modified schmidt games and diophantine approximation with weights abstract we show that the sets of weighted badly approximable vectors in r n are winning sets of certain games which are modifications of games introduced by w m schmidt in 1966 the latter winning property is stable with respect to countable intersections and is shown to imply full hausdorff dimension

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

shape optimization using reproducing kernel particle method and an enriched genetic algorithm combining reproducing kernel particle method rkpm with the proposed multi family genetic algorithm mfga a novel approach to continuum based shape optimization problems is brought forward in this paper taking full advantage of the features of meshfree method and the merits of mfga the new method solves shape optimization problems in such a unique way that remeshing is avoided and particularly the computation burden and errors caused by sensitivity analysis are eliminated completely the effectiveness versatility and performance of the proposed approach are demonstrated via three 2 d numerical examples

a nodal integration and post processing technique based on voronoi diagram for galerkin meshless methods a stabilized nodal integration technique is adopted by adding the square of the residual of the equilibrium equation to the potential energy functional an approach is proposed based on voronoi diagram to evaluate nodal volumes accurately it is a key issue for nodal integration of galerkin meshless methods furthermore as the dual of voronoi diagram the corresponding delaunay triangles are at hand provided voronoi diagram construction is completed a realistic technique is proposed based on background delaunay triangles for post processing of meshless results in the whole domain of interest two elastic stress analysis problems are investigated which demonstrate the approach is realistic versatile and robust

equidistribution of expanding translates of curves and dirichlet s theorem on diophantine approximation we show that for almost all points on any analytic curve on k which is not contained in a proper affine subspace the dirichlet s theorem on simultaneous approximation as well as its dual result for simultaneous approximation of linear forms cannot be improved the result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in k 1 the proof involves ratner s theorem on ergodic properties of unipotent flows on homogeneous spaces

a subdomain collocation method based on voronoi domain partition and reproducing kernel approximation a subdomain collocation reproducing kernel approximation method is proposed subdomains are constructed by nonoverlapping voronoi cells and a local weak form is defined over these subdomains the standard rkpm shape functions are used directly for approximation while weighting functions of the subdomain collocation method hold a constant value of unit only over a specific voronoi cell the body integration of the local weak form is now converted into much cheaper and efficient contour integration along the boundaries of voronoi cells it does not need to impose traction free boundary conditions explicitly in contrast to point collocation method furthermore the method provides a natural background structure for performing h adaptivity analysis straightforwardly all these features constitute the subdomain collocation method a promising alternative to standard galerkin method and point collocation method some elastostatics examples are presented to demonstrate the effectiveness the convergence property and the adaptivity performance of present method

robust impulse control of uncertain singular systems by decentralized output feedback this correspondence considers a problem of robust impulse control of uncertain singular systems by decentralized output feedback it derives a perturbation upper bound that ensures the existence of decentralized output feedback control law such that the closed loop system remains both regularity and impulse free for any allowable perturbations algebraic sufficient conditions are also presented that guarantee the existence of so called decentralized robust impulse controllers

local conjugacy classes for analytic torus flows abstract if a real analytic flow on the multidimensional torus close enough to linear has a unique rotation vector which satisfies an arithmetical condition y then it is analytically conjugate to linear we show this by proving that the orbit under renormalization of a constant y vector field attracts all nearby orbits with the same rotation vector

robust pole assignment in singular control systems abstract necessary conditions are given for the problem of pole assignment by state feedback in singular linear systems descriptor systems to have a solution which is regular and nondefective for a robust solution such that the assigned closed loop poles are insensitive to perturbations in the system data the same conditions must hold it can be shown that these conditions are also sufficient for the existence of a feedback which assigns the maximum possible number of finite poles with regularity these results provide the basis of a procedure for constructing closed loop semistate systems with given poles guaranteed regularity and maximum robustness

radial spline collocation method for static analysis of beams conventional spline collocation method scm uses equally spaced knots in the computation of approximation however when load distribution geometry material property of structure are discontinuous it will greatly reduce accuracy the goal of this paper is mainly to develop unequally spaced knots to improve the disadvantage of using equally spaced knots applying the concept of radial basis functions rbf the spline function is transformed into radial spline function rsf consequently rsf can conveniently calculate the values for unequally spaced knots so that the complicated beam problems with discontinuous characteristics can be solved accurately and efficiently by the proposed radial spline collocation method rscm in this paper the proposed rscm also can be extended to solve other structural problems such as frame plate shell etc

impulsive mode elimination for descriptor systems by a structured p d feedback the problem of impulsive mode elimination for descriptor systems via a structured output proportional and derivative p d feedback is investigated in this technical note the motivation of solving this problem arises from an observation that the impulse behavior for some descriptor systems can not be eliminated completely by a parameterized output p d feedback but it can be removed by a structured p d feedback elegantly designed in order to explain this phenomenon explicitly a concept of the structured p d feedback is first introduced then an explicit necessary and sufficient condition is constructively derived for the closed loop systems to be regular and impulse free by the structured p d feedback the main result is represented in terms of the original system parameters instead of decomposed subsystem parameters finally an example is provided to illustrate the effectiveness of the proposed result

on impulsive modes of linear singular systems subject to decentralized output feedback some algebraic properties of linear singular systems at infinity under static decentralized output feedback are studied in this note new concepts of algebraic multiplicity and geometric multiplicity of the impulsive decentralized fixed modes and the impulsive decentralized cycle index of the singular system are defined these concepts indicate how much a singular system can be made close to being impulse free by decentralized output feedback and the results are shown to be generic the geometric multiplicity and the impulsive decentralized cycle index are determined in terms of the system matrices explicitly the number of impulsive modes that can be eliminated is given in terms of these indexes the impulsive decentralized cycle index is shown to characterize generic properties on controllability and observability of the closed loop system through an individual channel or an external channel illustrative examples are provided

regularization of descriptor systems by output feedback conditions are given under which a descriptor or generalized state space system can be regularized by output feedback it is shown that under these conditions proportional and derivative output feedback controls can be constructed such that the closed loop system is regular and has index at most one this property ensures the solvability of the resulting system of dynamic algebraic equations a reduced form is given that allows the system properties as well as the feedback to be determined the construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way

brief structural properties and poles assignability of lti singular systems under output feedback this paper studies the structural properties of both finite poles and the infinite pole of linear time invariant singular systems under output feedback three main problems are studied namely 1 the algebraic structures of both finite poles and the infinite pole 2 the assignability of finite poles and the elimination of the infinite pole by output feedback and 3 the controllability and observability of the system with minimal number of inputs and outputs new generic solutions to these problems are presented in terms of some new concepts defined in this paper including the geometric multiplicity of the infinite pole the finite and impulsive output feedback cycle indices of the system determination of these multiplicities and indices are discussed an assignability equivalence is established between the variable finite poles and the poles of a controllable and observable non singular system the number of the independent infinite poles that can be reduced is given in terms of the system matrices the minimal number of inputs and outputs that guarantee controllability and observability are shown to be the output feedback cycle indices

technical communique on impulsive algebraic multiplicities of linear time invariant singular systems under feedback impulsive algebraic multiplicities of linear time invariant singular systems under output feedback or decentralized output feedback are characterized in terms of system matrices directly the multiplicities describe the number of impulsive modes that can be eliminated and indicate the number of finite poles that can be assigned by the respective feedback these characterizations also lead to new regularizability conditions for singular systems

2 dimensional badly approximable vectors and schmidt s game we prove that for any pair s t of nonnegative numbers with s t 1 the set of s t badly approximable vectors in r 2 is 34 p 2 1 winning for schmidt s game

bounded orbits of diagonalizable flows on finite volume quotients of products of sl2 r abstract we prove a number field analogue of a result of c mcmullen that the set of badly approximable numbers is absolute winning we also prove a weighted version we use this to prove an instance of a conjecture of an guan and kleinbock 4 namely let g sl 2 r sl 2 r and be a lattice in g we show that the set of points on g whose orbits under a one parameter ad semisimple subgroup of g are bounded form a hyperplane absolute winning set

numerical solution to the unsteady two dimensional schrodinger equation using meshless local boundary integral equation method a meshless local boundary integral equation lbie method is proposed to solve the unsteady two dimensional schrodinger equation the method is based on the lbie with moving least squares mls approximation for the mls approximation nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables a time stepping method is employed to deal with the time derivative an efficient method for dealing with singular domain integrations that appear in the discretized equations is presented finally numerical results are considered for some examples to demonstrate the accuracy efficiency and high rate of convergence of this method copyright 2008 john wiley sons ltd

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

a point collocation method based on reproducing kernel approximations a reproducing kernel particle method with built in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations the design and implementation of a galerkin based reproducing kernel particle method however faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions in this paper we present a point collocation method based on reproducing kernel approximations we show that in a point collocation approach the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point collocation method a true meshless approach the point collocation method based on reproducing kernel approximations however requires the calculation of higher order derivatives that would typically not be required in a galerkin method a correction function and reproducing conditions that enable consistency of the point collocation method are derived the point collocation method is shown to be accurate for several one and two dimensional problems and the convergence rate of the point collocation method is addressed copyright 2000 john wiley sons ltd

on regularizing singular systems by decentralized output feedback this paper considers linear time invariant decentralized singular systems which are either nonregular or if they are regular they have impulsive modes it derives algebraic necessary and sufficient conditions for making a singular system both regular and impulse free by decentralized output feedback control laws and decentralized proportional plus derivative output feedback control laws

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

on the convergence analysis stability and implementation of meshless local radial point interpolation on a class of three dimensional wave equations summary in this article the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt ut u uxx uyy uzz f x y z t xt x y z rd d 2 3 t 0 subject to given initial and dirichlet boundary conditions the main difficulty of the great number of methods in full 3 d problems is the large computational costs in meshless local radial point interpolation method it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions the point interpolation method with the help of radial basis functions is proposed to construct shape functions that have kronecker delta function property a weak formulation with the heaviside step function converts the set of governing equations into local integral equations on local subdomains a two step time discretization method is employed to evaluate the time derivatives this suggests crank nicolson technique to be applied on the right hand side of the equation the convergence analysis and stability of the method are fully discussed three illustrative examples are presented and satisfactory agreements are achieved it is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones copyright 2015 john wiley sons ltd

novel adaptive meshfree integration techniques in meshless methods summary we propose two strategies of novel adaptive numerical integration based on mapping techniques for solving the complicated problems of domain integration encountered in meshfree methods several mapping methods are presented in detail that map a complex integration domain to much simpler ones for example squares triangles or circles the techniques described in the paper can be applied to both global and local weak forms and the highly nonlinear meshfree integrands are evaluated with controlled accuracy the necessity of the clumsy procedure of background mesh or cell structures used for integration purpose in existing meshfree methods is avoided and many meshfree methods that require the domain integration can now become truly meshfree various numerical examples in two dimensions are considered to demonstrate the applicability and the effectiveness of the proposed methods and it shows that the accuracy is improved significantly their obtained results are compared with analytical solutions and other approaches and very good agreements are found additionally some three dimensional cases applied by the present methods are also examined copyright 2012 john wiley sons ltd

the meshless finite element method a meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n and the condition that the shape functions depend only on the node positions furthermore the method proposed also shares several of the advantages of the finite element method such as a the simplicity of the shape functions in a large part of the domain b c0 continuity between elements which allows the treatment of material discontinuities and c ease of introduction of the boundary conditions copyright 2003 john wiley sons ltd

non linear version of stabilized conforming nodal integration for galerkin mesh free methods a stabilized conforming sc nodal integration which meets the integration constraint in the galerkin meshfree approximation is generalized for nonlinear problems in sc nodal integration a strain smoothing stabilization is introduced and applied to the deformation gradient using a lagrangian meshfree discretization the assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems a significant gain in computational efficiency is achieved as well as enhanced accuracy in comparison to the solution of meshfree method using gauss integration method the performance of the proposed method appears to be quite robust when dealing with irregular discretization

implementation of meshless lbie method to the 2d non linear sg problem in this paper the meshless local boundary integral equation lbie method for numerically solving the non linear two dimensional sine gordon sg equation is developed the method is based on the lbie with moving least squares mls approximation for the mls nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables the approximation functions are constructed entirely using a set of scattered nodes and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes a time stepping method is employed to deal with the time derivative and a simple predictor corrector scheme is performed to eliminate the non linearity a brief discussion is outlined for numerical integrations in the proposed algorithm some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped sg equation is investigated the final numerical results confirm the ability of method to deal with the unsteady non linear problems in large domains copyright 2009 john wiley sons ltd

constant solutions of polynomial equations abstract a necessary and sufficient condition is given for the equation ax by c in polynomial matrices to have a constant solution pair x y and also for x to be non singular a sufficient condition is then established under which the equation with a and b fixed has a constant solution for each c from a given class applications to the construction of static state feedback or static output injection in linear systems are discussed

the natural neighbour petrov galerkin method for elasto statics in this paper an efficient and accurate meshless natural neighbour petrov galerkin method nnpg is proposed to solve elasto static problems in two dimensional space this method is derived from the generalized meshless local petrov galerkin method mlpg as a special case in the nnpg the local supported trial functions are constructed based on the non sibsonian interpolation and test functions are taken as the three node triangular fem shape functions the local weak forms of the equilibrium equation and the boundary conditions are satisfied in local polygonal sub domains these sub domains are constructed with delaunay tessellations and domain integrals are evaluated over included delaunay triangles by using gaussian quadrature scheme as this method combines the advantages of natural neighbour interpolation with petrov galerkin method together no stiffness matrix assembly is required and no special treatment is needed to impose the essential boundary conditions several numerical examples are presented and the results show the presented method is easy to implement and very accurate for these problems copyright 2005 john wiley sons ltd

further enhancement on robust h_ infty control design for discrete time singular systems this note deals with the robust h control design for uncertain discrete time singular systems necessary and sufficient conditions for the admissibility for these systems are formulated in strict linear matrix inequalities lmis formulation then both state feedback and static output feedback controllers are studied and robust necessary and sufficient design conditions are expressed in terms of strict bilinear matrix inequalities bmis robust h controllers are designed for uncertain singular systems including uncertainties on matrix e lmi design conditions are also proposed an illustrative example is provided to show the effectiveness of the proposed approach

eigenstructure assignment in descriptor systems coordinate free conditions are given for pole assignment by feedback in linear descriptor singular systems which guarantee closed loop regularity these conditions are shown to be both necessary and sufficient for assignment of the maximum possible number of finite poles transformation to special coordinates are not used and the results provide a robust algorithm for the computation of the required feedback

generic controllability theorems for descriptor systems this paper presents an extension of the structural controllability theorem to the case of structured descriptor systems sds a major difficulty in this problem is the loss of independence among the entries of the system matrices this difficulty is resolved by introducing the idea of column structured matrices csm first the paper defines and characterizes c and r controllability for descriptor systems then the generic controllability theorem is proved extending the structural controllability theorem to column structured systems css finally based on this result necessary conditions and sufficient conditions for generic r and c controllability are obtained the result gives a complete characterization of generic controllability for sds with nonsingular e and a partial solution for the case of singular e

badly approximable systems of affine forms fractals and schmidt games this is the final version of the article available from de gruyter via the doi in this record

hausdorff dimension of singular vectors we prove that the set of singular vectors in rd d 2 has hausdorff dimension d2d 1 and that the hausdorff dimension of the set of e dirichlet improvable vectors in rd is roughly d2d 1 plus a power of e between d2 and d as a corollary the set of divergent trajectories of the flow by diag et et e dt acting on sld 1 r sld 1 z has hausdorff codimension dd 1 these results extend the work of the first author

technical communique regularizability of linear time invariant descriptor systems under decentralized control regularizability conditions for linear time invariant descriptor systems under decentralized output feedback are presented in this paper the regularizability is characterized by rank conditions that involve the original system matrices only new rank conditions for regularity of the system are also derived

nearest neighbor distances on a circle multidimensional case we study the distances called spacings between pairs of neighboring energy levels for the quantum harmonic oscillator specifically we consider all energy levels falling between e and e 1 and study how the spacings between these levels change for various choices of e particularly when e goes to infinity primarily we study the case in which the spring constant is a badly approximable vector we first give the proof by boshernitzan dyson that the number of distinct spacings has a uniform bound independent of e then if the spring constant has components forming a basis of an algebraic number field we show that when normalized up to a unit the spacings are from a finite set moreover in the specific case that the field has one fundamental unit the probability distribution of these spacings behaves quasiperiodically in loge we conclude by studying the spacings in the case that the spring constant is not badly approximable providing examples for which the number of distinct spacings is unbounded

on regularizing descriptor systems by output feedback in this paper linear time invariant descriptor variable systems are considered which are either nonregular or if they are regular they have impulsive modes it is shown that if a certain necessary and sufficient condition is satisfied then both regularity can be established and impulsive modes can be eliminated by output feedback we have also considered the problem of achieving structural properties of the regularized system

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

closed orbits for actions of maximal tori on homogeneous spaces let g be a real algebraic group defined over mathbb q let gamma be an arithmetic subgroup and let t be any torus containing a maximal mathbb r split torus we prove that the closed orbits for the action of t on g gamma admit a simple algebraic description in particular we show that if g is reductive an orbit tx gamma is closed if and only if x 1 tx is a product of a compact torus and a torus defined over mathbb q and it is divergent if and only if the maximal mathbb r split subtorus of x 1 tx is defined over mathbb q and mathbb q split our analysis also yields the following there is a compact k subset g gamma which intersects every t orbit if rank mathbb q g rank mathbb r g there are no divergent orbits for t

dimension estimates for the set of points with non dense orbit in homogeneous spaces let x g gamma where g is a lie group and gamma is a lattice in g and let u be a subset of x whose complement is compact we use the exponential mixing results for diagonalizable flows on x to give upper estimates for the hausdorff dimension of the set of points whose trajectories miss u this extends a recent result of kadyrov et al dyn syst 30 2 149 157 2015 and produces new applications to diophantine approximation such as an upper bound for the hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms generalizing an estimate due to broderick and kleinbock int j number theory 11 7 2037 2054 2015

algebraic properties of singular systems subject to decentralized output feedback this paper investigates algebraic properties of decentralized singular systems dsss under local static output feedback new concepts of the geometric multiplicity gm of the finite decentralized fixed mode dfm the decentralized output feedback variable polynomial dvp and the finite decentralized output feedback cycle index dci of the dss are defined the formulas for determining the gm and the dci are given in terms of the dfm and the system matrices it is shown that almost any decentralized output feedback can make the zeros of the dvp i e the closed loop poles that are variable distinct and away from any given finite set in the complex plane it is also shown that the finite dfm of the dss are those uncontrollable and or observable finite modes of the closed loop dss through any single channel finally the minimal number of the inputs and outputs that guarantee the finite modes of the closed loop dss controllable and observable is shown to be the finite dci of the dss an illustrative example is provided

elasto plasticity revisited numerical analysis via reproducing kernel particle method and parametric quadratic programming aiming to simplify the solution process of elasto plastic problems this paper proposes a reproducing kernel particle algorithm based on principles of parametric quadratic programming for elasto plasticity the parametric quadratic programming theory is useful and effective for the assessment of certain features of structural elasto plastic behaviour and can also be exploited for numerical iteration examples are presented to illustrate the essential aspects of the behaviour of the model proposed and the flexibility of the coupled parametric quadratic programming formulations with the reproducing kernel particle method copyright 2002 john wiley sons ltd

badly approximable points on manifolds this paper is motivated by two problems in the theory of diophantine approximation namely davenport s problem regarding badly approximable points on submanifolds of a euclidean space and schmidt s problem regarding the intersections of the sets of weighted badly approximable points the problems have been recently settled in dimension two but remain open in higher dimensions in this paper we develop new techniques that allow us to tackle them in full generality the techniques rest on lattice points counting and a powerful quantitative result of bernik kleinbock and margulis the main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of mathbb r n has full dimension one of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree

a fictitious domain method using equilibrated basis functions for harmonic and bi harmonic problems in physics abstract in this paper we present a new meshless method to solve well known problems in physics and engineering with either constant or variable material properties in 2d space harmonic and bi harmonic problems are considered in this paper the method constructs a set of basis functions called equilibrated basis functions eqbfs through a weighted residual integration over a fictitious domain embedding the main one the bases can satisfy the governing partial differential equations approximately chebyshev polynomials of the first kind are employed to construct the eqbfs and exponential functions are used as the weights in the integrals the parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1d ones over a normalized intervals this reduces the computational efforts significantly either the eqbfs or the results of the integration process may be stored for further use the validity of the results is examined through an extended patch test a set of physical sample problems with variable non variable material properties as the potential flow over a cylinder steady state heat conduction problems in an anisotropic inhomogeneous functionally graded material potential and stokes flow through a channel of finite width obstructed by periodic cylinders and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method as a preliminary study we show that the method may effectively be used in a domain decomposition approach

modelling three dimensional piece wise homogeneous domains using the α shape based natural element method in this paper the application of the natural element method nem to the numerical analysis of two and three dimensional piece wise homogeneous domains is presented the nem differs from other meshless methods in its capability to accurately reproduce essential boundary conditions along convex boundaries the shape based extension of this method nem generalizes this behaviour to non convex domains enables us to construct models entirely in terms of the initial cloud of points and allows us to simulate material discontinuities in a straightforward manner in the following sections simple and effective algorithms are presented for the construction of shapes in domains composed of various materials examples are presented in two and three dimensional cases in the context of linear elastostatics showing good performance even with the simple numerical quadrature used copyright 2002 john wiley sons ltd

a meshfree method for transverse vibrations of anisotropic plates a meshfree approach called displacement boundary method for anisotropic kirchhoff plate dynamic analysis is presented this method is deduced from a variational principle which uses a modified hybrid functional involving the generalized displacements and generalized tractions on the boundary and the lateral deflection in the domain as independent variables the discretization process is based on the employment of the fundamental solutions of the static problem operator for the expression of the variables involved in the functional the stiffness and mass matrices obtained for the dynamic model are frequency independent symmetric and positive definite and their computation involves boundary integrals of regular kernels only due to its features the final resolving system can be solved with the classical approaches by using standard numerical procedures to assess the formulation the free vibrations of some anisotropic plates were calculated and the results compared with those obtained using other solution techniques the present results are in good agreement with those found in the literature showing the accuracy and effectiveness of the proposed approach

coupling three field formulation and meshless mixed galerkin methods using radial basis functions in this work we solve the elliptic partial differential equation by coupling the meshless mixed galerkin approximation using radial basis function with the three field domain decomposition method the formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper

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

least squares collocation meshless method a finite point method least squares collocation meshless method is proposed except for the collocation points which are used to construct the trial functions a number of auxiliary points are also adopted unlike the direct collocation method the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least squares sense the moving least squares interpolant is used to construct the trial functions the computational effort required for the present method is in the same order as that required for the direct collocation while the present method improves the accuracy of solution significantly the proposed method does not require any mesh so that it is a truly meshless method three numerical examples are studied in detail which show that the proposed method possesses high accuracy with low computational effort copyright 2001 john wiley sons ltd

on the regularizability of singular systems the property of regularity of singular systems is not a feedback invariant to correct this deficiency a new property of regularizability is defined and geometric tests are given for it in terms of system matrices regularizability is shown to be the natural extension of regularity which is a condition on the homogeneous system to controlled singular systems definitions of controllability and reachability are modified depending on regularizability rather than regularity a brief comparison of proportional and proportional plus derivative feedback laws in the context of making the closed loop system regular regular and reachable and regular and controllable is also given dynamical interpretations of these properties are also presented

applications of point interpolation method for spatial general shells structures the meshfree point interpolation method pim based on polynomial function is presented for the static analysis of spatial general shell structures the formulation of the discrete system equations is derived from stress resultant geometrically exact shell theory of shear flexible shells based on the cosserat surface the pim technique constructs its interpolation functions through a set of arbitrarily distributed points in the problem domain and its shape function has the delta function property hence the implementation of essential boundary conditions can be imposed with ease as in conventional finite element method an improved matrix triangularization algorithm mta is proposed to obtain proper node enclosure and basis selection in order to overcome the singularity problem in the point interpolation method using a polynomial basis it is found that the shape functions are generally not compatible to ensure the compatibility of the solution and lead to non conforming pim npim when the energy principles are used to formulate pim to obtain compatible shape functions the enforcement of compatibility is needed along the common edges of the background cells in the problem domain and a background cell based nodal selection is utilized which lead to conforming pim cpim both npim and cpim are evaluated in the paper several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the npim and cpim the phenomena of shear locking and membrane locking are illustrated by showing the membrane and shear energies as fractions of the total energy the convergences and performances are investigated for both conforming and non conforming pims it is also shown that the npim and cpim with the improved mta are very easy to implement and very efficient in constructing shape functions in comparison with the element free galerkin method

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

multidimensional continued fractions dynamical renormalization and kam theory the disadvantage of traditional multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors following ideas of dani lagarias and kleinbock margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space sl d mathbb z backslash sl d mathbb r the space of lattices of covolume one that indeed yields best possible approximations to any irrational vector the algorithm is ideally suited for a number of dynamical applications that involve small divisor problems as an example we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension

an output feedback precompensator for nonlinear dae systems with control dependent state space this note considers singular systems of nonlinear differential and algebraic equations daes whose constrained state space depends on the control inputs a state space realization of such systems cannot be derived independently of the controller design an output feedback precompensator is derived which results in a modified dae system whose state space is invariant under any feedback control law and can be used for output feedback controller synthesis its application is illustrated by a nonlinear electrical circuit example

robust admissibility of uncertain switched singular systems this article investigates the robust admissibility of uncertain discrete time switched singular systems first the admissibility is introduced by using the switched lyapunov function for singular systems sufficient conditions for robust admissibility of uncertain switched singular systems are presented in strict linear matrix inequalities formulation robust admissibility condition designs for both state feedback and static output control feedback are then derived numerical examples are provided to illustrate our approach

meshless local petrov galerkin method for two dimensional nonlinear water wave problems in this paper the meshless local petrov galerkin mlpg method is extended to dealing with nonlinear water wave problems the formulation is based on general fluid governing equations and a time marching procedure at each time step the boundary value problem for the pressure is solved using the mlpg method and the velocity and position of nodes are updated by numerical integration the newly extended method is applied to simulating water waves generated by a wave maker and good agreement with analytical solutions and finite element results is presented

new concepts in meshless methods meshless methods have been extensively popularized in literature in recent years due to their flexibility in solving boundary value problems two kinds of truly meshless methods the meshless local boundary integral equation mlbie method and the meshless local petrov galerkin mlpg approach are presented and discussed both methods use the moving least squares approximation to interpolate the solution variables while the mlbie method uses a local boundary integral equation formulation and the mlpg employs a local symmetric weak form the two methods are truly meshless ones as both of them do not need a finite element or boundary element mesh either for purposes of interpolation of the solution variables or for the integration of the energy all integrals can be easily evaluated over regularly shaped domains in general spheres in three dimensional problems and their boundaries numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable in essence the present meshless method based on the lswf is found to be a simple efficient and attractive method with a great potential in engineering applications copyright 2000 john wiley sons ltd

hausdorff dimension of the set of singular pairs in this paper we show that the hausdorff dimension of the set of singular pairs is 4 3 we also show that the action of diag e t e t e 2t on sl 3 r sl 3 z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates answering a question of a n starkov as a by product of the analysis we obtain a higher dimensional generalization of the basic inequalities satisfied by convergents of continued fractions as an illustration of the technique used to compute hausdorff dimension we reprove a result of i j good asserting that the hausdorff dimension of the set of real numbers with divergent partial quotients is 1 2

an extension of quantitative nondivergence and applications to diophantine exponents we present a sharpening of nondivergence estimates for unipotent or more gener ally polynomial like flows on homogeneous spaces applied to metric diophantine approxima tion it yields precise formulas for diophantine exponents of affine subspaces ofr n and their nondegenerate submanifolds

refined study on structural controllability of descriptor systems by means of matroids for a dynamical system described in the descriptor form fd bf x dt a bf x b bf u where the coefficients are classified into generic parameters and fixed constants the structural controllability is investigated under a physically reasonable assumption that can be justified by the dimensional analysis a necessary and sufficient condition for the structural controllability is given in matroid theoretic terms the condition can be tested by efficient algorithms for the matroid union intersection problem

local radial point interpolation mlrpi method for solving time fractional diffusion wave equation with damping the purpose of the current investigation is to determine numerical solution of time fractional diffusion wave equation with damping for caputo s fractional derivative of order 1 2 a meshless local radial point interpolation mlrpi scheme based on galerkin weak form is analyzed the reason of choosing mlrpi approach is that it does not require any background integrations cells instead integrations are implemented over local quadrature domains which are further simplified for reducing the complication of computation using regular and simple shape the unconditional stability and convergence with order o 6 2 are proved where is time stepping also several numerical experiments are illustrated to verify theoretical analysis

free mesh radial basis function collocation approach for the numerical solution of system of multi ion electrolytes the main objective of this work is to show the versatility of an innovative mesh free numerical scheme which allows the solution of the coupled transport equations governing the evolution of a system of multi ion electrolytes by mean of the use of radial basis functions interpolation and a nesting approach the nesting approach is used in order to capture the detail behaviour of the solution near a solid wall i e at the diffusion reaction layer as a numerical example the problem of silver deposition on a rotating disk is analysed the versatility of the proposed numerical scheme is proved by solving the proposed electrochemical phenomena imposing three different kinds of boundary conditions at the solid wall dirichlet neumann and robin the obtained numerical results are compared with simplified analytical approximation of the corresponding problems the effect of the ligand concentration on the others species is considered and a comparison between the obtained concentration profiles for the complex ags2o and ag s2o3 when the concentration of the ligand s2o takes different values is shown copyright 2005 john wiley sons ltd

random walks on homogeneous spaces and diophantine approximation on fractals we extend results of y benoist and j f quint concerning random walks on homogeneous spaces of simple lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily zariski dense but satisfies some expansion properties for the adjoint action using these dynamical results we study diophantine properties of typical points on some self similar fractals in mathbb r d as examples we show that for any self similar fractal mathcal k subseteq mathbb r d satisfying the open set condition for instance any translate or dilate of cantor s middle thirds set or of a koch snowflake almost every point with respect to the natural measure on mathcal k is not badly approximable furthermore almost every point on the fractal is of generic type which means in the one dimensional case that its continued fraction expansion contains all finite words with the frequencies predicted by the gauss measure we prove analogous results for matrix approximation and for the case of fractals defined by mobius transformations

flows on homogeneous spaces and diophantine approximation on manifolds we present a new approach to metric diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces this approach yields a new proof of a conjecture of mahler originally settled by v g sprindzuk in 1964 we also prove several related hypotheses of baker and sprindzuk formulated in 1970s the core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices

a local search algorithm for natural neighbours in the natural element method a local basis algorithm for searching natural neighbours in natural element method nem is presented for solving the elasticity problems in this paper comparison with the global sweep algorithm used in natural element method or natural neighbour method nnm for searching natural neighbours the proposed algorithm is more expedient and convenient in the constructions and computation of natural neighbour interpolations in the proposed nem based on local search the laplace non sibson interpolations are constructed with respect to the natural neighbour nodes of the given point which have been locally defined the shape functions from the laplace approximations have the delta function property and the laplace interpolants are strictly linear between adjacent nodes which facilitate imposition of essential boundary conditions and treatment of material discontinuity with ease as it is in the conventional finite element method the laplace interpolants derived from the local algorithm and the global algorithm in nem are identical because of the uniqueness of the voronoi diagram numerical results and convergence studies also show that the present nem based on local search algorithm possesses the same accuracy and rate of convergence as they are in previous nem

flows on homogeneous spaces and diophantine properties of matrices we generalize the notions of badly approximable resp singular systems of m linear forms in n variables and relate these generalizations to certain bounded resp divergent trajectories in the space of lattices in rm n

expanding translates of curves and dirichlet minkowski theorem on linear forms we show that a multiplicative form of dirichlet s theorem on simulta neous diophantine approximation as formulated by minkowski cannot be improved for almost all points on any analytic curve on r k which is not contained in a proper affine subspace such an investigation was initiated by davenport and schmidt in the late sixties the diophantine problem is then settled via showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in r k 1 gets equidistributed in the limit we use ratner s theorem on unipotent flows linearization techniques and a new observation about intertwined linear dynamics of various sl m r s contained in sl k 1 r

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov