a new representation for at s 2 3 a new representation of a t s 2 3 of a having prescribed range space t and null space s is derived using this representation the well known generalized inverses such as the moore penrose inverse the group inverse the drazin inverse the bott duffin inverse and the generalized bott duffin inverse are computed three numerical examples are worked out to demonstrate the computation of the moore penrose inverse the drazin inverse and the 2 3 inverse to show the efficacy of our approach

a higher order scheme for two dimensional quasi static crack growth simulations an efficient scheme for the simulation of quasi static crack growth in two dimensional linearly elastic isotropic specimens is presented the crack growth is simulated in a stepwise manner where an extension to the already existing crack is added in each step in a local coordinate system each such extension is represented as a polynomial of some user specified degree n the coefficients of the polynomial describing an extension are found by requiring that the mode ii stress intensity factor is equal to zero at certain points of the extension if a crack grows from a pre existing crack so that a kink develops the leading term describing the crack shape close to the kink will in a local coordinate system be proportional to x 3 2 we therefore allow the crack extensions to contain such a term in addition to the monomial terms the discontinuity in the crack growth direction at a kink the kink angle is determined by requiring that the mode ii stress intensity factor should be equal to zero for an infinitesimal extension of the existing crack to implement the scheme accurate values of the stress intensity factors and t stress are needed in each step of the simulation these fracture parameters are computed using a previously developed integral equation of the second kind c 2007 elsevier b v all rights reserved less

on holomorphic banach vector bundles over banach spaces let x be a banach space with a countable unconditional basis e g x l2 hilbert space x pseudoconvex open e a locally trivial holomorphic vector bundle with a banach space z for fiber type mathcal o e the sheaf of germs of holomorphic sections of e and z1 the banach space z 1 ell p z z z n z n in z z big sum n 1 infty z n p big 1 p infty 1 le p infty then e oplus omega times z 1 and z1 are holomorphically isomorphic mathcal o e is acyclic and e is so to speak stably trivial over in a generalized sense we also show that if e is continuously trivial over then e is holomorphically trivial over in particular if z l2 or is contractible then e is holomorphically trivial over some applications are also given

integral equation methods and numerical solutions of crack and inclusion problems in planar elastostatics we present algorithms for crack and inclusion problems in planar linear elastostatics the algorithms are based on new integral equations for the pure crack problem the integral equations are of fredholm s second kind our algorithms show great stability and allow for solutions to problems more complex than have previously been possible our results are orders of magnitudes more accurate than those of previous investigators which rely on integral equations of fredholm s first kind

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

an embedding method for modeling micromechanical behavior and macroscopic properties of composite materials this paper presents a numerical method for modeling the micromechanical behavior and macroscopic properties of fiber reinforced composites and perforated materials the material is modeled by a finite rectangular domain containing multiple circular holes and elastic inclusions the rectangular domain is assumed to be embedded within a larger circular domain with fictitious boundary loading represented by truncated fourier series the analytical solution for the complementary problem of a circular domain containing holes and inclusions is obtained by using a combination of the series expansion technique with a direct boundary integral method the boundary conditions on the physical external boundary are satisfied by adopting an overspecification technique based on a least squares approximation all of the integrals arising in the method can be evaluated analytically as a result the elastic fields and effective properties can be expressed explicitly in terms of the coefficients in the series expansions several numerical experiments are conducted to verify the accuracy and efficiency of the numerical method and to demonstrate its application in determination of the macroscopic properties of composite materials

the representation and approximation for drazin inverse we present a unified representation theorem for drazin inverse specific expression and computational procedures for drazin inverse can be uniformly derived numerical examples are tested and the application to the singular linear equations is also considered

representation and approximation of the outer inverse at s 2 of a matrix a abstract in this paper we establish a basic representation and a representation theorem for the outer inverse a t s 2 of a matrix a which is the matrix x satisfying xax x r x t and n x s we develop several specific representations and iterative methods for a t s 2 we show that this representation includes many of the traditional generalized inverses and outer inverses and the relation of our results to these outer inverses will be explored

corner singularities for elliptic problems integral equations graded meshes quadrature and compressed inverse preconditioning we take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points use of non standard integral equations graded meshes interpolatory quadrature and compressed inverse preconditioning are techniques that are explored developed mixed and tested on some familiar problems in materials science the recursive compressed inverse preconditioning the major novelty of the paper turns out to be particularly powerful and when it applies eliminates the need for mesh grading completely in an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 1 2 in another example five times as large but with a conductivity ratio of only a hundred we achieve an estimated relative error of 10 1 4

on the evaluation of layer potentials close to their sources when solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources close to the sources this is not so cancellation and nearly singular kernels may cause serious degradation this paper presents a new scheme based on a mix of composite polynomial quadrature layer density interpolation kernel approximation rational quadrature high polynomial order corrected interpolation and differentiation temporary panel mergers and splits and a particular implementation of the gmres solver criteria for which mix is fastest and most accurate in various situations are also supplied the paper focuses on the solution of the dirichlet problem for laplace s equation in the plane in a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close to touching contours densities are computed and potentials are evaluated rapidly and accurate to almost machine precision at points that lie arbitrarily close to the boundaries

on the numerical solution of the biharmonic equation in the plane the biharmonic equation arises in a variety of problems in applied mathematics most notably in plane elasticity and in viscous incompressible flow integral equation methods are natural candidates for the numerical solution of such problems since they discritize the boundary alone are easy to apply in the case of free or moving boundaries and achieve superalgebraic convergence rates on sufficiently smooth domains regardless of shape in this paper we follow the work of mayo and greenbaum and make use of the sherman lauricella integral equation which is a fredholm equation with bounded kernel we describe a fast algorithm for the evaluation of the integral operators appearing in that equation when combined with a conjugate gradient like algorithm we are able to solve the discretized integral equation in an amount of time proportional to n where n is the number of nodes in the discretization of the boundary

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

stress calculations on multiply connected domains the outstanding problem of finding a simple muskhelishvili type integral equation for stress problems on multiply connected domains is solved complex potentials are represented in a way which allows for the incorporation of cracks and inclusions several numerical examples demonstrate the generality and extreme stability of the approach the stress field is resolved with a relative error of less than 10 10 on a large yet simply reproducible setup with a loaded square plate containing 4096 holes and cracks comparison with previous results in the literature indicates that general purpose finite element software may perform better than many special purpose codes

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

open sets with the runge property in banach spaces we show that every pseudoconvex open subset of a banach space with an unconditional schauder basis has the f valued runge property for every banach space f

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

an iterative method for solving general restricted linear equations the newton iterative method for computing outer inverses with prescribed range and null space is used in the non stationary richardson iterative method to develop an iterative method for solving general restricted linear equations starting with any suitably chosen initial iterate our method generates a sequence of iterates converging to the solution the necessary and sufficient conditions for the convergence along with the error bounds are established the applications of the iterative method for solving some special linear equations are also discussed a number of numerical examples are worked out they include singular square rectangular randomly generated rank deficient matrices full rank matrices and a set of singular matrices given in matrix computation toolbox mctoolbox with the condition numbers ranging from order 1016 to 1050 the mean cpu time mct and the error bounds are the performance measures used our results when compared with the results obtained by chen 1997 leads to substantial improvement in terms of both computational speed and accuracy

an embedding theorem for pseudoconvex domains in banach spaces we show that a pseudoconvex open subset of a banach space with an unconditional basis is biholomorphic to a closed direct submanifold of a banach space with an unconditional basis

singularities in 2d anisotropic potential problems in multi material corners real variable approach an analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane the concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi material corner problems explicit forms of eigenequations for evaluation of singularity exponent in the case of multi material corners are derived both for all combinations of homogeneous neumann and dirichlet boundary conditions at faces of open corners and for multi material planes with singular interior points perfect transmission conditions at wedge interfaces are considered in both cases it is proved that singularity exponents are real for open anisotropic multi material corners and a sufficient condition for the singularity exponents to be real for anisotropic multi material planes is deduced a case of a complex singularity exponent for an anisotropic multi material plane is reported apparently for the first time in potential theory simple expressions of eigenequations are presented first for open bi material corners and bi material planes and second for a crack terminating at a bi material interface as examples of application of the theory developed here analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces and also for a special case of a crack perpendicular to a bi material interface a numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi material interface is presented

the singular finite element method for some elliptic boundary value problem with interface a kind of numerical method is proposed for some elliptic boundary value problems with interface it includes mainly two steps at the first step we obtain the approximation of the singularity near the singular points by solving a simple eigenvalue problem which is one dimension less than the original problem at the second step we apply the approximation of the singularity together with the standard finite element basis functions to construct the singular finite element space and finally solve the original problem on a conventional mesh some numerical examples show the effectiveness of this method

a numerical model of stress driven grain boundary diffusion the stress driven grain boundary diffusion problem is a continuum model of mass transport phenomena in microelectronic circuits due to high current densities electromigration and gradients in normal stress along grain boundaries the model involves coupling many different equations and phenomena and difficulties such as nonlocality stiffness complex geometry and singularities in the stress tensor near corners and junctions make the problem difficult to analyze rigorously and simulate numerically we present a new numerical approach to this problem using techniques from semigroup theory to represent the solution the generator of this semigroup is the composition of a type of dirichlet to neumann map on the grain boundary network with the laplace operator on the network to compute the former we solve the equations of linear elasticity several times once for each basis function on the grain boundary we resolve singularities in the stress field near corners and junctions by adjoining special singular basis functions to both finite element spaces 2d for elasticity 1d for grain boundary functions we develop data structures to handle jump discontinuities in displacement across grain boundaries singularities in the stress field complicated boundary conditions at junctions and interfaces and the lack of a natural ordering for the nodes on a branching grain boundary network the method is used to study grain boundary diffusion for several geometries

an accurate boundary value problem solver applied to scattering from cylinders with corners in this paper we consider the classic problem of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries the scattering problems are formulated as integral equations and solved using a nystrom scheme where the corners of the cylinders are efficiently handled by a method referred to as recursively compressed inverse preconditioning rcip this method has been very successful in treating static problems in nonsmooth domains and the present paper shows that it works equally well for the helmholtz equation in the numerical examples we focus on scattering of e and h waves from a cylinder with one corner even at a size kd 1000 where k is the wavenumber and d the diameter the scheme produces at least 13 digits of accuracy in the electric and magnetic fields everywhere outside the cylinder

comments on some recent results concerning 2 3 and 2 4 generalized inverses abstract we give a comment on some recent results concerning the representations of generalized 2 3 and 2 4 inverses shorter proofs of some previous results are presented

the representation of generalized inverse at s 2 3 and its applications this paper presents the explicit expression for matrix right symmetry factor with prescribed rang and null space moreover the explicit expression for generalized inverse a t s 2 3 which is a 2 3 inverse of a having the prescribed rang t and null space s is derived as an application two numerical examples are given

the dolbeault complex in infinite dimensions i infinite dimensional complex analysis was a popular subject in the sixties and seventies but in the last fifteen years enjoyed much less attention oddly enough this very same period began to see the emergence of examples of infinite dimensional complex manifolds in mathematical physics representation theory and geometry thus it appears to be a worthwhile undertaking to revisit infinite dimensional complex analysis and in particular to clarify fundamental properties of infinite dimensional complex manifolds the most fundamental questions at this point seem to be related to the solvability of the inhomogeneous cauchy riemann or 0 equations and more generally to the study of the dolbeault complex as an example in le3 we use a result on the a equation to describe the virasoro group as an infinite dimensional complex manifold up to now precious little has been known about the solvability of the infinite dimensional equation the available results pertain to solving the equation in domains in or over locally convex topological vector spaces and almost exclusively on the level of 0 1 forms postponing the precise definitions to sections 1 2 below we discuss those results that are of immediate relevance to this paper see d2 for more in li ligocka observes that h6rmander s proof of ehrenpreis theorem see e hor can be extended to infinite dimensions to solve the equation

full rank representations of outer inverses based on the qr decomposition abstract an efficient algorithm for computing a t s 2 inverses of a given constant matrix a based on the qr decomposition of an appropriate matrix w is presented correlations between the derived representation of outer inverses and corresponding general representation based on arbitrary full rank factorization are derived in particular cases we derive representations of 2 4 and 2 3 inverses numerical examples on different test matrices dense or sparse are presented as well as the comparison with several well known methods for computing the moore penrose inverse and the drazin inverse

the representation and computation of generalized inverse at s 2 this paper presents a novel representation for the generalized inverse a t s 2 based on this we give an algorithm to compute this generalized inverse as an application we use gauss jordan elimination to compute the weighted moore penrose inverse a m n and the drazin inverse a d

matrix left symmetry factor and its applications in generalized inverses a 2 4 t s in this paper we propose some definitions about matrix symmetry factors and discuss the construction of matrix left symmetry factor in detail the explicit expression of the generalized inverses at s 2 4 which is a 2 4 inverses of a having the prescribed rang t and null space s is derived furthermore using the expression at s 2 4 we get the corresponding results for six important generalized inverses

full rank representations of 2 4 2 3 inverses and successive matrix squaring algorithm abstract we present the full rank representations of 2 4 and 2 3 inverses with given rank as well as with prescribed range and null space as particular cases of the full rank representation of outer inverses as a consequence two applications of the successive matrix squaring sms algorithm from p s stanimirovic d s cvetkovic ilic successive matrix squaring algorithm for computing outer inverses appl math comput 203 2008 19 29 are defined using the full rank representations of 2 4 and 2 3 inverses the first application is used to approximate 2 4 inverses the second application after appropriate modifications of the sms iterative procedure computes 2 3 inverses of a given matrix presented numerical examples clarify the purpose of the introduced methods

a regularized domain integral formulation for inclusion problems of various shapes by equivalent inclusion method abstract a method of regularized domain integral formulation with inclusion is presented to calculate the elastic stress strain field in an infinite body characterized by inclusions of various shapes with isotropic orthotropic or anisotropic material properties in the given domain integral the integrand containing the r 1 singularity is overcome by the well known subdivide technique when the collocation point is located on the interface between the inclusion and the matrix fictitious inclusion elements similar to the method used by dong and antes engrg anal bound elem 22 2 1998 133 are added to obtain a regularized domain integral formulation the results obtained by the present method are compared with those from other formulations and good agreement is observed among these results

new proofs of two representations and minor of generalized inverse a 2 t s abstract in this paper we prove the limit representation for the generalized inverse a t s 2 of a given complex matrix a in a simpler way then using this representation two known results the generalized inverse a t s 2 as an analogs of convex combination with the minor of adja and the representation of the minor of the generalized inverse a t s 2 were derived anew as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained

elastostatic computations on aggregates of grains with sharp interfaces corners and triple junctions we present a fast algorithm for the calculation of elastostatic fields in two dimensional assemblies of elastic grains separated by sharp grain boundaries the algorithm uses an integral equation approach combined with the fast multipole method and recursive compression to resolve stress concentrations also very close to grain boundary junctions singular basis functions on analytic form are not required accurate results can be obtained at a cost roughly proportional to the number of grains in the assembly large scale problems with thousands of grains are solved using modest computational resources