positive solutions to a system of adjointable operator equations over hilbert c modules abstract we present necessary and sufficient conditions for the existence of a positive solution to the system of adjointable operator equations a 1 x c 1 xb 2 c 2 a 3 xb 3 c 3 over hilbert c modules we also derive a representation for a general positive solution to this system when the solvability conditions are satisfied the results of this paper extend some known results in the literature

the congruence class of the solutions of some matrix equations the congruence class of a least square solution for the following matrix equations ax b a xa d axb dand axxb ef is presented also we derive necessary and sufficient conditions for the existence of a least square solution and present a general form of such solutions using the singular value decomposition svd and generalized singular value decomposition gsvd

explicit solutions to the matrix inverse problem ax b abstract recently wu and cain l wu b cain linear algebra and its applications 236 1996 137 146 gave necessary and sufficient conditions for the matrix equationax b to have a solution with respect toa subject to the condition that the hermitian part ofa is non negative definite in this note we give an easier approach which also provides possible explicit expressions fora further necessary and sufficient conditions are derived when the hermitian part ofa is prescribed to be positive definite thus correcting a result in l wu b cain linear algebra and its applications 236 1996 137 146 theorem 5

the reflexive and anti reflexive solutions of the matrix equation ax b abstract an n n complex matrix p is said to be a generalized reflection matrix if p h p and p 2 i an n n complex matrix a is said to be a reflexive or anti reflexive matrix with respect to the generalized reflection matrix p if a pap or a pap this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti reflexive with respect to a generalized reflection matrix p solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided

geometric mapping properties of semipositive matrices abstract semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices nevertheless the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices it is shown that for a semipositive matrix a there exist a proper polyhedral cone k 1 of nonnegative vectors and a polyhedral cone k 2 of nonnegative vectors such that a k 1 k 2 the set of all nonnegative vectors mapped by a to the nonnegative orthant is a proper polyhedral cone as a consequence a belongs to a proper polyhedral cone comprising semipositive matrices when the powers a k k 0 1 have a common semipositivity vector then a has a positive eigenvalue if a has a sole peripheral eigenvalue and the powers of a have a common semipositivity vector with a non vanishing term in the direction of the left eigenspace of then a leaves a proper cone invariant

common hermitian solutions to some operator equations on hilber c modules abstract we establish necessary and sufficient conditions for the existence of the general common hermitian solution to the equations a 1 x c 1 xb 1 c 2 a 3 xa 3 c 3 a 4 xa 4 c 4 for adjointable operators between hilbert c modules and present an expression for the common hermitian solution to the equations in terms of moore penrose inverse of operators when the solvability conditions are satisfied the findings of this paper extend some known results in the literature

the common positive solution to adjointable operator equations with an application abstract we give necessary and sufficient conditions for the existence of the common positive solution to the equations a 1 x 1 c 1 x 1 b 1 d 1 a 2 x 2 c 2 x 2 b 2 d 2 a 3 x 1 a 3 a 4 x 2 a 4 c 5 for adjointable operators between hilbert c modules and present an expression of the positive solution to the system when the solvability conditions are satisfied as an application we consider the reflexive positive semidefinite solution to the system of complex matrix equations a x b x c d e x e f

shockless acceleration of thin plates modeled by a tracked random choice method a method for accelerating thin metallic plates to hypervelociti es has been proposed by g mccall in this method a shock in a propellant generates a strong expansion wave that smoothly accelerates the plate we have studied the hydrodynamics of this process in one dimension both analytically and computationally the metal was modeled as a stiffened gas and the corresponding riemann problem was solved the asymptotic behavior of the solution was determined analytically the one dimensional random choice method modified so that material boundaries are tracked and the spatial mesh is refined locally was used to compute the flow comparison with the asymptotic solution demonstrated its accuracy with this method shocks that form within the accelerating plate were accurately resolved so that possible structural damage to the plate could be evaluated

re nnd generalized inverses abstract in this paper we give some necessary and sufficient conditions for the existence of re nnd and nonnegative definite 1 3 and 1 4 inverses of a matrix a c n n and completely described these sets moreover we prove that the existence of nonnegative definite 1 3 inverse of a matrix a is equivalent with the existence of its nonnegative definite 1 2 3 inverse and present the necessary and sufficient conditions for the existence of re nnd 1 3 4 inverse of a

equations ax c and xb d in rings and rings with involution with applications to hilbert space operators abstract this paper reviews the equations ax c and xb d from a new perspective by studying them in the setting of associative rings with or without involution results for rectangular matrices and operators between different banach and hilbert spaces are obtained by embedding the rectangles into rings of square matrices or rings of operators acting on the same space necessary and sufficient conditions using generalized inverses are given for the existence of the hermitian skew hermitian reflexive antireflexive positive and real positive solutions and the general solutions are described in terms of the original elements or operators new results are obtained and many results existing in the literature are recovered and corrected

principal pivot transforms properties and applications the principal pivot transform ppt is a transformation of the matrix of a linear system tantamount to exchanging unknowns with the corresponding entries of the right hand side of the system the notion of the ppt is encountered in mathematical programming statistics and numerical analysis among other areas the purpose of this paper is to draw attention to the main properties and uses of ppts make some new observations and motivate further applications of ppts in matrix theory special consideration is given to ppts of matrices whose principal minors are positive 2000 elsevier science inc all rights reserved

dynamics of shear flow of a non newtonian fluid abstract viscoelastic materials with fading memory e g polymers suspensions and emulsions exhibit behavior that is intermediate between the nonlinear hyperbolic response of purely elastic materials and the strongly diffusive parabolic response of viscous fluids many popular numerical methods used in the computation of steady viscoelastic flows fail in important flow regimes and thus do not capture significant non newtonian phenomena a key to satisfactory explanation of these phenomena is the study of the full dynamics of the flow this paper studies the dynamics of shear flow presenting a description of non newtonian phenomena caused by a non monotone relation between the steady shear stress and shear strain rate analytical results for such phenomena are surveyed and three distinct numerical methods are developed to accurately compute the dynamics the computations reproduce experimental measurements of non newtonian spurt in shearing flow through a slit die they also predict related phenomena such as hysteresis and shape memory experiments are suggested to verify these predictions

on the symmetric solutions of linear matrix equations abstract the symmetric solutions of linear matrix equations ax c and axb c are considered the necessary and sufficient conditions for the consistency of the equations with a symmetric condition on solutions are derived using the singular value decomposition and the generalized singular value decomposition numerical algorithms for solving the equations are suggested

stokes second problem for a johnson segalman fluid the laminar flow of a johnson segalman fluid on oscillating plate in its own plane is considered the governing partial differential equations for unsteady motion are constructed analytical solution exists for the problem the differences between these solutions and the corresponding cases in the navier stokes theory are delineated

positive solutions to the equations ax c and xb d for hilbert space operators abstract the paper studies the equation a x c for bounded linear operators between hilbert spaces gives conditions for the existence of hermitian solutions and positive solutions and obtains the formula for the general form of these solutions then the common hermitian and positive solutions to the equations a x c and x b d are studied and new representations of the general solutions are given many results for matrices are recovered as special cases and the results of phadke and thakare s v phadke n k thakare generalized inverses and operator equations linear algebra appl 23 1979 191 199 are corrected

multiscale simulations for suspensions of rod like molecules we study the doi model for suspensions of rod like molecules the doi model couples a microscopic fokker planck type equation smoluchowski equation to the macroscopic stokes equation the smoluchowski equation describes the evolution of the distribution of the rod orientations it comes as a drift diffusion equation on the sphere at every point in physical space for sufficiently high macroscopic shear rates high deborah numbers the solution of the coupled system develops internal layers in the macroscopic strain rate the spurt phenomenon in the high deborah numbers regime the drift term in the smoluchowski equation is dominant we thus introduce a finite volume type discretization of the microscopic smoluchowski equation which is motivated by transport dominated pdes we carry out direct numerical simulations of the spurt phenomenon both in the dilute and concentrated regimes below the isotropic nematic transition the solution structure is identical to the one described by purely macroscopic models jso model for higher concentrations we observe the formation of microstructure coming from a position dependent tumbling rate we also investigate the 2 d stability of the spurted solution

positive solutions to operator equations axb c abstract the operator equation axb c has been studied by several authors but under the extra condition that the operators a and b have closed range in this article we present different results regarding the existence of solution and also the existence of positive solution to axb c without this extra hypothesis

thin film flow of magnetohydrodynamic mhd johnson segalman fluid on vertical surfaces using the adomian decomposition method abstract in this paper the magnetohydrodynamic mhd thin film flow of johnson segalman fluid for lifting and drainage problems on a vertical surfaces are investigated the nonlinear differential equations arising from the mathematical modeling of the johnson segalman fluid the laws of momentum mass and suitable boundary conditions are solved analytically using the adomian decomposition method adm a constant magnetic field is applied transversely to the direction of flow and particular attention is given to the effects of magnetic parameters on the velocity the influence on the velocity field of various non newtonian parameters such as the weissenberg number stokes number and applied magnetic field are investigated several graphs illustrate the influence of various pertinent parameters on the velocity it is observed that increasing the applied magnetic field will generally reduce the flow velocity in the lifting case while in the drainage case the magnetic field and the velocity field are shown to have a direct relation

solution of steady thin film flow of johnson segalman fluid on a vertical moving belt for lifting and drainage problems using adomian decomposition method abstract the steady thin film flow on a vertical belt of a non newtonian johnson segalman fluid for lifting and drainage problems are investigated in this paper the analytical solutions of the non linear problems are obtained by adomian decomposition method adm and homotopy perturbation method hpm expressions for the velocity profile average velocity volume flux the belt speed of the lifting and the shear stress at the belt have been derived for weissenberg number w e 0 we retrieve newtonian cases for both the problems we also obtain the results for maxwell fluid by taking slip parameter a 1 the manner in which the stokes number s t weissenberg number w e the ratio of viscosities and the slip parameter a affect the structure of the velocity profile for lifting and drainage problems are delineated comparison between the adm solutions and homotopy perturbation method hpm solutions are made

a mathematical description of peristaltic hydromagnetic flow in a tube abstract peristaltic motion of an incompressible non newtonian fluid in a deformable tube is studied under long wavelength and low reynolds number the effect of magnetic field on the motion of a johnson segalman fluid has been modelled analytical solutions are obtained for the stream function axial velocity and axial pressure gradient under the small weissenberg number the pressure rise and frictional force per wavelength have been computed through numerical integration the study shows that in addition to weissenberg number and the effect of magnetic field the occlusion and volume flux also affect the peristaltic flow these effects are noticeable in the stream function axial velocity axial pressure gradient the pressure rise and the frictional force graphs are plotted for axial pressure gradient pressure rise and frictional force and discussed in detail

symmetric solutions of linear matrix equations by matrix decompositions abstract the symmetric solutions of linear matrix equations like ax b have been considered using the kronecker product formulation 17 and the commutation matrix k 5 11 in this paper a novel approach is followed using the singular value generalized singular value real schur and real generalized schur decompositions solvability conditions for the symmetric and nonsymmetric solutions are obtained and the numerical properties of the related solution procedures are discussed briefly

products of projections and self adjoint operators abstract let h be a hilbert space and l h be the algebra of all bounded linear operators from h to h our goal in this article is to study the set p l h of operators in l h that can be factorized as the product of an orthogonal projection and a self adjoint operator we describe p l h and present optimal factorizations in different senses for an operator in this set

on the symmetric solutions of a linear matrix equation a formula for the partitioned minimum norm reflexive generalized inverse is applied to find the general symmetric solution x to the matrix equation ax b also the dimension of the space of symmetric solutions is established

the matrix equations ax c xb d for the pair of matrix equations ax c xb d this paper gives common solutions of minimum possible rank and also other feasible specified ranks

analysis of new phenomena in shear flow of non newtonian fluids phase plane and small parameter asymptotic techniques are used to analyze systems of ordinary differential equations that describe the transient behavior of non newtonian fluids in shear flow these systems approximate the partial differential equations that derive from three dimensional balance laws and from differential constitutive models for highly elastic liquids two models are considered one with a single relaxation time and small newtonian viscosity the other with two relaxation times and no newtonian viscosity both possess the key feature that the variation of steady shear stress with strain rate is not monotone the analysis shows that both models exhibit several distinctive phenomena spurt shape memory hysteresis latency and normal stress oscillations the predictions for the spurt phenomenon agree quantitatively with experimental results for polymer melts the other new phenomena which were discovered recently in numerical simulation should also be observable in rheological experiments

hyperbolicity and change of type in the flow of viscoelastic fluids the equations governing the flow of viscoelastic liquids are classified according to the symbol of their differential operators propagation of singularities is discussed and conditions for a change of type are investigated the vorticity equation for steady flow can change type when a critical condition involving speed and stresses is satisfied this leads to a partitioning of the field of flow into subcritical and supercritical regions as in the problem of transonic flow

on the hermitian solutions to a system of adjointable operator equations we establish necessary and sufficient conditions for the existence of a hermitian solution to the system of equations a1x1 c1 x1b1 d1 a2x2 c2 x2b2 d2 a3x1a3 a4x2a4 c5 for adjointable operators between hilbert c modules and provide an expression for the general hermitian solution to the system we present necessary and sufficient conditions for the existence of a unique hermitian solution to the systems a1x1 c1 x1b1 d1 and a3x1a3 c5 of adjointable operators between hilbert c modules several examples are given to explain the use of the theory some of the findings of this paper extend some known results in the literature

about the von neumann regularity of triangular block matrices abstract necessary and sufficient conditions are given for the von neumann regularity of triangular block matrices with von neumann regular diagonal blocks over arbitrary rings this leads to the characterization of the von neumann regularity of a class of triangular toeplitz matrices over arbitrary rings some special results and a new algorithm are derived for triangular toeplitz matrices over commutative rings finally the drazin invertibility of some companion matrices over arbitrary rings is considered as an application

on the matrix equation ax b with applications to the generators of a controllability matrix abstract given two matrices a and b over an arbitrary field for which ax b is consistent we determine all possible ranks for solutions x of ax b given one solution x we furthermore determine all possible ranks of other solutions of this matrix equation we then apply the results to determine when a given n nm matrix a with first m columns b occurs as a controllability matrix a b ab a n 1 b for some a we determine all matrices a that generate a and find all their possible ranks and some bounds on their sparsity structure

the skew symmetric orthogonal solutions of the matrix equation ax b abstract an n n real matrix x is said to be a skew symmetric orthogonal matrix if xt x and xtx i using the special form of the c s decomposition of an orthogonal matrix with skew symmetric k k leading principal submatrix this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the skew symmetric orthogonal solutions of the matrix equation ax b in addition in corresponding solution set of the equation the explicit expression of the nearest matrix to a given matrix in the frobenius norm have been provided furthermore the procrustes problem of skew symmetric orthogonal matrices is considered and the formula solutions are provided finally an algorithm is proposed for solving the first and third problems numerical experiments show that it is feasible

on matrices leaving invariant a nontrivial convex set abstract a real square matrix a leaves a nontrivial convex set invariant if there exists a convex set c which is not a linear subspace such that a c c it is shown that this is equivalent to the statement that a has an eigenvalue with 0 or 1

common hermitian and positive solutions to the adjointable operator equations ax c xb d let a be a c algebra for any hilbert a modules h and k let l h k be the set of adjointable operators from h to k let h k l be hilbert a modules a c l h k and b d l l h in this paper we propose necessary and sufficient conditions for the existence of common hermitian and positive solutions x l h to the equations ax c xb d and obtain the formulae for the general forms of these solutions some results known for finite matrices and hilbert space operators are extended to the adjointable operators acting on hilbert c modules

the re positive definite solutions to the matrix inverse problem ax b abstract an n n complex matrix a is termed re positive definite re pd if the real part of x ax is positive for every nonzero complex n vector x this paper is concerned with constructing complex re pd matrices and solving the following matrix inverse problem given complex matrices x and b find the set of all complex re pd matrices a such that ax b

geometric conditions for the existence of positive eigenvalues of matrices finite dimensional theorems of perron frobenius type are proved for a e e nn and a nonnegative integer k we let w t a be the cone generated by ai ak l in en we show that a satisfies the perron schaefer condition if and only if the closure wk a of w t a is a pointed cone this theorem is closely related to several known results if k po a the index of the eigenvalue 0 in spec a we prove that a has a positive eigenvalue if and only if w t a is a pointed nonzero cone or equivalently w t a is not a real subspace of en our proofs are elementary and based on a method of birkhoff s we discuss the relation of this method to pringsheim s theorem