about the group inverse and moore penrose inverse of a product abstract let a be a matrix over a ring suppose that p q p and q are matrices over the ring such that p pa a aqq if a has group inverse a then paq has a group inverse iff aa qpa 1 aa is invertible an analogous result for moore penrose inverses is also given

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

on the rate of convergence of the alternating projection method in finite dimensional spaces abstract using the results of smith solmon and wagner k smith d solomon s wagner practical and mathematical aspects of the problem of reconstructing objects from radiographs bull amer math soc 83 1977 1227 1270 and nelson and neumann s nelson m neumann generalizations of the projection method with application to sor theory for hermitian positive semidefinite linear systems numer math 51 1987 123 141 we derive new estimates for the speed of the alternating projection method and its relaxed version in r m these estimates can be computed in at most o m 3 arithmetic operations unlike the estimates in papers mentioned above that require spectral information the new and old estimates are equivalent in many practical cases in cases when the new estimates are weaker the numerical testing indicates that they approximate the original bounds in papers mentioned above quite well

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

the uniqueness and asymptotic stability of pyramidal traveling fronts in the allen cahn equations abstract this paper studies the uniqueness and the asymptotic stability of a pyramidal traveling front in the three dimensional whole space for a given admissible pyramid we prove that a pyramidal traveling front is uniquely determined and that it is asymptotically stable under the condition that given perturbations decay at infinity for this purpose we characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces moreover we characterize the pyramidal traveling front in another way that is we write it as a combination of two dimensional v form waves on the edges this characterization also uniquely determines a pyramidal traveling front

complexity of computations with matrices and polynomials this paper reviews the complexity of polynomial and matrix computations as well as their various correlations to each other and some major techniques for the design of algebraic and numerical algorithms

parallel computation of polynomial gcd and some related parallel computations over abstract fields several fundamental problems of computations with polynomials and structured matrices are well known for their resistance to effective parallel solution this includes computing the gcd the 1cm and the resultant of two polynomials as well as any selected entry of either the extended euclidean scheme for these polynomials or the pade approximation table associated to a fixed power series the solution of the berlekamp massey problem of recovering the n coefficients of a linear recurrence sequence from its first 2n terms the solution of a toeplitz linear system of equations computing the ranks and the null spaces of toeplitz matrices and several other computations with toeplitz hankel vandermonde and cauchy generalized hilbert matrices and with matrices having similar structures our algorithms enable us to reach new record estimates for randomized parallel arithmetic complexity of all these computations over any field of constants that is o log n h time and o n2 log log n log n h g arithmetic processors where n is the input size g varies from 1 to 2 and h varies from 2 to 4 depending on the computational problem and on the fixed field of constants the estimates include the cost of correctness tests for the solution the results have further applications to many related computations over abstract fields and rely on some new techniques in particular on recursive pade approximation and regularization of pade approximation via randomization which might be of some independent technical interest in particular an auxiliary algorithm computes over any field the coefficients of a polynomial from the power sums of its zeros which is an important problem of algebraic coding

complexity of parallel matrix computations abstract we estimate parallel complexity of several matrix computations under both boolean and arithmetic machine models using deterministic and probabilistic approaches those computations include the evaluation of the inverse the determinant and the characteristic polynomial of a matrix recently processor efficiency of the previous parallel algorithms for numerical matrix inversion has been substantially improved in pan and reif 1987 reaching optimum estimates up to within a logarithmic factor that work however applies neither to the evaluation of the determinant and the characteristic polynomial nor to exact matrix inversion nor to the numerical inversion of ill conditioned matrices we present four new approaches to the solution of those latter problems having several applications to combinatorial computations in order to extend the suboptimum time and processor bounds of pan and reif 1987 to the case of computing the inverse determinant and characteristic polynomial of an arbitrary integer input matrix in addition processor efficient algorithms using polylogarithmic parallel time are devised for some other matrix computations such as triangular and qr factorizations of a matrix and its reduction to hessenberg form

existence and global stability of traveling curved fronts in the allen cahn equations abstract this paper is concerned with existence and stability of traveling curved fronts for the allen cahn equation in the two dimensional space by using the supersolution and the subsolution we construct a traveling curved front and show that it is the unique traveling wave solution between them our supersolution can be taken arbitrarily large which implies some global asymptotic stability for the traveling curved front

algebraic and numerical techniques for the computation of matrix determinants we review modify and combine together several numerical and algebraic techniques in order to compute the determinant of a matrix or the sign of such a determinant the resulting algorithms enable us to obtain the solution by using a lower precision of computations and relatively few arithmetic operations the problem has important applications to computational geometry

the product of operators with closed range in hilbert c modules abstract suppose t and s are bounded adjointable operators with close range between hilbert c modules then ts has closed range if and only if ker t ran s is an orthogonal summand if and only if ker s ran t is an orthogonal summand moreover if the dixmier or minimal angle between ran s and ker t ker t ran s is positive and ker s ran t is an orthogonal summand then ts has closed range

applications of cut polyhedra ii abstract in this paper and in its continuation part ii this issue we group within a unified framework many applications of the following polyhedra cut boolean quadric hypermetric and metric polyhedra we treat in particular the following applications l1 and l1 metrics in functional analysis the max cut problem the boole problem and multicommodity flow problems in combinatorial optimization lattice holes in geometry of numbers density matrices of many fermions systems in quantum mechanics we present some other applications in probability theory statistical data analysis and design theory in this first part after introducing the main definitions and operations for cut polyhedra we describe the connections with l1 metrics and with other metric properties and we consider in particular the applications to some classes of metrics arising from graphs normed spaces and lattices

on disjoint range operators in a hilbert space for a bounded linear operator m in a hilbert space h various relations among the ranges r m r m r m m and the null spaces n m n m are considered from the point of view of their relations to the known classes of operators such as ep co ep weak ep gp dr or sr particular attention is paid to the range projectors of the operators m m and some further characteristics of these projectors are derived as well

parallel algorithms for dense linear algebra computations scientific and engineering research is becoming increasingly dependent upon the development and implementation of efficient parallel algorithms on modern high performance computers numerical linear algebra is an indispensable tool in such research and this paper attempts to collect and describe a selection of some of its more important parallel algorithms the purpose is to review the current status and to provide an overall perspective of parallel algorithms for solving dense banded or block structured problems arising in the major areas of direct solution of linear systems least squares computations eigenvalue and singular value computations and rapid elliptic solvers a major emphasis is given here to certain computational primitives whose efficient execution on parallel and vector computers is essential in order to obtain high performance algorithms

parametrization of newton s iteration for computations with structured matrices and applications abstract we apply a new parametrized version of newton s iteration in order to compute over any field f of constants the solution or at least squares solution to linear system b x v with an n n toeplitz or toeplitz like matrix b as well as the determinant of b and the coefficients of its characteristic polynomial det i b dramatically improving the processor efficiency of the known fast parallel algorithms our algorithms together with some previously known and some recent results of 1 5 as well as with our new techniques for computing polynomial god s and lcm s imply respective improvement of the known estimates for parallel arithmetic complexity of several fundamental computations with polynomials and with both structured and general matrices

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

on a resolvent estimate for bidomain operators and its applications abstract we study bidomain equations that are commonly used as a model to represent the electrophysiological wave propagation in the heart we prove existence uniqueness and regularity of a strong solution in l p spaces for this purpose we derive an l resolvent estimate for the bidomain operator by using a contradiction argument based on a blow up argument interpolating with the standard l 2 theory we conclude that bidomain operators generate c 0 analytic semigroups in l p spaces which leads to construct a strong solution to a bidomain equation in l p spaces

moore penrose invertibility of differences and products of projections in rings with involution abstract this article concerns the mp inverse of the differences and the products of projections in a ring r with involution some equivalent conditions are obtained as applications the mp invertibility of the commutator p q q p and the anti commutator p q q p are characterized where p and q are projections in r some related known results in c algebra are generalized

motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic allen cahn equation we consider the spatially inhomogeneous and anisotropic reaction diffusion equation ut m x 1 div m x ap x u e 2 f u involving a small parameter e 0 and a bistable nonlinear term whose stable equilibria are 0 and 1 we use a finsler metric related to the anisotropic diffusion term and work in relative geometry we prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces more precisely we show that within the time scale of order e2 ln e the unique weak solution u e develops a steep transition layer that separates the regions u e 0 and u e 1 then on a much slower time scale the layer starts to propagate consequently as e 0 the solution u e converges almost everywhere a e to 0 in t and 1 in t where t and t are sub domains of separated by an interface t whose motion is driven by its anisotropic mean curvature we also prove that the thickness of the transition layer is of order e

the drazin inverses of products and differences of orthogonal projections in this note the drazin inverses of products and differences of orthogonal projections on a hilbert space are established

on invertibility of combinations of k potent operators abstract some properties of combinations c 1 p 1 c 2 p 2 c 3 p 1 s p 2 k 1 s where p 1 and p 2 are two different nonzero k potents c 1 c 2 and c 3 are three nonzero complex numbers and positive integers k 2 and s k 1 are obtained furthermore the invertibility the group invertibility and the k potency of the linear combinations of k potents are investigated under certain commutativity properties imposed on them

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

bounded h infty calculus for a class of nonlocal operators the bidomain operator in the l_q setting the bidomain operator mathbb a a nonlocal operator is studied in a bounded domain omega subset mathbb r d with boundary partial omega of class c 2 within the l q setting for 1 q infty assuming a fairly general framework it is shown that this operator is sectorial invertible on functions with mean zero and admits a bounded h infty calculus with h infty angle 0 in particular it has the property of maximal l p l q regularity and the fractional power domains mathcal d mathbb a alpha are identified furthermore it is shown that mathbb a generates a strongly continuous compact analytic and exponentially stable semigroup on l q 0 omega with q independent purely countable point spectrum sigma mathbb a