mixed type reverse order laws for 1 3 4 generalized inverses over hilbert spaces abstract the reverse order laws for 1 3 4 generalized inverses of a product of two operators have been studied by wang et al j wang h zhang g ji a generalized reverse order law for the products of two operators journal of shaanxi normal university 38 4 2010 13 17 in this paper using a block operator matrix technique we study mixed type reverse order laws for 1 3 4 generalized inverses over hilbert spaces

covering singular linear semi groups abstract let v be a vector space of dimension n n over a field k k 4 let v v be a singular linear transformation and a similarity class of nullity 1 let k n k n 4 then is a product of k elements of

reverse order law for reflexive generalized inverses of products of matrices abstract the relationships between generalized inverses of the product of two matrices a b and the product of generalized inverses of a b have been studied in the literature by applying the product singular value decomposition p svd in this paper we derive equivalent conditions for b 1 2 a 1 2 ab 1 2 ab 1 2 b 1 2 a 1 2 b 1 2 a 1 2 ab 1 2 and equivalent conditions for ab 1 b 1 a 1

the generalized moore penrose inverse abstract we define the generalized moore penrose inverse and give necessary and sufficient conditions for its existence over an integral domain we also prove its uniqueness and give a formula for it which leads us towards a generalized cramer s rule to find the generalized moore penrose solution

dislocation and point force based approach to the special green s function bem for elliptic hole and crack problems in two dimensions in this paper we give the theoretical foundation for a dislocation and point force based approach to the special green s function boundary element method and formulate as an example the special green s function boundary element method for elliptic hole and crack problems the crack is treated as a particular case of the elliptic hole we adopt a physical interpretation of somigliana s identity and formulate the boundary element method in terms of distributions of point forces and dislocation dipoles in the infinite domain with an elliptic hole there is no need to model the hole by the boundary elements since the traction free boundary condition there for the point force and the dislocation dipole is automatically satisfied the green s functions are derived following the muskhelishvili complex variable formalism and the boundary element method is formulated using complex variables all the boundary integrals including the formula for the stress intensity factor for the crack are evaluated analytically to give a simple yet accurate special green s function boundary element method the numerical results obtained for the stress concentration and intensity factors are extremely accurate 1997 john wiley sons ltd

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

further results on the reverse order law for the moore penrose inverse in rings with involution abstract we present some equivalent conditions of the reverse order law for the moore penrose inverse in rings with involution extending some well known results to more general settings then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted moore penrose inverse in c algebras

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

reverse order laws in c algebras abstract in this paper we offer purely algebraic necessary and sufficient conditions for reverse order laws for generalized inverses in c algebras extending rank conditions for matrices and range conditions for hilbert space operators

group moore penrose core and dual core inverse in rings with involution abstract let r be a ring with involution the recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary ring case it is shown that the group moore penrose core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents the several characterizations of these inverses are given some new properties are obtained and some known results are generalized a number of characterizations of ep elements in r are obtained it is shown that core and dual core inverse belong to the class of inverses along an element and to the class of b c inverses

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

products of idempotent and square zero matrices abstract this paper presents necessary and sufficient conditions for a square matrix over an arbitrary field to be expressed as a product of idempotent and square zero matrices since the similarity structures of such factors are completely determined by their rank we also investigate the ranks that these factors may have the case where the factors are all idempotent or all square zero have been investigated by various authors some of whom are listed in the references of this paper

piezoelectric solid with an elliptic inclusion or hole the two dimensional problem of an elliptic hole in a solid of anisotropic piezoelectric material is studied the stroh formalism is adopted here real form solutions are obtained along the hole boundary in the case of an arbitrarily prescribed vector field on the hole surface for an elliptic rigid inclusion of electric conductor subjected to a line force a torque and a line charge a real form solution at the interface is obtained finally general solutions for an elliptic piezoelectric inclusion with uniform loading at infinity are investigated

a short proof of botha s theorem on products of idempotent linear mappings abstract we give an elementary self contained proof to botha s theorem on the factorization of a singular transformation into idempotent mappings

plane problems in piezoelectric media with defects abstract a two dimensional electroelastic analysis is performed on a transversely isotropic piezoelectric material containing defects a general solution is provided in terms of complex potentials with emphasis being placed on stress concentrations that arise in the vicinity of circular and elliptical holes it is shown that for this genre of problem both mechanical and electrical variables are responsible for the peak stresses

the absorption laws for the generalized inverses abstract in this paper we give necessary and sufficient conditions for the absorption laws in terms of 1 1 2 1 3 and 1 4 inverses also we consider the various types of mixed absorption law for the generalized inverses

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

the two dimensional electroelasticity problems for multiconnected bodies situated under electric potential difference action abstract solutions of the two dimensional electroelasticity problems for infinitely long piezoelectric cylinders with cavities and cracks under electrical potential difference action are obtained using the lekhnitskii s generalized complex potentials herewith piezoelectric bodies under consideration are free from mechanical loads some or all openings and maybe the exterior boundary of the cylinders are fully covered by thin electrodes voltage differences are supplied on the electroded surfaces the rest boundary surfaces are charge free in this case the complex potentials are investigated boundary conditions are satisfied by the least squares method there are carried out numerical investigations of behaviour of some basic electroelastic characteristics in a circular hollow cylinder and in an infinite body with two longitudinal cavities and a charge free plane crack new electromechanical regularities of influence of material properties geometrical characteristics of considered regions on values of electroelastic state characteristics and the stress electric displacements and tensions intensity factor k 1 are determined

an exact correspondence between plane piezoelectricity and generalized plane strain in elasticity we consider an anisotropic body bounded by a cylindrical surface which is infinitely long in the axial direction suppose the body is loaded in such a way that the field variables do not vary along the generators an exact correspondence is established between the plane piezoelectric equations and those of generalized plane strain in elasticity in particular we show that by setting a linkage between the two sets of material constants any problem of a plane deformation in piezoelectricity may be solved as a generalized plane strain in elasticity and vice versa the assertation is true for rectilinearly as well as for cylindrically anisotropic solids the equivalence is found for the most general anisotropic case which links the fields between a monoclinic piezoelectric body of class m and a fully anisotropic triclinic elastic solid a few degenerate systems are also identified in addition these correspondences can be extended to inhomogeneous media applied to composite materials or polycrystalline aggregates they imply that results for effective elastic tensors immediately give the formulae for the effective electroelastic tensors and vice versa we also demonstrate that the connection can be used to extend the scope of the invariant stress theorem proved by cherkaev lurie and milton in illustration we present solutions for the plane problem of an elliptical inhomogeneity in an unbounded piezoelectric medium subjected to a uniform loading at infinity lekhnitskii9s complex potential approach together with the conformal mapping technique are employed general solutions of the fields inside the inhomogeneity and the matrix are obtained the results are analytically proven to be identical with the existing solutions of the corresponding purely elastic boundary value problem

moore penrose inverse in rings with involution abstract we study the moore penrose inverse mp inverse in the setting of rings with involution the results include the relation between regular mp invertible and well supported elements we present an algebraic proof of the reverse order rule for the mp inverse valid under certain conditions on mp invertible elements applications to c algebras are given

when is b a a generalized inverse of ab in practice factorizations of a generalized inverse often arise from factorizations of the matrix which is to be inverted in addition to full rank factorizations normal factorizations and singular value decompositions svd there are other factorizations of particular matrices that are natural to certain problems for instance in statistics answers to questions like when is b a a generalized inverse of ab may thus be a computational tool besides they are of significant interest because they provide us with insights into the g inversion of matrix products

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

reverse order law for the moore penrose inverse abstract in this paper we present new results related to the reverse order law for the moore penrose inverse of operators on hilbert spaces some finite dimensional results are extended to infinite dimensional settings

reverse order laws for generalized inverses of multiple matrix products abstract in this article we study reverse order laws for generalized inverses and reflexive generalized inverses of the products of multiple matrices a 1 a n and the products of generalized inverses and reflexive generalized inverses of a n a 1 by applying the multiple product singular value decomposition we obtain necessary and sufficient conditions for one side inclusion relation of reverse order law for generalized inverses and necessary and sufficient conditions of reverse order law for reflexive generalized inverses

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

the reverse order law revisited abstract necessary and sufficient conditions are given for the triple reverse order law abc c b a to hold some special cases are considered

on the p q outer generalized inverse in banach algebra abstract in this note the p q outer generalized inverse in banach algebra is considered a new representation based on the 1 5 inverse is presented also we construct an iterative method for computing the p q outer generalized inverse in banach algebra

matrix division with an idempotent divisor or quotient abstract for given matrices f and g over an arbitrary field f necessary and sufficient conditions in terms of rank amongst others are presented for f to divide g where either f itself is idempotent or one of its quotients is idempotent or both formulae are also given by which these quotients can be constructed finally new proofs of some known results on idempotent factorization are presented in the present context to provide another perspective on the behaviour of such factorizations

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

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

mixed type reverse order law for products of three operators abstract we present new results related to the mixed type reverse order law for the moore penrose inverse of various products of three operators on hilbert spaces some finite dimensional results are extended to infinite dimensional settings

computation of moore penrose generalized inverses of matrices with meromorphic function entries abstract in this paper given a field with an involutory automorphism we introduce the notion of moore penrose field by requiring that all matrices over the field have moore penrose inverse we prove that only characteristic zero fields can be moore penrose and that the field of rational functions over a moore penrose field is also moore penrose in addition for a matrix with rational functions entries with coefficients in a field k we find sufficient conditions for the elements in k to ensure that the specialization of the moore penrose inverse is the moore penrose inverse of the specialization of the matrix as a consequence we provide a symbolic algorithm that given a matrix whose entries are rational expression over c of finitely many meromorphic functions being invariant by the involutory automorphism computes its moore penrose inverve by replacing the functions by new variables and hence reducing the problem to the case of matrices with complex rational function entries

reverse order law in c algebras we study equivalent conditions for the reverse order law a1a2 an an a1 a1a2 anan a1 in c algebras as corollaries we obtain some recent and special results

exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack based on the complex potential approach the two dimensional problems in a piezoelectric material containing an elliptic hole subjected to uniform remote loads are studied the explicit closed form solutions satisfying the exact electric boundary condition on the hole surface are given both inside and outside the hole when the elliptic hole degenerates into a crack the field intensity factors are obtained it is shown that the stress intensity factors are the same as that of isotropic material while the electric displacement intensity factor depends on both the material properties and the mechanical loads but not on the electric loads in other words the uniform electric loads have no influence on the field singularities it is also shown that the impermeable crack assumption used previously to simply the electric condition is not valid to crack problems in piezoelectric materials

identities concerning the reverse order law for the moore penrose inverse we prove some identities related to the reverse order law for the moore penrose inverse of operators on hilbert spaces extending some results from y tian s cheng linear multilinear algebra 52 2004 and r e cline siam rev vol 6 no 1 1964 to infinite dimensional settings

new developments concerning piezoelectric materials with defects abstract the problem of a piezoelectric body with an elliptic cavity is revisited within the frame work of two dimensional electro elasticity in contrast to our previous formulation the present analysis is based on the use of exact electric boundary conditions at the rim of the hole thus avoiding the common assumption of electric impermeability expressions for the elastic and electric variables induced inside and outside the cavity are derived in closed form in terms of complex potentials comparisons between present exact and previous approximate models are effected to establish the limitations associated with the assumption of the impermeable hole in particular when it becomes a slit crack

the moore penrose inverses of products and differences of projections in a c algebra abstract for two given projections p and q in a c algebra we investigate how to express moore penrose inverses of products pq and differences p q and pq qp moreover it is shown that pq qp is moore penrose invertible if and only if pq and p q are moore penrose invertible in addition some related topics are considered