a new reiterated structure with optimal macroscopic behavior we consider homogenization of sequences of integral functionals of the type mathcal f h u int omega f h x du x dx omega subset mathbf r n the functions fh are defined in the first variable via a new reiterated structure with m microlevels in particular our results imply that the corresponding homogenized functional becomes optimal as m turns to infinity both the scalar case the conductivity problem and the vector valued case the elasticity problem are considered

matrices a such that as 1r ra with rk i abstract we study matrices a c n n such that a s 1 r r a where r k i n and s k are nonnegative integers with k 2 such matrices are called r s 1 k potent matrices the s 0 case corresponds to matrices such that a r a r 1 with r k i n and is studied using spectral properties of the matrix r for s 1 various characterizations of the class of r s 1 k potent matrices and relationships between these matrices and other classes of matrices are presented

multigrid methods for propagators in lattice gauge theories multigrid methods were invented for the solution of discretized partial differential equations in ordered systems the slowness of traditional algorithms is overcome by updates on various length scales in this overview article we discuss generalizations of multigrid methods for disordered systems in particular for propagators in lattice gauge theories a discretized nonabelian gauge theory can be formulated as a system of statistical mechanics where the gauge field degrees of freedom are su n matrices on the links of the lattice these su n matrices appear as random coefficients in dirac equations we aim at finding an efficient method by which one can solve dirac equations without critical slowing down if this could be achieved monte carlo simulations of quantum chromodynamics the theory of the strong interaction would be accelerated considerably in principle however the methods discussed can be used in arbitrary space time dimension and for arbitrary gauge group moreover there are applications in multigrid monte carlo simulations and for the definition of block spins and blocked gauge fields in monte carlo renormalization group studies as a central result it was found that the geometric multigrid method works in principle in arbitrarily disordered gauge fields finally an overview is given of other approaches to the propagator problem in lattice gauge theories

invariant properties of the stress in plane elasticity and equivalence classes of composites attention is drawn to the invariance of the stress field in a two dimensional body loaded at the boundary by fixed forces when the compliance tensor scr g chi is shifted uniformly by ell text i lambda lambda where lambda is an arbitrary constant and scr g text i kappa mu is the compliance tensor of a isotropic material with two dimensional bulk and shear moduli kappa and mu this invariance is explained from two simple observations first that in two dimensions the tensor scr g text i frac 1 2 frac 1 2 acts to locally rotate the stress by 90 degrees and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain for composite materials the invariance of the stress field implies that the effective compliance tensor ell ast also gets shifted by scr g text i lambda lambda when the constituent moduli are each shifted by ell text i lambda lambda this imposes constraints on the functional dependence of ell ast on the material moduli of the components applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1 lambda and the inverse shear modulus is shifted by 1 lambda then the inverse effective bulk and shear moduli undergo precisely the same shifts in particular it explains why the effective young s modulus of a two dimensional media with holes does not depend on the poisson s ratio of the matrix material

convergence of an iterative method for total variation denoising in total variation denoising one attempts to remove noise from a signal or image by solving a nonlinear minimization problem involving a total variation criterion several approaches based on this idea have recently been shown to be very effective particularly for denoising functions with discontinuities this paper analyzes the convergence of an iterative method for solving such problems the iterative method involves a lagged diffusivity approach in which a sequence of linear diffusion problems are solved global convergence in a finite dimensional setting is established and local convergence properties including rates and their dependence on various parameters are examined

on disjoint range matrices abstract for a square complex matrix f and for f being its conjugate transpose the class of matrices satisfying r f r f 0 where r denotes range column space of a matrix argument is investigated besides identifying a number of its properties several functions of f such as f f f f ff f f and f f are considered particular attention is paid to the moore penrose inverses of those functions and projectors attributed to them it is shown that some results scattered in the literature whose complexity practically prevents them from being used to deal with real problems can be replaced with much simpler expressions when the ranges of f and f are disjoint furthermore as a by product of the derived formulae one obtains a variety of relevant facts concerning for instance rank and range

characterizations and linear combinations of k generalized projectors abstract we extend generalized projectors introduced by gros and trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 to k generalized projectors and we characterize them obtaining results in the aforesaid paper as a consequence moreover we list all situations when a linear combination of commuting k generalized projectors is a k generalized projector the method for obtaining this result permits to give a revisited version of the main result by baksalary and baksalary on linear combinations of generalized projectors linear algebra appl 388 2004 17 24 in addition the case of orthogonal projectors is also analyzed

heat flows and related minimization problem in image restoration abstract a new anisotropic diffusion model is proposed for image restoration and segmentation which is closely related to the minimization problems for the unconstrained total variation e u x u 2 u i 2 existence uniqueness and stability of the viscosity solutions of the equation are proved the experimental results are given and compared with the existing models in the framework of image restoration the improvement on preserving sharp edges by using the new model is visible

acceleration methods for total variation based image denoising in this paper we apply a fixed point method to solve the total variation based image denoising problem an algebraic multigrid method is used to solve the corresponding linear equations krylov subspace acceleration is adopted to improve convergence in the fixed point iteration a good initial guess for this outer iteration at finest grid is obtained by combining fixed point iteration and geometric multigrid interpolation successively from the coarsest grid to the finest grid numerical experiments demonstrate that this method is efficient and robust even for images with large noise to signal ratios

element free amge general algorithms for computing interpolation weights in amg we propose a new general algorithm for constructing interpolation weights in algebraic multigrid amg it exploits a proper extension mapping outside a neighborhood about a fine degree of freedom dof to be interpolated the extension mapping provides boundary values based on the coarse dofs used to perform the interpolation at the boundary of the neighborhood the interpolation value is then obtained by matrix dependent harmonic extension of the boundary values into the interior of the neighborhood we describe the method present examples of useful extension operators provide a two grid analysis of model problems and by way of numerical experiments demonstrate the successful application of the method to discretized elliptic problems

some results on partial ordering and reverse order law of elements of c algebras abstract in this paper we establish some results relating star left star right star minus ordering and the reverse order law under certain conditions on moore penrose invertible elements of c algebras

on a partial order defined by the weighted moore penrose inverse the weighted moore penrose inverse of a matrix can be used to define a partial order on the set of mxn complex matrices and to introduce the concept of weighted ep matrices in this paper we study the weighted star partial order on the set of weighted ep matrices in addition some properties that relate the eigenprojection at zero with the weighted star partial order are obtained

building blocks for odd even multigrid with applications reduced to systems abstract building blocks yielding an efficient implementation of the odd even multigrid method for the poisson problem in the reference domain 0 1 d d 2 3 are described modifications needed to transform these techniques to solve reduced linear systems representing boundary value problems in arbitrary domains are given a new way to define enriched coarser subspaces in the multilevel realization is proposed numerical examples demonstrating the efficiency of developed multigrid methods are included

fitzpatrick functions and continuous linear monotone operators the notion of a maximal monotone operator is crucial in optimization as it captures both the subdifferential operator of a convex lower semicontinuous and proper function and any not necessarily symmetric continuous linear positive operator it was recently discovered that most fundamental results on maximal monotone operators allow simpler proofs utilizing fitzpatrick functions in this paper we study fitzpatrick functions of continuous linear monotone operators defined on a hilbert space a novel characterization of skew operators is presented a result by brezis and haraux is reproved using the fitzpatrick function we investigate the fitzpatrick function of the sum of two operators and we show that a known upper bound is actually exact in finite dimensional and more general settings cyclic monotonicity properties are also analyzed and closed forms of the fitzpatrick functions of all orders are provided for all rotators in the euclidean plane

the star partial order and the eigenprojection at 0 on ep matrices abstract the space of n n complex matrices with the star partial order is considered in the first part of this paper the class of ep matrices is analyzed and several properties related to this order are given in addition some information about predecessors and successors of a given ep matrix is obtained the second part is dedicated to the study of some properties that relate the eigenprojection at 0 with the star and sharp partial orders

proof of a conjecture on the conductivity of checkerboards in 1985 mortola and steffe conjectured a formula for the effective conductivity tensor of a checkerboard structure where the unit cell of periodicity is square and subdivided into four equal squares each having a different conductivity in this article their conjecture is proven the key idea is to superimpose suitably reflected potentials to obtain the solution to the dual problem this is then related back to the original problem using a well known theorem of keller thereby proving the conjecture the analysis also yields formulas relating the potentials in the four squares independently craster and obnosov have obtained a completely different proof of the conjecture

on dissipative matrices abstract a matrix the imaginary part of whose toeplitz cartesian decomposition is positive semidefinite positive definite is said to be dissipative strictly dissipative characterizations for strictly dissipative matrices and related theory are given dissipative preserving linear transformations are characterized finally a natural generalization to linear transformations on matrices is developed in terms of the completely positive cone ten characterizations follow

on inverse eigenvalue problems for block toeplitz matrices with toeplitz blocks we propose an algorithm for solving the inverse eigenvalue problem for real symmetric block toeplitz matrices with symmetric toeplitz blocks it is based upon an algorithm which has been used before by others to solve the inverse eigenvalue problem for general real symmetric matrices and also for toeplitz matrices first we expose the structure of the eigenvectors of the so called generalized centrosymmetric matrices then we explore the properties of the eigenvectors to derive an efficient algorithm that is able to deliver a matrix with the required structure and spectrum we have implemented our ideas in a matlab code numerical results produced with this code are included

a multilevel algorithm for simultaneously denoising and deblurring images in this paper we develop a fast multilevel algorithm for simultaneously denoising and deblurring images under the total variation regularization although much effort has been devoted to developing fast algorithms for the numerical solution and the denoising problem was satisfactorily solved fast algorithms for the combined denoising and deblurring model remain to be a challenge recently several successful studies of approximating this model and subsequently finding fast algorithms were conducted which have partially solved this problem the aim of this paper is to generalize a fast multilevel denoising method to solving the minimization model for simultaneously denoising and deblurring our new idea is to overcome the complexity issue by a detailed study of the structured matrices that are associated with the blurring operator a fast algorithm can then be obtained for directly solving the variational model supporting numerical experiments on gray scale images are presented

new time dependent model for image restoration in this paper we propose a new time dependent model for solving total variation tv minimization problems in image restoration we present a proof of the existence uniqueness and stability of the viscosity solution of our model the results from our new model by explicit numerical schemes are compared with the results from those models introduced by several image researchers by simple and significant 2d numerical experiments experimental results demonstrate that our new model can get better results than the existing models

relation between effective conductivity and susceptibility of two component rhombic checkerboard the heterogeneity of composite leads to extra charge concentration at the boundaries of different phases which results in essentially nonzero effective electric susceptibility the relation between tensors of effective electric susceptibility ef and effective conductivity ef of the infinite two dimensional two component regular composite with rhombic cell structure has been established the degrees of electric field singularity at corner points of cells are found by constructing the integral equation for the effective conductivity problem the limits of weak and strong contrast of partial conductivities 1 2 are considered the results are valid for thin films and cylindrical samples

factorizations of ep operators abstract in this paper we characterize ep operators through the existence of different types of factorizations our results extend to ep operators existing characterizations for ep matrices and give new characterizations both for ep matrices and ep operators

periodic heterogeneous structures new explicit solutions and effective characteristics of refraction of an imposed field complex variable methods are applied to study three double periodic two phase planar heterogeneous structures explicit expressions of functionals of energy dissipation and of effective resistivities conductivities are found on the basis of strict analytical solutions of a boundary value problem for an arbitrary direction of an imposed external field comparisons with the results of bakhvalov and panasenko homogenization of processes in periodic media nauka moscow 1984 dykhne zh eksp teor fiz 59 1970 pp 110 113 soviet phys jetp 32 1971 pp 63 65 emets zh eksp teor fiz 96 1989 pp 701 711 soviet phys jetp 96 1989 keller j appl phys 34 1963 pp 991 993 and mendelson j appl phys 46 1975 pp 4740 4741 are made in particular the dissipation in the general case is shown essentially to depend on the direction of the external field for a rectangular checkerboard dissipation within the two phases does not coincide it is rigorously pr

the effective conductivity of random checkerboards an algorithm is presented for the fast and accurate solution of the electrostatic equation on multi component random checkerboards it relies on a particular choice of integral equation extended as to separate ill conditioning due to singular fields in corners from ill conditioning due to interaction of clusters of well conducting squares at large distances two separate preconditioners take care of the two separate phenomena in a series of numerical examples effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106 the achievable relative precision in these examples is on the order of 10 11

krylov subspace acceleration of nonlinear multigrid with application to recirculating flows this paper deals with the combination of two solution methods multigrid and gmres siam j sci comput 14 1993 pp 856 869 the generality and parallelizabili nonlinear pdes as the preconditioner for a nonlinear krylov subspace method we use the full approximation storage fas scheme math comp 31 1977 pp 333 390 a nonlinear multigrid method the nonlinear krylov acceleration is applied also on coarse grids so that recirculating incompressible flow problems discretized with a higher order upwind scheme can be solved efficiently

on the moore penrose inverse ep banach space operators and ep banach algebra elements abstract the main concern of this note is the moore penrose inverse in the context of banach spaces and algebras especially attention will be given to a particular class of elements with the aforementioned inverse namely ep banach space operators and banach algebra elements which will be studied and characterized extending well known results obtained in the frame of hilbert space operators and c algebra elements

matrices with multiple symmetry properties applications of centrohermitian and perhermitian matrices abstract twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system we investigate matrices satisfying more than one symmetry pattern we show how a group of operators on gl n c gives rise to distinct types of matrices which satisfy sets of patterns and which give unique decompositions of matrices into components of each type these give a new characterization of normal and unitary matrices we extend symmetry patterns to vectors to study spectral properties of these matrices when a skew symmetric basis of eigenvectors exist we can infer symmetry properties of these matrices

factorization of weighted ep elements in c algebras abstract we present characterizations of weighted ep elements in c algebras using different kinds of factorizations

characterizations of normal hyponormal and ep operators abstract in this paper normal and hyponormal operators with closed ranges as well as ep operators are characterized in arbitrary hilbert spaces all characterizations involve generalized inverses thus recent results of s cheng and y tian s cheng y tian two sets of new characterizations for normal and ep matrices linear algebra appl 375 2003 181 195 are extended to infinite dimensional settings

more on maximal and minimal ranks of schur complements with applications the maximal and minimal ranks of the schur complement d cgb are determined with respect to generalized inverse g of a matrix a where g is taken respectively as least squares g inverse a 1 3 minimum norm g inverse a 1 4 as well as a 1 2 3 a 1 2 4 and a 1 3 4 various consequences are also presented

involutions for matrices and generalized inverses abstract all involutions on the set of all matrices with complex entries are described it is established when every matrix possesses a generalized inverse for an involution this generalized inverse is then applied to the least squares solution to a system of linear equations using more general inner products

a nonlinear primal dual method for total variation based image restoration we present a new method for solving total variation tv minimization problems in image restoration the main idea is to remove some of the singularity caused by the nondifferentiability of the quantity nabla u in the definition of the tv norm before we apply a linearization technique such as newton s method this is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function which can be interpreted as the normal vector to the level sets of the image u our method can be viewed as a primal dual method as proposed by conn and overton a primal dual interior point method for minimizing a sum of euclidean norms preprint 1994 and andersen ph d thesis odense university denmark 1995 for the minimization of a sum of euclidean norms in addition to possessing local quadratic convergence experimental results show that the new method seems to be globally convergent

fourier analysis of gmres m preconditioned by multigrid this paper deals with convergence estimates of gmres m saad and schultz siam j sci statist comput 7 1986 pp 856 869 preconditioned by multigrid brandt math comp 31 1977 pp 333 390 hackbusch multi grid methods and applications springer berlin 1985 fourier analysis is a well known and useful tool in the multigrid community for the prediction of two grid convergence rates brandt math comp 31 1977 pp 333 390 stuben and trottenberg in multigrid methods lecture notes in math 960 k stuben and u trottenberg eds springer berlin pp 1 176 this analysis is generalized here to the situation in which multigrid is a preconditioner since it is possible to obtain the whole spectrum of the two grid iteration matrix a preconditioned krylov subspace acceleration method like gmres m implicitly builds up a minimal residual polynomial the determination of the polynomial coefficients is easily possible and can be done explicitly since from fourier analysis a simple block diagonal two grid iteration matrix results based on the gmres m polynomial sharp theoretical convergence estimates can be obtained which are compared with estimates based on the spectrum of the iteration matrix several numerical scalar test problems are computed in order to validate the theoretical predictions

drazin moore penrose invertibility in rings abstract characterizations are given for elements in an arbitrary ring with involution having a group inverse and a moore penrose inverse that are equal and the difference between these elements and ep elements is explained the results are also generalized to elements for which a power has a moore penrose inverse and a group inverse that are equal as an application we consider the ring of square matrices of order m over a projective free ring r with involution such that r m is a module of finite length providing a new characterization for range hermitian matrices over the complexes

characterizations of ep matrices and weighted ep matrices abstract a square complex matrix a is said to be ep if a and its conjugate transpose a have the same range in this paper we first collect a group of known characterizations of ep matrix and give some new characterizations of ep matrices then we define weighted ep matrix and present a wealth of characterizations for weighted ep matrix through various rank formulas for matrices and their generalized inverses

on a matrix decomposition of hartwig and spindelbock abstract the main task of the paper is to demonstrate that corollary 6 in r e hartwig k spindelbock matrices for which a and a commute linear and multilinear algebra 14 1984 241 256 provides a powerful tool to investigate square matrices with complex entries this aim is achieved on the one hand by obtaining several original results involving square matrices and on the other hand by reestablishing some of the facts already known in the literature often in extended and or generalized forms the particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings

on recombining iterants in multigrid algorithms and problems with small islands this article deals with nearly singular problems and with problems having small islands or other small features not visible on a coarse grid in which cases usual multigrid solvers converge slowly or even diverge in these and many other problems the main difficulty is related to a few badly approximated components the numerical experiments reported below show that by recombining multigrid iterants such components are eliminated effectively and the usual multigrid asymptotic convergence rate is restored

characterizations of ep and normal banach algebra elements and banach space operators abstract several characterizations of ep and normal moore penrose invertible banach algebra elements will be considered the banach space operator case will be also studied the results of the present article will extend well known facts obtained in the frames of matrices and hilbert space operators

four phase checkerboard composites two dimensional periodic rectangular checkerboard media are considered in the situation where mean fluxes are prescribed across the structure the closed form solution is obtained in the general case where the checkerboard is constructed using four rectangular cells each having a different constant resistivity this four cell structure then repeats doubly periodically to cover the whole plane this general solution is then used to calculate the effective properties thus this four phase checkerboard encapsulates many limiting and special cases as a starting point we develop a concise closed form solution to a basic problem involving four joined quarter planes each of a different resistivity subsequent manipulations yield solutions to problems posed in increasingly convoluted domains while retaining the essentially simple structure found for joined quarter planes

a geometric multigrid solver for the free surface equation in environmental models featuring irregular coastlines a recently developed multigrid method botto 2013 based on the concept of volume fraction has been tested for the inversion of the helmholz type equation for the free surface in the environmental public domain code coherens the volume fraction concept is particularly interesting for coarse grid cells that are agglomerated from both dry and wet fine grid cells at irregular coastlines at these locations modifying the prolongation operator and the coarse grid discretization using the volume fraction results in better convergence however as convergence deteriorates in the case of small elongated islands that tend to disappear by the multigrid coarsening procedure a correction is proposed yielding good convergence rates irrespective of the presence of small or large islands the method is tested extensively for the inversion of the academic poisson equation larger test cases solving the helmholz type equation prove the applicability for real life applications of environmental flows

a lattice boltzmann method for image denoising in this paper we construct a lattice boltzmann scheme to simulate the well known total variation based restoration model that is rof model the advantages of the lattice boltzmann method include the fast computational speed and the easily implemented fully parallel algorithm a conservative property of the lb method is discussed the macroscopic pde associated with the lb algorithm is derived which is just the rof model moreover the linearized stability of the method is analyzed the numerical computations demonstrate that the lb algorithm is efficient and robust even though the quality of the restored images is slightly lower than those by using the rof model the restored images of the lb method are satisfactory furthermore computational speed of the lb method is much faster than rof model in general cpu time of the lb method for restored images is about one tenth of rof model

an adaptive algorithm for tv based model of three norms lq q 12 1 2 in image restoration abstract in this paper we present an adaptive method for the tv based model of three norms l q q 1 2 1 2 for the image restoration problem the algorithm with the l2 norm is used in the smooth regions where the value of u is small the algorithm with the l 1 2 norm is applied for the jumps where the value of u is large when the value of u is moderate the algorithm with the l1 norm is employed thus the three algorithms are applied for different regions of a given image such that the advantages of each algorithm are adopted the numerical experiments demonstrate that our adaptive algorithm can not only keep the original edge and original detailed information but also weaken the staircase phenomenon in the restored images specifically in contrast to the l1 norm as in the rudin osher fatemi model the l2 norm yields better results in the smooth and flat regions and the l 1 2 norm is more suitable in regions with strong discontinuities therefore our adaptive algorithm is efficient and robust even for images with large noises

rank equalities and inequalities for kronecker products of matrices with applications some rank equalities and inequalities are established for kronecker products of matrices as applications upper and lower bounds are presented for the dimensions of the ranges of the two well known linear transformations t 1 x x axb and t 2 x ax xb

a high order accurate algorithm for electrostatics of overlapping disks a fast and accurate algorithm for the computation of effective electric and mathematically equivalent properties of composites with nonsmooth interfaces is reported the algorithm is based on an integral equation reformulation of the electrostatic partial differential equation and a fast hierarchical technique for potential field evaluation in a numerical example 200 large and strongly inhomogeneous aggregates of randomly overlapping disks are solved with a relative error of 0 0005

exact solution of a boundary value problem for a rectangular checkerboard field a class of two phase composite materials with a biperiodic structure is investigated by the methods of complex analysis two interface conditions continuity of normal component of a desired vector w and tangential component of w at the contact boundary as well as the double periodicity condition are involved in rigorous form the exact analytic solution of the corresponding generalized riemann boundary value problem is obtained the explicit values of the effective parameters namely effective resistivity and dissipation of energy of an elementary cell and resistivities along the symmetry axes are calculated in closed analytic form the coincidence of our formulae with the well known effective resistivity conductivity formula of keller 1964 dykhne 1970 and mendelson 1975 and the dissipation formula of dykhne 1970 is shown in the case of square checkerboard field the keller 1963 identity is generalized for the heterogeneous structure studied

conduction in a two phase plane with diamond shaped tiling the effective conductivity of a two phase two dimensional composite with diamond shaped tiling is considered this analysis based on a projection of the boundary conditions on linear combinations of solutions to the electrostatic equation that are orthonormal on the boundary generalizes results by keller j math phys 28 2516 1987 and others numerical results are given for several conductivity ratios of the two phases and for varying obtuse angles of the tiles special emphasis is given to very large and very small conductivity differences and very elongated tiles

a three phase tessellation solution and effective properties two dimensional doubly periodic three phase structures are considered in the situation where mean fluxes are applied across the structure the approach is to utilize complex variables and to use a mapping that reduces the doubly periodic problem to a m uch simpler one involving joined sectors this is a model composite structure in electrostatics and mathematically analogous areas such as porous media anti plane elasticit y heat conduction and we find various eectiv e parameters and investigate limiting cases the structure is also amenable to asymptotic methods in the case of highly varying composition and we provide these solutions partly as a check upon our analysis and partly as they are useful in their own right

normal matrices an update abstract a list of seventy conditions on an n n complex matrix a equivalent to its being normal published nearly ten years ago by grone johnson sa and wolkowicz has proved to be very useful hoping that in an extended form it will be even more helpful we compile here another list of about twenty conditions they either have been overlooked by the authors of the original list or have appeared during the last decade

reflectors on vector spaces abstract a function on a vector space v which is both additive and involutory is said to be a reflector on v for spec x case1 2 x x the range of spec is called the speculum of it naturally generalizes among many others the classes of hermitian perhermitian and real matrices this paper gives some basic results for reflectors followed by results for reflectors which preserve or invert the specula of other reflectors

r s conjugate solution to a pair of linear matrix equations abstract let r and s be m m and n n nontrivial real symmetric involutions an m n complex matrix a is termed r s conjugate if a ras where a denotes the conjugate of a in this paper necessary and sufficient conditions are established for the existence of the r s conjugate solution to the system of matrix equations ax c and xb d the expression is also presented for such solution to this system in addition the explicit expression of this solution to the corresponding optimal approximation problem is obtained furthermore the least squares r s conjugate solution with least norm to this system mentioned above is considered the representation of such solution is also derived finally an algorithm and numerical examples are given

expansion formulae for the homogenized determinant of anisotropic checkerboards in this paper some effective properties of anisotropic four phase periodic checkerboards are studied in two dimensional electrostatics an explicit low contrast second order expansion for the determinant of the effective conductivity is given in the case of a twophase checkerboard with commuting conductivities the expansion reduces to an explicit formula for the effective determinant valid for any contrast as soon as the second order term vanishes such an explicit formula cannot be extended to four phase checkerboards a counter example with high contrast conductivities is provided the construction of the counter example is based on a factorization principle due to astala nesi which allows us to pass from an anisotropic four phase square checkerboard to an isotropic one with the same effective determinant

relations between k s 1 potent matrices and different classes of complex matrices in this paper k s 1 potent matrices are considered a matrix a cn n is called k s 1 potent when kas 1k a where k is an involutory matrix and s 1 2 3 specifically k s 1 potent matrices are analyzed considering their relations to different classes of complex matrices these classes of matrices are s 1 generalized projectors k hermitian matrices normal matrices and matrices b cn n anti commuting with k or such that kb is involutory hermitian or normal in addition some new relations for k generalized centrosymmetric matrices have been derived

convergence of fixed point iteration for deblurring and denoising problem abstract to solve deblurring and denoising problem we present a proof of the global and linear convergence of fixed point iteration as well as an estimate for the rate of convergence is given we show the equivalence among three different iterative methods half quadratic regularization iteration based on bergman distance weiszfeld s method

weighted partial isometries and weighted ep elements in c algebras we investigate weighted partial isometries weighted ep elements weighted star dagger weighted normal and weighted hermitian elements of c algebras

moore penrose invertible normal and hermitian elements in rings in this paper we present several new characterizations of normal and hermitian elements in rings with involution in purely algebraic terms and considerably simplify proofs of already existing characterizations

an inverse eigenvalue problem and an associated approximation problem for generalized k centrohermitian matrices a partially described inverse eigenvalue problem and an associated optimal approximation problem for generalized k centrohermitian matrices are considered it is shown under which conditions the inverse eigenproblem has a solution an expression of its general solution is given in case a solution of the inverse eigenproblem exists the optimal approximation problem can be solved the formula of its unique solution is given

geometry of total variation regularized l p model in this paper the geometry and scale selection properties of the total variation tv regularized l p model are rigorously analyzed some intrinsic features different from the tv l 1 model are derived and demonstrated numerical algorithms based on recent developed augmented lagrangian methods are implemented and numerical results consistent with the theoretical results are provided

rank equalities for idempotent and involutary matrices abstract we establish several rank equalities for idempotent and involutary matrices in particular we obtain new formulas for the rank of the difference the sum the product and the commutator of idempotent or involutary matrices extensions to scalar potent matrices are also included our matrices are complex and are not necessarily hermitian

on the algebraic multigrid method new formulations for the algebraic multigrid amg method are presented a new interpolation operator is developed in which the weighting could be negative numerical experiments demonstrate that the use of negative interpolation weights is necessary in some applications new approaches to construct the restriction operator and the coarse grid equations are discussed two new amg methods are proposed theoretical study and convergence analysis of the amg methods are presented the main contributions of this paper are to improve the convergence rate and to extend the range of applications of an amg method numerical experiments are reported for matrix computations that resulted from partial differential equations signal processing and queueing network problems the success of the proposed new amg algorithms is clearly demonstrated by applications to non diagonally dominant matrix problems for which the standard amg method fails to converge

using rank formulas to characterize equalities for moore penrose inverses of matrix products a group of rank formulas related to the moore penrose inverses ab and b a are established various interesting consequences of these rank formulas are obtained the extensions of these results to the weighted moore penrose inverses ab m n and b p na m p are considered

some properties of generalized k centrosymmetric h matrices every nxn generalized k centrosymmetric matrix a can be reduced into a 2x2 block diagonal matrix see z liu h cao h chen a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices appl math comput 169 2 2005 1332 1345 this block diagonal matrix is called the reduced form of the matrix a in this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices finally exploiting these properties the development of structure preserving algorithms for certain computations for generalized k centrosymmetric h matrices is discussed

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

moore penrose inverses and commuting elements of c algebras abstract let a be an element of a c algebra a satisfying a a a a where a is the moore penrose inverse of a and let b a we study several spectral properties concerning the commutativity of a and b we deduce some new and old results concerning ep matrices

semigroups of ep linear transformations abstract some known results about ep matrices are studied in the setting of a finite dimensional inner product space over r or c a characterization of semigroups of ep linear maps is given the results obtained are generalized for bounded ep operators on hilbert spaces

reverse order laws for the generalized inverses of multiple matrix products by using the ranks of matrices this article gives necessary and sufficient conditions for the reverse product a n a 2a 1of the moore penrose inverses of a1 a2 an to a 1 inverse inner inverse 1 2 inverse reflexive inner inverse 1 3 inverse 1 4 inverse 1 2 3 inverse 1 2 4 inverse or moore penrose inverse of matrix product a a1a2 an

on the matrix reorderings γ and ψ abstract we discuss eight natural matrix reorderings motivated by the two which are central to the theory of 1 and 6 these mappings are used to relate dyad and kronecker products of matrices some of their algebraic properties and their properties as linear transformations are explored

ep elements in rings in this paper we present a number of new characterizations of ep elements in rings with involution in purely algebraic terms and considerably simplify proofs of already existing characterizations

a simple multigrid scheme for solving the poisson equation with arbitrary domain boundaries we present a new multigrid scheme for solving the poisson equation with dirichlet boundary conditions on a cartesian grid with irregular domain boundaries this scheme was developed in the context of the adaptive mesh refinement amr schemes based on a graded octree data structure the poisson equation is solved on a level by level basis using a one way interface scheme in which boundary conditions are interpolated from the previous coarser level solution such a scheme is particularly well suited for self gravitating astrophysical flows requiring an adaptive time stepping strategy by constructing a multigrid hierarchy covering the active cells of each amr level we have designed a memory efficient algorithm that can benefit fully from the multigrid acceleration we present a simple method for capturing the boundary conditions across the multigrid hierarchy based on a second order accurate reconstruction of the boundaries of the multigrid levels in case of very complex boundaries small scale features become smaller than the discretization cell size of coarse multigrid levels and convergence problems arise we propose a simple solution to address these issues using our scheme the convergence rate usually depends on the grid size for complex grids but good linear convergence is maintained the proposed method was successfully implemented on distributed memory architectures in the ramses code for which we present and discuss convergence and accuracy properties as well as timing performances

on k ep matrices abstract the concept of k ep matrix is introduced relations between k ep and ep matrices are discussed necessary and sufficient conditions are determined for a matrix to be k ep r

twosetsofnewcharacterizationsfornormaland ep matrices two sets of new characterizations for normal matrices and ep matrices are presented which are derived through ranks and generalized inverses of matrices

a note on computing matrix vector products with generalized centrosymmetric centrohermitian matrices to reduce the costs of computing matrix vector product ax related to a centrosymmetric matrix a as compared to the case of an arbitrary matrix a two algorithms were proposed recently one was designed by melman a melman symmetric centrosymmetric matrix vector multiplication linear algebra appl 320 2000 193 198 for symmetric centrosymmetric matrices another was presented by fassbender and ikramov h fassbender k d ikramov computing matrix vector products with centrosymmetric and centrohermitian matrices linear algebra appl 364 2003 235 241 for general centrosymmetric matrices in this note we further discuss this topic of computing ax where a is a generalized centrosymmetric matrix we firstly investigate the reducibility of a generalized centrosymmetric matrix and then provide an algorithm which can be viewed as a generalization of one h fassbender k d ikramov computing matrix vector products with centrosymmetric and centrohermitian matrices linear algebra appl 364 2003 235 241 we show that our algorithm is suitable for computing many matrix vector products with the same matrix furthermore we show that similar results can be obtained for certain subclasses of generalized centrosymmetric matrices as compared to the case of a centrosymmetric matrix and analogue gains are available for generalized skew centrosymmetric or generalized centrohermitian matrices

new characterizations of ep generalized normal and generalized hermitian elements in rings abstract we present a number of new characterizations of ep elements in rings with involution in purely algebraic terms then we study equivalent conditions for an element a in a ring with involution to satisfy a n a a a n or a n a n for arbitrary n n for n 1 we present some new characterizations of normal and hermitian elements in rings with involution

on κ real and κ hermitian matrices abstract a body of theory for real and hermitian matrices is developed some basic results for these matrices their spectral properties and characterizations of linear transformations which preserve them are given

effective medium approximation and exact formulae for electrokinetic phenomena in porous media electrokinetic phenomena in porous media are studied by application of the effective medium theory and the theory of duality transformation we deduce new exact relations and analytical formulae for the effective constants of the macroscopic tensor we also prove that the effective tensors obtained by these approaches coincide for 2d problems the obtained results for the electrokinetic processes are closely related to similar results derived for piezoelectric composites because of a common mathematical background