some new results on easy lambda terms abstract given two closed terms a and b we consider the question whether the equation a b is consistent with the calculus in general the problem is undecidable however if a is a 0 term we can give good sufficient conditions for the consistency of a b this allows us to prove some counterintuitive results such as 1 there is a closed term x which can be consistently equated to every closed term with the exception of the identity 2 there is a closed term which can be consistently equated to every closed normal form but not to the curry fixed point operator y

multidimensional viscous shocks ii the small viscosity limit in this paper we prove the existence of curved multidimensional viscous shocks and also justify the small viscosity limit starting with a curved multidimensional inviscid shock solution to a system of hyperbolic conservation laws we show that the shock can be obtained as a small viscosity limit of solutions to an associated parabolic problem viscous shocks the two main hypotheses are a natural evans function assumption on the viscous profile together with a restriction on how much the shock can deviate from flatness the main tools are a conjugation lemma that removes xn dependence from the linearization of the parabolic problem about the viscous profile new degenerate kreiss type symmetrizers used to prove an l2 estimate for the linearized problem and a finite regularity calculus of semiclassical and mixed type classical semiclassical pseudodifferential operators 2003 wiley periodicals inc

l2 contraction of large planar shock waves for multi dimensional scalar viscous conservation laws abstract we consider a l 2 contraction a l 2 type stability of large viscous shock waves for the multi dimensional scalar viscous conservation laws up to a suitable shift by using the relative entropy methods quite different from the previous results we find a new way to determine the shift function which depends both on the time and space variables and solves a viscous hamilton jacobi type equation with source terms moreover we do not impose any conditions on the anti derivative variables of the perturbation around the shock profile more precisely it is proved that if the initial perturbation around the viscous shock wave is suitably small in l 2 norm then the l 2 contraction holds true for the viscous shock wave up to a suitable shift function note that bv norm or the l norm of the initial perturbation and the shock wave strength can be arbitrarily large furthermore as the time t tends to infinity the l 2 contraction holds true up to a spatially homogeneous time dependent shift function in particular if we choose some special initial perturbations then l 2 convergence of the solutions towards the associated shock profile can be proved up to a time dependent shift

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

relaxation limit for piecewise smooth solutions to systems of conservation laws abstract in this paper we study the asymptotic equivalence of a general system of 1 d conservation laws and the corresponding relaxation model proposed by s jin and z xin 1995 comm pure appl math 48 235 277 in the limit of small relaxation rate it is shown that if the relaxation system satisfies the subcharacteristic condition and the solution of the hyperbolic conservation laws is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions of the relaxation systems that converge to the solution of the original conservation laws equilibrium system at a rate of order e as the rate of relaxation e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the nonlinear stability theory for viscous shock profiles

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

iterative solutions of the generalized sylvester matrix equations by using the hierarchical identification principle abstract in this paper by extending the well known jacobi and gauss seidel iterations for ax b we study iterative solutions of matrix equations axb f and generalized sylvester matrix equations axb cxd f including the sylvester equation ax xb f as a special case and present a gradient based and a least squares based iterative algorithms for the solution it is proved that the iterative solution always converges to the exact solution for any initial values the basic idea is to regard the unknown matrix x to be solved as the parameters of a system to be identified and to obtain the iterative solutions by applying the hierarchical identification principle finally we test the algorithms and show their effectiveness using a numerical example

an efficient adaptive analysis procedure for node based smoothed point interpolation method ns pim this paper presents an efficient adaptive analysis procedure being able to operate in the framework of the node based smoothed point interpolation method ns pim the ns pim uses three node triangular cells and is very easy to be implemented which make it an ideal candidate for adaptive analysis in the present adaptive procedure a new error indicator is devised for ns pim settings two ways are proposed to calculate the local critical value a simple h type local refinement scheme is adopted and delaunay technology is used for regenerating optimal new mesh a number of typical numerical examples involving stress concentration and solution singularities have been tested the results demonstrate that the present procedure achieves much higher convergence rate results compared to the uniform refinement and can obtain upper bound solution in strain energy

the residual based interactive stochastic gradient algorithms for controlled moving average models this paper derives a residual based interactive stochastic gradient isg parameter estimation algorithm for controlled moving average cma models and studied the performance of the residual based isg algorithm under weaker conditions on statistical properties of the noise compared with the residual based extended stochastic gradient algorithm for identifying cma models the proposed isg algorithm can give highly accurate parameter estimates by the simulation example

a class of bounded functions a database language and an extended lambda calculus abstract we develop an approach of partial computations for the lambda calculus it produces a class of bounded functions i e the co domains are finite while the domains are possibly infinite including self applicable functions we show that the bounded functions are recursive and have to be represented as lambda terms without head normal form in the lambda calculus in parallel we develop a language independently from the lambda calculus it also represents the class of bounded functions and therefore can produce whatever a turing machine produces provided that computation has finite time and space we call such a language a database language because we can use the language to construct and query business objects in database practice with the bounded functions we are able to extend the lambda calculus to effectively reduce terms having weak head normal form in a manner similar to how the standard lambda calculus reduces terms that have normal form

optimal l 1 rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks we derive optimal error bounds for the viscosity method and monotone difference schemes to an initial value problem of scalar conservation laws with initial data being a finite number of piecewise constants subject to the initial discontinuities satisfying the entropy conditions it is known that the entropy solution of the problem is piecewise constant with a finite number of interacting shocks satisfying the entropy conditions a rigorous analysis shows that both the viscosity method and monotone schemes to approach the initial value problem have uniform l 1 error bounds of o epsilon and o delta x for the time infty t 0 respectively where epsilon and delta x are their corresponding viscosity coefficient and discrete mesh length the results are improvements over the half order rates of l 1 convergence numerical experiments for the lax friedrichs scheme are presented and numerical results justify the theoretical analysis

graph easy sets of mute lambda terms among the unsolvable terms of the lambda calculus the mute ones are those having the highest degree of undefinedness in this paper we define for each natural number n an infinite and recursive set m n of mute terms and show that it is graph easy for any closed term t of the lambda calculus there exists a graph model equating all the terms of m n to t alongside we provide a brief survey of the notion of undefinedness in the lambda calculus

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

easiness in graph models we generalize baeten and boerboom s method of forcing to show that there is a fixed sequence uk k of closed untyped terms satisfying the following properties a for any countable sequence gk k of scott continuous functions of arbitrary arity on the power set of an arbitrary countable set there is a graph model such that x xx x xx uk represents gk in the model b for any countable sequence tk k of closed terms there is a graph model that satisfies x xx x xx uk tk for all k we apply these two results which are corollaries of a unique theorem to prove the existence of 1 a finitely axiomatized theory l such that the interval lattice constituted by the theories extending l is distributive 2 a continuum of pairwise inconsistent graph theories theories that can be realized as theories of graph models 3 a congruence distributive variety of combinatory algebras lambda abstraction algebras respectively

a point interpolation method for two dimensional solids a point interpolation method pim is presented for stress analysis for two dimensional solids in the pim the problem domain is represented by properly scattered points a technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points the pim equations are then derived using variational principles in the pim the essential boundary conditions can be implemented with ease as in the conventional finite element methods the present pim has been coded in fortran the validity and efficiency of the present pim formulation are demonstrated through example problems it is found that the present pim is very easy to implement and very flexible for obtained displacements and stresses of desired accuracy in solids as the elements are not used for meshing the problem domain the present pim opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures copyright 2001 john wiley sons ltd

zero dissipation limit to rarefaction waves for the one dimensional navier stokes equations of compressible isentropic gases we study the zero dissipation limit problem for the one dimensional navier stokes equations of compressible isentropic gases in the case that the corresponding euler equations have rarefaction wave solutions we prove that the solutions of the navier stokes equations with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity vanishes uniformly away from the initial discontinuities in the case that either the effects of initial layers are ignored or the rarefaction waves are smooth we then obtain a rate of convergence which is valid uniformly for all time our method of proof consists of a scaling argument and elementary energy analysis based on the underlying wave structure 1993 john wiley sons inc

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

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

relaxation approximations to front propagation abstract we introduce a relaxation model for front propagation problems our proposed relaxation approximation is a semilinear hyperbolic system without singularities it yields a direction depedent normal velocity at the leading term and captures in the chapman enskog expansion the higher order curvature dependent corrections including possible anisotropies

vanishing curvature viscosity for front propagation in this paper we study the front propagation with constant speed and small curvature viscosity we first investigate two related problems of conservation laws one of which is on the nonlinear viscosity methods for the conservation laws and the other one is on the structure of solutions to conservation laws with l 1 initial data we show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first order rate of l 1 convergence the solutions of conservation laws with l 1 initial data are shown to be bounded after t 0 if all singular points of initial data are from shocks these results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed as the small parameter goes to zero after the front breaks down the cusps will disappear promptly and corners will be formed 2000 academic press

viscous limits for piecewise smooth solutions of the p system abstract we study the zero dissipation problem for a one dimensional model system for the isentropic flow of a compressible viscous gas the so called p system with viscosity when the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof is given by a matched asymptotic analysis and an elementary energy method and we do not need the smallness condition on the shock strength

gradient based and least squares based iterative algorithms for matrix equations axb cxtd f abstract this paper develops a gradient based and a least squares based iterative algorithms for solving matrix equation axb cx t d f the basic idea is to decompose the matrix equation system under consideration into two subsystems by applying the hierarchical identification principle and to derive the iterative algorithms by extending the iterative methods for solving ax b and axb f the analysis shows that when the matrix equation has a unique solution under the sense of least squares the iterative solution converges to the exact solution for any initial values a numerical example verifies the proposed theorems

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

upper bound solution to elasticity problems a unique property of the linearly conforming point interpolation method lc pim it is well known that the displacement based fully compatible finite element method fem provides a lower bound in energy norm for the exact solution to elasticity problems it is however much more difficult to bound the solution from above for general problems in elasticity and it has been a dream of many decades to find a systematical way to obtain such an upper bound this paper presents a very important and unique property of the linearly conforming point interpolation method lc pim it provides a general means to obtain an upper bound solution in energy norm for elasticity problems this paper conducts first a thorough theoretical study on the lc pim we derive its weak form based on variational principles study a number of properties of the lc pim and prove that lc pim is variationally consistent and that it produces upper bound solutions we then demonstrate these properties through intensive numerical studies with many examples of 1d 2d and 3d problems using the lc pim together with the fem we now have a systematical way to numerically obtain both upper and lower bounds of the exact solution to elasticity problems as shown in these example problems copyright 2007 john wiley sons ltd

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

three anisotropic benchmark problems for adaptive finite element methods in this paper we provide three benchmark problems with known exact solutions that can be used to assess the ability of adaptive finite element algorithms to handle anisotropically behaved solutions the first one is a poisson equation with a smooth solution that only changes in one spatial direction the second one is a singularly perturbed linear elliptic equation whose solution exhibits a boundary layer and the last one is a two equation system that contains a boundary layer in one solution component only in an appendix we show sample results obtained with the open source library hermes http hpfem org hermes

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

viscosity methods for piecewise smooth solutions to scalar conservation laws it is proved that for scalar conservation laws if the flux function is strictly convex and if the entropy solution is piecewise smooth with finitely many discontinuities which includes initial central rarefaction waves initial shocks possible spontaneous formation of shocks in a future time and interactions of all these patterns then the error of viscosity solution to the inviscid solution is bounded by o log in the l 1 norm which is an improvement of the o upper bound if neither central rarefaction waves nor spontaneous shocks occur the error bound is improved to o

exact a posteriori error analysis of the least squares finite element method1this work was supported by nsc under grant nsc87 2115 m 009 005 taiwan 1 a residual type a posteriori error estimator is presented for the least squares finite element method the estimator is proved to equal the exact error in a norm induced by some least squares functional the error indicator of each element is equal to the exact error norm restricted to the element as well in other words the estimator is perfectly effective and reliable for error control and for adaptive mesh refinement the exactness property requires virtually no assumptions on the regularity of the solution and on the finite element order in the approximation or in the estimation the least squares method is in a very general setting that applies to various linear boundary value problems such as the elliptic systems of first order and of even order and the mixed type partial differential equations numerical results are given to demonstrate the exactness

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985