dichotomy for a linear evolutionary system on a banach space the theory we present here applies to general time varying linear equations in banach spaces as a result it gives a description of the behavior of the nonlinear dynamics generated by certain nonlinear evolutionary equations in the vicinity of a compact invariant set in the case of dissipative systems our theory applies to the study of the flow in the vicinity of the global attractor the theory formulated here holds for linear evolutionary systems which are uniformly contracting and applies to the study of the linearization of nonlinear equations of the following type a parabolic pdes including systems of reaction diffusion equations and the navier stokes equations b hyperbolic pdes including the nonlinear wave equation and the nonlinear schrodinger equation with dissipation c retarded differential equations and d certain neutral differential delay equations

a high order finite volume method with improved isotherms reconstruction for the computation of multiphase flows using the navier stokes korteweg equations abstract in this work we solve the navier stokes korteweg nsk equations to simulate a two phase fluid with phase change we use these equations on a diffuse interface approach where the properties of the fluid vary continuously across the interface that separates the different phases the model is able to describe the behavior of both phases with the same set of equations and it is also able to handle problems with great changes in the topology of the problem however high order derivatives are present in nsk equations which is a difficulty for the design of a numerical method to solve the problem here we propose the use of a high order finite volume method with moving least squares approximations to handle high order derivatives and solve the nsk equations moreover a new methodology to obtain accurate equations of state is presented in this method we use any accurate equation of state for the pure phases under the saturation curve a b spline reconstruction fulfilling a given set of thermodynamic criteria is performed the new eos can be used for computations using diffuse interface modeling several numerical examples to show the accuracy of the new approach are presented

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

dealing with extremely large deformation by nearest nodes fem with algorithm for updating element connectivity in the recently developed nearest nodes finite element method nn fem elements are mainly used for numerical integration while shape functions are constructed in a similar way as in meshless methods based on this strategy nn fem inherits major merits from both the classical finite element method and meshless methods one of them is that nn fem is nearly not affected by element distortion so nn fem is more efficient than the classical fem on dealing with large deformation problems nevertheless nn fem still has a requirement on finite element meshes that is elements in a mesh are required not to overlap or penetrate to each other to avoid difficulty in numerical integration to eliminate overlapped elements nn fem is supplemented with an algorithm for updating element connectivity with this supplement nn fem is able to deal with extremely large deformation in updating element connectivity element nodes are kept not changed and all information associated with nodes are not touched therefore there is no need to transfer solution data and error introduced by solution transfer is avoided

evolution semigroups translation algebras and exponential dichotomy of cocycles abstract we study the exponential dichotomy of an exponentially bounded strongly continuous cocycle over a continuous flow on a locally compact metric space acting on a banach space x our main tool is the associated evolution semigroup on c 0 x we prove that the cocycle has exponential dichotomy if and only if the evolution semigroup is hyperbolic if and only if the imaginary axis is contained in the resolvent set of the generator of the evolution semigroup to show the latter equivalence we establish the spectral mapping annular hull theorem for the evolution semigroup in addition dichotomy is characterized in terms of the hyperbolicity of a family of weighted shift operators defined on c 0 z x here we develop banach algebra techniques and study weighted translation algebras that contain the evolution operators these results imply that dichotomy persists under small perturbations of the cocycle and of the underlying compact metric space also exponential dichotomy follows from pointwise discrete dichotomies with uniform constants finally we extend to our situation the classical perron theorem which says that dichotomy is equivalent to the existence and uniqueness of bounded continuous mild solutions to the inhomogeneous equation

approximation of parametric curves by moving least squares method in this work we propose a method to approximate a parametric curve in r d some distinct points in r d are given we assume that these points belong to a parametric curve and our aim is to approximate these data by moving least squares method we mention several applications of the proposed method to emphasize the importance of the work also root mean squares errors and hausdorff distances between the exact curve and its approximation are presented to demonstrate the efficiency and reliability of the method

a generalized moving least square reproducing kernel method a generalization of moving least square reproducing kernel method is presented in this work the moving least square reproducing kernel method is obtained by using a moving least square scheme but not in the discrete version the resulted scheme provides a continuous basis which is able to reproduce any m th order polynomial and prepares a scheme that can approximate smooth functions with an optimal accuracy on the other hand considering the power of moving least square scheme in meshless approximation for the numerical solution of partial differential equations the generalized moving least square approximation is able to approximate l u just in terms of node values where l is an arbitrary linear operator in this paper a generalization of moving least square reproducing kernel method is presented which employs the generalized version of moving least square method the method approximates a test functional based on the values of nodes the convergence rate of the method is measured in terms of dilation parameter of window function the method is simpler and faster to implement than the classical ones where it does not use the shape function numerical tests are presented to confirm the theoretical results the numerical results establish the efficiency of the proposed method

a kriging based error reproducing and interpolating kernel method for improved mesh free approximations an error reproducing and interpolating kernel method erikm which is a novel and improved form of the error reproducing kernel method erkm with the nodal interpolation property is proposed the erkm is a non uniform rational b splines nurbs based mesh free approximation scheme recently proposed by shaw and roy comput mech 2007 40 1 127 148 the erkm is based on an initial approximation of the target function and its derivatives by nurbs basis functions the errors in the nurbs approximation and its derivatives are then reproduced via a family of non nurbs basis functions the non nurbs basis functions are constructed using a polynomial reproduction condition and added to the nurbs approximation obtained in the first step in the erkm the interpolating property at the boundary is achieved by repeating the knot open knot vector however for most problems of practical interest employing nurbs with open knots is not possible because of the complex geometry of the domain and consequently erkm shape functions turn out to be non interpolating in erikm the error functions are obtained through localized kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint interpolating error functions so obtained are then added to the nurbs approximant while enriching the erkm with the interpolation property the erikm naturally possesses all the desirable features of the erkm such as insensitivity to the support size and ability to reproduce sharp layers the proposed erikm is finally applied to obtain strong and weak solutions for a class of linear and non linear boundary value problems of engineering interest these illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent

reproducing kernel element method part iii generalized enrichment and applications in this part of the work a notion of generalized enrichment is proposed to construct the global partition polynomials or to enrich global partition polynomial basis with extra terms corresponding to the higher order derivatives of primary variable this is accomplished by either multiplying enrichment functions with the original global partition polynomials or increasing the order of global partition polynomials in the same mesh without refining mesh high order consistency in interpolation hierarchy with generalized kronecker delta property can be straightforwardly achieved in quadrilateral and triangular mesh in 2d by the proposed scheme comparing with the traditional finite element methods the construction proposed here has more flexibility and only needs minimal degrees of freedom the optimal element with high reproducing capacity and overall minimal degrees of freedom can be constructed by the generalized enrichment procedure two optimal elements in two dimensional space have been constructed t10p3i 4 triangular element satisfies third order consistency condition with only 10 degrees of freedom and q15p4i 4 quadrilateral element satisfies fourth order consistency condition with 15 degrees of freedom the performance of interpolation hierarchy is

schaffer spaces and exponential dichotomy for evolutionary processes a very general characterization of exponential dichotomy for evolutionary processes in terms of the admissibility of some pair of spaces which are translation invariant the so called schaffer spaces is given it includes as particular cases many interesting situations among which we note the results obtained by n van minh f rabiger and r schnaubelt and the authors concerning the connections between admissibility and dichotomy

super convergence of reproducing kernel approximation abstract the reproducing kernel particle methods rkpm are meshfree methods arising in mechanics especially in dealing with problems involving large deformation and singularities we provide a theoretical analysis of super convergence in sobolev norms for reproducing kernel rk approximations when the interpolation order p is even super convergence phenomenon means the convergence rate is higher than the order that is generally expected we distinguish the continuous rk approximation and the discrete rkp approximation while the continuous rk approximations are proven to be super convergent when p is even its discrete counterpart has super convergence only with uniform particle distribution and special choices of rk kernel functions and support sizes moreover super convergence does not exist for the discrete rkp approximation with general rk support sizes the concept of pseudo super convergence is then introduced to explain why in practice the super convergence phenomenon is sometimes observed for general cases although in theory it is not true our analysis is general for multi dimensional rk approximations

moving particle finite element method with global smoothness we describe a new version of the moving particle finite element method mpfem that provides solutions within a c 0 finite element framework the finite elements determine the weighting for the moving partition of unity a concept of general shape function is proposed which extends regular finite element shape functions to a larger domain these are combined with shepard functions to obtain a smooth approximation the moving particle finite element method combines desirable features of finite element and meshfree methods the proposed approach in fact can be interpreted as a moving partition of unity finite element method or moving kernel finite element method this method possesses the robustness and efficiency of the c 0 finite element method while providing at least c 1 continuity

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

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

results of von neumann analyses for reproducing kernel semi discretizations the reproducing kernel particle method rkpm has many attractive properties that make it ideal for treating a broad class of physical problems rkpm may be implemented in a mesh full or a mesh free manner and provides the ability to tune the method via the selection of a window function and its associated dilation parameter in order to achieve the requisite numerical performance rkpm also provides a framework for performing hierarchical computations making it an ideal candidate for simulating multi scale problems although the method has many appealing attributes it is quite new and its numerical performance is still being quantified with respect to more traditional discretization techniques in order to assess the numerical performance of rkpm detailed studies of the method on a series of model partial differential equations has been undertaken the results of von neumann analyses for rkpm semi discretizations of one and two dimensional first and second order wave equations are presented in the form of phase and group errors excellent dispersion characteristics are found for the consistent mass matrix with the proper choice of dilation parameter in contrast row sum lumping the mass matrix is demonstrated to introduce severe lagging phase errors a higher order mass matrix improves the dispersion characteristics relative to the lumped mass matrix but also yields significant lagging phase errors relative to the fully integrated consistent mass matrix published in 2000 by john wiley sons ltd

convergence estimates for the wavelet galerkin method this paper presents an analysis of the galerkin approximation of a time dependent initial value problem correctly posed in the petrovskii sense the approximating spaces mathcal v h are spanned by translations and dilations of a single function phi and fourier techniques are used to analyze the accuracy of the method this kind of procedure has already been applied in the literature for spline approximations our purpose here is to point out that the same methodology can be used for wavelet based methods since the hypotheses required are automatically satisfied in the context of wavelet analysis for instance the basic function phi is supposed to be regular which means that phi and all its derivatives up to a certain order should have fast decay at infinity phi also must satisfy the so called strang and fix condition which guarantees that smooth functions can be approximated from mathcal v h with good accuracy this class of functions includes not only the b splines but al

existence and robustness of exponential dichotomy of linear skew product semiflows over semiflows in this paper we investigate the exponential dichotomy of linear skew product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on banach bundles we characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy

nurbs based parametric mesh free methods a mesh free error reproducing kernel method erkm has recently been proposed by a shaw d roy a nurbs based error reproducing kernel method with applications in solid mechanics comput mech 40 1 2007 127 148 the erkm is based on an initial approximation of the target function by non uniform rational b splines nurbs followed by reproduction of the error via a family of non nurbs basis functions however specifications of the window supports of non nurbs basis functions remain a tricky issue moreover nurbs in higher dimensions 1 is generally defined over rectangular cuboidal in 3d grid structures and thus in many problems of practical interest the geometric complexity of the domain would prevent making use of nurbs in the erkm presently we develop a parametric reformulation of the erkm purely using nurbs this reformulation allows the method to be applicable to non rectangular non cuboidal in 3d physical domains in two or still higher dimensions without a need to explicitly specify the support size of the window a key feature of this development is a geometric map that provides a local bijection between the physical domain and rectangular cuboidal parametric domain the shape functions and their derivatives are then constructed over the parametric domain so that polynomial reproduction and interpolation properties are satisfied over the physical domain with the geometric map being simultaneously preserved a couple of new schemes are also proposed to empower the shape functions with the interpolation property that in turn enables a precise imposition of dirichlet boundary conditions we illustrate the parametric mesh free method in the context of strong weak solutions of a few linear and non linear boundary and initial value problems of engineering interest

deformations of quantum field theories and integrable models deformations of quantum field theories which preserve poincare covariance and localization in wedges are a novel tool in the analysis and construction of model theories here a general scenario for such deformations is discussed and an infinite class of explicit examples is constructed on the borchers uhlmann algebra underlying wightman quantum field theory these deformations exist independently of the space time dimension and contain the recently studied warped convolution deformation as a special case in the special case of two dimensional minkowski space they can be used to deform free field theories to integrable models with non trivial s matrix

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations abstract we prove that the admissibility of any pair of vector valued schaffer function spaces satisfying a very general technical condition implies the existence of a no past exponential dichotomy for an exponentially bounded strongly continuous cocycle over a semiflow roughly speaking the class of schaffer function spaces consists in all function spaces which are invariant under the right shift and therefore our approach addresses most of the possible pairs of admissible spaces complete characterizations for the exponential dichotomy of cocycles are also obtained moreover we involve a concept of a no past exponential dichotomy for cocycles weaker than the classical concept defined by sacker and sell 1994 in 23 our definition of exponential dichotomy follows partially the definition given by chow and leiva 1996 in 4 in the sense that we allow the unstable subspace to have infinite dimension the main difference is that we do not assume a priori that the cocycle is invertible on the unstable space actually we do not even assume that the unstable space is invariant under the cocycle thus we generalize some known results due to o perron 1930 14 j daleckij and m krein 1974 7 j l massera and j j schaffer 1966 11 n van minh f rabiger and r schnaubelt 1998 26

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

analyses of wrinkled and slack membranes through an error reproducing mesh free method a mesh free approximation of large deformations of flexible membrane structures within the tension field theory is considered in this paper a modification of the wrinkling theory originally proposed by roddeman et al 1987 roddeman d g drukker j oomens c w j janssen j d 1987 the wrinkling of thin membranes part i theory part ii numerical analysis asme j appl mech 54 884 892 is proposed to study the behaviour of an isotropic membrane under the mixed state of stress taut wrinkled and slack using the facts that the state of stress is not uniform across an element and that the deformation gradient is a spatially continuous and possibly non differentiable tensor the proposed model uses a continuously modified deformation gradient to capture the location and orientation of wrinkles more precisely while the deformation gradient need not be everywhere differentiable in a wrinkled membrane it is argued that the fictive non wrinkled non slack surface may be looked upon as an everywhere taut surface in the limit as the minor and major principal tensile stresses over the wrinkled slack portions go to zero accordingly the modified deformation gradient is thought of as the limit of a sequence of everywhere differentiable tensors the weighted residual from the governing equations are presently solved via a mesh free method where the entire domain is discretized only by a set of grid points a non uniform rational b spline nurbs based error reproducing kernel method erkm has been used to approximate the field variable over the domain the first step in the method is to approximate a function and its derivatives through nurbs basis functions however since nurbs functions neither reproduce any polynomial nor interpolate the grid points also referred to as control or nodal points the approximated functions result in uncontrolled errors over the domain including the grid points accordingly the error functions in the nurbs approximation and its derivatives are reproduced via a family of non nurbs basis functions the non nurbs basis functions are constructed using a polynomial reproduction condition and added to the nurbs approximation of the function obtained in the first step several numerical examples on wrinkled and or slack membranes are also provided

schaffer spaces and uniform exponential stability of linear skew product semiflows we study the exponential stability of linear skew product semiflows on locally compact metric space with banach fibers our main tool is the admissibility of a pair of the so called schaffer spaces this characterization is a very general one it includes as particular cases many interesting situations among them we can mention some results due to clark datko latushkin van minh montgomery smith randolph rabiger schnaubelt

the cost of continuity a study of the performance of isogeometric finite elements using direct solvers we study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis the problem of choice is the canonical laplace equation in three dimensions from this study we conclude that for a fixed number of unknowns and polynomial degree of approximation a higher degree of continuity k drastically increases the cpu time and ram needed to solve the problem when using a direct solver this paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause

implementation of boundary conditions for meshless methods abstract boundaries and boundary conditions are an aspect of the numerical solution of partial differential equations where meshless methods have had to surmount many initial difficulties due to the lack of a finite element like kronecker delta condition furthermore it is frequently desirable especially in fluid mechanics to impose general nonlinear boundary and interface constraints this paper describes a computationally efficient algorithm based on d alembert s principle that can be used for general constraints both in meshless methods and finite elements first a method of partitioning meshless shape functions suitable for imposing linear boundary conditions is developed subsequently an analogous method is developed for nonlinear constraints special attention is given to imposing general boundary and fluid structure interface conditions on the navier stokes equations in terms of conservative variables numerical results using d alembert s principle with the reproducing kernel particle method rkpm including viscous supersonic flow past a naca 7012 airfoil are shown

a meshless local petrov galerkin method for the time dependent maxwell equations abstract in this paper the meshless local petrov galerkin mlpg method is employed to solve the 2 d time dependent maxwell equations the mlpg method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces in the current work the moving least square reproducing kernel mlsrk scheme is chosen to be the trial function the method is applied for the unsteady maxwell equations in different media in the local weak form by employing the difference operator for evolution in time and simultaneously in time and space the semi discrete and fully discrete schemes are obtained respectively the error estimation is discussed for both the semi discrete and fully discrete numerical schemes for modelling the time dependent maxwell equations we show that provided that the time step size is sufficiently small the proposed scheme yields an error of o 2 m 1 2 in the l 2 norm for the square of error the new scheme is implemented and the numerical results are provided to justify our theoretical analysis

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics