wavelet based integral representation for solutions of the wave equation an integral representation of solutions of the wave equation as a superposition of other solutions of this equation is built the solutions from a wide class can be used as building blocks for the representation considerations are based on mathematical techniques of continuous wavelet analysis the formulae obtained are justified from the point of view of distribution theory a comparison of the results with those by g kaiser is carried out methods of obtaining physical wavelets are discussed

highly localized solutions of the wave equation simple explicit solutions of the linear wave equation in three dimensions are presented which describe wave packets exponentially localized near a point moving with the wave speed for large values of a certain free parameter these new solutions are localized in gaussian manner with respect to longitudinal and transverse variables and time this agrees with considerations by babich ulin and ralston who have presented an asymptotic description of solutions exhibiting such local behavior global estimates and large time asymptotics of these solutions are given

symmetric successive overrelaxation methods for solving the rank deficient linear least squares problem symmetric successive overrelaxation ssor methods are developed for finding the least squares solution of minimal norm to the linear systemax b where email protected c m x n is a matrix of rank r and email protected c m by first augmenting the system to a block 4x4 consistent system and then splitting the augmented coefficient matrix by subproper ssor splittings we obtain intervals for the relaxation parameter in which the subproper ssor iteration matrix is semiconvergent and a solution for the augmented system subsequently a method for transforming such a solution to the least squares solution of minimal 2 norm is presented

successive matrix squaring algorithm for computing the drazin inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 this paper derives a successive matrix squaring sms algorithm to approximate the drazin inverse which can be expressed in the form of successive squaring of a composite matrix t given an n by n matrix a the study shows that the drazin inverse of a can be computed in parallel time ranging from o logn to o log 2n provided that there are enough processors to support matrix multiplication in time o logn the sms algorithm is generalized to higher order schemes where the composite matrix is repeatedly raised to an integer power l 2 this form of expression leads to a simplified notation compared to that of earlier methods we argue that there is no obvious advantage in choosing l other than 2 our derived error bound for the approximation of a d is new

symmetric block sor methods for rank deficient least squares problems in this article we develop symmetric block successive overrelaxation s block sor methods for finding the solution of the rank deficient least squares problems we propose an s2 block sor and an s3 block sor method for solving such problems and the convergence of these two methods is studied the comparisons between the s2 block and the s3 block methods are presented with some numerical examples

symmetric successive overrelaxation methods for rank deficient linear systems in this paper we develop symmetric successive overrelaxation symmetric sor or ssor methods for finding the least square solution of minimal norm to the linear system ax b where a is an mxn matrix of rank r the methods are obtained by first augmenting the system to a block 4x4 consistent system the augmented coefficient matrix is then split by a subproper ssor splitting we state and prove some theorems and by some numerical examples we show the number of iterations for ssor is less than sor and accelerated overrelaxation methods for finding the least square solution of minimal norm to the linear system

the penrose polynomial of a plane graph let g v e be a connected plane graph with vertex set v edge set e set of regions r and let g r e be its dual graph the 4 color theorem 4 ct for short states that the regions of g can always be colored with 4 colors such that adjacent regions receive different colors 3 16 it is well known that it suffices to verify the 4 ct for connected 3 regular bridgeless plane graphs and that in this case the number of 3 edge colorings equals the number of 4 region colorings with the outer region colored 0 see e g 2 thus the number of 3 edge colorings is an important parameter to study in 15 penrose derived a number of remarkable expressions for this parameter defining implicitly what is now called the penrose polynomial p g in the variable among other things he showed for a 3 regular connected plane graph

wavelet methods for pdes some recent developments abstract this paper is concerned with recent developments of wavelet schemes for the numerical treatment of operator equations with special emphasis on two issues adaptive solution concepts and nontrivial domain geometries after describing a general multiresolution framework the key features of wavelet bases are highlighted namely locality norm equivalences and cancellation properties assuming first that wavelet bases with these properties are available on the relevant problem domains the relevance of these features for a wide class of stationary problems is explained in subsequent sections the main issues are preconditioning and the efficient adaptive application of wavelet representations of the involved operators we indicate then how these ingredients combined with concepts from nonlinear or best n term approximation culminate in an adaptive wavelet scheme for elliptic selfadjoint problems covering boundary value problems as well as boundary integral equations these schemes can be shown to exhibit convergence rates that are in a certain sense asymptotically optimal we conclude this section with some brief remarks on data structures and implementation interrelations with regularity in a certain scale of besov spaces and strategies of extending such schemes to unsymmetric or indefinite problems we address then the adaptive evaluation of nonlinear functionals of wavelet expansions as a central task arising in connection with nonlinear problems wavelet bases on nontrivial domains are discussed next the main issues are the development of fourier free construction principles and criteria for the validity of norm equivalences finally we indicate possible combinations of wavelet concepts with conventional discretizations such as finite element or finite volume schemes in connection with convection dominated and hyperbolic problems

comments on symmetric successive overrelaxation methods for rank deficient linear systems applied mathematics and computation 173 2006 404 420 abstract in this note we present a critical review on the paper symmetric successive overrelaxation methods for rank deficient linear systems by authors darvishi and khosro aghdam m t darvishi r khosro aghdam symmetric successive overrelaxation methods for rank deficient linear systems appl math comput 173 2006 404 420 we point out that a sufficient condition for the semiconvergence of ssor iteration matrix one of the main results in their paper is not true and its proof is unacceptable in fact this problem has been thoroughly solved and one nice set of necessary and sufficient conditions has been given out by us see b zheng k wang symmetric successive overrelaxation methods for solving the rank deficient linear least squares problem appl math comput 169 2005 1305 1323

twisted duality for embedded graphs we consider two operations on an edge of an embedded graph or equivalently a ribbon graph giving a half twist to the edge and taking the partial dual with respect to the edge these two operations give rise to an action of s3 je g j the ribbon group on g the action of the ribbon group on embedded graphs extends the concepts of duality partial duality and petrie duality we show that this ribbon group action gives a complete characterization of duality in that if g is any cellularly embedded graph with medial graph gm then the orbit of g under the group action is precisely the set of all graphs with medial graphs isomorphic as abstract graphs to gm we provide characterizations of special sets of twisted duals such as the partial duals of embedded graphs in terms of medial graphs and we show how dierent kinds of graph isomorphism give rise to these various notions of duality the ribbon group action then leads to a deeper understanding of the properties of and relationships among various graph polynomials via the generalized transition polynomial which interacts naturally with the ribbon group action ribbon group action on the set of embedded graphs by taking duals and twisting individual edges the ribbon group action completes the classical concept of duality for embedded graphs and we exploit this group action to unravel and explore connections between embedded graphs and their medial graphs and to determine implications of these new relations the ribbon group action allows us to classify the twisted duals of an embedded graph characterize all graphs with isomorphic medial graphs under various kinds of isomorphism extend the concept of a tait graph to all 4 regular graphs and to determine new properties of polynomials of embedded graphs there are many investigations into planar and abstract graphs that have not yet been fully extended to the context of graphs embedded in surfaces consider for example the classical relations among the medial graphs and the duals of plane graphs suppose that g is a plane graph g is its dual and gm is its medial graph the medial graph of g is exactly the medial graph of g i e g m gm where denotes equality as plane graphs in fact the connection between geometric duals and medial graphs is a little stronger than this the two graphs g and g are the only plane graphs that have gm as their plane medial graphs that is

the drazin inverse of a semilinear transformation and its matrix representation abstract the drazin inverse t of a semilinear transformation t on is studied a canonical form for the matrix as of td is given and some of its properties are derived 1 introduction if a is a square complex matrix the drazin 5 inverse ad of a is the unique matrix solution to the equations ap x at xax x ax xa for some p 0 the drazin inverse has applications in the theory of finite markov chains 4 the study of singular differential and difference equations 4 the investigation of cesaroneumann iterations 7 cryptography 8 g iterative methods in numerical analysis 14 151 quadratic dynamic program ming 18 and others it has also been used recently to show the equivalence of consimilarity and semiconsimihuity of complex matrices 2 given a square matrix a there are actually two ways to proceed we can uniquely induce a linear transformation a or on the other hand uniquely

convergence of a direct iterative method for large scale least squares problems abstract in 1975 chen and gentleman suggested a 3 block sor method for solving least squares problems based on a partitioning scheme for the observation matrix a into a a 1 a 2 where a1 is square and nonsingular in many cases a1 is obvious from the nature of the problem this combined direct iterative method was discussed further and applied to angle adjustment problems in geodesy where a1 is easily formed and is large and sparse by plemmons in 1979 recently niethammer de pillis and varga have rekindled interest in this method by correcting and extending the sor convergence interval the purpose of our paper is to discuss an alternative formulation of the problem leading to a 2 block sor method for this formulation it is shown that the resulting direct iterative method always converges for sufficiently small sor parameter in contrast to the 3 block formulation formulas for the optimum sor parameter and the resulting asymptotic convergence factor based upon a2a 11 2 are given furthermore it is shown that this 2 cyclic block sor method always gives better convergence results than the 3 cyclic one for the same amount of work per iteration the direct part of the algorithm requires only a sparse matrix factorization of a1 our purpose here is to establish theoretical convergence results in line with the purpose of the recent paper by niethammer de pillis and varga practical considerations of choosing a1 in certain applications and of estimating the resulting a2a 11 2 will be addressed elsewhere

on accelerate overrelaxation methods for rank deficient linear systems tian h tian accelerate overrelaxation methods for rank deficient linear systems appl math comput 140 2003 485 499 presented an accelerated overrelaxation aor method for solving the rank deficient linear least squares problem ax b by augmenting the system to a block 4x4 consistent system and splitting the coefficient matrix into a subproper splitting where a c m x n is a matrix of rank r and b c m with this splitting one set of sufficient conditions for the semiconvergence of the aor iteration matrix are presented by tian in this paper we further investigate the aor method for solving such linear problem by using an new subproper splitting and the necessary and sufficient conditions for the semiconvergence of the aor iteration matrix are explicitly given out a numerical example shows that the method is applicable and efficient for solving such rank deficient linear system

successive overrelaxation methods for solving the rank deficient linear squares problem abstract we develop successive overrelaxation sor methods for finding the least squares solution of minimal norm to the system ax b where a is an m n matrix of rank r the methods are obtained by first augmenting the system to a block 4 4 consistent system of linear equations the augmented coefficient matrix is then split by a subproper sor splitting an interval for the relaxation parameter in which the subproper sor iteration matrix is semiconvergent is determined along with the optimal relaxation parameter which minimizes the modulus of the controlling eigenvalue of the sor matrix since the scheme computes at first only a solution to the augmented system it is subsequently shown how to transform such a solution to the unique solution of minimal 2 norm analysis of the practical implementation of the algorithms developed here will be given elsewhere

new discrete time znn models for least squares solution of dynamic linear equation system with time varying rank deficient coefficient in this brief a new one step ahead numerical differentiation rule called six instant g cube finite difference 6i g cfd formula is proposed for the first order derivative approximation with higher precision than existing finite difference formulas i e euler and taylor types subsequently by exploiting the proposed 6i g cfd formula to discretize the continuous time zhang neural network model two new type discrete time znn dtznn models namely new type dtznnk and dtznnu models are designed and generalized to compute the least squares solution of dynamic linear equation system with time varying rank deficient coefficient in real time which is quite different from the existing znn related studies on solving continuous time and discrete time dynamic or static linear equation systems in the context of full rank coefficients specifically the corresponding dynamic normal equation system of which the solution exactly corresponds to the least squares solution of dynamic linear equation system is elegantly introduced to solve such a rank deficient least squares problem efficiently and accurately theoretical analyses show that the maximal steady state residual errors of the two new type dtznn models have an o g 4 pattern where g denotes the sampling gap comparative numerical experimental results further substantiate the superior computational performance of the new type dtznn models to solve the rank deficient least squares problem of dynamic linear equation systems

block sor methods for rank deficient least squares problems many papers have discussed preconditioned block iterative methods for solving full rank least squares problems however very few papers studied iterative methods for solving rank deficient least squares problems miller and neumann 1987 proposed the 4 block sor method for solving the rank deficient problem here a 2 block sor method and a 3 block sor method are proposed to solve such problem the convergence of the block sor methods is studied the optimal parameters are determined comparison between the 2 block sor method and the 3 block sor method is given also

convergence of block iterative methods applied to sparse least squares problems abstract recently special attention has been given in the mathematical literature to the problems of accurately computing the least squares solutions of very large scale overdetermined systems of linear equations such as those arising in geodetical network problems in particular it has been suggested that one solve such problems iteratively by applying the block sor successive overrelaxation iterative method to a consistently ordered block jacobi matrix that is weakly cyclic of index 3 here we obtain new results theorem 1 giving the exact convergence and divergence domains for such iterative applications it is then shown how these results extend and correct the literature on such applications in addition analogous results theorem 2 are given for the case when the eigenvalues of the associated block jacobi matrix are nonnegative

a necessary and sufficient condition for semiconvergence and optimal parameter of the ssor method for solving the rank deficient linear least squares problem let a c m n r be partitioned as a a 11 a 21 a 12 a 22 where a 11 c r r r write b a 21 a 1 11 and c a 1 11 a 12 suppose that b 0 for finding the minimum norm least squares solution a b of the linear systems ax b many authors studied the sor aor and ssor methods for solving the augmented systems az b 1 1 and obtained many results in this paper we deeply study the ssor method whose iteration matrix is written as j and prove the following new conclusions 1 if b 1 then j is semiconvergent 0 2 if b 1 then j is semiconvergent 0 0 2 u 1 2 where 2 1 b 1 b 1 and 1 1 b 1 b 1 2 the optimal parameters of j are min j min max j 1 1 1 2 1 2 2 b 2 1 1 b 2 in addition we obtain other results concerning the sor aor and ssor methods

on astigmatic exponentially localized solutions for the wave and the klein gordon fock equations highly localized explicit solutions to the wave and klein gordon fock multidimensional linear equations are presented their fourier transforms are also found explicitly solutions depend on a set of parameters and demonstrate astigmatic properties asymptotic analysis for large and moderate time shows that constructed particle like solutions have gaussian localization near a point moving with the group speed

determination of the optimal value of relaxation parameter in symmetric sor method for rectangular coefficient matrices in this paper we obtain the optimal value of relaxation parameter in symmetric sor ssor method by ssor method we find the least square solution of minimal norm to the linear system ax b where a is an m by n matrix

on the interplay between embedded graphs and delta matroids the mutually enriching relationship between graphs and matroids has motivated discoveries in both fields in this paper we exploit the similar relationship between embedded graphs and delta matroids there are well known connections between geometric duals of plane graphs and duals of matroids we obtain analogous connections for various types of duality in the literature for graphs in surfaces of higher genus and delta matroids using this interplay we establish a rough structure theorem for delta matroids that are twists of matroids we translate petrie duality on ribbon graphs to loop complementation on delta matroids and we prove that ribbon graph polynomials such as the penrose polynomial the characteristic polynomial and the transition polynomial are in fact delta matroidal we also express the penrose polynomial as a sum of characteristic polynomials

accelerate overrelaxation methods for rank deficient linear systems in this paper we apply accelerated overrelaxation aor methods to find the least square solution of minimal norm to the linear systemay bwhere a c m x n is a matrix of rank r and b c m we first augment the system to a block 4x4 consistent system and then split the augmented coefficient matrix by aor subproper splitting intervals for the two relaxation parameters where the aor iteration matrix is semiconvergent are presented also we provide a method to compute the least square solution of minimal norm to the system

new physical wavelet gaussian wave packet an exact solution of the homogeneous wave equation which was found previously is treated from the point of view of continuous wavelet analysis cwa if time is a fixed parameter the solution represents a new multidimensional mother wavelet for the cwa both the wavelet and its fourier transform are given by explicit formulae and are exponentially localized the wavelet is directional the widths of the wavelet and the uncertainty relation are investigated numerically if a certain parameter is large the wavelet behaves asymptotically as the morlet wavelet the solution is a new physical wavelet in the definition of kaiser it may be interpreted as a sum of two parts an advanced and a retarded part both being fields of a pulsed point source moving at a speed of wave propagation along a straight line in complex spacetime