outer inverses and jacobi type identities abstract we consider matrices over a commutative ring and characterize the class of outer inverses for which jacobi type identities can be extended we obtain a necessary and sufficient condition for the existence of rao regular drazin inverse in terms of sum of principal minors of a k for some k also we obtain determinantal formula for the rao regular drazin inverse conjectures are formulated which give expressions for outer inverse and the conjectures are proved in some special cases

outer inverses jacobi type identities and nullities of submatrices abstract according to the jacobi identity if a is an invertible matrix then any minor of a 1 equals up to a sign the determinant of a 1 times the complementary minor in the transpose of a the identity is extended to any outer inverse thereby generalizing several results in the literature for special generalized inverses a permanental analog of the jacobi identity is proved bounds are obtained for the difference between the nullity of a submatrix of a and that of the complementary submatrix in any generalized inverse or an outer inverse of a the result extends earlier work of fiedler markham and gustafson for the inverse and of robinson for the moore penrose inverse

a general class of infinite dimensional dirac operators and path integral representation of their index abstract a general class of infinite dimensional dirac operators is defined in an abstract boson fermion fock space and some of their properties are studied in particular a path integral representation of their index is established which gives a topologically invariant integer valued functional on a space of functionals with values in a hilbert space further it is shown that in concrete realizations the supersymmetric quantum theories associated with the dirac operators yield the wess zumino models in supersymmetric quantum field theory

three results in connection with inverse matrices abstract first we show that the moore penrose solution of an arbitrary system of linear equations is a convex combination of the solutions of all uniquely solvable partial systems the other two results concern the elements of inverse toeplitz band matrices namely the asymptotic behavior of a determinant appearing in a formula of d s meek and a modification of a formula of w d hoskins and p j ponzo for matrices with binomial coefficients in the limit case

the moore penrose inverse of a morphism with factorization abstract given an m by n matrix a of rank r over a field with an involutory automorphism it is well known that a has a moore penrose inverse if and only if rank a a r rank aa by use of the full rank factorization theorem this result may be restated in the category of finite matrices as follows if a 1 r a 2 is an epic monic factorization of a m n through r then a has a moore penrose inverse if and only if a a 1 r a 2 and a 1 r a 2 a are respectively epic monic factorizations of a a n n and aa m m through r this characterization of the existence of moore penrose inverses is extended to arbitrary morphisms with epic monic factorizations

generalized inverses of matrices over commutative rings abstract a rao regular matrix and the rao idempotent of a matrix over a commutative ring are defined we prove that a matrix a over a commutative ring is regular if and only if a is a sum of rao regular matrices with mutually orthogonal rao idempotents we find necessary and sufficient conditions for a matrix to have group inverse over a commutative ring also we give a method for computing minors of reflexive g inverse whenever it exists

on matrices over an arbitrary semiring and their generalized inverses abstract in this paper we consider matrices with entries from a semiring s we first discuss some generalized inverses of rectangular and square matrices we establish necessary and sufficient conditions for the existence of the moore penrose inverse of a regular matrix for an m n matrix a an n m matrix p and a square matrix q of order m we present necessary and sufficient conditions for the existence of the group inverse of qap with the additional property that p q a p q is a 1 2 inverse of a the matrix product used here is the usual matrix multiplication the result provides a method for generating elements in the set of 1 2 inverses of an m n matrix a starting from an initial 1 inverse of a we also establish a criterion for the existence of the group inverse of a regular square matrix we then consider a semiring structure m m n s made up of m n matrices with the addition defined entry wise and the multiplication defined as in the case of the hadamard product of complex matrices in the semiring m m n s we present criteria for the existence of the drazin inverse and the moore penrose inverse of an m n matrix when s is commutative we show that the hadamard product preserves the hermitian property and provide a schur type product theorem for the product a c c of a positive semidefinite n n matrix a and an n n matrix c

the moore idempotents of a matrix abstract let r be a commutative ring with 1 and with an involution and let mr be the category of finite matrices over r with the involution aij a ij a ji if a is in mr then there exists a unique list e0 es of pairwise orthogonal symmetric idempotents of r that are characterized by several properties involving the rank and squared volume of the eia in particular a has a moore penrose inverse in mr iff esa 0

symmetric morphisms and the existence of moore penrose inverses abstract let x y be a morphism of an additive category with an involution then has a moore penrose inverse with respect to iff is invertible for some annihilator n x of iff there exists an object z and morphisms x z z z z y such that 2 and 1 z and 1 z are invertible in this case 1 1 z 1 1 z 1

the classical adjoint abstract this paper summarizes the historical background of the notion of the classical adjoint as outlined by muir and provides applications of the adjoint to various studies of generalized invertibility of matrices over commutative rings specifically in this setting the classical adjoint is used to provide a novel proof of von neumann s 1936 observation that every matrix over a regular ring is regular and to provide a necessary and sufficient condition for the existence of the moore penrose inverse of a given matrix in particular a representation of the moore penrose inverse is given that leads to an immediate proof of moore s 1920 formula specifying the entries of his reciprocal in terms of determinants

index of a family of dirac operators on loop space we use methods of constructive field theory to generalize index theory to an infinite dimensional setting we study a family of dirac operatorsq on loop space these operators arise in the context of supersymmetric nonlinear quantum field models with hamiltoniansh q2 in these modelsq is self adjoint and fredholm a natural grading operator exists such that q q 0 we studyq p qp wherep 1 2 1 are the orthogonal projections onto the eigenspaces of we calculate the indexi q for wess zumino models defined by a superpotentialv herev is a polynomial of degreen 2 we establish thati q n 1 deg v in particular the field theory models have unbroken supersymmetry and forn 3 they have degenerate vacua we believe that this is the first index theorem for a dirac operator that couples infinitely many degrees of freedom

stochastic analysis on product manifolds dirichlet operators on differential forms abstract we define a de rham complex over a product manifold infinite product of compact manifolds and dirichlet operators on differential forms associated with differentiable measures in particular gibbs measures which generalize the notions of bochner and de rham laplacians we give probabilistic representations for corresponding semigroups and study properties of the corresponding stochastic dynamics

strict stationarity of generalized autoregressive processes in this paper we consider the multivariate equation x n 1 a n 1 x n b n 1 with i i d coefficients which have only a logarithmic moment we give a necessary and sufficient condition for existence of a strictly stationary solution independent of the future as an application we characterize the multivariate arma equations with general noise which have such a solution

the moore penrose inverse over a commutative ring abstract let r be a commutative ring with 1 and with an involution a and let mr be the category of finite matrices over r with the involution a ij a ij a ji a matrix a m n in mr of determinantal rank r such that u a q r m q r m det a det a has a moore penrose inverse u a in r is said to be moore invertible with moore idempotent u a u a if u a u a a a for every matrix a of mr a has a moore penrose inverse with respect to if and only if a is the sum of moore invertible matrices whose moore idempotents are pairwise orthogonal

convergence of two dimensional branching recursions abstract the asymptotic distribution of branching type recursions ln of the form l n d a l n 1 b l n 1 is investigated in the two dimensional case here l n 1 is an independent copy of ln 1 and a b are random matrices jointly independent of l n 1 l n 1 the asymptotics of ln after normalization are derived by a contraction method the limiting distribution is characterized by a fixed point equation the assumptions of the convergence theorem are checked in some examples using eigenvalue decompositions and computer algebra

group and moore penrose inverses of regular morphisms with kernel and cokernel abstract let y be a morphism with kernel k x and cokernel y l in an additive category c 1 if x y then has a group inverse iff is invertible and is regular 2 if c has an involution then has a moore penrose inverse with respect to iff and are invertible and is regular

the determinal rank idempotents of a matrix abstract let m r be the category of finite matrices over a commutative ring r with 1 for a m r of determinantal rank r let i r a be the ideal of r generated by the r r minors of a there exists a unique list e 1 e t of pairwise orthogonal idempotents of r that sum to 1 such that if r i rank e i a then rank a r 1 r 2 r t e i is the identity element of i r i e i a for 1 i t and either e t a 0 or i r t e t a does not possess an identity element characterizations are given for various generalized inverses of a e 1 a e t a in particular a is regular iff e t a 0

path integral representation of the index of kahler dirac operators on an infinite dimensional manifold abstract operators of kahler dirac type are defined in an abstract infinite dimensional boson fermion fock space and a path integral representation of their index is established as preliminaries to this end some trace formulas associated with gibbs states are derived in both an abstract boson and fermion fock space this is done by introducing euclidean bose and fermi fields at finite temperature in each case in connection with supersymmetric quantum field theories the result gives a path integral formula of the so called witten index in a model with cutoffs

generalized inverses over integral domains in an earlier paper one of the authors showed that a matrix of rank r over an integral domain has a generalized inverse if and only if a linear combination of all the r r minors of the matrix is one in the same paper a procedure for constructing a generalized inverse from such a linear combination was also given in the present paper we show that any reflexive generalized inverse can be obtained by that procedure we also show that if the integral domain is a principal ideal domain every generalized inverse can be obtained by that procedure it is also shown that a matrix a of rank r over an integral domain has moore penrose inverse if and only if the sum of squares of all r r minors of a is an invertible element of the integral domain

categories of matrices with only obvious moore penrose inverses abstract let r be an associative ring with 1 and with an involution a and let mr be the category of finite matrices over r with the involution aij aij ji then the following two statements are equivalent i if a in mr has a moore penrose inverse with respect to then a is permutationally equivalent to a matrix of the form b 0 0 0 with b invertible ii if 1 a in r then at most one of the a s is not zero

dirichlet forms and white noise analysis we use the white noise calculus as a framework for the introduction of dirichlet forms in infinite dimensions in particular energy forms associated with positive generalized white noise functionals are considered and we prove criteria for their closability if the forms are closable we show that their closures are markovian in the sense of fukushima

outer inverses of matrices abstract let r be a commutative ring with 1 and let g r n m if a r m n is of rank r 1 with tr c r ag 1 then aga a iff gcr a a that is a is an outer inverse of g iff a is the adjoint image of cr a under g

on dimensional reduction of functional measures abstract a notion of dimensional reduction for functional measures indexed by bounded rectangles in euclidean spaces is introduced for a general class of functional measures absolutely continuous with respect to a gaussian functional measure a sufficient condition for the dimensional reducibility is given it is shown that functional measures associated with the p v or the albeverio hoegh krohn models in euclidean quantum field theories are dimensionally reducible so that functional integrals with respect to them can be approximated by one dimensional integrals as an application a limit theorem for quantum partition functions of the models in the two dimensional space time is proved

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given